Download engineering of complex optical fields and its

ENGINEERING OF COMPLEX OPTICAL FIELDS AND ITS
APPLICATIONS
Dissertation
Submitted to
The School of Engineering of the
UNIVERSITY OF DAYTON
In Partial Fulfillment of the Requirements for
The Degree
Doctor of Philosophy in Electro-Optics
By
Wei Han
UNIVERSITY OF DAYTON
Dayton, Ohio
August, 2013
ENGINEERING OF COMPLEX OPTICAL FIELDS AND ITS
APPLICATIONS
Name: Han, Wei
APPROVED BY
Qiwen Zhan, Ph.D.
Advisory Committee Chairman
Professor
Electro-Optics Program
Joseph W. Haus, Ph.D.
Committee Member
Professor
Electro-Optics Program
Partha Banerjee, Ph.D.
Committee Member
Director and Professor
Electro-Optics Program
Guru Subramanyam, Ph.D.
Committee member
Chair and Professor
Electrical and Computer Engineering
John G. Weber, Ph.D.
Associate Dean
School of Engineering
Tony E. Saliba, Ph.D.
Dean, School of Engineering
& Wilke Distinguished Professor
ii
© Copyright by
Wei Han
All rights reserved
2013
iii
ABSTRACT
ENGINEERING OF COMPLEX OPTICAL FIELDS AND ITS
APPLICATIONS
Name: Han, Wei
University of Dayton
Advisor: Dr. Qiwen Zhan
Generation of optical fields with complex spatial distribution in the cross section
is of great interest in application areas where exotic optical fields are desired, including
particle manipulation, optical nanofabrication, beam shaping and optical imaging.
The dissertation is organized in two parts. In the first part, different aspects of the
optical field are controlled with different approaches used in four projects. First, a
diffractive optics element (DOE) Simulator consisting of 4-f imaging systems and
reflective spatial light modulator (SLM) is introduced, where the phase is modulated. In
the second project, a complex optical filter design using optical antennas is presented for
optical needle field generation, using the amplitude and binary phase modulation. In the
third example, a “Bull’s Eye” structure on fiber end is analyzed and demonstrated as a
polarization sensitive device for cylindrical vector beam generation, where the SOP can
be controlled. As the last demonstration, the vectorial beam is constructed by
superimposing x-polarized Gaussian and y-polarized Laguerre Gaussian is introduced as
iv
one type of second order full Poincaré beams, which can be realized using liquid crystal
(LC) based device.
The demands to arbitrarily tailor the properties of optical fields lead to our Vector
Optical Field Generator (VOF-Gen). In the second part, a vectorial optical field generator
capable of creating arbitrarily complex beam cross section is designed, built and tested.
Based on two reflective phase-only liquid crystal spatial light modulators, this generator
is capable of controlling all the parameters of the spatial distributions of an optical field,
including the phase, amplitude and polarization (ellipticity and orientation) on a pixel-bypixel basis. Various optical fields containing phase, amplitude and/or polarization
modulations are successfully generated and tested using Stokes parameter measurement
to demonstrate the its capability and versatility.
v
Dedicated to my wife and my parents
vi
ACKNOWLEDGEMENTS
I am greatly indebted to the people who have helped me over the last six years. Above all,
I would like to acknowledge my advisor, Dr. Qiwen Zhan, for his tremendous amount of
help and support through my Ph.D. period. I couldn’t thank him enough. I would like to
thank him for the great amount of time, effort and thoughts he put in my research and for
the inspiring discussions and interactions with him, which leads to the successful
conclusion of my dissertation. He is a knowledgeable and diligent professor and a leading
scientist in the fields of space variant polarization engineering and I am especially
grateful for his experienced guidance and vision in my research of complex optical fields
engineering.
I would also like to thank my committee members, Dr. Joseph Haus, Dr. Partha Banerjee
and Dr. Guru Subramanyam for their continuous support during my Ph.D. research. I am
very grateful for their valuable revision advice on my dissertation manuscript. Dr. Haus is
a very knowledgeable and passionate professor and always ready to start discussion on
any subject in optics. I am very grateful for his mentor as the advisor for my master’s
study. Dr. Banerjee is the Director of the Electro-Optics Program and is a very nice and
vii
knowledgeable professor with strong disciplines. Dr. Guru Subramanyam is the Chair of
the Electrical and Computer Engineering Department and is a very humble and
knowledgeable professor.
I also want to take this opportunity to express my sincere appreciation to all the
professors from whom I have had the honor to take the courses. My sincere thanks go to
Dr. Loomis, Dr. Duncan, Dr. Powers, Dr. Sarangan, Dr. Vorontsov, Dr. Watson and Dr.
McManamon as well as my committee members. My wishes go to Dr. Powers for a full
recovery. I have learned a lot of knowledge and experience from all the professors and
am deeply impressed with their love and passion for optics.
I would like to thank the former members of Dr. Zhan’s group, including Dr. Weibin
Chen, Dr. Zhi Wu, Dr. Jian Gao, Dr. Wenzao Li, Dr. Alain Tschimwangan, Xue Liu,
Shuangyang Yang and Mengshu Pan. I have received a lot of help from them since I
came to the USA to pursue the graduate study in 2007. I also want to thank the current
members of Dr. Zhan’s group, including Shiyi Wang, Zhijun Yang, Hongwei Chen,
Zhenyu Yang, Renjie Zhou and Chenchen Wan for their help and support.
My special thanks also go to my close friends and former ECE graduate, Dr. Hai Jiang
and his wife Xiaoyan Ruan. They are very nice people and always ready to help. They
have offered me countless help and valuable advice both inside and outside the
academics with their experience.
Last but not least, I would like to thank my family. My parents are always there for me.
My most special thanks goes to my wife, Wen Cheng. We met here in the Electro-Optics
viii
Program. None of this could have happened without her accompany and support. She is
always there for me whenever I am down.
ix
TABLE OF CONTENTS
ABSTRACT ...................................................................................................................... iv
ACKNOWLEDGEMENTS ........................................................................................... vii
LIST OF FIGURES ....................................................................................................... xiv
LIST OF TABLES .......................................................................................................... xx
LIST OF ABBREVIATIONS AND NOTATIONS ..................................................... xxi
CHAPTER 1 INTRODUCTION ..................................................................................... 1
CHAPTER 2 OVERVIEW OF RESEARCH BACKGROUND .................................. 4
1.
STATE OF POLARIZATION, STOKES PARAMETERS AND POINCARÉ SPHERE ........... 4
2.
INTRODUCTION TO CYLINDRICAL VECTOR BEAMS................................................. 7
3.
LIQUID CRYSTAL SPATIAL LIGHT MODULATOR AND ITS LIGHT MODULATION
CAPABILITY ..................................................................................................................... 9
4.
LIGHT MODULATION USING SUBWAVELENGTH METALLIC STRUCTURES ............ 13
4.1.
Wiregrid polarizer for polarization modulation ........................................... 13
4.2.
Periodic grating structure for amplitude and phase modulation................. 14
4.3.
Antenna structure for amplitude, phase and polarization modulation ....... 14
CHAPTER 3 DOE SIMULATOR USING PHASE MODULATION CREATED BY
A REFLECTIVE SLM ................................................................................................... 16
x
1.
INTRODUCTION ...................................................................................................... 16
1.1.
4-f imaging system ......................................................................................... 16
1.2.
Beam propagation and its paraxial approximation ...................................... 20
2.
EXPERIMENTAL SETUP OF DOE SIMULATOR ....................................................... 22
3.
COMPLEX SCALAR FIELD REALIZATION USING DOE SIMULATOR: BIFOCAL LENS
AND TORIC LENS ............................................................................................................ 24
4.
3.1.
DOE bifocal lens ............................................................................................ 24
3.2.
Toric lens ........................................................................................................ 28
SUMMARY .............................................................................................................. 31
CHAPTER 4 SECOND-ORDER FULL POINCARÉ BEAMS ................................. 32
1.
INTRODUCTION ...................................................................................................... 32
2.
FULL POINCARÉ BEAM AND ITS STATE OF POLARIZATION................................... 33
3.
SUMMARY .............................................................................................................. 39
CHAPTER 5 GENERATING CYLINDRICAL VECTOR BEAM WITH
SUBWAVELENGTH CONCENTRIC METALLIC GRATING FABRICATED ON
OPTICAL FIBER ........................................................................................................... 40
CHAPTER 6 COMPLEX OPTICAL FILTER FOR HIGH PURITY OPTICAL
NEEDLE FIELD GENERATION ................................................................................ 47
1.
INTRODUCTION ...................................................................................................... 47
2.
ELECTRIC FIELD CALCULATION AT PUPIL PLANE ................................................ 48
3.
DISCRETIZATION OF THE PUPIL FILTER................................................................ 54
xi
4.
SUB-WAVELENGTH METALLIC GRATING IMPLEMENTATION FOR OPTICAL NEEDLE
GENERATION ................................................................................................................. 56
4.1.
Introduction ................................................................................................... 56
4.2.
Pupil filter design and proposed experiment setup ...................................... 57
5.
PRELIMINARY RESULTS ......................................................................................... 59
6.
DESIGN BASED ON SUBWAVELENGTH METALLIC LINEAR ANTENNA .................... 60
7.
6.1.
Introduction ................................................................................................... 60
6.2.
Initial analysis................................................................................................ 60
6.3.
Complex optical filter design ......................................................................... 65
6.4.
Realization of complex optical filter with slot antennas .............................. 72
SUMMARY .............................................................................................................. 73
CHAPTER 7 VECTOR OPTICAL FIELD GENERATOR....................................... 74
1.
INTRODUCTION ...................................................................................................... 74
2.
PRINCIPLES ............................................................................................................ 77
2.1.
Spatial light modulator .................................................................................. 78
2.2.
Spatially variant polarization rotator ............................................................ 79
2.3.
System flow chart ........................................................................................... 82
2.4.
Modulation of Light....................................................................................... 83
2.4.1. Phase modulation (SLM Section 1) ......................................................... 83
2.4.2. Amplitude modulation (SLM Section 2) ................................................. 83
2.4.3. Polarization ratio modulation (SLM Section 3) ..................................... 84
2.4.4. Phase retardation modulation (SLM Section 4) ..................................... 85
3.
EXPERIMENTAL SETUP .......................................................................................... 87
xii
4.
3.1.
Experimental setup of the generator ............................................................. 87
3.2.
Gamma Curve Calibration Using Polarization Rotator ............................... 91
3.3.
4-f imaging system and its alignment procedure .......................................... 93
EXPERIMENTAL RESULTS ...................................................................................... 97
4.1.
Spatially variant phase modulation: vortex generation ............................... 97
4.2.
Spatially variant amplitude modulation ........................................................ 99
4.3.
Spatially variant polarization rotation: radially polarized beam ............... 102
4.4.
Spatially variant phase retardation ............................................................. 108
4.5.
Stokes parameters measurement and complex vectorial optical field
generation with multiple parameters ..................................................................... 111
4.5.1. Stokes parameters and its measurement .............................................. 111
4.5.2. Experimental results for complex vector field generation with multiple
parameters .......................................................................................................... 114
4.6.
5.
Realization of full Poincaré beam and the singularities ............................ 122
SUMMARY ............................................................................................................ 126
CHAPTER 8 CONCLUSIONS AND FUTURE WORK .......................................... 127
BIBLIOGRAPHY ......................................................................................................... 129
VITA............................................................................................................................... 137
xiii
LIST OF FIGURES
Fig. 2-1: Polarization ellipse. .............................................................................................. 5
Fig. 2-2: Poincaré sphere. ................................................................................................... 6
Fig. 2-3: Cylindrical vector beams. (a) Radial polarization; (b) azimuthal polarization; (c)
generalized CVB. ................................................................................................................ 7
Fig. 2-4. Twisted nematic LC molecules with external voltage applied. Here LC directors
are shown in cylinders. The front surface is rubbed in vertical direction while the back is
in horizontal direction. ...................................................................................................... 10
Fig. 2-5: Illustration of the extraordinary refractive index for uniaxial LC molecules .... 10
Fig. 3-1. Schematic diagram of 4-f imaging system. ........................................................ 17
Fig. 3-2. Schematic diagram of DOE Simulator. .............................................................. 23
Fig. 3-3. Experimental setup of the DOE Simulator. ........................................................ 24
Fig. 3-4: Phase pattern of the bifocal phase with the 20 D base power. ........................... 25
Fig. 3-5. Through focus PSF of ReSTOR with 20 D base power for 4 mm pupil size. ... 26
Fig. 3-6. Experimental results of 2D through focus PSF for 4mm pupil size................... 27
Fig. 3-7. Simulation results of 2D through focus PSF. ..................................................... 27
Fig. 3-8. 3 D toricity phase pattern. .................................................................................. 28
Fig. 3-9. Evolution of 3 D toricity near focus. The upper column is simulation results and
the lower is the experimental measurement. The defocus from left to right is 0 mm, -4
mm and -6.5 mm, respectively. ......................................................................................... 29
xiv
Fig. 3-10. Evolution of 0.5 D toricity near focus. The upper column is simulation results
and the lower is the experimental measurement. The defocus from left to right is 0 mm, 0.6 mm and -1.2 mm, respectively. ................................................................................... 30
Fig. 3-11. Evolution of 0.1 D toricity near focus. The upper column is simulation results
and the lower is the experimental measurement. The defocus from left to right is 0 mm, 0.12 mm and -0.24 mm, respectively. ............................................................................... 31
Fig. 4-1. Intensity and phase pattern for LG02 mode. ....................................................... 34
Fig. 4-2. SOPs evolution for second order FP beam at 1). z = -10zR, 2). z = -zR, 3). z = 0,
4). z = zR and 5). z = 10zR. ................................................................................................ 37
Fig. 4-3. One slice (shown in red) of the Poincaré sphere at fixed ϕ and r/w(z) with δ
spanning from 0 to 2π. ...................................................................................................... 38
Fig. 5-1. Experimental setup for generalized CV beam generation. [46] ......................... 41
Fig. 5-2. SEM pictures of the Bull's Eye structure. [46] Sample is prepared by Don
Abeysinghe using FIB....................................................................................................... 42
Fig. 5-3. Simulated energy density and local SOP. [46] ................................................... 44
Fig. 5-4. Experimental results for generalized CV beam generation. [46] ....................... 45
Fig. 6-1. High NA focusing of pupil plane field in the focal volume (shown in red). ..... 49
Fig. 6-2. Schematic configuration of reversing radiation of an electric dipole array. ...... 49
Fig. 6-3. Far field intensity along focus (upper) and its line scan (lower) [48]. The
longitudinal coordinate is normalized to the wavelength. ................................................ 52
Fig. 6-4. Intensity distribution in the pupil plane (upper) and its line scan (lower) [48].
The transverse coordinate is normalized to the pupil radius of the high NA objective. ... 53
Fig. 6-5. Discrete filter design. ρ is normalized to the pupil radius for the high NA
objective. ........................................................................................................................... 54
Fig. 6-6. Electric field along focus (a) and its line scan (b) for discrete pupil filter. The
horizontal axis is normalized by the wavelength. ............................................................. 55
Fig. 6-7. Grating structure (upper) and proposed experimental setup (lower). The radii are
normalized to the radius of the pupil. ............................................................................... 58
xv
Fig. 6-8. SEM of sample (by Don C Abeysinghe). ........................................................... 59
Fig. 6-9: Linear antenna oriented at angle θ from the incident polarization. .................... 61
Fig. 6-10. Normalized amplitude (upper) and phase (lower) of the longitudinally
polarized scattered field. ................................................................................................... 62
Fig. 6-11. The absolute value of the Ez component of the scattered field off the rod
antenna. ............................................................................................................................. 63
Fig. 6-12. The absolute value of the Eρ component of the scattered field off the rod
antenna. ............................................................................................................................. 63
Fig. 6-13. 2D amplitude plot of linear antenna versus length L and θ.............................. 65
Fig. 6-14. 2D phase plot of linear antenna versus length L and θ. ................................... 65
Fig. 6-15. Normalized amplitude (upper) and phase (lower) of the longitudinally
polarized scattered field for 25 nm radius rod antenna. .................................................... 67
Fig. 6-16. Normalized amplitude (upper) and phase (lower) of cross-polarized scattered
field as a function of L and θ for 25-nm radius rod antenna. ............................................ 68
Fig. 6-17. Rectangle linear antenna and its complementary design. ................................ 72
Fig. 6-18. SEM pictures of antenna structure as complex optical filter fabricated by Don
C Abeysinghe using FIB. .................................................................................................. 73
Fig. 7-1. Taiji pattern coded in circular polarization. The total field (left), the upper half
Taiji pattern (upper right) in RCP and the lower half Taiji pattern (lower right) in LCP. 78
Fig. 7-2. Illustration of the Polarization Rotator setup. (a) The Polarization Rotator
comprised of QWP and reflective SLM; (b) Illustration of the effective rotation of the
QWP fast axis for the incident (upper) and the reflected (lower) beams due to the mirror
imaging of the laboratory coordinates in dashed lines...................................................... 80
Fig. 7-3. Flow chart of the system. The VOF-Gen System consists of light source, 4
subsystems for control of all the aspects of light, 4-f imaging subsystems and detection
subsystem. PR: Polarization Rotator................................................................................. 82
Fig. 7-4. Schematic diagram of the VOF-Gen. ................................................................. 88
Fig. 7-5. Experiment Setup of the VOF-Gen. ................................................................... 89
xvi
Fig. 7-6. Driver circuits for the color channels of the VOF-Gen system. The upper one is
the red channel responsible for the control of SLM1 and the lower one is the green
channel for SLM 2. ........................................................................................................... 90
Fig. 7-7. Calibration setup consists of a PR and a polarizer. ............................................ 92
Fig. 7-8: Pattern with fine features for 4-f imaging system alignment. Each line is 100 μm
wide. .................................................................................................................................. 94
Fig. 7-9: 4-f imaging system alignment procedure ........................................................... 95
Fig. 7-10: Test results for (a) amplitude modulation: the intensity is directly captured by
the CCD camera; (b) polarization rotation: the intensity is captured after a linear polarizer;
(c) Retardation: the intensity is captured after a circular analyzer. .................................. 95
Fig. 7-11: Comparison of the same patterns generated by VOF-Gen system without and
with well aligned 4-f imaging systems. ............................................................................ 96
Fig. 7-12. Pure phase modulation with spiral phase of topological charge 1 (upper), 10
(middle) and 15 (lower). The images are captured in the focal plane of 75-mm planoconvex lens........................................................................................................................ 99
Fig. 7-13. x-polarized "EO" logo coded in amplitude (upper) and its phase pattern (lower).
......................................................................................................................................... 101
Fig. 7-14. Phase patterns for radially polarized beams without (upper) and with (with)
pre-compensation. For the latter phase pattern, the variation in green color indicates the
pre-compensation phase. ................................................................................................. 104
Fig. 7-15. Focused field by a 75-mm plano-convex lens. (a) Radially polarized beam
without proper phase cancellation; (b) radially polarized beam with phase cancellation.
......................................................................................................................................... 105
Fig. 7-16. Radially polarized beam generated by the VOF-Gen. Upper graphs show the
fields after a polarizer with polarization axis orientation indicated by black arrows at 0°,
45°, 90° and 135°, respectively; Lower graph shows a polarization map of radially
polarized beam overlapped with the intensity distribution and the local polarization
directions are indicated by the bars. ................................................................................ 107
Fig. 7-17. Phase pattern for "EO" logo generation VOF-Gen. ....................................... 108
Fig. 7-18. "EO" logo coded in circular polarization. The total field (left), the EO logo
(upper right) in RCP and the complimentary “EO” logo (lower right) in LCP. ............. 109
xvii
Fig. 7-19. Phase pattern for Taiji pattern generation. ..................................................... 110
Fig. 7-20. Taiji pattern coded in circular polarization. The total field (left), the upper half
(upper right) in RCP and the lower half (lower right) in LCP. ....................................... 110
Fig. 7-21. Full Stokes parameters measurement setup. .................................................. 113
Fig. 7-22. The ideal vector field where the SOP varying from azimuthal to radial as
radius increases. .............................................................................................................. 114
Fig. 7-23. The phase pattern for VOF-Gen. .................................................................... 115
Fig. 7-24. Ring structure with SOP continuously varying from azimuthal to radial
direction as radius increases. The upper set of graphs show the field components along 0°,
45°, 90° and 135° respectively and the lower graph shows the measured field overlapped
with polarization map. .................................................................................................... 116
Fig. 7-25. The ideal field for double ring pattern where inner ring has azimuthal
polarization and outer ring has radial polarization. ........................................................ 117
Fig. 7-26. The phase pattern for VOF-Gen. .................................................................... 118
Fig. 7-27. Double ring structure with amplitude and polarization rotation modulation:
(upper) the linear polarization components with polarizer at 0°, 45° and 90°; (lower) the
polarization map.............................................................................................................. 119
Fig. 7-28. Ideal field distribution with polarization map. ............................................... 120
Fig. 7-29. Phase pattern for VOF-Gen. ........................................................................... 121
Fig. 7-30. Experimental results (left) for optical field with constant ellipticity and
elevation angle along radial direction as well as the histogram (right) of the ellipticity in
the unit of π. .................................................................................................................... 121
Fig. 7-31. Simulation of FP beam with polarization map superimposed........................ 122
Fig. 7-32. Experimental result for FP beam with polarization map superimposed. Two
singularities are highlighted by green circles.................................................................. 123
Fig. 7-33. Phase pattern for retardation modulation for the FP beam............................. 124
Fig. 7-34. The phase pattern for the generation of the FP beam. .................................... 125
Fig. 7-35. Diffraction pattern after propagation distance of 1 mm. ................................ 125
xviii
Fig. 8-1. Experiment setup for linear antenna design. .................................................... 128
xix
LIST OF TABLES
Table 1: Parameters of dipole array for N = 2 .................................................................. 51
Table 2: Grating parameters for each concentric ring ...................................................... 57
Table 3: Design parameters .............................................................................................. 66
Table 4: Antenna design parameters for optical needle field generation .......................... 70
Table 5. Tolerance study of the antenna structure ............................................................ 71
xx
LIST OF ABBREVIATIONS AND NOTATIONS
CCD
CV
DOE
F
FIB
FP
HWP (λ/2)
LCP
LG
LP
NA
PR
QWP (λ/4)
RCP
RCWA
SF
SLM
SOP
VOF-Gen
PSF
DOF
NSOM
CP
OAM
SAM
EM
TE
TM
Charge coupled device
Cylindrical vector
Diffractive Optics Element
Focal length
Focused ion beam
Full Poincaré
Half wave plate
Left-hand circular polarization
Laguerre Gaussian
Linear polarizer
Numerical aperture
Polarization Rotator
Quarter wave plate
Right-hand circular
polarization
Rigorous coupled wave
analysis
Spatial filter
Spatial light modulator
State of polarization
Vector optical field generator
Point spread function
Depth of focus
Near-field scanning optical
microscope
Circular polarizer
Orbital angular momentum
Spin angular momentum
Electro-magnetic
Transverse electric
Transverse magnetic
HDTV
NPBS
LC
SEM
xxi
High-definition television
Non-polarizing beam splitter
Liquid crystal
Scanning electron microscope
CHAPTER 1
INTRODUCTION
As a form of electromagnetic field, optical fields governed by the Maxwell’s Equations
also exhibit vectorial nature in addition to phase and amplitude of the scalar field.
Engineering of complex optical fields has drawn tremendous amount of research in
various fields of applications. Optical trapping, or particle manipulation is made possible
by the laser beam shaping or focus shaping. Shaped optical fields have enabled
researchers to better understand the biophysics and colloidal dynamics through the
trapping, guiding or patterning of molecules or nano/micro particles. Spatial engineering
of focal field intensity has been studied to reach resolution far beyond diffraction limit in
microscope system. Vortex beam, also known as “twisted light”, has also drawn a lot of
interest owing to its spiral phase wavefront carrying orbital angular momentums. Better
integrity of vortex beam through propagation in turbulent atmosphere has been shown
and a lot of research has been done using the orbital angular momentum as information
carrier for free space communication due to its orthogonality and multiplexing capability.
Besides the manipulation of intensity and phase, as the vector nature of electromagnetic
1
wave, the state of polarization (SOP) also plays an important part in beam shaping for
flattop generation, focus shaping, optical tweezing and surface plasmon sensing using
cylindrical vectorial beams, ellipsometry. All the above applications require local control
over all variables of the optical fields. The need to generate arbitrarily complex optical
fields led to the development our vectorial optical field generator (VOF-Gen), which is
capable of tailoring all aspects of the light field.
Various techniques of manipulating degrees of freedom for complex optical fields are
investigated from Chapter 3 to Chapter 6, which demonstrated partial control of complex
optical field using SLM and subwavelength structure and in the meanwhile served as
motivations for our VOF-Gen. In Chapter 3, we show that a DOE Simulator using the
phase modulation capability of liquid crystal spatial light modulator (LC-SLM). In
Chapter 4, the modulation of polarization is added for the generation of a more complex
optical field. We show that, by adjusting the weighting of the x- and y-polarized
components and loading corresponding phase, the complex SOP can be realized using
SLM, which is then found to be one type of the full Poincaré beams. In Chapter 5, a
vectorial optical field whose polarization possesses a rotational symmetry is achieved
with a metallic grating structure. A concentric ring structure (Bull’s Eye) on fiber end is
shown to have polarization sensitivity. More specifically, a generalized cylindrical vector
(mostly radially polarized) beam is generated after the propagation of a circularly
polarized input beam through the fiber. A more complex engineering of optical field is
shown in Chapter 6, where modulations of phase, amplitude and polarization are
simultaneously realized using a complex filter design. An optical antenna structure is
proposed as a complex optical filter with amplitude, phase and polarization modulation to
2
achieve optical needle field in the focal volume of a high numerical aperture (NA)
objective. An optical needle field features a mostly longitudinal polarization and a flattop
intensity profile along axial direction with an extended depth of focus.
All these projects lead to the need for a universal system that’s capable of manipulating
all degrees of freedom for vectorial optical field, as the complexity of the conventional
system increases and fabrication of the individual filter design gets more and more
challenging. In Chapter 7, VOF-Gen that is capable of generating vectorial optical field
with arbitrary phase, amplitude and polarization on a pixel by pixel basis is proposed and
experimentally demonstrated. Various exotic vectorial optical fields are generated with
desired spatial distribution of amplitude, phase and polarization.
3
CHAPTER 2
OVERVIEW OF RESEARCH BACKGROUND
1. State of polarization, Stokes parameters and Poincaré sphere
Polarization in nature is the oscillation of electromagnetic wave and the SOP describes
the oscillation trajectory. The vectorial form of electric field after normalization can be
represented using Jones vector as:
 E0 x
E 
j
 E0 y e
  cos  

