Sub-Diffraction Imaging Using Superoscillatory Electromagnetic Waves by Alex Man Hon Wong A ...

164 Pages·2014·8.3 MB·English

Sub-Diffraction Imaging Using Superoscillatory Electromagnetic Waves
by
Alex Man Hon Wong
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering
University of Toronto
(cid:13)c Copyright 2014 by Alex Man Hon Wong
Abstract
Sub-Diffraction Imaging Using Superoscillatory Electromagnetic Waves
Alex Man Hon Wong
Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering
University of Toronto
2014
This thesis presents an investigation on imaging beyond the diffraction limit with superoscillatory elec-
tromagnetic waves. It begins with a survey on sub-diffraction imaging, which motivates the need for
a new sub-diffraction imaging device which is free from drawbacks in devices which have heretofore
been proposed. This work addresses this need through the design and demonstration of superoscillatory
wave-based focusing and imaging devices. Through establishing a relationship between superoscilla-
tion and superdirectivity, a design procedure is formulated whereby antenna array theory is leveraged
to design 1D and 2D superoscillatory waves. These waves are then physically synthesized or imple-
mented as a filter in focusing and imaging systems at frequencies ranging from microwave to optical.
Experimental characterization of these systems has led to successful demonstrations of sub-diffraction
focusing and imaging, with working distances orders of magnitude farther than most other electromag-
netic wave-based sub-diffraction imaging devices. In particular, the Optical Super-Microscope which
achieves far-field sub-diffraction focusing has attractive merits to become a tool for general-purpose
sub-diffraction optical microscopy. An investigation has also been conducted on superoscillatory waves
in the time domain, which has resulted in the first reported demonstration of a superoscillatory tem-
poral waveform, as well as a demonstrated improvement of radar range resolution beyond the Fourier
transform limit.
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My utmost for His Highest
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Acknowledgements
The completion of this degree is, in many sense, a joint effort of countless people and encounters. I
would like to express my heartfelt gratitude towards some such people who have contributed, directly
or otherwise, to this degree.
Let me begin by thanking my parents, whose love, respect, support and patience have sustained me
throughout my lengthy (but soon-to-end) days as a student. I could not have asked for more from any
earthly parent!
Surely, I must thank my supervisor Prof. George V. Eleftheriades. It has been my privilege and
pleasure to study under his supervision for both my master and doctoral degrees. His inspiration,
integrity, insightful perspectives and timely encouragements provided tremendous help throughout my
degree. I have learnt but a fraction of his knowledge and professionalism through several years of study
under him, but am positive that what I have learnt will serve me in good stead.
I also sincerely thank Prof. Costas D. Sarris and Prof. Sean V. Hum for their guidance as my
thesis committee members: they have provided expert guidance and valuable discussions which directly
contributed to this thesis. The same is true with my defense committee members, which also included
Prof. Piero Trivero and Prof. Adrian Nachman, as well as my external appraiser Prof. Christos G.
Christodoulou. I respectfully thank them for their time and contribution, and for triggering insightful
discussions in both defenses which ultimately improved the quality of my thesis from its state at the
initial defense.
Through several years as a member of the electromagnetics research group, I’ve very much enjoyed
interactions — academic or otherwise — with fellow graduate students, within the EM and photonics
groups and beyond. Particularly, I thank Lo¨ıc Markley and Yan Wang, my co-authors, for invaluable
collaborations on research projects. Lo¨ıc and Yan, along with senior members Ashwin Iyer, Marco
Antoniades, Jiang Zhu and Rubaiyat Islam have provided generous help, from simulation support to
insightful discussions to wise perspectives, especially in the early stage of my degree. I am also thankful
for the company of many colleagues throughout my degree. I’ve started to name some of them in this
acknowledgement, only to realize this space wouldn’t be sufficient to recollect our interactions over the
years. I wish them all the best in their own studies and their endeavors thereafter.
I’m grateful to have worked with undergraduate student Nima Yasrebi over the summer of 2012,
who helped marvelously in building the Eleftheriades optical lab and the Optical Super-Microscope.
I thoroughly enjoyed working with Nima and with other undergraduate students, through teaching,
mentorship and research involvements. I benefit tremendously from these interactions too: they helped
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keep me sane when my research isn’t working!
Last but most importantly I impart praise and thanksgiving to my Lord and Saviour Jesus Christ,
who has helped me learn and grow throughout this degree. His wondrous works often goes beyond my
imagination and comprehension. Nevertheless, I thank His loving and wise bestowment throughout my
degree, and look upon Him for guidance and provision beyond this degree.
With Gratefulness,
Alex M. H. Wong
April, 2014.
