Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
TESTING A GENERALIZED DOMAIN MODEL OF PHOTODEGRADATION AND SELF-HEALING USING NOVEL OPTICAL CHARACTERIZATION TECHNIQUES AND THE EFFECTS OF AN APPLIED ELECTRIC FIELD by BENJAMIN R. ANDERSON A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Physics and Astronomy DECEMBER 2013 c Copyright by BENJAMIN R ANDERSON, 2013 All Rights Reserved c Copyright 2013 BENJAMIN R. ANDERSON All Rights Reserved To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of BENJAMIN R. ANDERSON find it satisfactory and recommend that it be accepted. Mark G. Kuzyk, Ph.D., Chair Frederick Gittes, Ph.D. Matthew McCluskey, Ph.D. Philip Marston, Ph.D. ii ACKNOWLEDGMENTS “For His invisible attributes, namely, His eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. ” Romans 1:20 First, and foremost, I give thanks to God who has created, redeemed, and sustains me; whose infinite glory is revealed in the mysteries of the universe, which science seeks to understand. I would like to thank my parents for supporting me through all my schooling, nurturing my innate curiosity, and encouraging me through all these years to persist and never give up. I would like to thank my brother and sister for their support. I would especially like to thank my wife, Lindsay, for all the patience, support, care and love which she has shown through years of hard work and long hours. I would like to thank my advisor, Mark Kuzyk, for providing me this research opportunity, and believing in my abilities as a physicist. His mentorship and vast knowledge helped through many difficulties, and his “do-it-yourself” attitude has given me a much deeper insight and skill with both theoretical and experimental research. I would like to thank my committee members Matt McCluskey, Philip Marston, and Fred Gittes for their time and willingness to participate in my dissertation. I would like to thank Sue Dexheimer and Nicholas Cerruti for their willingness to serve before my committee was finalized. I would like to thank Steve Langford for his help and mentorship. iii I would like to thank the office staff, especially Sabreen Dobson, Laura Krueger, Kris Boreen, and Mary Guenther, without whom the physics department would fall apart. I would like to thank the technical staff including Tom Johnson, Dave Savage, Fred Schutze, and Tim Whitacare for their help with building and fixing of equipment. I would also like to thank the janitorial and facilities staff who maintained Webster and made the building a comfortable place to work. I would like to thank my coworkers past and present, especially Shiva Ramini, Sheng-Ting Hung, Xianjun Ye, Nathan Dawson, Elisao Deleon, and Elizabeth Bernhardt for their help with experiments and insightful discussions. I would like to thank my friends in the physics department who shared in classes and social activities, with special thanks to Chris Varney for many opportunities to de-stress. I would like to thank the Geezer’s hockey team, and Concordia Lutheran church, for providing a home and family for my time in grad school. Finally, I would like to thank the Air Force Office of Scientific Research (AFOSR), Wright-Patterson Air Force base, the National Science Foundation (NSF), and Washington State University for supporting my research. iv TESTING A GENERALIZED DOMAIN MODEL OF PHOTODEGRADATION AND SELF-HEALING USING NOVEL OPTICAL CHARACTERIZATION TECHNIQUES AND THE EFFECTS OF AN APPLIED ELECTRIC FIELD Abstract by Benjamin R Anderson, Ph.D. Washington State University December 2013 Chair: Mark G. Kuzyk Reversible Photodegradation is a relatively new phenomenon which is not well understood. Previous research into the phenomenon has focused primarily on nonlinear measurements such as amplified spontaneous emission (ASE) and two-photon fluorescence (TPF). We expand on this research by considering linear optical measurements, such as transmittance imaging and absorption spectroscopy, of disperse orange 11 (DO11) dye-doped (poly)methyl-methacralate (PMMA) thin films and find photodegradation to contain both a reversible component and irreversible component, with the irreversible component having a small nonlinear susceptibility. From absorption measurements, and the small nonlinear susceptibility of the irreversible component, we hypothesize that the reversible component corresponds to damage to the dye, and the irreversible component is due to damage to the polymer host. Also, we develop models of depth dependent photodegradation taking pump beam absorption and propagation into account. We find that pump absorption must be v taken into account, and that ignoring the effect leads to an underestimation of the true decay rate and degree of damage. In addition, we find pump propagation effects occur on large length scales, such that they are negligible when compared to absorption and typical sample thicknesses. Finally, we perform electric field dependent reversible photodegradation measurements and find that the underlying mechanism of reversible photodegradation is sensitive to the dye-doped polymer’s electrical properties. We develop an extension to the correlated chromophore domain model to include the effect of an applied field, and find the model to fit experimental data for varying intensity, temperature, and applied electric field with only one set of model parameters. vi Contents Table of Contents vii List of Figures xii List of Tables xxii 1 Introduction 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Uses of Organic Dyes . . . . . . . . . . . . . . . . . . . . . . . 2 Dye Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Organic light emitting diodes . . . . . . . . . . . . . . . . . . 3 Dye sensitized solar cells . . . . . . . . . . . . . . . . . . . . . 4 Fluorophores for biological applications . . . . . . . . . . . . . 4 1.2.2 History of Reversible Photodegradation . . . . . . . . . . . . . 5 1.2.3 New Physical Effect? . . . . . . . . . . . . . . . . . . . . . . . 6 Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Non-Interaction Model . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Correlated Chromophore Domain Model . . . . . . . . . . . . 9 1.3.3 Limitations of previous models 1.3 vii . . . . . . . . . . . . . . . . . 11 1.4 1.5 Depth Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Dose dependence effects . . . . . . . . . . . . . . . . . . . . . 12 Electric field effects . . . . . . . . . . . . . . . . . . . . . . . . 13 Brief description of experiments . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Imaging Experiments . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Grating Spectrometer Experiments . . . . . . . . . . . . . . . 15 1.4.3 Conductivity Experiments . . . . . . . . . . . . . . . . . . . . 16 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Experimental Methods 2.1 2.2 2.3 18 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.1 Samples prepared from monomer . . . . . . . . . . . . . . . . 19 2.1.2 Samples prepared from PMMA and solvent . . . . . . . . . . . 19 2.1.3 Making Thin Films . . . . . . . . . . . . . . . . . . . . . . . . 20 Bulk Pressing . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Spin Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Drop Pressing . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Conductivity Substrate Preparation . . . . . . . . . . . . . . . 22 Digital Camera Operation . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Photodetector theory . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Processing: gain and gamma factor . . . . . . . . . . . . . . . 23 2.2.3 Digital camera noise . . . . . . . . . . . . . . . . . . . . . . . 26 Major Noise Sources . . . . . . . . . . . . . . . . . . . . . . . 26 Minor Noise sources . . . . . . . . . . . . . . . . . . . . . . . 27 Digital Imaging Measurements . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 29 Digital imaging microscopy viii . . . . . . . . . . . . . . . . . . . 2.3.2 Confocal Digital Imaging Microscopy and Temperature Chamber 34 2.3.3 Relating scaled damaged population to intensity . . . . . . . . 35 2.4 Conductivity measurements . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Absorbance Measurements . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 White light interferometric microscope . . . . . . . . . . . . . . . . . 37 2.6.1 Interferometer Theory . . . . . . . . . . . . . . . . . . . . . . 41 Empty interferometer . . . . . . . . . . . . . . . . . . . . . . . 42 Samples in both interferometer arms . . . . . . . . . . . . . . 43 Effect of photodegradation of sample in one arm . . . . . . . . 45 Discrete Fourier Transforms . . . . . . . . . . . . . . . . . . . 46 Amplitude and phase . . . . . . . . . . . . . . . . . . . . . . . 47 2.6.3 Interferometer Alignment . . . . . . . . . . . . . . . . . . . . 49 2.6.4 WLIM Measurement Procedure . . . . . . . . . . . . . . . . . 52 2.6.5 WLIM Limitations . . . . . . . . . . . . . . . . . . . . . . . . 54 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Apodization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Mirror Misalignment . . . . . . . . . . . . . . . . . . . . . . . 59 Wavefront errors . . . . . . . . . . . . . . . . . . . . . . . . . 61 WLIM Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Sampling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Optical Jitter-Induced Noise . . . . . . . . . . . . . . . . . . . 65 2.6.2 2.6.6 3 Modeling Depth Effects 67 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Depth effects due to pump absorption . . . . . . . . . . . . . . . . . . 68 ix 3.3 3.2.1 Effect of depth on population decay . . . . . . . . . . . . . . . 68 3.2.2 Effect of absorption depth profile on recovery . . . . . . . . . 69 3.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.4 Comparison with Data . . . . . . . . . . . . . . . . . . . . . . 72 3.2.5 Absorptive effect summary . . . . . . . . . . . . . . . . . . . . 75 Effect of beam propagation on intensity . . . . . . . . . . . . . . . . . 76 3.3.1 Linear wave propagation . . . . . . . . . . . . . . . . . . . . . 76 3.3.2 Beam propagation in an isotropic and homogenous material . 78 3.3.3 Photodamage induced lensing . . . . . . . . . . . . . . . . . . 82 WLIM results . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Approximate steady state pump wave propagation through dam- 3.3.4 aged media . . . . . . . . . . . . . . . . . . . . . . . 87 Thermal self (de-)focusing . . . . . . . . . . . . . . . . . . . . 89 Heating due to CW Laser . . . . . . . . . . . . . . . . . . . . 91 Effect of thermally induced refractive index change on beam 3.3.5 3.4 propagation . . . . . . . . . . . . . . . . . . . . . . . 92 Coupled Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 Approximate Numerical Solution . . . . . . . . . . . . . . . . 96 Summary of propagation effects . . . . . . . . . . . . . . . . . 98 Summary of depth effects . . . . . . . . . . . . . . . . . . . . . . . . 4 Three-Population Model of Reversible Photodegradation 98 101 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Three population rate equations . . . . . . . . . . . . . . . . . . . . . 103 4.2.1 4.3 Three-population model of absorption . . . . . . . . . . . . . . 106 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 x 4.4 Absorbance cross sections . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Proposed energy level diagram . . . . . . . . . . . . . . . . . . . . . . 119 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5 Applied Electric Field Effects 123 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Conductivity 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Dark conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 124 Mechanisms of transient conductivity . . . . . . . . . . . . . . 124 Mathematical description of transient conductivity . . . . . . 125 5.2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Field strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Electric field history . . . . . . . . . . . . . . . . . . . . . . . 131 5.3 5.2.3 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2.4 Summary of conductivity measurements . . . . . . . . . . . . 139 Electric field effect on reversible photodegradation: noninteracting model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.4 5.3.1 Decay Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3.2 Recovery Results . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Extending the correlated chromophore domain model . . . . . . . . . 153 5.4.1 Domain model extended to include an irreversible component 5.4.2 Inclusion of depth effects . . . . . . . . . . . . . . . . . . . . . 154 5.4.3 Density of domains including dielectric energy . . . . . . . . . 155 5.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 xi 153 5.5 5.6 Fitting imaging data to the extended CCDM . . . . . . . . . . . . . . 162 5.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Conclusions 177 Depth effect results . . . . . . . . . . . . . . . . . . . . . . . . 177 Irreversible photodegradation results . . . . . . . . . . . . . . 178 Effect of an Electric Field . . . . . . . . . . . . . . . . . . . . 178 Extended correlated chromophore model . . . . . . . . . . . . 178 6.0.1 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 A Corrections to imaging population 180 A.1 Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 A.2 Approximating spectral convolution . . . . . . . . . . . . . . . . . . . 183 A.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 B Justification of zero-charge electromagnetic wave equation 189 B.1 Wave equation from Maxwell’s equations . . . . . . . . . . . . . . . . 189 B.2 Bound charge effect estimates for typical experiments . . . . . . . . . 192 C Ohmic vs blocking electrodes 193 D Reversible photodegradation in other anthraquinone derivatives 198 Bibliography 201 xii List of Figures 1.1 Schematic structure of dyes found to reversibly photodegrade. a: AF455, b: Pyrromethene, c: Anthraquinones, d: Rhodamine. . . . . . . . . . 1.2 Image of burn lines which showed full ASE probed recovery, but not full recovery of transmittance. Edited for clarity and contrast. . . . . 2.1 6 13 Effect of gamma factor correction on output signal for three γ’s. Increasing γ brightens dark regions, decreasing the contrast, while decreasing γ increases the contrast. . . . . . . . . . . . . . . . . . . . . 25 2.2 Normalized spectra of LEDs used for illumination. . . . . . . . . . . . 32 2.3 Confocal digital imaging microscope setup. The design is essentially the same as the digital imaging microscope with a collinear pump and probe beam focused onto the sample via a cylindrical lens (L2). The difference is in the additional confocal lens and iris to allow the camera to be a long distance from the sample. . . . . . . . . . . . . . . . . . 2.4 (a) Damage profile with x and y axes specified. (b) Pump profile as a function of position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 34 35 DIM apparatus with a high voltage (HV) power supply to apply the electric field and a picoammeter to measure current. . . . . . . . . . . xiii 36 2.6 Schematic diagram of the absorption setup. The probe light source is an Ocean Optics Xenon PX2 light source and the spectrometer is an Ocean Optics SD2000. The pump laser is a Verdi Nd:YAG CW laser operating at 532nm with power control via crossed polarizers. Both the pump and probe beams are focused onto the sample using a positive lens, such that the probe spot is much smaller than the pump spot. . 2.7 38 Schematic diagram of the WLIM. P1, P2 are crossed polarizers used to control the pump beam power. M1 is the stationary mirror in arm 1, which contains the sample that is damaged. M2 is the moving mirror on a piezo translation stage. The attenuating sample is in arm 2. The beam splitter and mirrors used are uncoated UV fused silica (UVFS), which has excellent transmission down to 300nm. Irises are used for alignment and blocking divergent white light incident on the interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 39 Image of the WLIM. Drawn blue arrows show the path of the white light. Two prisms are used to align the white light exiting the fiber (red) with the optical axis after being collimated. . . . . . . . . . . . 2.9 40 Example of an interferogram. The data are cropped to show the highest contrast fringes in detail. . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.10 (a) Wrapped phase; (b) Unwrapped phase. . . . . . . . . . . . . . . . 48 2.11 White light fringe patterns for different alignments. (a) misaligned in both directions, (b) misaligned in the horizontal direction, (c) misaligned in the vertical direction, (d) correct alignment . . . . . . . . . xiv 51 2.12 Geometry for detector off center from optical (beam) axis by an angle α0 . (a) General geometry for diverging rays incident on the detector, (b) spherical geometry of the angles α0 , α, and ρ. A0 connects the interferometer to center of the detector, with the angle between the optical axis, 0, and A0 being α0 , A connects the interferometer to any point on the detector with an angle α being made with the optical axis, r connects A0 and A in the detector plane, ρ is the angle between A0 and A, R is the radius of the detector, and φ is the azimuthal angle in the detector plane. The angles, α0 , α, and ρ are assumed to be related in a locally flat region on the sphere due to their small sizes. 3.1 . . . . 57 Predicted scaled damaged population as a function of time for different sample thicknesses. Note that as the sample thickness increases the decay rate appears to decrease. . . . . . . . . . . . . . . . . . . . . . 3.2 71 Predicted population as a function of time at various depths. The average population is what would be measured had the depth absorption profile not been taken into account. . . . . . . . . . . . . . . . . . . . 72 3.3 Predicted intensity as a function of depth at various times. . . . . . . 73 3.4 Scaled damaged population during decay for four 9g/l samples of differing thicknesses with an incident intensity of 120W/cm2 . . . . . . . 3.5 74 Diagram of beam propagating from air, into glass, and then into a dyedoped polymer half-space. The beam is assumed to have its minimum waist at the surface of the air-glass interface. . . . . . . . . . . . . . . 3.6 81 Change in WLIM phase due to photodegradation for a 9g/l, DO11/PMMA thin film. A pump beam of 488nm has a wavenumber of k0 = 12.875µm−1. 85 xv 3.7 Upper bound on the change in the refractive index due to photodegradation for a 9g/l DO11/PMMA thin film degraded at 40W/cm2 for 45 mins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Beam propagation profile for: (a) positive refractive index change, and (b) negative refractive index change. . . . . . . . . . . . . . . . . . . 3.9 86 88 Transverse beam profile at several depths for: (a) positive refractive index change, and (b) negative refractive index change. . . . . . . . . 89 3.10 Intensity profile at beam center as a function of depth for lensing due to damage without absorption, and a comparison to the intensity profile due to absorbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.11 Photothermal temperature change as a function of time and depth from numerical solutions of the heat equation with the laser as heat source. While each depth shows a slightly different time scale to reach the steady state, all depths reach the steady state within 100ms. . . . 92 3.12 Temperature change as a function of position in the steady state. The peak temperature occurs within the sample, at a depth of approximately 20µm, and the width of the temperature profile is much larger than the pump intensity beam width. . . . . . . . . . . . . . . . . . . 93 3.13 Cross section of the steady state temperature change at the incident surface of the sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.14 Calculated intensity profile as a function of depth and transverse position taking into account thermal lensing. . . . . . . . . . . . . . . . xvi 98 3.15 Intensity profile as a function of depth at the beam center (x = 0) for normal propagation, thermal lensing propagation, and including absorption. The beam is fully absorbed over a propagation distance that is short compared to the length scale where refractive effects come into play. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 99 Schematic three population model. The undamaged species (n0 ) can decay either to the reversibly damaged species (n1 ) or the irreversibly damaged species (n2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2 Scaled damaged population decay and recovery for a pump intensity of 90 W/cm2 . Both the reversible and irreversible portions are marked with arrows. 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Exponential amplitude as a function of intensity data (points) and the three level model prediction (curve). Data for 125 W/cm2 peak burn intensity of a 12g/l thin film exposed for 25 min. . . . . . . . . . . . . 109 4.4 Exponential offset as a function of intensity data (points) with the three level model prediction (curve). Data for 125 W/cm2 peak burn intensity of a 12g/l thin film exposed for 25 min. . . . . . . . . . . . . 110 4.5 Optical density data and model fits as a function of time at several energies (a) 2.33 eV, (b) 2.64 eV, (c) 3.25 eV, (d) 2.78 eV . . . . . . . 116 4.6 Molecular absorbance cross sections for undamaged species, damaged species, and irreversibly damaged material, as determined from absorbance decay and recovery measurements using 9g/l DO11/PMMA samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.7 Energy level diagram proposed by Embaye and coworkers. Reprinted with permission from [1]. Copyright 2008, AIP Publishing LLC. . . . 119 xvii 4.8 Proposed energy level diagram for the three population model, with the ground states of each population being marked by boxes. . . . . . 120 5.1 Time evolution of transient dark current as a function of applied field strength for a 9g/l sample. . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2 Transient dark current for a 9g/l sample after an applied electric field is abruptly turned off. . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.3 Transient current of a 9g/l DO11/PMMA sample in response to a step function voltage of 100V, as a function of electric field conditions time. 132 5.4 Extrinsic photoconductivity diagram, with the polymer states in blue and the dopant states in orange. Light is absorbed by the dopant and forms an exciton (1), which is then transferred to the polymer (2), where the electric field separates the electron and hole (3). The electron is free to move under the influence of the electric field (4), with some number becoming temporarily or permanently trapped in trap sites (6). Eventually electrostatic attraction leads to the recombination of the electrons with holes (5). . . . . . . . . . . . . . . . . . . . . . . . 136 5.5 Typical photocurrent response of DO11 dye doped in PMMA polymer. 137 5.6 Photocurrent for zero applied electric field before and after electric field conditioning. 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Fits of the scaled damaged population during decay and recovery(inset) of the burn center for various applied electric fields for a 9g/l sample burned with an intensity of 175W/cm2 . . . . . . . . . . . . . . . . . . 141 5.8 Decay rate as a function of intensity for several applied electric fields 5.9 Exponential amplitude as a function of intensity for several applied 142 electric fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xviii 5.10 Intensity independent decay rate for electric fields applied parallel (+) to the k−vector, and anti-parallel (-) found from fits to the noninteracting model give by Equation 5.10. The decay rate is found to decrease with applied field independent of direction. . . . . . . . . 143 5.11 Equilibrium scaled damaged population (ESDP) for electric fields applied parallel (+) to the k−vector, and anti-parallel (-) determined by fits to the non-interacting model give by Equation 5.10. The ESDP is found to be independent of the direction of the applied field. . . . . . 144 5.12 Recovery rates for electric fields applied during recovery parallel (+) to the field applied during decay, and anti-parallel (-) to the field applied during decay obtained from fits to the non-interacting model give by Equation 5.11. Maintaining polarity between decay and recovery reduces the recovery rate, while reversing the polarity increases the recovery rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.13 Recovery rate histograms for different applied fields with fits to a poissonian. The histograms are generated using the recovery rates of 1200 points in a burned area with binning of ∆β = 10−5 min−1 . As the electric field is increased the distribution narrows and the mean shifts towards smaller recovery rates. . . . . . . . . . . . . . . . . . . . . . 147 5.14 Recovery rate histogram for zero field reversible photodegradation both before and after electric field conditioning. The effect of conditioning is to narrow the distribution and shift the mean towards a slower recovery rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.15 Average recovery fraction as a function of applied electric field for both + and - polarities. The recovery fraction increases with applied field strength, but the increase is found to be asymmetric. . . . . . . . . . 149 xix 5.16 Scaled damaged population recovery for a sample that was burned with a 0.75 V/µm field applied. The applied field is increased during recovery.151 5.17 (a) Image of horizontal burn lines when 2.5 V/µm field is first applied (red line shows the location where the burn profile is measured). Two of the burn lines had recovered nearly 100%. (b) Image of burn lines after several days of 2.5 V/µm field conditioning. The two burn lines (marked by arrows), which had recovered to the background level, continued to recover leading to two dark lines. (c) The image line profile corresponding to the red line in a. (d) The image profile corresponding to the red line in b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.18 Linear array of equally spaced dipoles separated by grid spacing r. Each dipole has a polarizability α. . . . . . . . . . . . . . . . . . . . . 156 5.19 Molecular dipole moments at a given grid position for four different domain sizes, with α/r 3 = 10−3 . As the domain size increases the individual dipole moments become more homogenous, with only the boundary molecules having different dipole moments. . . . . . . . . . 158 5.20 Spectra for light sources used in the DIM and CDIM, with the change in absorbance during photodegradation in DO11/PMMA for comparison.165 5.21 Scaled damaged population during decay at the burn center for different applied fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.22 Scaled damaged population during recovery for different applied fields. 167 5.23 Exponential amplitude for recovery as a function of intensity. The amplitude scales with the reversibly damaged population n1 . . . . . 168 5.24 Exponential offset for recovery as a function of intensity. The offset scales with the irreversibly damaged population n2 . . . . . . . . . . . 168 xx 5.25 Scaled damaged population as a function of time during decay for several temperatures with fits using the new model. Inset shows recovery for T=298K and T=308K. . . . . . . . . . . . . . . . . . . . . . . . . 169 5.26 (a) Reversibly and (b) irreversibly damaged components as a function of time during decay at the surface of the sample for three different temperatures. As the temperature is increased the reversible component gets larger, while the irreversible component becomes smaller. . 171 5.27 (a) Reversibly and (b) irreversibly damaged components as a function of time during decay at the surface of the sample for three different field strengths. As the field is increased the reversible component gets larger, while the irreversible component becomes smaller. . . . . . . . 173 A.1 Normalized intensity spectra and camera sensitivity used for calculations, along with the pristine sample absorbance. . . . . . . . . . . . 181 A.2 Calculated scaled damaged population (SDP) using absorbance data and approximate camera sensitivity for several Gaussian intensities of differing bandwidths. SDP calculated from absorbance is added for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 A.3 Raw absorbance data compared to the scaled undamaged population for the widest spectral width, showing that the two overlap having the same decay rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 A.4 Normalized light spectra and sensitivity used to calculate scale factors. Broad light spectrum approximates a white light source centered at 500nm, and the narrow light spectrum approximates a LED centered at 400nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 xxi A.5 Difference between damaged and undamaged molecular absorbance cross sections for both the reversible, and irreversible components. . . 187 C.1 Band diagram of metal-semiconductor interface for the Schottky model. 194 C.2 Band diagram of metal-semiconductor interface with band bending. xD is the depth over which the interface effects are important. . . . . 195 C.3 Photocurrent as a function of applied voltage for Ohmic and Blocking electrodes. JSS is the space charge limited steady state current. . . . 197 D.1 Molecular structures of other anthraquinone derivatives tested with alphabetical coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 xxii List of Tables 3.1 Model parameters used for predicting the functional form of population and intensity as a function of depth during decay. . . . . . . . . . . . 3.2 70 Fit parameters for two-population depth model. β was held constant at the average recovery rate, and σ0 was held constant at the value determined from absorption measurements. . . . . . . . . . . . . . . . 4.1 75 Three population model fit parameters for a 12g/l DO11/PMMA sample assuming the thin sample approximation. . . . . . . . . . . . . . . 107 5.1 Parameters determined from self consistent fitting of the full data set. 170 A.1 Calculated scale factors for light spectra shown in Fig A.4. The scale factors have differing sign due to the spectral region which they probe. Additionally, the ratio between the scale factors differs between the light sources, with the narrow light source weighing the irreversibly damage component more than the broad light source does. . . . . . . 188 xxiii D.1 Tabulation of Anthraquinone decay and recovery parameters. λ is the pump wavelength, F is the CW pump fluence, α is the TPNIM intensity independent decay rate, n′0 is the peak equilibrium scaled damaged population, β is the recovery rate, and RF is the average recovery fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 xxiv Chapter 1 Introduction 1.1 Preliminaries The field of reversible photodegradation is relatively young and dominated by the nonlinear optics group at Washington State University. Over the past decade our understanding of reversible photodegradation has been incrementally expanding with each student adding new insights and perspective. This thesis builds on previous work, using the same sample type - disperse orange 11 (DO11) doped polymethacrylate (PMMA) - but with complimentary experimental methods in order to explore new properties affecting reversible photodegradation, including: sample thickness, intensity, and applied electric field. This thesis begins by motivating the study of reversible photodegradation with an overview of the history of reversible photodegradation. Chapter 2 describes the experimental methods, Chapter 3 presents modeling of the effects of thickness, intensity (Chapter 4), and applied electric field (Chapter 5) which are used to interpret experimental results that lends to a more nuanced understanding of the underlying mechanisms. 1 1.2 Motivation In this section we will provide motivation for this thesis beginning with a discussion of organic dye applications and photodegradation, followed by a brief history of reversible photodegradation research with a discussion of proposed mechanisms. 1.2.1 Uses of Organic Dyes Dye Lasers The first dye laser was realized in 1966 by Sorokin and Lankard using chloro-aluminum phthalocyanine in solution [2–4]. Just a year later in 1967 Soffer and McFarland developed a solid state dye laser (SSDL) by using rhodamine 6G doped PMMA as the lasing medium [5]. Recently SSDLs have been improved using new enhanced forms of PMMA [6] as well as the introduction of organic-inorganic dye-doped polymer nanoparticle compounds [7–9], allowing for improved efficiency and optical characteristics [10]. Dye lasers, both in solution and solid state, have several advantages over other types of lasers. Organic dyes have broad absorption and emission peaks [11] which means they can be used to create tunable lasers over a large range of frequencies. Along with large tunability, the broad bandwidth of organic dyes allows for the generation of ultra short pulses [12]. Finally, dyes in liquid solutions tend to have very large laser gains making them highly efficient as lasing media [10]. While organic dyes have many benefits as lasing media, they have one major drawback; namely, they irreversibly photodegrade over time leading to loss of efficiency, power, and stability. In order to combat loss of efficiency liquid dye lasers use a circulating reservoir of solution to ensure that new molecules are continually exposed and photodegraded molecules are circulated out of the excitation region [13–16]. This 2 method requires regular maintenance and replacement of the dye solution which is hazardous as most dyes are toxic and flammable. Solid state dye lasers circumvent theses hazards by placing the dyes safely in a polymer matrix, but given the inability to circulate out degraded molecules and replenish undamaged molecules the dye doped polymers quickly decay making it difficult for solid state dye lasers to be useful given their short lifetimes. Organic light emitting diodes Organic light emitting diodes (OLEDs) have become a common place technology finding use in many modern electronics such as TVs, monitors, smart phones, and handheld game consoles. To produce light, OLEDs utilize electroluminescence, an electro-optic phenomenon in which an applied electric field creates electron hole pairs, which eventually recombine and produce photons. The first studies of electroluminescence in organic materials, by Bernanose and co-workers, began in the 1950’s with experiments utilizing high voltage AC fields [17–20]. In the 1960’s this research was extended by Martin Pope and co-workers to DC fields utilizing anthracene and tetracene [21–24], and the model of electron hole recombination was proposed [25]. The first diode device based on electroluminescence was developed by Kodak scientists Tang and Slyke in 1987, which used a novel two-layer structure with one layer being hole transporting, and the other layer being electron transporting, which resulted in lower operating voltages and improved efficiency [26]. These developments lead to the invention of a high-efficency green light emitting poly (p-phenylene vinylene) OLED [27], which ushered in the current era of OLED device design. OLEDs have proven to be rather remarkable devices for consumer electronics. They are extremely lightweight, have fast response time (allowing for high image refresh rates), have better energy efficiency than CRTs, LCDs, and plasma displays, 3 and have the ability to make flexible polymer displays. Despite these benefits, OLEDs have the same major drawback as other organic photonic devices in that the organic dyes photodegrade irreversibly over time, which leads to shortened lifetimes when compared to other technologies such as LCD or inorganic LEDs. Dye sensitized solar cells In 1968 it was discovered that some organic dyes generate photocurrent when placed near oxide electrodes and illuminated [28]. This discovery led to experiments into dye sensitized solar cells (DSSCs) which consist of a porous oxide layer coated with a layer of organic dye. These early experiments found that the primary difficulty with the technique is the organic dye’s instability, in which the dye would irreversibly photodegrade due to the UV portion of sun light [29, 30]. In order to overcome this setback some modern DSSCs include a buffer layer of UV absorbing material which will emit the absorbed light at a longer wavelength [31] thus limiting the organic dyes exposure to UV radiation. Modern DSSCs are made using a nano porous TiO2 thin film sensitized with a variety of dyes including indolines [32], arylamines [33], ru-polypyridyl-complex sensitizers [34], and a whole host of other dyes. Currently the peak power conversion efficiency is 11% for commercial units [35,36], with the prototype record being 12.3% [37]. For comparison, similarly priced traditional solar cells made of silicon have efficiencies between 12% and 15%. Fluorophores for biological applications Fluorophores are are fluorescent dye molecules which have a long history of use in the biological sciences, primarily in microscopy [38]. The basic idea when using fluorophores for microscopy is to use fluorescing molecules which are designed to attach 4 to specific compounds in cells. When the specimen is exposed to light those cells containing the fluorescing dye will emit light at a different wavelength allowing for the differentiation of cell components. Advances in both techniques and dyes now allow for more precise measurements, including higher resolution confocal fluorescence microscopy [39], three dimensional fluorescence microscopy [40], and two photon fluorescence microsopy [41]. As with the other applications using organic dyes, prolonged use leads to photodegradation, which results in fluorescence quenching in imaging leading to decreased contrast over time as the dyes decay. 