j  ，
  sin  e 
(2-1)
where the tangent of χ represents the absolute value of the ratio between y- and xcomponents (E0y/E0x) and δ is the phase retardation. Polarization can also be viewed in
terms of a polarization ellipse, as shown in Fig. 2-1. An elliptically polarized electric
field is shown as an example, where E0x and E0y are given in the figure. A more intuitive
way to represent the SOP is through elevation angle α and ellipticity ε. The laboratory
coordinate (x, y) is rotated to (x’, y’) by the amount of α so that the horizontal axis ox’ is
aligned with the major axis of the polarization ellipse, as depicted in Fig. 2-1.
Mathematically, it is given as:
4
 cos  
E 
  .
jsin



(2-2)
ε is the ellipticity of the polarization ellipse, whose tangent is defined as b/a, as shown in
Fig. 2-1.
Fig. 2-1: Polarization ellipse.
Stokes parameters are all measurable quantities (intensities), which is an alternative of
the SOP representation discussed above. Stokes parameters are given as
2
2
 S 0  I x  I y  E0 x  E0 y
 S I I E 2 E 2
x
y
0x
0y
 1
 S 2  I   I   2 E0 x E0 y cos  .


4
4
 S  I  I  2 E E sin 
R
L
0x
0y
 3
5
(2-3)
After normalization, the Stokes parameters can be written as:
S0  1

 S  cos 2 cos 2  cos 2 

1
.

S
cos
2
sin
2
sin
2
cos






 2
 S3  sin 2   sin 2  sin 
(2-4)
Note that S12+S22+S32 = 1 after normalization. Thus any combination of S1, S2 and S3 can
be found on a sphere with constant radius (r = S12+S22+S32 = 1) using the coordinate (S1,
S2, S3). In other words, the sphere defined by Stokes parameters contains all possible
SOP on its surface. This is the so called Poincaré sphere, which is shown in Fig. 2-2.
From Eq. (2-4), it can be shown that in Fig. 2-2 the azimuth angle with respect to S1 axis
is equal to 2α and the elevation angle is equal to 2ε measured from the S1-S2 plane.
Fig. 2-2: Poincaré sphere.
6
2. Introduction to cylindrical vector beams
In order to engineer complex optical field, it is of fundamental importance to introduce
spatially inhomogeneous polarization where the distribution of SOPs adds another degree
of freedom for the manipulation of such beams. As the most well-known and investigated
inhomogeneously polarized optical field, cylindrical vector beams (CVBs) are solutions
to Maxwell’s Equations, whose SOPs possess rotational symmetry [1]. Radial and
azimuthal polarization are the most common CVBs (Fig. 2-3 (a) and (b)) where the SOP
follows radial and azimuthal direction at any point on the beam, respectively. Due to the
orthogonality, radial and azimuthal polarizations form the basis for CVBs. And for any
generalized CVB as shown in Fig. 2-3 (c), it can always be represented as a linear
combination of radial and azimuthal modes.
Fig. 2-3: Cylindrical vector beams. (a) Radial polarization; (b) azimuthal polarization; (c)
generalized CVB.
The unique properties of CVBs have attracted much research interest recently. Plasmonic
focusing [2] [3] can be realized using radial polarization due to the TM polarization
direction with respect to the nanostructure. The focal spot of radially polarized beam can
7
be much smaller than the diffraction limited spot size of spatially homogeneously
polarized beams under using high NA objective [4] [5] [6]. High resolution imaging [7]
can thus be realized using CVBs. When tightly focused, the radially polarized beam also
exhibits a strong axial component with a smaller spot size centered on optic axis. The
strong axial component provides a large gradient force while due to the non-propagating
property, the axial scattering and absorption forces will be reduced [8] [9], which leads to
the successful trapping of gold nanoparticles with a higher transverse trapping stiffness
[10] and micrometer-sized dielectric particles with a higher axial and transverse trapping
efficiency for radially and azimuthally polarized beams, respectively [11]. Evanescent
Bessel beam generation has also been confirmed via surface plasmon resonance using
tightly focused radially polarized beam [12]. In 4Pi microscopy, both dark and bright
spherical focal spot can be created using radial polarization with spatially engineered
amplitude and phase distribution [13] [14].
By carefully designing pupil apodization function, optical fields with unique amplitude,
phase and polarization distribution can be obtained in the 3D focal volume of a high NA
objective. It is possible to achieve a flattop focal shape by balancing the radial and
azimuthal components and using a high-pass filter in the pupil apodization function [15].
By designing a DOE pupil mask with binary phase, optical needle field can be generated
with significant longitudinal polarization component and an extended depth of focus [16].
Optical chain along longitudinal direction for stably trapping and delivering particles has
been proposed by using a DOE with spatially engineered amplitude and phase
modulation for different concentric ring regions [17]. Three-dimensional optical cage for
8
optical trapping can also be obtained by using spatially patterned generalized CVBs as
the illumination for a high NA objective [18].
3. Liquid crystal spatial light modulator and its light modulation capability
3.1. Liquid Crystal and Spatial light modulators
In order to engineer complex optical fields, phase-only liquid crystal spatial light
modulator (LC-SLM) is introduced for the realization of phase modulation. Liquid crystal
(LC) is a phase of matter where the properties of the molecular order are between liquid
and crystal. The molecules of LC differ from liquid as they demonstrate anisotropy as
seen in crystalline structures. However, at the same time, unlike crystals, LC molecules
show a flow behavior of liquid with randomly positioned and oriented molecules.
The nematic LC phase can be characterized by rod-like molecules with no positional
order but tend to self-align to have long range directional order. For LC display (LCD),
the LC cells are spatially separated in cell (pixel) structures with carefully chosen
dimensions. The transvers direction of LC molecules can be surface-aligned, which is
done be rubbing the surface to introduce micro-grooves. Such direction is called rubbing
direction. The longitudinal orientation of LC molecules (director) can be manipulated by
externally applied electric field. The dipole-like director is excited by the external electric
field and tends to align parallel to the electric field. Due to the long-range order of the
nematic LC molecules, the LC cells will exhibit a voltage-dependent birefringence. In
general, for twisted nematic-LC cells (TN-LC), the LCD structure with directors
indicated by cylinders is shown in Fig. 2-4.
9
V-
V+
Fig. 2-4. Twisted nematic LC molecules with external voltage applied. Here LC directors
are shown in cylinders. The front surface is rubbed in vertical direction while the back is
in horizontal direction.
For uniaxial LC molecule, if the wave vector k of the incident beam has an angle θ with
respect to the director as depicted in Fig. 2-5, the extraordinary refractive index ne can be
found by Eq. (2-5).
Fig. 2-5: Illustration of the extraordinary refractive index for uniaxial LC molecules
10
  cos  2  sin  2
ne    

