Light Interaction with Human Retinal Photoreceptor:
Finite-Difference Time-Domain Analysis
Amir Hajiaboli
Department of Electrical & Computer Engineering
McGill University,
Montreal, Quebec H3A 2A7, Canada
2008
A thesis submitted to McGill University in partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
©2008 Amir Hajiaboli
Abstract
Approximately 70% of the total sensory receptors in the human body reside in the eye
and they provide 80% of the acquired information during life. The initial stage of
activating these sensory receptors is the light interaction with photoreceptors, which are
the cells covering the retinal surface at the back of the eye. Photoreceptors are different in
physical dimension and also in chemical composition, which defines their role in vision.
While the absorption of the light is governed by the chemical composition inside the
outer-segment of photoreceptors, due to the wave guiding nature of the photoreceptor, the
shape and dimensions of outer-segment may also affect the absorbed spectrum.
To the best of our knowledge, this thesis is the first comprehensive numerical study of the
light interaction with the human retinal photoreceptor to understand the role of diversity
in the physical dimensions of the photoreceptor. Due to the inherent complexity of
experimental studies, the well known finite-difference time-domain (FDTD) method has
been adopted to investigate the optical characteristics of the human photoreceptor.
Previous numerical studies of the photoreceptor were limited to 2-dimensional (2-D)
FDTD analysis due to the higher computational complexity of 3-dimensional (3-D)
simulations. This limitation can affect the number of identified electromagnetic modes
propagating inside the outer-segment. Consequently, although 2-D models offer valuable
insight, they lack in accuracy of physical representation. In this thesis, we conduct a 3-D
FDTD study of the light interaction with the human retinal outer-segment in order to fully
assess its optical properties.
Our results, which are based on a parametric study, show the significant role of
photoreceptor outer-segment morphology and its filtering effect on the spectrum of the
perceived colors.
i
Sommaire
Environ 70% de tous les récepteurs sensoriels dans le corps humain résident dans l'oeil et
ils fournissent 80% d'informations acquises pendant la vie. L'étape initiale d'activer ces
récepteurs sensoriels est l'interaction de la lumière avec les photorécepteurs qui sont des
cellules qui couvrent la surface rétinienne au fond de l'œil. Les photorécepteurs sont
différents dans les dimensions physiques et également en composition chimiques, qui
définit leur rôle dans la vision. Tandis que l'absorption de la lumière est régie par la
composition chimique à l'intérieur du segment externe, dû à la nature du guidage des
ondes électromagnétiques dans le photorécepteur, la forme et les dimensions du segment
externe peuvent également affecter le spectre de la lumière absorbé.
Au meilleur de notre connaissance, cette thèse est la première étude numérique complète
de l'interaction de la lumière avec le photorécepteur humain pour comprendre le rôle de la
diversité dans les dimensions physiques du photorécepteur. En raison de la complexité
inhérente des études expérimentales, la méthode bien connue des différences finies dans
le domaine temporel (FDTD) a été adoptée pour étudier les caractéristiques optiques du
photorécepteur humain. Précédemment, les études numériques du photorécepteur ont été
limitées à des analyses FDTD en deux dimensions (2-D) due à une complexité plus élevé
des codes de calculs en trois dimensions (3-D). Cette limitation peut affecter le nombre
de modes électromagnétiques identifiés propageant à l'intérieur du segment externe du
photorécepteur. En conséquence, bien que les modèles 2-D offrent la perspicacité valable,
ils manquent l'exactitude de la représentation physique. Dans cette thèse, nous
entreprenons une étude FDTD à trois dimensions de l'interaction de la lumière avec le
segment externe de la rétine humaine afin d'évaluer entièrement ses propriétés optiques.
Nos résultats, qui sont basés sur une étude paramétrique, montrent le rôle significatif de
la morphologie du segment externe du photorécepteur et son effet de filtrage sur le
spectre des couleurs perçues.
ii
Table of Contents
Chapter 1
Introduction .................................................................................................................................... 1
1.1 Photoreceptor Optics: Challenges and Solutions .................................................................... 2
1.2 Objective and Motivation........................................................................................................ 2
1.3 Methodology........................................................................................................................... 3
1.4 Dissertation Overview ............................................................................................................ 3
1.5 Original Contribution of the Thesis ........................................................................................ 4
1.5.1 Technical contributions........................................................................................................ 4
1.5.2 Discoveries related to the biophysics of light interaction with photoreceptor ..................... 5
1.6 Contributions Related to Antenna Modeling .......................................................................... 5
Chapter 2
Human Eye: Physiology, Photochemistry and Optical Properties............................................. 7
2.1 Gross Anatomy of the Human Eye and the Retinal Layer ...................................................... 7
2.2 Photoreceptor Structure .......................................................................................................... 9
2.3 Trichromatic Color Vision .................................................................................................... 11
2.4 Photochemistry of Vision ..................................................................................................... 12
2.5 Hyper-Polarization................................................................................................................ 13
2.6 Retina: a Field Sampler......................................................................................................... 15
2.7 Wave-Guiding Effects in Photoreceptor ............................................................................... 16
2.9 Dimensions of Photoreceptors .............................................................................................. 17
Chapter 3
Finite-Difference Time-Domain Technique:
Basics and Special Topics in Modeling Biological Tissues ........................................................ 19
3.1 Full-Wave FDTD Technique ................................................................................................ 20
3.2 Absorbing Boundary Condition ............................................................................................ 21
3.3 Scalar FDTD Technique ....................................................................................................... 22
3.4 Rotationally Symmetric Scalar FDTD .................................................................................. 23
3.5 Total-Field Scattered-Field (TF/SF) Formulation................................................................. 24
3.6 Biological Tissues: Requirements for Detail Modeling ........................................................ 25
3.7 Graphical Processing Unit Programming.............................................................................. 26
3.8 Different Types of Dispersive Material ................................................................................ 27
3.9 Modeling Dispersive Materials in FDTD ............................................................................. 28
Preface to Chapter 4..................................................................................................................... 30
Chapter 4
Photoreceptor Outer-Segment:
2-D vs. 3-D FDTD Optical Waveguide Models .......................................................................... 31
4.1 Introduction and Methodology.............................................................................................. 31
4.2 Averaged-Permittivity Model of the Retinal Cone Outer-Segment ...................................... 32
4.3 Excitation Technique ............................................................................................................ 33
4.4 Simulation Parameters .......................................................................................................... 34
4.5 Monochromatic Excitation.................................................................................................... 35
4.6 White Light Excitation.......................................................................................................... 37
4.7 Conclusion ............................................................................................................................ 38
Preface to Chapter 5..................................................................................................................... 41
Chapter 5
FDTD Analysis of Retinal Photoreceptor Outer-Segment:
Effect of Stacking Nano-Layers of Membrane and Cytoplasm................................................ 42
5.1 Introduction........................................................................................................................... 42
5.2 Model of Photoreceptor Outer-Segment ............................................................................... 43
5.3 FDTD Simulation Parameters............................................................................................... 45
5.4 Results and Discussion ......................................................................................................... 47
5.5 Future Work.......................................................................................................................... 47
Preface to Chapter 6..................................................................................................................... 52
Chapter 6
FDTD Analysis of Light Propagation in the Human
Photoreceptor Cells ...................................................................................................................... 53
iii
6.1 Introduction........................................................................................................................... 53
6.2 Outer-Segment Physical Structure ........................................................................................ 55
6.3 Simulation Parameters and Results ....................................................................................... 56
6.5 Conclusion ............................................................................................................................ 61
Preface to Chapter 7..................................................................................................................... 63
Chapter 7
Human Retinal Photoreceptors: Electrodynamic Model of Optical Micro-Filters ................ 64
7.1 Introduction........................................................................................................................... 64
7.2 Outer-Segment Physical Structure ........................................................................................ 67
7.3 Simulation Parameters .......................................................................................................... 68
7.4 Electric Field in the Outer-segment ...................................................................................... 69
7.5 Irradiant Flux Calculation ..................................................................................................... 73
7.6 Conclusion ............................................................................................................................ 76
Preface to Chapter 8..................................................................................................................... 77
Chapter 8
Rhodopsin Absorption Spectrum:
Effect of the Photoreceptor Outer-Segment Morphology ......................................................... 78
8.1 Introduction........................................................................................................................... 78
8.2 Outer-Segment Physical Structure ....................................................................................... 79
8.3 Rhodopsin Absorption Spectrum .......................................................................................... 80
8.4 Simulation Parameters .......................................................................................................... 81
8.5 The Transmitted and Absorbed Electric Field ...................................................................... 82
8.6 Conclusion ............................................................................................................................ 84
Chapter 9
Conclusion and Future Work...................................................................................................... 85
9.1 Conclusion ............................................................................................................................ 85
9.2 Future Work.......................................................................................................................... 86
Appendix I
3D FDTD Source Code for Modeling
Photoreceptor Using Acceleware Software ................................................................................ 88
A1.2: Validation of the FDTD software ................................................................................... 108
Appendix II
MATLAB Source Code for Rotationally Symmetric Scalar FDTD Formulation ................ 111
Preface to Appendix III.............................................................................................................. 115
Appendix III
FDTD Sub-Cell Modeling of the Inner Conductor of the Coaxial Feed:
Accuracy and Convergence Analysis ........................................................................................ 116
A3.1 Introduction...................................................................................................................... 116
A3.2 Antenna Structure ............................................................................................................ 117
A3.3 Sub-Cell FDTD Modeling................................................................................................ 118
A3.4 Feeding Structure and Excitations.................................................................................... 119
A3.5 First Category: the Coaxial Inner Conductor not Extended into UPML .......................... 120
A3.6 Second Category: Extending the Coaxial Inner Conductor to UPML ............................. 121
A3.7 Convergence Analysis...................................................................................................... 125
A3.8 Conclusion ....................................................................................................................... 126
Preface to Appendix IV.............................................................................................................. 127
Appendix IV
Analysis of a Singly-Fed Circularly Polarized Electromagnetically
Coupled Patch Antenna ............................................................................................................. 128
A4.1 Introduction...................................................................................................................... 129
A4.2 Antenna Structure ............................................................................................................ 130
A4.3 Return Loss ...................................................................................................................... 131
A4.4 Axial Ratio ....................................................................................................................... 133
A4.5 Modal Analysis ................................................................................................................ 135
A4.6 Conclusion ....................................................................................................................... 137
References ................................................................................................................................... 138
iv
List of Figures
Fig. 2.1: Eye structure with an enlarged section of the retinal layer………………………….…….8
Fig. 2.2: The structure of rod photoreceptor cells…………………………….. ………………….10
Fig. 2.3: The outer-segment of the photoreceptor…………………………………….…………...11
Fig. 2.4: The spectral sensitivities of the 3 cone types and the rod. Source of data…………….....12
Fig. 2.5: Isomerisation of 11-cis-Retinal to all-trans-Retinal……………………………………..13
Fig. 2.6: The process of hyper-polarization by rhodospin decomposition……………………..….14
Fig. 2.7: The diagram to study the Stiles-Crawford effect………………………………..…….…16
Fig. 2.8: Three segment model of photoreceptor………………………………………………….17
Fig. 3.1: Yearly FDTD-related publications..……………………………………………………..20
Fig. 3.2: Yee’s FDTD cell……………………………………………………..…………………..21
Fig. 3.3: Total-field scattered-field technique……………………………………………………..25
Fig. 3.4: The exponential increase in the performance of graphics processing units compare to
Intel processors………………………………………………………………………………..….. 26
Fig. 4.1: Averaged-permittivity model: the cone photoreceptor outer-segment………...…..…….32
Fig. 4.2: Normalized |Ey| distribution in the excitation plane….……………………………….....33
Fig. 4.3: The excitation technique………………………………………………………..………..34
Fig. 4.4: Transmitted Ey at the center of the narrow end of the outer-segment, for blue light
(λ=480nm)………………………………………………………………………………..………..35
Fig. 4.5: Transmitted Ey at the center of the narrow end of the outer-segment, for green light
(λ=520nm)………………………………………………………………………………..……......36
Fig. 4.6: Transmitted Ey at the center of the narrow end of the outer-segment, for red light
(λ=650nm)………………………………………………………………………………..………..36
Fig. 4.7: Transmitted Ey , broadband excitation. Excited mode diameter is 5µm………………....38
Fig. 4.8: Transmitted Ey , broadband excitation. Excited mode diameter is 3µm……....................38
Fig. 4.9: The modal pattern at λ=697nm. The Gaussian beam diameter is 5µm…….....................39
Fig. 4.10: The modal pattern at λ=697nm. The Gaussian beam diameter is 3µm…….................. 40
Fig. 5.1: Model of the geometry of the cone outer-segment............................................................44
Fig. 5.2: Bulk averaged model of the outer-segment.......................................................................45
Fig. 5.3: Spectrum of the excitation signal versus the wavelength..................................................46
Fig. 5.4: Outer-segment illuminated by a plane wave......................................................................46
Fig. 5.5: Poynting power distribution across surface SF for three incident angles………………...50
Fig. 5.6: Irradiance for Φ= π /12, ψ=90o and ψ=0o for two different structures: ............................50
Fig. 5.7: Irradiance for Φ= π /10, ψ=90o and ψ=0o for two different structures.............................51
Fig. 6.1: The outer-segment of the photoreceptor............................................................................55
Fig. 6.2: Model of the geometry of the cone outer-segment............................................................56
Fig. 6.3: |Ey| in the outer-segment when D2=4.2µm.........................................................................58
Fig. 6.4: |Ey| in the outer-segment when D2=1.2µm.........................................................................59
Fig. 6.5: The block diagram of the post-processing technique used to obtain the energy
spectrum...........................................................................................................................................60
Fig. 6.6: Energy spectrum versus free-space wavelength for different structures...........................61
Fig. 7.1: The outer-segment of the photoreceptor for the a) rod and b) cone cells..........................66
Fig. 7.2: Model of the geometry of the cone outer-segment............................................................67
Fig. 7.3: Incident electric field (|Ey,inc|) versus wavelength..............................................................68
Fig. 7.4: |Ey| in the outer-segment when D2=1.2µm.........................................................................69
Fig. 7.5: Ey field distribution along the cone radius at f=622 THz, when D2=1.2µm......................70
Fig. 7.6: Ex and Ez components of the scattered electric field versus wavelength....... ...................71
Fig. 7.7: Ey component of scattered electric field (|Ey,scat|) versus wavelength................................72
Fig. 7.8: Amplification factor (|Ey,scat|/ |Ey,inc|) versus wavelength...................................................72
v
Fig. 7.9: Three dimensional view of the cone inside the total-field scattered-field domain............73
Fig. 7.10: Back-scattered irradiant flux versus wavelength for D2=1.2µm......................................74
Fig. 7.11: Lateral-scattered irradiant flux versus wavelength for D2=1.2µm..................................74
Fig. 7.12: Forward-scattered irradiant flux versus wavelength for D2=1.2µm................................75
Fig. 7.13: Normalized forward-scattered irradiant flux versus wavelength.....................................75
Fig. 8.1: Model of the geometry of the cone outer-segment............................................................79
Fig. 8.2: The Lorentz dispersive dielectric model of membrane layers...........................................81
Fig. 8.3: Incident electric field (|Ey,inc|) versus wavelength..............................................................82
Fig. 8.4: Transmitted electric field when D2=1.2µm........................................................................83
Fig. 8.5: Absorbed electric field versus wavelength for different values of D2...............................83
Fig. A1.1: Cross section of the FDTD space..................................................................................109
Fig. A1.2: The field propagation at different time steps inside the FDTD domain.......................110
Fig. A1.3: The Ey versus time sampled at different points inside the FDTD domain....................110
Fig. A3.1: Antenna Structure.........................................................................................................118
Fig. A3.2: Two main categories of modeling reported in this paper focus on the region around the
inner conductor of the coaxial probe……………………………………………………………..120
Fig. A3.3: Input voltage of the modeled excitation for the inner conductor not extended into
UPML……………………………………………..…..………………………………………….122
Fig. A3.4: Input impedance for the probe model category depicted in Fig. A3.2(a) and excitation
sources of Fig. A3.3……..……………………………………………….……….…….………..123
Fig. A3.5: Input voltage for hard source excitation for the inner conductor of the coaxial probe
extended to UPML. …………..……………………………………………………………….....124
Fig. A3.6: Input impedance when the thin wire extended into UPML. ………………….…...…124
Fig. A3.7: Error based on Eq. (2) calculated for the thin wire model not extending into the UPML
and for four different distances between the resistive source and the termination of the inner
conductor of the coaxial cable (Fig. A3.2(a))……………………………………………………125
Fig. A3.8: Error based on Eq. (2) calculated for the thin wire model extended into the UPML (Fig.
A3.2(b))…..…………………………………………………………………………..…………..126
Fig. A4.1: Three-dimensional view of the electromagnetically coupled patch antenna………....130
Fig. A4.2: Lower patch of antenna in Fig. 1, with indicated coordinates of the feed point……...131
Fig. A4.3: Return loss for the separation between the lower and upper patch (foam spacer)
d=18mm…………………………………………………………………….................................131
Fig. A4.4: Return loss for the separation between the lower and upper patch (foam spacer)
d=22mm………………………………………………………………………………..………...132
Fig. A4.5: Percentage of bandwidth for different values of the separation between the lower and
upper patch (thickness of the foam spacer), d…………………………………………………....133
Fig. A4.6: Axial ratio versus frequency for different values of d………………………………..134
Fig. A4.7: Axial ratio at φ=0° versus θ for different values of d at 1.24GHz……………………134
Fig. A4.8: Current distribution at 1.3GHz on the patches, illustrating the simultaneous excitation
of TM01 and TM10 modes……..…...……………………………………………………………..138
vi
List of Tables
Table 2.1: Refractive indices of each section of the eye…………………………………………....9
vii
To my parents
viii
Acknowledgments
I would like to thank my adviser, Dr. Milica Popović, for her guidance throughout my
Ph.D. program. I would also like to express my appreciation to Dr. Dennis
Giannacopoulos and Dr. Lawrence Chen for their interest in my research and for serving
on my committee.
I am grateful to my friends in the Computational Electromagnetic group at McGill
University for providing the friendly environment to conduct this research. Last but not
the least I would like to thank my family for all their support and encouragement
throughout the Ph.D. program.
ix
Preface to the Thesis
Concerning the Format of This Thesis
This thesis was prepared in the manuscript-format in the form of five self-contained
research papers designated Chapters 4, 5, 6, 7, and 8.
Chapter 4, entitled “Photoreceptor Outer-Segment: 2-D vs. 3-D FDTD Optical
Waveguide Models”, is submitted to the IEEE Transactions on Magnetics. The results
and techniques included in the paper have already been presented in the IEEE Conference
on Electromagnetic Field Computation (CEFC2008). Chapter 5 entiled “FDTD Analysis
of Retinal Photoreceptor Outer-Segment: Effect of Stacking Nano-Layers of Membrane
and Cytoplasm “is a paper under preparation for journal publication. The major content
of the paper has been published in the conference proceeding of the General Assembly of
International Union of Radio Science (URSI2008). Chapter 6, entitled “FDTD Analysis
of Light Propagation in the Human Photoreceptor Cells,” was published in the IEEE
Transactions on Magnetics. Chapter 7,
entiled “Human Retinal Photoreceptors:
Electrodynamic Model of Optical Micro-Filter” was published in the IEEE Journal on
Selected Topics in Quantum Electronics: Special Issue on Biophotonic. Chapter 8,
entitled “Rhodopsin Absorption Spectrum:Effect of the Photoreceptor Outer-Segment
Morphology” is a paper under prepartion for journal pulication. The technique and the
results of the paper have already been presented in the Photonics North 2008 Conference.
Four more chapters are added to the above papers to develop a coherent dissertation.
Chapter 1 is the thesis introduction to overview the motivations, objective, challenges,
methodolody and original contrubtion of the thesis. Chapter 2 provides the background
and litertures about the photoreceptor optics. Chapter 3 provides a quick reveiw of
computational methods exploited in this thesis. Finally, in Chapter 9 we will present the
Conclusion and Future Work of the thesis.
Contributions of Authors
The applicant, Amir Hajiaboli, is the primary author of each chapter in this thesis. Prof.
Milica Popović has contributed to this work by introducing the initial topic of this
research and by providing guidance, support and manuscript editing.
x
Guidelines Concerning a Manuscript-Format Thesis Preparation
According to the Guidelines Concerning Thesis Preparation published by the Faculty of
Graduate Studies and Research at McGill University, a thesis can be prepared in the
format of published or to-be-published papers. The following excerpts are taken from the
guidelines:
“Candidates have the option of including, as part of the thesis, the text of one or more
papers submitted or to be submitted for publication, or the clearly duplicated text of one
or more published papers. These texts must be bound together as an integral part of the
thesis.
The thesis must be more than a collection of manuscripts. All components must be
integrated into a cohesive unit with a logical progression from one chapter to the next by
connecting texts that provide logical bridges between the different papers.
The thesis must conform to all other requirements of the Guidelines for Thesis
Preparation. The thesis must include: a table of contents; an abstract in English and
French; an introduction which clearly states the rational and objectives of the research; a
comprehensive review of the literature; and a final conclusion and summary.
Additional material must be provided where appropriate, (e.g., in appendices) and in
sufficient detail to allow a clear and precise judgment to be made of the importance and
original of the research reported in the thesis.
In the case of manuscripts co-authored by the candidate and others, the candidate is
required to make an explicit statement in the thesis as to who contributed to such work
and
to
what
extent.
This
statement
should
appear
in
a
single
section
entitled ”Contribution of Authors” as a preface to the thesis. The supervisor must attest to
the accuracy of this statement at the doctoral oral defense. Since the task of the examiner
is made more difficult in these cases, it is in the candidate’s interest to clearly specify the
responsibilities of all the authors of the co-authored papers.”
xi
INTRODUCTION
Chapter 1
Introduction
Approximately 70% [1, 2] of the total sensory receptors in the human body reside in the
eye and they provide 80% [3] of the acquired information during life. In human beings,
the visual perception involves three distinctive processes:
•
The image formation: The first stage of visual process involves the formation of
the object’s image at the back of the eye on the retinal layer. To study this process,
the gross anatomy of the eye structure is considered as a camera. The link
between the image formation in a camera and that of the human eye has been
investigated since the seventeenth century [4]. Knowing the refractive index and
the dimensions of the average eye structure, different models have been presented
to describe the analogy of the eye structure to that of a camera [5].
•
Transduction of optical energy to electrical signal: In the second stage, the optical
signal needs to be transduced to an electrical signal. The transduction of optical
energy to electrical signal occurs in the photoreceptor cells. Photoreceptors are
pigmented micro-size cells covering the retinal layer and the proof of their
existence dates back to 1835 after the initial work done by Gottfried Reinhold
Treviranus [4].
A pico-second chemical process which is initiated by light interaction with a
human photoreceptor results in changing the optical energy to an electrical signal.
The process is triggered by the decomposition of a specialized light-sensitive
pigment, rhodopsin [6] .
•
Data processing: In the final stage, the electric signal will be sent to the brain for
further processing that will result in resolving the image.
1
INTRODUCTION
In this research our focus is on the second process which explains the mechanism of light
interaction with the photoreceptors. This is related to the area of photoreceptor optics
which dates back to 1961 and Enoch’s observation of the optical modal pattern in the
visual photoreceptors [7].
1.1 Photoreceptor Optics: Challenges and Solutions
“Photoreceptor optics is the science that investigates how the optical properties of
photoreceptors – their arrangement, orientation, shape, size, refractive index, and
membrane properties- influence their absorption and establish many of their specialized
functions ” [8]. This is a multidisciplinary branch of science that integrates the knowledge
of biology and optics, and it can potentially result in the development of novel biomimetic optical devices, which have applications in navigation systems, specialized
detectors and surveillance cameras [9]. However, research in this area of science has
been hindered by some challenges. The highlights of these challenges are:
•
The investigation, which is mostly done in experimental work, requires real
biological tissues. Preparing biological samples can be very challenging, especially
with respect to making measurements on a single photoreceptor. The equipment
necessary to make such measurements is expensive and not readily available since it
depends on the use of custom-designed platforms.
•
The analysis for support of the experimental results has been done through the
analytical solution of the electromagnetic wave equations. The analytical solution can
be complicated, and the approximations involved reduce the accuracy of the solution.
In the recent years, the advances in numerical techniques, together with the advances in
computer architecture, have provided researchers with promising opportunities to
investigate different areas of life sciences. The complex structure of biological cells and
tissues has been investigated through numerical simulation, and the results obtained have
shown good agreement with the measurement.
1.2 Objective and Motivation
Each photoreceptor cell consists of an inner-segment and also a rod or cone-shaped outersegment which contains the photo-sensitive visual pigment, rhodopsin. Although the light
2
INTRODUCTION
absorption is governed by the visual pigments in the outer-segment, due to the waveguiding effect in photoreceptors, the shape and dimensions of the outer-segment may also
play a role in vision. For example, previous studies [10] show that the absorption
spectrum of the in situ rhodopsin is slightly shifted to higher wavelengths relative to the
absorption spectrum of the rhodopsin which are extracted form the outer-segment. Our
objective in this work is to study the effect of the photoreceptor outer-segment’s physical
dimensions and its filtering effect on the spectrum of the perceived light.
1.3 Methodology
Due to the complexity in measuring this phenomenon we have adopted a very welldeveloped computational electromagnetic technique, finite-difference time-domain
(FDTD) to analyze the detail structure of the photoreceptor. We choose the FDTD
technique mainly due to its flexibility in modeling inhomogeneous media of biological
tissue. Furthermore, due to its time-domain nature, it is a suitable method for modeling
the broadband excitation of a natural light source, which is required in our particular
application of photoreceptor optics.
1.4 Dissertation Overview
In Chapter 2, we first review the gross anatomy of the eye structure with an emphasis on
the photoreceptor morphology and its photochemistry. A camera model of the eye is
presented through which we review some of the previously obtained experimental results
about the existence of an electrodynamic process in the human vision.
In Chapter 3, we review the main computational techniques used in this work which are
based on the finite-difference time-domain (FDTD) technique. We propose two main
FDTD techniques based on 1) the rotationally-symmetric scalar FDTD technique and 2)
the full wave 3- dimensional (3-D) FDTD technique.
In Chapter 4, which is based on a submitted paper to the IEEE Transactions on
Magnetics, we compare the two different computational electromagnetic techniques
based on the FDTD method which are presented in Chapter 3. One of the techniques,
based on the rotationally symmetric FDTD technique, has been previously exploited in
photoreceptor optics. We compare this technique with a 3-D full-wave FDTD technique
3
INTRODUCTION
and present some of the drawbacks in using rotationally symmetric FDTD techniques in
the special application of photoreceptor optics.
In Chapter 5, which is included as a published paper in the conference proceeding of
General Assembly of International Union of Radio Science, we compare the bulk average
model of the photoreceptor with its precise model. This helps us to develop a model of
the photoreceptor outer-segment, which is then investigated further in Chapters 6, 7 and 8.
In Chapter 7, which is included as a paper published in the IEEE Transactions on
Magnetics we offer a parametric study of the photoreceptor and also propose an imageprocessing technique to obtain the spectrum of the light propagating in the outer-segment.
In Chapter 8, which is included as a paper published in the IEEE Journal on Selected
Topics in Quantum Electronics: Special Issue on Bio-Photonics, we present more details
about the light interaction with the photoreceptor outer-segment. The significant filtering
effect of the photoreceptor outer-segment on the electric field will be presented.
Furthermore, the discovery of the field enhancement and mode conversion along the
photoreceptor will be presented.
Chapter 9, which is included as a journal manuscript under preparation, presents a novel
technique to extract the absorption spectrum of the in situ rhodospin molecules. Then, we
will study the effect of the outer-segment morphology on the absorbed electric field.
1.5 Original Contribution of the Thesis
The original contribution of the thesis can be divided into two distinct categories. In the
first category, we show some of the technical contributions in modeling the light
interaction with a photoreceptor. In the second category, the discoveries about the nature
of light interaction with human retinal photoreceptors are shown.
1.5.1 Technical contributions
•
Developing the required numerical code based on predefined libraries for large
scale and high performance electromagnetic simulation. The libraries have been
provided by Acceleware Inc. (www.acceleware.com), which is the pioneer in
developing FDTD codes based on a graphical-processing unit (GPU). Appendix I
shows the developed computer code to analyze in detail the structure of the
4
INTRODUCTION
photoreceptor. Setting up the computational resources and then programming for
the specific application of photoreceptor modeling required roughly a two year
time frame.
•
Large-scale data management for developing the required post-processing
technique for visualization and modal analysis of the light interaction with the
human photoreceptor.
•
Developing a rotationally symmetric FDTD code. The code has been included in
Appendix II.
•
Comparison between the rotationally symmetric FDTD technique and 3-D FDTD
technique in order to show some drawbacks in utilizing the rotationally symmetric
FDTD technique for the special application of photoreceptor optics.
•
The novel technique developed to extract the absorption spectrum of the in situ
rhodopsin molecules.
1.5.2 Discoveries related to the biophysics of light interaction with photoreceptor
This part of the contributions is related to some exciting discoveries about the nature of
light interaction with human photoreceptor:
•
The role of the stacking nano-layers in the photoreceptor outer-segment on the
propagated electromagnetic field and the higher irradiance in the stacked layers
which is the main difference between the absorption spectrum of the in situ
rhodopsin molecules and the extracted rhodopsin molecule.
•
The role of the diversity in the photoreceptor outer-segment dimensions, which
can alter the spectrum of the propagated electromagnetic field in the outersegment.
•
The effect of the outer-segment morphology on the rhodopsin absorption
spectrum.
1.6 Contributions Related to Antenna Modeling
Apart form the contributions related to the main focus of the thesis, which are mentioned
above, we present some additional contributions in Appendices 3 and 4. These
contributions are mainly related to a novel antenna structure based on an
5
INTRODUCTION
electromagnetically coupled patch (EMCP) antenna, which has a broadband behavior
together with a circular polarization at the L-band. The antenna and its FDTD modeling
had been studied rigorously. The contributions in this area can be identified as:
•
A novel formulation for modeling the junction of a coaxial probe excitation and
thin dielectric materials sheets, which can be exploited also in the via-hole
structure used frequently in microwave printed circuits.
•
Extracting and visualizing equal mode excitation on the patches of the antenna
which results in circular polarization behavior.
Although these contributions are not related to the bio-photonics topics directly, the work
on these contributions involved the development of numerical methods later used for the
main topic of the thesis.
6
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Chapter 2
Human Eye: Physiology, Photochemistry and Optical Properties
In this chapter, we first study the gross anatomy of the eye with an emphasis on the
photoreceptor structure, which is the focus of this research. The tri-chromatic color vision
and the existence of different photoreceptors, sensitive to a distinctive fraction of the
visible range of the light spectrum, are presented. Furthermore, the photochemistry of
vision is briefly reviewed. Later in this chapter, a camera model of the eye will be
presented and through that we review the Stiles-Crawford effect, which explains the
directional sensitivity in human vision. Finally, the physical dimensions of the
photoreceptor and its variation will be discussed.
2.1 Gross Anatomy of the Human Eye and the Retinal Layer
In Fig. 1.1, we briefly outline the general structure of the human eye to situate the
photoreceptors and their role in the overall system of the human vision. For additional
reading on the physiology of vision, the reader is advised to consult [6, 11, 12].
As the light propagates in the eye, it first hits the cornea and aqueous humour, and then,
after passing through the lens and vitreous humour, it reaches the retina on the back of
the eye. Table 1.1 shows the average values of refractive indices of each section of a
typical human eye. These values may change to some extent based on age or other
conditions. When the light hits the cornea, some part of its energy reflects back because
of the differences in the refractive indices at the air-cornea interface. This is the
maximum reflection which occurs along the light path to the final image formation on the
retinal surface.
7
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Fig. 2.1: Eye structure with an enlarged section of the retinal layer. Pictures were adopted from [13, 14].
8
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
After the light reaches the retina, the photoreceptors change the optical energy to an
electrical signal, and send it to the optic nerves and the brain for processing.
The retinal layer consists of around 100-120 million rods and around 6-7 million cones,
which are responsible for transduction of the optical energy to electrical signals. The rods
play a role in black and white vision and are very sensitive cells which can work under
dim light conditions. Cones, however, are photoreceptor cells whose function enables the
human color vision, and they are not as sensitive as rods.
Table 2.1: Refractive indices of each section of the eye [15].
Cornea
1.3371
Aqueous humour
1.3374
Crystalline lens
1.42
Vitreous humour
1.336
2.2 Photoreceptor Structure
The photoreceptors are responsible to capture and transduce the optical energy into
electrical signals. Fig. 2.2 shows the structure of a rod photoreceptor [6]. The structure
can be divided into two main segments: the outer-segment and the inner-segment.
Unlike the general morphology of living animal cells, in the photoreceptors, the nucleus
is not surrounded by the membrane and cytoplasm. The stacking layers of cytoplasm and
pigmented membranes form a cylinder farther away from the nucleus, which is called
outer-segment.
The transduction of optical energy to electrical signal occurs at the outer-segment, which,
is a major part of the photoreceptor length. The morphology of the cones and rods mainly
differs in the outer-segment shape as shown in Fig. 2.3.
9
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Fig. 2.2: The structure of rod photoreceptor cells.
10
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Fig. 2.3: The outer-segment of the photoreceptor for the a) rod and b) cone cells. Adopted from Fig. 3.4,
pp. 34 in [12].
2.3 Trichromatic Color Vision
Trichromatic color vision theory, which explains the mechanism of color vision in
humans and some other animals, was introduced by Young in 1802 [12]. Based on his
suggestion, the human retina contains three different kinds of “sensitive particles,” each
maximally, but not exclusively, sensitive to different regions of the visible spectrum.
Young’s tri-chromatic theory of human color vision was finally validated by Wald in
1964 [12]. Young’s suggested particles are the visual pigments which are three different
kinds segregated into separate cones [12]. Fig. 2.4 shows the spectral sensitivity of these
three cone type photoreceptors. The spectral sensitivity of the rods is also shown. As it
can be observed in Fig. 2.4, the three cones are specialized for maximum absorption at
different wavelengths: which are called long-wavelength (L-cone), medium-wavelength
(M-cone) and short-wavelength (S-cone) respectively. The combination of these basic
colors then provides the full spectrum of color vision.
11
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
The photo-chemical component in the rods is called rhodopsin. Although it is not
identical, it is similar to the photo-chemical component in cones. The photochemical in
cones are erythrolabe, chlorolabe and cyanolabe and they are responsible for red, green
and blue light absorption, respectively [15].
Fig. 2.4: The spectral sensitivities of the 3 cone types and the rod. Source of data [16]
2.4 Photochemistry of Vision
As noted previously, the molecule which triggers the absorption of light in a
photoreceptor is rhodopsin. Rhodopsin consists of a vitamin A derivative called retinal
and a protein called opsin. Retinal is the light sensitive part of the rhodopsin and can exist
in different forms which are called isomers. Although many isomers of retinal can react
with opsin, it is the 11-cis retinal which binds to opsin in the natural state (dark-adapted
photoreceptor) [12].
When separated, opsin can only absorb light below 280nm and 11-cis retinal absorbs
around 380nm which is still below the visible range. However, when these two combine
to form rhodopsin, the absorption band shifts to the visible range [12].
12
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
The process which initiates the vision process is the photo-isomerisation of 11-cis-Retinal
into all-trans-retinal. As shown in Fig. 2.5, all-trans-retinal isomer has the same
composition as 11-cis-retinal isomer but with a different geometrical shape [12, 17]. This
photo-isomerisation causes the photoreceptor hyper-polarization, the state that triggers
the optic nerves and it is the basis of vision. The detail of hyper-polarization is explained
in the next section.
Fig. 2.5: Isomerisation of 11-cis-Retinal to all-trans-Retinal.
2.5 Hyper-Polarization
Excitation of the rod causes increased negativity of the membrane potential, which is a
state of hyper-polarization, rather than decreased negativity, which is the process of
“depolarization” that is characteristic of almost all other sensory receptors [6].
The process is based on the fact that rhodospin decomposition decreases the membrane
conductance for sodium ions (Na+) in the outer segment of the rod, and this in turn,
causes hyper-polarization of the entire rod membrane.
Fig. 2.6 illustrates movement of sodium ions in a complete electrical circuit through the
inner and outer segments of the rod. The inner segment continually pumps sodium from
inside the rod to the outside, thereby creating a negative potential on the inside of the
entire cell. However, the membrane of the outer segment, in the dark state, is very leaky
to sodium. Therefore, sodium continually leaks back to the inside of the rod and thereby
neutralizes much of the negativity on the inside of the entire cell. Thus, under normal
conditions, when the rod is not excited there is a reduced amount of electro-negativity
inside the membrane of the rod, normally about -40 mV. This is often called the resting
membrane potential.
13
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Fig. 2.6: The process of hyper-polarization by rhodospin decomposition.
14
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
The rhodopsin in the outer segment of the rod is exposed to light and begins to
decompose, this decreases the outer segment conductance of sodium to the interior of the
rod even though sodium ions continue to be pumped out of the inner segment. Thus,
more sodium ions now leave the rod than leak back in. Because these are positive ions,
their loss from inside the rod creates increased negativity inside the membrane. At
maximum light intensity, the membrane potential approaches -70mv to -80 mV [6].
2.6 Retina: a Field Sampler
The optical power inside the outer-segment of a photoreceptor determines the
photoreceptor’s response and ultimately the signal sent to the brain. However, the general
idea of considering photoreceptor as an intensity sampler, like a digital camera, is not a
correct assumption. “It is the field describing the retina image - including the phase and
not only the magnitude - that determines how much light propagates along the
photoreceptors” [18].
The initial experiment that triggered this hypothesis was done by Stiles and Crawford in
1933. Stiles and Crawford observed that changing the entrance point on the pupil so that
a beam of light strikes the same point on the fovea at different angles results in changes
in the perceived brightness. This was later called the Stiles-Crawford effect of first kind.
Fig. 2.7 shows the diagram to study the Stiles and Crawford effect [19].
As we can see, all rays of light emitted from source A are imaged at point A' on retina.
These rays propagate at various angles of incident at the image plane (or retina). A small
aperture at B is used to limit the bundle of rays reaching the image. The angle of
incidence θ is determined by the displacement d of the aperture. The cone space angle φ
is controlled by the size of the aperture. The retina does not respond equally to a stimulus
incident at different angles of incidence θ.
15
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
Fig. 2.7: The diagram to study the Stiles-Crawford effect [19].
Quantitively, Stiles and Crawford showed that a change from 0° to 5° in the angle of
incident field requires an increase of 70% in the intensity to obtain the same brightness
[18]. On the other hand, based on the Poynting theorem, the power of the plane wave
crossing a flat surface at an angle θ, is dependent on cos(θ) . Considering this fact, a 5°
change can decrease the light intensity by 1%, and an increase of 70% is a considerably
high value sufficient to compensate this decrease in intensity.
This experiment has shown that the retina cannot be considered just as an intensity
sampler. The underlying mechanism behind this phenomenon is the wave-guiding effect
in photoreceptors, which makes them work like an optical fiber.
2.7 Wave-Guiding Effects in Photoreceptor
In 1961, Enoch’s observation of the characteristic modal patterns proved that the
vertebrate receptor outer-segments were, indeed, optical waveguides [7]. Previously in
1949, Toralos Di Francia has predicted, on theoretical grounds, that the outer-segments
should be waveguides [20].
16
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
The first optical waveguide model to explain the Stiles-Crawford effect was proposed by
Snyder and Pask in 1973 [21]. They assumed that photoreceptors form a waveguide,
consisting of three distinctive sections. Fig. 2.8 shows these sections and the typical
waveguide model of the photoreceptor. The incident light enters the aperture of the inner
segment and is guided through it. Then, it funnels through the tapered ellipsoid, and
finally it propagates to the outer-segment, in which the energy transduces to the electrical
signal.
Fig. 2.8: Three segment model of photoreceptor [21, 22].
The values of refractive index for each section can vary among species. Also, due to the
bleaching of photo-pigment in the outer-segment, the refractive index can be time
dependent and this can cause a nonlinear effects. However, in low illumination levels the
percentage of bleaching is small and we can assume a linear medium [22]. Furthermore,
there has been a debate on the entrance aperture of the photoreceptor and some authors
believe that it is plausible to assume the outer-segment as the entrance aperture due to its
higher refractive index [23] relative to the surrounding medium.
2.9 Dimensions of Photoreceptors
The physical dimensions of the vertebrate photoreceptor vary among species. These
dimensions can also be dependent on the age and the location of the photoreceptor. For
example, the outer-segment of central fovea photoreceptors is very similar to rods which
17
HUMAN EYE: PHYSIOLOGY, PHOTOCHEMISTRY AND OPTICAL PROPERTIES
are thin and slightly tapered. However, moving toward the pre-foveal region, the
photoreceptors are shorter, wider and more tapered.
For example, the length of M and L cones for central fovea is around 38µm and for the
peripheral photoreceptor, it is around 6.5-13µm [24]. The diameter of the outer-segment
to inner-segment tapers from 1.5µm to 1µm in the central fovea and for the
periphery it tapers from 8µm to 3µm [15].
In this work, as will be seen later, we have assumed typical values for the photoreceptor
model in accordance with the available physiological data.
18
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Chapter 3
Finite-Difference Time-Domain Technique:
Basics and Special Topics in Modeling Biological Tissues
In this chapter, we describe the theoretical background of the full-wave finite-difference
time-domain (FDTD) method in a Cartesian coordinate system, which is the main
computational tool in this thesis. To overcome the computational burden of the full-wave
FDTD, we exploit a simplified FDTD formulation based on the rotationally symmetric
scalar FDTD technique.
Our main concern is the right choice of the computational electromagnetic technique.
Due to the broadband excitation signal, time-domain methods which have shown to be
reliable and precise are preferred. The frequency-domain techniques such as, FEM and
MOM, are computationally intensive for modeling this range of frequency. Other
techniques, such as beam propagation method (BPM), which are generally used in
modeling optical fibers, are developed for modeling forward propagation and they are not
very suitable for modeling reflections.
Later in this chapter, we overview the basic challenge in modeling biological tissues
which is to perform the large scale FDTD simulation and to model the detailed structure
of biological tissues. Further, we present a recent novel solution for large-scale
simulations using a desktop computer. In the end, we present different types of dielectric
materials and techniques for modeling them using the FDTD method.
19
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
3.1 Full-Wave FDTD Technique
Finite-difference time-domain was originally introduced by Yee in 1966 [25]. Due to its
simplicity and its explicit approach to the solution of Maxwell’s equations, it has been in
use and under careful investigation in recent years [26-28]. Fig. 3.1 shows the growth in
the number of publications in the area of FDTD technique since its inception in 1966.
The exponential growth in this area is mainly due to the advances in computational
resources.
Fig. 3.1: Yearly FDTD-related publications, sources from [29] and [30].
The basic idea of the FDTD technique is based on a discretized form of the time-domain
Maxwell’s equations shown in (1).
r
r 1 r
r
∂H
1
= − ∇ × E − ( M source + σ * H )
∂t
µ
µ
r
r 1 r
r
∂E 1
= ∇ × H − ( J source + σ E )
∂t ε
ε
(1)
Here, ε [F/m] represents the dielectric constant of the medium and σ [S/m] represents the
electric loss of the medium. E [V/m] and H [A/m] are the electric and magnetic vector
fields respectively. Msource [V/m2] and Jsource[A/m2] are the magnetic and electric current
sources respectively.
After decomposing E and H in (1) into three main components (Ex, Ey, Ez) and (Hx, Hy,
Hz) and replacing the time and space derivatives with the central difference
approximation we will obtain the basic updating equation for the FDTD technique as
shown in (2) for Ex.
20
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
σ
∆t