1.2.2 History of Reversible Photodegradation Reversible photodegradation was first reported by Peng and coworkers in rhodaminedoped and pyrromethene-doped polymer optical fibers using fluorescence as a probe [42]. The discovery was secondary to the study, and no further research developed from the observation. Several years later, while studying photodegradation of amplified spontaneous emission (ASE) in 1-amino-2-methylanthraquinone (disperse orange 11, DO11) dye doped polymer, Howell and Kuzyk observed full recovery even when the ASE signal was nearly 100% degraded [1, 43]. However, when measuring the dye in liquid solution no recovery was observed for any amount of degradation, suggesting that the polymer plays a crucial role in self-healing [44]. About the same time Kobrin et.al. observed partial self healing in photodegraded organic LEDs with the dye 8-hydroxyquinoline aluminum(Alq) [45]. Several years later full recovery was measured with two photon fluorescence in AF455 doped into PMMA [46, 47]. AF455/PMMA was also studied by DeSautels and coworkers using femtosecond laser oblation to damage samples to varying degrees to the point of drilling holes in the sample. The samples were observed optically to display recovery of burned areas, and more surprisingly closing of the holes in the sample. Studies in undoped PMMA 5 Figure 1.1: Schematic structure of dyes found to reversibly photodegrade. a: AF455, b: Pyrromethene, c: Anthraquinones, d: Rhodamine. showed that the holes did not heal, but, instead grew larger and caused cracking of the polymer sample. This suggests that not only does the polymer host help heal the damaged dye molecule, but that the dye molecule can lead to the healing of the polymer host [48]. Recently reversible photodegradation has been measured in other anthraquinones [49, 50] and DO11 doped into a random copolymer of PMMA and polystyrene (PS) [51, 52]. 1.2.3 New Physical Effect? As discussed in the previous section, reversible photodegradation has been observed in a wide variety of dyes, from small anthraquinone derivatives, to large octopolar molecules such as AF455 (see Figure 1.1). The wide variation of the dyes that display self healing suggests that there may exists some underlying process which is common to many dyes. Diffusion and orientational hole burning were proposed early on as possible expla- 6 nations, but measurements of linear dichroism [1], and measurements of the spatial profile of damage [53–55] showed that self healing could not be due to either mechanism. Using the structure of DO11 Embaye and coworkers proposed that the polymer changes the photodegradation mechanism to phototautomerization and the formation of dimers, with recovery occurring when then dimer pairs break apart [1]. Also using DO11, Westfall and Dirk proposed that photodegradation induces the formation of a higher-energy twisted internal charge transfer (TICT) state, and recovery occurs when the sample relaxes back to the ground state [56]. While both phototautomerization and the formation of a TICT state is a plausible explanation for DO11 and other anthraquinones, they may not explain reversible photodegradation for other dyes. Given this difficulty, DesAutels and coworkers proposed that photocharge ejection and recombination could be the underlying process based on their measurements of AF455 doped into PMMA [48]. While the exact nature of self healing is unknown, this thesis seeks to advance our understanding of the process by primarily considering the effect of an applied electric field on reversible photodegradation, and proposing a simple model to explain the observed effects. This model will be seen to suggest particular microscopic mechanism suggesting domains of interacting molecules. 1.3 Previous Models Without considering the underlying mechanisms, there have been two proposed empirical models of reversible photodegradation. The first model is a simple two level model in which one species is converted into another. The parameters of the model are purely phenomenological, with no prediction as to their dependence on any variables such as temperature, concentration, or applied electric field. While it is simplistic, it 7 is the go to approach when first fitting data [1]. The second model adds a degree of complexity by using statistical mechanics to account for interactions between molecules in order to predict the concentration and temperature dependence of the phenomena [52, 54]. The model predicts observed concentration and temperature dependence, and suggests that domains of molecules play a role in the recovery process. However the model makes no claim as to the nature of the interactions. 1.3.1 Non-Interaction Model The first model of reversible photodegradation was developed by Embaye et. al. to explain decay and recovery in ASE signal from DO11 doped into PMMA [1]. It was later used by Zhu et.al. to describe two photon fluorescence decay and recovery in AF455 doped into PMMA [46], as well as optical transmittance of anthraquinone derivatives doped into PMMA [49]. The model assumes that there are two different and distinct populations, one that corresponds to undamaged molecules, nu , and one that corresponds to damaged molecules, nd , with the total number of molecules in the system being N = nu + nd . The model also assumes that the damage process occurs at a rate proportional to intensity and that recovery proceeds at a rate independent of dose during damage. The rate equations for the two populations are dnu (t) = −αInu (t) + βnd , dt dnd (t) = αInu (t) − βnd , dt (1.1) (1.2) where I is the intensity, α is the intensity independent decay rate, and β is the recovery rate. Using the total number of molecules, N, and letting nu → n, we can write a single differential equation to describe the time dependent undamaged population, 8 dn(t) = −αIn(t) + β(N − n(t)). dt (1.3) Assuming I 6= 0 and integrating Equation 1.3 using the boundary condition n(0) = N gives the undamaged population during decay as N β + αIe−(β+αI)t . n(t) = β + αI (1.4) For recovery we assume that I = 0 and the initial population at the start of recovery is n0 . Integrating Equation 1.3 using this assumptions gives the population during recovery as n(t) = N + (n0 − N)e−βt . (1.5) Equations 1.4 and 1.5 have been found to fit decay and recovery data very well for a large range of systems, often with only a small adjustment to Equation 1.5 with the addition of a constant offset. This constant offset accounts for an irreversibly damaged species. 1.3.2 Correlated Chromophore Domain Model While the non-interaction model has proven to be quite robust and useful, the model makes no predictions on the behavior of its parameters, α and β, when varying conditions such as temperature, concentration, dye and polymer. To model these effects Ramini developed the correlated chromophore domain model (CCDM) [54,55], which was later modified to correctly predict dose dependent behavior [52]. The model is based on a condensation domain model [57–61] where molecules aggregate to form domains of interacting molecules. The exact nature of these domains is currently unknown, but several hypothesis exist: (1) the molecules form aggregates 9 via electrostatic interactions, nanocrystallite formation, or hydrogen bonding and that these aggregates interact to lead to the observed phenomenon; (2) another possibility is that the process is an entirely new phenomena with the molecules being correlated via exchange statistics so that they behave similar to a Bose-Einstein condensate. Currently the most likely candidate is that domains are molecules correlated with each other through the polymer chains via charge or phonon transfer. The CCDM assumes that the decay and recovery rates in Equation 1.3 are modified such that αI → αI N and β → βN, where N is re-interpreted as the number of molecules in a domain. This modification implies that the larger domains decay slower and recover faster than domains of smaller size. Making these adjustments Equation 1.3 becomes αI dn = − n + βN(N − n). dt N (1.6) Integrating Equation 1.6 with the boundary condition n(0) = n0 we get n(t) = βN 2 + βN + αI n0 − βN 2 e[−(βN + N )t] . βN + αI N αI N (1.7) For recovery we let I = 0 and set the boundary condition to be n(0) = n1 , where we have adjusted the times such that the recovery begins at t = 0. Integrating Equation 1.6 with these assumptions we find n(t) = N + (n1 − N) exp [−βNt] . (1.8) Equation 1.7 and 1.8 describe the dynamics of the undamaged population of a single domain. In order to obtain the measurable population dynamics we must take the ensemble average over a distribution of domains. The distribution of domains of size N, Ω(N), is formally derived from the Helmholtz free energy and is given 10 by [52, 54, 55]: N √ 1 (1 + 2ρz) − 1 + 4ρz Ω(N) = , z 2ρz where z = exp λ kT (1.9) , λ is the energy advantage of a molecule being in a domain, T is the temperature, k is the boltzmann constant and ρ is the total number of molecules P in the system, given by ρ = ∞ N =1 NΩ(N). The mean number of undamaged molecules in a domain, n, is given by the en- semble average, n(t; ρ, T, I, n0 ) = ≈ ∞ X n(t; N, I)Ω(N; ρ, T ), N =1 Z ∞ n(t)Ω(N)dN. (1.10) 1 which represents the population measured by optical techniques. So far the CCDM has successfully modeled reversible photodegradation with changing concentration and temperature for ASE measurements. 1.3.3 Limitations of previous models In the previous two sections we provided an overview of proposed models for reversible photodegradation. These models were developed to explain ASE measurements of reversible photodegradation, and were found to be in excellent agreement with experiment. However, research using transmission imaging and absorbance spectroscopy have found discrepancies and new data not explained by these models. The three areas of interest in this thesis are: depth effects, intensity dependence effects, and electric field effects. 11 Depth Effects Both the non-interaction model and the CCDM assume that the intensity of light propagating in a sample is constant throughout the depth of the sample. This is false for thick samples, as light propagating through a material is absorbed by the material and is also subject to beam divergence. These two effects lead to the intensity changing with depth, and therefore our previous models are incomplete. In Chapter 3 we use the non interaction model as a starting point to consider depth effects and show that pump absorption results in a large effect on the decay characteristics, while the effect of beam divergence is negligible for the samples used. Dose dependence effects For small doses and high enough concentrations, DO11/PMMA samples show fully reversible decay of ASE, suggesting that there are two species involved in which one is converted to the other. This observation served as the basis for both the non interaction model and the CCDM. While ASE data supported the existence of two populations, transmittance imaging and absorbance measurements have shown that there exists at least one more irreversibly decayed species which emits ASE light. As an example, Figure 1.2 shows a picture of a burn line after ASE fully recovers. Even though the ASE returns to pre-decay levels, the sample still has a visible burn mark. This suggests that there is a third species, which linear measurements see, but does not emit ASE, which is a nonlinear process. In Chapter 4 we develop a three level model to explain the irreversibly decayed species and discuss possible reasons why ASE is unaffected by it. 12 Figure 1.2: Image of burn lines which showed full ASE probed recovery, but not full recovery of transmittance. Edited for clarity and contrast. Electric field effects Given the proposal of photocharge ejection and recombination as a possible mechanism for reversible photodegradation, we perform measurements of conductivity and photoconductivity during decay and recovery and find that an applied electric field has a large effect on decay and recovery characteristics. In Chapter 5 we explain the observed effect by expanding the CCDM to include the energetics of an applied electric field and the irreversibly decayed species discussed in the previous section. 1.4 Brief description of experiments While Chapter 2 covers the various experimental techniques used in detail, we provide here a brief description of experiments and their various benefits. 13 1.4.1 Imaging Experiments The main experimental methods utilized in this thesis are digital imaging techniques including: digital imaging microscopy (DIM), confocal digital imaging microscopy (CDIM), and white light interferometric microscopy (WLIM). The primary benefit of each of these techniques is the ability to resolve decay and recovery spatially, and thereby correlate damage to intensity, which previously was difficult and time consuming as only one intensity could be measured per run. Imaging techniques allow us to measure a wide range of intensities simultaneously. The simplest imaging technique is the DIM, which uses a micro-imaging camera in close proximity to the sample in order to measure the transmitted intensity during decay and recovery. A slightly more complicated method, the CDIM, moves the camera a long distance from the sample and uses a lens and microscope objective to project the image onto the camera. This method is used in cases where the sample is to undergo temperature dependent studies, as digital cameras operate poorly at elevated temperatures. Both methods provide a quick and easy method for measuring intensity dependent reversible photodegradation, but suffer the same flaw; since digital cameras operate by convoluting the whole spectrum of light into a signal, neither of these methods provides spectral resolution, which is important for determining the underlying mechanisms behind reversible photodegradation. In order to combine spatial and spectral resolution into one experiment we developed the WLIM. The WLIM combines a Michelson interferometer and digital camera in order to resolve the complex refractive index of a sample as a function of space and frequency. The Michelson interferometer works to produce an interference pattern on the digital camera, which is read as the path length difference in each arm is changed by moving one of the mirrors in the interferometer. As the interference pattern changes, the camera measures the changing intensity at each pixel, creating 14 an interferogram that is Fourier transformed into a complex spectral intensity, which is related to the complex index of refraction. While the WLIM provides the complex index of refraction as a function of position, it is an extremely complicated method which requires judicious alignment, calibration, and analysis. Along with its difficulty of operation, the WLIM is a very slow and data consuming method as it takes 10,000 images to produce good clean spectra for a single time slice during decay or recovery. This process typically requires five minutes to complete, making the time resolution of the WLIM poor in comparison to other measurements which take under a second. Given the low repetition rate of the WLIM, it is primarily used to find the complex index of refraction as a function of dose after decay in order to determine how dose affects the absorbance spectrum and index of refraction. 1.4.2 Grating Spectrometer Experiments Given the (C)DIM’s lack of spectral resolution, and the WLIM’s poor repetition rate, we also utilize grating spectrometers in order to measure the absorbance spectrum during decay and recovery for time resolution. Modern grating spectrometers utilize a diffraction grating to separate light into its spectral components, which are imaged by a linear CCD array. Using the grating slit width and position on the CCD array, the spectrometer is able to reconstruct the spectrum based on the intensity measured at each pixel. A downside to grating spectrometers is that they measure a wide spatial range and thereby are unable to effectively measure the intensity dependence of reversible photodegradation, as the intensity is spatially dependent, and spatial integration results in measuring a convolution over the intensity profile. In this thesis grating spectrometers are primarily used to measure the optical absorption spectrum as a function of time during decay and recovery. Using the change 15 in absorbance due to photodegradation we are able to determine the absorbance cross sections of the various species involved, which allows for inferences about the energetics involved. 1.4.3 Conductivity Experiments Previous work into reversible photodegradation has neglected the semi-conductor nature of dye doped polymers, which in itself is a large field of study. Dye-doped polymers have complex electrical properties, as both the dye and polymer are polarizable, and there exists free and trapped charge in the polymer matrix. These electrical properties affect the fundamental properties of dye doped polymers, such as alignment, dye and charge mobility, and homogeneity. Given that electrical properties can have a large influence on other aspects of dye doped polymers, it is likely that the electrical properties may affect reversible photodegradation. The effect of dye doped polymer electrical properties on reversible photodegradation is studied with a variety of experiments utilizing an applied electric field and a picoammeter. The experiments ranged from simple transient conductivity measurements to photoconductivity measurements, and a combination of conductivity measurements and DIM during decay and recovery. 1.5 Summary Organic dyes are used in many applications such as lasing media, organic light emitting diodes, dyes sensitized solar cells, and fluorescence microscopy. All of these applications share one common setback: organic dyes tend to irreversibly decay under exposure to light, and thus become less efficient to the point of inoperability. Recently several dye types have been found to reversibly photodegrade when placed 16 into a polymer matrix. These findings are exciting as they suggest the possibility of designing optical devices with organic dyes which decay in efficiency during use, but when turned off, given enough time, recover to their pre-damaged efficiency. Research into the self healing mechanism has primarily been performed by the nonlinear optics group at WSU, using DO11 as a template molecule to test various hypotheses. Currently two models exist to describe reversible photodegradation: the noninteracting two population model, and the correlated chromophore domain model. These models were primarily developed using data from ASE measurements, and found to fit ASE data very well. However, linear measurements of the process suggest that the current models are incomplete in their description of reversible photodegradation. This thesis aims to further advance our understanding of reversible photodegradation by addressing the effects of sample thickness, the inclusion of an irreversibly decayed species, and an applied electric field. Using imaging and conductivity we find that the electrical properties have a large influence on the reversible photodegradation process, hinting that the underlying mechanism may be related to electrostatic interactions between the dye and polymer. While this study is far from a comprehensive explanation, it will hopefully be the starting point for further advancing our understanding of reversible photodegradation. 17 Chapter 2 Experimental Methods In this chapter we describe the process of sample preparation and the various experimental methods used to measure reversible photodegradation. Several different techniques were developed in order to measure a sample’s population of dye molecules during decay and recovery, including simple and confocal digital imaging microscopy, white light interferometric microscopy, conductivity measurements, and optical absorbance measurements. 2.1 Sample Preparation When preparing samples for testing, there are two different material processing techniques used to dope disperse orange 11 (DO11) into poly(methyl methacrylate)(PMMA). The first approach utilizes monomer, methyl methacrylate (MMA), and dye (DO11) as the starting constituents, while the other approach begins with polymer (PMMA) and dye (DO11) dissolved in solvent. All compounds are purchased from SigmaAldrich. 18 2.1.1 Samples prepared from monomer When starting from MMA, the monomer must first be filtered through a column flask filled with alumina powder, as commercially available MMA is mixed with an inhibitor to prevent polymerization during shipping and storage. The filtering process removes the inhibitor and leaves pure MMA. DO11 is added to the filtered monomer in ratios to reach set concentrations, such as 7g/l, 9g/l and 12g/l. The solution is then sonicated for a half hour to insure that all dyes are in solution, at which point an initiator (butanethiol) and a chain transfer agent (Tert-butyl peroxide) are added in amounts of 33 µl per 10 ml of MMA, and the solution is returned to the sonicator for another 30-60 min for additional mixing. The chain transfer agent (CTA) controls the length of polymer chains by stopping polymerization once the chains reach a certain length, and the initiator catalyzes the polymerization process. After sonication the solution is filtered through 0.2µm disk filters into vials in order to remove dust particles or pre-polymerized chunks larger than 0.2µm. The vials are then placed in an oven at 95◦ C to initiate polymerization. Typically full polymerization occurs within 48 hrs. To separate the solid dye-doped polymer from the glass vial, the vial is placed in a freezer for several hours, allowing the cylinder of polymer to be separated from the glass vial through differential expansion. 2.1.2 Samples prepared from PMMA and solvent The second preparation method begins by dissolving PMMA and DO11 into a solution of 33% γ-butyrolactone and 67% propylene glycol methyl ether acetate (PGMEA), with the PMMA and DO11 in the desired ratio for the final concentration, while maintaining the ratio of 15% solids to 85% solvents. The solution is stirred for 24 hrs in a magnetic stirrer in order to fully dissolve the dye and polymer in solution, at 19 which point the solution is filtered into vials using 0.2µm disk filters to remove any remaining solids. The filtered solution is placed in an opaque container and stored in a refrigerator until samples are made. 2.1.3 Making Thin Films We use three different methods for making thin films: thermal pressing from bulk, spin coating, and drop pressing. Also, depending on the application, there are two substrate types: plain glass and indium tin oxide (ITO) coated glass. Bulk Pressing The simplest method for making thin films is pressing a chunk of bulk dye-doped polymer between two glass substrates. While simple, bulk pressing has several disadvantages: relatively thick samples (≈ 60 − 100µm) with nonuniform thickness across the sample; anisotropic chain alignment; the formation of micro bubbles; and the inclusion of dust and other contaminates from the bulk sample. Bulk pressing begins by placing a small amount of the bulk dye-doped polymer onto a cleaned glass substrate, and then sandwiching the polymer between it and a second substrate. The sandwich structure is pressed in a custom oven/sample press, with an Omega CN-2010 temperature controller maintaining a temperature of 150◦C, well above the glass transition of the polymer, allowing it to flow. The uniaxial stress is gradually increased perpendicular to the sample until reaching 90psi, and kept constant for an hour allowing the polymer melt to uniformly flow from the center, at which point the stress is removed and the sample is allowed to cool. After cooling, the samples are placed in an opaque airtight container and stored in a cool, dry place. 20 Spin Coating While bulk pressed samples work well for most of our experiments, some experiments require extremely thin (≈ 2 − 10µm) uniform samples, which cannot be made using bulk pressing. In such cases, we use spin coating. For spin coating, a PMMA/solvent solution (Section 2.1.2) is used instead of bulk dye-doped polymer. While the solvent is being prepared, glass substrates are cleaned and cut to 1.5 cm × 1.5 cm squares or 1.5 cm× 3 cm rectangles. Once the solution and substrates are ready, the substrate is placed in a Headway Research Inc. spin coater (PM101D- R790) and the substrate surface is coated with the viscous polymer, solvent, dye solution. The sample is then spun at 1200 rpm for 30s/layer, with thicker films requiring multiple layers. The coated samples are placed in an 85◦ C oven overnight to force solvent evaporation, after which they are cooled and then stored in an opaque airtight container. Drop Pressing As a method to obtain intermediate film thicknesses (≈ 20 − 40µm), we use drop pressing. Drop pressing begins by placing a cleaned glass substrate on a hot plate at 50◦ C for 10 mins at which point several drops of PMMA/solvent solution (Section 2.1.2) are placed on the heated substrate with the hot plate temperature being raised to 95◦ C to induce solvent evaporation. The heated sample is allowed to sit for half an hour before it is removed and placed in an evacuated chamber overnight, at room temperature, to ensure the sample is dry. The dried sample is used to make a sandwich structure with another cleaned glass substrate and is placed in the thermal pressing oven for 135 mins at a pressure of 72psi and a temperature of 130◦ C. After pressing the samples are allowed to cool and then placed in an opaque evacuated chamber. 21 Conductivity Substrate Preparation For conductivity and photoconductivity measurements, thin films are prepared using the previous methods discussed in Section 2.1.3, but instead of using plain glass substrates, glass coated with etched ITO electrodes are used. The ITO etching process is as follows: glass substrates (25 × 75 × 1.1 mm) covered with ITO on one surface are purchased from Delta Technologies. A 1mm wide strip of acid resistant tape is applied over the length of the ITO substrate to protect it as the substrate is etched in an aqueous solution of 20% HCl heated to 50◦ C for 20 mins or longer if needed to remove the exposed ITO. The tape is subsequently removed leaving a strip of ITO and the substrate is cleaned to remove residual acid and adhesive. 2.2 2.2.1 Digital Camera Operation Photodetector theory The primary optical detectors used in the experiments are CCD (charged coupled device) and CMOS (complementary metal-oxide-semiconductor) cameras. Both CCD and CMOS cameras consist of an array of semiconductor pixels and associated electronics. The basic concept behind both devices is hole-electron pair generation in a semiconductor. When a photon of sufficient energy enters a semiconductor, it excites an electron into the conduction band, producing a current. This current then is detected and used to determine the intensity of light incident on the semiconductor. For intensities below the saturation limit we can write the number of charges generated, Ne− , for a monochromatic incident photon flux, Nγ as 22 Ne− = Nγ ΦA, = 1 Iτ ΦA, h̄ω (2.1) (2.2) where Φ is the quantum efficiency, I is the intensity, h̄ω is the photon energy, τ is the exposure time, and A is the detector area. In the case where the incident light is not monochromatic we must sum over all the electrons produced by photons of all energies, which can be written as a convolution integral to give the total number of electrons produced: Ne− = Z ∞ 0 1 I(ω)Φ(ω)τ Adω, h̄ω (2.3) where Φ and I now both depend on frequency. Once the charge is produced in a pixel, the camera converts it into a voltage which is used to define the signal for that pixel, V = Z ∞ I(ω)S(ω)τ dω, (2.4) 0 where S(ω) is the sensitivity, which wraps together the quantum efficiency, fundamental constants and properties of the electronics. 2.2.2 Processing: gain and gamma factor In the previous section, we considered the basic functioning of a digital camera in order to convert the intensity at a pixel to the measured signal. In addition to the basic processing of an image recorded by a camera, there are two signal processing techniques commonly used to enhance the recorded image: gain and gamma factor correction. 23 Gain correction considers the digitization of incident light in order to make dim images brighter. The gain can be simply written in terms of the number of counts, Nc , for a given number of electrons produced, Ne , g= Ne . Nc (2.5) This definition is counter to the usual meaning of gain used in photomultiplier tubes and other detectors, where gain is simply amplification. As a camera’s gain decreases the amplification increases. Given the differences from application to application, the meaning of gain is not intuitive. To make matters worse most cameras report gain as its inverse. For example, consider a detector whose base gain is 8e− /count, which is reported as 1x. If the reported gain is increased by 4x, the true gain becomes 2e− /count, not 32e− /count . For our calculations we use the camera convention, but remember that the amplification is actually the reciprocal of gain. The second image processing technique is gamma factor correction, which is an exponential correction used to change the contrast of an image. Given an initial signal, V0 , gamma correction converts the input signal to a new signal 1/γ V = AV0 , (2.6) where A is a scaling factor and γ is known as the gamma factor. Figure 2.1 shows V versus V0 for several γ’s. Values of γ greater than 1 brighten dark regions decreasing the contrast, while values smaller than 1 increase the contrast. Taking gain and gamma factor correction into account we can write the digital signal, C, from a pixel as Z ∞ 1/γ 1 C= I(ω)SC (ω)∆tdω , g 0 24 (2.7) Figure 2.1: Effect of gamma factor correction on output signal for three γ’s. Increasing γ brightens dark regions, decreasing the contrast, while decreasing γ increases the contrast. 25 where SC is the sensitivity for a specific type of pixel. A monochrome camera has only one type and a color camera has three type: red (R), green (G), and blue (B). 2.2.3 Digital camera noise With the prolific use of digital cameras in scientific research, there has been much research into quantifying and minimizing noise associated with their operation [62,63]. There are several components of digital cameras which contribute to the noise: the CCD/CMOS array, on chip electronics, and off chip electronics. Noise in the actual semiconductor array is primarily due to quantum and thermal fluctuations, while noise associated with the electronics is typically related to the process of amplification and digitization of the signal from the semiconductor array. This section is organized based on the relative magnitude of the noise sources. Major Noise Sources Assuming a constant incident intensity, the three primary sources of noise for digital cameras are read noise, shot noise, and dark noise. Read noise (Nr ) is a combination of noise due to the electronics of the camera, the preamplifier, and analog-to-digital converters. It is typically specified by the manufacturer in the camera’s documentation. Given that read noise is an intrinsic characteristic of the camera unit, there is little that can be done by the end-user to minimize its effect, other than purchasing a camera with a smaller read noise value. Shot noise (Ns ) is due to the statistical nature of the photoconversion process and behaves as a counting error Ns = √ C. (2.8) where C is the signal. Given the statistical nature of shot noise, there is no method to eliminate the effect. However, since shot noise goes as the square root of the signal, 26 we can minimize its relative effect by increasing signal strength. The signal to noise √ ratio for shot noise is proportional to C. The final major noise source is dark noise (Nd ). Dark noise is due to thermal fluctuations which result in a finite number of electrons being excited into the conduction band of the semiconductor without any incident light. These fluctuations lead to a dark current given by [62] D = αAT 1.5e−Eg (T )/2kT , (2.9) where α is a scaling constant, A is the pixel area, T is the temperature, and Eg is the band gap of the semiconductor at temperature T . Dark noise behaves similarly to shot noise, but dark noise is the dark current integrated over the exposure time τ Nd = √ Dτ , (2.10) where we have assumed that the dark current is constant over the exposure time. Dark noise is the simplest noise source to limit as it depends nonlinearly on the temperature; small changes in the temperature will yield large changes in the dark noise level. For instance, cooling one of our cameras by 20◦ C, nearly eliminates the dark noise. Minor Noise sources There are several sources of noise which are small compared to shot, dark and read noise and are related to pixel non-uniformity, vibrational noise, light fluctuations, and electronic noise. When CCD/CMOS arrays are fabricated, much effort is made to make sure that all pixels have the same properties, yet there is always some inherent pixel to pixel variation. This variation results in the image produced by the array to vary spatially under uniform illumination. Typically this variation is very small 27 across the array, and the fluctuations due to shot, dark, and read noise overwhelm the noise introduced by pixel non-uniformity. Vibrational noise and light fluctuations are highly dependent on the camera mounting and light source used; therefore, much care is taken to mount the camera securely on a vibrationally isolated optical bench to minimize vibrations. Light sources are chosen for their uniformity and consistency. Electronic noise can be separated into four subtypes; 1. Off-chip electronic noise and electronic interference arise due to other electronics related to, or near, the CCD array. Off-chip electronic noise is introduced by the electronics supplementing the CCD array in the camera. Electronic interference is noise originating from other electronics and electromagnetic radiation. 