2
 no 2
n
e





1/2
,
(2-5)
where angle θ is dependent on voltage V, ne and no are the properties of the LC material.
Therefore, assuming the extraordinary direction is aligned along the x-axis, a single LC
molecule can be regarded as a thin wave plate, whose Jones matrix is given by,
W e
 jno
2

d
  j ne  no  2 d
e

0


0 ,
1 
(2-6)
where d is the thickness of the LC molecule. A twisted nematic LC cell (in a LC layer)
can thus be modeled as a succession of thin wave plates whose optic axes follow the
molecular axes of the LC molecules as shown in Fig. 2-4. The overall Jones Matrix can
be written as [19],
WTN  LC


cos


j
sin 


 j  
 R   e  0  



sin 





sin 


,


cos   j sin  


(2-7)
where R is the rotation matrix, α is the overall twist angle, β is the overall birefringence
of the LC molecule defined as πdΔn/λ, Φ0 is the common phase and γ2 = α2 + β2. For the
phase-only LC-SLM, the twist angle α is set to 0 so the Jones matrix can be simplified as
WPhaseonly SLM  e
 j    0 
11
 e  j

 0
0 
.
e j 
(2-8)
It can be shown for LC cells with parallel rubbing directions, the LC cells become a
variable retarder whose birefringence is a function of voltage applied.
For a reflective-type phase-only LC-SLM, the phase pattern is imposed onto the reflected
beam for incident beam with polarization parallel to the fast axis of the LC-SLM (or the
rubbing direction of LC molecules). Thus phase modulation can be readily realized using
a LC-SLM. However, for arbitrarily polarized input beam, the parallel component will
carry the prescribed phase pattern while the orthogonal component remains unaffected
after reflection. In other words, simple polarization control can also be realized where the
phase retardation between the orthogonal polarization components is modulated using a
single reflective-type LC-SLM for arbitrarily polarized incident beam.
3.2. Light modulation using LC-SLM based systems
Tremendous amount of research has been conducted by scientists in the community
developing versatile systems to generate optical field with exotic properties. Arbitrary
vector fields with inhomogeneous distribution of linear polarization were realized using a
LC-SLM and an interferometric arrangement [20]. A recent study showed that in order
to fully control the SOP, two spatially addressable retarders need to be used [21].
Researchers constructed such system consisting of two LC-SLMs with the fast axes 45°
from each other and were able to generate optical fields that cover the entire Poincare
Sphere. Another non-interferometric method for vector field generation was proposed
[22]. However, the complete control of both polarization elevation angle and ellipticity
can only be realized by modifying the experimental setup. Complete amplitude, phase
and polarization control was reported [23] with the help of a double modulation system
12
which requires two transmissive LC-SLMs whose modulation depth can be controlled to
achieve amplitude modulation for certain diffraction order. However, this approach
comes with the limitation that each area must contain a large number of periods to
achieve the ideal diffraction efficiency, which leads to optical field with limited spatial
resolution for practical applications. Very recently, a technique for generating arbitrary
intensity and polarization was reported with interesting results using transmissive LCSLMs and Mach-Zehnder interferometry setup [24]. Due to the nature of the technique,
the absolute phase of each electric field component does not cover an entire 2π range. As
a result, a complete phase control cannot be fully realized. The limitations of this
technique also include limited transmittance and relatively low spatial resolution. All the
present techniques mentioned above have limitations when it comes to generating a
spatially-invariant arbitrary vectorial field with high spatial resolution on a pixel basis.
4. Light modulation using subwavelength metallic structures
The interactions between optical fields and specially engineered structures have long
been investigated and demonstrated for the generation of beams with spatially
inhomogeneous polarization such as CVBs or with prescribed amplitude or phase
modulation. For instance, subwavelength metallic structures with exotic subwavelength
geometries have been extensively studied for polarization control as well as amplitude
and phase modulation.
4.1. Wiregrid polarizer for polarization modulation
It has been demonstrated that a wiregrid grating with a period smaller than the
wavelength of incident light strongly reflects TE polarized light and allows TM polarized
13
light to transmit through [25]. Subwavelength concentric metallic slits have been used for
creating two-lobe shaped radially polarized or azimuthally polarized beam with linearly
polarized illumination [26]. When an azimuthally polarized light is focused onto the
axially symmetric metallic grating, the entire beam is TE polarized with respect to the
interface and therefore experiences high reflection. In contrary, a radially polarized beam
is entirely TM polarized with respect to the metallic grating and can pass through the
device with higher transmittance than the azimuthal polarization. Therefore such
structure can behave as a TM polarizer.
4.2. Periodic grating structure for amplitude and phase modulation
Periodic metallic grating structure can also be used for the modulation of amplitude and
phase of the diffracted orders. Rigorous coupled wave analysis (RCWA) is used to
evaluate the reflected and diffracted waves [27] [28] [29] [30]. Based on Fourier series
expansion and using Floquet’s theorem, the optical field in each layer of the grating is
calculated by matching boundary conditions for each layer and each order. RCWA can in
principle provide analytical solution if the infinite number of orders are kept. In practical
application, finite number of orders are kept based on the accuracy and speed
requirements.
4.3. Antenna structure for amplitude, phase and polarization modulation
The scattered field off V-shaped nano antenna has recently been theoretically studied for
its ability to control the amplitude and phase of the cross-polarized component [31] [32].
Thanks to the greater-than-2π phase coverage, the design of plasmonic nano V-shaped
antenna array to introduce a phase gradient from 0 to 2π at the interface of two media in
14
order to steer the beam has been proposed and experimentally demonstrated. The
amplitude and phase of the scattered field can be engineered by appropriately choosing
the design parameters, the length of the rod antenna and the angle between the two arms.
5. High NA focusing of vectorial optical field
The focusing of electromagnetic field over a three dimensional volume has always been
an interesting and important area for both theoretical and applied optics. The focusing
properties of linearly polarized (scalar) field have been well established by Richards and
Wolf [33]. However, for spatially inhomogeneously polarized beams, the focusing
properties hadn’t been thoroughly investigated until recent years. The vectorial optical
field distribution in the focal volume of a high NA objective for radially and azimuthally
polarized illumination can be found by the following equations respectively [4].
 Ex 
 2
 E   C sin  cos  L  ,  e jk  Z


r
 y
0 0
 Ez 
S cos    S sin  cos S  
 Ex 
 2
 E   C sin  cos  L  ,  e jk  Z


 
 y
0 0
 Ez 
 cos  cos  
 cos sin   d d (2-9)


 sin  
S cos    S sin  cos S  
  sin  
 cos   d d (2-10)


 0 
Coordinates (zS, ρS, ϕS) describe the locations in the image space measured from the focal
point. Lr and Lϕ are the field distribution in the pupil plane of the high NA objective for
radial and azimuthal illumination, respectively, where ϕ and θ are the azimuth angle in
the pupil plane and polar angle.
15
CHAPTER 3
DOE SIMULATOR USING PHASE MODULATION CREATED
BY A REFLECTIVE SLM
1. Introduction
In this chapter, phase modulation using a LC-SLM based DOE Simulator is demonstrated
as the start point for modulations of complex optical field. Two types of LC-SLMs have
been investigated and used for DOE Simulator. The BNS XY series is a reflective LCSLM with a resolution of 512 x 512 and 15 μm pitch size. The SLM is set up to work
with a He-Ne laser at 632.8 nm. The HOLOEYE HEO 1080P is another reflective phaseonly LC-SLM with a resolution of 1920x1080 and 8 μm pitch size and is set up to work
with a He-Ne laser at either 543 nm or 633 nm.
1.1. 4-f imaging system
4-f imaging system is widely used for holography reconstruction [34]. The optical field at
the image plane can be a reconstruction of that at the object plane without diffraction
effect or quadratic wavefront. In our DOE simulator, 4-f imaging system is also
16
introduced to relay the optical fields between the object and image planes. The schematic
diagram of 4-f imaging system is depicted below in Fig. 3-1.
Fig. 3-1. Schematic diagram of 4-f imaging system.
Assuming L1 and L2 have the same focal length denoted as f, the incident transparency
t(x’, y’) is located at one f before L1. Based on Fresnel diffraction, the field T 1 right in
front of Lens L1 (in Plane (ξ, η)) can be found by Eq. (3-1):
T1  ,   t  x ', y '   h  , ; f 
  t  x ', y '  e
e
 jk
 2  2
2f
 jk
  x '2   y '2
2f
 t  x ', y ' e
 jk
dx ' dy '
x '2  y '2
2f
e
jk
(3-1)
 x '  y '
f
dx ' dy '
The field T2 immediately after Lens L1 can be calculated by multiplying the
transformation function of the lens, giving rise to the following expression (Eq. (3-2)):
17
T2  ,   T1  ,  e
jk
 2  2
2f
(3-2)
The field T3 at one f distance from L1 (the focal plane Plane (x, y)) is then given by Eq.
(3-3):
T3  x, y    T2  ,  e
e
e
 jk
x2  y 2
2f
x2  y 2
 jk
2f
 jk
 x  2  y  2
 T  ,  e
1
T  ,  
1
d  d
2f
jk
x  y
f
d  d
k 
kx
f
k 
ky
f
(3-3)
Remember in spatial frequency domain,
T  ,     t  x ', y '; z  f     t  x ', y '  e
1
j
k
2
2
x ' k y '
2k
f
(3-4)
By equating the spatial frequencies kx’ and kξ, ky’ and kη, respectively, Eq. (3-3) can be
simplified as:
18
T3  x, y   e

x2  y 2
 jk
2f
 t  x ', y '  e
 t  x ', y ' 
  kx  2  ky  2 
      f
 f   f  

j
2k
kx' 
kx
f
ky' 
ky
f
kx
kx' 
f
ky' 
(3-5)
ky
f
The field T3(x, y) at Plane (x, y) is proportional to the Fourier transform of the
transparency t(x’, y’), without quadratic phase term. Thus, the output field Tout at Plane
(x1, y1) can be determined similarly, which turns out to be proportional to the Fourier
transform of the field at T3(x, y). As a result, Tout at Plane (x1, y1) can be written as
T  x, y  
Tout  x1 , y1  
3


 



kx 
kx1
f
ky 
ky1
f
kx1
kx 
k

x
f 
f
 t  x ', y ' 
ky 
ky
ky'   ky  1
f 
f
kx' 
 f kx1
f ky1 
,
t

k f 
 k f
 t   x1 ,  y1 

f
k
19
(3-6)
Through rigorous derivation, one can find that for a 4-f imaging system, the output field
Tout is directly proportional to the inverse of the original transparency t as shown in Eq.
(3-6) without any approximation or quadratic phase term. In other words, the object
(transparency) can be reimaged at the image plane in terms of both amplitude and phase.
1.2. Beam propagation and its paraxial approximation
From Helmholtz Equation, we have

2
 k 2  E  r, z   0
(3-7)
For scalar case, E field can be Fourier transformed into plane wave decomposition in
angular frequency space.
E  r, z  
1
 2 
2
 U  kr , z  e
 jkr r
dkr 2
(3-8)
Plug into the Helmholtz Equation, yielding,
d2
U  kr , z   kr 2U  kr , z   k 2U  k r , z   0
2
dz
(3-9)
Assuming U (kr, z) has a exp (-j kz z) dependence, we then can rewrite the equation as
k
z
2
 k r 2  k 2 U  k r , z   0
(3-10)
Therefore,
k z  k 2  kr 2
U  kr , z   U 0  kr  e  j
20
(3-11)
k 2  kr 2 z
(3-12)
In paraxial case, kr/k is much smaller than 1, thus,
U  kr , z   U 0  kr  e  jkz e
kr 2
z
2k
j
(3-13)
In space domain, the input E field can be viewed as a collection of point sources located
at different 2D coordinates. The diffraction of point source will introduce a spherical
wavefront to the field at the point source. Therefore, we have the following expression Eq.
(3-14) for the diffraction in the space domain.
E  r , z   E0  r  
 
1  jkR
e
R
E0  x0 , y0 
e
 jk
1
 x  x0 
 x  x0   y  y0 
2
2
 z2
2
  y  y0   z
2
2
(3-14)
dx0 dy0
In paraxial case, r/z is much smaller than 1, yielding,
E  r , z    E0  x0 , y0  e
  x  x0 2   y  y0 2 
 jkz  1



2 z2


 x  x0   y  y0 
2
  E0  x0 , y0  e  jkz e
 E0  x, y   e
 jkz
e
 jk
 jk
2z
dx0 dy0
2
dx0 dy0
(3-15)
x2  y 2
2z
The propagation of optical field be viewed as a linear, time-invariant system problem
where the output (response) is equal to the convolution of the input with the impulse
response (point spread function) in space domain or the multiplication of the input with
the system function (amplitude transfer function, ATF) in angular frequency domain.
Mathematically, it follows,
21
U  kr , z  z   U  kr , z  ATF  z 
where
ATF  z   e
2
2
 j k  kr z
 e  jkz e
j
kr 2
z
2k
.
E  r , z  z   E  r , z   PSF  z 
where
PSF  z   e jkR  e  jkz e
 jk
(3-16)
(3-17)
x2  y 2
2 z
2. Experimental setup of DOE Simulator
Based on the 4-f imaging system previously discussed and the phase-only LC-SLM, we
designed the following DOE Simulator consisting of a 4-f imaging system and the SLM
with schematic diagram shown in Fig. 3-2. Here the field right after the reflection from
the SLM, which is the input transparency multiplied by the phase loaded onto the SLM,
can be reimaged at the output plane, so that precise measurements can be performed
without introducing any phase aberration. The experimental setup is also shown in Fig.
3-3.
22
Output
(x2, y2)
Input
(x1, y1)
f
SLM (x’, y’)
f
L1
(ξ, η)
f
(x, y)
f
L2
Fig. 3-2. Schematic diagram of DOE Simulator.
23
Fig. 3-3. Experimental setup of the DOE Simulator.
3. Complex scalar field realization using DOE Simulator: bifocal lens and toric
lens
As an illustration of the capability of our DOE Simulator, two types of lens functions
have been realized and tested.
3.1. DOE bifocal lens
A bifocal DOE design is investigated to test the performance of our DOE Simulator
system. A bifocal power of 3 diopters (D) is added to a base power of 20 D. The phase
pattern is shown in Fig. 3-4.
24
Fig. 3-4: Phase pattern of the bifocal phase with the 20 D base power.
Optical powers of 20 D and 23 D are expected and the spacing d between two foci can be
calculated as:
d
1
1
1
1