 1 − i , j + 1 / 2 ,k + 1 / 2
2ε i , j + 1 / 2 ,k + 1 / 2

E x in,+j +1 1/ 2/ 2 ,k + 1 / 2 = 
 1 + σ i , j + 1 / 2 ,k + 1 / 2 ∆t

2ε i , j + 1 / 2 ,k + 1 / 2



∆t


ε


i , j + 1 / 2 ,k + 1 / 2
n −1 / 2
 E x i , j + 1 / 2 ,k + 1 / 2 + 
σ

 1 + i , j + 1 / 2 ,k + 1 / 2 ∆t


2ε i , j + 1 / 2 ,k + 1 / 2





×



(2)


n
n
n
n
 H z i , j + 1,k + 1 / 2 − H z i , j + 1,k + 1 / 2 H y i , j + 1 / 2 ,k + 1 − H y i , j + 1 / 2 ,k

n

−
− J source i , j + 1 / 2 ,k + 1 / 2 
∆y
∆z






Here, (i,j,k) corresponds to (i∆x, j∆y, k∆z) coordinate. Half a cell displacement has been
shown by adding 1/2 to the corresponding coordinate location. The arrangements of E
and H fields are on the edges and centers of a cube shown in Fig. 3.2. The cube is called
Yee’s cell and the simulation space is divided into many of these Yee’s cells.
Fig. 3.2: Yee’s FDTD cell.
3.2 Absorbing Boundary Condition
Modeling open boundary problems in FDTD requires the implementation of some type of
absorbing boundary condition (ABC) which can absorb the out-going wave. This helps to
truncate the simulation space and allocate it in the computer memory. Different types of
boundary conditions can be categorized into two main groups 1) analytical ABC and 2)
material ABC.
In the analytical ABC, the wave components on the boundary are replaced by some
values which are consistent with the updating FDTD equations. Examples of this type of
21
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
boundary condition are Mur’s first-order and Mur’s second-order ABCs which have been
successfully implemented in different applications [31]. Mur’s ABC predicts the value of
electromagnetic field on the boundary based on a Taylor’s series approximation.
On the other hand, the material ABC requires the implementation of absorbing materials
on the outer-boundary of the FDTD space. This is analogous to the absorbing walls inside
an anechoic chamber well-known for experimental study in antenna technology. The
initial implementation of material ABC dates back to Berenger’s work in 1994 [32]. It
was based on a hypothetical medium and did not have a physical interpretation. Later in
1995, Sack’s et al. [33] presented the uniaxial perfectly matched layer (UPML), which
was a physical and realistic type of material ABC. In 2000, a new type of material ABC,
the convolutional perfectly matched layer (CPML), was successfully implemented [34].
The main advantage of CPML is its flexibility to truncate a generalized inhomogeneous
medium, such as a dispersive and lossy medium. However, it is computationally more
expensive than UPML.
Material ABCs are difficult to implement and they generally require more computational
resources, but their performance is better than that of the analytical ABCs. For example,
the value of reflected field from the outer boundary obtained in material ABC is 1/3000th
less than the value of reflection in analytical ABCs [32, 35].
We have chosen UPML as the absorbing boundary condition in our work to obtain
minimal of boundary reflection with acceptable computational burden. The details of its
formulation and the procedure for an efficient implementation of UPML in computer
programs have been discussed in [26].
3.3 Scalar FDTD Technique
Unlike the full-wave FDTD, the scalar FDTD method is based on a simplified model of
Maxwell’s equations and its application is limited to some applications in which the
electromagnetic field components are not strongly coupled to each other. An example of
those applications is the weakly-guiding waveguide. It has been shown that scalar FDTD
technique can be successfully applied to leaky waveguides in which the refractive index
variation between the core and cladding is less than 10% [36-39].
22
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
The advantage of this technique is the reduced computational burden, while still
providing a good insight into the transmitted and reflected field in a wave-guiding
structure. However, investigating some aspects of wave propagation cannot be easily
incorporated into this modeling technique and requires special treatment. For example,
the polarization effect which is related to the coupling of electromagnetic fields needs a
semi-vectorial formulation [39].
In addition to the utilization of the scalar FDTD formulation, we can use the rotationallysymmetric FDTD formulation. This is a plausible assumption when the modes
propagating in the structure have the same symmetrical condition around the central axis
of the waveguide in φ-direction [40, 41].
As the photoreceptor structure acts as a weakly-guiding waveguide, we have developed a
rotationally symmetric scalar FDTD to investigate the structure. Later, in Chapter 4, we
show some examples in which the rotationally-symmetric formulation cannot support
broadband excitation required in photoreceptor optics.
3.4 Rotationally Symmetric Scalar FDTD
The scalar wave function can be written as shown in (3) [36]
∇ 2ψ −
n2 ∂ 2
ψ =0
c 2 ∂t 2
(3)
Here, ψ is the scalar field and n is the refractive index of the medium and c [m/s] is the
velocity of light in the vacuum.
In cylindrical coordinates ψ can be written as
ψ ( r ,φ , z ,t ) = ψ ( r , z ,t )e jlφ
(4)
Here, l is an integer that defines the order of modes in φ-direction. Substituting (4) into
(3) we obtain (5) which is the scalar FDTD formulation in a cylindrical coordinate
system.
∂ 2ψ n 2 ∂ 2ψ
∂ 2ψ 1 ∂ψ l 2
=0
−
ψ
+
−
+
∂z 2 c 2 ∂t 2
∂r 2 r ∂r r 2
23
(5)
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
After applying a central difference scheme in time and space, analogous to the one done
in the full-wave FDTD, we can obtain (6), which is the discretized form of the (5) and it
is the main updating equation in a rotationally symmetric scalar FDTD formulation.