2. Reset noise arises when the photo-induced charge in a pixel is converted into a voltage via an amplifier for measurement. Before each measurement the voltage is reset to a fixed reference level which varies slightly due to thermal fluctuations. The fluctuation of the reference level is known as reset noise. Typically higherend CCDs are designed such that this noise source is negligible, but in lower-end CCDs reset noise can dominate. 3. White noise comes from the operation of the output amplifier, which raises the voltage from the CCD for further processing. Similar to the conversion amplifier, the output amplifier has a resistance which generates thermal noise that follows the Johnson white noise equation Nwhite = √ 4kT BRout , Sa γ (2.11) where k is Boltzmann’s constant, B is the noise power bandwidth, Rout is the amplifier impedance, Sa is the amplifier sensitivity, and γ is the amplifier gain. Since the noise is independent of frequency the noise is known as white noise. 28 4. 1/f noise is associated with the output amplifier and is fundamentally related to quantum effects in solid state electronics. While the exact mechanism of 1/f noise is not well understood, it is found to empirically follow an inverse relationship given by [64–68] N(f ) = γ 2+β VDC , nc f α (2.12) where α, β, and γ are constants and nc is the number of charge carriers. Most of the electronic noise sources can not be minimized by the end user. Therefore the primary method of minimizing those noise sources is to purchase cameras designed to minimize those effects. Noise due to interference is minimized by using shielding and keeping the cameras away from noisy electronics. 2.3 Digital Imaging Measurements When performing transmittance imaging experiments we use two different imaging geometries: direct imaging, in which the camera is near the sample, and confocal imaging in which the camera is set some distance from the sample and an arrangement of lenses is used to project the image onto the camera. 2.3.1 Digital imaging microscopy The most simple experimental technique to characterize change in population of a molecule in a sample is digital imaging microscopy. The base setup consists of a digital camera with an attached microscope objective, probe light source, and a sample holder attached to a three axis translation stage. Also used is a pump light source, temperature controller, and/or a voltage source to apply an electric field to a sample. 29 Past reversible photodegradation measurements used amplified spontaneous emission, absorption spectroscopy, fluorescence and two photon fluorescence as a probe [1,43,46–48]. These techniques provided time resolved spectral data, which is related to population dynamics during decay and recovery. However, these techniques lack spatial resolution, and in experiments using focused light sources, yield a spatially convolved signal. This leads to a convolution between pump and probe intensity making it difficult to determine the effect of pump intensity on signal. Digital imaging allows for measuring the spatial profile of photodamage as a function of time, and therefore allows for precise measurement of intensity dependence. However, digital cameras convolute the incident light’s spectrum with a sensitivity function, making it impractical, if not impossible, to obtain spectral information. To relate the damaged population to the imaged transmittance, consider the color channel intensity at time t and position (x, y), Z ∞ 1/γ 1 C(x, y, t) = I(x, y, t; ω)SC (ω)τ dω , g 0 (2.13) where x = y = 0 corresponds to the center of the burned area. Recognizing that the intensity incident on the camera is the light transmitted through the sample given by I(x, y, t; ω) = I0 (x, y; ω)e−A(x,y,t;ω), (2.14) where I0 (x, y; ω) is the intensity incident on the sample and assumed to be constant in time, and A(x, y, t; ω) is the absorbance of the sample as a function of space, time, and frequency. Both the noninteracting model (Section 1.3.1) and the CCDM (Section 1.3.2) assume there are two species of absorbing molecules in DO11/PMMA samples, one undamaged and the other damaged, with normalized densities 1 − n and n, respectively. The two species are assumed to have absorption per unit length, σ(ω) for 30 the undamaged, and σ ′ (ω) for the damaged species. Assuming that samples are sufficiently thin such that damage is constant through their thickness, the absorbance may be written as A(x, y, t; ω) = [1 − n(x, y, t)]σ(ω)L + n(x, y, t)σ ′ (ω)L, (2.15) = σ(ω)L + n(x, y, t)[σ ′ (ω) − σ(ω)]L, (2.16) = σ(ω)L + n(x, y, t)∆σ(ω)L, (2.17) where L is the sample thickness and ∆σ(ω) = σ ′ (ω) − σ(ω). The absorbance may be rewritten in terms of the transmittance as T (x, y, t; ω) = I(x, y, t; ω) , I0 (ω) (2.18) = e−σ(ω)L−n(x,y,t)∆σ(ω)L , (2.19) = e−σ(ω)L e−n(x,y,t)∆σ(ω)L , (2.20) = T0 (ω)∆T (x, y, t; ω), (2.21) where T0 (ω) = e−σ(ω)L and ∆T (x, y, t; ω) = e−n(x,y,t)∆σ(ω)L . (2.22) Substituting into Equation 2.14 and 2.13, the color channel intensity may be written as, Z ∞ 1/γ τ C(x, y, t) = I0 (ω)SC (ω)T0 (ω)∆T (x, y, t; ω)dω . g 0 (2.23) Experimentally an LED light source is used for probe light, and has a narrow spectrum as seen in Figure 2.2. Given the LED’s narrow spectrum, we make a 31 Figure 2.2: Normalized spectra of LEDs used for illumination. simplifying approximation that the LED spectrum is a delta function centered at ω0 . Doing so, the integral in Equation 2.23 collapses to τ C(x, y, t) = I0 (ω0 )SC (ω0 )T0 (ω0 )∆T (x, y, t; ω0) g 1/γ . (2.24) For the case of the fresh sample with no damage, the change in transmittance is ∆T = 1. We can therefore use the fresh sample color channel intensity, C0 , to normalize the signal and get the change in transmittance directly according to ∆T (x, y, t; ω0) = C(x, y, t; ω0) C0 γ . (2.25) Comparing Equation 2.22 and 2.25, the damaged population is related to the measured color channel intensity as: − n(x, y, t)∆σ(ω0 )L = γ ln 32 C(x, y, t; ω0) C0 , (2.26) from which we define the scaled damage population n′ (x, y, t) = n(x, y, t)∆σ(ω0 )L. (2.27) In theory, it is a simple matter to retrieve the actual damaged population from Equation 2.27; in practice, ∆σ(ω0 )L varies with position on the sample due to variations in thickness and concentration from point to point. Thus the scaled damaged population is what is typically measured. It is important at this point to take a moment to consider the various approximations used to define the scaled damaged population. They are 1. Probe light intensity incident on the sample is uniform in space and constant in time. 2. The camera’s sensitivity is uniform in space and constant in time. 3. The absorbance is due to only two molecular species. 4. The samples are thin enough such that the damage is uniform in depth. 5. The probe light source has a delta function spectrum. The first two approximations have been found to hold, as much pain has been taken to ensure that the probe light and detector are stable over prolonged use. Chapter 3 will discuss the effects of sample thickness, and Chapter 4 will consider the effect of a third species. Appendix A discusses how the width of the probe light’s spectrum changes the measured scaled damaged population. 33 Figure 2.3: Confocal digital imaging microscope setup. The design is essentially the same as the digital imaging microscope with a collinear pump and probe beam focused onto the sample via a cylindrical lens (L2). The difference is in the additional confocal lens and iris to allow the camera to be a long distance from the sample. 2.3.2 Confocal Digital Imaging Microscopy and Temperature Chamber In the previous section, we discussed a basic setup for digital imaging microscopy, which involves placing a digital camera in close proximity to the sample being measured. This arrangement is suitable for isothermal measurements, but given the adverse effects that can occur to the camera at an elevated temperature, this simple setup is unsuitable. In order to use imaging to measure the effects of temperature on decay and recovery, a confocal apparatus was developed to remove the camera from close proximity to the sample. Figure 2.3 shows a schematic of the confocal imaging apparatus. The confocal digital imaging microscope (CDIM) uses a microscope objective and lens that share the same focal point (hence the name confocal) so that the image produced from the objective is projected to the camera a long distance from the sample. The use of an iris placed in between the lens and the camera blocks all out of focus light, allowing for higher resolution of the focal plane. For temperature dependent studies, a custom built temperature chamber is used, which includes an aluminum sample holder, resistive heating element, power supply, 34 (a) (b) Figure 2.4: (a) Damage profile with x and y axes specified. (b) Pump profile as a function of position. K-type thermocouple, and a CN Omega temperature controller. The controller is programmed to ramp the temperature to a set value, and then to hold it constant. 2.3.3 Relating scaled damaged population to intensity Both the DIM and CDIM are used to measure the spatial profile of damage as a function of time. Typically the damage profile is from a TEM00 cylindrical Gaussian pump beam as shown in Figure 2.4. Therefore we can use the known pump profile imaged by the CCD using ND filters - to relate the position in the image to intensity. Thus, we are able to resolve damage parameters as a function of intensity. 2.4 Conductivity measurements The effects of dye concentration and temperature on reversible photodegradation have been studied using amplified spontaneous emission [52,54,55,69], but the effects of an applied electric field were not studied. Given the wide variety of dyes found to exhibit reversible photodegradation, see Figure 1.1, DesAutels et.al. proposed that photocharge ejection and recombination may be the underlying mechanism of 35 Figure 2.5: DIM apparatus with a high voltage (HV) power supply to apply the electric field and a picoammeter to measure current. self healing [48]. If the underlying mechanism involves charged species, then the application of an electric field should change the decay and recovery characteristics. In order to test the effect of applying an electric field, a simple apparatus was developed consisting of an SRS 350 high voltage power supply and an RBD picoammeter connected in series with the sample. The conductivity apparatus is used to measure both dark conductivity and photoconductivity; and in conjunction with our DIM apparatus, measures the optical changes in reversible photodegradation due to an applied electric field. The full imaging and conductivity setup is shown in Figure 2.5. The samples used for applied electric field measurements are made of etched ITO glass substrates in contact with the dye doped polymer film, giving them exposed conductive surfaces. In order to minimize current leakage, an acrylic sample holder is used to position the sample. To apply the voltage, small flat aluminum strips are clamped to the exposed ITO, and the power supply wires are attached to the strips 36 using alligator clips. The alligator clips are not directly attached to the ITO, as direct attachment tends to scratch the ITO surface, diminishing contact reliability. As there are exposed contacts and high voltages, care must be taken not to touch any of the conducting surfaces while the voltage is applied. 2.5 Absorbance Measurements One of our most useful measurements for understanding the underlying mechanisms of reversible photodegradation is simple absorption spectroscopy. The main components of the absorbance setup are an Ocean Optics SD2000 spectrometer, an Ocean Optics Xenon PX2 probe light source, and a Verdi Nd:YAG CW laser. We utilize several positive lenses to focus the pump and probe beam such that the probe beam spot is smaller than the beam waist of the focused pump beam (approximately 750µm and 2000µm). Figure 2.6 shows a schematic of the absorbance setup. 2.6 White light interferometric microscope Simple spectrometers for reversible photodegradation studies pose two difficulties; first, they integrate over space, limiting the ability to relate damage to intensity, and secondly they only measure the imaginary part of the index of refraction. In order to improve on the spectrometer method, a white light interferometric microscope (WLIM) was developed and consists of a Michelson interferometer with a white light source (Thorlabs HPLS-30-03) and a CCD detector (EO 0813M). The WLIM allows for the measurement of both the real and imaginary parts of the index of refraction of a sample as a function of wavelength, position, and time. Figure 2.7 is a schematic representation of the WLIM, and Figure 2.8 is a photograph of the apparatus. 37 Figure 2.6: Schematic diagram of the absorption setup. The probe light source is an Ocean Optics Xenon PX2 light source and the spectrometer is an Ocean Optics SD2000. The pump laser is a Verdi Nd:YAG CW laser operating at 532nm with power control via crossed polarizers. Both the pump and probe beams are focused onto the sample using a positive lens, such that the probe spot is much smaller than the pump spot. 38 Figure 2.7: Schematic diagram of the WLIM. P1, P2 are crossed polarizers used to control the pump beam power. M1 is the stationary mirror in arm 1, which contains the sample that is damaged. M2 is the moving mirror on a piezo translation stage. The attenuating sample is in arm 2. The beam splitter and mirrors used are uncoated UV fused silica (UVFS), which has excellent transmission down to 300nm. Irises are used for alignment and blocking divergent white light incident on the interferometer. 39 Figure 2.8: Image of the WLIM. Drawn blue arrows show the path of the white light. Two prisms are used to align the white light exiting the fiber (red) with the optical axis after being collimated. 40 Figure 2.9: Example of an interferogram. The data are cropped to show the highest contrast fringes in detail. 2.6.1 Interferometer Theory1 When using the WLIM, each pixel of the camera measures an interferogram, I(x), as a function of path length difference, x (see Figure 2.9 for an example interferogram). The interferogram is converted into the spectral intensity, I(k0 ), by a Fourier transform, I(k0 ) = Z ∞ I(x)e−ik0 x dx, (2.28) −∞ where k0 = ω/c is the wavenumber in vacuum. The spectral intensity found this way is the interference intensity, given by 1 This section is an expanded version found in a manuscript submitted to Journal of Applied Physics 41 I(k0 ) ∝ E1∗ (k0 )E2 (k0 ) + E1 (k0 )E2∗ (k0 ), (2.29) where Ei (k0 ) is the electric field in arm i. For an electric field incident on the interferometer, E0 (k0 ), the complex electric field in each arm is: Ei (k0 ) = E0 (k0 )Si (k0 )eiΦi , (2.30) where i = {1, 2}, denotes either arm, Si (k0 ) is the spectral response of the optics in arm i, and Φi is the phase in arm i. Assuming E0 (k0 ) and Si (k0 ) are real quantities and substituting equation 2.30 into 2.29 we find the interference intensity to be I(k0 ) ∝ E0 (k0 )2 S1 (k0 )S2 (k0 ) exp{iΦ(k0 )} + c.c. (2.31) where Φ(k0 ) = Φ2 (k0 ) − Φ1 (k0 ) and c.c denotes the complex conjugate. Empty interferometer For the empty interferometer the zero path length difference phase, Φ(x = 0), in each arm can be written Φ1 = 2k0 y + φ1 (k0 ), (2.32) Φ2 = 2k0 y + φ2 (k0 ), (2.33) where y is the balanced arm length multiplied by two for roundtrip travel, and φi (k0 ) is the phase introduced due to the light not being a perfect plane wave and the optics not being perfectly flat. Combining the phases, we find that for the empty interferometer the phase difference (Φ(k0 ) = Φ2 (k0 ) − Φ1 (k0 )) between the arms is simply 42 Φ(k0 ) = φ2 (k0 ) − φ1 (k0 ). (2.34) Samples in both interferometer arms Given the highly absorbing nature of our samples and their relatively large index of refraction (n ≈ 1.5 [70]), nearly identical samples must be placed in each arm to maintain fringe contrast; “nearly identical” meaning each samples composition is identical, and therefore their complex index of refraction is the same, but their thickness and roughness may be different. The zero path length difference phase of each arm is Φ1 = 2k0(y − d1 − a1 ) + 2kg (k0 )a1 + 2k̃(k0 )d1 + ψ1 (k0 ) + φ1 (k0 ), (2.35) Φ2 = 2k0 (y − d2 − a2 ) + 2kg (k0 )a2 + 2k̃(k0 )d2 + ψ2 (k0 ) + φ2 (k0 ), (2.36) where di is the sample thickness, ai is the glass substrate thickness, kg (k0 ) is the real wavenumber of the glass, where we assume the imaginary portion is negligible, k̃(k0 ) = k0 ñ(k0 ) is the complex wavenumber of the dye doped polymer, and ψi (k0 ) is a phase factor introduced due to the samples not being perfectly flat and aligned, and φi comes from the empty interferometer phase. Combining phases and separating into the real, Φ′ , and imaginary, Φ′′ , parts we find Φ′ = 2(d2 − d1 )[k ′ (k0 ) − k0 ] + 2(a2 − a1 ) [kg (k0 ) − k0 ] + ψ1 (k0 ) − ψ2 (k0 ) + φ2 (k0 ) − φ1 (k0 ), Φ′′ = 2(d2 + d1 )k ′′ (k0 ). (2.37) (2.38) Equations 2.37 and 2.38 can be rewritten in terms of the absorbance per unit length, α(k0 ), and the real index of refraction, n′ (k0 ), by using the definitions of the real and imaginary parts of k̃(k0 ), 43 k ′ (k0 ) = k0 n′ (k0 ), (2.39) α(k0 ) . 2 (2.40) k ′′ (k0 ) = Substituting these into Equations 2.37 and 2.38 gives Φ′ = 2(d2 − d1 )k0 [n′ (k0 ) − 1] + 2(a2 − a1 )k0 [ng (k0 ) − 1] + ψ1 (k0 ) − ψ2 (k0 ) + φ2 (k0 ) − φ1 (k0 ), Φ′′ = (d2 + d1 )α(k0). (2.41) (2.42) Therefore the spectral intensity measured by the interferometer with samples in both arms is I(k0 ) ∝ E0 (k0 )2 S1 (k0 )S2 (k0 )e−(d2 +d1 )α(k0 ) × exp{i[2(d2 − d1 )k0 [n′ (k0 ) − 1] + 2(a2 − a1 )k0 [ng (k0 ) − 1] + ψ1 (k0 ) − ψ2 (k0 ) + φ2 (k0 ) − φ1 (k0 )]}. (2.43) While white light interferometry has been used to measure the absolute complex index of refraction of many materials including glass [71], gases [72], and liquids [73–75]; inspection of Equation 2.43 shows that for a sample in each arm, the measured phase difference is proportional to the thickness difference d2 − d1 . Therefore if the samples are of the same thickness, the phase change due to one sample’s index of refraction, is canceled by the other samples phase change, so that the net phase difference is zero. Even in the case of nonidentical sample thickness, there are problems finding the absolute index of refraction due to substrate thickness differences. Simple absorbance spectroscopy measurements have shown that for spin 44 coated thin films, the thickness variation from point to point is typically on the order of 1µm, implying that d2 − d1 ≤ 1µm, while measurements of the glass substrates using micrometers show a substrate thickness difference of a2 − a1 ≤ 40µm. Since the real index of refraction of the thin film is similar to the glass substrate, the phase due to substrate differences is approximately 40 times as large as that due to the dye doped polymer. These complications make measuring the absolute index of refraction of the dye doped polymer extremely difficult using the WLIM. Effect of photodegradation of sample in one arm While measuring the absolute index of refraction using the WLIM is difficult, it is relatively simple to measure the change in the index of refraction due to photodegradation. One of the assumptions in our previous section was that the samples in each arm have the same complex index of refraction and only vary in thickness. While this is a good approximation for fresh samples, this assumption breaks down once one of the films starts to photodegrade. Letting n′0 (k0 ) and n′d (k0 ) denote the undamaged and damaged index of refraction, respectively, and αu (k0 ) and αd (k0 ) to denote the undamaged and damaged abosrbance per unit length, respectively, and letting the sample in arm one be exposed to the pump to induce photodegradation, we can rewrite Equations 2.41 and 2.42 as Φ′ = 2k0 (d1 − d2 + n′0 (k0 )d2 − n′d (k0 )d1 ) + 2(a2 − a1 )k0 [ng (k0 ) − 1] + ψ1 (k0 ) − ψ2 (k0 ) + φ2 (k0 ) − φ1 (k0 ), Φ′′ = d2 αu (k0 ) + d1 αd (k0 ). (2.44) (2.45) If we take the difference of the undamaged and damaged phases we find that 45 Φ′u (k0 ) − Φ′d (k0 ) = 2k0 d1 [nd (k0 ) − n0 (k0 )], (2.46) Φ′′u (k0 ) − Φ′′d (k0 ) = d1 [αd (k0 ) − αu (k0 )]. (2.47) Equation 2.46 may be rewritten to give the change in the index of refraction due to photodegradation as ∆n(k0 ) = ∆Φ′ , 2k0 d1 (2.48) where ∆Φ′ = Φ′u (k0 ) − Φ′d (k0 ). 2.6.2 Discrete Fourier Transforms(DFT’s)2 In the previous section the interferogram was transformed using the continuous Fourier transform which requires an infinite path length domain. In practice, an interferogram is measured in N discrete steps of size δx over a length L. Therefore when analyzing interferograms, we must use a discrete Fourier transform, which is given by N −1 1 X I(km ) = I(xn )e−ikm xn , N n=0 (2.49) where xn = n∆x, km = m∆k0 , m and n are integers, and ∆x = 2δx as a step of δx corresponds to a path length difference of 2δx, and the value ∆k0 is determined from reciprocity relations for Fourier transforms [76], 2 Notation and indexing conventions for DFT’s vary widely in literature. See Briggs and Henson The DFT: An Owner’s Manual for the Discrete Fourier Transform for a comparison of different notations and indexing conventions [76]. My notation is based on the conjugate variable pair k0 and x. 46 N∆x∆k0 = 2π, (2.50) 2π , N∆x π ∆k0 = , L ∆k0 = (2.51) (2.52) Therefore the maximum k0 that may be measured for a given sampling interval δx is kM AX = N∆k0 , (2.53) πN , L (2.54) kM AX = Amplitude and phase The Fourier transform of an interferogram gives a complex spectral intensity with I ′ (km ) = N −1 1 X I(xn ) cos(−ikm xn ), N n=0 (2.55) N −1 1 X I(xn ) sin(−ikm xn ), I (km ) = N n=0 ′′ (2.56) where I(km ) = I ′ (km ) + iI ′′ (km ). From the real and imaginary parts we can define the amplitude and phase: A= p |I ′ |2 + |I ′′ |2 , (2.57) ΦDF T = Arg(I ′ + iI ′′ ), = 2 tan−1 I ′′ p |I ′ |2 + |I ′′ |2 + I ′ (2.58) ! . (2.59) where the phase is given by the Arg function, which is related to the two term arctangent function. 47 Figure 2.10: (a) Wrapped phase; (b) Unwrapped phase. The amplitude found with the DFT is easily related to the spectral intensity, but the use of the two term arctangent function to determine the DFT phase, ΦDF T , makes finding the physical phase, Φ′ , more complicated, as the two term arctangent function is bounded on the interval [−π, π]. The process of extracting the physical phase from the DFT phase is call phase unwrapping and works by systematically adding or subtracting multiples of 2π every time the wrapped phase jumps by π. The simplest way to understand phase unwrapping is by looking at it graphically. Figure 2.10 shows the wrapped phase and the unwrapped phase side by side. When the wrapped phase exceeds π, it jumps to −π. This discontinuity arises from the finite domain of the arctangent function and is called “wrapping”. To unwrap the jump, 2π is added at the discontinuity, which makes the phase continuous. In theory the process of phase unwrapping gives the exact physical phase, but in practice there are several subtle issues due to the discrete nature of measurements and how this is taken into account. For the discrete case, phase unwrapping algorithms add an integer multiple of 2π if the phase changes by > π when going through each ∆k0 step. So long as the change 48 in the true phase for each step is bounded by | ∆Φ | ≤ 2π, ∆k0 (2.60) the algorithm will produce the correct behavior. If the change in true phase for a step is greater than 2π, the algorithm will miss the jump and produce the incorrect phase. In order to avoid phase uncertainty we use small enough step sizes such that the phase should not change by more than 2π for each wavenumber step. Along with algorithm limitations due to finite step size, noise poses difficulties for the phase unwrapping algorithm. When measuring an interferogram, the underlying spectrum has a finite bandwidth; therefore, the signal-to-noise ratio changes drastically over the spectrum, having a large SNR where the intensity is high, and small in the wings where the spectrum is dominated by noise. Since phase unwrapping is an iterative process starting with k0 = 0 and making steps in intervals of ∆k0 , the noise in the wrapped phase due to the low SNR regions is added into the unwrapped phase in the large SNR regions, which leads to the introduction of arbitrary phase shifts. As an example, 10000 interferograms are taken using the empty WLIM and then converted into the unwrapped phase. The phases are found to vary greatly in regions of low SNR, while in the regions of high SNR the phases are found to have the same functional form, but offset from each other by a constant amount. The constant offset is found to vary randomly in both the positive and negative direction, therefore in order to extract the true physical phase we average over many unwrapped phases to find the phase with zero offset. 2.6.3 Interferometer Alignment In order to produce a usable interferogram the mirrors of the Michelson Interferometer must be aligned and positioned such that the path length difference between the arms 49 is within the coherence length of the light source (typically 40-60 µm for white light sources). The alignment and positioning procedure is as follows. Positioning begins by adjusting the scanning arm so that the distance from the mirror surface to beamsplitter face is within 1mm of the stationary mirror’s distance from surface to beamsplitter. The scanning mirror is then moved inward so scanning in the outward direction will pass through the point where the arms are balanced. Once the initial position is set, a beamsplitter is placed before the irises so that a HeNe laser can be aligned collinear with the collimated white light and a white piece of paper is placed in the output arm of the interferometer to make the interference pattern more visible. Initially, the unaligned mirrors and beam splitter will produce a pattern of spots on the paper as the light is reflected multiple times in different directions due to misalignment. Cardboard beam blocks are placed in each arm to remove reflections from the mirrors and then the beam splitter is aligned such that the back reflection of the incoming laser beam is aligned with the original beam. After aligning the beamsplitter one beam block is removed to uncover one mirror, producing two spots on the paper. Using the tilt mount controls, the mirror’s reflected beam is adjusted so that it overlaps with the beamsplitter’s beam. Once the first mirror is aligned the second mirror’s beam block is removed producing a new spot on the paper, which is moved using the second mirror’s tilt controls to align the second mirror’s beam with the beam from the first mirror and beamsplitter. As the beams get close to being aligned an interference pattern appears, typically with bright and dark lines. If the pattern is a set of straight horizontal lines the mirrors are aligned in the horizontal direction, and only movement in the vertical direction is needed. If the pattern is a set of straight vertical lines the mirrors are aligned in the vertical direction and only horizontal adjustments are needed to bring the interferometer into alignment. 50 Figure 2.11: White light fringe patterns for different alignments. (a) misaligned in both directions, (b) misaligned in the horizontal direction, (c) misaligned in the vertical direction, (d) correct alignment A pattern of skewed lines implies that adjustment is needed in both directions. The ideal aligned pattern is a set of concentric rings in the form of a bullseye. Figure 2.11 shows the various patterns for white light. Once a centered bullseye interference pattern is produce the laser coupling beam splitter is removed so that only the white light is incident on the interferometer and a systematic search for the white light coherence range begins. Since the moveable mirror was originally moved closer to the beamsplitter, the mirror is moved outward in increments of 15 − 25µm. When far from the coherence range the interferometer 51 output is a spot of steady white light. When approaching the coherence range the pattern will begin to show faint colors that change with mirror movement if aligned properly. If the alignment is still slightly off the pattern will consist of light and dark fringes, in which case the alignment procedure should be repeated, but with the white light pattern. At this point the step size is decreased to approximately 5µm and steps are performed systematically in both the inward and outward direction until the pattern contrast is greatest. At this point any residual misalignment is corrected. For a perfect interferometer the pattern should be a bullseye with colorful fringes. In order to produce such a pattern, all optics should be within λ/10 flatness. While the mirrors in the interferometer are of the correct order of flatness, the beam splitter is rougher, on the order of λ/5. The beam splitter’s imperfection introduces wavefront distortions which result in the actual interference pattern being distorted into a saddle point shape. Repeated trials and measurements have shown that the effect of the distortion primarily arises in the absolute phase. In general this is acceptable for our purposes as we are primarily interested in changes in the measured values, and not their absolute values. 2.6.4 WLIM Measurement Procedure The interferograms produced by the WLIM are recorded using code developed for LabVIEW 2011 full developer suite that controls the piezo translation stage and camera. Analysis was done using code written for Igor Pro. A typical data collection run proceeds as follows. The light source is initially turned on one hour before collecting data to insure that the light source reaches a stable output. After finding the center of the coherence range - where the interference pattern has the highest contrast - the moveable mirror is positioned so that a single sweep of the mirror will cover most of the coherence range with the highest contrast 52 in the center of the interferogram. With the interferometer aligned and positioned, the white light interferogram is measured by moving the piezo stage in 20nm steps over 1000 steps, the full range of the piezo crystal. After each step the mirror pauses while the camera takes ten images. which are then averaged together to produce one output image for each step. After measuring the white light’s spectrum with the above procedure, nearly identical samples are placed in each arm of the interferometer. Since the samples are neither perfectly flat nor perfectly aligned they distort the interference pattern requiring slight readjustment and repositioning of the mirrors. Once the adjustments have been made the same data acquisition process is repeated with the samples in the interferometer. The two measurements give the background white light spectrum and the base sample spectrum, after which the sample in arm 1 is damaged using an ArKr laser focused to a line with a cylindrical lens (Iavg ≈ 60 W/cm2 , ∆t = 30min). After the damage cycle is complete, the pump laser is blocked and the sample is remeasured to find the spectrum after photodegradation. As a final consistency check the sample is removed and the while light is remeasured to ensure that the spectrum of the white light has not changed over the measurement period. Once the images for the empty interferometer, interferometer with fresh sample, and interferometer with damaged sample are collected, they are imported into Igor pro. The damaged region is isolated, and Igor converts the corresponding pixels into interferograms which are then Fourier transformed using Igor’s FFT algorithm to find the magnitude and wrapped phase. The wrapped phase is unwrapped utilizing custom code designed to minimize the effects of noise on the unwrapping process. The resulting magnitude and unwrapped phase is relatively noisy with large pixel to pixel variations; therefore, an averaging procedure is used to determine the mean magnitude and phase over adjacent pixels. Given the use of a cylindrical lens, the burn 53 has a long axis, x, over which the damage gradient is small, and a short axis, y, over which the damage gradient is large. For the averaging procedure, 20 adjacent pixels along the burn line are averaged together to produce the magnitude and phase for that x and y coordinate. Performing the procedure across the burn line produces dose dependent measurements of the amplitude and phase taking advantage of the large intensity gradient. The amplitudes from the various cases (empty interferometer, fresh sample, damaged sample) are used to find the change in absorbance due to photodegradation; with the absorbance of the fresh, α0 , and damaged, αd , samples being A0 α0 = − ln , AW L Ad , αd = − ln AW L (2.61) (2.62) where AW L is the intensity of the white light, A0 is the transmitted intensity through the fresh sample, and Ad is the transmitted intensity of the damaged sample. For the change in the complex index of refraction due to photodegradation, we take the difference in the averaged unwrapped phase for the fresh and damaged samples, and use Equation 2.48 to find the change in the index of refraction. 