Download testing a generalized domain model of photodegradation and self

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.
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