 6.52mm
p1 p2 20 23
The experiment was done with 4 mm pupil size. The one dimensional (1D) through-focus
point spread function (PSF) simulation and experimental data for 4 mm pupil size is
shown in Fig. 3-5. Fresnel propagation simulation is used for PSF evaluation in the
MATLAB environment. The locations of the peaks appear as predicted with the spacing
of about 6.49 mm. The difference between experimental data and simulation could be due
to scattering from the 87% filled SLM pixels, vibrations and other perturbations.
25
1
Through focus PSF of ReSTOR with 20D for 4 mm pupil size
Simulation
Experimental results
0.9
Normalized Intensity
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-8
-7
-6
-5
-4
-3
-2
Defocus (mm)
-1
0
1
2
Fig. 3-5. Through focus PSF of ReSTOR with 20 D base power for 4 mm pupil size.
The two dimensional (2D) through-focus PSF are shown in Fig. 3-6 for 4 mm pupil size.
Fig. 3-7 shows the simulation result. Good agreement has been obtained. Note that the
captured image is not affected by other diffraction orders since the separation in the far
field is large.
26
40
30
20
x (m)
10
0
-10
-20
-30
-40
-7
-6
-5
-4
-3
-2
Defocus z (mm)
-1
0
1
Fig. 3-6. Experimental results of 2D through focus PSF for 4mm pupil size.
40
30
20
x (m)
10
0
-10
-20
-30
-40
-7
-6
-5
-4
-3
-2
Defocus z (mm)
-1
0
Fig. 3-7. Simulation results of 2D through focus PSF.
27
1
3.2. Toric lens
In order to verify the astigmatism effect from toric lens and the spacing between its
sagittal and tangential foci, we tested toric lenses consisting of a 20 D base power with
different toricities of 3 D, 0.5 D and 0.1 D on x-axis, respectively.
As one example, toric phase pattern of 3 D toricity is shown in Fig. 3-8. This combined
with base power of 20 D is then loaded onto the SLM and tested on our DOE Simulator
with 4-f system.
Fig. 3-8. 3 D toricity phase pattern.
The spacing between the tangential and sagittal foci is found to be 6.52 mm, 1.2 mm and
0.24 mm, respectively and the evolution is in Fig. 3-9, Fig. 3-10 and Fig. 3-11. In each
figure, the tangential focus, mid-point between two foci and sagittal focus are shown with
the simulation results for each toricity. Even for 0.1 D toricity, low space-bandwidth
28
product is present due to the small beam size and limited pixel size of the detector.
However, even at 0.1 D, we were able to observe the transition of the energy
concentration from along horizontal direction to along vertical direction as the CCD is
translated from the tangential focus to the sagittal focus.
Fig. 3-9. Evolution of 3 D toricity near focus. The upper column is simulation results and
the lower is the experimental measurement. The defocus from left to right is 0 mm, -4
mm and -6.5 mm, respectively.
29
Fig. 3-10. Evolution of 0.5 D toricity near focus. The upper column is simulation results
and the lower is the experimental measurement. The defocus from left to right is 0 mm, 0.6 mm and -1.2 mm, respectively.
30
Fig. 3-11. Evolution of 0.1 D toricity near focus. The upper column is simulation results
and the lower is the experimental measurement. The defocus from left to right is 0 mm, 0.12 mm and -0.24 mm, respectively.
4. Summary
The DOE Simulator is proposed and experimentally verified. The DOE phase has been
imaged on the output plane of the DOE Simulator and the through-focus PSF for both
bifocal and toric lens designs show very good agreement with the simulation results
based on beam propagation method. The DOE Simulator has well demonstrated the phase
modulation capability using the SLM.
31
CHAPTER 4
SECOND-ORDER FULL POINCARÉ BEAMS
1. Introduction
Phase modulation capability has been previously demonstrated using LC-SLM based
DOE Simulator. As discussed before, for incident beams with arbitrary polarization
direction, phase retardation can be introduced between the polarization components along
the fast and slow axes of the LC-SLM. Thus, more complex modulation can be achieved
by spatially engineering the retardation within each LC-SLM pixel as prescribed by the
phase pattern.
It is well known that electromagnetic waves carry both energy and momentum and the
interactions between light and matter involves both energy and momentum exchange. As
a fundamental nature of light, the momentum can be further divided into two categories,
spin angular momentum (SAM) and orbital angular momentum (OAM). SAM is
associated with polarization while OAM describes the angular distribution of the phase or
wave front. If the EM wave has an azimuthal dependence of exp(ilφ), where φ is the
azimuthal angle, then it’s said to have an OAM of lħ or a topological charge of l. The
32
origin has an abrupt change in phase, which is defined as a singularity point. OAM has
attracted a lot of research interest recently due to its potential applications in optical
communications [35], optical trapping [36], optical tweezers, Full Poincaré (FP) Beams
generation [37] [38] and laser beam shaping [39].
2. Full Poincaré beam and its state of polarization
Full Poincaré (FP) beam is a new class of beams that have been proposed and studied
recently [38] [37]. Beams whose SOPs within the cross-section span the entire Poincaré
sphere are defined as FP beams. FP beams can be generated through superimposing
orthogonally polarized beams with spatially different intensity distributions [1].
Generation of the first order FP beams has been previously studied using fundamental
Gaussian (LG00) and first order Laguerre Gaussian (LG01) beams of right hand circular
polarization (RCP) and left hand circular polarization (LCP), respectively [38]. 2D flattop
beam shaping has been reported in one previous paper using the linear combination of xpolarized fundamental Gaussian and y-polarized LG01 mode [40], which in fact generated
the first order FP beam. Smooth flattop profile has been obtained for chosen parameters.
However, the edge roll-off is not steep due to the gentle roll-off of the transverse profile
for the focused LG01 component. This can be improved through the using of LG modes
with higher topological charges. The interest in flattop generation leads to our
investigation on the second order FP beam. Here we report the second order FP beams
generated through linear combination of horizontally polarized (x-polarized) fundamental
Gaussian (LG00) and vertically polarized (y-polarized) second order Laguerre Gaussian
(LG02) beams.
33
Vortex beams with spiral wavefront carry OAM, or phase singularity in the center. In the
far-field, the vortex beam takes the form of Laguerre Gaussian mode where the phase
singularity evolves into a doughnut distribution. The far-field intensity and phase pattern
of vortex beam with topological charge of 2 are shown in Fig. 4-1. In this example, LG02
is expected. LG beams are a set of solutions to Maxwell’s Equations that carry OAM
with rotational symmetry expressed in cylindrical coordinates. P denotes radial nodes
while l represents the topological charge associated with the beam. Generally speaking,
as l increases, the dark center of the donut-shaped beam also increases.
Fig. 4-1. Intensity and phase pattern for LG02 mode.
Generation of scalar vortex beam (LG02) is reported in this section. A novel beam with
spatially variant polarization has been demonstrated theoretically and experimentally by
superimposing orthogonally polarized fundamental Gaussian (LG00) and second order
Laguerre Gaussian (LG02) beams [39]. This beam in fact belongs to the family of the
second order FP beams. The evolution of the SOPs along propagation will be discussed
in details.
34
The fundamental Gaussian beam can be represented as:
LG00  r , z   A0
 A0
w0
e
w z 
w0
e
w z 
 j  kz   z   
jkr 2
2 q z 






1
jk
,
 j  kz   z    r 2 


  z  2   z02  
 w0 2  1    2 z   
z 
  z0   





(4-1)
where A0 is the amplitude, w0 is the beam waist, q(z) is the Gaussian beam propagation
parameter as a function of axial distance z, w(z) is the beam size, φ(z) is the Gouy phase
for fundamental Gauss and z0 is the Rayleigh range. A more detailed discussion on
general Laguerre Gaussian modes LGpl can be found on Chapter 7, Section 4.1. Similarly,
second-order Laguerre Gaussian (LG02) beams can be written as:
LG02  r , , z   A0 2
r2
w z 
2
w0  j kz 3  z  j 2 
e
w z 






 2
 ,
jk
1
exp   r 


2
2




z

 2
 z 
2  z  0  


w
1


 0    
z   


z

  0 

 
(4-2)
where ϕ is introduced as the azimuthal angle due to the spiral wave front. Note that the
Gouy phase for LG0,2 is 3 times larger than the one for the fundamental Gauss.
35
A superposition of orthogonally polarized LG00 and LG02 can be expressed as follows:
EFP  r , z   cos  LG00 xˆ  sin  LG02 yˆ
 1 
 C
j 
 0e 
，
(4-3)
Here we denote the second term in Jones vector as ρ0ejδ, where ρ0 is the ratio between y
and x components and δ is the phase difference. Angle γ is the angle between the
polarization direction and the horizontal axis, which is used to adjust the weighting
between the x and y components of the FP beam. Then we would have the following
expressions:
0  2 tan 
r2
w z 
2
,   2  z   2 ，
(4-4)
Simple algebra shows that close to axis, the polarization is mainly along x-axis while
away from axis the polarization evolves to y-polarization. The phase delay is twice the
difference of azimuthal angle and Gouy phase shift. Therefore at any cross section of the
superimposed beam along propagation, the phase delay between two components will
range from 0 to 4π. Thus, the SOP of in the superimposed beam cross section second will
span the entire surface of Poincaré sphere twice. Hence, we call this the second order FP
beam.
Along propagation, the Gouy phase shift will further introduce an additional phase delay
of 0 to -π, which will cause the SOP to rotate by 2π as it propagates. Numerical
simulation agrees well with the prediction, as illustrated in Fig. 4-2.
36
Fig. 4-2. SOPs evolution for second order FP beam at 1). z = -10zR, 2). z = -zR, 3). z = 0,
4). z = zR and 5). z = 10zR.
In the above figure, the local SOP is denoted as lines, ellipses or circles. The ± 10zR
represent z at ± infinity. Between adjacent plots, a change of π/2 in the phase difference δ
has been introduced for the local polarization as the beam propagates. For instance, the
polarization of the red circled area evolves from 45° linear to RCP, 135° linear, LCP and
finally back to 45° linear as the beam propagates from minus infinity to plus infinity. In
other word, as the beam propagates from negative infinity to positive infinity, the local
polarization at each relative position (azimuth φ and normalized radius r/w(z))
experiences a continuous rotation in phase difference δ from 0 to 2π.
37
Recall that normalized Stokes Vector can be written as
 S1  cos  2  cos  2   cos  2  

 S 2  cos  2  sin  2   sin  2   cos  

S3  sin  2   sin  2   sin  

(4-5)
where tan(χ) = ρ0 = 2tan(γ)(r/w(z))2, describing the ratio of the amplitudes of y and x
components. The evolution of SOP at fixed azimuthal angle φ and relative radius r/w(z)
along propagation covers one slice (shown in red) of the Poincaré sphere perpendicular to
the S1 axis as shown in Fig. 4-3 where δ spans from 0 to 2π due to Gouy phase at a
constant S1 (cos(2χ)). However, the intensity, as a function of γ and r/w(z), remains
unchanged during propagation.
Fig. 4-3. One slice (shown in red) of the Poincaré sphere at fixed ϕ and r/w(z) with δ
spanning from 0 to 2π.
38
3. Summary
Second-order full Poincaré beams are introduced. As one type of 2nd order FP beams, the
vectorial beam as a superposition of x-polarized Gaussian and y-polarized LG02 is
investigated in details. The SOPs in the cross section are shown to cover the entire
Poincaré sphere twice, hence the 2nd order FP beams. The evolution of the SOP during
propagation is also investigated. Potential applications of such beam include laser beam
shaping, where the beam has been experimentally generated both using the SLM [39]
[41]and compact beam shaper [42]. The flattop beam profile has been demonstrated in
both cases.
39
CHAPTER 5
GENERATING CYLINDRICAL VECTOR BEAM WITH
SUBWAVELENGTH CONCENTRIC METALLIC GRATING
FABRICATED ON OPTICAL FIBER
Subwavelength grating structures have long been used to modify optical fields [43] [44].
In this chapter, a metallic grating structure is used to realize polarization selection by
generating generalized cylindrical vector beams, as an example of more complex control
of complex vectorial optical field.
As one example of polarization control using subwavelength metallic structures, we
report the generation of cylindrical vector (CV) beam with a subwavelength concentric
metallic grating fabricated on an optical fiber, eliminating the widely used conical
devices or birefringent crystals. It has been demonstrated that a wiregrid grating with a
period smaller than the wavelength of incident light strongly reflects TE polarized light
and allows TM polarized light to transmit through [25]. Subwavelength concentric
metallic slits have been used for creating two-lobe shaped radially polarized or
azimuthally polarized beam with linearly polarized illumination [26]. When an
40
azimuthally polarized light is focused onto the axially symmetric metallic grating, the
entire beam is TE polarized with respect to the interface and therefore experiences high
reflection. In contrary, a radially polarized beam is entirely TM polarized with respect to
the metallic grating and can pass through the device with higher transmittance than the
azimuthal polarization. It has been shown that circular polarization can be decomposed
into the combination of radial polarization and azimuthal polarization components with a
spiral phase wavefront [45]. Therefore, if a circularly polarized beam is coupled into a
fiber with a subwavelength concentric metallic grating integrated on the core region, the
radial polarization component has much higher transmittance than the azimuthal
polarization component. Consequently, a doughnut-shaped CV beam can be created.
LP QW Lens
Metallic rings
LP
Laser
CCD
Fiber
Fig. 5-1. Experimental setup for generalized CV beam generation. [46]
The diagram of the experimental setup is illustrated in Fig. 5-1. A subwavelength
concentric metallic grating fabricated on the core of an optical fiber was used as a
polarization selector for generating CV beams. First, a 200 nm gold film was deposited
onto a cleaved fiber (Thorlabs 630HP) facet with e-beam evaporation. This thickness was
chosen to prevent high direct transmission of the laser through the gold layer. Then
twenty periods of concentric annular slits were fabricated into the gold film with focused
41
ion beam milling (FIB, FEI dual beam SEM-FIB NOVA 200 Nanolab system) with the
center of the concentric rings coincides with the center of the fiber. The number of rings
is chosen to ensure that the concentric grating is larger than the core region of the fiber.
Fig. 5-2 shows the scanning electron microscope (SEM) images of the fiber based
concentric grating (Bull’s Eye structure) fabricated by Don Abeysinghe with focused ion
beam (FIB) and a zoom-in of the metallic annular structure. The grating has a period of
200 nm, which is smaller than the laser wavelength (Nd:YAG second harmonic 532 nm
green laser). The duty cycle was chosen to be 50%. A circularly polarized Gaussian
beam was focused onto the fiber end with the concentric grating structure. The output
beam at the other end of the fiber was imaged onto a CCD camera. A linear analyzer was
placed in front of the camera to investigate the SOP of the light.
Fig. 5-2. SEM pictures of the Bull's Eye structure. [46] Sample is prepared by Don
Abeysinghe using FIB.
42
A left-hand or right-hand circularly polarized beam (LCP or RCP) is a linear
superposition of uniformly distributed radially polarized and azimuthally polarized vortex
beams with a topological charge of 1 [45].
j
E LHC 
P ( r ) e ( er  je )
2
, E RHC 
P ( r )e
 j
( er  je )
2
,
(5-1)
where P(r) is the amplitude distribution of the beam in cylindrical coordinate, er and eφ
are amplitude of unit vectors in radial and azimuthal directions, respectively. Due to
much higher transmittance of radial polarization component, a mostly radially polarized
beam can be obtained at the output end. A finite element method model (COMSOL
Multiphysics) was developed to numerically investigate the polarization selection
properties of the subwavelength concentric metallic grating fabricated on optical fiber
under circularly polarized illumination. 3D model was used and the thickness is chosen at
200 nm to balance the polarization conversion efficiency and the milling time using FIB.
The simulated energy density and local electric vector field distributions are illustrated in
Fig. 5-3. The extinction ratio between radial polarization and azimuthal polarization was
found to be 50.7, therefore the electric vector field orientation of the output CV beam has
a small angle (~8 degrees) deviating from the radial direction.
43
Fig. 5-3. Simulated energy density and local SOP. [46]
44
Fig. 5-4. Experimental results for generalized CV beam generation. [46]
45
Fig. 5-4 (upper) shows the total intensity distribution of the output laser beam captured
by the CCD camera. A doughnut spot with a dark center was obtained at the output end.
Fig. 5-4 (lower) shows the pictures of the output laser beam after it passes through a
linear analyzer oriented at different angles indicated by the arrows. The two-lobe spot
follows the rotation of the linear analyzer, indicating the generation of a CV beam. The
beam can be conveniently converted into an azimuthally polarized beam or radially
polarized beam with two cascaded half-wave plates [47]. The transmission efficiency of
the CV beam was measured to be 1.96%. This efficiency can be improved by adjusting
the thickness of the metal film and annular ring parameters. The integrated
subwavelength structure on the fiber end provides a compact and convenient way for CV
beam generation in fiber laser cavity design. If the concentric metallic grating is used as
one end mirror coupler, the axially symmetry of the metallic rings ensures that the
oscillation mode has axially polarization symmetry and radial polarization output will be
resulted due to its polarization selectivity.
Summary
In this project, the polarization-sensitive Bull’s Eye structure is designed and
experimentally verified. The generation of generalized cylindrical vector beam with
mostly radial polarization is confirmed with the all-fiber device. This approach eliminates
the needs of conventional conical devices or birefringent crystals in fiber laser cavity
design and enables a compact all-fiber laser design. It also avoids the requirement of
precise alignment, which could improve the stability of the CV beam generation.
46
CHAPTER 6
COMPLEX OPTICAL FILTER FOR HIGH PURITY OPTICAL
NEEDLE FIELD GENERATION
1. Introduction
In this chapter, a more complex engineering of the optical field is proposed, where
modulations of phase, amplitude and polarization are simultaneously realized using
complex filter designs with subwavelength grating and optical antennas. The engineered
optical field will then be focused by a high NA objective and high-purity optical needle
field is expected with flattop distribution along longitudinal direction and an extended
depth of focus.
The investigation of focused electromagnetic field over a three dimensional volume has
always been an interesting and important area of theoretical and applied optics. The
focusing properties of linearly polarized (scalar) field have been well established by
Richards and Wolf [33]. However, beams that possess polarization axial symmetry, i.e.
radial and azimuthal polarization, hadn’t been thoroughly investigated until recent years.
These beams characterized by a spatially inhomogeneous SOP, are also called cylindrical
47
vector beams (CVBs). Of particular interest, it is found that under tight focusing, an
azimuthally polarized beam maintains its polarization property while a radially polarized
beam acquires a significant longitudinal polarization component [4] [6]. By using a
carefully designed pupil apodization function, it is possible to achieve a tighter focusing
spot and increase the resolution of the system by a factor of two [47] [15]. By using a
DOE pupil mask with binary phase coding, it is also possible to achieve a flattop profile
at focal plane with extended depth of focus [16].
The generation of an optical needle field proposed in [48] provides a new approach to
obtain an electrical field with uniform axial intensity profile, extended depth of focus and
high-purity longitudinal polarization. The unique electric field distribution opens the door
to applications in polarization sensitive imaging [49], light-matter interaction on the
nanometer scale and particle trapping and acceleration [50]. For instance, the uniform
axial intensity profile along with extended depth of focus ensures a uniform and
elongated trapping force in longitudinal direction for increased trapping efficiency.
2. Electric field calculation at pupil plane
Reversing radiation of an electric dipole array located around the focus of a high NA lens
is used to calculate the incident field at the pupil plane [48]. A high NA focusing of
engineered optical field at the pupil plane is depicted in Fig. 6-1, where the vector field in
the focal volume near focal plane shown in red is of interest. An array of equally spaced
electric dipoles is placed in the vicinity of the focal plane, which is shown in Fig. 6-2 [48].
48
Fig. 6-1. High NA focusing of pupil plane field in the focal volume (shown in red).
Fig. 6-2. Schematic configuration of reversing radiation of an electric dipole array.
49
where N pairs of dipoles (N = 2) shown in red in Fig. 6-2 are located symmetrically with
respect to the focal point along the optical axis. Due to the symmetry, we only need to
adjust the amplitude An, the spacing between two dipoles in each pair dn and additional
phases ±βn associated with both dipoles in each pair. The superposition is written as,
N
AFn   An  e