2.6.5 WLIM Limitations3 While in principle, the WLIM has the ability to measure the complex index of refraction, there are inherent limitations due to the finite measurement range, beam divergence, mirror misalignment, and imperfect wavefronts. For the purpose of deriving the inherent limitations, we will assume that the detector is circular with radius 3 The mathematics in this section is based on a technical report by D.R. Hearn on Fourier trans- form interferometry [77] 54 R. Resolution Ideally when using interferometric techniques we would measure an infinite interferogram with −∞ < x < ∞. However, in reality, we are limited to a finite range, namely −L < x < L, so a Fourier transform of an experimental interferogram, Im (k0 ), is an approximation of the true spectral function, Im (k0 ) = Z L I(x)e−ik0 x dx, Z−L ∞ x I(x)e−ik0 x dx. = Π 2L −∞ (2.63) (2.64) where Π is the symmetric unit rectangle window function. Using the product properties of the Fourier transform we can write the measured spectral function as h x i Im (k0 ) = F Π I(k0 ), 2L where I(k0 ) is the true spectral function and h x i F Π = 2Lsinc(2Lk0 ), 2L (2.65) (2.66) is the Fourier transform of the window function. From Equations 2.65 and 2.66, the effect of a finite range on the WLIM’s measurement is to convolute the true spectrum with an instrument resolution function. This limits the spectral resolution of the measured spectral function. By convention the first zero of the instrument resolution function occurs at ∆k0 = 1 2L is called the unapodized spectral resolution. Apodization So far when considering the incoming light beam we have assumed that it has a negligible angular divergence (i.e. the rays are parallel to the beam axis). The 55 effect of a nonzero angular divergence in the WLIM is known as apodization and is related to the solid angle of the detector. The geometry of the problem is shown in Figure 2.12, with angles being exaggerated for clarity. For the point connected by the vector A, the optical path length difference(OPD) is modified from the on axis OPD by x → x cos α, where x is the on axis OPD. In this geometry, the interferogram measured by the detector is I(x) → = Z Z ∞ I(k0 ) −∞ ∞ 1 Ω Z π −π Z ρ0 −ik0 x cos α(ρ) e sin ρdρdφ dk0 , 0 I(k0 )G(k0 , ρ0 )dk0 , (2.67) (2.68) −∞ where ρ0 is the angle formed between A0 and the rim of the detector, the interferogram is integrated over field of view of the detector which has a solid angle: Ω= Z π −π Z ρ0 sin ρdρdφ = 2π(1 − cos ρ0 ), 0 (2.69) and the G function is given by 1 G(k0 , ρ0 ) = Ω Z π −π Z ρ0 e−ik0 x cos α(ρ,φ) sin ρdρdφ. (2.70) 0 The general solution for Equation 2.70 will not be dealt with here, but can be found in Hearn [77]. Instead, we will consider the case where α0 = 0 which gives ρ = α, and the G function becomes 2π G(k0 , ρ0 ) = Ω Z ρ0 e−ik0 x cos α sin αdα. (2.71) 0 To solve Equation 2.71, we define y = cos α, which leads to, 2π G(k0 , ρ0 ) = Ω Z 56 1 eik0 xy dy, cos α0 (2.72) Figure 2.12: Geometry for detector off center from optical (beam) axis by an angle α0 . (a) General geometry for diverging rays incident on the detector, (b) spherical geometry of the angles α0 , α, and ρ. A0 connects the interferometer to center of the detector, with the angle between the optical axis, 0, and A0 being α0 , A connects the interferometer to any point on the detector with an angle α being made with the optical axis, r connects A0 and A in the detector plane, ρ is the angle between A0 and A, R is the radius of the detector, and φ is the azimuthal angle in the detector plane. The angles, α0 , α, and ρ are assumed to be related in a locally flat region on the sphere due to their small sizes. 57 which upon integration gives G(k0 , ρ0 ) = 2π eik0 x − eik0 x cos α0 . iΩk0 x (2.73) Using the definition of the solid angle in Equation 2.69 and trig identities we can simplify Equation 2.73 to give: Ω Ωk0 x 4π exp ik0 x 1 − , sin G(k0 , ρ0 ) = Ωk0 x 4π 4π (2.74) Ω Ωk0 x = sinc exp ik0 x 1 − . 4π 4π (2.75) Substituting Equation 2.75 into 2.68 the apodized interferogram is given by: ∞ Ω Ωk0 x exp ik0 x 1 − dk0. I(x) = I(k0 )sinc 4π 4π −∞ Z (2.76) The effect of a non-zero field of view can be seen by inspection of equation 2.76: the apparent optical path length difference is modified by 1 − Ω , 4π and the fringes are modulated by a sinc function, which results in decreased fringe contrast as |x| is increased. Throughput The optical throughput of the interferometer is related to the intensity incident on the detector, and is found to be AΩ, where A is the modulation factor and Ω is the detector solid angle. In the following derivation we show that the optical throughput is inversely related to the spectral resolution, and therefore there is a tradeoff when increasing either throughput or spectral resolution. A measure of the spectral resolution of an interferometer is it’s spectral resolving power, Pr = k0 = 2k0 L. ∆k0 58 (2.77) Using the spectral resolving power we can rewrite the modulation factor determined in the previous section as Ωk0 L ΩPr sinc = sinc . 4π 4π (2.78) The optical throughput may therefore be written as 4π Pr Ω Ωk0 L = . sin Ωsinc 4π Pr 4π (2.79) If we consider only the central lobe of the sinc function, we can find the solid angle which gives the maximum throughput as ΩM AX = 2π , Pr (2.80) which shows that as we increase the spectral resolution (i.e. make Pr bigger), the throughput decreases. Mirror Misalignment While much care is taken in the alignment of the mirrors there is always some small error in mirror alignment. Assuming that the moving mirror is misaligned by an angle ǫ, the optical pathlength difference at the detector will be x′ = x + 2ǫr sin φ, (2.81) where (r, φ) represents polar coordinates on the detector. The interferogram therefore becomes I(x) = Z ∞ I(k0 )Hǫ (k0 , x)dk0 , −∞ where, 59 (2.82) Z πZ R 1 e−ik0 (x+2ǫr sin φ) rdrdφ, Hǫ (k0 , x) = 2 πR −π 0 Z Z e−ik0 x π R −2ik0 ǫr sin φ) = e rdrdφ, πR2 −π 0 Z Z e−ik0 x π R = [cos(2k0 ǫr sin φ)) − i sin(2k0 ǫr sin φ))] rdrdφ, πR2 −π 0 (2.83) (2.84) (2.85) with R being the radius of the detector. Equation 2.85 may be integrated over the angular coordinate, recalling the following identities: Z π cos(x sin φ)dφ = 2πJ0 (x), −π Z π sin(x sin φ)dφ = 0, −π Z xn Jn−1 (x)dx = xn Jn (x), (2.86) (2.87) (2.88) where Jn (x) is the nth Bessel function. Equation 2.85 therefore becomes Z e−ik0 x R Hǫ (k0 , x) = 2πJ0 (2k0 ǫr)rdr, πR2 0 1 = J1 (2k0 ǫR)e−ik0 x . k0 ǫR (2.89) (2.90) The interferogram is thus found to be modulated by a function, 2 M(y) = J1 (y), y (2.91) where y = 2k0 ǫR. Assuming y << 1, the modulation function can be expanded using a second-order taylor series as: 2 y2 J1 (y) ≈ 1 − . y 8 (2.92) which gives the approximate modulation for small angles. From the condition y << 1, the small angle regime is given by 60 ǫ << λ 1 = . 2k0 R 4πR (2.93) Wavefront errors Michelson interferometers are sensitive to changes in optical path length difference between the two arms. In the previous section we considered how mirror misalignment effects the optical path length difference. Another source of optical path length difference is wavefront errors introduced by the beamsplitter, samples, and mirrors. To analyze the effect a similar procedure to the previous section is used, with the interferogram being given by, I(x) = Z ∞ I(k0 )Hδ (k0 , x)dk0 , (2.94) −∞ where, Z πZ R 1 Hδ (k0 , x) = e−ik0 (x+δ(r,φ)) rdrdφ, πR2 −π 0 Z Z e−ik0 x π R cos(k0 δ(r, φ)) − i sin(k0 δ(r, φ))rdrdφ, = πR2 −π 0 (2.95) (2.96) and k0 δ(r, φ) is the wavefront error at radius r and azimuth φ on the detector. Since it is possible to measure an effective interferogram, the wavefront error must be small, so the sine and cosine can be expanded as a power series to second order 1 cos(k0 δ(r, φ)) ≈ 1 − (k0 δ(r, φ))2 , 2 (2.97) sin(k0 δ(r, φ)) ≈ k0 δ(r, φ). (2.98) Given that the wavefront error is random and can either be positive or negative, the mean value over the detector is assumed to be zero, or: 61 Z π −π Z R δ(r, φ)rdrdφ = 0. (2.99) 0 Applying these approximations to Equation 2.96 the H function becomes Z Z e−ik0 x π R Hδ (k0 , x) = cos(k0 δ(r, φ)) − i sin(k0 δ(r, φ))rdrdφ, πR2 −π 0 Z Z e−ik0 x π R 1 (k0 δ(r, φ))2 )rdrdφ, ≈ (1 − πR2 −π 0 2 k02 2 −ik0 x 1 − hδ i , =e 2 (2.100) (2.101) (2.102) where hδ 2 i is the mean-squared value of the wavefront error. As in the previous sections, the result of a non ideal case is to introduce a modulation factor into the interferogram which reduces fringe contrast by a factor k02 2 hδ i. 2 2.6.6 (2.103) WLIM Noise Section 2.6.5 described the inherent limitations of the WLIM due to alignment errors and optical imperfections. Added to the inherent limitations of the WLIM there are also experimental limitations that arise as noise. Section 2.2.3 described the noise sources of a digital camera. Errors in the moving mirror, which will be discussed in the following sections also adds to the noise. To begin the discussion of WLIM noise we consider how noise in the interferogram converts to noise in the spectral domain. From Fourier analysis it is know that we can relate a function, f (x), and its Fourier transform, F (k0 ), using the Rayleigh power theorem. For the discrete case of a function measured at steps of equal spacing, ∆x, the Rayleigh power theorem may be written as 62 ∆x N −1 X ∗ fm fm = ∆k0 m=0 N −1 X Fj Fj∗ , (2.104) j=0 where ∆k0 is the frequency spacing as discussed in Section 2.6.2. Equation 2.104 may be expressed in terms of the mean values of the functions as N∆xh|f |2 i = N∆k0 h|F |2i, (2.105) 2Lh|f |2 i = kM ax h|F |2i, (2.106) where the brackets denote averaging over the whole function; and ∆x and ∆k0 from Section 2.6.2 are used. Assuming that the function is a measurement of the noise in an interferogram, the relationship between noise in the path length domain and nose in the spectral domain is given by 2Lσx2 = kmax σk2 , (2.107) where σx2 is the mean variance in the path length domain, and σk2 is the mean variance in the spectral domain. After some rearrangement and substitutions, the spectral error may be written as σk = r 2L2 σx . πN (2.108) Sampling Errors Ideally the WLIM should take measurements in identically-spaced increments over the whole range of the interferometer. In practice though, there is always some small positioning uncertainty, which may be either systematic, or random. For the case of systematic errors, they may simply be addressed by adjusting the interferogram 63 spacing using interpolation methods. However, random sampling error is more complicated, and has been modeled by Bell and Sanderson [78] and is reproduced below. For small optical path errors, the measured interferogram, Im , at nominal position xn will be Im (xn ) = I(xn ) + ǫn dI(x) dx , (2.109) x=xn where ǫn is the small positioning error, and I(x) is the true interferogram. Therefore the Fourier transform of the measured interferogram will yield N −1 dI(x) 1 X ikp xn ikp xn e I(xn )e + ǫn , Im (kp ) = N n=0 dx x=xn (2.110) Since the errors are random, the average value of the spectral error over many interferograms at every spectral position is dI(x) , hσk i = ǫn dx x=xn dI(x) , = hǫn i dx x=xn (2.111) (2.112) = 0. (2.113) While the mean spectral error is zero, the mean square spectral error is nonzero and may be written as σk2 2 = ǫ (∆x) 2 N −1 X n=0 2 where ǫ = hǫ2n i. dI(x) dx x=xn 2 , (2.114) Using Rayleigh’s spectral theorem, the sum in Equation 2.114 maybe expressed as (∆x) 2 X dI(x) n=0 dx x=xn 2 64 = ∆x∆k0 N −1 X p=0 kp2 |I(kp )|2 , (2.115) where the Foruier transform of a derivative is given by dI(x) = ikI(k). F dx (2.116) Thus the root mean square spectral error may be written as σ = ǫ[kI(k)]rms , (2.117) where, 2 ([kI(k)]rms ) = ∆x∆k0 N −1 X p=0 = kp2 |I(kp )|2 , N −1 2π X 2 k |I(kp )|2 . N p=0 p (2.118) (2.119) Optical Jitter-Induced Noise Optical jitter is a tilt error in the mirrors due to a variety of factors such as vibrations, variations due to mirror velocity, and thermal expansion/contraction of the mirrors. The tilt error can either be systematic, ǫ0 , or random, α, with the random effect being optical jitter. The net tilt error therefore is ǫ = ǫ0 + α. (2.120) Using the equation for fringe modulation due to misalignment, Equation 2.92, the total optical modulation is M =1− k02 R2 (ǫ0 + α)2 . 2 (2.121) The difference in optical modulation due to jitter alone is therefore ∆M = − k02 R2 (2ǫ0 α + α2 ). 2 65 (2.122) Using Equation 2.122 the variance in the optical modulation due to optical jitter is given by: 2 σM = h(∆M)2 i − h∆Mi2 , = k04 R4 (4hǫ20 iσα2 + σα4 ), 4 (2.123) (2.124) where σα2 = hα2 i, and hαi = 0. Therefore the noise due to optical jitter is given by σjitt = k02 R2 σα 2 q 66 4hǫ20 i + σα2 . (2.125) Chapter 3 Modeling Depth Effects 3.1 Introduction Previously, reversible photodegradation studies have assumed that the pump laser beam has a constant intensity along the beam axis. For a sample with thickness much less than the 1/e absorption length, this approximation is valid. However, with the majority of samples tested having thickness larger than the 1/e absorption length, corrections for pump absorption must be made. In addition to absorption, the pump beam also experiences propagation effects, which change the beam profile as a function of depth, and may be separated into three classes: normal linear wave propagation, photodamage-induced lensing, and thermally-induced self (de-)focusing. In this chapter we will extend the two population noninteracting model (TPNIM) of reversible photodegradation to include depth effects. This procedure may be used in conjunction with other population models (such as the correlated chromophore domain model), but the simplest mathematical population model suffices to describe the phenomenon. 67 3.2 3.2.1 Depth effects due to pump absorption Effect of depth on population decay Pump absorption leads to a decreasing intensity as a function of the depth in a sample, which results in the amount of damage varying with depth. Thus the population at the entry surface of the pump beam is far more damaged than the population at the exit surface. The undamaged population as a function of depth during decay is modeled using the differential Beer-Lambert law, and the TPNIM equations: ∂Ip (z, t) = − {n(z, t)σ0 (ωp ) + [1 − n(z, t))σ1 (ωp ]} Ip (z, t), ∂z ∂I(z, t; ω) = − {n(z, t)σ0 (ω) + [1 − n(z, t))σ1 (ω)]} I(z, t; ω), ∂z ∂n(z, t) = −αI(z, t)n(z, t) + β [1 − n(z, t)] , ∂t (3.1) (3.2) (3.3) where Ip (z, t) is the pump intensity, I(z, t; ω) is the probe intensity, n(z, t) is the undamaged population, σ0 (ω) and σ1 (ω) are the undamaged and damaged absorbance per unit length, respectively, ωp is the pump frequency, α is the intensity independent decay rate, and β is the recovery rate. To solve Equations 3.1 - 3.3 we assume that the undamaged population is constant as a function of depth at t = 0, and that the intensity at z = 0 is constant in time, Ip (0, t) = Ip,0 , (3.4) I(0, t; ω) = I0 (ω). (3.5) n(z, 0) = 1. (3.6) Equations 3.1-3.6 have no closed form solution, but are simple to solve numerically. Section 3.2.3 uses fixed parameters to predict the functional form of the population 68 and intensity during decay, and Section 3.2.4 fits experimental decay data to the model. 3.2.2 Effect of absorption depth profile on recovery For recovery, beginning at t = t0 , the probe intensity and population may be expressed as ∂I(z, t; ω) = −[n(z, t)σ0 (ω) + (1 − n(z, t))σ1 (ω)]I(z, t), ∂z ∂n(z, t) = β[1 − n(z, t)], ∂t (3.7) (3.8) which have solutions Z L I(z, t; ω) = I0 (ω) exp − [n(z, t)σ0 (ω) + (1 − n(z, t)σ1 (ω)]dz , (3.9) 0 n(z, t) = 1 + (n(z, t0 ) − 1)e−βt , (3.10) where I0 is the incident probe intensity, and n(z, t0 ) is the undamaged population when the pump is turned off. Since the recovery rate is constant as a function of population and therefore depth, a sample’s absorption profile with depth will only affect the probe intensity measured during recovery, and therefore it is a minor effect when compared to the depth effect during decay. The following sections thus consider only photodegradation since the primary influence of pump absorption is observed during decay. 3.2.3 Numerical Results Since Equations 3.1-3.6 have no closed form solution we consider numerical solutions using parameters that are consistent with experiments, but chosen to best demon- 69 strate the effect of pump absorption, Table 3.1 shows the parameter values. Model Parameters × 10−3 cm2 /W min α 0.47 β 1.935 × 10−3 min−1 Ip 132 × W/cm2 σ0 (ωp ) 2.66 × 10−2 µm−1 σ1 (ωp ) 0.798 × 10−2 µm−1 Table 3.1: Model parameters used for predicting the functional form of population and intensity as a function of depth during decay. While the model predicts the population and intensity as a function of depth and time, optical transmittance experiments do not directly measure the population and intensity at a given depth, but instead measure the probe intensity transmitted through the sample, which may be related to the scaled damage population (SDP), n′ , by ′ n = − ln I(t) − I(0) , I(0) (3.11) where I(t) is the measured intensity at time t, I(0) is the intensity measured before photodegradation, and the SDP in this case is understood to be an average population over the thickness of the sample. Figure 3.1 shows the SDP computed for several sample thicknesses using the numerical solutions to the model. There are two important features of the SDP as the thickness is increased: first, the apparent damage is greater as the thickness is increased, which is consistent with the earlier definition of the SDP as n′ = n∆σL. Secondly, the simple exponential decay rate decreases as the thickness is increased. 70 Figure 3.1: Predicted scaled damaged population as a function of time for different sample thicknesses. Note that as the sample thickness increases the decay rate appears to decrease. Figure 3.2 shows the numerically predicted population during decay for several depths as well as the population determined from the scaled damaged population. At the entrance surface the population decays quickly down to 10% of its initial value, while deeper into the sample the decay is slower and to a smaller degree. Using the predicted transmitted probe intensity, the scaled damaged population is calculated and converted into the population that would be determined if depth effects are not taken into account. As can be seen, the decay of the “average” population is slower then the decay at the surface, and the amount of decay for the “average” population is smaller than the amount of decay for the population at the surface. Therefore, neglecting the pump absorption effect will underestimate the true decay rate and damage amount. In addition to modeling the population, we also model the pump beam intensity 71 Figure 3.2: Predicted population as a function of time at various depths. The average population is what would be measured had the depth absorption profile not been taken into account. profile as a function of depth for several times as shown in Figure 3.3. As the population decays the pump beam profile deviates from the exponential form predicted by the Beer-Lambert law, with the intensity within the sample increasing with time. Both the intensity and population are found to follow the functional form f (z, t) = A − B tanh(γ(t)z + ζ(z)t + η), (3.12) where the parameters A, B, and η are constants, the parameter γ changes with time, and the parameter ζ changes with depth. 3.2.4 Comparison with Data Experiments using thicker samples have been found to decay more quickly and to a higher degree, which is consistent with the numerical results in Section 3.2.3. How- 72 Figure 3.3: Predicted intensity as a function of depth at various times. ever, quantitative results are difficult to determine as there are many variables that are difficult to control from sample to sample leading to large variations when comparing samples. As a rough test of the depth model, four 9g/l samples are chosen with average thicknesses of 8µm, 22 µm, 35 µm, 83 µm, with the thickness being determined by measuring the absorption spectrum at several spots on the sample and converting to thickness using L= A , ǫc (3.13) where A is the absorbance, ǫ is the molecular absorbance cross section, and c is the molecular concentration. The samples are placed in the DIM and burned for ten minutes with an ArKr laser operating at 488nm, focused to a peak intensity of 120 W/cm2 . Images are taken in one minute intervals and the scaled damaged population is computed as a function of 73 Figure 3.4: Scaled damaged population during decay for four 9g/l samples of differing thicknesses with an incident intensity of 120W/cm2 . position, and correlated to the intensity profile assuming a perfect TEM00 elliptical Gaussian pump beam. While the SDP for each sample is fit at multiple intensities, for simplicity the SDP and model fits are shown in Figure 3.4 at the burn center (I=120 W/cm2 ). In order to obtain good fits and well defined parameters several assumptions are enforced: (1) the recovery rate is fixed at an average value determined from recovery measurements, (2) the undamaged absorbance per unit length is fixed, and (3) the intensity independent decay rate, α, and the absorbance per unit length of the damaged species, σ1 , are constrained to be constant across fits (see Table 3.2 for fit parameters). Initially only the thickness and intensity were changed between fits but problems arose when fitting the 83µm film data, and therefore an adjustable amplitude factor is included in the model to account for the deviation. For the 8µm, 22 µm, and 35 µm data the adjustable amplitude factor is within experimental uncer- 74 Fit Parameters × 10−3 cm2 /W min α 2.18 (± 0.92) β 0.001 min−1 σ0 56 × 10−3 µm−1 σ1 51.81 (± 0.67) × 10−3 µm−1 Table 3.2: Fit parameters for two-population depth model. β was held constant at the average recovery rate, and σ0 was held constant at the value determined from absorption measurements. tainty of unity, suggesting that the two population (pristine and degraded) depth model fits those data sets well; however, the 83µm data set requires an amplitude factor of 3.02 ± 0.12, which is too large to be explained simply by detector variations from experiment to experiment. Most likely the large amplitude factor is due to the differences in sample preparation, as reversible photodegradation is found to change drastically when varying the preparation method, and the 8-35µm samples are prepared using the polymer solution method and the 83µm sample is prepared using the monomer bulk pressing method. 3.2.5 Absorptive effect summary The depth effect due to pump intensity absorption is found to be an important effect during decay, as averaging the population over the sample’s thickness results in an underestimation of the decay rate and decay amount; and the absorptive effect is found to be negligible in recovery measurements as the recovery rate is dose independent. Using the differential Beer-Lambert law and the TPNIM, we developed a mathematical model which is found to fit experimental data well for four different sample thicknesses, with only the thickest sample requiring an arbitrary scale fac- 75 tor. The deviation of the thickest sample is most likely due to the different sample preparation method required to produce such thick samples. 3.3 Effect of beam propagation on intensity During sample decay, the pump beam not only experiences intensity modification due to absorption, but also modification due to refractive propagation effects that may be separated into three categories: normal Gaussian beam propagation, photodamageinduced lensing, and thermal self (de-)focusing. Normal Gaussian beam propagation effects are due to universally-present diffraction, which leads to beam divergence. Photodamage-induced lensing originates in a change in the refractive index gradient as the dye-doped polymer is damaged. Finally, thermal self (de-)focusing is a thermally induced nonlinear effect, in which the pump beam heats the dye-doped polymer leading to a thermally induced refractive index gradient that can either focus or defocus the pump beam. In this section we will first discuss the fundamental physics of linear wave propagation, deriving a master wave equation to describe beam propagation, and then we will apply the master wave equation to the three propagation effects. 3.3.1 Linear wave propagation Electromagnetic wave propagation through a linear nonmagnetic dielectric is governed by the nonmagnetic wave equation derived from Maxwell’s equations, ∇×∇×E= − n(r, t)2 ∂ 2 E , c2 ∂t2 (3.14) where E is the electric field, n = n0 + n1 (r, t) is the linear index of refraction, and c is the speed of light. In preparation for Section 3.3.3 the refractive index is split into 76 two parts: a homogenous isotropic index, n0 , and an inhomgenous index, n1 (r, t). We assume that there are no charges or currents, such that the divergence of the electric field is zero, ∇ · E = 01 ; and we assume that the electric field may be written as E = Ẽ(r, t)e⊥ , where the polarization vector, e⊥ , is perpendicular to direction of propagation, ẑ, and Ẽ(r, t) is a scalar. With these approximations Equation 3.14 becomes ∇2 Ẽ(r, t) = n2 ∂ 2 Ẽ(r, t) . c2 ∂t2 (3.15) To further simplify, Ẽ(r, t) is assumed to be of the form Ẽ(r, t) = E(r, t)ei(kz−ωt) , (3.16) where the wave vector k = n0 k0 , with k0 = ω/c being the free space wave vector, n0 is the homogenous refractive index, ω is the angular frequency of the radiation, and E(r, t) is the field envelope. Substituting Equation 3.16 into 3.15 and simplifying yields: ∇2⊥ E(r, t) + 2ik ∂E(r, t) ∂ 2 E(r, t) + = k02 (n20 − n2 )E(r, t) ∂z ∂z 2 2iωn2 ∂E(r, t) n2 ∂ 2 E(r, t) + 2 , − 2 c ∂t c ∂t2 (3.17) where ∇2⊥ is the laplacian in the x, y-plane, and k = n0 ω/c. Equation 3.17 represents the master wave propagation equation for the electric field envelope, which will be used in the following sections to derive the electric field envelope as a function of depth. In general we are mainly concerned with the beam intensity which requires the full scalar electric field to compute, 1 See Appendix B for a detailed justification of this assumption. 77 1 I = Ẽ ∗ Ẽ, 2 1 = |E|2 exp[i(k̃ − k̃ ∗ )z], 2 1 = |E|2 exp[−σz], 2 (3.18) (3.19) (3.20) where |E|2 is the envelope intensity, and we use the definition of the absorbance per unit length σ = 2k ′′ , with k ′′ being the imaginary part of the complex wave vector. For dye-doped polymers the absorbance term is typically large in the visible range, so it is difficult to separate propagation effects from absorptive effect when considering the full intensity. Therefore, to compare propagation effects to the absorptive effect, in the following sections we will compute the envelope intensity separately, and compare that to the purely absorptive effect (e−σz ). 3.3.2 Beam propagation in an isotropic and homogenous material In the case of a homogenous medium (n = n0 ), the electric field envelope will be time independent and the right hand side of Equation 3.17 will be zero, resulting in the electric field envelope equation becoming: ∇2⊥ E(r, t) ∂E(r, t) ∂ 2 E(r, t) + 2ik + = 0. ∂z ∂z 2 (3.21) In the slowly varying envelope approximation the envelope function is assumed to vary slowly along the propagation direction on wavelength scales, or 2 ∂ E(r) << k0 ∂E(r) . ∂z 2 ∂z 78 (3.22) For optical frequencies this approximation is typically very good, as k is large, and allows us to write Equation 3.21 as ∇2⊥ E(r, t) + 2ik ∂E(r, t) = 0. ∂z (3.23) The lowest order solution to Equation 3.23 is the TEM00 Gaussian mode which is given in cylindrical coordinates as [79–81]: w0 exp E(r, z) = E0 w(z) −r 2 r2 − ik + iζ(z) , w(z)2 2R(z) (3.24) where r is the radial distance from the optical axis, z is the propagation distance measured relative to the minimum beam waist (narrowest part of the beam), w(z) is the beam width at position z, w(0) = w0 is the minimum beam waist, R(z) is the radius of curvature of the beam’s wavefronts, ζ(z) is known as the Gouy phase shift, and E0 is the peak electric field. The beam width, radius of curvature, and Gouy phase are s 2 z w(z) = w0 1 + , zR z 2 R , R(z) = z 1 + z z −1 , ζ(z) = tan zR (3.25) (3.26) (3.27) where zR is the Rayleigh range: 1 zR = kw02 . 2 (3.28) For a TEM00 Gaussian beam it is often unnecessary to directly solve Equation 3.23 as a boundary value problem, as a geometrical optics formalism has been developed 79 to describe beam propagation in terms of an ABCD ray tracing matrix and a complex beam parameter, q(z), 1 1 2i = − . q(z) R(z) kw(z)2 (3.29) For the case of a Gaussian beam incident on a glass substrate, followed by propagation into the dye-doped polymer the ABCD matrix may be written as: 1 M = 0 1 = 0 A = C ′ z 1 0 1 a 1 . . . ng 0 0 1 0 n 1 ng z ′ 1 a+ n ng , 0 , 1 ng (3.30) (3.31) 1 n B , D (3.32) where ng is the refractive index of glass, a is the glass thickness, n is the refractive index of the dye-doped polymer, and z ′ is the distance propagated into the dye doped polymer as shown in Figure 3.5. Assuming that the minimum beam waist is at the air-glass interface, the incident complex beam parameter there is 1 2i , =− qi k0 w02 k0 w02 . qi = i 2 (3.33) (3.34) where w0 is the beam’s radius at the glass-air interface, and the output complex beam parameter is found using the ABCD matrix to be 80 Figure 3.5: Diagram of beam propagating from air, into glass, and then into a dyedoped polymer half-space. The beam is assumed to have its minimum waist at the surface of the air-glass interface. qi A + B , qi C + D an 1 = + z ′ + ik0 nw02 . ng 2 qo = (3.35) (3.36) The beam width in the dye-doped polymer is found from the output complex beam parameter, nk0 1 1 , Im = − 2 wout 2 qo k 2 n2 n2g w02 = , 4 (an + z ′ ng )2 + k 2 n2 n2g w04 (3.37) (3.38) which upon inversion and taking the square root gives the beam width in the dyedoped polymer: 81 wout = s w02 + = w0 s 4 (an + z ′ ng )2 , k 2 n2 n2g w02 1+ 4 (an + z ′ ng )2 . k 2 n2 n2g w04 (3.39) (3.40) Using Equation 3.40 we can find the position, zs , where the beam waist will be √ w(zs ) = sw0 , where s = I(zI0s ) . Solving Equation 3.40 for zs gives zs = p 1 2 2 (s − 1) − 2ann knn w g . g 0 2n2g (3.41) As an example, we estimate the propagation distance required for the beam intensity to decrease by 5%. Using the experimental conditions of λ = 488nm, w0 = 15µm, and assuming the glass has a refractive index of ng = 1.5 and the dye-doped polymer has a refractive index of n = 1.48, the distance the beam propagates before decreasing to I = 0.95I0 due to beam divergence is zs ≈ 393µm. For comparison the 1/e absrobance length for a 9g/l sample at 488nm is approximately 40µm. Therefore when the beam’s divergence decreases the intensity by 5%, the sample’s absorbance will have made the intensity essentially zero. Since the absorbance dominates and typical sample thicknesses are smaller than 50µm, we ignore the effect of simple beam propagation on population decay as a function of depth. 3.3.3 Photodamage induced lensing During photodegradation it is well known that the absorbance of dye doped polymers change, therefore it is reasonable to assume that the refractive index also changes. Assuming the two-population noninteracting model, and letting χ̃0 and χ̃1 be the complex electric susceptibility for the undamaged and damaged populations respectively, and χ̃poly be the complex electric susceptibility for the polymer, the complex 82 index of refraction is ñ(t) = q 1 + m0 (t)χ̃0 + m1 (t)χ̃1 + χ̃poly , (3.42) where the population is denoted by mi to allow the index of refraction to be denoted by n. Substituting equation 3.42 into the linear wave Equation and rewriting Equation 1.3 in terms of the electric field, we find a set of coupled nonlinear partial differential equations, ∂E(r, t) ∂ 2 E(r, t) ω2 + = (m(r, t)χ̃0 + (1 − m(r, t))χ̃1 + χ̃poly )E(r, t) ∂z ∂z 2 c2 ∂E(r, t) 2iω − 2 (1 + m(r, t)χ̃0 + (1 − m(r, t))χ̃1 + χ̃poly ) c ∂t ∂ 2 E(r, t) 1 , (3.43) + 2 (1 + m(r, t)χ̃0 + (1 − m(r, t))χ̃1 + χ̃poly ) c ∂t2 α ∂m(r, t) = − |E(r, t)|2m(r, t) + β(1 − m(r, t)), (3.44) ∂t 2 ∇2⊥ E(r, t) + 2ik0 which describe the population and electric field as a function of time and space during photodegradation. Without approximation, Equations 3.43 and 3.44 have no analytical solutions, and obtaining a numerical solution is difficult. Given the complexity of the problem we will consider an approximate “worst case scenario”. First, we assume that the population throughout the sample has become so damaged that ∂m(r, t) = 0, ∂t (3.45) and therefore the system is in a steady state. In this scenario the change in the refractive index will be greatest, and will therefore give the largest lensing effect. Since experimentally we measure the index of refraction directly, and not the susceptibility, we will use Equation 3.17 for numerical modeling with the index of refraction being given by 83 n → n0 + n1 (r), (3.46) where n0 is the pristine index of refraction, n1 (r) is the damage induced change in the index of refraction. Finally, we once again use the slowly varying envelope approximation, as in Section 3.3.2. After all these approximations and substitutions Equation 3.17 becomes, ∇2⊥ E(r, t) + 2ik ω2 ∂E(r, t) = 2 (2n0 n1 (r) + n1 (r)2 )E(r, t). ∂z c (3.47) In the next section we will describe WLIM measurements of the change in refractive index, n1 , and using those results we model lensing due to photodamage in Section 3.3.3. WLIM results To measure the change in the refractive index due to photodegradation, 9g/l, DO11/ PMMA spin coated thin films are prepared, with an average thickness of 10µm, as determined from spectrometer absorbance measurements. Multiple 45 min burns are performed using a line focused ArKr laser operating at 488nm with a peak intensity of 40W/cm2 ; and spatially resolved interferograms are measured using the WLIM both before and after decay in the burn region. The interferograms are analyzed using the procedure outlined in Section 2.6.4 to find the spectral magnitude and phase, at which point the change in absorbance and phase due to burning is determined. The WLIM’s change in absorbance being found consistent with spectrometer measurements. The change in phase is calculated for several points near the peak damage and is found to be within experimental uncertainty to vanish as shown in Figure 3.6. Finding the measured change in phase to be within experimental uncertainty of 84 Figure 3.6: Change in WLIM phase due to photodegradation for a 9g/l, DO11/PMMA thin film. A pump beam of 488nm has a wavenumber of k0 = 12.875µm−1. 85 Figure 3.7: Upper bound on the change in the refractive index due to photodegradation for a 9g/l DO11/PMMA thin film degraded at 40W/cm2 for 45 mins. zero for the burn center implies that the actual change in phase must be less than the WLIM phase error, σΦ (k0 ), for all regions of the burn. Using the WLIM phase error and Equation 2.48 we can set an upper limit on the change in index of refraction ∆n(k0 ) < σΦ (k0 ) , 2k0 d1 (3.48) where σΦ (k0 ) is found by averaging the measured error from experiment to experiment, and d1 is assumed to be the average thickness, L = 10µm. Figure 3.7 shows the upper bound on the change in refractive index as a function of wavenumber as determined by the WLIM. 86 Approximate steady state pump wave propagation through damaged media With Equation 3.47, and an approximate experimental value for n1,max , we can now model the lensing effect due to photodegradation. In order to numerically solve Equation 3.47 several assumptions are made: 1. Since the pump beam is much wider in one direction, wx = 15µm and wy = 600µm, we assume that the system is approximately two dimensional with transverse dimension x and longitudinal direction z. 2. The sample is assumed to fill the half-space, x ∈ (−∞, ∞) and z ∈ [0, ∞). 