1 l 2  c∆t
c∆t  m
) (
)−(
)ψ ( i , k ) − ψ m −1( i ,k )
2 i − 1  n∆r
n∆z 


ψ m +1( i ,k ) = 2 1 − 1 + (

 c∆t 
+

 n∆r 
2
2



1  m
1  m
ψ ( i − 1,k )
ψ ( i + 1, k ) + 1 −
1 +
 2( i − 1 ) 
 2( i − 1 ) 

[
(6)
]
 c∆t  m
m
+
 ψ ( i ,k + 1 ) + ψ ( i ,k − 1 )
 n∆z 
The nodes on the central axis need to be treated differently and the interested reader is
referred to [36] for further reading on this topic.
Like for the full-wave FDTD, we still need to truncate the simulation space. We have
exploited Mur’s second order formulation due to its straight-forward programming
approach.
3.5 Total-Field Scattered-Field (TF/SF) Formulation
In most cases, such as calculating the radar cross section of receiving antennas, it is
important to model the source as a plane wave.
Different procedures have been implemented to model the plane wave source condition.
The initial attempt to model the incident plane wave condition dates back to Yee’s work
in 1966 [25] and it was based on a defining the initial condition of the electric field in
space. However, this technique requires a large simulation space to model a sinusoidal
wave or a long-duration pulse accurately. To overcome this problem, techniques based on
the scattered-field (SF) [27, 42] formulation or the total-field/scattered-field (TF/SF) [31,
43, 44] formulation have been proposed. Both these techniques are based on the linearity
of Maxwell’s equations and the possibility to decompose the total E and total H into a
known incident and an unknown scattered field. In SF formulation we exploit a new
formulation for updating scattered fields, in which the scattered fields in the whole space
are updated instead of total fields. However, in the TF/SF formulation, the simulation
space is divided into two distinct regions of total fields and scattered fields which are
connected with a hypothetical boundary as shown in Fig. 3.3.
24
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
The disadvantage of the SF technique is the need for significant computational resources
required to evaluate the scattered field at any point on or inside the scatterer. However, in
this formulation, we exploit the analytical solution of incident field everywhere in space,
there are no phase errors due to numerical dispersion such as those that occur on the
connecting surface of the original TF/SF formulation [26]. However, the advanced
formulation introduced in TF/SF technique has suppressed the numerical dispersion [45,
46].
Fig. 3.3: Total-field scattered-field technique.
The choice of SF or TF/SF technique is mainly problem-dependent. The TF/SF technique
is well-suited to the applications such as guided wave simulations, including waveguides,
transmission lines and microwave circuits. In this thesis, we have exploited the TF/SF
formulation.
3.6 Biological Tissues: Requirements for Detail Modeling
Biological tissues consist of a number of minuscule scattering particles. In the lowfrequency spectrum of the electromagnetic field where the wavelength is an order of
magnitude larger than the size of these particles, it is a plausible approximation to
consider the bulk average properties of biological tissues. However, as the wavelength of
the electromagnetic wave shrinks and in the optical regime it is necessary to model the
25
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
detailed geometry of biological tissues including the detailed structure of the scatterers
[47, 48]. The requirement to model the detailed structure of biological tissues has
motivated researchers to develop advanced computational methods. One of these
methods, recently under careful investigation, is the pseudo-spectral time-domain (PSTD)
[47, 49].
Although this technique can provide an acceptable results, its stability can be challenged
as shown in the recent publication [48].
In this thesis, we have considered a more reliable computational method, based on the
classical FDTD technique, the numerical behaviour of which has already been wellstudied. Meanwhile, to overcome the requirement of large-scale and high-performance
simulation, we have chosen to accelerate our simulation based on a graphic processing
unit, which is explained next.
3.7 Graphical Processing Unit Programming
An idea to achieve high performance computation for a repetitive algorithm such as
FDTD is to exploit the power of graphic cards [50, 51]. As shown in Fig. 3.4, in the
recent years, the performance of GPUs, which is shown based on billion floating-point
operations per second (GFLOPS), has been several times the performance of general
purpose processors (CPU) developed by Intel.
Fig. 3.4: The exponential increase in the performance of graphics processing units compare to Intel
processors (source of data [50]).
26
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
The idea has been successfully exploited by some of the leading software companies in
the area of FDTD simulation. In this thesis, we have used the software development kit
(SDK) provided by Acceleware Corp. (www.acceleware.com). The basic idea behind the
SDK is to provide users with some predefined libraries, and then the user can develop the
required code, for the specific application, based on them. Appendix I shows the
developed computer code and some of the benchmarked solutions to test the accuracy of
the code.
3.8 Different Types of Dispersive Material
While Maxwell’s equations presented in Eq. (1) can model the wave interaction with
frequency-independent materials, as the frequency increases, the material properties
become frequency-dependent and this effect is called dispersion. Dispersion phenomena
are related to the finite response of the electric or magnetic dipoles inside the material to
align with the external electromagnetic field [52]. In this work, the magnetic properties
are frequency-independent and it is the electric properties which are frequencydependent.
To incorporate this microscopic phenomenon into the macroscopic Maxwell equations,
we can rewrite Ampere’s law as presented in (7).
r
r
r
∂E ∂P r
+J
∇ × H = ε0
+
∂t ∂t
(7)
Here, P is the volume density of electric dipoles called the polarization vector. We can
calculate the polarization vector using (8).
r
r
r
P = ( ε - ε 0 )E = χε 0 E
(8)
Where, χ is susceptibility. In the special case where χ is frequency-independent, (7) is the
same as Ampere’s law presented in (1).
Depending on χ(ω), we can define different types of dispersive materials. Debye, Lorentz
and Drude are different types of dispersive materials mostly known in the literature.
Biological tissues have different permittivity values and dispersion models [52-54],
depending on the operating frequency. In the low-frequency range, the dielectric
permittivity is mainly determined by the cell membrane and/or the free ions reside in the
27
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
cell membrane and the intracellular medium. However, in the gigahertz region the cell
membrane is transparent to the electromagnetic wave and the electric permittivity is
dominated by the polarized water molecule [53, 54].
As the frequency increases to the optical regime, the cellular membrane stays transparent
and the effect of water content on the wave propagation decreases significantly [17].
Consequently, in the optical frequency, the dielectric properties of biological tissues are
mainly determined by the electromagnetic interaction with molecules such as proteins
and their isomers dissolved in the water.
As we will see later in this thesis, the absorption of rhodopsin molecules inside the
membrane layers of the photoreceptor outer-segment follows a Lorentz model. χ(ω)
exploited to present the Lorentz dispersion in this work is as follows:
χ( ω ) =
Ap ( ε static − ε ∞ )ω 2p
ω 2p − 2iωδ damp − ω 2
(9)
Where, Ap is the lorentz pole, ωp [rad/s] is the lorentz resonance frequency, δdamp [rad/s] is
the damping factor, εstatic is the static relative permittivity and ε ∞ is the relative
permittivity at infinity.
3.9 Modeling Dispersive Materials in FDTD
The initial applications of FDTD techniques were mostly focused on single frequency
simulation and, consequently, there was no need to consider the dispersive nature of
materials. In recent years, however, the dispersive materials and their modeling
techniques have been under careful investigation. The main feature of dispersion is that
the material properties, and in most applications the relative permittivity (εr), will be
frequency dependent. Dispersion is a frequency domain phenomenon and can be easily
incorporated into the frequency domain techniques. However, incorporating this effect
into time-domain techniques such as FDTD method can be cumbersome. There are two
main techniques which have been proposed to incorporate the dispersion effect into the
FDTD method. These techniques are 1) Recursive convolution and 2) Auxiliary
differential equation. In the next section, we explain in more detail these formulations
and the pros and cons of each technique.
28
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
3.9.1 Recursive Convolution (RC)
The idea was originally proposed by Lubbers et al. in 1990 [55]. The technique is based
on the fact that the multiplication in the frequency domain can be presented as a
convolution in the time domain. The convolution integral in the time domain was simply
calculated by discretizing the integral by a running sum and assuming that susceptibility
can be modeled as a decaying exponential in the time domain. Different dispersive
materials, such as, Debye, Drude and Lorentz media have been modeled using this
technique. Although RC provides a computationally efficient algorithm to model a
dispersive material, its accuracy, in terms of numerical dispersion, is not as competitive
as that of the auxiliary differential equation (ADE) method [56], which is explained in the
next section. To overcome the numerical dispersion in the original RC, the piecewiserecursive convolution has been proposed by Kelley et al. [57] and later by Schuster et al.
[58].
3.9.2 Auxiliary Differential Equations
Another technique to model the dispersion is to use ADE. In this method, we use the
time-domain differential equations to link the polarization vector (shown in (7)) and the
electric flux density. The method was originally proposed by Kashiwa et al. [59] and
separately at the same time by Joseph et al. [60]. Although the ADE method provides less
numerical dispersion relative to the RC technique, it requires more back-storage of
variable relative to the RC scheme. However, the recent improvement has reduced the
memory usage of this technique considerably [61, 62].
In this thesis, we have exploited the ADE method to model the dispersive nature of
rhodopsin molecules. The detail of its formulation and implementation has been covered
in [26].
29
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Preface to Chapter 4
The following chapter is included as paper submitted to the IEEE Transactions on
Magnetics. In this chapter, we will introduce two different numerical techniques that are
exploited to investigate the electrodynamics of light interaction with human retinal
photoreceptor: the two-dimensional rotationally symmetric scalar FDTD, and the fullwave three-dimensional FDTD technique. The advantages and disadvantages of each
technique will be discussed to justify the choice of the three-dimensional FDTD
technique as the exploited simulation tool in the subsequent chapters.
30
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Chapter 4
Photoreceptor Outer-Segment:
2-D vs. 3-D FDTD Optical Waveguide Models
Amir Hajiaboli and Milica Popović
Department of Electrical and Computer Engineering, McGill University
3480 University Street Montreal, Quebec H3A 2A7, CANADA
[email protected], [email protected]
Abstract— In this work, we present two- and three-dimensional (2-D and 3-D) finitedifference time-domain (FDTD) models of the human photoreceptor cell outer-segment.
The paper provides the first comparison between the models that are based on the same
physical assumptions. For each of the two models, we characterize the wave propagation
in the human photoreceptor outer-segment. Our study shows sufficient discrepancy
between the 2-D and the 3-D results, implying that one should exercise caution when
using 2-D analysis in this area of bio-photonics.
Index Terms—Photoreceptor outer-segment, finite-difference time-domain
4.1 Introduction and Methodology
In 1973, Snyder and Pask proposed a waveguide model of the photoreceptor to explain
the Stiles-Crawford effect (directional sensitivity) in the center-foveal photoreceptors
31
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
[21]. The model was based on the analytical solution with the plausible assumption of
negligible mode coupling along center-foveal photoreceptors. However, the pre-foveal
cones can behave like tapered multi-mode waveguides, with possible mode coupling, and
such behaviour leads to the usage of numerical techniques as an effective tool in
characterizing their optical properties. Finite-difference time-domain (FDTD) method has
been successfully exploited to investigate wave propagation in the photoreceptors [63,
64]. Although their results offer valuable insight into the problem, the constraint of the
computational resources has limited their simulations to 2-D models.
Here, we study FDTD modeling of the pre-foveal cone outer-segment, using both 2-D
rotationally symmetric scalar and 3-D full-wave FDTD. Although rotationally symmetric
algorithms have a low computational burden, the modes are limited to those imposed by
the symmetry of the structure, and this, in turn, can limit the proper representation of the
mode coupling. The 3-D formulation can model all the modes and their coupling
simultaneously, but at a higher computational cost. As the refractive indices of the
photoreceptor and its surrounding medium have close values, the scalar FDTD algorithm,
used in our work, offers a good representation of 2-D rotationally symmetric algorithms
[65].
4.2 Averaged-Permittivity Model of the Retinal Cone Outer-Segment
The cone outer-segment consists of stacking layers of cytoplasm and membrane which
are having different refractive indices [12]. However, in here, we have considered an
average bulk structure of photoreceptor ignoring the effect of negligible reflection at each
boundary of stacked layers. In the next chapters, we will present a model including the
stacking layers of cytoplasm and membrane. Fig. 4.1 shows the model and the values of
the refractive indices and dimensions.
32
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Fig. 4.1: Averaged-permittivity model: the cone photoreceptor outer-segment.
4.3 Excitation Technique
Numerically, the spatial excitation is executed through a Gaussian-like distribution in
space. Fig. 4.2 shows the electric field distribution at the excitation region. As the radius
of the focused light on the retina is several times wider than the radius of the entrance
aperture of the photoreceptor, we have considered a Gaussian beam excitation with a
large mode diameter.
Two different types of time-domain excitations have been considered in our simulations.
The first excitation is based on a monochromatic excitation. The second method is based
on a broadband Gaussian pulse, which covers the spectrum of the white light.
Fig. 4.3 illustrates the excitation mechanism exploited in this work The FDTD space has
been separated into two distinct regions: the total-field and the reflected-field. At the
plane of excitation, we excite the y-component of the electric field (Ey) using a Gaussianlike distribution in space. Assuming that the speed of light in the host medium is c, the kplane nodes have been updated by applying Ey(n∆t-dz/(2c)) to each node. To evaluate the
field at the (k-1)-plane, the fields at the k-plane, which are in the total field regions, are
required. To ensure the consistency we subtract the value of Ey(n∆t+dz/(2c)), the delayed
form of the electric field at k-plane, from the nodes at the (k-1) plane.
33
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Fig. 4.2: Normalized |Ey| distribution in the excitation plane.
Fig. 4.3: The excitation technique.
4.4 Simulation Parameters
In both 2-D scalar FDTD and 3-D FDTD simulations, the FDTD spatial discretization in
all directions is set to 30nm. Considering the courant stability condition ∆t=6.3640×10-2fs
is set for the 2-D rotational symmetry scalar FDTD, and ∆t=5.5426×10-2fs is considered
for the full wave 3-D FDTD. In both simulations, the number of FDTD cells in the
34
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
z-direction is 300. In 3-D full wave, the number of FDTD cells in x and y direction is 180
resulting in a total of 9.72M cells. By exploiting the rotational symmetry of the structure,
the 2-D scalar FDTD, the number of cells has reduced to 300×90=27000 Cells. As it can
be observed, the number of FDTD cells required for rotational symmetric scalar FDTD is
360 times less than what is required by 3D full wave FDTD. Furthermore, in 3-D full
wave FDTD, it is required to save six quantities of electromagnetic fields at each time
step, while in scalar FDTD this reduces to only one quantity. Considering this, we can
estimate that the computational resources for the 3D FDTD are 2160 times higher than
the rotationally symmetric scalar FDTD.
In the next sections, we are going to present the transmitted field obtained in each
simulation, and explore some special cases were the 2-D rotational symmetric FDTD
cannot result a valid result.
4.5 Monochromatic Excitation
Figs. 4.4-4.6 show the transmitted field for monochromatic excitation of wavelength
corresponding to the blue, green and red light, respectively. We observe the increased
divergence of the 2-D from the 3-D simulation with the increase in wavelength.
Fig. 4.4: Transmitted Ey at the center of the narrow end of the outer-segment, for blue light (λ=480nm).
35
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Fig. 4.5: Transmitted Ey at the center of the narrow end of the outer-segment, for green light (λ=520nm).
Fig. 4.6: Transmitted Ey at the center of the narrow end of the outer-segment, for red light (λ=650nm).
36
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Further, the adopted outer-segment dimensions favour transmission of the green light
(relative to the blue or red), a result that is consistent with our previous results on the
filtering effect of the photoreceptor outer-segment.
4.6 White Light Excitation
Although monochromatic excitation can provide a good insight into the problem of light
interaction with the photoreceptors, it is important to study the behavior of the
photoreceptor under the white light illumination, which is the natural light source. Here,
we excite the outer-segment with a wideband Gaussian pulse of a 5µm excitation mode
diameter.
Fig. 4.7 shows the normalized transmitted signal. We note the similarity of results
between the 2-D and the 3-D formulations for the wide frequency range of the white light
spectrum. However, for a small region around 700nm, the 2-D formulation has a deep
null. Fig. 4.8 shows the improved agreement between the transmitted field calculated
with 2-D and 3-D models, when the smaller mode diameter is 3µm.
To better analyze the behaviour of the transmitted field around λ=700nm, we have
performed modal analysis on the fields along the exiting cross-section of the outersegment. Figs. 4.9 and 4.10 show the mode at 700nm for the Gaussian beam diameters of
5µm and 3µm, respectively. We observe that the two modal patters completely differ in
symmetry across the exiting cross-section of the outer-segment.
We here briefly discuss the results of Fig. 4.6. As the 2-D formulation supports only
rotationally symmetric modes, the mode at 700nm cannot be supported by the 2-D
formulation when the Gaussian beam diameter is 5µm. This can explain the deep null at
700nm for the exciting Gaussian beam diameter of 5µm.
37
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Fig. 4.7: Transmitted Ey , broadband excitation. Excited mode diameter is 5µm.
Fig. 4.8: Transmitted Ey , broadband excitation. Excited mode diameter is 3µm.
4.7 Conclusion
In this paper, we presented a comparison between the results of the 2-D and the 3-D
models of the retinal cone outer-segment. The outer-segment was modeled in both cases
38
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
with averaged-permittivity values of the membrane and the cytoplasm, which both occur
within the outersegment geometry.
As the output and measure of comparison of the two models, we consider the transmitted
electric fields. For the chosen outer-segment dimensions, our findings indicate the
following: 1) for the monochromatic light excitation, the difference between the 2-D and
3-D model results increases with the increased excitation wavelength. 2) For the wideband spectrum of the white-light excitation, the 2-D formulation fails to transmit the
modes that do not exhibit rotational symmetry, while the 3-D model does not demonstrate
this limitation.
Fig. 4.9: The modal pattern at λ=697nm. The Gaussian beam diameter is 5µm.
39
PHOTORECEPTOR OUTER-SEGMENT:2-D VS. 3-D FDTD OPTICAL WAVEGUIDE MODELS
Fig. 4.10: The modal pattern at λ=697nm. The Gaussian beam diameter is 3µm.
Acknowledgment
This work was funded by Natural Science and Engineering Research Council (NSERC)
of Canada. The first author would like to thank Ms. Deena Donia for her help in
reviewing the initial version of this work.
40
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
Preface to Chapter 5
The following chapter is included as a paper under preparation for journal publication.
The major content of the paper has already been published in the conference proceeding
of the General Assembly of the International Union of Radio Science which was held in
Aug. 2008 in Chicago. In this chapter, we will investigate the effect of stacking nanolayers (shown in Fig. 2.3) on the wave-guiding properties of the retinal photoreceptor
outer-segment. Since these layers are considerably thinner than the wavelength of the
light propagating through them, they may introduce some anisotropic effect [66]. Under
normal incidence the effect of these nano-stacking layers may not be very significant.
However, the light may not always reach the outer-segment at normal angles due to the
existence of many scattering objects in the light path as shown in Fig. 2.1. Consequently,
we have proposed a model that includes these stacking layers. This model is developed
and investigated in this and the subsequent chapters.
41
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
Chapter 5
FDTD Analysis of Retinal Photoreceptor Outer-Segment:
Effect of Stacking Nano-Layers of Membrane and Cytoplasm
Amir Hajiaboli, Milica Popović
Department of Electrical and Computer Engineering, McGill University
3480 University Street, Montreal, Quebec H3A 2A7
[email protected], milica.popovich@ mcgill.ca
Abstract- this paper presents finite-difference time-domain (FDTD) analysis of the retinal
photoreceptor outer-segment. Our model simulates, in an optical sense, the reaction of the
photoreceptor to the visible light spectrum. We consider a broadband pulse excitation to
examine the mode coupling. Our investigations include the variations of Poynting power
versus wavelength at exiting cross-sections of the outer-segment, and for varying incident
angles of the impinging wave. In order to study the role of the stacking nano-layers of the
cytoplasm and membrane which comprise the inner photoreceptor anatomy, the results
are compared with those obtained from an averaged-permittivity model of the outersegment.
5.1 Introduction
In 1963, Enoch’s observation of characteristic modal patterns in human photoreceptor
proved that the human photoreceptors are, indeed, optical waveguides and they are
capable of funneling and guiding light [7]. Later in 1973, and based on the available
physiological data at the time, Snyder and Pask proposed an optical fiber model of the
photoreceptor [21]. They exploited the analytical solution of wave propagation and their
results successfully revealed some aspects of directional sensitivity in human vision