n 1
where
 kdn cos n 
j  kd n cos   n  /2
e
 j  kd n cos    n  /2
,

(6-1)
is the phase difference between both dipoles in each pair and it has
been divided by 2 to achieve equal distribution on both dipoles.
For an objective lens that follows sine condition, the incident field can be represented as
[4] [33],
Ei  i ,   C sin  / cos AFn  cos  xi  sin  yi  ,
(6-2)
where φ is the azimuthal angle in the pupil plane.
For a radially polarized beam, the radial and longitudinal electric field near focus may be
expressed using the vectorial Debye theory [4] as
Er  r , z   C
 max
 AF   sin
n
2
 cos  J1  kr sin   eikz cos d
， (6-3)
0
Ez  r , z   C
 max
 AF   sin
n
3
0
50
 J 0  kr sin   eikz cos d
，
(6-4)
respectively, where C is a constant. Generally, the higher the number of dipole pairs N is,
the longer the DOF of the resulting field will be. However, the complexity of the
corresponding pupil plane distribution will also increase. A trial and error methodology is
employed and a DOF of 5λ with a uniform intensity in the longitudinal direction around
the focus has been achieved using the transparency reconstructed from an array of 2 pairs
of dipoles. The parameters of the dipole array are shown in Table 1 [48].
Table 1: Parameters of dipole array for N = 2
An
dn
βn
1
1.39 λ
π
0.87
4.10 λ
3π
The far field intensity along focus is shown in Fig. 6-3 (upper) and the line scan of the
normalized intensity in Fig. 6-3 (lower) [48]. This demonstrates very high longitudinal
field purity and a nearly flat top axial distribution in focal volume. Therefore, the optical
needle field is expected near the focal volume. Beam purity of 86%, which is defined as
the intensity of the longitudinally polarized component confined within the optical needle
over the total intensity, has been achieved.
51
Fig. 6-3. Far field intensity along focus (upper) and its line scan (lower) [48]. The
longitudinal coordinate is normalized to the wavelength.
The depth of focus is found to be 7.9 λ. In order to generate such optical needle field, the
required intensity distribution of the optical field in the pupil plane and its line scan are
calculated via Eq. (6-2) and Table 1 and shown in Fig. 6-4 [48]. We noticed that the
highest transmittance occurs at the outermost ring.
52
Fig. 6-4. Intensity distribution in the pupil plane (upper) and its line scan (lower) [48].
The transverse coordinate is normalized to the pupil radius of the high NA objective.
53
3. Discretization of the pupil filter
The desired field at pupil plane can be considered as the transmitted field of a radially
polarized electric field from a filter located at the pupil plane. The filter has a complex
transmittance for both amplitude and phase.
To realize the optical needle field in practice, we introduce a simplified discrete filter
design with non-continuous transmittance and binary phase. The amplitude and phase of
the filter are plotted along radial direction as shown in Fig. 6-5 [48].
Fig. 6-5. Discrete filter design. ρ is normalized to the pupil radius for the high NA objective.
54
The resulting field is further calculated and the performance is shown in Fig. 6-6 [48].
Fig. 6-6. Electric field along focus (a) and its line scan (b) for discrete pupil filter. The
horizontal axis is normalized by the wavelength.
The results show a good agreement with the continuous complex filter. Implementation is
made possible with the discretized design discussed here and liquid crystal devices, such
as SLMs and subwavelength metallic structures are good candidates to realize such pupil
filter.
55
Nikon CFI Plan Apo VC is used as the high NA objective. The lens has a magnification
of 100X and NA of 1.4 in oil. The focal length is found to be 2 mm and NA in air is
0.924. The radius of pupil is found to be 1.85 mm and the diameter is 3.7 mm.
4. Sub-wavelength
metallic
grating
implementation
for
optical
needle
generation
4.1. Introduction
Sub-wavelength metallic gratings have been thoroughly investigated for its reflection and
transmittance properties by researchers in this field [34]. The degrees of freedom in the
grating design include the choice of metal, depth of grating, periodicity and duty cycle of
the grating. By carefully choosing the parameters, the amplitude and phase of the zeroth
order of the transmitted electric field can be adjusted to the desired combination in each
annular ring. For each annular ring, the structure can be approximated as a rectangular
grating with infinite number of periods. Rigorous Coupled Wave Analysis (RCWA) [27]
[28] [29] is used to evaluate the diffraction of electric field by a periodic grating structure.
A thorough search is performed to generate the desired amplitude and phase of the
transmitted electric field for the three inner rings with lower transmittance. The zero
transmittance is simply obtained by coating a layer of Al film and high transmittance is
approximated by the transparent glass substrate.
56
4.2. Pupil filter design and proposed experiment setup
In our proposed design, aluminum (Al) is chosen as the metal and the highest amplitude
at the outmost ring is provided by using a glass substrate. Al gratings on the glass
substrate are engineered such that the three inner rings share the same period of 1 um and
50% duty cycle. By varying the depth of the grating, different amplitude and phase can
be achieved. Fig. 6-7 has a schematic plot of the pupil filter. The black region is covered
by the Al film and is opaque, the white region is the transparent glass substrate, the green
region has a 180 nm thick Al grating structure and the blue region has a 675 nm thick Al
grating structure. The parameters for different regions of the pupil mask are shown in
Table 2, where Ring 1 refers to the innermost ring and Ring 7 is the outermost area of the
pupil. FIB is to be used for the fabrication of the subwavelength metallic structure. The
technique limits the radius of the pupil filter to 250 μm and in order to cover the entrance
pupil of the Nikon objective, a telescope relay is required. The radius in Table 2 is given
in microns and scaled with respect to the radius of the pupil filter.
Table 2: Grating parameters for each concentric ring
Ring
1
2
3
4
5
6
7
Radius (μm)
37.5
87.5
132.5
162.5
180
215
232.5
The proposed experimental setup is also included in Fig. 6-7. A linearly polarized laser
beam is converted to radial polarization by a circular polarizer, a phase plate and a radial
analyzer before passing through the pupil filter. With a telescope relay setup discussed
57
above, the beam can be expanded to match the NA of the 1.4-NA oil immersion objective
lens. The resulting longitudinal field along the focus then can be detected using near-field
scanning optical microscope (NSOM) setup.
Filter
λ=1.064μm
Pinhole
with 0.5
mm in
dia.
Radial
Analyzer
Telescope
1.4 NA oil
immersion
objective
Fig. 6-7. Grating structure (upper) and proposed experimental setup (lower). The radii are
normalized to the radius of the pupil.
58
5. Preliminary results
Initial FIB sample is shown in Fig. 6-8. The depth of the trench is 100 nm. However, for
grating thickness as much as 675 nm, even with the largest available current, it would
take days of milling time to fabricate a deep trench. Therefore, a different fabrication
technique needs to be adopted for the optical needle field generation using subwavelength
metallic grating structure.
Fig. 6-8. SEM of sample (by Don C Abeysinghe).
59
6. Design based on subwavelength metallic linear antenna
6.1. Introduction
As an alternative approach, we also investigate the properties of subwavelength linear
antenna design for its much smaller cross section, which as a result reduces the amount of
metal removing. The scattered field off V-shaped nano antenna has recently been
theoretically studied for its ability to control the amplitude and phase of the crosspolarized component [31] [32]. Thanks to the greater-than-2π phase coverage, the design
of a plasmonic nano V-shaped antenna array to introduce a phase gradient from 0 to 2π at
the interface of two media in order to steer the beam has been proposed and
experimentally demonstrated. The amplitude and phase of the scattered field can be
engineered by appropriately choosing the design parameters, the length of the rod
antenna and the angle between the two arms. One cell of the nano V-shaped array
consists of 8 V-shaped antennas providing a phase shift from 0 to 2π in one direction
while exhibiting constant amplitude of the cross-polarized component. This is also
referred to as generalized laws of reflection and refraction, or light bending using
plasmonic nano-antenna.
6.2. Initial analysis
Since our design of the discrete pupil filter only requires two phase values, 0 and π, we
decided to investigate a linear rod antenna for its simplicity in numerical simulation and
fabrication.
60
Fig. 6-9: Linear antenna oriented at angle θ from the incident polarization.
A linear antenna with a certain length and angle θ from the incident polarization (as
illustrated in Fig. 6-9) has the potential to spatially modify the amplitude and phase of
cross-polarized scattering field. The Hallen and Pocklington Integral Equations [51] are
given as

4
l /2

I  z '  G  z  z '  dz '   j   2z  k 2  Ein  z  ，
1
(6-5)
 l /2

4
l /2
 I  z '  
2
z
 k 2  G  z  z '  dz '   j Ein  z  ，
(6-6)
 l /2
where G  z  z '  
1
2
2

0
e  jkR
d ' and R 
R
 z  z '
2
 2a 2  2a 2 cos  ' .
Based on these equations, the normalized amplitude and phase of the scattered field
polarized in the axial direction (z direction) off the linear rod antenna can be calculated
numerically and plotted as a function of the length L as shown in Fig. 6-10.
61
1
0.9
Normalized amplitude
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.2
0.3
0.4
0.5
L/0
0.6
0.7
0.8
0.3
0.4
0.5
L/0
0.6
0.7
0.8
0
-20
Phase (degree)
-40
-60
-80
-100
-120
-140
-160
0.2
Fig. 6-10. Normalized amplitude (upper) and phase (lower) of the longitudinally
polarized scattered field.
The z (along antenna) and ρ components of the scattered field of the rod antenna placed
along z direction with L = 0.5 λ are shown in Fig. 6-11 and Fig. 6-12, respectively.
COMSOL Mulitphysics is a finite element method (FEM) based simulation tool.
62
Compared with COMSOL results, good agreement is observed between the two
simulation tools.
Fig. 6-11. The absolute value of the Ez component of the scattered field off the rod antenna.
Fig. 6-12. The absolute value of the Eρ component of the scattered field off the rod antenna.
63
Here we assume the antenna is parallel to the polarization of the incident light, λ0 denotes
the wavelength and the radius of the rod antenna is λ0/100. By putting the antenna at an
angle from the incident polarization, the projection of the incident light to the antenna
direction follows a cosine rule. Consider the cross-polarized component of the scattered
field, another projection of the longitudinally excited electric field to the orthogonal
direction of the incident polarization needs to be performed in order to calculate the
resulting field. Assuming the angle between the antenna and the incident polarization is θ,
the cross-polarized component of the scattered field and the original scattered field
follows such relationship:
scattered
scattered
Ecross
cos sin 
 polarized  E
，
(6-7)
A phase jump of 180 degrees can be achieved if the product of cosine and sine functions
becomes negative. The amplitude and phase of the cross-polarized scattered field off
linear rod antennas then can be calculated as a function of both antenna length (L) and
orientation (θ) from the incident polarization in Fig. 6-13 and Fig. 6-14, respectively. In
both figures, the dashed line refers to a constant 0.5 amplitude contour plot. As can be
seen from Fig. 6-14, the phase coverage is greater than π but less than 2π
64
Fig. 6-13. 2D amplitude plot of linear antenna versus length L and θ.
Fig. 6-14. 2D phase plot of linear antenna versus length L and θ.
6.3. Complex optical filter design
The maximum amplitude occurs at L = 0.4744 λ. Since we only need a phase change of 0
and π, a horizontal line at L = 0.4744 λ is the area of interest. In our proposed pupil filter
65
design, both the amplitude and phase can be regarded as relative quantities for different
annular rings. Amplitude modulation can be obtained by rotating the linear antenna with
respect to the local polarization, in other words, to change the angle θ. The parameters are
shown in the following table (Table 3) for different annular rings on the pupil mask.
Table 3: Design parameters
Amplitude
0.07
0.077
0.132
0.48
1
Phase
0 (π)
π (0)
0 (π)
π(0)
π(0)
Orientation (θ)
2.0
2.2
3.8
14.3
45
The orientation angle θ listed above all have phase of 0. To change the phase from 0 to π,
one can use a simple transformation as follows.
 new   