3. The pump beam is assumed to be at its minimum beam waist at the interface of the sample such that the initial electric field at z = 0 is a Gaussian of width √ w0 = wx 2 = 21.21µm 4. The spatial profile of the index of refraction change is assumed to follow the intensity profile and therefore to be of the form " # 2 x n1 (x, z) = n1,max exp − , wx (3.49) where n1,max is the upper bound of the refractive index change calculated using the phase error of the WLIM, which is always positive. Thus calculations are performed for both the case of an increase and decrease of the refractive index. Figure 3.8 shows the beam profile as a function of depth and transverse position due to photodamaged lensing. In the case of positive n1 the beam is focused with a focal length of roughly z = 900µm, and in the case of a negative n1 the beam diverges. Note that the divergence due to lensing due to damage is different than the 87 (a) (b) Figure 3.8: Beam propagation profile for: (a) positive refractive index change, and (b) negative refractive index change. divergence in the isotropic case, with damaged lensing producing two weaker beams. To better understand the beam profile, Figure 3.9 shows profile cross sections for different depths. The positive refractive index change causes the beam to narrow and produces higher intensity near the beam center, but decreased intensity in the “wings” of the Gaussian. The negative refractive index change causes the beam to separate into two peaks, which become more separated as a function of depth. For the numerical calculations of the damaged lensing effect, the index of refraction was assumed to be wholly real, with no absorption, which is not realistic. Figure 3.10 compares the peak intensity at x = 0 for the damaged lensing cases, and the case with absorption. Within the absorptive 1/e length (approximately 18µm) the damaged lensing effect is negligible, with the peak intensity remaining approximately constant. The lensing effect due to photodamage is not apparent until approximately 100µm of propagation, at which point the intensity with absorption is almost zero. Since the true intensity controlled by a combination of lensing and absorption, it is 88 (a) (b) Figure 3.9: Transverse beam profile at several depths for: (a) positive refractive index change, and (b) negative refractive index change. apparent that lensing has a negligible effect on the pump beam intensity. 3.3.4 Thermal self (de-)focusing Along with refractive index changes due to photodamage, laser-induced heating leads to changes in the refractive index, typically due to thermal expansion, causing self (de) focusing of the pump beam. For small changes in temperature, ∆T , the temperature dependent refractive index is n = n0 + ! dn ∆T, dT T =T0 (3.50) where n0 is the refractive index at temperature T0 . Since photo thermal heating depends on the intensity of the pump, Equation 3.50 may be reframed in terms of the intensity dependent refractive index in the absence of thermal diffusion as n = n0 + nT2 H I, 89 (3.51) Figure 3.10: Intensity profile at beam center as a function of depth for lensing due to damage without absorption, and a comparison to the intensity profile due to absorbance. 90 where I is the intensity and nT2 H is the thermal nonlinear index of refraction. Heating due to CW Laser In general, photo thermal heating of a homogenous material is effected by heat diffusion and is described by the heat equation: ρ0 C ∂T − κ∇2 T = σI(x, z), ∂t (3.52) where T is the temperature change from the ambient temperature, ρ0 is the density of the material, C is the heat capacity of the material, κ is the heat transfer coefficient (assumed to be constant in space) and α is the absorbance per length of the sample. Since DO11/PMMA samples are primarily PMMA with a very small amount of dye, it is safe to assume the sample’s thermal properties are the same as pure PMMA. At room temperature PMMA has a density of ρ0 = 1.18 g/cm3 , a heat capacity of C = 1466J/kg/K, a heat transport coefficient of κ ≈ 0.002W/Kcm and the dye doped samples have α ≈ 2.5 × 10−2 µm−1 . Assuming that dn dT = 0, and that the intensity within the sample is, x2 I(x, z) = I0 exp − 2 − σz , w (3.53) with I0 = 50W/cm2 and w = 15µm, Equation 3.52 can be solved using numerical methods. We choose contact conductance boundary conditions such that, ∂T (x, z) C = (T (x, 0) − T0 ), ∂z κ z=0 ∂T (x, z) C = − (T (x, L) − T0 ), ∂z κ z=L (3.54) (3.55) where T0 is the ambient temperature, which for simplicity is assumed to be zero and C is the contact conductance term, which is estimated to be C = 3.575 × 103 W/(m2 K), 91 Figure 3.11: Photothermal temperature change as a function of time and depth from numerical solutions of the heat equation with the laser as heat source. While each depth shows a slightly different time scale to reach the steady state, all depths reach the steady state within 100ms. based on Dawson’s measurement of a similar sample [82, 83]. For the x boundaries, we assume that the temperature is symmetric with T (x) = T (−x). The numerical calculations show that the temperature quickly reaches a steady state, with the temperature profile being far wider than the pump intensity, and peaking some distance into the sample. Figure 3.11 shows the temperature as a function of depth and time at the beam center (x = 0), with the temperature reaching the steady state within 100 ms of turning on the pump. Figure 3.12 shows the temperature as a function of depth and transverse direction at t = 0.5s, and Figure 3.13 is the transverse temperature profile at z = 0µm, and t = 0.5s. Effect of thermally induced refractive index change on beam propagation Since the thermally induced refractive index change is a nonlinear process, we can use the framework of nonlinear optics to model how thermal effects change beam propagation. The nonlinear effect enters into the nonmagnetic wave equation as a 92 Figure 3.12: Temperature change as a function of position in the steady state. The peak temperature occurs within the sample, at a depth of approximately 20µm, and the width of the temperature profile is much larger than the pump intensity beam width. Figure 3.13: Cross section of the steady state temperature change at the incident surface of the sample. 93 nonlinear polarization, PN L , which transforms Equation 3.14 into: ∇×∇×E=− 1 ∂ 2 PN L n20 ∂ 2 E − . c2 ∂t2 c2 ∂t2 (3.56) Using the same assumptions as in Section 3.3.1, and assuming that the nonlinear polarization may be written as a scalar, P̃ N L (x, z, t) = 3 (3) χ |E(x, z, t)|2 E(r, z, t)e−i(kz−ωt) , 4 TH = n0 nT2 h |E(x, z, t)|2 E(x, z, t)e−i(kz−ωt) , dn T Eei(kz−ωt) , = n0 dT (3.57) (3.58) (3.59) the nonlinear wave propagation equation becomes ∇2⊥ E(r, t) + 2ik Assuming that n0 2iωn2 ∂E(r, t) n2 ∂ 2 E(r, t) ∂E(r, t) ∂ 2 E(r, t) + = − + ∂z ∂z 2 c2 ∂t c2 ∂t2 1 ∂2 dn +e−i(kz−ωt) 2 2 n0 T Eei(kz−ωt) . c ∂t dT dn dT ∇2⊥ E(r, t) + 2ik (3.60) is time independent, Equation 3.60 may be simplified to give ∂E(r, t) ∂ 2 E(r, t) 2iωn2 ∂E(r, t) n2 ∂ 2 E(r, t) + = − + 2 2 ∂z c2 ∂t c ∂t2 ∂z 2 1 ∂(T E) ∂ (T E) dn + 2 n0 −ω 2 T E − 2iω + . c dT ∂t ∂t2 (3.61) Coupled Equations Equations 3.52 and 3.61 form a coupled set of differential equations that describe thermally induced self (de-)focusing. Assuming once again that the system is effectively two dimensional, the equations become: 94 ∂ 2 E(x, z, t) ∂E(x, z, t) ∂ 2 E((x, z, t) 2iωn2 ∂E(x, z, t) n2 ∂ 2 E(x, z, t) + 2ik + = − + 2 ∂x2 ∂z 2 c2 ∂t c ∂t2 ∂z 1 ∂(T (x, z, t)E(x, z, t)) dn + 2 n0 − ω 2T (x, z, t)E(x, z, t) − 2iω c dT ∂t 2 ∂ (T (x, z, t)E(x, z, t)) + , (3.62) ∂t2 2 σe−σz ∂ T (x, z, t) ∂ 2 T (x, z, t) κ ∂T (x, z, t) = − + |E(x, z, t)|2 , (3.63) ∂t ρ0 C ∂x2 ∂z 2 2ρ0 C Upon inspection of Equations 3.62 and 3.63 we find two coupling factors, α and dn . dT In the case that either coupling factor is zero, the two equations are decoupled and easily solved. However in reality the coupling factors are nonzero and finding a full solution is nontrivial. Section 3.3.4 will go through an approximate numerical solution, but at this point we consider a rough treatment to estimate the magnitude of the self (de-)focusing effect. We begin by considering the steady state case where the heat transfer equation is given by − κ∇2 T = σI. (3.64) For this ballpark treatment we assume a flat top step beam profile with infinite width in y and width in the x direction of w. The maximal change in temperature occurs at the beam center, at which point the laplacian may be approximated as ∇2 T ≈ T (max) /(2w)2, which substituting into Equation 3.64 yields a maximum temperature change of ∆T (max) 4σIw 2 . = κ (3.65) Substituting Equation 3.65 into Equation 3.50 the maximum change index of refraction will be ∆n = dn dT 95 4σIw 2 . κ (3.66) Recalling that the change in the index of refraction is related to n2 by ∆n = n2 I, the approximate thermally induced n2 is (T h,max) n2 = dn dT 4σw 2 . κ (3.67) Using approximate experimental values of σ ≈ 10−2µm−1 and w = 15µm, along with material parameters κ ≈ 0.002W/K cm, and (T h,max) n2 dn dT ≈ −1.4 ×10−4 /K [84,85] we find that ≈ −6.3 × 10−5 cm2 /W. For comparison typical nonlinear refractive indices due to electronic effects are on the order of 10−16 cm2 /W [86], therefore the thermally induced nonlinear refractive index is relatively large. Using the estimated nonlinear refractive index with a a pump intensity of 100W/cm2 gives a refractive index change of ∆n = n2 I ≈ −6.3 × 10−3 , which is of the same order of magnitude as the refractive index change due to photodegradation. Since the depth effects due to photodamage induced refractive index change are found to be negligible when compared to absorption effects, it is reasonable to assume that thermal lensing, which is on the same order as photodamaged lensing, will also be negligible. Approximate Numerical Solution To confirm that we may safely neglect the thermal lensing effect, we consider an approximate solution to Equations 3.62 and 3.63. For our approximate solution we use an iterative approach to solve the equations, rather then attempting to solve them simultaneously, which is far more complicated. The process is as follows: 1. Solve for the electric field in the non-coupled case where dn dT = 0. 2. Use the electric field found in the previous step to solve for the temperature profile. 96 3. Solve for the electric field assuming the temperature profile found in previous step. 4. Repeat steps 2 and 3 until the desired accuracy is reached. While the steps may be repeated to obtain higher accuracy, we will limit ourselves to one iteration and making two other approximations: (1) we once again use the paraxial approximation in which ∂2E ∂z 2 = 0, and (2) when calculating the electric field in step 3 we assume a steady state temperature profile as that will result in the largest lensing effect. With these approximations Equation 3.62 may be written as ∂2E ∂E k2 + 2ik = − ∂x2 ∂z n0 dn dT T E, (3.68) where T is the temperature profile from step 2, which was already calculated in Section 3.3.4. To perform the numerical computation of Equation 3.68 we assume that the intensity at the surface is Gaussian in the transverse direction with width w = 15µm, and peak intensity of 50 W/cm2 . To solve we utilize a Crank-Nicolson numerical method with matrix inversion. Figure 3.14 shows the envelope intensity as a function of depth and transverse position due to thermal lensing. For a better sense of the depth dependence Figure 3.15 shows the envelope intensity at the beam center(x = 0) as a function of depth for thermal lensing and normal propagation, as well as absorption for length scale comparison. From Figures 3.14 and 3.15, the effect of thermal lensing appears to counteract the normal propagation divergence, making the beam maintain it’s intensity deeper into the sample. However, comparing the absorptive length scale to the thermal lensing length scale, the thermal lensing effect is found to be negligible when compared to absorption. 97 Figure 3.14: Calculated intensity profile as a function of depth and transverse position taking into account thermal lensing. 3.3.5 Summary of propagation effects The three propagation effects on the pump beam are normal Gaussian beam propagation, photodamage induced lensing, and thermal self (de-)focusing. In each case the depth required for a noticeable effect on the envelope intensity was found to be on the order of hundreds of microns, and well beyond the point where absorptive effects will have made the beam intensity negligible. Also, even if the absorbance per unit length were smaller, the majority of propagation effects would only be noticeable outside of the sample, as the samples tend to be thinner than 50µm. 3.4 Summary of depth effects In previous literature [1, 43, 44, 49, 50, 52–54, 87–91] the effects of pump absorption and propagation were neglected when calculating measured damaged populations, 98 Figure 3.15: Intensity profile as a function of depth at the beam center (x = 0) for normal propagation, thermal lensing propagation, and including absorption. The beam is fully absorbed over a propagation distance that is short compared to the length scale where refractive effects come into play. 99 with the assumption being that the intensity, and therefore damage, was uniform throughout the sample. In this chapter a model of the absorptive effect is developed, which is found to fit experimental data well for samples of varying thickness, and predicts that the absorptive effect is large, and that neglecting it will lead to an underestimation of the decay rate and degree of damage. Along with the absorptive effect, depth effects due to refraction and diffraction effects were considered and found to be negligible in comparison to the absorptive effect. Therefore in the following chapters, when considering depth effects we will only consider the effect of pump absorption. 100 Chapter 4 Three-Population Model of Reversible Photodegradation 4.1 Introduction From linear optical measurements, such as transmittance imaging and absorption spectroscopy, reversible photodegradation does not appear to be fully reversible. Evidence suggests that there are at least two damage processes, one reversible, and the other permanent. The irreversible process is not observed with nonlinear measurements such as amplified spontaneous emission (ASE) and two-photon fluorescence (TPF). The inability of ASE and TPF to probe the irreversible process suggests that damage to the polymer host is responsible, as the nonlinear properties are primarily due to the dye molecules. Therefore, damage to the polymer would be detected in linear measurements, but not nonlinear measurements. We assume the irreversible process originates from damaging the polymer. Damage of the molecules and the polymer will occur simultaneously, with decay rates being proportional to the undamaged population. These two processes are mediated 101 Figure 4.1: Schematic three population model. The undamaged species (n0 ) can decay either to the reversibly damaged species (n1 ) or the irreversibly damaged species (n2 ). by absorption of the pump light by the dopant molecules, which results in damage of the molecule which can self heal; or the energy absorbed by the molecule is deposited in nearby polymer, causing damage. The probe light then is altered through absorption by both damaged molecules and damaged polymer. However, since the polymer has a negligible nonlinearity, the irreversible damage to the polymer is not observed by a nonlinear probing technique. The simplest model to describe this process, that is consistent with data, has three parallel processes. The system is assumed to initially be in the undamaged state with molecular population n0 . When the pump is turned on, the undamaged population, n0 , is converted to the into the reversibly damaged species, n1 , or irreversibly damaged polymer population, n2 . Figure 4.1 shows a schematic of the process with decay and recovery rates indicated. Note that the pristine dye molecule of population n0 is not converted to damaged polymer, but rather provides a local center of absorption that deposits energy into the polymer. 102 More complicated models are possible1 , but the parallel three-population model fits the data, so it is assumed to be a good approximation to the underlying process. In the following chapter we will derive the mathematics of the three-population model and compare the results to intensity-dependent measurements of the scaled reversiblydamaged population and the scaled irreversibly-damaged population, concluding with estimates of the different population’s absorbance cross sections, and a proposed energy level diagram. 4.2 Three population rate equations Assuming that the intensity does not change over time, the three level population model (shown in Figure 4.1) can be written mathematically as three coupled first order linear differential rate equations, dn0 = −(α + ǫ)In0 + βn1 , dt dn1 = αIn0 − βn1 , dt dn2 = ǫIn0 , dt (4.1) (4.2) (4.3) where ni is the population of the ith state, I is the intensity, β is the recovery rate, and α and ǫ are intensity independent decay parameters. Equations 4.16-4.18 can be rewritten in matrix form as dn = An, dt 1 (4.4) It is possible that more than three populations are involved, and that they each can be converted to the other ones. This leads to an overly complex mathematical model that is no better at fitting the data than the one we propose. 103 where n = {n1 , n2 , n3 } is the column vector representing the populations, and A is a matrix given by −(α + ǫ)I β 0 A= αI −β 0 ǫI 0 0 The solution to Equation 4.4 can be written as . (4.5) n = c0 v0 eλ0 t + c1 v1 eλ1 t + c2 v2 eλ2 t , (4.6) where λi are the eigenvalues of A, vi are it’s eigenvectors, and ci are constants determined by the initial conditions. To simplify the derivation, we introduce three parameters which appear in the eigenvectors and eigenvalues: B= A = β + (α + ǫ)I, (4.7) C = β + αI − ǫI, (4.8) p −4βǫI + (β + (α + ǫ)I)2 . (4.9) Using these parameters the eigenvalues can be written as λ0 = 0, (4.10) −A − B , 2 −A + B = , 2 λ1 = (4.11) λ2 (4.12) and the eigenvectors are 0 v0 = 0 1 −A − B 1 C +B v1 = 2ǫI 2ǫI 104 −A + B 1 C −B . v2 = 2ǫI 2ǫI Assuming no initially-damaged population, the boundary conditions are n0 (0) = 1, (4.13) n1 (0) = 0, (4.14) n2 (0) = 0. (4.15) The ci parameters in Equation 4.6 are solved for and substituted into Equation 4.6 along with the eigenvectors and eigenvalues to give the population dynamics 1 e− 2 (A+B)t (A + B)(B − C) + (A − B)(B + C)eBt , n0 (t) = 2B(A − C) (4.16) 1 (B − C)(B + C)e− 2 (A+B)t −1 + eBt , n1 (t) = 2B(A − C) (4.17) n2 (t) = − 1 1 ǫI e− 2 (A+B)t C −1 + eBt + B 1 + eBt − 2e 2 (A+B)t . (4.18) B(A − C) For the case of recovery, where I = 0, the population dynamics are simplified with the solutions being n0 (t) = n0 (t0 ) + n1 (t0 )(1 − e−β(t−t0 ) ), (4.19) n1 (t) = n1 (t0 )e−β(t−t0 ) , (4.20) n2 (t) = n2 (t0 ), (4.21) where ni (t0 ) is the ith population at the time the pump beam is turned off, and β is once again the recovery rate. 105 4.2.1 Three-population model of absorption In the previous section the population dynamics of a three-state system were derived; leading to Equations 4.16-4.18. The absorbance of the material is given by a linear combination of the populations that in the thin sample approximation is given by: A = σ0 L + n1 ∆σ1 L + n2 ∆σ2 L, (4.22) where ∆σ1 = σ1 − σ0 , ∆σ2 = σ2 − σ0 , and σi is the absorbance per unit length of the ith species. For a thick sample, the depth effect analysis of Section 3.2 is applied to the three populations, and is briefly discussed in Section 4.4. Since measurements probe a linear combination of the populations, it is challenging to use Equations 4.16-4.18 to isolate each damage pricess during photodegradation. However, during recovery, only the damaged DO11 molecular population changes as a function of time, allowing for the two processes to be differentiated. 4.3 Data Given the difficulties separating the two damaged processes using decay data, the recovery data are used as follows: transmittance imaging is performed during decay and recovery to obtain images of photodamage as a function of time, from which the color channel intensity is determined as a function of time at 1400 different pixels across the burn, and converted to the scaled damaged population using Equation 2.26. The scaled damaged population at each pixel is then related to the pump intensity (assuming a perfect TEM00 elliptical Gaussian beam) and the recovery data is fit to a simple exponential: n′ (t; I, t0 ) = n′IR (I; t0 ) + n′R (I; t0 )e−β(t−t0 ) , 106 (4.23) where t0 is the time at which the pump was turned off, I is the pump intensity during burning; n′IR and n′R are the exponential offset and the exponential amplitude, respectively, which in the thin sample approximation are given by n′IR (I; t0 ) = n2 (I; t0 )∆σ2 L, (4.24) n′R (I; t0 ) = n1 (I; t0 )∆σ1 L. (4.25) As an example, we consider a 12g/l thin film sample burned with a peak intensity of 125 W/cm2 for 25 mins. Figure 4.2 shows data for 90 W/cm2 with a simple exponential fit, while Figure 4.3 and Figure 4.4 show the exponential amplitude and offset as functions of intensity, respectively. The data shown are smoothed from the full data set2 , as the full set is noisy due to point to point variations in intensity and sample properties. The full data sets are fit to Equations 4.24 and 4.25 holding all parameters fixed for both the amplitude and offset fits. The only parameters varied are the ∆σL terms. Table 4.1 shows the fit parameters. Fit Parameters α β ǫ 1.80 (± 0.74) × 10−4 cm2 /W min × 10−3 min−1 2.9 (± 0.5) 2.43 (± 0.66) × 10−4 cm2 /W min × 10−2 ∆σ1 L 4.6 (± 1.6) ∆σ2 L 0.79 (± 0.17) Table 4.1: Three population model fit parameters for a 12g/l DO11/PMMA sample assuming the thin sample approximation. Here we present only one representative data set: The three-population model, as represented in Figure 4.1, is found to fit transmittance imaging data as a func2 Smoothing is done using Igor Pro’s binomial smoothing algorithm using 50 points. 107 Figure 4.2: Scaled damaged population decay and recovery for a pump intensity of 90 W/cm2 . Both the reversible and irreversible portions are marked with arrows. tion of intensity for all data on all samples, with only one adjustable parameter the cross section dependent amplitude is different for the reversible and irreversible processes. Despite finding values for the cross section differences using transmittance imaging, their usefulness is limited as they represent an average value over a range of wavelengths, as discussed in Appendix A. In the next section we discuss absorbance spectroscopy measurements which are used to determine the individual cross sections of each species as a function of energy. 4.4 Absorbance cross sections To determine the absorbance cross sections of the three species involved, the absorbance spectrum for 9g/l samples is measured during decay and recovery for a variety of pump doses using a Verdi Nd:YAG CW laser operating at 532nm. The pump 108 Figure 4.3: Exponential amplitude as a function of intensity data (points) and the three level model prediction (curve). Data for 125 W/cm2 peak burn intensity of a 12g/l thin film exposed for 25 min. 109 Figure 4.4: Exponential offset as a function of intensity data (points) with the three level model prediction (curve). Data for 125 W/cm2 peak burn intensity of a 12g/l thin film exposed for 25 min. 110 is focused to circular spot with a diameter of 2mm. A broad spectral pulsed Ocean Optics xenon PX-2 light source focused to a circular spot with diameter 0.75mm serves as the probe beam. The probe beam is focused to the center of the burned area where the damage profile is most uniform. Since the pump intensity is not uniform over the probe’s spatial profile, the probe measures the average degree of damage within its waist. The signal measured by the spectrometer, is therefore the spatially integrated incident intensity; S(t; ω) = C Z Z I(r, φ, t; ω)rdrdφ, (4.26) where C is a constant accounting for the properties of the spectrometer, I(r, φ, t; ω) is the intensity at the detector, and the limits of integration span the detector area. Assuming that the probe beam size is much smaller than the detector area, the limits of integration of the radial part spans 0 to ∞, and since both the probe and pump beams are radially symmetric, the angular integral is simply 2π. Using Equation 4.26, the differential Beer-Lambert law and the three population model, the signal detected by the spectrometer is therefore given by S(t; ω) = Z Z ∞ 2πC I0 (r; ω) exp − 0 L [σ0 (ω) + n1 (r, z; t)∆σ1 (ω) + n2 (r, z, t)∆σ2 (ω)] dz rdr, 0 (4.27) where I0 (r; ω) is the probe beam intensity at the surface of the sample. Equation 4.27 is difficult to evaluate, to simplify first we assume that the sample is thin, or σ0 L << 1, where L is the sample thickness. Then, Equation 4.27 may be approximated using a taylor series expansion to first order, Z ∞ S(t; ω) ≈ 2πC I0 (r) [1 − σ0 L − n1 (r; t)∆σ1 L − n2 (r; t)∆σ2 L] rdr. 0 111 (4.28) Next we assume that the damaged populations are proportional to the pump intensity, Ip (r), at radial coordinate r, ni (r; t) = ni (0; t)Ip (r) , Ip (0) (4.29) where n1 (0; t) is the reversibly damaged population at r = 0, n2 (0; t) is the polymer damage at r = 0,and Ip is the pump intensity. Substituting Equation 4.29 into 4.28 yields S(t; ω) = 2πC Z ∞ 0 Ip (r) [n1 (0; t)∆σ1 + n2 (0; t)∆σ2 ] L rdr, I0 (r) 1 − σ0 L − Ip (0) (4.30) = 2πC Z ∞ I0 (r) (1 − σ0 L) rdr Z ∞ 2πC I0 (r)Ip (r)rdr [n1 (0; t)∆σ1 + n2 (0; t)∆σ2 ] L, (4.31) − Ip (0) 0 ! R ∞ Z ∞ 2πC 0 I0 (r)rdr R∞ = SF − I0 (r)Ip (r)rdr [n1 (0; t)∆σ1 + n2 (0; t)∆σ2 ] L, Ip (0) 0 I0 (r)rdr 0 0 (4.32) = SF − f ∆S, (4.33) where SF is the spectrometer signal of the fresh sample, ∆S is the change in signal assuming uniform damage across the probe beam, and f is the transverse correlation factor, which accounts for the variation of damage over the probe beam area. The three parameters are given by 112 SF (t; ω) = 2πC Z ∞ 0 I0 (r) (1 − ∆σL) rdr, (4.34) Z ∞ ∆S(t; ω) = 2πC [n1 (0; t)∆σ1 + n2 (0; t)∆σ2 ] L I0 (r)rdr, 0 Z ∞ 1 R∞ I0 (r)Ip (r)rdr. f= Ip (0) 0 I0 (r)rdr 0 (4.35) (4.36) In practice we are concerned not with the signal, but with the absorbance, which is expressed in terms of the signal as A(t; ω) = − ln S(t; ω) , S0 (ω) (4.37) where S0 is the spectrometer reading with no sample, given by: S0 (ω) = 2πC Z ∞ I0 (r)rdr. (4.38) 0 Using Equation 4.38, we can rewrite the fresh sample absorbance and the change in absorbance as, SF (ω) = S0 (1 − σ0 (ω)L), ∆S(t; ω) = S0 [n1 (0, t)∆σ1 (ω) + n2 (0, t)∆σ2 (ω)] L, (4.39) (4.40) which upon substitution into Equation 4.37 gives the absorbance: A(t; ω) = − ln SF − f ∆S S0 , (4.41) = − ln (1 − σ0 (ω)L − f [n1 (0, t)∆σ1 + n2 (0, t)∆σ2 (ω)] L) . (4.42) Recalling that σ0 L << 1, we expand the natural logarithm in Equation 4.42 as a taylor series to find the thin sample absorbance: 113 A(t; ω) = σ0 (ω)L + f [n1 (0, t)∆σ1 (ω) + n2 (0, t)∆σ2 (ω)] L, (4.43) where the correlation factor f accounts for the damage profile not being uniform across the probe beam. For pump spot diameter of dpump = 2mm, and probe spot diameter, dprobe = 0.750mm, we estimate the correlation factor for a TEM00 mode Gaussian pump beam with width w = dpump 2 for two probe beam cases: (1) the probe beam is a uniform step with diameter dprobe , and (2) the probe beam is a TEM00 mode Gaussian beam with width w = dprobe . 2 Performing the integrals in Equation 4.36 yields f = 0.87 for case 1, and f = 0.93 for case 2. The correlation factor most likely lies between these two values, so we assume f = 0.9. As the mathematics in the previous derivation are complicated, we pause to consider the physical meaning of the correlation factor. To begin we consider the limiting cases of Equation 4.36: if the pump beam is uniform across the probe beam, the damage will be uniform and the correlation factor will be unity. On the other hand, as the pump beam gets narrower, the probe beam measures a large portion of pristine sample, and the correlation factor approaches zero. Essentially, the correlation factor is the ratio of the effective damaged area to the probe area; if the probe measures mostly a uniformly damaged area, the correlation factor is near unity; and, if the probe beam measures mostly pristine sample, the correlation factor is nearly zero. Neglect of the correlation factor when it is less than unity will lead to a bias of the absorbance data toward a pristine sample, underestimating the damage. In addition, the damaged population decreases as a function of depth due to absorption of the pump. This effect, along with absorption of the probe as a function of depth also needs to be taken into account. Using Equation 4.43 as an example of how to correct the change in absorbance for pump probe overlap, we can write a set of coupled differential equations to describe the population and pump/probe intensities 114 as a function of depth, ∂n0 ∂t ∂n1 ∂t ∂n2 ∂t ∂Ip ∂z ∂I(ω) ∂z = −(α + ǫ)Ip n0 + βn1 , (4.44) = αIp n0 − βn1 , (4.45) = ǫIp n0 , (4.46) = − (σ0 (ωp ) + n1 ∆σ1 (ωp ) + n2 ∆σ2 (ωp )) I (4.47) = − (σ0 (ω) + f [n1 ∆σ1 (ω) + n2 ∆σ2 (ω)]) I, (4.48) where Ip is the pump intensity, ωp is the pump frequency, I(ω) is the probe intensity at frequency ω, ∆σi (ω) = σi (ω) −σ0 (ω), with σ0 (ω) being the undamaged absorbance per unit length, σ1 (ω) the reversible damaged absorbance per unit length, σ2 (ω) the irreversibly damaged absorbance per unit length, f is the correlation factor, and the intensity and populations are assumed to have their peak value (i.e. at r=0). Note that Equations 4.44-4.46 here are the same as Equations 4.16-4.18. In Equations 4.44-4.48, the populations and pump intensity are coupled such that they must be solved simultaneously; however, Equation 4.48 may be straightforwardly integrated to find the probe intensity, Z I(ω, t) = I0 (ω) exp σ0 (ω)L + f ∆σ1 (ω) L n1 (z, t)dz + ∆σ2 (ω) 0 L n2 (z, t)dz 0 from which the absorbance is found to be I(ω, t; z = L) , A(ω, t) = − ln I0 (ω) Z L Z = σ0 (ω)L + f ∆σ1 (ω) n1 (z, t)dz + ∆σ2 (ω) 0 Z 0 , (4.49) (4.50) L n2 (z, t)dz , (4.51) where I0 (ω) is the probe intensity incident on the sample. Fitting absorbance data, as a function of frequency and time, to Equation 4.51 requires that we determine the reversibly decayed molecular population, n1 , and the 115 Figure 4.5: Optical density data and model fits as a function of time at several energies (a) 2.33 eV, (b) 2.64 eV, (c) 3.25 eV, (d) 2.78 eV irreversibly decayed polymer population, n2 , as a function of depth and time. To determine the populations, we fit the absorbance decay and recovery at the pump frequency, ωp , to Equations 4.44-4.48. The pump frequency is considered individually as the depth effect of pump absorption is only due to the absorbance at ωp , with other frequencies effecting the probe intensity, but not the population dynamics. Once the populations as a function of depth and time are determined, they are used with Equation 4.51 to fit the absorbance at all frequencies measured. To demonstrate that this method works well, Figure 4.5 shows absorbance data and the model for several energies using the population determined from the absorbance data at the pump energy. 116 Figure 4.6: Molecular absorbance cross sections for undamaged species, damaged species, and irreversibly damaged material, as determined from absorbance decay and recovery measurements using 9g/l DO11/PMMA samples. Fitting the absorbance decay and recovery data to Equation 4.51 at each frequency determines the absorbance per unit length for each population, denoted by σi . In order to determine the molecular absorbance cross section, ǫi , the absorbance per unit length is divided by the concentration, ǫi = σi , c (4.52) where c is the dye concentration of the pristine sample in molecules/unit volume, which for 9g/l DO11/PMMA is c = 2.285 × 107µm−3 . Figure 4.6 shows the estimated molecular absorption cross sections for the three species determined from experiment. From Figure 4.6, it appears that the reversibly damaged species has a very similar absorption cross section to the fresh sample, with both peaking near 2.64eV. However, the irreversibly damaged species has a drastically different absorption cross section, 117 with the visible peak nearer the UV(peaking at 2.88 eV). In addition to the peaks in the visible region, there appears to be a peak in the UV, as suggested by the high energy shoulder in the spectrum. Unfortunately this peak lies outside of the spectral range of the experimental apparatus. Since the shoulder near the UV in the cross section of the irreversible species is higher than the shoulder in either of the other species, it is likely that the irreversible species has a higher absorption cross section in the UV regime. This observation is consistent with the irreversible species being related to polymer damage, as damaging neat PMMA typically results in a yellowing of the polymer, due to bond breaking that increases the absorbance in the deep blue/UV region. Photodamage to the polymer alone does not fully explain the species associated with irreversible damage, as there is a visible peak in addition to the UV peak. One hypothesis for the visible peak is that it is due to photocharge ejection and recombination [48], which plays a role in creating correlated chromophores [52,54,55]. The argument is as follows. Domains are hypothesized to consist of dye molecules which are correlated through their interactions with polymer chains, and decay is associated with the ejection of an electron/ion from the dye. In this picture it is possible for the damaged dye fragment to break free from the polymer chain, leaving the charge on the chain, and making it impossible for the molecule to heal under the influence of a domain when the charge fragment is missing. The smaller molecule’s absorbance spectrum will be shifted to higher energy, and the free charge fragment will interact with the polymer chains, leading to a change in the absorbance spectrum. To test this hypothesis auxiliary measurements such as FTIR, Raman spectroscopy, NMR, and UV spectroscopy should be used to determine the structures of the molecules after photodegradation. Other tests of the nature of the irreversible process include, linear spectroscopic measurements of the effect of varying the host polymer, which 118 Figure 4.7: Energy level diagram proposed by Embaye and coworkers. Reprinted with permission from [1]. Copyright 2008, AIP Publishing LLC. should result in a similar reversibly damaged species, but a drastically different irreversibly damaged species. However, changing the polymer host to test this hypothesis may also change the self healing process of the molecule if the polymer host plays an important role in self healing, as suggested by past studies [1, 43, 44, 52, 54, 55]. 4.5 Proposed energy level diagram With the three absorbance cross sections determined of the species involved, we can now deduce an approximate energy level diagram for the DO11/PMMA system. In 2008, Embaye and coworkers proposed an energy level diagram for the system using ASE and absorbance measurements and assumed a single reversibly decayed species shown in Figure 4.7 [1]. The proposed energy level diagram based on our new work expands on Embaye’s diagram with the inclusion of a third irreversibly damaged species as shown in Figure 4.8. The new diagram has the same undamaged states, where states 0,3,4, and 5 are 119 Figure 4.8: Proposed energy level diagram for the three population model, with the ground states of each population being marked by boxes. 