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Contents
1 Introduction 1
1.1 Sub-Diffraction Imaging: Historical Perspectives and Motivation . . . . . . . . . . . . . . 1
1.2 Avenues to Sub-Diffraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Slightly Sub-Diffraction Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Evanescent Field Based Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.3 Non-Linear and Non-Electromagnetic Imaging Devices . . . . . . . . . . . . . . . . 7
1.3 Superoscillation and Sub-diffraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Objectives and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Mathematical Preludes 13
2.1 The Plane Wave Formulation of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . 13
2.2 The Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Abb´e’s Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Rayleigh’s Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Summary and Comparison on Diffraction Limits . . . . . . . . . . . . . . . . . . . 18
2.2.4 Angular Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Evanescent-Wave Based Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Schelkunoff’s Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 A Fourier Perspective on Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.3 Superdirectivity and the Spatial Diffraction Limit . . . . . . . . . . . . . . . . . . 24
2.5 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 An Antenna-Based Theory on Superoscillatory Waves 29
3.1 An introduction to Spatial Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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3.1.1 Characterizing Spatial Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 Relating Superoscillations to Superdirectivity . . . . . . . . . . . . . . . . . . . . . 31
3.2 Superdirectivity-Inspired Superoscillation Design . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 A Schelkunoff Approach to Superoscillation Design . . . . . . . . . . . . . . . . . . 32
3.2.2 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.3 Comparing Superoscillation with Superdirectivity . . . . . . . . . . . . . . . . . . . 35
3.2.4 Connection with Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . . 36
3.3 Controlling Sidebands of Superoscillatory Waveforms . . . . . . . . . . . . . . . . . . . . . 37
3.4 Deducing the Limit to Sideband Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Superoscillatory 1D Sub-Diffraction Focusing Devices 45
4.1 Sub-Diffraction-Focused Waveform Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 A Superoscillatory Sub-Diffraction Focusing Screen . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Superoscillatory Sub-Diffraction Focusing in a Waveguide Environment . . . . . . . . . . 52
4.3.1 Waveguide Excitation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.2 Fabrication and Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4.1 Salient Features in Comparison to Evanescent-Wave-Based Devices . . . . . . . . . 62
4.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.3 Comparison to other Superoscillatory Focusing Devices . . . . . . . . . . . . . . . 66
4.4.4 Implementing the Superoscillatory Focusing Screen . . . . . . . . . . . . . . . . . . 66
4.5 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Superoscillatory 2D Sub-Diffraction Imaging: The Optical Super-Microscope 68
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Conceptualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.1 From Focusing to Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.2 Implementing a Superoscillatory PSF . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Designing a 2D Superoscillation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 OTF Implementation with an SLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
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5.5 Constructing the Optical Super-Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.1 Numerical Parameters and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.2 Assembling the Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Calculation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.1 Point Imaging Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.2 Two-Point Resolution Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6.3 Real-Time Imaging for Moving Objects . . . . . . . . . . . . . . . . . . . . . . . . 88
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.7.1 Connection to Super-Resolving Apertures . . . . . . . . . . . . . . . . . . . . . . . 90
5.7.2 Connection to Resolution Restoration Methods . . . . . . . . . . . . . . . . . . . . 90
5.7.3 Comparison to Concurrent Developments . . . . . . . . . . . . . . . . . . . . . . . 93
5.7.4 Imaging in the Presence Superoscillatory Sidebands . . . . . . . . . . . . . . . . . 93
5.7.5 Further Resolution Improvements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.7.6 Extending the Field of View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.7.7 Sub-wavelength Optical Super-Microscopy . . . . . . . . . . . . . . . . . . . . . . . 96
5.8 Summarizing Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 Temporal Superoscillations 98
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 Designing Temporal Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.1 A Design Procedure for Temporal Superoscillations . . . . . . . . . . . . . . . . . . 99
6.2.2 The Rapid Oscillation Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.3 The Sharp Pulse Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Synthesizing Superoscillatory Electromagnetic Waveforms . . . . . . . . . . . . . . . . . . 102
6.3.1 Rapid Oscillation Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.2 Superoscillatory Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Superoscillatory Radar Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.2 Superoscillatory Radar Pulse Design . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4.3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.4 Single Scatterer Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.5 Resolving Two Scatterers in Close Range . . . . . . . . . . . . . . . . . . . . . . . 115
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6.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5 Summarizing Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Conclusion 119
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A The Heisenberg Uncertainty Principle 124
A.1 Defining Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.2 Deriving the Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B The Design of Dolph-Tschebyscheff Superdirective Antenna Arrays 126
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C Calibrating the Spatial Light Modulator 132
C.1 Visibility for Grating Diffraction Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
C.2 Performing SLM Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Bibliography 136
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List of Tables
3.1 Summary of comparison between superdirectivity and spatial superoscillation . . . . . . . 36
4.1 Summary of parameters for the Superoscillatory Sub-Diffraction Focusing Screen . . . . . 49
4.2 Summary of parameters for the SFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 CalculatedcurrentexcitationcoefficientsfortheSuperoscillatorySub-DiffractionFocusing
Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 Summary of parameters for a test 2D superoscillatory waveform. . . . . . . . . . . . . . . 73
5.2 Summary of superoscillatory filter design parameters for OSM. . . . . . . . . . . . . . . . 81
5.3 ComparisonbetweentheOSMandtwoconcurrentsuperoscillation-basedimagingdevices
[85,86]. Inthistable∆w and∆s respectivelydenotethespotwidthandtwo-pointres-
dl dl
olution distance of a conventional, diffraction-limited imaging system. Table annotations
are hereby given. 1. For [86], NA represents the numerical aperture of the superoscilla-
i
tory illumination, which is relevant for evaluating the focal quality, while NA represents
o
thenumericalapertureofthelightcollectioncomponent,relevantforevaluatingtheimag-
ing performance of the device. 2. The term ROI (region of interest) adopts the definition
taken in this work, not the slightly different definition used by [85]. 3. Since an explicit
comparison was not given in [86], the “improvement” figures are derived by comparison
with the calculated diffraction limits of a simple optical system with illumination wave-
length λ and numerical aperture NA. 4. Th resolution improvement is calculated and
quoted as the center-to-center distance between two apertures. . . . . . . . . . . . . . . . 92
6.1 Summary of parameters for the rapid oscillation waveform . . . . . . . . . . . . . . . . . . 102
6.2 Summary of parameters for a designing a superoscillatory sharp temporal pulse . . . . . . 104
6.3 Summary of parameters for a designing a superoscillatory sharp temporal pulse . . . . . . 110
6.4 Singleobjectrangemeasurementusingasuperoscillatoryradarpulseandasincradarpulse115
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