known as the Stiles-Crawford effect of the first kind.
42
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
For the waveguide model to be valid, a finite aperture, which limits the bundle of light
entering the photoreceptor, is required [67]. However, this may not be a plausible
assumption because the diameter of the focused light on the retina is several times larger
than the entrance aperture of the photoreceptor [6]. Moreover, the existence of several
scattering objects along the light path can diffract the light reaching the photoreceptor.
Considering these facts, we here propose a model of the photoreceptor outer-segment,
similar to that of a dielectric resonator antenna [20], in which light incident from all
directions can enter and propagate within the photoreceptor.
The numerical tool chosen in our work is the finite-difference time-domain(FDTD)
method, because of its flexibility in modeling the heterogeneous medium required to
model the detailed morphology of living cells, such as photoreceptors [63, 64]. Although
researchers have been mostly limited to the 2-dimensional FDTD formulation due to the
computational burden of 3-dimensional simulations, the results have provided a good
insight into the problem of light propagation in the retinal rods and cones. For example,
in [5], researchers have been able to model the Stiles-Crawford effect through the FDTD
simulations.
As the outer-segment of the human photoreceptor is the site of the photon transduction to
an electrical signal, we suspect that the detailed morphology of the outer-segment can
alter the spectral sensitivity of the photoreceptor. To investigate this, we have calculated
and compared the spectrum of light guided by different models of the photoreceptor
outer-segment for varying angles of incident light.
5.2 Model of Photoreceptor Outer-Segment
The outer-segment consists of stacking nano-layers of cytoplasm and folded membrane.
The thickness of the layers is several times smaller than the light wavelength, so it is
plausible, in the waveguide model, to consider them as geometries of a uniform, averaged
refractive index [21]. However, when the angle of incidence is not normal to the outersegment aperture, the presence of these stacking nano-layers can introduce a significant
effect. To explore this phenomenon, we compare the results obtained from a model that
consists of stacking nano-layers of cytoplasm and membrane and a model which consists
of a bulk, permittivity-averaged model.
43
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
The outer-Segment consists of very thin (~15nm) stacked disks of cytoplasm and
membrane [68, 69] which can be modeled by different refractive index that alternates in
value between that of the cytoplasm and that of the folded cell membrane. For relative
permittivity, the values of accepted physiological data [3, 6], for the cytoplasm εrcytoplasm=1.85,
the membrane εr-membrane=2.22 and the intercellular medium εr-intercellular
=1.79 have been adopted. Due to bleaching of the rhodospin molecules, the refractive
index can change over time [70], and to account for this fact, we choose the membrane
refractive index in accordance with the physiological data for the dark-adapted and
unbleached photoreceptors [71, 72]. Furthermore, as discussed in [73], it is safe to
neglect the effects of dispersion. Therefore, we have considered a constant electric loss of
σmembrane=37.38 S/m [71] over the spectrum of visible light. The length of the outersegment (6.75µm) is, approximately, that of the reported length of the peripheral cones
[7].
Fig. 5.1: Model of the geometry of the cone outer-segment. In our study, L=6.75µm and D1=4.2µm and
D2=1.2µm. The structure consists of very thin (~15nm) alternating stacked layers of different relative
permittivity values. For the cytoplasm, τ1=15nm, εr-cytoplasm=1.85; for the folded membrane, τ2=15nm, εrmembrane=2.22 σmembrane=37.38; for the medium surrounding the outer-segment, εr-intercellular=1.79 [4, 6].
44
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
Fig. 5.2: Bulk averaged model of the outer-segment.
In the averaged bulk model shown in Fig. 5.2, the average value of refractive indices of
cytoplasm and membrane has been considered. The dimensions of the model are shown
in Fig. 5.2.
5.3 FDTD Simulation Parameters
The geometry is modeled by a uniform FDTD space with ∆x=5nm and ∆y=∆z=50nm.
This discretization results in a total of 1600×140×140=31.36 million FDTD cells. The
time-step, based on the Courant-stability condition, is set to ∆t=1.49×10-2fs. The incident
signal is a y-polarized plane wave propagating in the x-direction, and it is a sinusoidmodulated wideband Gaussian pulse centered at 622THz :
E y = Ae
−
( t-t0 )2
2σ 2
sin ( 2πf(t − t0 ))
(1)
Here, A=1V/m is the amplitude of the signal, σ=100∆t is the mean variation of the signal,
t0=400∆t is the delay introduced to preserve the causality of the signal and f=622THz is
the frequency of the modulation signal (the region of visible light, λ=482nm).
Fig. 5.3 shows the spectrum of the incident electric field (|Ey,inc|) versus wavelength (λ).
The peak value of the incident field is 125V/m, centered at λ=482nm. The full width at
half maximum (FWHM) of the incident plane wave is 250nm and, as shown in Fig. 5.3,
this covers the required bandwidth to properly model the white light spectrum.
45
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
Fig. 5.3: Spectrum of the excitation signal versus the wavelength.
In our work, we adopt the total-field/scattered-field (TF/SF) formulation, which allows us
to examine the amount of electromagnetic power coupled into and guided by the outersegment. Fig. 5.4 illustrates the illumination of the outer-segment by a plane wave. k̂ inc
is the incident k-vector and Ê inc represents the polarization vector. Φ is the angle of
incidence relative to the x-axis and ψ is the polarization angle based on the model
discussed in [26].
Fig. 5.4: Outer-segment illuminated by a plane wave.
46
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
The forward-scattered fields of Ey, Ez, Hy and Hz on the SF surface shown in Fig. 5.4 have
been sampled at each time step. After applying the Fourier-transform, the time-average
Poynting power (Sx) versus frequency has been calculated using the following equation:
S x = Real{E y H z − E z H y }
(2)
5.4 Results and Discussion
Figs. 5(a) – 5(c) show the Poynting power distribution on the exiting aperture of
photoreceptor (SF), shown in Fig. 5.4, for different values of Φ, at λ=482nm. As it can be
observed, as the incidence angle increases, the forward-scattered power, which represents
the power coupled into the outer-segment, decreases. Further, the peak of the scattered
power will shift to the lateral side of the outer-segment. Consequently, negligible power
will be available to the rhodopsin (chemical in photoreceptors responsible for the photochemical process) to trigger its visual cycle. This result agrees with the observations on
the directional sensitivity in human vision.
To investigate the total power available to the rhodopsin molecules, we integrate
the forward-scattered power over the SF surface (Fig. 5.4) for the complete range of
visible wavelengths. Figs. 5.6 and 5.7 show the spectrum of irradiance scattered power
on SF surface of Fig. 5.4 for angles of incidence Φ=π/10, Φ= π/12, and two different
polarization of ψ=0o and ψ=90o. We can observe that the value of guided power is higher
in the laminar structure, which consists of stacking nano-layers of cytoplasm and
membrane. Furthermore, in all cases the wavelength of the peak value will be shifted
slightly to higher wavelengths in laminar structures, which can explain the small
difference (towards higher wavelengths), in the spectral sensitivity of the intact
photoreceptors and its solution [10].
5.5 Future Work
We are currently investigating the possibility of incorporating the random alignment of
rhodopsin molecules, inside the membrane, into the FDTD model of the outersegment[10]. This model can eventually help us to investigate and explain some aspects
of polarization sensitivity in some types of vertebrate photoreceptors [74].
47
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
Acknowledgments
This research was funded by Natural Science and Engineering Research Council of
Canada (NSERC) and Le Fonds québécois de la recherche sur la nature et les
technologies (FQRNT). We are grateful for their support.
48
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
(a)
(b)
49
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
(c)
Fig. 5.5: The white circle shows the boundary of SF defined in Fig. 5.4. Graphs show the Poynting power
distribution across surface SF for three incident angles: a) Φ=π/45, b) Φ=π/12 and c) Φ=π/10.
Fig. 5.6: Irradiance for Φ= π /12, ψ=90o and ψ=0o for two different structures: one characterized by a bulk
averaged refractive index and the other (laminar) by refractive indices of stacking nano-layers of cytoplasm
and membrane.
50
FDTD ANALYSIS OF RETINAL PHOTORECEPTOR OUTER-SEGMENT: EFFECT OF STACKING NANO-LAYERS OF MEMBRANE …
.
Fig. 5.7: Irradiance for Φ= π /10, ψ=90o and ψ=0o for two different structures: one characterized by a bulk
averaged refractive index and the other (laminar) by refractive indices of stacking nano-layers of cytoplasm
and membrane.
51
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Preface to Chapter 6
The following chapter is included as a paper published in the IEEE Transactions on
Magnetics. We investigate the spectrum of the light “trapped” due to total internal
reflection, inside the cone outer-segment. Our main focus is to provide a framework in
which we can analyze the spectrum of the light propagating inside the outer-segment. The
framework is a modified version of the image processing technique exploited in [64] to
extract the spatial frequency of the light propagating in the photoreceptor. We then use
this framework to investigate different structures of the photoreceptors, the outer-segment
and their filtering properties.
52
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Chapter 6
FDTD Analysis of Light Propagation in the Human
Photoreceptor Cells
Amir Hajiaboli and Milica Popović
Department of Electrical and Computer Engineering,
McGill University, Montréal, QC H3A 2A7, Canada
Abstract— This paper presents a numerical study of the light interaction with the
human retinal photoreceptor. The photoreceptors, cells in the retina responsible for
monochromatic vision and color discrimination, differ in shape and size. The cell type
(rod or cone) and the variety among the cone cells (S, M and L) condition their peak
absorption wavelength (blue, green or red). In our work, we observe the outer-segment of
the photoreceptor as a micro-size optical filter by modeling its interaction with the light
wave. Our finite-difference time-domain results suggest that the physical structure of the
photoreceptor outer-segment can affect its filtering properties.
Index terms: Human photoreceptor, finite-difference-time domain (FDTD), spectral
analysis
6.1 Introduction
The retinal surface of the human visual system contains specialized cells, known as rods
and cones, responsible for detection of light and its transduction to electrical signals,
which are then sent for processing in the visual cortex. The rod-shaped cells play the
53
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
main role in monochromatic vision by detecting the intensity of light. The human color
perception is a result of the absorption of the three constituent colors (red, green and
blue) by three cone-cell photoreceptors (L, M and S).
Each human photoreceptor consists of two distinctive segments: 1) inner-segment and 2)
outer-segment. It is the outer-segment that dominates the geometry of the photoreceptor
and within which the incident light wave changes into the electrical signal. The shape
and dimensions of the outer-segment determine the type of the photoreceptor (M, L, S
cone or rod). Fig. 6.1 shows a typical structure of the rod and cone outer-segment. The
cone outer-segment varies in length and conical angle among the cone photoreceptor
cells. Our study is motivated by the similarity of the outer-segment structure to that of the
optical filters and its goal is to investigate the effect of the physical structure of the
photoreceptor cell on its optical filtering properties. In order to study the retinal cones as
optical filters, we here investigate their electrodynamic properties using the finitedifference time-domain (FDTD) method. Literature to date reports on two-dimensional
FDTD models deployed for studies on electromagnetic wave propagation within the
photoreceptor structure as an optical waveguide [63, 64]. For example, results of [64]
showed that the rod outer-segment is a frequency-independent structure, i.e. that the
spectrum of the electric field inside it is the same for all three wavelengths of blue, red
and green. A recently reported study exploits body of revolution FDTD (BOR-FDTD)
to model the cylindrical shape of the photoreceptor. Considering the dominant
electromagnetic mode propagating inside the photoreceptor, the investigators
successfully modeled the Stiles-Crawford effect, which explains the intensity change of
the received electromagnetic power as the direction of the incident wave changes.
54
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Fig. 6.1: The outer-segment of the photoreceptor for the a) rod and b) cone cells. Adopted from Fig. 3.4,
pp. 34 in [12].
The methodology of both [63] and [64] was limited to two-dimensional space due to
heavy computational burden needed for three-dimensional modeling. This limitation
could affect the number of identified electromagnetic modes propagating inside the outersegment. Consequently, although it offers valuable insight, the model lacks in accuracy
of physical representation.
In this paper, we conduct a three-dimensional parametric study of the outer-segment in
order to fully assess its optical filtering properties by identifying all possible modes
“trapped” by the outer-segment geometry.
6.2 Outer-Segment Physical Structure
The physical structure of the photoreceptor outer-segment is shown in Fig. 6.1. It consists
of very thin (~15nm) stacked disks of different refractive index that alternates in value
between that of the cytoplasm and that of the folded cell membrane. For relative
permittivity, we adopt the values of generally accepted physiological data [63], [67] for
the cytoplasm εr-cytoplasm=1.85, the membrane εr-membrane=2.01 and the intercellular medium
εr-intercellular=1.79. The length of the outer-segment considered for our work (6.75µm) is
close to the reported length of the peripheral cones [24]. In the rod outer-segment, the
55
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
disks’ radii are constant. In the cone outer-segment, the disks’ radii linearly decrease
along the central axis of the photoreceptor.
ε r-intercellular
L
D2
D1
y
τ1
τ2
z
x
εr-cytoplasm
ε r-membrane
Fig. 6.2: Model of the geometry of the cone outer-segment. In our study, L=6.75µm and D1=4.2µm, while
D2 is a variable parameter. The structure consists of very thin (~15nm) alternating stacked layers of
different relative permittivity values. For the cytoplasm, τ1=15nm, εr-cytoplasm=1.85; for the folded
membrane, τ2=15nm, εr-membrane=2.01; for the medium surrounding the outer-segment, εr-intercellular=1.79 [63,
67].
6.3 Simulation Parameters and Results
The geometry is modeled by a uniform FDTD space with ∆x=5nm and ∆y=∆z=30nm.
This discretization results in a total of 1500×200×200=60 million FDTD cells. The timestep, based on the Courant-stability condition, is set to ∆t=1.46×10-2fs. Each side of the
FDTD simulation space is terminated with a 10-layer UPML absorbing boundary
condition. The excitation signal is a y-polarized plane wave propagating in the x-
56
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
direction. This a sinusoid-modulated wideband Gaussian pulse centered at 622THz and
given by
E y = Ae
−
( t-t0 ) 2
2σ 2
sin ( 2πf(t − t 0 ))
(1)
Here A=0.2V/m is the amplitude of the signal, σ=100∆t is the mean variation of the
signal, t0=400∆t is the delay introduced to preserve the casualty of the signal and
f=622THz is the frequency of the modulation signal (the region of visible light,
λ=482nm).
Our study was conducted for four values of D2 (Fig. 6.2), the smaller base of the cone.
First, it was assumed that D2=4.2µm which resemble a rod outer-segment with its
cylindrical shape. For the three conical structures, the values are D2=2.4µm, 1.8µm,
1.2µm, and these values imply cone outer-segments with three different tapering angles.
Figs. 6.3 and 6.4 show the light propagation inside the outer-segment for two different
values of D2=4.2µm, and D2=1.2µm, respectively. As can be observed, the excitation
signal passes through the structure after approximately t=3000∆t=43.8fs. After this
period elapses, there remains only the scattered energy which moves back and forth
within the outer-segment.
We further observe that, in the tapered structure (Fig. 6.4) and as the time passes, the
“trapped” pulse “stretches” more in space when compared to its spatial distribution in the
cylinder (Fig. 6.3). In the next section, we describe a technique used to quantify this
“pulse-stretching” phenomenon.
57
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Fig. 6.3: |Ey| in the outer-segment when D2=4.2µm, for four different snapshots in time, from left to right: t
=21.9fs, 43.8fs, 87.6fs, 131.1fs.
58
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Fig. 6.4: |Ey| in the outer-segment when D2=1.2µm for four different snapshots in time, from left to right t
=21.9fs, 43.8fs, 87.6fs, 131.1fs.
59
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
6.4 Post-Processing Technique
To quantify the “pulse-stretching” phenomenon (the spatial distribution of the pulse
energy values) for various structures of the photoreceptor outer-segment, we propose a
post-processing technique which calculates the energy spectrum “trapped” inside the
outer-segment versus spatial frequency or wavelength. Fig. 6.5 shows the block diagram
of the technique followed to obtain the energy spectrum of the pulse.
In each time step after 43.8fs (needed for the excitation pulse to pass through the
structure), we calculate the
V(x, t) =
D/ 2
∫ E (x, y, z = 0, t)dy
y
− D/ 2
Induced voltage at each time step
after 43.8fs along outer-segment
V(x, f) = F{V ( x, t )}
Induced voltage
versus frequency
S(x) =
1081THz
∫ | V ( x, f ) |
2
df
211THz
Energy spectrum versus
spatial frequency
S (v) = F {S ( x)}
Signal energy
over 211-1081THz
Fig. 6.5: The block diagram of the post-processing technique used to obtain the energy spectrum.
voltage induced across the outer-segment using the following expression:
D/ 2
V(x, t) =
∫ E (x, y, z = 0, t)dy
y
(2)
− D/ 2
Here, D is the diameter of outer-segment at the specific x along the outer-segment.
Ey(x,y,z=0,t) is the component of the electric field parallel to the polarization of the
incident plane-wave sampled at the central cross-section of the outer-segment at each
time step. After applying (2) in each time step after 43.8fs and for all points along the
outer-segment, we Fourier-transform the induced voltage to obtain the induced voltage as
a function of frequency V(x,f). Using Parseval’s theorem and integrating |V(x,f)|2 over the
211THz – 1081THz bandwidth, we define the signal energy S(x) “trapped” inside the
outer-segment. Using the Fourier transform of S(x), the energy spectrum versus the
spatial frequency S(υ) is defined. Considering an average value of nr-average= 1.38 for the
refractive index of the outer-segment, we can then graph the energy spectrum versus the
free- space wavelength.
60
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
Fig. 6.6 shows the energy spectrum of the scattered field inside the outer-segment for
different values of D2 versus the free space wavelength. These results are normalized to
the maximum value of the energy spectrum obtained for D2=4.2µm. As we can observe,
the cylindrical (rod) shape contains the largest spectral energy range. This result confirms
the higher sensitivity of the rods to various wavelengths, as they are specialized for black
and white vision even in very dim light conditions. For the remaining three structures
which resemble the cone outer-segment, we note that there is a maximum value of the
spectral energy, localized in the wavelength sense, and that this wavelength,
corresponding to the peak energy value, changes considerably with the changes in D2.
Fig. 6.6: Energy spectrum versus free-space wavelength for different structures. The results are normalized
to the maximum value of energy spectrum at D2=4.2µm.
6.5 Conclusion
The interaction of EM wave with the photoreceptor (retinal rods and cones) outersegment has been investigated using the FDTD method.
The results of our investigations imply that the electromagnetic pulse propagates with
multiple reflections inside the outer-segment scatters and also experiences widening of its
spatial distribution within the outer-segment. A technique used to quantify this behaviour
and obtain the spectrum of the pulse has been proposed and used to obtain the spectrum
of the pulse propagating inside the outer-segment for the outer-segment of a rod and three
different cone shapes.
61
FDTD ANALYSIS OF LIGHT PROPAGATION IN THE HUMAN PHOTORECEPTOR CELLS
The results can be summarized as follows: 1) the electrodynamic model of the light
interaction with the retinal rod outer-segment suggests that this geometry contains the
maximum energy spectrum of all cases under investigation, confirming the role of rods in
highly sensitive monochromatic vision; 2) as the tapering angle of the cone structure
changes, the results show energy spectrum peaks for different wavelengths, suggesting
that the variation in cone shape and size plays a role in color discrimination of human
visual system.
Acknowledgments
The authors would like to thank Natural Sciences and Engineering Research Council
(NSERC) of Canada for their funding support.
62
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Preface to Chapter 7
The following chapter is included as a paper published in the IEEE Journal on Selected
Topics on Quantum Electronics: Special Issue on Bio-photonics. In this chapter, we
present additional studies of the biophysics of light interaction with a human retinal
photoreceptor. We show the filtering effect of the photoreceptor outer-segment on the
transmitted electromagnetic field. We also investigate the effect of field enhancement in
the tapered section of the outer-segment. The mode conversion along the photoreceptor
outer-segment is also presented.
63
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Chapter 7
Human Retinal Photoreceptors: Electrodynamic Model of Optical
Micro-Filters
Amir Hajiaboli and Milica Popović
Abstract— This paper presents a numerical study of the light interaction with the human
retinal photoreceptor using the finite-difference time-domain (FDTD) method. The
photoreceptors, cells in the retina responsible for monochromatic vision and color
discrimination, differ in shape and size. The cell type (rod or cone) and the variety among
the cone cells (S, M and L) condition their peak absorption wavelength (blue, green or
red). In our work, we observe the outer-segment of the photoreceptor as a micro-size
optical filter by modeling its interaction with the light wave. Several aspects of wideband pulse propagation, such as field enhancement and filtering properties, have been
investigated.
Index terms: Human photoreceptor, finite-difference-time domain (FDTD), field
enhancement, irradiant flux.
7.1 Introduction
The retinal surface of the human visual system contains specialized cells, known as rods
and cones, responsible for detection of light and its transduction to electrical signals,
which are then sent for processing in the visual cortex. The rod-shaped cells play the
main role in monochromatic vision by detecting the intensity of light. The human color
64
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
perception is a result of the absorption of the three constituent colors (red, green and