2
or   
，
(6-8)
Since the required electric field before tight focusing needs to be radially polarized, a
radial analyzer is required in front of the high NA objective. The incident field onto the
antenna pupil mask needs to be azimuthally polarized to ensure correct field distribution
in the cross-polarized component, which is a radial polarization in this case. The
illumination wavelength is 1.064 μm so the length of the rod antenna is set to be 0.505
μm.
To ensure the fabrication is feasible, we set the radius of the cross section of the rod
antenna to be 25 nm. The 1D plots (Fig. 6-15) of normalized amplitude and phase of
66
longitudinally polarized scattered field and 2D plots (Fig. 6-16) of the cross-polarized
scattered field as a function of length L and orientation θ have been regenerated for 25
nm radius rod antenna.
1
0.9
Normalized amplitude
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
L/0
0.6
0.7
0.8
0.9
0.2
0.3
0.4
0.5
L/0
0.6
0.7
0.8
0.9
0
-20
Phase (degree)
-40
-60
-80
-100
-120
-140
-160
0.1
Fig. 6-15. Normalized amplitude (upper) and phase (lower) of the longitudinally
polarized scattered field for 25 nm radius rod antenna.
67
Fig. 6-16. Normalized amplitude (upper) and phase (lower) of cross-polarized scattered
field as a function of L and θ for 25-nm radius rod antenna.
Note that the maximum amplitude occurs at L = 0.4739 λ, only with a negligible shift.
Since the amplitude modulation is achieved by adjusting angle θ, the design is identical
as the previous model. The complete design is shown in Table 4, where the antenna
68
direction is measured from the local polarization direction (azimuthal direction). A
tolerance study of all of the design parameters for +/- 10% fluctuation in the orientations
of the linear antennas is performed in
69
Table 5.
Table 4: Antenna design parameters for optical needle field generation
Annular ring #
1
2
3
4
5
2°
92.2°
3.8°
104.3°
135°
Amplitude
0.007
0.0077
0.0133
0.0481
0.1005
Phase (radians)
-2.096
1.048
-2.094
1.048
1.050
0.070
0.077
0.132
0.479
1
Antenna
direction
Normalized
Amplitude
70
Table 5. Tolerance study of the antenna structure
Upper limit
Lower limit
Phase (-93.4 ◦)
+0.2π (-57.4◦)
-0.2π
Length (0.4744)
0.4256
0.5663
Amplitude1 (0.07)
+10% (0.077)
-10% (0.063)
Orientation (2.0)
2.2
1.8
Amplitude2 (0.077)
+10% (0.0847)
-10% (0.0693)
Orientation (2.2)
2.4
2.0
Amplitude3 (0.132)
+10% (0.145)
-10% (0.119)
Orientation (3.8)
4.2
3.4
Amplitude4 (0.48)
+10% (0.528)
-10% (0.432)
Orientation (14.3)
15.9
12.8
Amplitude5 (1)
+10% (1.1)
-10% (0.9)
Orientation (45.0)
n.a.
32.0
71
6.4. Realization of complex optical filter with slot antennas
The linear antenna design may offer the ultimate method to achieve optical needle field
after tight focusing. Rectangle cross section needs to numerically simulated and
compared with circular one. We expect a cross section of 100 nm x 50 nm rectangle to
behave similarly as the rod antenna with 50 nm radius. COMSOL simulations are
performed for both original design and the complimentary design (slot in metal) based on
Babinet’s Principle as depicted in Fig. 6-17.
Fig. 6-17. Rectangle linear antenna and its complementary design.
Due to the vector Babinet’s principle [52] [53], the transmitted polarization will be
orthogonal to the long axis of the slot antenna. The orthogonal component of the
transmitted field will follow the same relationship as the original design of the linear
antenna. In other word, for azimuthally polarized input field, the radial component of the
transmitted field will have the designed amplitude/phase distribution. The new slot
antenna design requires much less metal to be removed compared to the grating, which
72
thus made the FIB a suitable and reliable fabrication technique. The antenna sample is
fabricated by Don Abeysinghe using FIB shown in Fig. 6-18 Very good uniformity of the
antenna in terms of direction and shape has been achieved.
Fig. 6-18. SEM pictures of antenna structure as complex optical filter fabricated by Don
C Abeysinghe using FIB.
7. Summary
The complex optical filter design with optical antennas is presented for optical needle
field generation. The tolerance study is performed and it shows a reasonable tolerance to
the fabrication error. The slot antenna design is proposed and investigated based on
Babinet’s principle and the fabrication is made possible using FIB. It has been
theoretically shown that with the complex optical filter, we are able to obtain the desired
amplitude and binary phase distribution at the pupil plane of a high NA objective for
optical needle field generation.
73
CHAPTER 7
VECTOR OPTICAL FIELD GENERATOR
1. Introduction
All these projects discussed in previous chapters lead to the need for a universal system
that’s capable of manipulating all degrees of freedom for vectorial optical field, as the
complexity of the conventional system increases and fabrication of the individual filter
design gets more and more challenging. In this chapter, VOF-Gen that is capable of
generating vectorial optical field with arbitrary phase, amplitude and polarization on a
pixel by pixel basis is proposed and experimentally demonstrated.
Optical trapping, or particle manipulation in colloidal and biomedical sciences, is made
possible by the shaping of light [54]. Shaped optical fields have enabled researchers to
better understand the biophysics and colloidal dynamics through the trapping, guiding or
patterning of molecules or nano/micro particles. Spatial engineering of focal field
intensity has been studied to reach resolution far beyond diffraction limit in microscope
system [55]. Vortex beam, also known as “twisted light”, has also drawn a lot of interest
owing to its spiral phase wavefront carrying orbital angular momentums [45]. Better
74
integrity of vortex beam through propagation in turbulent atmosphere has been shown [56]
and a lot of research has been done using the orbital angular momentum as information
carrier for free space communication due to its orthogonality and multiplexing capability
[57]. Besides the manipulation of intensity and phase, as the vector nature of
electromagnetic wave, the SOP also plays an important part in beam shaping for flattop
generation [39], focus shaping [15], optical tweezing and surface plasmon sensing using
cylindrical vectorial beams [58], ellipsometry [59]. All the above applications require
modulations of certain aspect of optical fields.
Tremendous amount of research has been conducted by scientists in the community
developing versatile systems for the generation of the optical fields with exotic properties.
Arbitrary vector fields with inhomogeneous distribution of linear polarization were
realized using a spatial light modulator (SLM) and an interferometric arrangement [20].
A recent study showed that in order to fully control the SOP, two spatially addressable
retarders need to used [21]. Researchers constructed such system consisting of two SLMs
with the fast axes 45° from each other and were able to generate optical fields that cover
the entire Poincare Sphere. Another non-interferometric method for vector field
generation was proposed by Tripathi and Toussaint [22]. However, the complete control
of both polarization elevation angle and ellipticity can only be realized by modifying the
experimental setup. Complete amplitude, phase and polarization control was reported by
Moreno et al. [23] with the help of a double modulation system which requires two
transmissive SLMs whose modulation depth can be controlled to achieve amplitude
modulation for certain diffraction order. However, this approach comes with the
limitation that each area must contain a large number of periods to achieve the ideal
75
diffraction efficiency, which leads to optical field with limited spatial resolution for
practical applications. Very recently, a technique for generating arbitrary intensity and
polarization was reported with interesting results using transmissive SLMs and MachZehnder interferometry setup [24]. However, due to the nature of the technique, the
absolute phase of each electric field component does not cover an entire 2π range. As a
result, complete phase control cannot be fully realized. The limitations of this technique
also include relatively low transmittance and poor spatial resolution. All the existing
techniques have limitations and cannot be used to generate a spatially-invariant arbitrary
vectorial field with high spatial resolution on a pixel basis. In this work, we propose and
demonstrate a Vectorial Optical Field Generator (VOF-Gen) that is capable of creating an
arbitrary beam with independent controls of phase, amplitude and polarization on the
pixel level utilizing high resolution reflective phase-only LC-SLM.
The scalar optical wave is typically written in the following complex form:
E  x, y   E0  x, y  e

 j k r  t 

,
(7-1)
where Eo(x, y) denotes the amplitude and the exponential part carries phase information.
However, in most cases, desired optical field takes vector form, which can be represented
as a superposition of two orthogonal polarization components. Using Jones vector
representation, the desired field is usually written as:
E d  x, y   Ad  x, y  e
jd  x , y 
76
E xd  x, y 




 E  x, y  e j d  x , y  
 yd

，
(7-2)
where Ad(x, y) represents the amplitude distribution, ϕd(x, y) is the common phase for
both x and y components and the Jones vector contains the polarization information
where Exd and Eyd are both real and normalized (Exd2 + Eyd2 = 1). δd(x, y) is the desired
phase retardation between the y and x components. As we know the polarization state can
be represented either in terms of linear polarization rotation χ and retardation δ or in
terms of polarization elevation angle α and ellipticity ε as shown in Eq. (7-3).
 cos  
 sin  e j 


 cos  
 sin   at  ，


or
(7-3)
These two representations are equivalent and can both describe any SOP on the
Poincare’s Sphere. They can be converted using Eq. (7-4).
  tan  e j 
tan   j tan 
1  j tan  tan 
，
(7-4)
Clearly four degrees of freedom, namely the phase, amplitude, polarization ratio and
retardation between the x and y components are necessary in order to fully characterize a
vectorial optical field. Thus a true vectorial optical field generator needs to be able to
control all of these four parameters on a pixel-by-pixel basis for the generation of
arbitrarily complex vectorial optical field. The principles of our proposed VOF-Gen will
be discussed in details in the following.
2. Principles
As one example, a Taiji pattern is generated using VOF-Gen with one half polarized in
right-hand circular polarization (RCP) and the other in left-hand circular polarization
77
(LCP) as depicted in Fig. 7-1. The principles of the VOF-Gen will be discussed in details
in this section.
Fig. 7-1. Taiji pattern coded in circular polarization. The total field (left), the upper half
Taiji pattern (upper right) in RCP and the lower half Taiji pattern (lower right) in LCP.
2.1. Spatial light modulator
As a key component for the VOF-Gen, the SLM, Holoeye HEO 1080P, is used as a
variable and addressable retarder. The SLM is a phase-only, reflective liquid crystal (LC)
device featuring a HDTV resolution of 1920 x 1080 with pixel pitch of 8 μm and fill
factor of 87%. The retardation for each pixel on the SLM can be described as a function
of the voltage (V) applied:  (V )  2 /  ne V   no d , where d is the thickness of the LC
layer, ne and no are the extraordinary and ordinary refractive indices of the LC retarder,
respectively. Due to the birefringent nature, the SLM in our system only responds to the
horizontal polarization parallel to the LC directors, meaning that the horizontal
78
component of the reflected beam will carry the wavefront specified by the SLM while the
vertical one will be reflected unaffected. Since four degrees of freedom in Eq. (7-2) need
to be independently controlled in the system, four reflections are required where each
SLM section is loaded with one of the phase patterns for the modulations of phase,
amplitude, polarization rotation and retardation.
First we show in detail how the modulations of the four degrees of freedom in the VOFGen can be achieved step by step with the introduction of the key components and the
calculation of the required phase patterns. Then we discuss the system integration,
experiment setup and extra care we took in building the system to ensure excellent results.
2.2. Spatially variant polarization rotator
Our proposed system relies on one key component called Polarization Rotator (PR) based
on the concept of a pure polarization rotator in order to realize the amplitude modulation
and linear polarization rotation. Pure polarization rotator that consists of a quarter-wave
plate (QWP), variable optical retarder with fast axis at 45° and anther QWP with its fast
axis perpendicular to that of the first QWP has been proposed to achieve fast, nonmechanical polarization rotation [60].
79
Fig. 7-2. Illustration of the Polarization Rotator setup. (a) The Polarization Rotator
comprised of QWP and reflective SLM; (b) Illustration of the effective rotation of the
QWP fast axis for the incident (upper) and the reflected (lower) beams due to the mirror
imaging of the laboratory coordinates in dashed lines.
In our design, the variable retarder is replaced with the LC-SLM to realize spatially
variant polarization rotation function on a pixel-by-pixel basis as shown in Fig. 7-2(a).
The fast axis of the QWP is 45° with respect to the horizontal axis. The incident light
passes the QWP with fast axis oriented at 45° (upper part in Fig. 7-2(b)) in the laboratory
coordinate (x, y) for the incident beam, then gets reflected off the SLM surface. The
reflected light goes through the same QWP for the second time (lower part in Fig. 7-2(b)).
However, due to the opposite propagation direction, the new laboratory coordinate (x’, y’)
is a mirror image of coordinate (x, y) about the y axis. Therefore, the fast axis of the
QWP has been effectively rotated to 135° in the coordinate (x’, y’). Thus, the Jones
matrix representation of the PR can be calculated as:
80
 3 
 3
M PR  R    J QWP R 
 4 
 4
  1 0 
 
 

 M SLM R    J QWP R  
 0 1 
 4
4
i x , y
1  1  i 1  i  1 0   e  
 


4  1  i 1  i  0 1   0
0  1  i 1  i 


1  1  i 1  i 

   x, y  
   x, y   

sin

cos





  x, y 
2
2
i





e 2 
   x, y  
   x, y   
 cos 
  sin 
 

2
2





e
   x, y 
i

 2 
， (7-5)
 3   x, y  
R


2 
 2
Equation (7-5) describes the Jones matrix of PR in terms of a rotation matrix R with an
extra phase term. The rotation matrix R indicates an effective polarization rotation of
3 / 2    x, y  / 2 at each pixel. In other words, the counter-clockwise polarization rotation
at each pixel is found to be   x, y  / 2   / 2 , which depends on the phase loaded onto the
SLM.
The PR setup is also used to calibrate the look up table (gamma curves) for both SLM
panels. By precisely measuring the amount of rotation based on the nulling effect with a
linear analyzer for each gray level, we are able to calibrate the gamma curves so that the
gray level and the actual phase imposed by the SLM are more concisely correlated.
81
2.3. System flow chart
Fig. 7-3. Flow chart of the system. The VOF-Gen System consists of light source, 4
subsystems for control of all the aspects of light, 4-f imaging subsystems and detection
subsystem. PR: Polarization Rotator.
The VOF-Gen consists of the light source, four SLM sections as subsystems to realize the
modulation of all degrees of freedom for arbitrary optical field generation, 4-f imaging
subsystems [34] to relay the optical field from one SLM section to the next or to the
image plane and a CCD camera for detection. The 4-f imaging system is introduced to
minimize the diffraction effect after careful alignment. The SLM sections (subsystems)
are integrated to achieve the full control functionality as described in Fig. 7-3. For the
desired complex optical field that we want to generate, the phase patterns for the SLMs
are calculated based on a collimated Gaussian input beam. Then the phase patterns are
loaded onto the generator and the modulations of phase, amplitude, polarization ratio and
retardation are realized through SLM Sections 1 to 4 as shown in the flow chart,
respectively. At the output plane, the beam is expected to have the desired complex
optical field distribution and can be analyzed thereafter. The 4-f imaging systems are
82
introduced to relay between SLM sections to minimize the diffraction effects. Details of
the modulations and phase pattern calculations are given in the following.
2.4. Modulation of Light
2.4.1.
Phase modulation (SLM Section 1)
Phase modulation can be readily realized as the phase information loaded on the SLM
will be directly imposed on the horizontal component of the reflected beam. This is done
in SLM Section 1 with a horizontally polarized, well collimated Gaussian input beam.
The resulting field can be represented in terms of Jones Vector as:
J1  x, y   e
i1  x , y 
1
E0  x, y    ,
0
(7-6)
where Eo(x, y) is the amplitude of the input field and ϕ1(x, y) is the phase pattern loaded
onto SLM Section 1 of the VOF-Gen. As will be discussed later, the phase ϕ1(x, y) will
contain not only the desired phase ϕd(x, y) according to Eq. (7-2), but also a precompensation phase that are due to the geometric phase effect for the polarization rotator
described in Section 2.2 above.
2.4.2.
Amplitude modulation (SLM Section 2)
Amplitude modulation is achieved by putting a linear polarizer with transmission axis
oriented along horizontal direction after a PR setup utilizing the second SLM section. For
horizontally polarized input field defined in Eq. (7-6), the resulting output field can be
represented in Jones Vector form as:
83
J 2  x, y   e
  x, y 

i  1  x , y  + 2
+ 
2


   x, y  
1
sin  2
E
x
,
y
  ,
 0
2
0


(7-7)
where ϕ2(x, y) is the phase pattern for SLM Section 2. Equation (7-7) shows that
amplitude modulation can be achieved with the sine function while the output is still
horizontally polarized. For ϕ2(x, y) = 0 radian, 0 amplitude can be obtained and unit
amplitude is expected for ϕ2(x, y) = π. Recall the general expression for desired field as
shown in Eq. (7-2). Compared to the definition for the desired field, ϕ2(x, y) can be given
via the following expression:
2  x, y   2 sin 1  Ad  x, y   ，
2.4.3.
(7-8)
Polarization ratio modulation (SLM Section 3)
As described in Eq. (7-7), the output field of SLM Section 2 is horizontally polarized.
Using another PR setup consisting of the third SLM section, the SOP at each location can
be linearly rotated to any direction prescribed by the local phase pattern to realize the
desired polarization ratio distribution between the x- and y- polarization components.
Assuming the phase pattern for SLM Section 3 is ϕ3(x, y), we have the output field of
SLM Section 3 given by:
84
J 3  x , y   E0  x , y  e
  x , y  3  x , y  

i  1  x , y   2

 
2
2



 3  x, y    
cos
 


2
2 
   x, y   

，
sin  2


2
   x, y    


sin  3
 

2
2  


(7-9)
Similarly, ϕ3(x, y) can be found from the desired field distribution given by Eq. (7-2):
 E yd  x, y 
3  x, y   2 tan 
 E xd  x, y 

1
2.4.4.