120 involved in the ASE cycle. In Embaye’s diagram, the energy levels of the damaged species were determined by assuming that the change in absorbance was due to the conversion of one species into the other one where the absorbance peak amplitudes associated with the new species increase during decay, while the peak amplitudes associated with the undamaged species decrease. Two new peaks were observed at 2.23 eV and 3.18 eV during degradation, which led to the conclusion that these peaks were excited states of the damaged population. However, if the conversion between three species is responsible, determining which peak is associated with which state is more difficult. For the proposed energy diagram, we assume that each peak in the damaged population’s cross section (Figure 4.6) corresponds to only one transition. However degenerate/near-degenerate transitions and vibronic/rotational states may be present which our measurements are unable to differentiate, and therefore we ignore at this time. From absorbance measurements the reversibly damaged species is found to have a transition with an energy of 2.64eV, and since it recovers, it’s ground state must have a higher energy than the ground state of the undamaged species, though the exact energy difference can not be determined. The transition from the undamaged population to the permanently damaged one is assumed to go from 0 → 6, with 6 relaxing into state 2, with the transition 2 → 6 having an energy of 2.88 eV, and state 2 having a lower energy than state 0, thus making recovery energetically impossible. The inclusion of the irreversibly damaged species in the energy level diagram is slightly misleading, as the current hypothesis holds that the irreversible species is actually a combination of polymer and dye, whereas the undamaged state and reversibly damaged state are assumed to originate in molecules. Most likely there are intermediate steps in the transition between 0 → 6 (i.e. the ground state is excited to the second excited state, which then either decays back to the ground state or state 121 6), but with no other evidence besides absorbance decay and recovery measurements the details are indeterminate. Further experiments, such as temperature dependent absorbance spectroscopy, FTIR, UV spectroscopy, and Raman spectroscopy, should be able to refine our understanding of the transitions involved. 4.6 Summary While the details are unknown, the three-population model as diagramed in Figure 4.1 is found to be in good agreement with decay and recovery measurements as a function of time and intensity. In addition, the absorbance decay and recovery measurements can be used to calculate the molecular absorption cross sections for the three populations and to suggest an energy level diagram that describes the energetics of the DO11/PMMA system during reversible and irreversible photodegradation. 122 Chapter 5 Applied Electric Field Effects 5.1 Introduction One of the proposed mechanisms of reversible photodegradation is photocharge ejection and recombination [48,90], which posits the mechanism of decay to be the ejection of a charged particle (be it an electron, proton or larger ionic molecule) creating an ion-hole pair, with recovery occurring when a hole and ion recombine. As both the ion and hole are charged, changing the electrical properties of the environment (e.g. dielectric constant, free and trapped charge densities, applied external field, etc.) should change the decay and recovery characteristics. To test the effects on reversible photodegradation of changing the electrical properties, decay and recovery measurements are performed on DO11/PMMA thin film samples with an applied electric field. In this chapter the results of the electric field studies are reported with a discussion of conductivity and optical measurements, and concludes with a model of electric field dependent reversible photodegradation based on the correlated chromophore domain model. 123 5.2 Conductivity To better understand the electrical properties of the DO11/PMMA system both dark conductivity and photoconductivity experiments are preformed. Dark conductivity is measured by connection a sample in series with a power supply and picoammeter to measure the current as a function of time for a step applied voltage. Photoconductivity experiments use the same setup as dark conductivity, but with the addition of a laser light source, with the picoammeter now measuring both the dark current and photo-induced current. The behavior of the current over time can be used to understand the underlying charge dynamics and polarization effects, which in turn will help us understand reversible photodegradation if charge products are formed. 5.2.1 Dark conductivity Mechanisms of transient conductivity The change in conductivity in response to a sudden change in voltage is known as transient conductivity, and has been used extensively to study the known mechanisms in dye doped polymers including [92–94]: 1. Capacitive charging. 2. Fast electron cloud distortion. 3. Change in sample capacitance due to electromechanical thickness change, and electric field dependence of the dielectric constant. 4. Electrode polarization due to complete or partial electrode blocking. 5. Charge injection due to ohmic electrodes. 6. Flow of conduction current caused by motion of charges. 124 7. Charge trapping in the bulk of the polymer. 8. Charge hopping from impurity sites or dopants. 9. Polarization and hopping effects from the motion of polymer chains. 10. Elastic reorientation of the polymer and dopants. 11. Slow viscous reorientation of polymer chains. The first three are typically faster then the other processes and contribute less to the transient current. Electrode effects (4 and 5) at the metal-semiconductor (polymer) interface can either allow or block the flow of electrons from the metal into the polymer, as discussed in Appendix C. The final six mechanisms can be split into two groups: charge effects (5-8) and polarization effects (8-10), with polymer chain motion affecting both polarization and charge movement. In dye doped polymers, polarization effects tend to dominate the transient current response with charge trapping and hopping leading to small currents that change over long periods of time. Mathematical description of transient conductivity To treat the mathematics of transient conductivity, we assume that the primary source of current is the polarization field, as charge hopping and trapping mechanisms tend to be far smaller. The changing polarization of the dye-polymer system in response to an applied field leads to a polarization current, j, j= dP (t) , dt (5.1) where P (t) is the time varying polarization. For the case of a step function applied field, a single molecule’s response can be described as a Debye poling process [92–95], 125 dP (t) + γP (t) = γP∞ , dt (5.2) where γ is the relaxation rate, and P∞ is the final polarization after an infinite amount of time. When the field is turned off the molecule relaxes via a Debye relaxation process with the same rate as in Equation 5.2: dP (t) + γP (t) = 0. dt (5.3) Debye’s model of dielectric poling and relaxation [95] assumes that the system responds at a single rate, γ. For most dielectrics this has been found to be an incomplete description of the poling and relaxation process, with the bulk poling and relaxation processes involve a distribution of relaxation rates [92–94, 96–104]. Mathematically this can be expressed as P (t) = P0 Z ∞ g(γ)e−γt dγ, (5.4) ∞ where g(γ) is the rate distribution function. For most dielectrics the rate distribution is found to correspond to one of two distribution functions of the Curie-Von Schweidler model or the Williams-Watts model. The Curie-Von Schweidler model is used to describe transient currents with the polarization current of the form j ∝ t−β , where 0 < β < 1 for most dielectrics [92, 96–102]. The Williams-Watts polarization current is a stretched exponential given by j ∝ exp[−αtβ ], where α is a constant and β is related to the width of the distribution in Equation 5.4 [93, 94, 103, 104]. Comparing the measured transient current with the Curie-Von Schweidler and Williams-Watts models determines the underlying charge and polarization dynamics of the dye-doped polymer system. The parameters so determined will shed light on the underlying mechanisms that may lead to better understanding of the electric field 126 effects on reversible photodegradation. 5.2.2 Data Transient dark conductivity measurements with varying concentration, applied electric field strength, and electric field histories can be used to test the scales of the responses and determine the species involved. Concentration Dark conductivity is measured for four concentrations(0g/l(undoped PMMA), 7g/l, 9g/l, and 12g/l) with a bias to higher concentrations as they typically have the best decay and recovery characteristics. Undoped PMMA is used as a baseline for conductivity. Current is measured, which has a response to a step voltage of I(t) = G(t)V0 , (5.5) where V0 is the applied voltage, and G(t) is the conductance as a function of time, which depends on the sample geometry and material composition. The conductance can be expressed in terms of the geometry-independent conductivity as G(t) = A I(t) = σ(t) , V0 L (5.6) where I is the current, V0 is the applied voltage, A is the electrode area, L is the sample thickness, and σ(t) is the conductivity. Rearranging Equation 5.6, the conductivity is σ(t) = I(t)L . AV0 127 (5.7) For each concentration 50V is applied using electrodes with a cross-sectional area of 1 mm2 . The sample thicknesses varies from sample to sample ranging from 30 − 60µm, due to dependence of viscosity on concentration. Within experimental uncertainty, no correlation is found between the conductivity and concentration. However, two different samples of the same concentration can yield drastically different conductivities, which suggests that there are other parameters, related to sample preparation, which have a large effect on sample conductivity, making it difficult to isolate any one cause. Proposed mechanisms for sample-tosample conductivity differences, aside from concentration, are absorbed water (whose concentration depends on humidity), the amount of residual solvent, free and trapped bulk charge densities, and free and trapped surface charge densities. Field strength To test the effect of applied electric field strength on transient dark current, several 9g/l samples are tested at room temperature under different applied electric field strengths. In order to minimize effects due to electric field history, the applied fields are only turned on for short time periods, with a much longer resting period in between runs to allow changes to the dye-doped polymer system due to the field to dissipate. Figure 5.1 shows the dark current as a function of time for applied fields in one 9g/l sample. Figure 5.1 shows that increasing the applied electric field has two noticeable effects on the transient dark current: 1) the time to reach a steady state current increases, and 2) the spacing between adjacent curves increases, implying that the current is a nonlinear function of applied field. In addition to qualitative observations, fits are attempted with both the Curie-Von Schweidler model and the Williams-Watts model, finding that the Curie-Von Schweidler model is unable to fit the data, and the 128 Figure 5.1: Time evolution of transient dark current as a function of applied field strength for a 9g/l sample. Williams-Watts model is found to fit the current for fields of 2.0 V/µm or less with one rate constant. Fits for higher fields requires an additional rate constant, which suggests that there are processes that are activated at higher applied fields, but are negligible at lower applied fields. In addition to measuring the transient current response to a step function applied field, the current is also measured in response to the applied field being turned off. Figure 5.2 shows that when the system relaxes the current is observed to flow in the direction opposite to the current induced by the applied field. This is expected as the sample is essentially a capacitor, and capacitors discharge with the current in the opposite direction to the current during charging. Fits to the data in Figure 5.2 show that for fields between 1.0 and 1.5V/µm the current discharge follows a single exponential with a time constant of τ = 2.6 ± 0.3s. For fields above 1.5 V/µm, the current follows a double exponential, with one time constant of τ = 2.6 ± 0.3 s, and the other on the order of 20-30 s. As with the transient current measurements for the 129 Figure 5.2: Transient dark current for a 9g/l sample after an applied electric field is abruptly turned off. step turned on, the addition of a second time constant for higher fields suggests that there are processes which are negligible for smaller fields, but become important for higher applied fields, leading to a secondary discharge time constant. Since discharge is assumed to be due to the sample’s capacitave nature, we can estimate the capacitive discharge time constant by assuming that the sample acts as an ideal parallel plate capacitor with capacitance, A C = ǫr ǫ0 , d (5.8) where ǫr is the relative permittivity, which can be approximated as ǫr = n2 ≈ 2.25, d is the sample thickness ( ≈ 20µm), and A is the area of the electrodes (5mm × 5mm), which gives C ≈ 25pF. From current measurements, the resistance is on the order of 100GΩ, which leads to an estimated capacitive discharge time being τ = 2.5 130 s, which is within experimental uncertainty of the measured value. Electric field history We have found that the electric field history affects the transient current in a sample in response to an applied field. Hysterises effects become apparent when the time the electric field has been applied to a sample is long compared with the resting time when no field is applied between conductivity measurements. As an example we consider transient dark current measurements of a 9g/l sample using an applied voltage of 100V (E0 =2.5V/µm). Several days after pressing, the sample is placed in the conductivity apparatus, the field is turned on for two hours, and the current is recorded. After two hours the field is turned off and the sample rests until this discharging current is negligible (approximately 10 mins). The field is then reapplied again for 24hrs, and the process of turning the field off and back on is repeated 24 hr later. Figure 5.3 shows the current response for the 0hr, 2hr, and 24hr field application times. The longer a sample is conditioned by an electric field, the smaller the current response to an applied electric field. Additionally, the rate at which the initial spike decays to the steady state increases as the electric field conditioning time increases. These results suggest that the applied electric field changes the electrical properties of the system over time, with the resulting properties being semi-stable, requiring days or even weeks for the sample to return to it’s initial state. The effect of electric field conditioning a sample is observed not only in dark conductivity measurements, but also in photoconductivity and reversible photodegradation measurements as well. In each case it is found that after several days of conditioning, it takes a week or more for the sample to return to its initial state. This suggests that the process of electric field conditioning is related to slow processes 131 Figure 5.3: Transient current of a 9g/l DO11/PMMA sample in response to a step function voltage of 100V, as a function of electric field conditions time. such as viscous reorientation of the polymer/dyes, and free/trapped charge exchange with the surrounding environment. In this hypothesis, the applied electric field slowly aligns the polymer and dyes, while also stripping out free and trapped charges from the sample. When the field is turned off the sample slowly relaxes removing the electric field-induced order, and charges from the environment are able to be slowly re-absorbed, returning the sample to its initial state with a free charge density. 5.2.3 Photoconductivity Photoconductivity is a phenomena in dielectrics and semiconductors in which a material’s conductivity increases when illuminated by light of sufficient energy. The theory of photoconductivity in polymers is complex, with a full discussion beyond 132 the scope of this thesis1 . Instead, we will provide an overview of the relevant physics and discuss how photoconductivity and reversible photodegradation may be related. There are two types of photoconductive polymers: intrinsic and extrinsic. In an intrinsic photoconductive polymer, the polymer itself absorbs the light leading to the generation of a photo excited electron and hole, which then move through the polymer. The dopants In an extrinsic photoconductive absorb light producing an exciton (bound electron-hole pair), which then transfers charge to the polymer where it is free to move. Thus in an extrinsic photoconductive polymer, the polymer acts solely as the charge transporting media. Almost all polymer’s have band gaps in the UV regime, including PMMA, so most photoconductive polymers in the visible range are extrinsic. The process of extrinsic photoconductivity occurs by the following steps [106, 108, 113, 114]: 1. The sensitizer (DO11 in our studies) absorbs light and forms an exciton. 2. The exciton is captured at a donor/acceptor site on the polymer, which polarizes the polymer forming an electron-hole pair. 3. The applied electric field separates a fraction of the electron-hole pairs, with the rest undergoing geminate recombination. 4. Either electrons/holes, or both move in the applied electric field, with random diffusion resulting in zero current. 5. Electrostatic forces will eventually cause the free electrons and holes to recombine at recombination sites within the circuit. 6. Alternatively, the moving charges can become temporarily or permanently trapped at trap sites within the polymer. 1 For further detail see references [105–112]. 133 7. Additionally, depending on the electrical contacts used, charges may be injected into the polymer from the electrodes adding to the current. Contacts that allow full charge injection are called ‘Ohmic’, and contacts that partially, or totally block charge injection are called ‘blocking’. See Appendix C for more details. Figure 5.4 shows a schematic of the process with the assumption that electrons are the mobile charge. While photogeneration of electron-hole pairs is the primary mechanism of photoconductivity, there are also several other effects which contribute to photoconductivity, including: 1. Dark conductivity change due to photothermal heating. 2. Capacitance change due to photomechanical effects. 3. Photoinduced reorientation of dye molecules and polymer chains. 4. (hypothesized)Photocharge ejection due to photodegradation of dyes. The first three additional effects are known to occur, with the final effect hypothesized as a mechanism for reversible photodegradation [48, 90]. The original motivation for for photoconductivity studies was to use it as a probe of reversible photodegradation. However, there are many processes which contribute to photoconductivity making it difficult to isolate any one mechanism. Figure 5.5 shows a typical photocurrent response for DO11/PMMA thin films, which follows a double exponential, with the larger magnitude component having a time constant on the order of minutes, and the smaller magnitude component having a time constant on the order of tens of minutes. For a single sensitizer species, the accepted explanation for the double exponential response is that the large and fast response corresponds to the the photo-generation of electron-hole pairs by the sensitizer molecule, and the smaller 134 and slower response corresponds to the other mechanisms [105, 106, 109]. Relaxation when the light source is turned off is found to follow a stretched exponential response, similar to the dark conductivity relaxation. 135 136 Figure 5.4: Extrinsic photoconductivity diagram, with the polymer states in blue and the dopant states in orange. Light is absorbed by the dopant and forms an exciton (1), which is then transferred to the polymer (2), where the electric field separates the electron and hole (3). The electron is free to move under the influence of the electric field (4), with some number becoming temporarily or permanently trapped in trap sites (6). Eventually electrostatic attraction leads to the recombination of the electrons with holes (5). Figure 5.5: Typical photocurrent response of DO11 dye doped in PMMA polymer. Given the proposed mechanism for photodegradation being ejection of charged fragments, we also simultaneously use optical measurements with photoconductivity to correlate photodegradation with the current response. The measured optical decay constant is found to be of the same order of magnitude as the slow current response, but, given all the other possible mechanisms associated with the slow current response, it is impossible to make a direct correlation between the photocurrent and optically probed photodegradation. While the complex nature of photoconductivity makes it a poor tool to measure reversible photodegradation, it is still a very useful technique for probing different aspects of the electro-optic properties of dye-doped polymers. For example, one of the unexpected results of the photoconductivity study was discovered when considering the effect of electric field conditioning on samples. In general the steady state photocurrent may be written as [105, 106] 137 Figure 5.6: Photocurrent for zero applied electric field before and after electric field conditioning. JSS = eφI0 µτ E, L (5.9) where e is the electron charge, φ is the charge generation efficiency, I0 is the incident intensity, µ is the carrier mobility, τ is the recombination time, L is the sample thickness, and E is the applied field. Equation 5.9, predicts that without an applied electric field the photocurrent vanishes. When measuring fresh samples, this is found to be the case. However, samples that have gone through electric field conditioning are found to have a nonzero photocurrent when the applied field is zero, as shown in Figure 5.6. This effect is found to persist for many days, eventually dissipating to zero, suggesting that electric field conditioning produces a quasi-stable internal electric field, which eventually relaxes. 138 5.2.4 Summary of conductivity measurements In general dye-doped polymers are electrically active, so reversible photodegradation will be affected by changes in the electrical properties of the system. From darkand photo- conductivity measurements we find that the electrical properties of the DO11/PMMA system are complex, with many different factors coming into play such as free and trapped charge densities, absorbed water from ambient humidity, trapped solvent, and electric field conditioning. Since many of these factors are difficult to control, it becomes understandable that electrical properties can vary drastically between two ‘identical’ samples. This variation of factors, such as humidity and free charge density, may also provide an explanation for why there is such difficulty in reproducing the parameters characterizing reversible photodegradation from sample to sample. 5.3 Electric field effect on reversible photodegradation: noninteracting model results In addition to measuring dark and photo conductivity, we also use digital imaging microscopy to probe the effect of an applied electric field on reversible photodegradation. Tests of numerous samples finds that there are multiple influences on reversible photodegradation due to an applied electric field, with some effects observed in every sample, and some limited to specific samples. While many different samples are used, for simplicity we consider the results from one sample, which was tested more thoroughly than all others. The sample of interest is of 9g/l concentration made by drop pressing to a thickness of 22µm. It is damaged for 25 minutes using an ArKr laser operating at 488nm 139 and focused to a line with peak intensity of 175 W/cm2 . The electric field is applied during both decay and recovery at five different field strengths, and different combinations of polarity are tested. Optical measurements are performed using the DIM apparatus, with the scaled damaged population (SDP), n′ , being fit to the thin-film noninteracting two population model (TPNIM); which predicts that the SDP during decay is given by, n′ (t; I) = and during recovery is given by ∆σLαI 1 − e−(β+αI)t , β + αI (5.10) n′ (t; I) = ∆σL n′IR (I) + n′R (I)e−βt , (5.11) where β is the recovery rate, I is the intensity, α is the intensity independent decay rate, ∆σ is the absorbance per unit length difference between the two populations, L is the sample thickness, n′IR is the ad hoc irreversibly damaged portion, and n′R is the reversibly damaged portion. Figure 5.7 shows the model fits during decay and recovery at the beam center for an electric field applied at different strengths, but constant polarity. Using the DIM’s spatial resolution, and assuming that the pump beam has a perfect elliptic gaussian shape, the SDP can be determined at a wide range of intensities, allowing for comparisons of the exponential rates and amplitudes in Equations 5.10 and 5.11 as functions of intensity. 5.3.1 Decay Results To determine the effect of an applied electric field on photodegradation, we consider the decay rate γ = β + αI and exponential amplitude, A(I) = ∆σLαI , β+αI as functions of intensity for different applied fields. Figure 5.8 shows the decay rate, and Figure 5.9 140 Figure 5.7: Fits of the scaled damaged population during decay and recovery(inset) of the burn center for various applied electric fields for a 9g/l sample burned with an intensity of 175W/cm2 . shows the decay amplitude as functions of intensity. As the raw data is extremely noisy, only three of the smoothed data sets are shown, with the others being represented by their model fits. Figure 5.8 shows that both the slope and intercept of the decay rate decreases as the applied field increases. Thus, the intensity independent decay rate, α, and the recovery rate, β, become smaller as the field strength increases. Figure 5.9 shows that the amount of decay decreases for all intensities as the applied field increases. The electric field dependent parameters α and β are found to be consistent between Figure 5.9 and Figure 5.8 for all five field strengths. In addition to testing the effect of the electric field magnitude on decay, the polarity of the applied field is also reversed, with positive polarity being defined as the direction of pump beam propagation, and negative polarity being in the opposite direction. Figure 5.10 shows the intensity independent decay rate for several field magnitudes applied in opposite polarities, and Figure 5.11 shows the equilibrium 141 Figure 5.8: Decay rate as a function of intensity for several applied electric fields Figure 5.9: Exponential amplitude as a function of intensity for several applied electric fields. 142 Figure 5.10: Intensity independent decay rate for electric fields applied parallel (+) to the k−vector, and anti-parallel (-) found from fits to the non-interacting model give by Equation 5.10. The decay rate is found to decrease with applied field independent of direction. scaled damage population at the burn center for several field values. Both the rate and amplitude are found to be independent of the polarity of the applied field. 5.3.2 Recovery Results For recovery measurements there are two parameters of interest: the recovery rate, β, and the recovery fraction, which is defined using equation 5.11 to be RF = n′R . n′IR + n′R (5.12) where once again n′IR is the irreversibly damaged portion of the SDP, and n′R is the reversibly damaged portion of the SDP. When applying the electric field during recovery, we define the positive polarity to be parallel to the field applied during 143 Figure 5.11: Equilibrium scaled damaged population (ESDP) for electric fields applied parallel (+) to the k−vector, and anti-parallel (-) determined by fits to the non-interacting model give by Equation 5.10. The ESDP is found to be independent of the direction of the applied field. 144 decay, and the negative polarity to be anti-parallel to the field applied during decay. Figure 5.12 shows the mean recovery rate as a function of field strength for both polarities, with the positive polarity resulting in a decreasing recovery rate, and the negative polarity resulting in an increasing recovery rate. To find the mean value, recovery rates are measured at 1200 points on the burn spot and the weighted mean is computed. In addition to finding the weighted mean, we also use a histogram to determine the distribution of recovery rates, with a bin size of ∆β = 10−5 min−1 . Figure 5.13 shows the distribution of recovery rates for the different applied field strengths in the positive direction, with fits to a poisson plotted as a guide to the eye. As the electric field is increased the distribution of rates becomes narrower with the mean value becoming smaller, suggesting that the applied field acts to narrow the distribution of properties of the recovery sites. While Figure 5.13 considers the recovery rate distribution with the applied field on, the effect of narrowing the distribution of properties at recovery sites persists even with the applied field is turned off. As an example, zero field reversible photodegradation is measured in a 7g/l sample both before and after a 2V/µm electric field is applied for eight days, with all other experimental conditions remaining the same. Figure 5.14 shows the rate distributions both before and after electric field conditioning, with the conditioned rate distribution being drastically narrower than the fresh distribution. The effect of conditioning is found to persist for many days, sometimes even weeks, but eventually the sample will return to the pre-conditioned state with a broad zero field recovery rate histogram. Along with measuring the recovery rates over the sample, we also measure the recovery fraction over the burn in order to determine the average recovery fraction. Figure 5.15 shows the average recovery fraction as a function of applied field for both polarities. The recovery fraction is found to increase regardless of the applied field’s 145 Figure 5.12: Recovery rates for electric fields applied during recovery parallel (+) to the field applied during decay, and anti-parallel (-) to the field applied during decay obtained from fits to the non-interacting model give by Equation 5.11. Maintaining polarity between decay and recovery reduces the recovery rate, while reversing the polarity increases the recovery rate. 146 Figure 5.13: Recovery rate histograms for different applied fields with fits to a poissonian. The histograms are generated using the recovery rates of 1200 points in a burned area with binning of ∆β = 10−5 min−1 . As the electric field is increased the distribution narrows and the mean shifts towards smaller recovery rates. 147 Figure 5.14: Recovery rate histogram for zero field reversible photodegradation both before and after electric field conditioning. The effect of conditioning is to narrow the distribution and shift the mean towards a slower recovery rate. polarity. The most peculiar observation is that of a recovery fraction greater than one in some samples2 . When the sample is damaged with no, or small, applied field the damage is observed to recover to a level less than unity. If the applied field is increased, recovery continues and for large enough fields the recovery fraction exceeds unity as shown in Figure 5.16. In Figure 5.16, the electric field is incrementally increased over time and the scaled damaged population continues to recover until it becomes negative, which corresponds to a recovery fraction greater than one. At first glance this result is paradoxical, as a recovery fraction greater than one suggests that there are more molecules which recover, then were damaged. However, the paradox may be resolved by studying actual images of recovering samples as shown in Figure 5.17. From the images and line profiles, it appears that the process 2 Currently the phenomenon has only been observed in two samples out of ten, with no clear difference between the samples which display the effect, and samples which do not. 148 Figure 5.15: Average recovery fraction as a function of applied electric field for both + and - polarities. The recovery fraction increases with applied field strength, but the increase is found to be asymmetric. 149 of reaching a recovery fraction greater than one involves the burned areas, which are usually bright lines, to dim into dark lines. For this to occur the change in transmittance, ∆T = exp {n∆σL} , (5.13) must change from being greater than one, to being less than one, which either requires the damaged population, n, to become negative, or the difference in absorbance per unit length, ∆σ, to become negative. Since the damaged population is defined to be non-negative, the cross section difference must change sign in order for the burn line to darken. At present an explanation of the burn line changing due to an increasing field is difficult to formulate precisely, as the phenomenon has only been observed in two samples, and there is no clear difference between them and the samples which do not display the phenomenon. However, taking absorbance spectra of the pristine samples under the influence of an electric field finds that the pristine absorbance cross section is unaffected by typical experimental electric field strengths.. This suggests that the darkening of the burn line is therefore due to the absorbance cross section of the damaged species changing due to the application of an applied electric field, but the reason why this occurs in some samples and not others still remains unresolved. 5.3.3 Summary Digital imaging measurements of electric field dependent reversible photodegradation shows that an applied electric field changes the decay and recovery characteristics of DO11/PMMA. Experimentally we find that unipolar measurements, in which the applied field direction is maintained between decay and recovery, result in highly reproducible quantitative results where increasing the field strength decreases both 150 Figure 5.16: Scaled damaged population recovery for a sample that was burned with a 0.75 V/µm field applied. The applied field is increased during recovery. the recovery and decay rate, decreases the amount of decay, and increases the recovery fraction. In addition the applied field is found to condition the sample such that the distribution of properties at recovery sites narrows. When considering the case of bipolar measurements, in which the applied field direction is changed between decay and recovery, we find consistent results with the recovery rate increasing, but not always to the same degree, and often depending on how long the electric field had been applied to the sample. This inconsistency leads us to hypothesize that the recovery mechanism is sensitive to changes in the local electrical properties(e.g. chain alignment, free/trapped charges, etc.), and that when changing the polarity abruptly, the local electrical properties change drastically resulting in the recovery characteristics changing unpredictably. 151 Figure 5.17: (a) Image of horizontal burn lines when 2.5 V/µm field is first applied (red line shows the location where the burn profile is measured). Two of the burn lines had recovered nearly 100%. (b) Image of burn lines after several days of 2.5 V/µm field conditioning. The two burn lines (marked by arrows), which had recovered to the background level, continued to recover leading to two dark lines. (c) The image line profile corresponding to the red line in a. (d) The image profile corresponding to the red line in b. 152 5.4 Extending the correlated chromophore domain model While the TPNIM is found to fit the decay and recovery data, it requires different parameters for each field strength, with no explanation for why the parameters vary with applied field. Currently, the best model of reversible photodegradation is the correlated chromophore domain model (CCDM) which accurately describes how temperature and concentration affect decay and recovery [52, 54, 55]. In this section we will extend the CCDM to include an irreversible component, absorbance depth effects, and the most consistent electric field effects, which are that an increasing unipolar electric field will 1. Decrease the NIM decay rate. 2. Decrease the amount of damage. 