blue) by three cone-cell photoreceptors (L, M and S).
Each human photoreceptor consists of two distinctive segments: 1) inner-segment and 2)
outer-segment. It is the outer-segment that dominates the geometry of the photoreceptor
and within which the incident light wave changes into the electrical signal. The shape
and dimensions of the outer-segment determine the type of the photoreceptor (M, L, S
cone or rod). Fig. 7.1 shows a typical structure of the rod and cone outer-segment. The
cone outer-segment varies in length and conical angle among the cone photoreceptor
cells. Our study is motivated by the similarity of the outer-segment structure to that of the
optical filters and its goal is to investigate the effect of the physical structure of the
photoreceptor cell on its optical filtering properties. In order to study the retinal cones as
optical filters, we here investigate their electrodynamic properties using the finitedifference time-domain (FDTD) method [26]. Literature to date reports on twodimensional FDTD models deployed for studies on electromagnetic wave propagation
within the photoreceptor structure as an optical waveguide [64] and [63]. For example,
results of [64] showed that the rod outer-segment is a frequency-independent structure,
i.e. that the spectrum of the electric field inside it is the same for all three wavelengths of
blue, red and green. A recently reported study exploits body of revolution FDTD (BORFDTD) [63] to model the cylindrical shape of the photoreceptor. Considering the
dominant electromagnetic mode propagating inside the photoreceptor, the investigators
successfully modeled the Stiles-Crawford effect, which explains the intensity change of
the received electromagnetic power as the direction of the incident wave changes.
65
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Fig. 7.1: The outer-segment of the photoreceptor for the a) rod and b) cone cells. Adopted from Fig. 3.4, pp.
34 in [12].
The methodology of both [64] and [63] was limited to two-dimensional space due to the
heavy computational burden of three-dimensional modeling. This limitation could affect
the number of identified electromagnetic modes propagating inside the outer-segment.
Consequently, although it offers valuable insight, the model lacks in accuracy of physical
representation.
In this paper, similar to the approach of [75], we conduct a three-dimensional parametric
study of the outer-segment in order to assess its optical properties. The methodology
exploited in [75] to assess the filtering effects was based on evaluating the spectrum of
the wave “trapped” inside the outer-segment. This energy remains inside the outersegment due to total internal reflections and it is indeed proportional to the energy
scattered by the outer-segment. Here, we look at the spectrum of the scattered field to
investigate the filtering properties of the outer-segment.
In our study, the structure has been illuminated by a wide-band Gaussian pulse centered
at the middle of the visible range of the electromagnetic spectrum. This pulse is
representative of the white light with its wide-band spectrum. We then study different
aspects of the pulse propagation inside a typical model of the cone outer-segment such as
field enhancement and filtering properties. A parametric study based on the different
66
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
values of D2 (the diameter of the smaller cone base in Fig. 7.2) is conducted to evaluate
and assess the filtering effects.
7.2 Outer-Segment Physical Structure
The physical structure of the photoreceptor outer-segment is shown in Fig. 7.2. It consists
of very thin (~15nm) stacked disks of different refractive index that alternates in value
between that of the cytoplasm and that of the folded cell membrane. For relative
permittivity, we adopt the values of generally accepted physiological data [63], [67] for
the cytoplasm εr-cytoplasm=1.85, the membrane εr-membrane=2.01 and the intercellular medium
εr-intercellular=1.79. The membrane of the structure contains electric loss of σmembrane=37.38
S/m [67]. The length of the outer-segment considered for our work (6.75µm) is,
approximately, that of the reported length of the peripheral cones [24].
In the rod outer-segment, the disks’ radii are constant. In the cone outer-segment, the
disks’ radii linearly decrease along the central axis of the photoreceptor.
ε r-intercellular
L
D2
D1
y
τ1
τ2
z
x
εr-cytoplasm
ε r-membrane
Fig. 7.2: Model of the geometry of the cone outer-segment. In our study, L=6.75µm and D1=4.2µm, while
D2 is a variable parameter. The structure consists of very thin (~15nm) alternating stacked layers of
different relative permittivity values. For the cytoplasm, τ1=15nm, εr-cytoplasm=1.85; for the folded membrane,
τ2=15nm, εr-membrane=2.01 σmembrane=37.38 S/m; for the medium surrounding the outer-segment, εrintercellular=1.79 [63, 67].
67
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
7.3 Simulation Parameters
The geometry is modeled by a uniform FDTD space with ∆x=5nm and ∆y=∆z=30nm.
This discretization results in a total of 1600×200×200=60 million FDTD cells. The timestep, based on the Courant-stability condition, is set to ∆t=1.46×10-2fs. To study the
scattered electromagnetic field and the filtering effects of outer-segment structure totalfield/scattered-field (TF/SF) formulation has been adopted. The incident signal is a ypolarized plane wave propagating in the x-direction, and it is a sinusoid-modulated
wideband Gaussian pulse centered at 622THz:
E y = Ae
−
( t-t0 ) 2
2σ 2
sin ( 2πf(t − t 0 ))
(1)
Here A=1V/m is the amplitude of the signal, σ=100∆t is the mean variation of the signal,
t0=400∆t is the delay introduced to preserve the causality of the signal and f=622THz is
the frequency of the modulation signal (the region of visible light, λ=482nm).
Fig. 7.3 shows the spectrum of the incident electric field (|Ey,inc|) versus wavelength (λ).
The peak value of the incident field is 125V/m which is centered at λ=482nm. The full
width at half maximum (FWHM) of the incident plane wave is 250nm and as shown in
Fig. 7.3 this covers the required bandwidth to properly model the white light spectrum.
Fig. 7.3: Incident electric field (|Ey,inc|) versus wavelength.
68
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
7.4 Electric Field in the Outer-segment
Fig. 7.4 plots the magnitude of the Ey component of electric field inside the outersegment at different snapshots in time for D2=1.2µm. As can be observed, as the incident
field propagates through the structure, it is enhanced and confined to the center of the
structure.
The field distribution shown at t=43.80(fs) is the scattered field after the excitation passes
the full length of the outer-segment. We can see that the forward-scattered fields include
mostly the fields guided by the outer-segment.
Fig. 7.4: |Ey| in the outer-segment when D2=1.2µm for three different snapshots in time, from top to down
t=21.9fs, 32.85fs, 43.80fs.
Figs. 7.5a and 7.5b show the field distributions along the cone outer-segment at x=3.5µm
and x=6µm for f=622THz when D2=1.2µm. As it can be observed, the modal distribution
changes considerably along the cone outer-segment, showing that the tapered structure of
the outer-segment can act like a mode converter [76]
69
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
(a)
(b)
Fig. 7.5: Ey field distribution along the cone radius at f=622 THz, when D2=1.2µm a) x=3.5µm b)x=6µm.
70
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Fig 7.6 shows the spectrum of Ex and Ez components of scattered electric field (|Ex,scat|
and |Ez,scat|) when D2=1.2µm. Fig. 7.7 shows the Ey component of scattered electric field
(|Ey,scat|) versus wavelength for different values of D2. The scattered electric fields are
sampled at a point on the central axis of the outer-segment and close to the TF/SF
domain. As one can observe, the |Ey,scat| has a considerably higher value than |E,x,scat| and
|Ez,scat| and this shows that the scattered field largely retains the same polarization as the
incident field.
Because of the physiological orientation of the rhodopsin molecules inside the membrane
layers, the components of the electric field which are parallel to the layers of membrane
cause the highest absorption rate [12]. Considering this fact, the vividly smaller value of
the Ex component of the scattered electric field shown in Fig. 7.6 is a physiologically
important effect since this is the component normal to the membrane layers.
Fig. 7.7 shows that as D2 decreases, the peak value of |Ey,scat| increases. By changing D2
from 4.2µm to 1.8µm, the wavelength at the peak value of |Ey,scat| shifts downward while
decreasing D2 from 1.8µm to 1.2µm shifts the peak-value wavelength upward.
Fig. 7.6: Ex and Ez components of the scattered electric field (|Ex,scat| and |Ez,scat|) versus wavelength.
71
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Fig. 7.7: Ey component of scattered electric field (|Ey,scat|) versus wavelength.
To evaluate the field enhancement effect, we have divided the spectrum of the |Ey,scat| by
the spectrum of the incident field |Ey,inc| and calculated the amplification factor versus
wavelength. Fig. 7.8 shows the amplification factor and it can be observed as D2
decreases the amplification factor increases. It is also interesting to note that the
minimum of the amplification (around 500nm) for D2=1.2µm moves toward higher
wavelengths as D2 increases.
Fig. 7.8: Amplification factor (|Ey,scat|/ |Ey,inc|) versus wavelength.
72
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
7.5 Irradiant Flux Calculation
To calculate the irradiant flux, we consider a box-like surface outside the TF/SF domain
and very close to it (Fig. 7.9). We then calculate the Poynting power crossing this surface
at each time-step. SB, SL and SF shown in Fig. 7.9 are the surfaces to calculate the backscattered, lateral-scattered, and forward-scattered irradiance flux, respectively. After
applying the Fourier transform, we obtain the spectrum of the scattered irradiant flux
versus wavelength as depicted in Figs. 7.10, 7.11 and 7.12 for D2=1.2µm.
As one can observe, the forward-scattered irradiant flux shown in Fig. 7.12, which
includes the waves guided by the structure, is the dominant scattered irradiant flux. The
lateral-scattered irradiant flux shown in Fig. 7.11 is the smallest scattered irradiant flux.
The spectrum of the back-scattered and forward-scattered irradiant flux has mostly a
Gaussian-like distribution while the spectrum of the lateral-scattered irradiant flux
contains several local minima at different wavelengths.
Fig. 7.9: Three dimensional view of the cone inside the total-field scattered-field domain. SB, SF and SL are
the surfaces to calculate the back-scattered, forward-scattered and lateral-scattered irradiance flux,
respectively.
For a better comparison, Fig. 7.13 shows the normalized forward-scattered irradiate flux
for different values of D2. The graphs are normalized to the peak value of forwardscattered irradiant flux obtained at D2=4.2µm, which can be considered as a rod outersegment. We can observe that by changing D2, the peak value of the scattered irradiant
flux changes; also, the spectrum of the scattered irradiant flux deviates from a Gaussianlike distribution. Both of these effects are of physiological interest, as they show that the
dimension can change the spectrum of the incident field, thus having a filtering effect
which can play a role in the absorption spectrum in each of the specialized photoreceptor
structures (blue, green or red cones or rods).
73
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Fig. 7.10: Back-scattered irradiant flux versus wavelength for D2=1.2µm.
Fig. 7.11: Lateral-scattered irradiant flux versus wavelength for D2=1.2µm.
74
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
Fig. 7.12: Forward-scattered irradiant flux versus wavelength for D2=1.2µm.
Fig. 7.13: Normalized forward-scattered irradiant flux versus wavelength. The graphs are normalized to the
maximum of the irradiant flux calculated for D2=4.2µm.
75
HUMAN RETINAL PHOTORECEPTORS: ELECTRODYNAMIC MODEL OF OPTICAL MICRO-FILTERS
7.6 Conclusion
Several aspects of the wide-band Gaussian pulse propagation inside the human
photoreceptors’ outer-segment have been investigated using the FDTD method.
The results show the field enhancement effect for the component of the scattered electric
field parallel to the incident polarization. It is observed that changes in D2 dramatically
change the peak value of the scattered electric field that is parallel to the incident
polarization as well as its corresponding wavelength. We calculated the amplification
factor versus wavelength and observed that, as D2 decreases, the amplification factor
increases. Further, the minimum of the amplification factor shifts toward higher
wavelengths as D2 increases.
We computed the scattered irradiant flux on each side of the outer-segment and
investigated the filtering effects of different cone and rod structures on the irradiant flux.
The differences in the structures under investigation have been shown to be able to shift
the peak value of the Gaussian pulse and deform its distribution from a Gaussian
distribution.
Finally, it is important to point out the significance of the FDTD method in the area of
bio-photonics. With evolution and fine-tuning of the FDTD algorithm itself, as well as
the increased availability of high-end computational resources, the previously intense and
demanding FDTD calculations no longer present a prohibitive obstacle to bio-photonic
studies. In fact, the reported work here demonstrates an investigation that would be
practically impossible to conduct experimentally on retinal cell samples. For example,
capturing the electric field distribution in a retinal cell in space and time is, with presently
available instrumentation, an extremely difficult task. Hence, accurate and precise FDTD
simulations of such phenomena represent a unique analysis tool and can therefore be
essential for the overall advancement of the bio-photonics area.
Acknowledgments
The authors would like to thank Natural Sciences and Engineering Research Council
(NSERC) of Canada for their funding support.
76
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
Preface to Chapter 8
The following chapter in included as a paper under preparation. A novel technique has
been exploited in this paper to extract the absorption spectrum of the rhodopsin. The
results obtained have already been presented in the proceeding of the Photonics North
conference which was held in June 2008 in Montreal. Based on this technique, we extract
the absorption spectrum of rhodospin molecules, which is dependent on the imaginary
part of the dielectric constant of membrane layers of the photoreceptor outer-segment.
We then investigate the effect of outer-segment morphology on the absorbed electric
field.
77
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
Chapter 8
Rhodopsin Absorption Spectrum:
Effect of the Photoreceptor Outer-Segment Morphology
Amir Hajiaboli, Milica Popović, McGill University
[email protected], [email protected]
Abstract- In this paper, we study the effect of morphology of the photoreceptor outersegment on the absorption spectrum of rhodopsin molecules, which exist in the
membrane layers of the outer-segment. We have modeled different structures of the
photoreceptor outer-segment using the finite-difference time-domain (FDTD) technique.
To compute the absorption spectrum, for each of the geometries, we subtract the FDTD
simulation results with and without the rhodopsin absorption spectrum. Our study shows
a significant change in the peak absorption and the wavelength of peak absorption for
different geometries of the outer-segment.
8.1 Introduction
The human retina consists of rods (black and white vision) and cones (color vision). The
shape and dimensions of the cones are different and they contain different photochemical
components specialized for maximal absorption at different wavelengths. The photochemical in rods, rhodopsin, is almost the same as the photo-chemical in the cones, called
chlorolabe (green-sensitive), erythrolabe (red-sensitive) and cyanolabe (blue-sensitive) [6,
15]. Although these photo-chemicals are specialized in sensing distinctive colors, the
wave-guiding effect in photoreceptors can also affect their absorption spectrum as
previously discussed by Snyder in 1969 [77]. The Snyder’s investigation was based on
78
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
the changes in modal pattern with respect to the angle of the incident wave in a simplified
model of the photoreceptor.
In our work, we study the effect of photoreceptor outer-segment morphology on the
absorption of these specialized photo-chemicals. To study this effect we have adopted
finite-difference time-domain (FDTD) numerical techniques. In our previous work [68,
69], we have investigated the effect of outer-segment morphology on the propagated and
transmitted electromagnetic field. Considering a constant absorption coefficient, the
transmitted field is proportional to the absorbed electromagnetic field. This
approximation can provide a good insight into the problem of light interaction with the
human retinal photoreceptor, especially for monochromatic light excitation. However, the
absorption spectrum of rhodopsin molecules varies with frequency and that affects the
overall absorption spectrum of the photoreceptor under the natural white light excitation
[78]. In this paper, we model the absorption band of the rhodopsin molecule using a
Lorentz dispersion model as previously discussed in [8]. We then compute the absorbed
electric field by subtracting two FDTD simulations with and without the absorption band
of rhodopsin molecules. The technique is similar to the approach adopted by Enoch to
explain the role of the photoreceptors wave-guiding effect on the absorbed power [79].
Our results show a significant effect of the photoreceptor morphology on the spectrum of
the absorbed light in the rhodopsin molecules.
8.2 Outer-Segment Physical Structure
The model of the physical structure of the photoreceptor outer-segment is shown in Fig.
8.1. It consists of very thin (~15nm) stacked disks of different refractive index that
alternates in value between that of the cytoplasm and that of the folded cell membrane.
For relative permittivity, we have adopted the values of generally accepted physiological
data [63, 67] for the cytoplasm εr-cytoplasm=1.85 and the intercellular medium εr-intercellular=1.79. As we will show in the next section, the dielectric constant of the membrane,
which is the site of rhodopsin molecules, follows a Lorentz dispersive model and the
precise dielectric constant of membrane layers including the rhodopsin molecules will be
discussed in the next section. The dielectric constant of the membrane can vary with time
due to bleaching of rhodopsin molecules [70] , however, when the effect of bleaching is
79
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
negligible we can consider an average value of εr-membrane_average=1.96025. The length of
the outer-segment considered for our work (6.75µm) is, approximately, that of the
reported length of the peripheral cones [7].
ε r-intercellular
L
D2
D1
y
τ1
τ2
z
x
εr-cytoplasm
ε r-membrane
Fig. 8.1: Model of the geometry of the cone outer-segment. L=6.75µm and D1=4.2µm, while D2 is a
variable parameter (D2=1.2µm, D2=4.2µm). The structure consists of very thin (~15nm) alternating stacked
layers of different relative permittivity values. For the cytoplasm, τ1=15nm, εr-cytoplasm=1.85; for the folded
membrane, τ2=15nm, εr-membrane-average=1.96025; for the medium surrounding the outer-segment, εrintercellular=1.79.
8.3 Rhodopsin Absorption Spectrum
As it was discussed previously, the dielectric constant of membrane layer which is the
site of rhodopsin molecules follows a Lorentz dispersive expression as given in (1).
ε ( ω ) = ε inf +
Ap ( ε static − ε inf )ω 2p
ω 2p − 2iωδ damp − ω 2
(1)
Here, εstatic=1.9605 is the value of dielectric permittivity at low frequency and εinf=1.9600
is the dielectric permittivity at high frequency. Ap=1 is the pole amplitude,
δdamp=395×1012 Rad/sec is the damping factor which defined the bandwidth of the
Lorentz pole and ωp=3770×1012 Rad/sec is the resonance frequency of the Lorentz pole.
Fig. 8.2 shows the variation of real and imaginary part of permittivity versus wavelength
for membrane layers. As it can be observed, it is centered at λ=490nm which is the peak
absorption level for rods photoreceptors.
80
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
Fig. 8.2: The Lorentz dispersive dielectric model of membrane layers.
8.4 Simulation Parameters
The geometry is modeled by a uniform FDTD space with ∆x=5nm and ∆y=∆z=50nm.
This discretization results in a total of 1600×140×140=31.36 million FDTD cells. The
time-step, based on the Courant-stability condition, is set to ∆t=1.49×10-2fs. To study the
scattered electromagnetic field and the filtering effects of outer-segment structure, the
total-field/scattered-field (TF/SF) formulation has been adopted. The incident signal is a
y-polarized plane wave propagating in the x-direction, and it is a sinusoid-modulated
wideband Gaussian pulse centered at 622THz:
E y = Ae
−
( t-t0 ) 2
2σ 2
sin ( 2πf(t − t 0 ))
(2)
Here A=1V/m is the amplitude of the signal, σ=100∆t is the mean variation of the signal,
t0=400∆t is the delay introduced to preserve the causality of the signal and f=622THz is
the frequency of the modulation signal (the region of visible light, λ=482nm).
Fig. 8.3 shows the spectrum of the incident electric field (|Ey,inc|) versus wavelength (λ).
The peak value of the incident field is 125V/m which is centered at λ=482nm. The full
width at half maximum (FWHM) of the incident plane wave is 250nm and as shown in
Fig. 8.3 this covers the required bandwidth to properly model the white light spectrum.
81
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
Fig. 8.3: Incident electric field (|Ey,inc|) versus wavelength.
8.5 The Transmitted and Absorbed Electric Field
Fig. 8.4 shows the transmitted electric field when D2=1.2µm with and without the
rhodopsin molecules included. As it can be observed, although the effect of rhodopsin
absorption is very small, the transmitted field when the rhodopsin molecules are included
is slightly smaller than when rhodopsin molecules are excluded from the structure. The
difference is due to the losses associated with the imaginary part of dielectric constant.
By subtracting these two values, one can find the absorption spectrum of rhodopsin.
Fig. 8.5 shows the absorbed electric field for different values of D2. As it can be observed,
the peak of absorption and also the wavelength of the peak absorption vary dramatically
for different values of D2. Furthermore, the field enhancement effect which occurs in the
tapered structure [68] results in a higher value of absorption. We also show the absorbed
electric field when the structure is a cylinder which resembles the rod structure. It can be
observed, the absorbed spectrum of the electric field changes significantly in both
structures. This shows the filtering effect of the outer-segment shape and dimensions on
the absorption spectrum of the rhodpsin molecules.
82
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
Fig. 8.4: Transmitted electric field when D2=1.2µm.
Fig. 8.5: Absorbed electric field versus wavelength for different values of D2.
83
RHODOPSIN ABSORPTION SPECTRUM: EFFECT OF THE PHOTORECEPTOR OUTER-SEGMENT MORPHOLOGY
8.6 Conclusion
This paper presents the numerical analysis of light interaction with human retinal
photoreceptor to investigate the role of diversity in the photoreceptor shape and
dimensions. We have developed a technique to approximately find the absorption
spectrum of the in situ rhodopsin molecules in different photoreceptor structures. Our
results show the significant effect of photoreceptor shape and dimensions in the spectrum
of the absorbed light by the rhodopsin molecules.
84
CONCLUSION AND FUTURE WORK
Chapter 9
Conclusion and Future Work
9.1 Conclusion
The main focus in this thesis was to investigate the role of diversity in the photoreceptor