 


，
(7-10)
Phase retardation modulation (SLM Section 4)
Phase retardation can be introduced by directly shining the linearly polarized output field
of Section 3 as shown in Eq. (7-9) to the last SLM section due to the birefringence nature
of the LC molecules. Assuming the phase pattern is ϕ4(x, y) for SLM Section 4, the final
output field of the VOF-Gen can be written as:
J 4  x , y   E0  x , y  e
  x , y  3  x , y  

i  1  x , y   2

 
2
2



 3  x, y    i4  x , y  
cos
 e



2
2
 2  x , y   


,
sin 


2
 3  x, y   



 
sin 


2
2



85
(7-11)
where
4  x, y    d  x, y  ，
(7-12)
as given by the desired field distribution Eq. (7-2).
As we previously discussed, the first SLM section is responsible for the phase modulation.
As we can see in Eq. (7-11), the phase of the final output will have the following
expression:
output  x, y   1  x, y  
2  x, y  3  x, y 
2

2

，
(7-13)
Additional phase information is acquired throughout the steps of amplitude and
polarization ratio modulations due to the geometrical phase effects arising from the two
PRs used in the setup. Therefore, in order to correctly generate the desired phase in the
final output, ϕ1(x, y) must contain both the desired phase information ϕd(x, y) and a precompensation phase that compensates the accumulated geometrical phases. By equating
ϕoutput(x, y) to ϕd(x, y), we have:
1  x, y   d  x, y   c  x, y  ,
(7-14)
where the pre-compensation phase c  x, y   2  x, y  / 2  3  x, y  / 2   . The
verification of the need for this phase pre-compensation will be shown in Section 4.3.
Note that the phase patterns ϕ1(x, y), ϕ2(x, y), ϕ3(x, y) and ϕ4(x, y) can all be spatially
inhomogeneously distributed. For any desired output field with arbitrary spatial
86
distributions of phase, amplitude and polarization, Eqs. (7-8), (7-10), (7-12) and (7-14)
can be used to calculate the required phase patterns. By loading the phase patterns onto
each of the SLM sections of the VOF-Gen, arbitrarily complex desired output field can
thus be generated.
3. Experimental setup
3.1. Experimental setup of the generator
From the discussions above, in general four SLMs would be needed in order to fully
control all of the degree of freedoms to create an arbitrarily complex optical field.
However, taking the advantage of the HDTV format of the Holoeye HEO 1080P SLM, in
our VOF-Gen setup two SLM panels are used with each of the SLM panel divided into
two halves. Each half of the SLM panels is used to realize the control of one degree of
freedom. This architecture utilizes the high resolution of the SLM panel while keeps the
complexity of the experimental setup is kept manageable.
87
Fig. 7-4. Schematic diagram of the VOF-Gen.
88
Fig. 7-5. Experiment Setup of the VOF-Gen.
The schematic diagram and actual experiment setup of the VOF-Gen are shown in Fig.
7-4 and Fig. 7-5, respectively. He-Ne laser of 632.8 nm wavelength is used as the input.
Polarizer P1 and half wave plate λ/2 are used in combination to adjust input Gaussian
beam intensity. Non-polarizing beam splitters (NPBSs) are used to properly direct the
beam, thanks to its insensitivity of the polarization direction of the input beam. The SLM
panels are divided into 4 sections, as shown in Fig. 7-4. The input beam first incidents on
SLM Section 1 where phase modulation can be directly obtained. Lens L1 and Mirror M1
are used as a 4-f system. The optical field at the SLM surface in SLM Section 1 can be
relayed to SLM Section 2 by carefully controlling the distances from the SLM to L1 and
from L1 to M1 to be both equal to the focal length of L1, which is 300 mm. The QWP
λ/4 combined with SLM Section 2 works as a PR discussed in Section 2.2. Amplitude
modulation is achieved in this SLM section by using the PR setup and a polarizer P2
89
while the output beam is still polarized horizontally. The second 4-f system comprised of
lenses L2 and L3 is used to image the optical field from SLM Section 2 to SLM Section 3,
where polarization rotation is obtained with a second PR setup. Lens L4 and Mirror M2
work as another 4-f system to relay the field at SLM Section 3 to SLM Section 4.
Retardation is added to the optical field after being reflected from SLM Section 4. Finally,
lenses L5 and L6 are used to relay the field from SLM Section 4 to the Detector (LBAFW-SCOR by Spiricon) as the last 4-f imaging system in the entire VOF-Gen.
Fig. 7-6. Driver circuits for the color channels of the VOF-Gen system. The upper one is
the red channel responsible for the control of SLM1 and the lower one is the green
channel for SLM 2.
As previously mentioned, the VOF-Gen consists of two SLM panels as shown in Fig. 7-6.
Therefore a simultaneous and independent control of both panels is required. This is
realized through a color channel coding scheme such that the phase pattern for SLM 1 is
coded into the green color (green channel) while the pattern for SLM 2 is coded in the red
color (red channel). Then the two colors are combined to generate a color image as the
90
overall phase pattern as shown in Fig. 7-4. Note that we have divided each SLM panel
into two halves. Therefore the control signal (overall phase pattern) is also multiplexed
spatially into right and left halves to control the left and right sections of both SLMs,
respectively. Thus the entire VOF-Gen can be operated with one computer that is capable
of outputting 1920 x 1080 resolution color graphics. In order to generate arbitrary beams,
diffraction effects have to be taken into consideration. The diffraction needs to be
minimized so that sharp edges or high frequency information in phase, amplitude and
polarization can survive. This is achieved by the four 4-f imaging systems used in our
setup. Spatial filters SF1 and SF2 located in the Fourier planes of the 4-f systems are used
to suppress the interference caused by bulk cube beam splitters. Opaque cardboards
(shown as black bars between NPBSs in Fig. 7-4) are placed to block the direct
illumination. Before proceeding to the complex optical field generation, gamma curves of
both SLM panels need to be calibrated for accurate phase generation. 4-f systems also
need to be well aligned to minimize the diffraction effects.
3.2. Gamma Curve Calibration Using Polarization Rotator
In order to calibrate the gamma curve of the SLM, a calibration setup is proposed to
perform the fine measurement of the actual phase-grey level relationship. The calibration
setup consists of a PR, a beam splitter (BS), a polarizer with transmission axis at angle θ
with respect to horizontal direction and a detector as depicted in Fig. 7-7.
91
Fig. 7-7. Calibration setup consists of a PR and a polarizer.
The input beam is horizontally polarized (x-polarized) and it goes through the PR setup.
The beam goes through a polarizer after passing through the BS and is collected by the
detector. The polarizer is rotated by angle θ so that the transmission is minimized, which
is the so-called nulling effect. For grey level X, based on the property of the PR the actual
phase value imposed by the SLM pixel δ(X) can be found by
 

  X   2    X      2  X  ，
2 2

(7-15)
where θ(X) is the amount of rotation of the polarizer as a function of grey level X.
The same calibration procedure is performed for many grey levels and a precise
calibration of the gamma curve can then be realized by generating the one to one
mapping of actual phase-grey level relationship. For any phase value to be generated by
92
the SLM, a grey level is found by interpolating the phase value in the nonlinear
relationship as opposed to using the linear relation: Grey _ level   mod  , 2   255 .

2

3.3. 4-f imaging system and its alignment procedure
In order to generate arbitrary beams, diffraction effects have to be taken into
consideration. The diffraction needs to be minimized so that sharp edges or high
frequency information in phase, amplitude and polarization can survive. In our setup,
four 4-f imaging systems [34] are used as discussed previously.
In order to test the resolution of our system, the alignment of the 4-f imaging system is
crucial in our experiment as shown in Fig. 3-1.
In a well aligned 4-f imaging system, the image is an inverted replica of the object
without any diffraction or any additional phase introduced. Thanks to the 4-f systems, the
optical field at the SLM surface in SLM Section 1 with the phase modulation is imaged to
SLM Section 2, adding the amplitude modulation, then to SLM Section 3 with
polarization rotation, to SLM Section 4 with retardation information, and finally to the
camera without the introduction of diffraction.
Since the input beam is well collimated in our system, a shearing interferometer can be
used as a collimation checker to align the 2f distance between the two lenses. Once the
output beam is well collimated based on the collimation checker, the 2f distance can be
precisely determined.
In order to align the entire 4-f imaging system, mostly the f distance before the first lens
and f distance after the second, we designed a pattern with fine features and used the
93
CCD camera to resolve the pattern as a figure of merit for the alignment. In our
experiment, 3 horizontal lines crossed with 3 vertical lines are generated where each of
the lines is only 100 μm wide shown in Fig. 7-8.
Fig. 7-8: Pattern with fine features for 4-f imaging system alignment. Each line is 100 μm wide.
The alignment is performed in a sequential order, as shown by the arrows in Fig. 7-9. The
first 4-f system needs to be aligned is the one between CCD and SLM Section 4, where
the retardation modulation is achieved. A beam is designed so that the pattern with fine
features as shown in Fig. 7-8 is polarized in RCP while the rest of the beam polarized in
LCP. The longitudinal distances in the 4-f system are adjusted until the sharpest pattern is
detected by the CCD camera in front of which a circular analyzer is used to resolve the
RCP component. At this point the first 4-f imaging system is well aligned. Then the
alignment needs to be performed for each following 4-f imaging system in a similar way
where different sets of patterns with fine features and the complementary are generated.
The patterns are designed so that the degree of freedom to be realized in the specific SLM
94
section can be revealed. Then the 4-f imaging system’s alignment is determined by
adjusting the distances to obtain the maximum sharpness where diffraction is eliminated.
Fig. 7-9: 4-f imaging system alignment procedure
The results for the alignment are shown in Fig. 7-10 below.
Fig. 7-10: Test results for (a) amplitude modulation: the intensity is directly captured by
the CCD camera; (b) polarization rotation: the intensity is captured after a linear polarizer;
(c) Retardation: the intensity is captured after a circular analyzer.
In all cases, the rectangular dark areas of 100 μm x 100 μm are visible with fairly sharp
edges. This shows that the 4-f systems are well calibrated and the VOF-Gen is free of
diffraction.
95
To test the 4-f imaging subsystems, ring and “EO” logo patterns are generated by the
VOFGEN system without and with the 4-f imaging systems as shown in the left and right
parts of Fig. 7-11, respectively. As we can see without the 4-f imaging system, the field
captured by the CCD camera is blurred or smoothed out, where the high frequencies are
not present. This is an indication of the diffraction effect. By introducing and aligning the
4-f imaging subsystems, the diffraction effects have been eliminated and sharp edges are
observed with the help of the reconstruction from the 4-f imaging systems.
Fig. 7-11: Comparison of the same patterns generated by VOF-Gen system without and
with well aligned 4-f imaging systems.
96
4. Experimental results
4.1. Spatially variant phase modulation: vortex generation
We first demonstrate the phase modulation capability with the generation of optical
vortex beams. Optical vortices with spiral wavefront carry orbital angular momentum
(OAM). If the phase of the beam has an azimuthal dependence of exp(ilφ), where φ is the
azimuthal angle, then it’s said to have an OAM of lħ or a topological charge of l. Vortex
beam with topological charge l in the far-field will have Laguerre Gaussian distribution
of LG0,l. The mathematical representation of a general Laguerre Gaussian beam LGp,l can
be shown as:
l
2
w0 
r  l 
r2 
E p ,l  r ,  , z  
2
 L p  2 2

w  z   w2  z  
w
z




2
 
kr 2 
r 2  , (7-16)
exp   j  kz   pl  z   l 



2 R  z   w2  z  
 
where
2
 z 
 z 
 w02
z2
 pl  z    2 p  l  1 tan   , z0 
, w  z   w0 1    and R  z   z  0 .

z
 z0 
 z0 
1
Lpl is the associated Laguerre polynomials and satisfies the following equation:
x
d 2 Llp  x 
dx
2
 (l  1  x )
dLlp  x 
97
dx
 pLlp  x   0 .
(7-17)
Here p gives the number of nodes in radial direction and l gives the topological charge of
the phase pattern. For the case where p = 0, the associated Laguerre polynomials reduces
to 1. Further assuming z = 0 (no propagation), Eq. (7-16) can then be simplified as:
l
2
 r 