3. Decrease the NIM recovery rate. 4. Increase the recovery fraction. 5.4.1 Domain model extended to include an irreversible component To begin extending the CCDM model to include an irreversible component, we introduce the domain size into Equations 4.16-4.18 of the three population model such that the parameters are consistent with the domain model proposed by Ramini et. al. [52]. From experimental observations we find the decay rate into the irreversibly damaged population transforms as ǫ → ǫN. With these substitutions, the coupled differential equations describing the three population model become 153 α dn0 = − + ǫN In0 + βNn1 , dt N αI dn1 = n0 − βNn1 , dt N dn2 = ǫNIn0 , dt (5.14) (5.15) (5.16) which have the same solutions as found in Section 4.2 but with the transformation of the parameters into the domain model form. The solutions to Equations 5.14-5.22 represent the dynamics of a single domain containing N molecules. In order to find the macroscopic dynamics of each species we must take the ensemble average which is given by n0 (t) = ∞ Z 1 n1 (t) = Z n2 (t) = (5.17) n1 (N, t)Ω(N)dN, (5.18) n2 (N, t)Ω(N)dN, (5.19) ∞ 1 Z n0 (N, t)Ω(N)dN, ∞ 1 where Ω(N) is the density of domains of size N. In Section 5.4.3 we will use a simple thermodynamic model to derive Ω(N). 5.4.2 Inclusion of depth effects In addition to including the irreversible component, the CCDM can be further extended to take into account that damage is greater at the surface, where the intensity is higher, and less in the interior, where the intensity of the pump is less due to absorption, and therefore the degree of damage is small. We refer to this behavior as a “depth effect”. To do so, we recognize that the differential Beer-Lambert law in this case will depend on the ensemble average over domains, and not the individual 154 domains themselves. Thus the coupled differential equations for the populations and the pump/probe intensities are ∂n0 ∂t ∂n1 ∂t ∂n2 ∂t ∂Ip ∂z ∂I(ω) ∂z = − = α N + ǫN Ip n0 + βNn1 , αIp n0 − βNn1 , N = ǫNIp n0 , Z ∞ = −Ip [n0 σ0 (ωp ) + n1 σ1 (ωp ) + n2 σ2 (ωp )] Ω(N)dN, 1 Z ∞ = −I(ω) [n0 σ0 (ω) + n1 σ1 (ω) + n2 σ2 (ω)] Ω(N)dN, (5.20) (5.21) (5.22) (5.23) (5.24) 1 where Ip is once again the pump intensity, σi (ω) is the absorbance per unit length of the ith species at frequency ω, and ωp is the pump beams frequency. Equations 5.20-5.24 have no closed form solution, and so numerical methods must be used to solve them. 5.4.3 Density of domains including dielectric energy In the original derivation of the density of domains a simple condensation model was used in which the energy of a domain of size N is given by −λ(N −1) [52,54,55]. This type of condensation is called isodesmic3 aggregation, and is found to correspond to linear arrays of molecules [115, 116]. Therefore when deriving the effect of an applied electric field on the domain energy we assume that the domain consists of a linear array of equally spaced molecules with step size r, and polarizability α, as shown in Figure 5.18. In realty the domains are far more complex with molecules spread out unequally in all three dimensions, but we find that despite it’s simplicity, the linear array of molecules is a good approximation to real systems that we have studied. 3 Meaning the interaction energy is independent of domain size. 155 Figure 5.18: Linear array of equally spaced dipoles separated by grid spacing r. Each dipole has a polarizability α. In the dilute case, where the molecules are noninteracting, we can write the dipole moment of the domain as N X pi , (5.25) = NαE0 , (5.26) P = i=1 where N is the size of the domain, and pi is the dipole moment of the ith molecule in the domain. In this case their is no energy advantage for a molecule being in a domain, and therefore we find that this level of approximation leads to no change in the distribution of domains. In order for the dielectric energy to affect the distribution of domains we must consider molecular interactions, which create a difference in energy for a molecule being in a domain, versus outside of the domain. 156 The simplest model for molecular interactions in a domain, is to assume that each molecule is sufficently spaced (r 3 >> α) such that each molecule behaves as a point dipole, producing an electric field that changes the effective field that the other molecules experience [117, 118]. Assuming that the interactions occur only between molecules in the same domain we can write the dipole moment of the ith molecule as " pi = α E0 − i−1 X j=1 # N X pj pj , − ((i − j)r)3 j=i+1 ((j − i)r)3 (5.27) where the summations account for the effective electric field due to the other molecules. Summing over the dipole moments in Equation 5.27 we find the total dipole moment for the domain, P = N X pi , (5.28) i=1 = NαE0 − N X N X i=1 j6=i pj . (|i − j|r)3 (5.29) In order to find the individual dipole moments we can rewrite Equation 5.27 as a matrix equation P = αE0 1 − α MP, r3 (5.30) where the column vector P = {p1 , p2 , · · · , pN }, the column vector 1 = {1, 1, · · · , 1} and M is an N × N matrix with elements given by Mij = Solving for P we obtain 0 1 |i−j|3 157 if i = j if i 6= j (5.31) Figure 5.19: Molecular dipole moments at a given grid position for four different domain sizes, with α/r 3 = 10−3 . As the domain size increases the individual dipole moments become more homogenous, with only the boundary molecules having different dipole moments. α −1 P = αE0 I + 3 M 1, r (5.32) where I is the identity matrix, and the superscript −1 represents the matrix inverse. The solution to Equation 5.32 depends on the matrix size, meaning there is no closed form solution, but the equation is easily solved numerically, and Figure 5.19 shows the solutions for several domain sizes. When considering the exact solutions to Equation 5.32, we find that molecular interactions work to decrease the individual dipole moments, and that as the number of molecules increases the individual dipole moments become more homogenous, with only the boundary dipole moments deviating. In the infinite domain limit the effect of 158 interactions is exactly given by 2ζ(3) rα3 , where ζ is the zeta function with ζ(3) ≈ 1.202. Using this limit, and considering the numerical solutions to Equation 5.32, one finds that the total dipole moment of a domain of size N may be approximated as P (N) ≈ NαE0 − 2ζ(3)(N − 1) α2 E0 , r3 = Nαf (N)E0 , (5.33) (5.34) where f is the local field factor given by, f (N) = 1 − 2ζ(3)(N − 1) α , N r3 (5.35) which represents the modification of the applied electric field due to the dipole interactions. With the local field factor determined we can now calculate the dielectric energy. For a collection of N dipoles with polarizability α, the dielectric energy is U(N) = −NαEL2 , (5.36) where EL is the local field given by EL = f E0 . Using the local field factor in Equation 5.35 and assuming α2 /r 6 ≈ 0, the dielectric energy of a domain of size N from Equations 5.34 and 5.36 is U(N) = −Nαf (N)2 E02 , ≈ −NαE02 + 4ζ(3)(N − 1) (5.37) α2 2 E . r3 0 (5.38) Using the dielectric energy and the domain energy used by Ramini and coworkers, the total energy of a domain of size N is: 159 E(N) = −λ(N − 1) − NαE02 + 4ζ(3)(N − 1) α2 2 E . r3 0 (5.39) With the full domain energy in equation 5.39 we can now derive the density of domains using the method of Ramini and coworkers [54, 55], which uses the grand canonical partition function to minimize the Helmholtz free energy. We begin by writing the partition function for a single domain of size N: zN = exp γ(N − 1) + Nα′ E02 − η(N − 1)E02 4ζ(3)α2 , kT r 3 where γ = λ/kT , α′ = α/kT , η = (5.40) k is Boltzmann’s constant, and T is the temperature. The global partition function of the ensemble is a product of the individual partition functions given by Z= Y z ΩN N N ΩN ! , (5.41) where ΩN is the number of domains of size N in a given volume. Using the global partition function we can write the Helmholtz free energy, F , as F = −kT ln Z, X = −kT (ΩN ln zN − ln(ΩN !)) , (5.42) (5.43) N ≈ kT X N ΩN ΩN ln −1 , zN (5.44) where we have used Stirling’s approximation to simplify the expression. To minimize the Helmholtz free energy we consider the chemical potential of a domain of size N, µN = ∂F , ∂ΩN = kT ln 160 (5.45) ΩN , zN (5.46) which in equilibrium is simply related to the chemical potential of a single molecule by µN = Nµ1 . Using this fact we can find a relation between the distribution of domains of size N, ΩN , and the distribution of single molecules, Ω1 , µN = Nµ1 , kT ln ΩN zN ΩN zN ΩN Ω1 = NkT ln , z1 N Ω1 = , z1 N Ω1 = zN , z1 2 = e(γ−ηE0 )(N −1) ΩN 1 . (5.47) (5.48) To determine the density of single molecule domains, Ω1 , we consider the total number of molecules in a fixed volume, ρ, ρ= ∞ X NΩ(N), (5.49) N =1 = (1 − Ω1 . 2 (γ−ηE 0 ) Ω )2 e 1 (5.50) which solving for Ω1 gives p 2 2 (1 + 2ρe(γ−ηE0 ) ) − 1 + 4ρe(γ−ηE0 ) Ω1 = . 2 2ρe2(γ−ηE0 ) (5.51) Substituting Equation 5.51 into Equation 5.48 the density of domains of size N is found to be √ N 1 (1 + 2ρz) − 1 + 4ρz . Ω(N) = z 2ρz where z = exp(γ − ηE02 ). 161 (5.52) 5.4.4 Summary In this section we extended the correlated chromophore domain model to include the effects of irreversible photodegradation, pump absorption, and the application of an electric field. The full model consists of the coupled differential equations given by Equations 5.20-5.24, and the density of domains given by Equation 5.52. 5.5 5.5.1 Fitting imaging data to the extended CCDM Results The original CCDM model was developed based on measurements of reversible photodegradation as probed by ASE in DO11/PMMA bulk samples at varying temperatures and concentrations with excellent results. For imaging and electric field experiments, bulk samples cannot be used as their large thickness makes it difficult to image and produce viable electric field strengths. Instead we use thin films which are typically on the order of 20-40µm in thickness, allowing for clear imaging and MV/m electric fields. While thin films are excellent for electric field dependent measurements and imaging, the process of producing them introduces wide variations from sample to sample, making it difficult to compare data between samples. Since concentration dependent measurements require comparisons of different samples, which is unreliable, we choose to focus on electric field and temperature dependent reversible photodegradation on single samples. For quantitative tests of the extended model we perform decay and recovery imaging measurements on numerous samples of differing concentrations with different applied fields and temperatures and find consistency. But given sample to sample variations, the quantitative results are highly variable. For a comprehensive test of 162 the model we perform a rigorous set of experiments on a 9g/l DO11/PMMA sample (designated sample number 120525D09I2), measuring a larger set of field strengths and temperatures than in any other sample. The electric field-dependent measurements are performed using the DIM and conductivity apparatus at room temperature (T = 293K). A single polarity electric field is applied during decay and recovery for all field strengths. An ArKr laser operating at 488nm, focused using a cylindrical lens to a line with a peak intensity of 175W/cm2 , burns the sample for 25 min in each run. For the temperature-dependent studies, the confocal DIM is used, with no electric field applied, and the burning laser is focused using a positive spherical lens to give a peak intensity of 96W/cm2 , for burn duration of 40 min to compensate for the lower intensity. Ideally, the temperature-dependent and electric field-dependent experiments should produce identical results for the same burn parameters; however, differences are observed between DIM and confocal DIM images. From repeated measurements the overall magnitude of the scaled damaged population is found to be different between experiments, but the decay and recovery rates are found to be the same, suggesting that the underlying populations are the same, within an overall scale factor. To understand this difference we consider the effect of the probe beam’s spectrum on a camera’s color channel intensity. From Appendix A the color channel intensity to first order including frequency integration may be written as τ C(t) = g Z ∞ I0 (ω)SC (ω)(1 − σ0 (ω)L + n1 (t)∆σ1 (ω)L + n2 (t)∆σ2 (ω)L)dω , Z ∞ Z ∞ τ τ = C0 + n1 (t)L I0 (ω)SC (ω)∆σ1 (ω)dω + n2 (t)L I0 (ω)SC (ω)∆σ2 (ω)dω, g g 0 0 τ τ (5.53) = C0 + n1 (t)S1 + n2 (t)S2 , g g 0 where τ is the exposure time, g is the gain, L is the sample thickness, ∆σi = σi − σ0 , 163 where σi is the absorbance per unit length of the ith species, C0 is the fresh color channel intensity given by C0 = Z ∞ I0 (ω)SC (ω)[1 − σ(ω)L]dω, 0 (5.54) and Si are the scale factors for the reversible(i = 1) and irreversible components(i = 2) given by S1 = L Z 0 S2 = L Z ∞ I0 (ω)SC (ω)∆σ1 (ω)dω, (5.55) I0 (ω)SC (ω)∆σ2 (ω)dω, (5.56) ∞ 0 which depend on the sample thickness, probe spectrum, and camera sensitivity. Given these dependencies it is apparent that changing cameras and/or light sources will effect the amplitude of the scaled damaged population, leading to differing amplitudes between experiments, as they use different cameras and light sources. Figure 5.20 shows the normalized spectra for the light sources used in the DIM and CDIM experiment, as well as an average change in absorbance due to burning. The CDIM light source is spectrally broader than the DIM light source, and therefore probes a wider range of the spectrum. Thus the integrals in Equations A.6 and A.7 yield different values than for the DIM light source. This difference in the scaling of the SDP is accounted for experimentally by including a free amplitude parameter that multiplies the calculated damaged populations. In order to fit the measured SDP, we use custom fitting functions, which numerically solve Equations 5.20-5.24 in order to calculate the scaled damaged population as a function of time and pump intensity, Ip given by I(z = L, t; Ip ) n (t; Ip ) = A ln , I(z = L, t = 0; Ip ) ′ 164 (5.57) Figure 5.20: Spectra for light sources used in the DIM and CDIM, with the change in absorbance during photodegradation in DO11/PMMA for comparison. 165 where I is the probe beam intensity, and A is the free amplitude parameter. Depending on the coding, Equation 5.57 can be used to either fit the SDP during decay and recovery as a function of time, or used to fit the reversible and irreversible components of the SDP as a function of intensity. With these capabilities in mind the full data set used for fitting is: 1. The scaled damaged population during decay at the beam center for five electric field strengths (Figure 5.21), and six different temperatures (Figure 5.25). 2. The scaled damaged population during recovery at the beam center for five electric field strengths (Figure 5.22) and two different temperatures4 (Figure 5.25 inset). 3. The noninteracting model exponential amplitude and offset as a function of intensity for the five electric field strengths (Figures 5.23 and 5.24). The model is fit to the entire data set simultaneously using Igor Pro’s global fit routine with parameters constrained to be consistent across all the data with the only free parameter being the adjustable amplitude factor. Table 5.1 lists the model parameters determined from self consistent fitting of the full data set. 4 Equipment malfunctions and time constraints resulted in only two good temperature measure- ments of the chosen sample, but the behavior of recovery as a function of time was observed consistently in every sample tested. 166 Figure 5.21: Scaled damaged population during decay at the burn center for different applied fields. Figure 5.22: Scaled damaged population during recovery for different applied fields. 167 Figure 5.23: Exponential amplitude for recovery as a function of intensity. The amplitude scales with the reversibly damaged population n1 . Figure 5.24: Exponential offset for recovery as a function of intensity. The offset scales with the irreversibly damaged population n2 . 168 Figure 5.25: Scaled damaged population as a function of time during decay for several temperatures with fits using the new model. Inset shows recovery for T=298K and T=308K. 169 Model Parameters α(10−2 cm2 W−1 min−1 ) 2.48 ± 0.66 β(10−5 min−1 ) 2.49 ± 0.21 ǫ(10−6 cm2 W−1 min−1 ) 3.12 ± 0.10 ρ(10−2 ) 1.19 ± 0.25 λ(eV) 0.282 ± 0.015 η(10−13 m2 V−2 ) 1.19 ± 0.15 Table 5.1: Parameters determined from self consistent fitting of the full data set. 5.5.2 Discussion In the previous section the extended CCDM was used to fit imaging results of electric field and temperature dependent reversible photodegradation. For the temperature dependent data the results are consistent with Ramini and coworkers, with the free energy advantage, λ, and density parameter, ρ, being within experimental uncertainty of their results [52]. Additionally, the effect of temperature on the amount of decay and recovery predicted by the extended CCDM is found to be in agreement with experimental data. To better understand this effect we use the measured model parameters to predict the underlying damaged populations as the measured data corresponds to the scaled damaged population. Figure 5.26 shows both damaged populations as a function of time during decay for three temperatures, at the surface of the sample. From Figure 5.26 we see that the effect of increasing the temperature is to increase the degree of reversible decay, and to decrease the amount of irreversible decay. The interpretation of this result is as follows: as the temperature is increased, it becomes entropically unfavorable for molecules to form large domains, this leads 170 (a) (b) Figure 5.26: (a) Reversibly and (b) irreversibly damaged components as a function of time during decay at the surface of the sample for three different temperatures. As the temperature is increased the reversible component gets larger, while the irreversible component becomes smaller. to larger domains breaking apart, which results in a decrease of the average domain size. Since the conversion rate from the undamaged species to the reversibly damaged species is proportional to αI , N the decreasing domain size causes the undamaged molecular species to convert more quickly into the reversibly damaged species, causing the damage amount and rate to increase. The rate of damage to the polymer is proportional to ǫNI, which becomes smaller as N decreases, so damage to the polymer will decrease as the temperature increases. This result can be explained physically by recalling that the irreversibly damaged component is hypothesized to be damage to the polymer mediated by the dye molecules. At higher temperatures, the mobility of both the dye and polymer increase so interactions between them will be weakened; additionally, at higher temperatures energy deposited from the dye into the polymer is more likely to be dissipated via phonons/excitons, limiting the damage to the polymer. 171 For the electric field dependent measurements the model is found to fit the experimental data well, with the applied field mitigating irreversible damage, increasing reversible damage, decreasing the recovery rate, and increasing the recovery fraction5 . As with the temperature dependence, to better understand the effect of the applied field, Figure 5.27 plots the calculated damaged populations during decay for three different applied field strengths. Comparing Figures 5.26 and 5.27 we find that increasing either temperature or the applied electric field mitigates the irreversible damage, while also simultaneously increasing the degree of reversible damage. This increase in reversible damage and decrease in irreversible damage leads to the measured increase in the recovery fraction. The explanation for the electric field effects in the framework of the model is as follows. The energy advantage of a molecule being in a domain versus outside of the domain is kT (γ − ηE02 ), where the electric field dependence arises due to dielectric interactions between molecules in the domain. By increasing the electric field these interactions intensify, causing the energy advantage of being in a domain to decrease. The decreased energy advantage decreases the likelihood of molecules to be in domains, thus decreasing the averaged domain size. As with the temperature case, the decreased average domain size leads to the reversible decay rate increasing 5 There is a slight experimental oddity at this point. In the temperature dependent case as shown in Figure 5.25, the increase in reversible damage leads to the total scaled damaged population increasing. However, in the case of the electric field measurements, shown in Figure 5.21, the increase in the reversible damage component leads to a decrease in the total scaled damaged population. The explanation for this oddity is that the experimental setup in the electric field case weighs the irreversibly damage component more heavily than the reversible component due to the spectral convolution in Equations A.6 and A.7. In the case of the temperature-dependnent measurements with the CDIM, the two species are weighted more equally, hence the increase in total scaled damaged population. 172 (a) (b) Figure 5.27: (a) Reversibly and (b) irreversibly damaged components as a function of time during decay at the surface of the sample for three different field strengths. As the field is increased the reversible component gets larger, while the irreversible component becomes smaller. and the irreversible decay rate decreasing. While the extended CCDM accurately describes reversible photodegradation as a function of intensity, temperature and applied electric field, it does not describe the nature of a domain, or the source of interactions/correlations between molecules in a domain. In order to develop a working hypothesis for the underlying features of the CCDM, we consider several important results: 1. The model is found to fit experimental data well with a domain energy that assumes a linear array of molecules. 2. Reversible photodegradation is found to be very sensitive to the electrical properties of the dye-doped polymer, and electric field conditioning is found to make the process more isotropic across a sample. 3. One component of transient photocurrent during photodegradation with an 173 applied electric field, is found to increase at a similar rate as probed by optical methods, suggesting that both methods may be probing the same process. With these results in mind we propose that domains consist of molecules which are correlated with each other through their interactions with a polymer chain, so that aggregation is linear, and the correlations between molecules may be mediated via phonons and excitons moving along the polymer chain. Currently, the nature of correlation is unknown, but it is speculated that hydrogen bonding between DO11 and the polymer, or a DO11 tautomer and polymer, may be responsible, as the measured free energy advantage of λ = 0.28eV, is close to the approximate energies for the hydrogen bonds: OH-O (0.22 eV) and OH-N (0.30 eV) [119, 120]. We model reversible photodegradation based on extrinsic photoconductivity. In our proposed model, light is absorbed by a DO11 molecule forming an excited state. In the case of extrinsic photoconductivity, this state is an exciton (a bound electronhole pair), for the case of photodegradation we assume that the excitation forms an unbound ion-hole pair, where the ion may correspond to an electron, proton, or larger molecular ion. The important difference between the two processes is that in extrinsic photoconductivity the electron and hole are bound at the DO11 site and are immobile; in photodegradation however, the excitation generates a free ion.6 Once generated the mobile ion can 1) recombine with the hole, 2) move along the polymer chain, 3) trapped at a trap site, or 4) escape into another polymer chain or voids in the polymer. In this picture, recovery occurs when ions recombine with holes, returning the degraded molecules back to their initial state. The irreversible component of decay arises through several possible processes. 1) Ions can interact with oxygen in the polymer to produce polymer oxide radicals, which lead to chain scisson and 6 For simplicity of our argument, we assume that the ion formed in photodegradation is mobile, while the hole is stationary, though in reality this may be switched or both are mobile. 174 cross linking, thereby changing the polymer structure. 2) Photo thermal heating of the polymer (due to the correlated dye molecules) can damage the polymer. 3) The ejected ion can become permanently trapped, keeping holes from recombining with the ejected ions. 4) The holes can combine with a different type of charged particle, limiting the recombination of holes and ions. We hypothesize that rate equations of the domain model of Equations 5.20-5.24 describe photocharge ejection and recombination in a domain, with the assumption that the probability of photocharge ejection is dependent on the interactions of molecules with each other, perhaps mediated by the polymer. To understand this hypothesis we consider the case where the domain size is increased. Increasing the domain size results in more molecules being associated with the same polymer chain, resulting in three effects: 1. The correlated molecules will interact more strongly limiting the probability of photocharge ejection. 2. With more molecules attached to the polymer chain there is a greater likelihood that energy will be deposited from the molecule into the polymer, resulting in damage to the polymer. 3. Once a charge is ejected, the larger number of molecules on the chain will result in a greater probability of finding a hole to recombine with, thus increasing the recovery rate. These three effects correlate to the domain size scaling used in the extended CCDM, where the reversible damage rate is decreased with domain size, and the irreversible damage rate and recovery rate increase with domain size. 175 5.6 Summary Measurements of reversible photodegradation with an applied electric field show that the underlying mechanism is sensitive to the electrical properties of the dye-doped polymer, suggesting that the species involved are either charged or polarizable. To better understand the electrical properties of the system we used dark- and photoconductivity measurements and find: (1) the system’s transient conductivity follows a Williams-Watt’s model with more than one rate being activated at higher applied fields, (2) the system is highly sensitive to the applied electric field history, (3) electrical properties can vary drastically between two similarly prepared samples, and (4) electric field conditioning appears to smooth out alignment and free charge density such that the dye-doped polymer becomes more electrically isotropic. In addition to measuring the electrical properties of the DO11/PMMA system we also extended the correlated chromophore model to include an irreversibly damaged component, the effect of pump absorption as a function of depth, and the effect of an applied electric field. Using the extended model we fit data as a function of intensity, temperature, and applied electric field, finding that the model accurately describes the physical process. 176 Chapter 6 Conclusions Previously, research into reversible photodegradation had primarily used amplified spontaneous emission as a probe of damage in DO11/PMMA. We have expanded the tool set used as a probe to include digital imaging, white light interferometry, absorbance spectroscopy, and conductivity measurements. Our findings can be separated into three categories: depth effects, irreversibility, and electric field effects. Depth effect results Using numerical calculations we find that differential pump absorption as a function of depth has a large effect on the decay dynamics, and that ignoring the effect results in underestimating the decay rate and degree of decay. In addition, we calculated the effect on the pump beam due to propagation effects including linear wave propagation, damaged lensing, and thermal lensing, finding that all three effects are negligible when compared to absorptive effects. 177 Irreversible photodegradation results While nonlinear measurements of reversible photodegradation often find full reversibility, our linear optical measurements find that for all tested cases there is always an irreversible component, even when nonlinear measurements show the system fully recovering. This leads us to believe that the irreversible component is related to chromophore mediated photodamage of the polymer, which does not contribute directly to nonlinear measurements. With this in mind we developed a three population parallel process model to incorporate both the reversible and irreversible component and find that the model fits experimental data well. Using the same model we also fit pristine, decayed, and recovered sample’s absorbance data to find the molecular absorption cross sections of the damaged species finding that the cross section of the reversible species is similar to the undamaged species, but the cross section of irreversibly damaged species is drastically different peaking 0.24eV higher in energy, and having a large UV absorbance, which is consistent with polymer damage. Effect of an Electric Field While applied electric field studies has yielded many results, the primary finding is that reversible photodegradation is highly sensitive to changes in the dye-doped polymer’s electrical properties and to an applied field, implying that the underlying process involves charged and/or polarizable species. Extended correlated chromophore model Taking all our results into account, we developed a model based on the correlated chromophore domain model (CCDM), which takes pump absorption, irreversible degradation, and electric field effects into account. To account for the electric field we assumed that a domain consists of a linear array of interacting point dipoles. Despite 178 being a very simple model of molecular dielectric interactions, we find that the energy derived in this way predicts the behavior in the presence of an electric field that is consistent with experimental observations. The universal success of an isodesmic aggregation model to predict temperature, concentration, and electric field-dependent behavior, suggests that domains are in fact linear. With this assumption, and our hypothesis that the irreversible component is polymer damage, we propose that domains are molecules correlated through a polymer chain, with hydrogen bonding connecting the dye to polymer. In addition, to explain the form of the model’s rate equations (Equations 5.20-5.24), we hypothesize a qualitative model of photocharge ejection and recombination within a domain, such that the rate equations are consistent. 6.0.1 Prospects While not providing definitive answers concerning the mechanism of reversible photodegradation, our study has revealed two fundamentally important aspects of the effect; namely, it has shown that there is an irreversible component to which nonlinear measurements are insensitive and more importantly, that the species involved are charged and/or polarizable. These discoveries open up a new path of investigation which will hopefully one day find the definitive answers concerning the underlying mechanism, with the next step being to determine the exact nature of the charged/polarizable species involved, and the nature of the irreversible species. To this end techniques such as FTIR, micro Raman spectroscopy, UV spectroscopy, and NMR should reveal further details about the species, where visible optical measurements fall short. 179 Appendix A Corrections to imaging population In Section 2.3.1 we derived the color channel intensity, C, assuming that our incident probe beam is a delta function in spectrum. While our LEDs are spectrally narrow, they do possess some spectral width. In this section we consider the effect of a non zero spectral width on the measured color channel intensity. A.1 Numerical Solutions To model the effect of the probe having some spectral width, ∆E, we use absorption data taken during photodegradation and perform numerical integration assuming a Gaussian probe spectrum, and Gaussian camera sensitivity, see Figure A.1. We use Equation 2.23 Z ∞ 1/γ τ C(t) = I0 (ω)SC (ω)T0 (ω)∆T (t; ω)dω , g 0 assuming that the gamma factor, gain, and exposure time are unity. After integration we convert the color channel intensity as a function of time into the scaled damage population, ′ n (t) = ln 180 C(t) C0 , (A.1) Figure A.1: Normalized intensity spectra and camera sensitivity used for calculations, along with the pristine sample absorbance. where C0 is the color channel intensity at time t = 0. Figure A.2 shows the scaled damage damaged population computed directly from the absorbance data as well as the numerical integration results. We find that as the spectral width increases the measured scaled damaged population becomes smaller. The reason being that integration over the spectrum essentially looks at the average change in absorbance across the spectrum, which includes a spectral region of increasing absorbance, and a region of decreasing absorbance, leading to a smaller net effect than measuring just at one energy. While the magnitude of the scaled damaged population decreases as we integrate over more of the spectrum, we find that the rates are unchanged. To demonstrate this we compare the scaled undamaged population (1 − n′ ) calculated for the widest 181 Figure A.2: Calculated scaled damaged population (SDP) using absorbance data and approximate camera sensitivity for several Gaussian intensities of differing bandwidths. SDP calculated from absorbance is added for comparison. 