shapes and dimensions. While the spectral sensitivity of photoreceptors is governed by
the photo-chemicals inside the outer-segment of the photoreceptor, we have shown that
the shapes and dimensions of photoreceptor can also affect this spectrum. These results
also support the previously suggested hypothesis on the filtering effect of the
photoreceptor shapes and dimensions [80]. Due to the inherent complexity to carry out
experimental investigation, we have conducted the research using a well-trusted
computational electromagnetic technique, finite-difference time-domain (FDTD).
Different FDTD techniques for investigating the structure have been developed and
compared and a framework for parametric analysis of the photoreceptor based on 3D
FDTD technique was developed. As a part of the thesis, some new aspects of light
interaction with human photoreceptor have been investigated. The conclusions of the
thesis which were presented in previous manuscript based chapters can be summarized as
follows:
•
The possible FDTD techniques such as the two-dimentional (2-D) rotationally
symmetric technique or the three-dimensional (3-D) technique have been
compared. Due to inherent limitation of the 2-D rotationally symmetric technique
in supporting all the waveguide modes propagating in the outer-segment, the
required computational resources to carry out a 3-D investigation have been
developed.
85
CONCLUSION AND FUTURE WORK
•
A precise model of the photoreceptor, including the stacking layers of cytoplasm
and membrane, has been compared with its bulk average model. The results show
that the wideband pulse, which covers the spectrum of visible light, is shifted to
higher wavelengths when it is propagating in the stacking layers. This result is in
accordance with the available physiological data, exhibiting the slightly shifted
absorption spectrum for the in situ rhodopsin.
•
The filtering effect of the photoreceptor outer-segment on the propagating electric
field inside the photoreceptor has been shown. To this end, we have developed a
novel signal processing technique to extract the average shift in the spectrum of
the propagating electric field inside the photoreceptor outer-segment. The results
obtained show a significant filtering effect of the photoreceptor outer-segment on
the electric field trapped due to total internal reflections.
•
Furthermore, we have investigated the filtering effect of the photoreceptor outersegment of the transmitted electric field. Our results show the significant shift in
the spectrum of the electric field by changing the photoreceptor outer-segment.
Also, we have observed that the electric field is enhanced as it propagates along
the outer-segment which can help with a better absorption of the electric field in
the rhodopsin molecules. Furthermore, by extracting the modal distribution along
different cross-section of the outer-segment we have observed the mode
conversion along the outer-segment.
•
In the end, we have developed a novel technique to extract the absorption
spectrum of the rhodopsin molecules. The technique is based on subtracting two
FDTD simulations including and excluding the rhodopsin molecules. The results
show a small portion of the electromagnetic energy will be absorbed in the
rhodopsin molecules. Finally, we have observed that the absorbed spectrum will
be altered significantly by changing the photoreceptor structure.
9.2 Future Work
Our investigations have resulted in some intriguing findings, and, further, they open new
avenues of possible research. The following suggested directions would represent a
natural continuation of the research presented in this thesis:
86
CONCLUSION AND FUTURE WORK
•
The white light source has been modeled using a coherent broad-band source.
With a random phase distribution, a better model of light source can be
considered and this may help with a better physical representation of the problem.
•
A full parametric investigation which incorporates the effect of the bleaching on
the dielectric constant of the membrane layers can be conducted.
•
Rhodopsin molecules orient in such a way that they maximally absorb in the
plane parallel to the membrane layers. This will cause an anisotropic absorption
effect which can be modeled using anisotropic losses in the FDTD model.
•
Using the model described above it would be possible to investigate the
polarization sensitivity which has been observed in some types of visual systems.
•
The main focus in this thesis was about the parametric analysis of the outersegment which is the site of photo-pigments. A more comprehensive model
including the inner segment and the mitochondria distribution can also be
developed. This model helps to better understand the mitochondria scattering
effect on the total absorption spectrum.
87
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
Appendix I
3D FDTD Source Code for Modeling
Photoreceptor Using Acceleware Software
/***********************************************************************
*******
* This is program is the main program to simulate the photoreceptor outer-segmnet. The
program is based on FDTD and it uses some of the predefined libraries developed by
acceleware company (www.acceleware.com)
Developer: Amir Hajiaboli, PhD student, CADlab, McGill University
************************************************************************
******/
#include <AccelewareFDTD.h>
#include <iostream>
#include <cmath>
#include <string>
#include <time.h>
#include <io.h>
#include <stdio.h>
#include <stdlib.h>
#include <fcntl.h>
#include <sys/types.h>
#include <sys/stat.h>
#include <errno.h>
#include <share.h>
static const ax_float_t axmath_pi = 3.14159265358979323846f;
static const ax_float_t axmath_c = 2.99792458e8f;
static const ax_float_t axmath_eps0 = 8.854187817e-12f;
static const ax_float_t axmath_mu0 = axmath_pi * 4e-7f;
static const ax_float_t axmath_sigma0 = 0.0;
static const ax_float_t axmath_sigma1 = 0.0;
88
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
static const ax_float_t axmath_sigma2 = 37.34f;
static const ax_float_t axmath_rho0 = 0.0;
static const ax_field_t field;
static const ax_axis_t axis;
void CheckAx(void)
{
if(AXE_SUCCESS!=ax_error)
{
std::cout << std::string(AxGetErrStr(ax_error)) + ": " + AxGetErrMsg()
<< std::endl;
exit(1);
}
}
int main()
{
std::cout << "Retina Simulation" << std::endl;
std::cout << "Setting up simulation..." << std::endl;
// Initialize the system. Each process can have only one system,
// but each system may contain multiple simulations.
AxOpenSystem(AX_LOG_ENABLED, "D:\Retina simulation ");
CheckAx();
// Opening the output file for the data
int
fileHandle = 0;
unsigned bytesWritten = 0;
if ( _sopen_s(&fileHandle, "write.o", _O_RDWR | _O_CREAT,
_SH_DENYNO, _S_IREAD | _S_IWRITE) )
return -1;
//////////////////////////////////////////////////////////////////////
// Simulation Set Up
//////////////////////////////////////////////////////////////////////
// Create a simulation, with logging output going to standard out.
ax_dim_t sizeX = 1600;
ax_dim_t sizeY = 200;
ax_dim_t sizeZ = 200;
float SIZE_X = sizeX,
SIZE_Y = sizeY,
SIZE_Z = sizeZ;
float Cone_Length=1350;
float Start_Radius = 70,
End_Radius = 20,
RADIUS=0;
int Start_Point = 51,
End_Point =Cone_Length+Start_Point;
89
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
float Cone_Slope=(Start_Radius-End_Radius)/(Start_Point-End_Point);
float
TFSF_X0=20,
TFSF_Y0=20,
TFSF_Z0=20,
TFSF_X1=Cone_Length+Start_Point+50,TFSF_Y1=180,TFSF_Z1=180;
ax_simhandle_t simHandle;
AxCreateSimulation(AX_LOG_ENABLED,
"D:\Retina ",
// Logging folder, NULL->stdout
&simHandle,
AX_SOLVER_HW,
sizeX, sizeY, sizeZ,
// Simulation size
AX_BND_UPML, AX_BND_UPML,
AX_BND_UPML, AX_BND_UPML,
AX_BND_UPML, AX_BND_UPML);
CheckAx();
// Use a uniform grid, where each cell in the grid is a cube.
ax_float_t deltax = 0.000000005f;
ax_float_t deltay = 0.00000003f;
ax_float_t deltaz = 0.00000003f;
AxSetUniformDelta(simHandle, AX_AXIS_X, deltax);
AxSetUniformDelta(simHandle, AX_AXIS_Y, deltay);
AxSetUniformDelta(simHandle, AX_AXIS_Z, deltaz);
CheckAx();
// Calculate the time step size from the spatial discritization
// using the Courant criteria.
ax_float_t timeDelta = 0.9f /( axmath_c * sqrt(1.0f / (deltax * deltax)+1.0f / (deltay *
deltay)+1.0f / (deltaz * deltaz)));
AxSetDeltaT(simHandle, timeDelta);
CheckAx();
ax_mathandle_t freeSpace, freeMagSpace, glass, dielSpace1,dielSpace2;
AxCreateLdmMaterial(&freeSpace, 1.7956*axmath_eps0, axmath_sigma0);
AxCreateLdmMaterial(&dielSpace1, 1.85*axmath_eps0, axmath_sigma1);
AxCreateLdmMaterial(&dielSpace2, 1.95*axmath_eps0, axmath_sigma2);
AxCreateLdmMaterial(&glass, 8.2f*axmath_eps0, axmath_sigma0);
AxCreateMagneticMaterial(&freeMagSpace, 3.14f*4e-7f, 0.0f);
// Cerating regions to evaluate the scattered power
//Top plane
ax_rgnhandle_t obsPlaneTopEx,obsPlaneTopEz,obsPlaneTopHx,obsPlaneTopHz;
AxCreateRegion(
&obsPlaneTopEx,
simHandle,
AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_X,
1, TFSF_Y1+5, 1, SIZE_X, TFSF_Y1+6, SIZE_Z);
90
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
AxCreateRegion(
&obsPlaneTopEz,
simHandle,
AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Z,
1, TFSF_Y1+5, 1, SIZE_X, TFSF_Y1+6, SIZE_Z);
AxCreateRegion(
&obsPlaneTopHx,
simHandle,
AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_X,
1, TFSF_Y1+5, 1, SIZE_X, TFSF_Y1+6, SIZE_Z);
AxCreateRegion(
&obsPlaneTopHz,
simHandle,
AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Z,
1, TFSF_Y1+5, 1, SIZE_X, TFSF_Y1+6, SIZE_Z);
unsigned numObsTopXHx, numObsTopYHx, numObsTopZHx,numObsTopXHz,
numObsTopYHz, numObsTopZHz;
unsigned numObsTopXEx, numObsTopYEx, numObsTopZEx,numObsTopXEz,
numObsTopYEz, numObsTopZEz;
AxGetRegionSize(obsPlaneTopEx,
&numObsTopXEx,
&numObsTopYEx,
&numObsTopZEx);
AxGetRegionSize(obsPlaneTopEz,
&numObsTopXEz,
&numObsTopYEz,
&numObsTopZEz);
AxGetRegionSize(obsPlaneTopHx,
&numObsTopXHx,
&numObsTopYHx,
&numObsTopZHx);
AxGetRegionSize(obsPlaneTopHz,
&numObsTopXHz,
&numObsTopYHz,
&numObsTopZHz);
unsigned
obsArraySizeTopEx=numObsTopXEx*numObsTopYEx*numObsTopZEx;
unsigned
obsArraySizeTopEz=numObsTopXEz*numObsTopYEz*numObsTopZEz;
unsigned
obsArraySizeTopHx=numObsTopXHx*numObsTopYHx*numObsTopZHx;
unsigned
obsArraySizeTopHz=numObsTopXHz*numObsTopYHz*numObsTopZHz;
float* ExPlaneTop;
float* EzPlaneTop;
float* HxPlaneTop;
float* HzPlaneTop;
ExPlaneTop = NULL;
EzPlaneTop = NULL;
HxPlaneTop = NULL;
HzPlaneTop = NULL;
ExPlaneTop = new float[obsArraySizeTopEx];
EzPlaneTop = new float[obsArraySizeTopEz];
HxPlaneTop = new float[obsArraySizeTopHx];
HzPlaneTop = new float[obsArraySizeTopHz];
//Right plane
91
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
ax_rgnhandle_t
obsPlaneRightEx,obsPlaneRightEy,obsPlaneRightHx,obsPlaneRightHy;
AxCreateRegion( &obsPlaneRightEx, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_X,
1, 1, TFSF_Z1+5, SIZE_X, SIZE_Y, TFSF_Z1+6);
AxCreateRegion( &obsPlaneRightEy, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Y,
1, 1, TFSF_Z1+5, SIZE_X, SIZE_Y, TFSF_Z1+6);
AxCreateRegion( &obsPlaneRightHx, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_X,
1, 1, TFSF_Z1+5, SIZE_X, SIZE_Y, TFSF_Z1+6);
AxCreateRegion( &obsPlaneRightHy, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Y,
1, 1, TFSF_Z1+5, SIZE_X, SIZE_Y, TFSF_Z1+6);
unsigned
numObsRightXHx,
numObsRightYHx,
numObsRightZHx,numObsRightXHy, numObsRightYHy, numObsRightZHy;
unsigned
numObsRightXEx,
numObsRightYEx,
numObsRightZEx,numObsRightXEy, numObsRightYEy, numObsRightZEy;
AxGetRegionSize(obsPlaneRightEx,
&numObsRightXEx,
&numObsRightYEx,
&numObsRightZEx);
AxGetRegionSize(obsPlaneRightEy,
&numObsRightXEy,
&numObsRightYEy,
&numObsRightZEy);
AxGetRegionSize(obsPlaneRightHx, &numObsRightXHx, &numObsRightYHx,
&numObsRightZHx);
AxGetRegionSize(obsPlaneRightHy, &numObsRightXHy, &numObsRightYHy,
&numObsRightZHy);
unsigned
obsArraySizeRightEx=numObsRightXEx*numObsRightYEx*numObsRightZEx;
unsigned
obsArraySizeRightEy=numObsRightXEy*numObsRightYEy*numObsRightZEy;
unsigned
obsArraySizeRightHx=numObsRightXHx*numObsRightYHx*numObsRightZHx;
unsigned
obsArraySizeRightHy=numObsRightXHy*numObsRightYHy*numObsRightZHy;
float* ExPlaneRight;
float* EyPlaneRight;
float* HxPlaneRight;
float* HyPlaneRight;
ExPlaneRight = NULL;
EyPlaneRight = NULL;
HxPlaneRight = NULL;
HyPlaneRight = NULL;
ExPlaneRight = new float[obsArraySizeRightEx];
EyPlaneRight = new float[obsArraySizeRightEy];
HxPlaneRight = new float[obsArraySizeRightHx];
HyPlaneRight = new float[obsArraySizeRightHy];
92
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
// Front plane
ax_rgnhandle_t
obsPlaneFrontEy,obsPlaneFrontEz,obsPlaneFrontHy,obsPlaneFrontHz;
AxCreateRegion( &obsPlaneFrontEy, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Y,
TFSF_X0-6, 1, 1,TFSF_X0-5, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneFrontEz, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Z,
TFSF_X0-6, 1, 1,TFSF_X0-5, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneFrontHy, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Y,
TFSF_X0-6, 1, 1,TFSF_X0-5, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneFrontHz, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Z,
TFSF_X0-6, 1, 1,TFSF_X0-5, SIZE_Y, SIZE_Z);
unsigned
numObsFrontXHy,
numObsFrontYHy,
numObsFrontZHy,numObsFrontXHz, numObsFrontYHz, numObsFrontZHz;
unsigned
numObsFrontXEy,
numObsFrontYEy,
numObsFrontZEy,numObsFrontXEz, numObsFrontYEz, numObsFrontZEz;
AxGetRegionSize(obsPlaneFrontEy,
&numObsFrontXEy,
&numObsFrontYEy,
&numObsFrontZEy);
AxGetRegionSize(obsPlaneFrontEz,
&numObsFrontXEz,
&numObsFrontYEz,
&numObsFrontZEz);
AxGetRegionSize(obsPlaneFrontHy,
&numObsFrontXHy,
&numObsFrontYHy,
&numObsFrontZHy);
AxGetRegionSize(obsPlaneFrontHz,
&numObsFrontXHz,
&numObsFrontYHz,
&numObsFrontZHz);
unsigned
obsArraySizeFrontEy=numObsFrontXEy*numObsFrontYEy*numObsFrontZEy;
unsigned
obsArraySizeFrontEz=numObsFrontXEz*numObsFrontYEz*numObsFrontZEz;
unsigned
obsArraySizeFrontHy=numObsFrontXHy*numObsFrontYHy*numObsFrontZHy;
unsigned
obsArraySizeFrontHz=numObsFrontXHz*numObsFrontYHz*numObsFrontZHz;
float* EyPlaneFront;
float* EzPlaneFront;
float* HyPlaneFront;
float* HzPlaneFront;
EyPlaneFront = NULL;
EzPlaneFront = NULL;
HyPlaneFront = NULL;
93
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
HzPlaneFront = NULL;
EyPlaneFront = new float[obsArraySizeFrontEy];
EzPlaneFront = new float[obsArraySizeFrontEz];
HyPlaneFront = new float[obsArraySizeFrontHy];
HzPlaneFront = new float[obsArraySizeFrontHz];
//Back plane
ax_rgnhandle_t
obsPlaneBackEy,obsPlaneBackEz,obsPlaneBackHy,obsPlaneBackHz;
AxCreateRegion( &obsPlaneBackEy, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Y,
TFSF_X1+5, 1, 1,TFSF_X1+6, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneBackEz, simHandle, AX_ACCESS_READ,
AX_FIELD_E, AX_AXIS_Z,
TFSF_X1+5, 1, 1,TFSF_X1+6, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneBackHy, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Y,
TFSF_X1+5, 1, 1,TFSF_X1+6, SIZE_Y, SIZE_Z);
AxCreateRegion( &obsPlaneBackHz, simHandle, AX_ACCESS_READ,
AX_FIELD_H, AX_AXIS_Z,
TFSF_X1+5, 1, 1,TFSF_X1+6, SIZE_Y, SIZE_Z);
unsigned
numObsBackXHy,
numObsBackYHy,
numObsBackZHy,numObsBackXHz, numObsBackYHz, numObsBackZHz;
unsigned
numObsBackXEy,
numObsBackYEy,
numObsBackZEy,numObsBackXEz, numObsBackYEz, numObsBackZEz;
AxGetRegionSize(obsPlaneBackEy,
&numObsBackXEy,
&numObsBackYEy,
&numObsBackZEy);
AxGetRegionSize(obsPlaneBackEz,
&numObsBackXEz,
&numObsBackYEz,
&numObsBackZEz);
AxGetRegionSize(obsPlaneBackHy,
&numObsBackXHy,
&numObsBackYHy,
&numObsBackZHy);
AxGetRegionSize(obsPlaneBackHz,
&numObsBackXHz,
&numObsBackYHz,
&numObsBackZHz);
unsigned
obsArraySizeBackEy=numObsBackXEy*numObsBackYEy*numObsBackZEy;
unsigned
obsArraySizeBackEz=numObsBackXEz*numObsBackYEz*numObsBackZEz;
unsigned
obsArraySizeBackHy=numObsBackXHy*numObsBackYHy*numObsBackZHy;
unsigned
obsArraySizeBackHz=numObsBackXHz*numObsBackYHz*numObsBackZHz;
float* EyPlaneBack;
float* EzPlaneBack;
float* HyPlaneBack;
float* HzPlaneBack;
94
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
EyPlaneBack = NULL;
EzPlaneBack = NULL;
HyPlaneBack = NULL;
HzPlaneBack = NULL;
EyPlaneBack = new float[obsArraySizeBackEy];
EzPlaneBack = new float[obsArraySizeBackEz];
HyPlaneBack = new float[obsArraySizeBackHy];
HzPlaneBack = new float[obsArraySizeBackHz];
// Creating region for voltage across cone and sampling along cone
ax_rgnhandle_t obsPnt1,obsPnt2,obsPnt3;
AxCreateRegion( &obsPnt1, simHandle, AX_ACCESS_READ, AX_FIELD_E,
AX_AXIS_Y,
1, 1, SIZE_Z/2, SIZE_X, SIZE_Y, SIZE_Z/2+1 );
AxCreateRegion( &obsPnt2, simHandle, AX_ACCESS_READ, AX_FIELD_E,
AX_AXIS_X,
1, SIZE_Y/2, SIZE_Z/2, SIZE_X, SIZE_Y/2+1, SIZE_Z/2+1 );
AxCreateRegion( &obsPnt3, simHandle, AX_ACCESS_READ, AX_FIELD_E,
AX_AXIS_Z,
1, SIZE_Y/2, SIZE_Z/2, SIZE_X, SIZE_Y/2+1, SIZE_Z/2+1 );
// Get the region size of Ex-field Point Observation at (4,4,4)
unsigned numObsXEy, numObsYEy, numObsZEy;
unsigned numObsXEx, numObsYEx, numObsZEx;
unsigned numObsXEz, numObsYEz, numObsZEz;
AxGetRegionSize(obsPnt1, &numObsXEy, &numObsYEy, &numObsZEy);
AxGetRegionSize(obsPnt2, &numObsXEx, &numObsYEx, &numObsZEx);
AxGetRegionSize(obsPnt3, &numObsXEz, &numObsYEz, &numObsZEz);
unsigned obsArraySizeEy = numObsXEy*numObsYEy*numObsZEy;
unsigned obsArraySizeEx = numObsXEx*numObsYEx*numObsZEx;
unsigned obsArraySizeEz = numObsXEz*numObsYEz*numObsZEz;
float* m_data1;
m_data1 = NULL;
m_data1 = new float[obsArraySizeEy];
float* m_data2;
m_data2 = NULL;
m_data2 = new float[obsArraySizeEx];
float* m_data3;
m_data3 = NULL;
m_data3 = new float[obsArraySizeEz];
// Create electric and magnetic materials representing the cone structure
ax_mathandle_t hFreespace,eFreespace;
AxCreateMagneticMaterial(&hFreespace, axmath_mu0, axmath_rho0);
AxCreateLdmMaterial(&eFreespace, axmath_eps0, axmath_sigma0);
CheckAx();
AxSetUniformMaterial(simHandle, AX_FIELD_E, AX_AXIS_X, eFreespace);
95
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
AxSetUniformMaterial(simHandle, AX_FIELD_E, AX_AXIS_Y, eFreespace);
AxSetUniformMaterial(simHandle, AX_FIELD_E, AX_AXIS_Z, eFreespace);
CheckAx();
AxSetUniformMaterial(simHandle, AX_FIELD_H, AX_AXIS_X, hFreespace);
AxSetUniformMaterial(simHandle, AX_FIELD_H, AX_AXIS_Y, hFreespace);
AxSetUniformMaterial(simHandle, AX_FIELD_H, AX_AXIS_Z, hFreespace);
CheckAx();
unsigned numSimX, numSimY, numSimZ;
{
AxGetMaterialMeshSize(simHandle,
AX_FIELD_E,
AX_AXIS_X,
&numSimX,&numSimY,&numSimZ);
CheckAx();
int matArraySize = numSimX * numSimY * numSimZ;
ax_mathandle_t *matData = new ax_mathandle_t[matArraySize];
for(int i=0; i<numSimX; i++)
{
printf("%d\n",i);
printf("%f\n",RADIUS);
RADIUS = Cone_Slope*(i-End_Point)+End_Radius;
for(int j=0; j<numSimY; j++)
for(int k=0; k<numSimZ; k++)
{
ax_mathandle_t
&m
=
matData[(i*numSimY+j)*numSimZ+k];
float POS_Y
=
SIZE_Y/2,
POS_Z
=
SIZE_Z/2;
float dy = jPOS_Y,
dz = k-POS_Z,
distance
=
sqrt(dy*dy + dz*dz);
if
(distance<=RADIUS && i>=Start_Point && i<=End_Point)
{
if((iStart_Point) % 6==0 || (i-Start_Point) % 6==1 || (i-Start_Point) % 6==2)
{
m = dielSpace1;
else
Start_Point) % 6==3 || (i-Start_Point) % 6==4 || (i-Start_Point) % 6==5)
96
if
}
((i-
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
{
m = dielSpace2;
}
}
else if (i<Start_Point ||
i>End_Point || distance>RADIUS)
{
m=freeSpace;
}
}
}
AxSetNonUniformMaterial(simHandle,
AX_FIELD_E,
AX_AXIS_X,
matData);
CheckAx();
delete matData;
}
{
AxGetMaterialMeshSize(simHandle,
AX_FIELD_E,
AX_AXIS_Y,
&numSimX,&numSimY,&numSimZ);
CheckAx();
unsigned matArraySize = numSimX * numSimY * numSimZ;
ax_mathandle_t *matData = new ax_mathandle_t[matArraySize];
for(int i=0; i<numSimX; i++)
{
printf("%d\n",i);
printf("%f\n",RADIUS);
RADIUS = Cone_Slope*(i-End_Point)+End_Radius;
for(int j=0; j<numSimY; j++)
for(int k=0; k<numSimZ; k++)
{
ax_mathandle_t
&m
=
matData[(i*numSimY+j)*numSimZ+k];
float POS_Y
=
SIZE_Y/2,
POS_Z
=
SIZE_Z/2;
float dy = jPOS_Y,
dz = k-POS_Z,
distance
=
sqrt(dy*dy + dz*dz);
97
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
if
(distance<=RADIUS && i>=Start_Point && i<=End_Point)
{
if((iStart_Point) % 6==0 || (i-Start_Point) % 6==1 || (i-Start_Point) % 6==2)
{
m = dielSpace1;
else
if
}
((i-
Start_Point) % 6==3 || (i-Start_Point) % 6==4 || (i-Start_Point) % 6==5)
{
m = dielSpace2;
}
}
else if (i<Start_Point ||
i>End_Point || distance>RADIUS)
{
m=freeSpace;
}
}
}
AxSetNonUniformMaterial(simHandle,
AX_FIELD_E,
AX_AXIS_Y,
matData);
CheckAx();
delete matData;
}
{
AxGetMaterialMeshSize(simHandle,
AX_FIELD_E,
AX_AXIS_Z,
&numSimX,&numSimY,&numSimZ);
CheckAx();
unsigned matArraySize = numSimX * numSimY * numSimZ;
ax_mathandle_t *matData = new ax_mathandle_t[matArraySize];
for(int i=0; i<numSimX; i++)
{
printf("%d\n",i);
printf("%f\n",RADIUS);
RADIUS = Cone_Slope*(i-End_Point)+End_Radius;
for(int j=0; j<numSimY; j++)
98
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
for(int k=0; k<numSimZ; k++)
{
ax_mathandle_t
matData[(i*numSimY+j)*numSimZ+k];
float
SIZE_Y/2,
&m
=
POS_Y
=
POS_Z
=
SIZE_Z/2;
float dy = jPOS_Y,
dz = k-POS_Z,
distance
=
sqrt(dy*dy + dz*dz);
if
(distance<=RADIUS && i>=Start_Point && i<=End_Point)
{
if((iStart_Point) % 6==0 || (i-Start_Point) % 6==1 || (i-Start_Point) % 6==2)
{
m = dielSpace1;
else
if
}
((i-
Start_Point) % 6==3 || (i-Start_Point) % 6==4 || (i-Start_Point) % 6==5)
{
m = dielSpace2;
}
}
else if (i<Start_Point ||
i>End_Point || distance>RADIUS)
{
m=freeSpace;
}
}
}
AxSetNonUniformMaterial(simHandle,
matData);
CheckAx();
delete matData;
}
99
AX_FIELD_E,
AX_AXIS_Z,
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
// AccelewareFDTD requires that each field component be set to a particular material
// independently. This allows greater accuracy in modelling the edges of materials
// compared to treating materials on a per-voxel basis.
float
SigmaX_Max[]
=
{0,4.931054804302191e-002,6.634410093371842e001,3.034901040652234e+000,8.926162663743197e+000,
2.060996933552876e+001,4.083259849174795e+001,7.278761334647653e+001,1.2009
56509746119e+002,1.867881151618081e+002,
2.772935893237631e+002,3.964262449141463e+002,5.493757711552813e+002,7.4169
84996665232e+002,9.793094902068087e+002,
1.268475404686907e+003,1.615808038273798e+003,2.028258405825149e+003,2.5131
11303224999e+003,3.077980279022936e+003,
3.730802963763008e+003,4.479836713619111e+003,5.333654532172759e+003,6.3011
41239892575e+003,7.391489865467554e+003,
8.614198236872662e+003,9.979065753098224e+003,1.149619031999883e+004,1.3175
96543582159e+004,1.502907741374068e+004,
1.706650273021962e+004,1.929950548929458e+004,2.173963499396118e+004,2.4398
72341678483e+004,2.728888356266532e+004,
3.042250671739291e+004,3.381226057624788e+004,3.747108724743652e+004,4.1412
20132563225e+004,4.564908803131059e+004,
5.019550141193996e+004,5.506546260142161e+004,6.027325813446861e+004,6.5833
43831287863e+004,7.176081562089287e+004,
7.807046318704914e+004,8.477771329012828e+004,9.189815590697025e+004,9.9447
63730009329e+004,1.074422586431938e+005,
1.158983746827375e+005,1.248325924339695e+005,1.342617699097825e+005,1.4420
30148809831e+005,1.546736836665835e+005,
1.656913799528406e+005,1.772739536398275e+005,1.894394997144113e+005,2.0220
63571485617e+005,2.155931078219876e+005,
2.296185754681506e+005,2.443018246427559e+005,2.596621597138752e+005,2.7571
91238728960e+005,2.924924981655389e+005,
3.100023005422165e+005,3.282687849270595e+005,3.473124403049480e+005,3.6715
39898259404e+005,3.878143899265057e+005,
100
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
4.093148294670041e+005,4.316767288848800e+005,4.549217393630616e+005,4.7907
17420130819e+005,5.041488470724567e+005,
5.301753931158838e+005,5.571739462798284e+005,5.851672995001104e+005,6.1417
84717620852e+005,6.442307073630631e+005};
float
SigmaX_Min[]
=
{0*1000000,0.00001988452364*1000000,0.00026753319442*1000000,0.001223826623
20*1000000,0.00359948326637*1000000,
0.00831098900370*1000000,0.01646578272548*1000000,0.02935167172159*1
000000,0.04842868120731*1000000,0.07532247845010*1000000,
0.11181889376685*1000000,0.15985924620382*1000000,0.22153628269123*1
000000,0.29909059904162*1000000,0.39490744851822*1000000,
0.51151386827389*1000000,0.65157607076306*1000000,0.81789705909098*1
000000,1.01341443386737*1000000,1.24119836551532*1000000};
float
sigmayz[]
=
{0*1000000/6,0.00001988452364*1000000/6,0.00026753319442*1000000/6,0.0012238