r2 
E0,l  r , ,0    2 2  exp   j  kz  l   2  ，
w0 
 w0 

2
(7-18)
Since phase cannot be directly measured using a camera without resort to interferometric
method, we first generated optical beams with pure spiral phase in order to demonstrate
the phase modulation capability. Here only the first SLM section is controlled with a
spiral phase and the phase patterns for the rest of VOF-Gen remain flat. The generated
field is focused by a lens and the intensity is recorded at the focal plane by the CCD
camera. Vortex beams with topological charges 1, 10 and 15 are generated here and the
far-field intensities are as shown in Fig. 7-12. As we can see, LG0,1, LG0,10 and LG0,15 are
observed in the focal plane of the lens. As the topological charge increases, the size of the
dark center also increases according to the property of Laguerre Gaussian beams as
described in Eq. (7-18) for LG0,l modes. This phenomenon can be understood as the
larger phase singularity (higher topological charge) will inevitably lead to a larger dark
center. This example shows the phase modulation capability of our VOF-Gen and we are
able to generate vortex beams with topological charge up to 15 with good beam quality
and integrity.
98
Fig. 7-12. Pure phase modulation with spiral phase of topological charge 1 (upper), 10
(middle) and 15 (lower). The images are captured in the focal plane of 75-mm planoconvex lens.
A weak central spot is also observed and it may be caused by the direct reflection due to
the filling factor, finite pixel size and level of quantization. The relative amplitude of the
central spot is more pronounced for higher topological charges due to the enlarged ring
area.
4.2. Spatially variant amplitude modulation
In order to demonstrate the functionality of amplitude modulation, we designed an “EO”
logo binary amplitude pattern. SLM Section 2 is loaded with “EO” shaped pattern for 100%
99
transmission and zero for the rest of the window. SLM Section 1 is loaded with a precompensation phase and SLM Section 3 and 4 both have flat phase. The output is directly
captured by the CCD camera shown in Fig. 7-13 along with the phase pattern loaded onto
VOF-Gen. It shows that fine features such as sharp edges in amplitude modulation are
well preserved in the output beam. Note that the entire window would have been
illuminated by the input Gaussian beam without the amplitude modulation. The result
shows that fine features such as sharp edges in amplitude modulation are well preserved
in the output beam.
100
Fig. 7-13. x-polarized "EO" logo coded in amplitude (upper) and its phase pattern (lower).
101
However, as one can see, the interference pattern has not been entirely removed. This is
due to the fact that reducing the pinhole size in the Fourier plane of the 4-f system will
remove both the interference pattern and high frequency components. In order to
maintain a high resolution, we balanced the pinhole size so that high frequency terms are
well preserved while only allowing a minimum amount of interference pattern.
4.3. Spatially variant polarization rotation: radially polarized beam
Cylindrical Vector (CV) beams are a group of beams whose spatially variant SOP
possesses cylindrical symmetry [1]. Due to the unique properties when focused by a high
NA objective [15], there has been a great increase in the research of the CV beams
recently, which has led to applications in high-resolution imaging [7], plasmonic focusing
[2] and particle manipulation [61]. Numerous approaches have been proposed to generate
such beams, including both active [62], [63] and passive methods [64], [59].
We illustrate the capability of using our generator to create these CV beams through the
generation of a radially polarized beam, one important subset of CV beams. Essentially
the CV beams such as the radial polarization can be generated with the polarization
rotation function of the SLM Section 3. A horizontally polarized input beam can be
locally rotated pixel-by-pixel to the desired polarization direction. However, as we
pointed out in Eq. (7-5), the spatially variant polarization rotation introduces an
additional geometric phase. It is important to pre-compensate this additional phase by the
phase pattern for SLM Section 1, as shown in Eq. (7-14).
When radially polarized beam is focused by low NA lens, a doughnut distribution will be
resulted in the focal plane owing to the polarization singularity at the center. In order to
102
generate radially polarized beam, the phase pattern for SLM Section 3 ϕ3(x, y) will have
an azimuthal dependence of 2φ according to Eq. (7-10), where φ is the azimuth angle.
Based on Eq. (7-11), we know that the extra phase to be introduced by the following
modulations will carry a phase with azimuthal dependence of φ, in other words, a spiral
phase with topological charge l = 1. In order to generate the radial polarization with flat
phase, a spiral phase with topological charge l = -1 needs to be incorporated in the precompensation phase. Therefore, to verify the phase pre-compensation, we generate
radially polarized beams without and with pre-compensation phase. The required phase
patterns for radial polarization without and with phase pre-compensation are shown in the
upper and lower parts of Fig. 7-14, respectively. Then the far field intensities are captured
at the focal plane of a lens, as shown in Fig. 7-15.
103
Fig. 7-14. Phase patterns for radially polarized beams without (upper) and with (with)
pre-compensation. For the latter phase pattern, the variation in green color indicates the
pre-compensation phase.
104
Fig. 7-15. Focused field by a 75-mm plano-convex lens. (a) Radially polarized beam
without proper phase cancellation; (b) radially polarized beam with phase cancellation.
In the phase pattern for the radially polarized beam with pre-compensation phase, the
introduction of and variation in green color indicate the existence of the precompensation phase and its spiral nature as shown in Fig. 7-14. From Fig. 7-15(a), we
can see that without proper phase pre-compensation, a bright spot is obtained when such
beam is focused. This can be understood as the additional spiral phase cancelled the
polarization singularity at the center of the focused radially polarized beam. Once precompensation phase is introduced a doughnut distribution is achieved as expected (shown
in Fig. 7-15(b)). This confirms that the geometrical phase generated due to the operation
of SLM Section 3 is successfully compensated. In general, the phase pre-compensation
scheme can be used to compensate any additional phases that are introduced in the
following modulation steps from SLM Sections 2, 3 and 4, as shown in Eq. (7-14).
Moreover, this confirmation of the phase pre-compensation also serves as another
evidence of the phase modulation capability discussed in Section 4.1.
105
The generation of radially polarized beam is then shown in Fig. 7-16. The arrows in the
upper set of graphs indicate the directions of the linear analyzer in front of the camera
and the intensity of each linear polarization component (0°, 45°, 90° or 135°) is shown
respectively. The polarization map is given in the lower graph that is calculated based on
partial Stokes parameter measurement of S0, S1 and S2. The orientation of the lines
indicates the local polarization direction while the length of lines indicates the local
intensity. As shown in the figure, radial polarization in the output field is generated. Thus
the polarization rotation capability is demonstrated.
106
Fig. 7-16. Radially polarized beam generated by the VOF-Gen. Upper graphs show the
fields after a polarizer with polarization axis orientation indicated by black arrows at 0°,
45°, 90° and 135°, respectively; Lower graph shows a polarization map of radially
polarized beam overlapped with the intensity distribution and the local polarization
directions are indicated by the bars.
107
4.4. Spatially variant phase retardation
Right-hand circular polarization (RCP) and left-hand circular polarization (LCP) have
phase retardation of +π/2 and -π/2, respectively. To demonstrate the capability of optical
field with spatially variant phase retardation, we designed patterns with zero phase, unit
amplitude modulation and spatially patterned retardation distribution. In the first example,
we designed the beam where the “EO” logo is polarized in RCP while the rest of the
window polarized in LCP. The required phase pattern for VOF-Gen and the total field as
well as the RCP and LCP components are shown in and Fig. 7-17 and Fig. 7-18,
respectively.
Fig. 7-17. Phase pattern for "EO" logo generation VOF-Gen.
108
Fig. 7-18. "EO" logo coded in circular polarization. The total field (left), the EO logo
(upper right) in RCP and the complimentary “EO” logo (lower right) in LCP.
In the second demonstration, we designed the Taiji pattern with one half coded in RCP
and the other in LCP. The required phase pattern and the output beam are shown in Fig.
7-19 and Fig. 7-20, respectively.
109
Fig. 7-19. Phase pattern for Taiji pattern generation.
Fig. 7-20. Taiji pattern coded in circular polarization. The total field (left), the upper half
(upper right) in RCP and the lower half (lower right) in LCP.
110
From the results shown in Fig. 7-18 and Fig. 7-20, it shows that the polarization
retardation modulation can also be achieved using our VOF-Gen. At this point, the
modulation of each individual degree of freedom in describing an optical field has been
successfully demonstrated. In the next two examples, we would like to generate optical
fields that require controls in different aspects of light.
4.5. Stokes parameters measurement and complex vectorial optical field
generation with multiple parameters
4.5.1.
Stokes parameters and its measurement
Arbitrary optical field can be represented in Jones vector form as shown in Eq. (7-2),
where the polarization can also be expressed using elevation angle and ellipticity,
 cos  
SOP  
 at  ,
sin



(7-19)
where α is the elevation angle and ε is the ellipticity. The conversion between the
elevation angle and ellipticity form and the polarization rotation and retardation form can
be found in Eq. (7-4). As previously discussed, Stokes parameters are all measurable
quantities (intensities), which is an alternative of the SOP representation. Stokes
parameters are given as
2
2
 S 0  I x  I y  E0 x  E0 y
 S I I E 2 E 2
x
y
0x
0y
 1
 S 2  I   I   2 E0 x E0 y cos 


4
4
 S  I  I  2 E E sin 
R
L
0x
0y
 3
111
，
(7-20)
For normalized Stokes parameters, we have
S0  1

 S  cos 2 cos 2  cos 2 

1

 S 2  cos 2 sin 2  sin 2  cos 
 S3  sin 2   sin 2  sin 
(7-21)
，
Both polarized and partially polarized light can be characterized by the Stokes parameters
in terms of degree of polarization p, given as:
p
S12  S 22  S32
S0
(7-22)
，
For partially polarized light, the Stokes parameters can be shown as:
 S0   pS 0   1  p  S0 

S   S  
0
1
1
S S

S  
P
UP

 S2   S2  
0

  
 
S
S
0
 3  3  

，
(7-23)
In order to compensate the absorption from the quarter-wave plate, the quarter-wave plate
and polarizer are used in combination to perform the full Stokes Parameters measurement
as shown in Fig. 7-21. Here θ is the angle between the polarizer’s transmission axis and
the horizontal direction and φ is the angle of the quarter-wave plate’s fast axis and the
horizontal direction.
112
Fig. 7-21. Full Stokes parameters measurement setup.
The Stokes parameters can be measured in terms of I(θ, φ) as:
 S 0  I  0,0   I  90,90 

 S1  I  0,0   I  90,90 

 S 2  2 I  45, 45   S 0

 S3  2 I  45,0   S 0
，
(7-24)
Then Eq. (7-21) can be used to calculate the local SOP via either α and ε or χ and ϕ at the
cross section of the beam provided that the intensities are well aligned as the polarizer
and QWP are rotated.
113
4.5.2.
Experimental results for complex vector field generation with
multiple parameters
At this point, the modulation of each individual degree of freedom in describing an
optical field has been successfully demonstrated. In the next two examples, we would
like to generate optical fields that require controls in different aspects of light. First we
propose a ring structure with the local SOP continuously varying from azimuthal to radial
direction as the radius increases. The desired ideal field and the required phase pattern are
shown in Fig. 7-22 and Fig. 7-23, respectively. The experimental result is shown in Fig.
7-24. The filed distribution after polarizer at 0°, 45°, 90° and 135° and the total field
overlapped with SOP map are shown in the upper and lower portions of Fig. 7-24,
respectively. It clearly shows the gradual change from azimuthal to radial for local SOP.
The weak pattern in the center part of the output beam is due to the interference.
Fig. 7-22. The ideal vector field where the SOP varying from azimuthal to radial as
radius increases.
114
Fig. 7-23. The phase pattern for VOF-Gen.
115
Fig. 7-24. Ring structure with SOP continuously varying from azimuthal to radial
direction as radius increases. The upper set of graphs show the field components along 0°,
45°, 90° and 135° respectively and the lower graph shows the measured field overlapped
with polarization map.
116
In the next example, we generate a double ring pattern where the inner ring is azimuthally
polarized with 0.5 amplitude while the outer ring is radially polarized with unit amplitude.
The ideal field and the required phase pattern for VOF-Gen are shown in Fig. 7-25 and
Fig. 7-26. The linear polarization components along 0°, 45° and 90° are shown in the
upper set of graphs in Fig. 7-27 and the polarization map is given in the lower part of Fig.
7-27. As we can see in the figure, the outer ring has a higher intensity than the inner one
as opposed to Gaussian illumination and the polarization distribution follows the design.
Fig. 7-25. The ideal field for double ring pattern where inner ring has azimuthal
polarization and outer ring has radial polarization.
117
Fig. 7-26. The phase pattern for VOF-Gen.
118
Fig. 7-27. Double ring structure with amplitude and polarization rotation modulation:
(upper) the linear polarization components with polarizer at 0°, 45° and 90°; (lower) the
polarization map.
As another demonstration, complex vector optical field with local polarization elevation
angle along radial direction and constant ellipticity π/10 is designed. In other words, the
SOP at each location is elliptical with constant ellipticity and the major axis of the ellipse
is always along the radial direction. The ideal field distribution with polarization map, the
required phase pattern for VOF-Gen and the experimental result with the histogram of the
119
ellipticity (in unit of π radian) generated experimentally are shown in Fig. 7-28, Fig. 7-29
and Fig. 7-30, respectively. In this case, the full Stokes parameter measurement of S0, S1,
S2 and S3 is performed to reveal the spatial distribution of the SOP of the generated
beam. It can be shown that the experiment results generally agree with the design. The
histogram of the ellipticity peaks around 0.1π, which shows the generation of the
designed ellipticity. At some points the local SOP is slightly different from the expected
and we think it is due to the fact that the interference patterns with vertical polarization
coming from the SLM surface will change the local SOP and the vibration also affects
the accuracy of the full Stokes parameters measurement.
Fig. 7-28. Ideal field distribution with polarization map.
120
Fig. 7-29. Phase pattern for VOF-Gen.
Fig. 7-30. Experimental results (left) for optical field with constant ellipticity and
elevation angle along radial direction as well as the histogram (right) of the ellipticity in
the unit of π.
121
4.6. Realization of full Poincaré beam and the singularities
As a continued research from Chapter 4, an example of the full Poincaré beam is shown
in Fig. 7-31. In the cross section of the beam profile, the elevation angle follows the
radial direction while the ellipticity varies from π/4 to –π/4 so that the SOP changes from
RCP to LCP from the center to the edge of the beam. In the polarization map of Fig. 7-31,
the blue color indicates positive ellipticity while the black indicates negative ellipticity.
Fig. 7-31. Simulation of FP beam with polarization map superimposed.
In the radial direction, from the center to the edge, the polarization evolves from RCP to
LCP, which covers from the north pole to the south pole on the Poincaré sphere. The
elevation angle α ranges from 0 to 2π, which translates to a coverage of 4π for the
122
azimuth angle 2α in the Poincaré sphere as shown in Fig. 4-3. Therefore, the SOP on the
cross section of the beam covers the entire Poincaré sphere two times.
Using the VOF-Gen, we are able to generate such FP beam experimentally. The intensity
profile as well as the polarization map is shown in Fig. 7-32. Two singularity points were
discovered and highlighted by green circles in the figure.
Fig. 7-32. Experimental result for FP beam with polarization map superimposed. Two
singularities are highlighted by green circles.
The singularities can be understood by investigating the phase pattern used for the
generation of such beam. It can be shown that the phase retardation shown in Fig. 7-33
has four vortex-like singularity points at the mid-points on both the horizontal and the
123
vertical direction, where the SOPs are horizontal and vertical polarizations. However, due
to the fact that the SLM only responds to the horizontal polarization, the retardation
singularities on the vertical direction are not reflected in the generated beam. Therefore,
only the retardation singularities on the horizontal direction affect the beam as shown
experimentally. The phase pattern used for the VOF-Gen generation is shown in Fig.
7-34. Numerical simulation of such beam also confirms the beam singularities. The
diffraction pattern is shown in Fig. 7-35 after a propagation distance of 1 mm.
It has been found that the two is the least number of singularities for different approaches
of generating such FP beam using VOF-Gen, which coincides with how many times the
SOP on the cross section of the beam covers the entire Poincaré sphere.
Fig. 7-33. Phase pattern for retardation modulation for the FP beam.
124
Fig. 7-34. The phase pattern for the generation of the FP beam.
Fig. 7-35. Diffraction pattern after propagation distance of 1 mm.
125
5. Summary
We reported a novel and versatile vectorial optical field generator (VOF-Gen) that is
capable of generating arbitrary optical fields by spatially modulating all aspects of optical
field (including phase, amplitude and polarization) on a pixel-by-pixel basis. Various
complex vector fields are generated and tested to demonstrate the functionality and
flexibility of the proposed VOF-Gen. To the best of our knowledge, it is the first
successful experimental demonstration of a beam generation system with high spatial
resolution that is capable of tailoring all the aspects of optical fields. This arbitrary
complex optical field generator may find extensive applications in areas where exotic
input fields are required, such as particle manipulation and beam shaping.
126
CHAPTER 8
CONCLUSIONS AND FUTURE WORK
VOF-Gen is discussed in details and successfully demonstrated as a promising arbitrary
vectorial complex optical field generator. Exciting results have been obtained and
discussed which show the capability of arbitrary phase, amplitude and polarization
modulation on a pixel by pixel basis. If a single piece device can be made to replace the
optical components of the VOF-Gen, the alignment can be greatly simplified, which may
lead to potential commercialization. In addition Labview or other automation technique
can be introduced to realize the automation of the vectorial optical field generation. The
motivations for such a full control beam generation system are also introduced with
examples focused on the modulations of different aspects of light.
As the future work, the experimental verification of optical needle field needs to be
achieved with the antenna based complex optical filter design. The experimental setup to
generate optical needle field is proposed here shown in Fig. 8-1. In order to generate
desired radially polarized field as the desired pupil plane field for needle field generation,
azimuthal input polarization is required for the complex filter (shown in green) due to the
127
fact that the amplitude and phase of the cross-polarized component are modulated. Recall
that the circular polarization can be decomposed into uniform distribution of radial and
azimuthal polarization with a spiral phase. Therefore, in order to generate azimuthal
polarization, a circular polarizer (CP), a spiral phase plate and an azimuthal analyzer are
used in the setup. After the complex filter, a radial analyzer is used, only allowing the
cross-polarized component to go through. Then the beam is expanded to match the NA of
the objective and focused by the high NA objective. A near-field scanning optical
microscope is proposed to perform the detection of the generated optical needle field.
Fig. 8-1. Experiment setup for linear antenna design.
An alternative to generate azimuthally polarized beam is to use fiber to perform
automatic mode selection. Careful alignment is required to ensure the generation of
azimuthal (radial) polarization.
128
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VITA
Wei Han was born on April 3rd, 1984 in Hefei, Anhui Province, People’s Republic of
China, the son of Jie Han and Zhongping Wang. After completing his high school at
Hefei No. 1 High School, he entered Nanjing University for undergraduate study majored
in Electrical Engineering in 2002. He graduated in 2007 with a bachelor’s degree and
came to the Electro-Optics program at the University of Dayton for graduate study. He
received his master’s degree in 2009 under the supervision of Dr. Joseph W. Haus. He
received his Ph.D. degree in August 2013 under the supervision of Dr. Qiwen Zhan. His
research is focused on the engineering of complex optical fields and its applications.
137