182 Figure A.3: Raw absorbance data compared to the scaled undamaged population for the widest spectral width, showing that the two overlap having the same decay rate. spectral width, ∆E = 0.288 eV, and the raw absorbance signal using axis scaling to over lap the data as shown in Figure A.3. We find that the two overlap showing that they have the same rate, but given the differences in scaling we know that the magnitudes are different. A.2 Approximating spectral convolution In the previous section we simulated the color channel intensity as a function of time using actual absorbance decay data, approximate camera sensitivities, and approximate probe beam spectra of differing bandwidths. The simulation found that while the magnitude of the decay changed with different probe bandwidths, the rates of de- 183 cay remained the same, suggesting a linear relationship between the scaled damaged population and the actual population. As a first order approximation to account for the spectral convolution we use the three-population model developed in Chapter 4 to write the change in transmission found in Equation 2.23 as ∆T (x, y, t; ω) = e−σ0 (ω)L+n1 ∆σ1 (ω)L+n2 ∆σ2 (ω)L , (A.2) ≈ 1 − σ0 (ω)L + n1 ∆σ1 (ω)L + n2 ∆σ2 (ω)L, (A.3) where ∆σi = σi − σ0 , σ0 is the undamaged populations’ absorbance per unit length, σ1 and σ2 are the absorbance per unit lengths of the reversible and irreversible components, respectively, n1 is the reversibly damaged molecular population, n2 is the irreversibly damaged polymer population, L is the sample thickness, and we have assumed the thin film approximation and expanded the exponential as a first-order Taylor series. Substituting Equation A.3 into Equation 2.23 and assuming γ = 1 we can write the color channel intensity to first order as τ C(t) = g Z ∞ I0 (ω)SC (ω)(1 − σ0 (ω)L + n1 (t)∆σ1 (ω)L + n2 (t)∆σ2 (ω)L)dω , Z ∞ Z ∞ τ τ = C0 + n1 (t)L I0 (ω)SC (ω)∆σ1 (ω)dω + n2 (t)L I0 (ω)SC (ω)∆σ2 (ω)dω, g g 0 0 τ τ (A.4) = C0 + n1 (t)S1 + n2 (t)S2 , g g 0 where C0 is the pristine color channel intensity given by C0 = Z 0 ∞ I0 (ω)SC (ω) [1 − σ(ω)L] dω, (A.5) and Si are the scale factors for the reversible (i = 1) and irreversible components (i = 2) given by, 184 S1 = L Z 0 S2 = L Z ∞ I0 (ω)SC (ω)∆σ1 (ω)dω, (A.6) I0 (ω)SC (ω)∆σ2 (ω)dω. (A.7) ∞ 0 which can be drastically different for different light sources. As an example, we calculate the scale factors for a broad Gaussian spectrum centered at 500nm and a narrow Gaussian spectrum centered at 400nm, which approximates a white light source, and deep blue LED, respectively. Figure A.4 shows the normalized intensity for both light sources, as well as the camera sensitivity curve. For the difference in absorbance per unit length for each species we assume a concentration of 9g/l and use the molecular absorption cross section difference, ∆ǫ, as shown in Figure A.5 and calculated from Section 4.4. Numerically integrating Equations A.6 and A.7 with these values and for a sample thickness of 1µm, we find the scale factors for each species for each light source, as well as their ratio, see Table A.1. From Table A.1 we see that the scale factors for the narrow light source are both negative, in practice this would lead to the color channel intensity decreasing due to burning. On the other hand, the broad light source has positive scale factors, meaning that the color channel intensity will increase due to burning. In addition, comparing the ratio of S1 and S2 gives the relative weighting between the reversible and irreversible damage components, with a ratio greater than unity implying the reversible component is weighted more heavily, and a ratio less than unity meaning that the irreversible component is weighted more. Comparing the ratios between the light sources, shows that the narrow light source more heavily weights the irreversible component, while the broad light source has a more even weighting, with the ratio being nearer to unity. Experimentally, the difference in weighting between light sources means that a 185 Figure A.4: Normalized light spectra and sensitivity used to calculate scale factors. Broad light spectrum approximates a white light source centered at 500nm, and the narrow light spectrum approximates a LED centered at 400nm. burn probed by the narrow light source will appear as a dark line, and will appear to only recover a small amount. However, the same burn probed by the broad light source will appear as a bright line, and will recover more fully than in the case of probing with the narrow light source. Thus it is important to consider the spectral characteristics of the probe light when performing imaging studies, as the magnitudes of decay and recovery depend on the spectral convolution of the probe beam spectra. On the other hand, in the thin film approximation, it is clear from Equation A.4 that the time dependence of decay and recovery is independent of the spectral convolution, as the color channel intensity is linear in the populations. 186 Figure A.5: Difference between damaged and undamaged molecular absorbance cross sections for both the reversible, and irreversible components. 187 Model Scale Factors S1 S2 Ratio(S1/S2 ) Narrow Light -1.27 -4.38 0.290 Broad Light 2.43 0.603 4.03 Table A.1: Calculated scale factors for light spectra shown in Fig A.4. The scale factors have differing sign due to the spectral region which they probe. Additionally, the ratio between the scale factors differs between the light sources, with the narrow light source weighing the irreversibly damage component more than the broad light source does. A.3 Summary By using real absorbance data and estimated camera sensitivity curves we find that the effect of light source broadening is to change the magnitude of the scale damaged population, but have no effect on the time dependence. This is further shown by considering a first order approximation to the color channel intensity in the thin film approximation. While we used an approximated color channel sensitivity, as there are variations in every camera due to manufacturing tolerances, a careful characterization of each camera can yield it’s real color channel sensitivity. Once the true sensitivity is found, it can be used to calculate the actual scale factor’s, which will help to estimate the true population from imaging measurements. Doing so will allow for better comparisons with other experimental methods, such as absorbance spectroscopy and fluorescence, where spectral convolution is absent. 188 Appendix B Justification of zero-charge electromagnetic wave equation In Section 3.3.1 we considered the propagation of an electromagnetic wave through a material with the assumption that ∇ · E = 0. This assumption is used often when considering propagation effects, however, in most cases of interest ∇ · E 6= 0. In this section we will consider the precise details, and show that in most situations the assumption is a good approximation. B.1 Wave equation from Maxwell’s equations We begin with Maxwell’s equations in a dielectric media: 189 ∇ · D = 4πρf , ∇×E=− (B.1) ∂B , ∂t ∇ · B = 0, (B.2) (B.3) ∇ × H = 4πJf + ∂D , ∂t (B.4) Assuming that the medium has a uniform linear magnetic susceptibility (µ(r) = 1, p n(r) = ǫ(r)), and that there are no free charges, ρf = 0, and no free currents, Jf = 0, we can rewrite Equations B.1-B.4 as: ∇·D=0 ∇×E=− (B.5) ∂B ∂t (B.6) ∇·B=0 ∇×B= (B.7) ∂D ∂t (B.8) Taking the curl of Equation B.6 and simplifying using Equation B.8 yields the familiar form of the wave equation1 , ∇ × ∇ × E = −∇ × =− 1 ∂B , ∂t ∂ (∇ × B) , ∂t ∂2D =− 2 , ∂t (B.9) (B.10) (B.11) In the case of the magnetic properties varying spatially there will be an added term taking the gradient of the magnetic permeability into account, however for most optical materials this may be safely neglected. 190 where D = ǫ(r)E + PN L is the electric displacement with ǫ(r) being the spatially varying dielectric constant, and PN L being the nonlinear polarization. Using vector identities and substituting in the electric displacement we can rewrite Equation B.11 as: ∂2D , ∂t2 ∂2 2 ∇(∇ · E) − ∇ E = − 2 (ǫE + PN L ). ∂t ∇×∇×E= − (B.12) (B.13) At this point in most derivations the assertion is made that since there are no free charges ∇ · E = 0. However, this is false as there are bound charges formed by the polarization of the material, which means ∇ · E 6= 0. To account for this we begin with Equation B.5 and simplify, ∇ · D = 0, (B.14) ∇ · (ǫE + PN L ) = 0, (B.15) ∇ǫ · E + ǫ∇ · E + ∇ · PN L = 0, (B.16) which upon rearranging becomes, ∇·E=− 1 ∇ǫ · E + ∇ · PN L . ǫ (B.17) In the case of a linear homogenous material the right hand side of Equation B.17 will be zero, however, if there is a nonlinear polarization and/or spatially dependent electric susceptibility the right hand side will no longer be zero. Using Equation B.17 we can expand the first term in Equation B.13 as ∇(∇ · E) = −∇ 1 1 NL , ∇ǫ · E − ∇ ∇·P ǫ ǫ 191 (B.18) where, ∇ 1 ∇ǫ · E ǫ ∇ǫ (∇ǫ · E) + ǫ2 ∇ǫ = − 2 (∇ǫ · E) + ǫ =− 1 ∇ (∇ǫ · E) , (B.19) ǫ 1 ([∇ǫ · ∇]E + [E · ∇]∇ǫ + ∇ǫ × ∇ × E) , (B.20) ǫ and, ∇ 1 ∇ · PN L ǫ =− 1 ∇ǫ ∇ · PN L + ∇ ∇ · PN L . 2 ǫ ǫ (B.21) Substituting Equations B.20 and B.21 into Equation B.13 we find the full wave equation to be: 1 ∇ǫ ∇ǫ NL (∇ǫ · E) − ([∇ǫ · ∇]E + [E · ∇]∇ǫ − ∇ǫ × ∇ × E) + ∇ · P ǫ2 ǫ ǫ2 2 2 N L 1 ∂ (ǫE) ∂ P − ∇ ∇ · PN L − ∇2 E = − − . (B.22) ǫ ∂t2 ∂t2 B.2 Bound charge effect estimates for typical experiments In the previous section we derived the full wave equation taking bound charges into account. Inspecting Equation B.22 we find that the terms accounting for bound charges are either proportional to ∇ǫ ǫ or proportional to PN L . Using the change in index of refraction discussed in Section 3.3.3, we can compute the dielectric constant as ǫ = (n0 + n1 )2 where n0 is the homogenous refractive index, and n1 is the in homogenous refractive index. Assuming the same spatial profile as in Section 3.3.3 we find that ∇ǫ ǫ ∼ 10−5 µm−1 , which is negligible compared to the terms in the typical wave equation. Additionally, as PN L is proportional to the nonlinear susceptibility, which is small in our case, we may safely neglect its effect. 192 Appendix C Ohmic vs blocking electrodes When performing dark- and photo- conductivity measurements on dye-doped polymer samples, the junction of the dye-doped polymer and the ITO electrode forms a metal-semiconductor interface, or Schottky barrier, which has been studied extensively [121–128]. Schottky barriers are primarily characterized by their SchottkyBarrier height, eΦB , which depends on both the semiconductor and metal used. Figure C.1 shows a schematic band diagram of the metal-semiconductor interface for a n-type semiconductor1 developed by Schottky, where EF is the Fermi level, EV and EC are the valence and conduction bands respectively2 , eΦm is the work function of the metal, eχs is the electron affinity of the semiconductor, and eΦB is the Schottky barrier height. The Schottky barrier height is easily calculated in this model, recalling that the Fermi level’s of the two materials in contact are equal in equilibrium, which gives, 1 For our discussion we will only consider an n-type semiconductor, with the results for p-type simply found with the relevant sign changes. 2 The valence band corresponds to the highest occupied molecular orbital(HOMO) level of the semiconductor, and the conduction band corresponds to the lowest unoccupied molecular orbital(LUMO) level. 193 Figure C.1: Band diagram of metal-semiconductor interface for the Schottky model. eΦB = eΦM − eχs . (C.1) If eΦB is positive the contact is called blocking, as energy is required for an electron in the metal to jump into the conduction band of the semiconductor and if eΦB is zero or negative the contact is called Ohmic, as the conduction band of the semiconductor lies at/below the Fermi level and charge is free to travel from the metal into the semiconductor’s conduction band. While Schottky’s model is good for understanding the basic physics of the interface, reality is more complicated. The primary complication arises due to interface states that bend the semiconductor’s band structure at the interface. Band bending occurs because the wavefunction of an electron in the semiconductor must be continuous at the interface leading to electron’s occupying states which are forbidden in the bulk of the semiconductor. The result of occupying these forbidden states is to bend the band structure. Figure C.2 shows the band structure in equilibrium at the metal-semiconductor interface taking band bending into account. The depth in the semiconductor over which the interface effects are important is denoted by xD . 194 Figure C.2: Band diagram of metal-semiconductor interface with band bending. xD is the depth over which the interface effects are important. Effects due to the Schottky barrier are of particular interest in semiconductor electronics, especially in the physics of diodes, however, the primary importance for our research is their effect on photo-induced current. In the case of a large positive Schottky barrier, such that the contact’s are blocking, we find that photocurrent is not a linear function of voltage as for an ohmic material. Instead the current eventually saturates and stays the same as the voltage increases. The saturated current is known as space charge limited current (SCLC). The physical picture of this effect is as follows. Light is absorbed by the semiconductor generating a charge density, which results in a photocurrent with an applied voltage. Eventually the applied voltage will result in a field strength such that the transit time, ∆t, of the charge through the sample will be smaller than the recombination time, τ , which can be expressed as ∆t < τ, L < τ, v L < τ, µE0 195 (C.2) (C.3) (C.4) where L is the sample thickness, v = µE0 is the charge velocity, where µ is the charge mobility, and E0 is the applied field. When this voltage is reached, the majority of the mobile charges will be stuck at the electrodes, limiting the photocurrent through the semiconductor, as there is only a finite number of charges in the semiconductor. The current is therefore limited by the amount of charge in a given space, and is therefore called space charge limited current. However for Ohmic contacts, where charge is free to move between the semiconductor and the metal, the current is not limited by the charge confined in the semiconductor as electrons can be injected into the semiconductor allowing for the photocurrent to exceed the SCLC. Figure C.3 shows a comparison of the photocurrent with Ohmic and Blocking electrodes. Both electrodes produce a linear current below the saturation limit, but diverge near the limit. For our experimental field strengths the current is universally Ohmic, which suggests that either the ITO-DO11/PMMA interface is Ohmic, or that we are well below the saturation limit. 196 Figure C.3: Photocurrent as a function of applied voltage for Ohmic and Blocking electrodes. JSS is the space charge limited steady state current. 197 Appendix D Reversible photodegradation in other anthraquinone derivatives In addition to DO11, we also use the DIM to measure reversible photodegradation of PMMA thin films doped with other anthraquinone (AQ) derivatives. Figure D.1 shows the molecular structures of the other AQ derivatives tested, with the alphabetical code we gave each dye. Each thin film sample is made using bulk pressing, with the bulk polymer made to have a dye concentration of 3g/l. Different pump wavelengths and intensities are used as the absorbance spectrums of each dye vary greatly, changing the energy deposition characteristics. Table D.1 compiles the results for all dyes tested, where we have used the twopopulation noninteracting model (TPNIM) for fitting. For a concentration of 3g/l, dyes C, H, and P did not display recovery. However, tests at higher concentrations have shown dye H to recover, suggesting that C and P may also recover at higher concentrations. These results are preliminary with the suggestion that the symmetry of the molecule plays a role in determining the decay and recovery characteristics. Further study 198 Figure D.1: Molecular structures of other anthraquinone derivatives tested with alphabetical coding. should be able to determine which molecular characteristics make a molecule be more resistant to damage, and able to recover better. This fundamental understanding should shed light on the underlying mechanisms of domain formation and lead to guidelines for determining which materials will have greater photostability. 199 200 Dye Chemical λ(nm) F(kJ/cm2 ) α(10−3 cm2 /Wmin) n′0 β(10−3 min−1 ) RF A 1,2 AMAQ 488 6.1 40.6 ± 7.4 0.50 0.708 ± 0.017 0.15 B 1-A 2,4 BAQ 488 10.5 23.9 ± 4.4 0.65 2.4 ± 1.4 0.5 C 1,4 AHAQ 514 10.1 23.9 ± 6.5 0.27 0 0 H 1,4 DAAQ 514 8.8 8.0 ± 5.6 0.46 0 0 I 1,5 DAAQ 488 5 23.6 ± 4.3 0.39 2.04 ±0.42 0.18 J 1,2 DHAQ 488 14.1 13.6 ± 2.6 0.10 6.5 ± 3.2 0.35 K 1,8 DHAQ 488 8.8 8.1 ± 1.7 0.32 10.0 ± 1.4 0.47 L 1,4 DHAQ 488 16.0 3.2 ± 1.5 0.10 0.91 ± 0.26 0.14 P 1-AMAQ 514 11.7 16.5 ± 8.1 0.40 0 0 Table D.1: Tabulation of Anthraquinone decay and recovery parameters. λ is the pump wavelength, F is the CW pump fluence, α is the TPNIM intensity independent decay rate, n′0 is the peak equilibrium scaled damaged population, β is the recovery rate, and RF is the average recovery fraction. Bibliography [1] N. Embaye, S. K. Ramini, and M. G. Kuzyk, “Mechanisms of reversible photodegradation in disperse orange 11 dye doped in PMMA polymer,” J. Chem. Phys. 129, 054504 (2008). [2] P. Sorokin and J. Lankard, “Stimulated emission observed from an organic dye, chloro-aluminum phthalocyanine.” IBM J. Res. Develop. 10, 162–163 (1966). [3] P. Sorokin and J. Lankard, “Flashlamp excitation of organic dye lasers: a short communication.” IBM J. Res. Develop. 11, 148 (1967). [4] P. Sorokin, J. Lankard, E. Hammond, and V. Moruzzi, “Laser-pumped stimulated emission from organic dyes: experimental studies and anlytical comparisons.” IBM J. Res. Develop. 11, 130–147 (1967). [5] B. Soffer and B. McFarland, “Continuously tunable, narrow-band organic dye lasers,” App. Phys. Lett. 10, 266 (1967). [6] A. Maslyukov, S. Sokolov, M. Kaivola, K. Nyholm, and S. Popov, “Solid-state dye laser with modified poly(methyl methacrylate)-doped active elements,” Applied Optics 34, 1516–1518 (1995). [7] F. Duarte and R. James, “Tunable solid-state lasers incorporating dye-doped polymer-nanoparticle gain media,” Opt. Lett. 28, 2088–90 (2003). 201 [8] A. Costela, I. Garcia-Moreno, and R. Sastre, Tunable Laser Applications (CRC Press, Boca Raton, 2009), chap. Solid state dye lasers, pp. 97–120, 2nd ed. [9] F. Duarte and R. James, Tunable Laser Applications (CRC Press, Boca Raton, 2009), chap. Tunable lasers based on dye-doped polymer gain media incorporating homogeneous distributions of functional nanoparticles, pp. 121–142, 2nd ed. [10] F. Duarte and L. Hillman, Dye laser principles, with applications (Academic Press Inc., 1990). [11] H. Kuhn, “A quantum-mechanical theory of light absorption of organic dyes and similar compounds,” J. Chem. Phys. 17, 1198–1212 (1949). [12] R. Fork, B. Greene, and C. Shank, “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking,” App. Phys. Lett. 38, 671–672 (1981). [13] J. Mialocq, M. Meyer, P. Hebert, X. Armand, and D. Lambert, “Photostability of aqueous micellar rhodamine 6g laser dye solutions under xecl laser pumping,” Optics Commnunications 77, 185–191 (1990). [14] G. Jones, W. Bergmark, and W. Jackson, “Products of photodegradation for coumarin laser dyes,” Optics Commnunications 50, 320–323 (1984). [15] F. Rosselli, W. Quirino, C. Legnani, V. Calil, K. Teixeira, A. Leitao, R. Capaz, M. Cremona, and C. Achete, “Experimental and theoretical investigation of tris-(8-hydroxy-quinolinate) aluminum (alq3) photo degradation,” Oragnic Electronics 2009, 1417–1423 (10). 202 [16] L. Song, E. Hennink, I. Young, and H. Tanke, “Photobleaching kinetics of fluorescein in quantitative fluorescence microscopy,” Biophysical Journal 68, 2588–2600 (1995). [17] A. Bernanose, M. Comte, and P. Vouaux, J.Chem. Phys 50, 64 (1953). [18] A. Bernanose and P. Vouaux, J. Chim Phys. 50, 261 (1953). [19] A. Bernanose, J. Chim. Phys. 52, 396 (1955). [20] A. Bernanose and P. Vouaux, J. Chim. Phys. 52, 509 (1955). [21] H. Kallmann and M. Pope, “Positive hole injection into organic crystals,” J. Chem. Phys. 32, 300 (1960). [22] H. Kallmann and M. Pope, “Bulk conductivity in organic crystals,” Nature 186, 4718 (1960). [23] P. Mark and W. Helfrich, “Space-charge-limited currents in organic crystals,” J. Appl. Phys. 33, 205 (1962). [24] M. Pope, H. Kallmann, and P. Magnante, “Electroluminescence in organic crystals,” J. Chem. Phys. 38, 2042 (1963). [25] M. Sano, M. Pope, and H. Kallmann, “Electroluminescence and band gap in anthracene,” J. Chem. Phys. 43, 2920 (1965). [26] C. Tang and S. Vanslyke, “Organic electroluminescent diodes,” App. Phys. Lett. 51, 913 (1987). [27] J. Burroughes, D. Bradley, A. Brown, R. Marks, K. Mackay, R. Friend, P. Burns, and A. Holmes, “Light-emitting diodes based on conjugated polymers,” Nature 347, 539–541 (1990). 203 [28] H. Gerischer, M. Michel-Beyerle, E. Rebentrost, and H. Tributsch, “Sensitization of charge-injection into semiconductors with large band gap,” Electrochemica Acta 13, 1509–1515 (1968). [29] H. Tributsch, “Reaction of excited chlorophyll molecules at electrodes and in photosynthesis,” Photochemistry and Photobiology 16, 261–269 (1972). [30] M. Matsumura, S. Matsudaira, H. Tsubomura, M. Takata, and H. Yanagida, “Dye sensitization and surface structures of semiconductor electrodes,” Ind. Eng. Chem. Prod. Res. Dev. 19, 415–421 (1980). [31] M. Krunks, A. Katerski, T. Dedova, A. Mere, and I. O. Acik, “Photovoltaic cell based on zinc oxide nanorods and method for making the same,” (2009). [32] Y. Wu, M. Marszalek, S. M. Zakeeruddin, Q. Zhang, H. Tian, M. Gratzel, and W. Zhu, “High-conversion-efficiency organic dye-sensitized solar cells: molecular engineering on daπ-a featured organic indoline dyes,” Energy Environ. Sci. 5, 8261 (2013). [33] M. Liang and J. Chen, “Arylamine organic dyes for dye-sensitized solar cells,” Chem. Soc. Rev. 42, 3453–3488 (2013). [34] K. Hara and N. Koumura, “Organic dyes for efficent and stable dye-sensitized solar cells,” Material Matters 4.4, 92 (2009). [35] F. Gao, Y. Wang, J. Zhang, D. Shi, M. Wang, R. Humphry-Baker, P. Wang, S. Zakeeruddin, and M. Gratzel, “A new heteroleptic ruthenium sensitizer enhances the absorptivity of mesoporous titania film for a high efficiency dyesensitized solar cell,” Chemical Communications 23, 2635 (2008). 204 [36] “Ultrathin, dye-sensitized solar cells called most efficient to date,” ScienceDaily (2006). [37] A. Yella, H.-W. Lee, H. N. Tsao, C. Yi, A. K. Chandiran, M. Nazeeruddin, E. W.-G. Diau, C.-Y. Yeh, S. M. Zakeeruddin, and M. Gratzel, “Porphyrinsensitized solar cells with cobalt (ii/iii)based redox electrolyte exceed 12 percent efficiency,” Science 334, 629–634 (2011). [38] J. Lichtman and J.Conchello, “Fluorescence microscopy,” Nature Methods 2, 910–919 (2005). [39] Y. Dai, R. Whittal, and L. Li, “Confocal fluorescence microscopic imaging for investigating the analyte distribution in maldi matrices,” Anal. Chem. 68, 2494–2500 (1996). [40] D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, “Flourescence microscopy in three dimensions,” in “Methods in cell biology,” , vol. 30, Y.-L. Wang and D. Taylor, eds. (Academic Press, 1989), pp. 353–377. [41] P. So, C. Dong, B. Masters, and K. Berland, “Two-photon excitation fluorescence microscopy,” Annual Review of Biomedical Engineering 2, 399–429 (2000). [42] G. D. Peng, Z. Xiong, and P. L. Chu, “Fluorescence Decay and Recovery in Organic Dye-Doped Polymer Optical Fibers,” J. Lightwave Technol. 16, 2365– 2372 (1998). [43] B. Howell and M. G. Kuzyk, “Amplified Spontaneous Emission and Recoverable Photodegradation in Disperse-Orange-11-Doped-Polymer,” J. Opt. Soc. Am. B 19, 1790 (2002). 205 [44] B. Howell and M. G. Kuzyk, “Lasing Action and Photodegradation of Disperse Orange 11 Dye in Liquid Solution,” Appl. Phys. Lett. 85, 1901–1903 (2004). [45] P. Kobrin, R. Fisher, and A. Gurrola, “Reversible photodegradation of organic light-emitting diodes,” Appl. Phys. Lett. 85, 2385 (2004). [46] Y. Zhu, J. Zhou, and M. G. Kuzyk, “Two-photon fluorescence measurements of reversible photodegradation in a dye-doped polymer,” Opt. Lett. 32, 958–960 (2007). [47] Y. Zhu, J. Zhou, and M. G. Kuzyk, “Self-Healing and Laser Hardening of Nonlinear-Optical Materials,” Optics and Photonics News 18, 31 (2007). [48] L. DesAutels, M. G. Kuzyk, and C. Brewer, “Femtosecond bulk transparent material processing and recovery,” Opt. Express 17, 18808–18819 (2009). [49] B. R. Anderson, S. K. Ramini, and M. G. Kuzyk, “Imaging studies of photodamage and self- healing of anthraquinone derivative dye doped polymers.” in “SPIE Laser Damage Symposium Proc.”, , G. Exarhos, ed. (SPIE, Boulder, CO, 2011), [8190-16]. [50] B. R. Anderson and M. G. Kuzyk, “Digital imaging studies of reversible photodegradation in select anthraquinone derivatives doped into pmma thin films.” To be published (2013). [51] S.-T. Hung, S. K. Ramini, D. G. Wyrick, K. Clays, and M. G. Kuzyk, “The role of the polymer host on reversible photodegradation in disperse orange 11 dye,” in “SPIE Optics and Photonics: Organic Photonics + Electronics,” (San Diego, CA), 84741A. 206 [52] S. K. Ramini, B. R. Anderson, S. T. Hung, and M. G. Kuzyk, “Experimental tests of a new correlated chromophore domain model of self-healing in a dyedoped polymer,” Polymer Chemistry (2013). [53] B. Anderson, S. K. Ramini, and M. G. Kuzyk, “Imaging studies of photodamage and self-healing in disperse orange 11 dye-doped pmma,” J. Opt. Soc. Am. B 28, 528–32 (2011). [54] S. K. Ramini and M. G. Kuzyk, “A self healing model based on polymermediated chromophore coerrelations,” J. Chem Phys. 137, 054705 (2012). [55] S. K. Ramini, “Experimental investigations of a proposed chromophore correlation model of self healing of disperse orange 11 doped in poly(methyl methacrylate),” Ph.D. thesis, Washington State University (2012). [56] N. J. Westfall and C. W. Dirk, “The photochemistry of the self-healing chromophore disperse orange 11,” Journal of Physical Organic Chemistry 25, 704– 712 (2012). [57] D. Duque and P. Tarazona, “Exact results and mean field approximation for a model of molecular aggregation,” J. Chem Phys. 107, 10207 (1997). [58] P. Collings, A. Dickinson, and E. Smith, “Molecular aggregation and chromonic liquid crystals,” Liq. Cryst. 37, 701–710 (2010). [59] M. Cates and S. Candau, “Statics and dynamics of worm-like surfactant micelles,” J. Phys.: Condens. Matter 2, 6869–6892 (1990). [60] C. B. McKitterick, N. L. Erb-Satullo, N. D. LaRacuente, A. J. Dickinson, and P. J. Collings, “Aggregation properties of the chromonic liquid crystal benzopurpurin 4b,” J. Phys. Chem. B 114, 1888–1896 (2010). 207 [61] M. Coopersmith and R. Brout, “Statistical mechanical theory of condensation,” Phys. Rev. 130, 2539 (1963). [62] T. J. Fellers and M. W. Davidson, “Concepts in digital imaging technology: Ccd noise sources and signal-to-noise ratio,” . [63] D. Dussault and P. Hoess, “Noise performance comparison of iccd with ccd and emccd cameras,” in “Infrared Systems and Photoelectronic Technology,” (2004), no. 195 in Proc. SPIE 5563. [64] S. Kogan, Electronic Noise & Fluctuations in Solids (Cambridge University Press, 1996). [65] P. Dutta and P. M. Horn, “Low-frequency fluctuations in solids: 1 f noise,” Rev. Mod. Phys. 53, 497–516 (1981). [66] M. B. Weissman, “ f1 noise and other slow, nonexponential kinetics in condensed matter,” Rev. Mod. Phys. 60, 537–571 (1988). [67] F. Hooge and A. Hoppenbrowers, Physica 45, 386 (1969). [68] F. Hooge, Phys. Lett. A 29, 139 (1969). [69] S. K. Ramini and M. G. Kuzyk, “Reversible photodegradation through chromophore-polymer interactions in disperse orange 11 dye-doped pmma thin films,” in “Nanophotonics and Macrophotonics for Space Environments VI,” , vol. 8519, E. W. Taylor, ed. (2012), vol. 8519. [70] F. Yakuphanoglu and B. Senkal, “Electrical conductivity, photoconductivity, and optical properties of poly(1,4-diaminoanthraquinone) organic semiconductor for optoelectronic applications,” Polymers for Advanced Technologies 19, 1193–1198 (2008). 208 [71] T. Dennis, E. M. Gill, and S. L. Gilbert, “Interferometric measurement of refractive-index change in photosensitive glass,” Applied Optics 40, 1663 (2001). [72] J. G. John Chamberlain and H. Gebbie, “Determination of refractive index spectra by fourier spectrometry,” Infrared Physics 9, 185–209 (1969). [73] A. G. V. Engen, S. A. Diddams, and T. S. Clement, “Dispersion measurements of water with white-light interferometry,” Applied Optics 37, 5679 (1998). [74] M. Honijk D.D.; Passchier, W.F.; Mandel, “The determination of complex refreactive indices with fourier-transform interferometry,” Physica 68, 457–474 (1973). [75] K. Naganuma, K. Mogi, and H. Yamada, “Group-delay measurement using the fourier transform of an interferometric cross coerrelation generated by white light,” Optics Letters 15, 393–395 (1990). [76] W. L. Briggs and V. E. Henson, The DFT: An Owners’ Manual for the Discrete Fourier Transform (Society for Industrial and Applied Mathematics, 1995). [77] D. Hearn, “Fourier transform interferometry,” Tech. rep., MIT (1999). [78] E. Bell and R. Sanderson, “Spectral erros resulting from random samplingposition errors in fourier transform scpectroscopy,” App. Optics 11, 688 (1972). [79] A. Yariv, Quantum Electronics 3rd Ed (Wiley, 1975). [80] B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 1991), 1st ed. [81] R. Menzel, Photonics 2nd Ed (Springer, 2007). 209 [82] N. Dawson, “Photo-induced deformations of nematic liquid crystal elastomers,” Ph.D. thesis, Washington State University (2010). [83] N. J. Dawson, M. G. Kuzyk, J. Neal, P. Luchette, and P. Palffy-Muhoray, “Experimental studies of the mechanisms of photomechanical effects in a nematic liquid crystal elastomer,” JOSA B 28, 1916–1921 (2011). [84] J. Cariou, J. Dugas, L. Martin, and P. Michel, “Refractive-index variations with temperature of pmma and polycarbonate,” App. Optics 25, 334–336 (1986). [85] R. M. Waxler, D. Horowitz, and A. Feldman, “Optical and physical parameters of plexiglas 55 and lexan,” App. Optics 18, 101–104 (1979). [86] R. W. Boyd, Nonlinear Optics (Academic Press, 1992). [87] B. R. Anderson, S. T. Hung, and M. G. Kuzyk, “Testing theories of self healing using photoconductivity as a probe of photodegradation and recovery,” in “SPIE Optics and Photonics: Optical Engineering + Applications,” , E. W. Taylor, ed. (San Diego, CA, 2012), 8519-16. [88] B. R. Anderson, E. Bernhardt, and M. G. Kuzyk, “Studies of mechanisms of decay and recovery in organic dye-doped polymers using spatially-resolved white light interferometry,” in “Optical Processes in Organic Materials and Nanostructures,” , vol. 8474, R. Jakubiak, ed. (2012), vol. 8474. [89] B. R. Anderson, S. T. Hung, and M. G. Kuzyk, “Electric field dependent decay and recovery of do11 doped into pmma thin films: beyond 100% recovery?” in “Laser-Induced Damage in Optical Materials: 2012,” , vol. 8530, G. J. Exarhos, V. E. Gruzdev, J. A. Menapace, D. Ristau, and M. J. Soileau, eds. (Boulder, CO, 2012), vol. 8530. 210 [90] B. R. Anderson, S.-T. Hung, and M. G. Kuzyk, “The influence of an electric field on photodegradation and self healing in disperse orange 11 dye-doped pmma thin films.” J. Opt. Soc Am. B (2013). [91] B. R. Anderson, E. Bernhardt, and M. G. Kuzyk, “A white light interferometric microscope for measuring dose-dependent reversible photodegradation,” J. Apl. Phys. (2013). [92] J. Vanderschueren and A. Linkens, “Nature of Transient Currents in Polymers,” J. Appl. Phys. 49, 4195–4205 (1978). [93] K. Zimmerman, “Thin film conductivity measurements as a probe of polarization relaxation in solid polymer solutions,” Master’s thesis, Washington State University (1992). [94] K. Zimmerman, F. Ghebremichael, M. G. Kuzyk, and C. W. Dirk, “ElectricField-Induced Polarization Current Studies in Guest-Host Polymers,” J. Appl. Phys. 75, 1267 (1994). [95] P. Debye, Ver. Deut. Phys. Gesell. 15, 777 (1913). [96] J. Curie, Ann. Chim. Phys 17, 385 (1889). [97] J. Curie, Ann. Chim. Phys 18, 203 (1889). [98] E. V. Schweidler, Ann. Phys. 24, 711 (1907). [99] K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics i. alternating currents,” J. Chem. Phys. 9, 341–51 (1941). [100] K. S. Cole and R. H. Cole, “Dispersion and absorption in dielectrics ii. direct current characteristics,” J. Chem Phys. 10 (1942). 211 [101] T. Guo and W. Guo, “A transient-state theory of dielectric relaxation and the curie-von schweidlerlaw,” J. Phys. C: Solid State Phys. 16, 1955–1960 (1983). [102] M. Schumacher and R. Waser, “Curie-von schweidler behavior observed in ferroelectric thin films and comparison to superparaelectric thin film materials,” Integrated ferroelectrics 22, 109–121 (1998). [103] G. Williams and D. C. Watts, “Non-Symmetrical Dielectric Relaxation Behavior Arising from a Simple Empirical Decay Function,” Transactions of the Faraday Society 66, 80–85 (1970). [104] M. Cook, D. Watts, and G. Williams, “Correlation function approach to the dielectric behavior of amporphous polymers,” Transactions of the Faraday Society 66, 2503–2511 (1970). [105] N. Joshi, Photoconductivity: art, science, and technology (Dekker, 1990). [106] J. Chilton and M. T. Goosey, eds., Special Polymers for Electronics & optoelectronics (Chapman & Hall, 1995). [107] J. Carnes and P. Warter, “Photoinduced current in polyvinylcarbazole,” Phys. Rev. B 5, 1557 (1972). [108] T. K. Daubler, L. Kulikovsky, D. Neher, V. Cimrov, J. Hummelend, E. Mecher, R. Bittner, and K. Meerholz, “Photoconductivity and charge-carrier photogeneration in photorefractive polymers,” in “Photoconductivity and Charge-Carrier Photogeneration in Photorefractive Polymers,” , vol. 4462, C. M. Lawson and K. Meerholz, eds. (SPIE, 2002), vol. 4462, p. 206. 212 [109] V. Kishore, “Development of photorefractive polymers: Evaluation of photoconductivity and electro-optic properties,” Ph.D. thesis, Cochin University of Science and Technology (2008). [110] B. Jones, “Photoconductivity and grating response time of a photorefractive polymer,” J. Opt. Soc Am. B 11, 1064–1072 (1994). [111] W. N. Sisk, K.-S. Kang, M. Y. A. Raja, and F. Farahi, “Matrix and donoracceptor dependence of polymer dispersed pyrromethene dye photoconductivity,” International Journal of Optoelectronics 10, 95–103 (1995). [112] A. Hirao and H. Nishizawa, “Measurement of diffusion and drift of charge carriers from photocurrent transients,” Phy. Rev. B 54, 4755 (1996). [113] J. S. Schildkraut and A. V. Buettner, “Theory and simulation of the formation and erasure of spacecharge gratings in photoconductive polymers,” J. Appl. Phys. 72, 1888 (1992). [114] J. S. Schildkraut and Y. Cui, “Zeroorder and firstorder theory of the formation of spacecharge gratings in photoconductive polymers,” J. Appl. Phys. 72, 5055 (1992). [115] J. Henderson, “Linear aggregation beyond isodesmic symmetry,” J. Chem Phys. 130, 045101 (2009). [116] P. Maiti, Y. Lansac, M. Glaser, and N. Clark, “Isodesmic self-assembly in lyotropic chromonic systems,” Liquid Crystals 29, 619–626 (2002). [117] N. J. Dawson, B. R. Anderson, J. L. Schei, and M. G. Kuzyk, “A classical model of the cascading contribution to the second hyperpolarizability.” Phys. Rev A 84 (2011). 213 [118] N. J. Dawson, B. R. Anderson, J. L. Schei, and M. G. Kuzyk, “A quantum mechanical model of the cascading contribution to the second hyperpolarizability,” Phys. Rev A 84 (2011). [119] M. G. Kuzyk and S. K. Ramini, “Correlated aggregate model of self healing in dye-doped polymers,” in “Nanophotonics and Macrophotonics for Space Environments VI,” , vol. 8519, E. W. Taylor, ed. (2012), vol. 8519. [120] M. G. Kuzyk and S. K. Ramini, “The role of polymer-mediated dopant correlations in damage moderation and self healing,” in “Laser-Induced Damage in Optical Materials: 2012,” , vol. 8530, G. J. Exarhos, V. E. Gruzdev, J. A. Menapace, D. Ristau, and M. J. Soileau, eds. (2012), vol. 8530. [121] W. Schottky, “Zur halbleitertheorie der sperrschicht- und spitzengleichrichter,” Zeitschrift fur Physik 113, 367–414 (1939). [122] N. Mott, “The theory of crystal rectifiers,” Proc. R. Soc. London Ser. A 171, 27 (1939). [123] J. Bardeen, “Surface states and rectification at a metal semi-conductor contact,” Phys. Rev. 71, 717–727 (1947). [124] J. Tersoff, “Theory of semiconductor heterojunctions: The role of quantum dipoles,” Phys. Rev B 30, 4874 (1984). [125] V. Heine, “Theory of surface states,” Phys. Rev. A 138, 1689 (1965). [126] J. Tersoff, “Schottky barriers and semiconductor band structures,” Phys. Rev. B 32, 6968 (1985). [127] W. Monch, “Electronic properties of ideal and interface modified metalsemiconductor interfaces,” J. Vac. Sci. Technol. B 14, 2985 (1996). 214 [128] W. Monch, “Mechanisms of schottky barrier formation in metal-semiconductor contacts,” J. Vac. Sci. Technol. B 6, 1270 (1988). 215