2662320*1000000/6,0.00359948326637*1000000/6,
0.00831098900370*1000000/6,0.01646578272548*1000000/6,0.0293516717215
9*1000000/6,0.04842868120731*1000000/6,0.07532247845010*1000000/6,
0.11181889376685*1000000/6,0.15985924620382*1000000/6,0.2215362826912
3*1000000/6,0.29909059904162*1000000/6,0.39490744851822*1000000/6,
0.51151386827389*1000000/6,0.65157607076306*1000000/6,0.8178970590909
8*1000000/6,1.01341443386737*1000000/6,1.24119836551532*1000000/6};
float kappa[] = {1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f};
float KappaX_Max[] = {1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f,1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f,1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f,1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f,
1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f, 1.0f};
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_X_MIN, 10, SigmaX_Min,
kappa);
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_Y_MIN, 10, sigmayz, kappa);
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_Z_MIN, 10, sigmayz, kappa);
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_X_MAX, 40, SigmaX_Max,
KappaX_Max);
101
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_Y_MAX, 10, sigmayz, kappa);
AxSetUpmlSigmaProfile(simHandle, AX_BNDLOC_Z_MAX, 10, sigmayz, kappa);
// Creating the plane wave excitation
ax_float_t theta = axmath_pi/2.0f;
ax_float_t phi = 0.0f;
ax_float_t psi = 0.0f;
ax_vector3_t kInc, eInc;
//unit wave vector
kInc.x=sin(theta)*cos(phi);
if (abs(kInc.x)<0.0001)
{
kInc.x=0;
}
kInc.y=sin(theta)*sin(phi);
if (abs(kInc.y)<=0.0001)
{
kInc.y=0;
}
kInc.z=cos(theta);
if (abs(kInc.z)<0.0001)
{
kInc.z=0;
}
//unit e-field vector (Taflove implementation)
eInc.x=cos(psi)*sin(phi)-sin(psi)*cos(theta)*cos(phi);
if (abs(eInc.x)<0.0001)
{
eInc.x=0;
}
eInc.y=-cos(psi)*cos(phi)-sin(psi)*cos(theta)*sin(phi);
if (abs(eInc.y)<0.0001)
{
eInc.y=0;
}
eInc.z=sin(psi)*sin(theta);
if (abs(eInc.z)<0.0001)
{
eInc.z=0;
}
// Creating a time domain excitation
// ax_timeexc_t gaussExcitation;
ax_timeexc_t timeEx;
102
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
AxCreateFreqShiftedGaussTimeExcitation(&timeEx, simHandle, 1.0f, 622e+12f,
timeDelta * 100,
0.0f, timeDelta* 400);
AxCreatePlaneWaveExcitation( simHandle, // simulation handle
timeEx,TFSF_X0,
TFSF_Y0,
TFSF_Z0,TFSF_X1,
TFSF_Y1,
TFSF_Z1,&kInc,&eInc );
// Ready the simulation. This performs a number of checks on the
// simulation set up. Any errors that are found are logged.
AxReadySimulation(simHandle);
CheckAx();
//////////////////////////////////////////////////////////////////////
// Perform the updates
//////////////////////////////////////////////////////////////////////
std::cout << "\n\nRunning simulation..." << std::endl;
FILE *Outputdata1, *Outputdata2, *Outputdata3, *Outputdata4, *Outputdata5,
*Top_Pln_Poy, *Right_Pln_Poy, *Front_Pln_Poy, *Back_Pln_Poy, *Modaldis;
if( (Outputdata1 = fopen("Vacross.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Outputdata2 = fopen("Eysurf.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Outputdata3 = fopen("Ey.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Outputdata4 = fopen("Ex.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Outputdata5 = fopen("Ez.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Top_Pln_Poy = fopen("Top_Pln_Poy.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Right_Pln_Poy = fopen("Right_Pln_Poy.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Front_Pln_Poy = fopen("Front_Pln_Poy.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
if( (Back_Pln_Poy = fopen("Back_Pln_Poy.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
103
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
if( (Modaldis = fopen("Modaldis.mat" , "w" )) == NULL )
printf( "The file 'datafile' was not opened\n" );
else {}
// Defining time steps
int const timeSteps = 3000;
int const Ydimension=End_Point-Start_Point;
float RADIUS1=0;
float* outData;
float
sum=0,Int_Top_Pln_Poy=0,Int_Right_Pln_Poy=0,Int_Front_Pln_Poy=0,Int_Back_Pln_P
oy=0 ;
outData=new float[1];
clock_t start = clock();
for (unsigned int i = 0; i <= timeSteps; i++)
{
AxDoE(simHandle);
std::cout << "\n\nRunning simulation..." << std::endl;
CheckAx();
std::cout << "\n\nRunning simulation..." << std::endl;
AxDoH(simHandle);
std::cout << "\n\nRunning simulation..." << std::endl;
CheckAx();
std::cout << "\n\nRunning simulation..." << std::endl;
// animating the field and 4,4,4
AxRead(obsPnt1, m_data1, obsArraySizeEy);
AxRead(obsPnt2, m_data2, obsArraySizeEx);
AxRead(obsPnt3, m_data3, obsArraySizeEz);
CheckAx();
//
}
// Writing Vacross file
for (int j=Start_Point;j<=End_Point;j++)
{
RADIUS1 = (Cone_Slope*(j-End_Point)+End_Radius);
for
(int
k=(SIZE_Y/2RADIUS1);k<=SIZE_Y/2+RADIUS1;k++)
{
sum=sum+m_data1[j*(numObsYEy)+k];
}
fprintf( Outputdata1, "%f ",sum);
sum=0;
}
fprintf( Outputdata1, "\n");
104
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
for (int j=Start_Point;j<=End_Point;j++)
{
if
(j==Start_Point+200
j==Start_Point+1150)
{
RADIUS1
End_Point)+End_Radius);
for
RADIUS1);k<=SIZE_Y/2+RADIUS1;k++)
{
||
j==Start_Point+675
=
(int
||
(Cone_Slope*(jk=(SIZE_Y/2-
fprintf(Modaldis,
"%f
",m_data1[j*(numObsYEy)+k]);
}
fprintf(Modaldis, "\n");
}
}
// End of Vacross file
// Writing the Ey field on the surface
if (i==1500 || i==3000 || i==4500 || i==6000|| i==2250 || i==5250)
{
for (int j=0;j<=SIZE_X-1;j++)
{
for (int k=0;k<=SIZE_Y-1;k++)
{
fprintf(
Outputdata2,
",m_data1[j*(numObsYEy)+k]);
}
}
fprintf( Outputdata2, "\n");
}
// End of writing Ey field on surface
// Writing Ey on the center axis
for (int j=0;j<=SIZE_X-1;j++)
{
fprintf(
Outputdata3,
",m_data1[j*(numObsYEy)+sizeZ/2]);
}
fprintf( Outputdata3, "\n");
// End of writing Ey on the center axis
// Writing Ex on the center axis
for (int j=0;j<=SIZE_X-1;j++)
{
fprintf( Outputdata4, "%f ",m_data2[j]);
}
fprintf( Outputdata4, "\n");
// End of writing Ex on the center axis
105
"%f
"%f
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
// Writing Ez on the center axis
for (int j=1;j<=SIZE_X;j++)
{
fprintf( Outputdata5, "%f ",m_data3[j]);
}
fprintf( Outputdata5, "\n");
// End of writing Ex on the center axis
// Calculating instantaneous poyting vector on plans
// covering the structure
//Top plane
AxRead(obsPlaneTopEx, ExPlaneTop, obsArraySizeTopEx);
AxRead(obsPlaneTopEz, EzPlaneTop, obsArraySizeTopEz);
AxRead(obsPlaneTopHx, HxPlaneTop, obsArraySizeTopHx);
AxRead(obsPlaneTopHz, HzPlaneTop, obsArraySizeTopHz);
for (int j=TFSF_X0-5;j<=TFSF_X1+5;j++)
{
for (int k=TFSF_Z0-5;k<=TFSF_Z1+5;k++)
{
Int_Top_Pln_Poy=Int_Top_Pln_Poy+(EzPlaneTop[j*(numObsTopZEz)+k]*HxPl
aneTop[j*(numObsTopZHx)+k]ExPlaneTop[j*(numObsTopZEx)+k]*HzPlaneTop[j*(numObsTopZHz)+k]);
}
}
fprintf( Top_Pln_Poy, "%f\n",Int_Top_Pln_Poy);
printf("%f\n",Int_Top_Pln_Poy);
Int_Top_Pln_Poy=0;
//Right plane
AxRead(obsPlaneRightEx, ExPlaneRight, obsArraySizeRightEx);
AxRead(obsPlaneRightEy, EyPlaneRight, obsArraySizeRightEy);
AxRead(obsPlaneRightHx, HxPlaneRight, obsArraySizeRightHx);
AxRead(obsPlaneRightHy, HyPlaneRight, obsArraySizeRightHy);
for (int j=TFSF_X0-5;j<=TFSF_X1+5;j++)
{
for (int k=TFSF_Y0-5;k<=TFSF_Y1+5;k++)
{
Int_Right_Pln_Poy=Int_Right_Pln_Poy+(ExPlaneRight[j*(numObsRightYEx)+k
]*HyPlaneRight[j*(numObsRightYHy)+k]EyPlaneRight[j*(numObsRightYEy)+k]*HxPlaneRight[j*(numObsRightYHx)+k]);
106
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
}
}
fprintf( Right_Pln_Poy, "%f\n",Int_Right_Pln_Poy);
printf("%f\n",Int_Right_Pln_Poy);
Int_Right_Pln_Poy=0;
//Front plane
AxRead(obsPlaneFrontEy, EyPlaneFront, obsArraySizeFrontEy);
AxRead(obsPlaneFrontEz, EzPlaneFront, obsArraySizeFrontEz);
AxRead(obsPlaneFrontHy, HyPlaneFront, obsArraySizeFrontHy);
AxRead(obsPlaneFrontHz, HzPlaneFront, obsArraySizeFrontHz);
for (int j=TFSF_Y0-5;j<=TFSF_Y1+5;j++)
{
for (int k=TFSF_Z0-5;k<=TFSF_Z1+5;k++)
{
Int_Front_Pln_Poy=Int_Front_Pln_Poy+(EyPlaneFront[j*(numObsFrontZEy)+k]
*HzPlaneFront[j*(numObsFrontZHz)+k]EzPlaneFront[j*(numObsFrontZEz)+k]*HyPlaneFront[j*(numObsFrontZHy)+k]);
}
}
fprintf( Front_Pln_Poy, "%f\n",Int_Front_Pln_Poy);
printf("%f\n",Int_Front_Pln_Poy);
Int_Front_Pln_Poy=0;
AxRead(obsPlaneBackEy, EyPlaneBack, obsArraySizeBackEy);
AxRead(obsPlaneBackEz, EzPlaneBack, obsArraySizeBackEz);
AxRead(obsPlaneBackHy, HyPlaneBack, obsArraySizeBackHy);
AxRead(obsPlaneBackHz, HzPlaneBack, obsArraySizeBackHz);
for (int j=TFSF_Y0-5;j<=TFSF_Y1+5;j++)
{
for (int k=TFSF_Z0-5;k<=TFSF_Z1+5;k++)
{
Int_Back_Pln_Poy=Int_Back_Pln_Poy+(EyPlaneBack[j*(numObsBackZEy)+k]*
HzPlaneBack[j*(numObsBackZHz)+k]EzPlaneBack[j*(numObsBackZEz)+k]*HyPlaneBack[j*(numObsBackZHy)+k]);
//printf("OK");
}
}
fprintf( Back_Pln_Poy, "%f\n",Int_Back_Pln_Poy);
107
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
printf("%f\n",Int_Back_Pln_Poy);
Int_Back_Pln_Poy=0;
printf("%f\n",m_data1[0]);
printf("%f\n",m_data1[1]);
printf("%d\n",i);
}
clock_t end = clock();
float timeElapsed = (end - start) / (double)CLOCKS_PER_SEC;
float mcps = 1e-6f * timeSteps * sizeX * sizeY * sizeZ / timeElapsed;
std::cout << "\n\nSimulation ran " << timeSteps << " time steps in "
<< timeElapsed << " seconds,\nachieving a speed of "
<< mcps << " MCell/s." << std::endl;
///////////////////////////////////////////////////////////
// Clean up
///////////////////////////////////////////////////////////
AxDelete(timeEx);
AxDelete(hFreespace);
AxDelete(eFreespace);
AxDelete(simHandle);
CheckAx();
// Close the system and free any resources that may be held.
AxCloseSystem();
CheckAx();
return 0;
fclose( Outputdata1 );
fclose( Outputdata2 );
fclose( Outputdata3 );
fclose( Outputdata4 );
fclose( Outputdata5 );
}
A1.2: Validation of the FDTD software
In this section, some benchmarks will be presented to validate the computational
electromagnetic code utilized in this research.
The schematic of the FDTD space is shown in Fig. A1.1. The boundary conditions are
Uniaxial Perfectly Matched Layers (UPML) to ensure the minimum of reflections from
the outer boundary. The last layer of the FDTD space is set to be a prefect electric
108
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
conductor (PEC). Each surface of the FDTD space is covered by 10 layers of UPML
except the forward surface which is 40 layers as shows in Fig. A1.2. This is mainly due to
strong forward scattering behaviour of the object, and to reduce the reflection from the
forward boundary. The excitation point is Ey hard source which is placed in the middle of
the FDTD space. The Ey is also sampled at points shown in Fig. A1.1.
Fig. A1.1: Cross section of the FDTD space.
To test the boundary conditions we used the method originally developed by Taflove et.
al. [26]. In this method we compare the simulation result for the original FDTD space
with the simulation results obtained from a considerably larger simulation space. The
second simulation space has been chosen larger than the first FDTD space to ensure that
the outer-boundary reflections will reach the sampling point at a later time step than in
the original space. In our case, we have selected the original FDTD space to be
100×100×600, and the larger FDTD space to be 200×200×1200. In both cases,
∆z=∆y=30nm and ∆x=5nm and ∆t=1.47×1e-17, which is obtained based on the courant
stability condition. For both spaces the sampling points are the same and they are located
two FDTD cells inside the UPML. Fig. A1.2 shows the wave propagation inside the
FDTD space. As it can be observed, the wave in the smaller domain passes the whole
domain after 26.46fs and the remaining field will be zero inside it.
109
3D FDTD SOURCE CODE FOR MODELING PHOTORECEPTOR USING ACCELEWARE SOFTWARE
Fig. A1.2: The field propagation at different time steps inside the FDTD domain. Top row is the original
domain and the lower row shows the larger domain.
Fig. A1.3 shows the sampled data at different location of FDTD domain corresponding to
the locations shown in Fig. A1.1. Ey is obtained at the same locations for two different
FDTD spaces. The solid curve shows the data obtained for the small FDTD space and the
dotted points shows the Ey component in the bigger FDTD space. As it can be observed,
the curves match to each other very well so that it is impossible to distinguish them. This
shows a very small reflection from the UPML layers.
Fig. A.3: The Ey versus time sampled at different points inside the FDTD domain. In all cases, the solid line
shows the sampled Ey at small domain and the dots show the Ey obtained at larger domain.
110
MATLAB SOURCE CODE FOR ROTATIONALLY SYMMETRIC SCALAR FDTD FORMULATION
Appendix II
MATLAB Source Code for Rotationally Symmetric Scalar FDTD
Formulation
clear all
global r Nz nr radius1 IncPlane
Lorder=0 % zero-order symmetry condition
c=3*1e8 % speed of light
deltar=50*1e-9
deltaz=50*1e-9
deltat=0.9/(c*sqrt((1/deltar)^2+(1/deltaz)^2));
lambda=485e-9/1.35 % wavelength in the dielectric medium
freq=c/lambda/1.35 % frequency of operation
beta=2*pi/lambda;
Nr=400 % Number of cells in r-direction
Nz=200 % Number of cells in z-direction
radius1=32;
IncPlane=19;
step=0;
StrInit2
sigh(Nr+1,Nz+1,3)=0;
Cr(1:Nr,1:Nz)=c*deltat./(nr.*deltar);
Cz(1:Nr,1:Nz)=c*deltat./(nr.*deltaz);
u01=1.8289
w01=2.8389
for ri=1:Nr+1
if ri<=radius1
SighInc(ri,1)=bessel(0,u01*ri*deltar/(radius1*deltar))/bessel(0,u01);
end
if ri>radius1
SighInc(ri,1)=besselk(1,w01*ri*deltar/(radius1*deltar))/besselk(1,w01);
end
end
111
MATLAB SOURCE CODE FOR ROTATIONALLY SYMMETRIC SCALAR FDTD FORMULATION
Sigma=deltat*100;
delayTInc=400*(deltat)-deltaz/c/2;
delayT=400*(deltat);
delayTRef=400*(deltat)+deltaz/c/2;
for step=1:6000
step;
Sampleinc(step)=sigh(1,IncPlane,2);
Exc_Sig(step)=sin(2*pi*freq.*(step*deltat-delayT)).*exp(-(step*deltatdelayT).^2./(2.*(Sigma).^2));
for k=2:Nz
sigh((2:Nr),k,3)=2.*(1-(1+0.5.*(Lorder./((1:Nr-1)')).^2).*Cr((2:Nr),k).^2Cz((2:Nr),k).^2).*sigh((2:Nr),k,2)-sigh((2:Nr),k,1)+...
Cr((2:Nr),k).^2.*((1+1./(2.*((1:Nr-1)'))).*sigh((3:Nr+1),k,2)+(1-1./(2*((1:Nr1)'))).*sigh((1:Nr-1),k,2))+Cz((2:Nr),k).^2.*(sigh((2:Nr),k+1,2)+sigh((2:Nr),k-1,2));
end
for k=IncPlane-1
sigh((2:Nr),k,3)=2.*(1-(1+0.5.*(Lorder./((1:Nr-1)')).^2).*Cr((2:Nr),k).^2Cz((2:Nr),k).^2).*sigh((2:Nr),k,2)-sigh((2:Nr),k,1)+...
Cr((2:Nr),k).^2.*((1+1./(2.*((1:Nr-1)'))).*sigh((3:Nr+1),k,2)+(1-1./(2*((1:Nr1)'))).*sigh((1:Nr-1),k,2))+Cz((2:Nr),k).^2.*(sigh((2:Nr),k+1,2)+sigh((2:Nr),k-1,2)SighInc((2:Nr)).*sin(2*pi*freq.*(step*deltat-delayTRef)).*exp(-(step*deltatdelayTRef).^2./(2.*(Sigma).^2)));
end
for k=IncPlane
sigh((2:Nr),k,3)=2.*(1-(1+0.5.*(Lorder./((1:Nr-1)')).^2).*Cr((2:Nr),k).^2Cz((2:Nr),k).^2).*sigh((2:Nr),k,2)-sigh((2:Nr),k,1)+...
Cr((2:Nr),k).^2.*((1+1./(2.*((1:Nr-1)'))).*sigh((3:Nr+1),k,2)+(1-1./(2*((1:Nr1)'))).*sigh((1:Nr-1),k,2))+Cz((2:Nr),k).^2.*(sigh((2:Nr),k+1,2)+sigh((2:Nr),k1,2)+SighInc((2:Nr))*sin(2*pi*freq.*(step*deltat-delayTInc)).*exp(-(step*deltatdelayTInc).^2./(2.*(Sigma).^2)));
end
for k=2:Nz
sigh(1,k,3)=2*(1-(2-Lorder^2/2)*Cr(1,k)^2-Cz(1,k)^2).*sigh(1,k,2)-sigh(1,k,1)+...
Cr(1,k)^2*(2Lorder^2/2)*(1+exp(j*pi*Lorder))*sigh(2,k,2)+Cz(1,k)^2*(sigh(1,k+1,2)+sigh(1,k-1,2));
end
for k=IncPlane-1
sigh(1,k,3)=2*(1-(2-Lorder^2/2)*Cr(1,k)^2-Cz(1,k)^2).*sigh(1,k,2)-sigh(1,k,1)+...
Cr(1,k)^2*(2Lorder^2/2)*(1+exp(j*pi*Lorder))*sigh(2,k,2)+Cz(1,k)^2*(sigh(1,k+1,2)+sigh(1,k-1,2)SighInc(1)*sin(2*pi*freq.*(step*deltat-delayTRef)).*exp(-(step*deltatdelayTRef).^2./(2.*(Sigma).^2)));
end
for k=IncPlane
sigh(1,k,3)=2*(1-(2-Lorder^2/2)*Cr(1,k)^2-Cz(1,k)^2).*sigh(1,k,2)-sigh(1,k,1)+...
112
MATLAB SOURCE CODE FOR ROTATIONALLY SYMMETRIC SCALAR FDTD FORMULATION
Cr(1,k)^2*(2Lorder^2/2)*(1+exp(j*pi*Lorder))*sigh(2,k,2)+Cz(1,k)^2*(sigh(1,k+1,2)+sigh(1,k1,2)+SighInc(1)*sin(2*pi*freq.*(step*deltat-delayTInc)).*exp(-(step*deltatdelayTInc).^2./(2.*(Sigma).^2)));
end
% boundary condition Nr+1
for k=2:Nz
sigh(Nr+1,k,3)=(1+(1+1/(4*Nr-2))*Cr(Nr,k))^-1*(-(1+(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,1)-...
(1-(1+1/(4*Nr-2))*Cr(Nr,k))*sigh(Nr+1,k,1)-(1-(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,3)+...
(2-0.5*(Lorder/(Nr-0.5))^2*Cr(Nr,k)^2Cz(Nr,k)^2)*(sigh(Nr+1,k,2)+sigh(Nr,k,2))+...
0.5*Cz(Nr,k)^2*(sigh(Nr+1,k+1,2)+sigh(Nr,k+1,2)+sigh(Nr+1,k-1,2)+sigh(Nr,k1,2)));
end
for k=IncPlane-1
sigh(Nr+1,k,3)=(1+(1+1/(4*Nr-2))*Cr(Nr,k))^-1*(-(1+(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,1)-...
(1-(1+1/(4*Nr-2))*Cr(Nr,k))*sigh(Nr+1,k,1)-(1-(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,3)+...
(2-0.5*(Lorder/(Nr-0.5))^2*Cr(Nr,k)^2Cz(Nr,k)^2)*(sigh(Nr+1,k,2)+sigh(Nr,k,2))+...
0.5*Cz(Nr,k)^2*(sigh(Nr+1,k+1,2)+sigh(Nr,k+1,2)+sigh(Nr+1,k-1,2)+sigh(Nr,k1,2)-2*SighInc(Nr+1)*sin(2*pi*freq.*(step*deltat-delayTRef)).*exp(-(step*deltatdelayTRef).^2./(2.*(Sigma).^2))));
end
for k=IncPlane
sigh(Nr+1,k,3)=(1+(1+1/(4*Nr-2))*Cr(Nr,k))^-1*(-(1+(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,1)-...
(1-(1+1/(4*Nr-2))*Cr(Nr,k))*sigh(Nr+1,k,1)-(1-(1-1/(4*Nr2))*Cr(Nr,k))*sigh(Nr,k,3)+...
(2-0.5*(Lorder/(Nr-0.5))^2*Cr(Nr,k)^2Cz(Nr,k)^2)*(sigh(Nr+1,k,2)+sigh(Nr,k,2))+...
0.5*Cz(Nr,k)^2*(sigh(Nr+1,k+1,2)+sigh(Nr,k+1,2)+sigh(Nr+1,k-1,2)+sigh(Nr,k1,2)+2*SighInc(Nr+1)*sin(2*pi*freq.*(step*deltat-delayTInc)).*exp(-(step*deltatdelayTInc).^2./(2.*(Sigma).^2))));
end
% end of boundary condition Nr+1
%boundary condition Nz+1 and 1
for i=2:Nr
sigh(i,Nz+1,3)=-sigh(i,Nz,1)+(1+Cz(i,Nz))^-1*(-(1Cz(i,Nz))*(sigh(i,Nz,3)+sigh(i,Nz+1,1))+...
(2-(1+0.5*(Lorder/(i-1))^2)*Cr(i,Nz)^2)*(sigh(i,Nz+1,2)+sigh(i,Nz,2))+...
113
MATLAB SOURCE CODE FOR ROTATIONALLY SYMMETRIC SCALAR FDTD FORMULATION
0.5*Cr(i,Nz)^2*((1+0.5/(i-1))*(sigh(i+1,Nz+1,2)+sigh(i+1,Nz,2))+...
+(1-0.5/(i-1))*(sigh(i-1,Nz+1,2)+sigh(i-1,Nz,2))));
end
sigh(1,Nz+1,3)=-sigh(1,Nz,1)+(1+Cz(1,Nz))^-1*(-(1Cz(1,Nz))*(sigh(1,Nz,3)+sigh(1,Nz+1,1))+...
(2-(2-0.5*Lorder^2)*Cr(1,Nz)^2)*(sigh(1,Nz+1,2)+sigh(1,Nz,2))+...
Cr(1,Nz)^2*(20.5*Lorder^2)*(1+exp(j*pi*Lorder))/2*(sigh(2,Nz+1,2)+sigh(2,Nz,2)));
for i=2:Nr
sigh(i,1,3)=-sigh(i,2,1)+(1+Cz(i,1))^-1*(-(1-Cz(i,1))*(sigh(i,2,3)+sigh(i,1,1))+...
(2-(1+0.5*(Lorder/(i-1))^2)*Cr(i,1)^2)*(sigh(i,1,2)+sigh(i,2,2))+...
0.5*Cr(i,1)^2*((1+0.5/(i-1))*(sigh(i+1,1,2)+sigh(i+1,2,2))+...
+(1-0.5/(i-1))*(sigh(i-1,1,2)+sigh(i-1,2,2))));
end
sigh(1,1,3)=-sigh(1,2,1)+(1+Cz(1,1))^-1*(-(1-Cz(1,1))*(sigh(1,2,3)+sigh(1,1,1))+...
(2-(2-0.5*Lorder^2)*Cr(1,1)^2)*(sigh(1,1,2)+sigh(1,2,2))+...
Cr(1,1)^2*((2-0.5*Lorder^2))*(1+exp(j*pi*Lorder))/2*(sigh(2,1,2)+sigh(2,2,2)));
%end of boundary condition Nz+1 and 1
%visualization
sigh(:,:,1)=sigh(:,:,2);
sigh(:,:,2)=sigh(:,:,3);
end
% end of the main loop
114
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
Preface to Appendix III
This appendix is included as a paper published in the IEEE Transactions on Magnetics.
In this work, we have shown a successful implementation of the sub-cell model of the
coaxial probe. The model will help us to analyze the feeding structure of the antenna,
where the electromagnetic field has a singular behaviour. We have also shown the
successful termination of the coaxial probe by extending the thin wire into the UPML
boundary condition. Furthermore, the novel formulation of modeling the junction of thin
wire and thin dielectric feed, which is presented in this paper, can be exploited in the via
hole structure in high frequency analysis of microwave circuits.
115
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
Appendix III
FDTD Sub-Cell Modeling of the Inner Conductor of the Coaxial Feed:
Accuracy and Convergence Analysis
Abstract – In this paper, we analyze the method of exciting and truncating the coaxial
probe used to excite an electromagnetically coupled patch antenna. The goal of the
analysis is to avoid the unstable and unreliable behaviour of finite-difference timedomain (FDTD) solution. The model considered here is based on the sub-cell thin wire
modeling of inner conductor of coaxial probe. Two main categories of structures and
excitation models are considered. In the first category, the truncated thin wire is separated
from the absorbing boundary by two FDTD cells, for several excitation models and their
location along the wire. In the second category, the thin wire is extended into the
absorbing boundary, and both the hard and the resistive source model are investigated.
Comparison of results for the two above-noted categories of FDTD coaxial probe feeds
may be of particular interest to computational electromagnetics community, as it may
yield insight into limitations of most commercial software packages which do not allow
for extension of the coaxial probe model into the absorbing boundary.
Index Terms—Coaxial feed, FDTD methods, microstrip antennas, sub-cell modeling, thin
wire
A3.1 Introduction
Coaxial feed modeling has historically been a challenge in numerical simulation [81].
Various modeling techniques such as gap modeling, magnetic frill modeling and
116
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
transmission line modeling have been applied successfully for specific applications. In
finite-difference time-domain (FDTD) method, a common approach to an accurate
solution involves mesh refinement around the probe. This, in turn, allows for accurate
observation of the effect of the electric field variation around the coaxial probe. However,
this method requires a very small time step (based on the Courant stability condition),
resulting in a prohibitively large number of iterations needed for convergence to the final
solution. One option to avoid this issue is to use sub-cell modeling technique for the inner
conductor of the coaxial probe, but this approach can yield unreliable computation of the
input impedance if not treated with precision. In this paper, we report an effective
modeling technique based on sub-cell modeling of the inner conductor as a solution to the
noted problems. In particular, to compensate for a deficiency encountered in several
commercial software packages, we extend the thin wire into the uniaxial perfectly
matched layer (UPML) absorbing boundary [82].
A3.2 Antenna Structure
Fig. A3.1 shows the antenna comprised of two stacked patches, separated by a foam layer
(relative permittivity εr-foam=1). Both patches are etched on a 0.81mm substrate (relative
permittivity εr-substrate=3.38, loss tangent tanδsubstrate=0.0027). Another substrate layer is
added to the fed-patch, where the two substrate layers are glued to result in the total
thickness of 1.74mm, with assumed 0.12mm gap, modeled electrically as air with
Maloney-Smith method for a thin dielectric sheet [83]. Antenna size is given by the
following dimensions: fed-patch 64mm×68mm, upper patch 72mm×72mm, finite ground
plane 100mm×100mm. The radii of the outer and inner conductor of the coaxial probe
are 5mm and 1mm, respectively. To preserve the characteristic impedance of 50Ω for the
coaxial probe, the dielectric constant of the material filling the space between the inner
and outer conductor is set to εr-probe=3.73. For all the simulations reported in this paper,
using 12 layers of UPML on each side of the FDTD simulation space resulted in a total of
90×90×90 grid cells.
117
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
Fig. A3.1: Antenna Structure
A3.3 Sub-Cell FDTD Modeling
Sub-cell modeling is often used in FDTD to model structures with dimensions much
smaller than those of other elements in the overall FDTD model grid. Examples include
thin material sheets (generally modeled with Maloney-Smith method [26, 83]) and thin
wires (modeled by the method initially developed by Umashankar and Taflove [26, 84]).
Sub-cell modeling resides on the idea of the contour path modeling technique which uses
the integral form of Faraday’s and Ampere’s laws to incorporate the effect of presence of
material into the normal FDTD Yee cell used for the rest of the problem geometry. To
precisely model the antenna structure here, we have applied two sub-cell modeling
techniques: the Maloney-Smith method for thin material sheets (used to model the thin
air gap beneath the fed patch) and the thin wire model (to model the inner conductor of
coaxial probe. Eq. (1) shows the updating equation of the y-component of magnetic field
at the junction of thin wire with the thin material sheets. We use the contour path
modeling technique of thin wire but we consider the discontinuity of the electric field in
the z-direction along the thin air layer.
n+
Hy
1
2
i , j ,k
n−
= Hy
1
2
i , j ,k

E
z


+
∆t
∆zµ0
n
1
i − , j ,k
2

E
 x

n
i , j ,k −
1
2
− Ex
(∆z − ∆g ) − E z ,in
118

∆t
+
×
1
∆
∆x
x
i , j ,k + 
µ0
∆z ln( )
2
2
r0
n
n
1
i + , j ,k
2

(∆g )


(1)
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
Here, r0 is the radius of the inner conductor of the coaxial probe and ∆g is the air gap
thickness beneath the lower patch. The remaining standard FDTD notation can be found
in [26].
A3.4 Feeding Structure and Excitations
The modeled probes considered in this study can be classified in two main categories:
In the first category, the inner conductor has been truncated in such a way that two FDTD
cells separate the end of the conductor and the UPML, as illustrated in Fig. A3.2 (a).
Most of the currently available commercial software does not allow extension of thin
wire model into the absorbing boundary condition. This can cause the accumulated
charge on the tip of the wire to distort the convergence of the final solution. With this in
mind, we find it important to study this category of coaxial probe models in order to
assess their limitations.
The second category involves models with the inner conductor extended into the UPML,
as depicted in Fig. A3.2 (b).
For both of the above-noted categories, the length of the coaxial probe is considered to be
40 FDTD-cells and the outer conductor is extended into the UPML in all the simulations.
For each category, hard source and resistive voltage source are used for comparison and,
ultimately, for determining the optimal configuration from the standpoint of model
accuracy.
In all the simulations considered in this paper, the input impedance is obtained using
Zin(f)=Vin(f)/Iin(f) , where the voltage Vin(f) and the current Iin(f) are Fourier transforms of
the time-dependent voltage Vin(t) and current Iin(t) calculated at the edge of the coax-toantenna transition.
119
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
Thin air gap
Thin air gap
Vin
Coaxial inner
conductor
Vin
Coaxial inner
conductor
d
Feed point
Coaxial outer
conductor
UPML Layer
Feed point
Coaxial outer
conductor
(a)
UPML Layer
(b)
Fig. A3.2: Two main categories of modeling reported in this paper focus on the region around the inner
conductor of the coaxial probe. (a) In the first category, the inner conductor does not extend into the
UPML. The distance d marks the distance between the modeled source and the end of inner conductor
along the inner conductor (b) In the second, the inner conductor of the coaxial probe extends into the
UPML.
A3.5 First Category: the Coaxial Inner Conductor not Extended into UPML
Within this category, the inner conductor is truncated so that it is separated by two FDTD
cells from the UPML. Fig. A3.3 shows the simulated input voltage for three different
excitation models:
Fig. A3.3(a): hard source, placed d=20cells from the inner conductor termination. The
result clearly demonstrates lack of convergence even after a relatively large number of
time-steps.
Fig. A3.3(b): resistive voltage source, placed d=2cells (refer to Fig. A3.2(a)) away from
the inner conductor termination.
Fig. A3.3(c): resistive voltage source, placed d=20cells (refer to Fig. A3.2(a)) away from
the inner conductor termination.
Figs. A3.4(a) and A3.4(b) graph the input impedance of the antenna calculated using the
three noted excitation models. These results suggest that the location of the modeled
resistive source along the inner conductor of the coaxial probe plays a key role in
achieving a converging solution.
120
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
A3.6 Second Category: Extending the Coaxial Inner Conductor to UPML
Our second category of simulations considers the inner conductor being extended into the
UPML. Again, we investigate and observe results for two excitation signals: a hard
source and a resistive voltage source. For both cases, the excitation point is seven cells
away from the UPML. As can be observed, and in contrast with the cases studied in the
category of the previous section, this model yields a stable solution that practically
overlaps for the two sources and is graphed in Fig. A3.5. These models resulted in
practically equal simulated input impedance shown in Fig. A3.6.
1.5
Input voltage (mV)
1
0.5
0
−0.5
−1
−1.5
0
2500
5000
7500
10000
Time steps, ∆t=1.5138 (psec)
(a)
121
12500
15000
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
1.2
1
Input voltage (mV)
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0
2500
5000
7500
10000
12500
15000
10000
12500
15000
Time steps, ∆t=1.5138 (psec)
(b)
1
0.8
Input Voltage (mV)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2500
5000
7500
Time steps, ∆t=1.5138 (psec)
(c)
Fig. A3.3: Input voltage of the modeled excitation for the inner conductor not extended into UPML: (a)
hard source, (b) resistive source d=2 cells (c) resistive source d=20 cells, where d is defined as in Fig.
A3.2(a).
122
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
180
Hard excitation
Resistive excitation d=2 cells
Resistive excitation d=20 cells
160
140
Re [Zin] (Ω)
120
100
80
60
40
20
0
−20
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Frequency (GHz)
(a)
150
Hard excitation
Resistive excitation d=2 cells
Resistive excitation d=20 cells
Imag[Zin] (Ω)
100
50
0
−50
−100
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Frequency (GHz)
(b)
Fig. A3.4: Input impedance for the probe model category depicted in Fig. A3.2(a) and excitation sources of
Fig. A3.3. (a) Real part (b) Imaginary part.
123
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
1
Input voltage (mV)
0.5
0
−0.5
−1
−1.5
0
2,500
5,000
7500
10,000
Time steps, ∆t=1.5138 (psec)
12,500
15,000
Fig. A3.5: Input voltage for hard source excitation for the inner conductor of the coaxial probe extended to
UPML. A practically overlapping result is obtained for the resistive voltage source excitation.
140
Re[Z ] hard excitation
in
Imag[Z ] hard excitation
120
in
Re[Z ] resistive excitation
in
Input impedance (Ω)
100
Imag[Z ] resistive excitation
in
80
60
40
20
0
−20
−40
−60
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Frequency (GHz)
Fig. A3.6: Input impedance when the thin wire extended into UPML. The difference between the hard
excitation and resistive excitation is so negligible that the graphs are well-matched.
124
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
A3.7 Convergence Analysis
In order to quantify the rate of convergence to the final result for all the cases of models
and excitations considered, we define the error function as
Error( N ) =
[
] 2
2GHz  Re[Z in ( f , N )] − Re Z in ,ref ( f ) 
∑

f = 1GHz 
[
]
Re Z in ,ref ( f )
(2)

Here, Zin(f,N) is the input impedance calculated after N time-domain iterations, Zin,ref is
the reference calculated input impedance obtained after a large number of time domain
iterations (i.e. 15000), f is frequency (1 GHz < f < 2GHz) with the frequency-increment
of ∆f = 2.7525×105 Hz. The error calculated using Eq. (2) and as a function of a number
or iterations used for calculation is graphed in Fig. A3.7 (the model of Fig. A3.2(a)) and
in Fig. A3.8 (the model in Fig. A3.2(b)).
40
d=20 cells
d=15 cells
d=10 cells
d=5 cells
30
Error (dB)
20
10
0
−10
−20
−30
4
5
6
7
8
9
10
11
Number of iterations ×103
12
13
14
Fig. A3.7: Error based on Eq. (2) calculated for the thin wire model not extending into the UPML and for
four different distances between the resistive source and the termination of the inner conductor of the
coaxial cable (Fig. A3.2(a)).
125
FDTD SUB-CELL MODELING OF THE INNER CONDUCTOR OF THE COAXIAL FEED: ACCURACY AND CONVERGENCE ANALYSIS
30
Hard Source
Resistive Source
25
20
Error (dB)
15
10
5
0
−5
−10
−15
−20
4
5
6
7
8
9
10
11
12
13
14
Number of iterations × 103
Fig. A3.8: Error based on Eq. (2) calculated for the thin wire model extended into the UPML (Fig.
A3.2(b)).
A3.8 Conclusion
This paper presents analysis of several FDTD approaches to the modeling of the coaxial
probe feed. The results and convergence analysis suggest several points of interest:
•
With the inner conductor of the coaxial probe not extended into the UPML, the
hard source excitation yields unstable solutions.
•
For the inner conductor not extended to the UPML, the location of the resistive
voltage source model relative to the inner conductor termination plays a key role
in the accuracy of the solution. Results suggest that the model of the resistive
source located in the central location along the inner conductor yields the fastest
convergence to an accurate impedance estimate.
•
For the thin wire extended into the UMPL, both hard source and resistive source
result in comparable convergence rates and simulated impedance value.
126
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
Preface to Appendix IV
This appendix is included as a published paper in the Journal of Applied Electromagnetic
Society (ACES). In this work we present a novel implementation of a circularly polarized
EMCP antenna for satellite communication. It is well known that the circular polarization
behaviour is results of excitation of two modes which are 90° out-phase. We also analyze
the antenna structure using measurement and also electromagnetic simulation tools.
Furthermore, we have done a modal analysis on the antenna to extract the standing modes
on the patches and illustrate their equal excitation. By changing the distance between the
patches, the phase difference between the modes can be controlled which results to a
circularly polarized behaviour.
127
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
Appendix IV
Analysis of a Singly-Fed Circularly Polarized Electromagnetically
Coupled Patch Antenna
Amir Hajiaboli and Milica Popović
Department of Electrical and Computer Engineering, McGill University, Montreal,
Canada
[email protected], [email protected]
Abstract- This paper presents analysis of a broadband circularly polarized
electromagnetically coupled patch antenna (EMCP) fed thorough a coaxial probe. The
bandwidth of the antenna is investigated numerically and through experiment. The 12%
bandwidth of the measured return loss implies broadband behaviour considering the
operating frequency and the inherent bandwidth limitations of the microstrip antenna
structure. The changes in circular polarization bandwidth were investigated using finiteelement-method (FEM) software and the results suggest that the increase in separation
between the patches causes a decrease of 3dB bandwidth and the degradation of the
minimum value of axial ratio. An axial ratio bandwidth of 2% is achieved in the reported
EMCP structure. In addition, we have applied a modal analysis using finite-difference
time-domain (FDTD) simulation to reveal the simultaneous excitation of TM01 and TM10
modes at around 1.3GHz. These results help explain the broadband and circular
polarization characteristics of the EMCP structure under investigation.
Keywords: EMCP antenna, bandwidth, return loss, axial ratio, modal analysis
128
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
A4.1 Introduction
Achieving circular polarization is a challenge in microstrip antennas. Various single-layer
microstrip structures have been tested for this purpose. In general, these can be
categorized into two groups: multiple-feed (e.g. dual-feed) and single-feed structures
(with modified patches, e.g. corner truncated structures). The noted configurations share
the same principle for obtaining the circular polarization: the dimensions are modified in
order to provide propagation of two orthogonal modes with close resonant frequencies.
Then, the antenna is excited at a frequency between the two resonant frequencies to
ensure approximately equal amplitude for both modes. The location of the feed point is
chosen strategically to result in a phase difference of 90° between the two modes [85].
It has been shown [85] that, in the case of dual-feed microstrip structures, the trade-off
for the wide axial ratio (AR) bandwidth is the narrow bandwidth in the return loss.
Further, the feeding structure would increase the complexity and overall size of the
antenna. On the other hand, in the single-feed structures, such as corner truncated
configurations, the AR bandwidth is narrow (on the order of 1%) [85]. The noted
antennas, therefore, cannot be considered as desirable structures with optimized
characteristics for applications that necessitate circular polarization.
Electromagnetically coupled patch (EMCP) antenna was first introduced in 1983 [86] for
broadband applications. Further investigations revealed its circularly polarized
characteristics [87-89] and the possibility of an EMCP design with acceptable AR and
return loss bandwidth. Most of the previously tested EMCP structures are based on the
dual-feed excitation, truncated corner patches or the circular patch. A recent work [87]
reports a systematic design procedure that results in the return loss bandwidth of 43% and
the AR bandwidth of 8% in C-band. The reported procedure is based on optimizing the
dimensions of a corner truncated square patch microstrip antenna.
In this paper, we analyze and report on a singly-fed coaxially fed EMCP antenna for
circular polarization applications. Unlike the structure discussed in [87], the patches are
not corner truncated and the feed is a coaxial probe placed off the patch diagonal. The
bandwidth of the antenna has been measured and compared with the simulation results.
The axial ratio has been calculated with finite-element tool (HFSS, Ansoft Co.). Our
parametric study reports on the variation of the return loss and axial ratio with the change
129
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
is separation between two antenna patches.
Finally, a modal analysis using finite-
difference time-domain (FDTD) simulation explains the broadband and the polarization
characteristics of the antenna by revealing the simultaneous excitation of standing modes
on the patches around 1.3GHz.
A4.2 Antenna Structure
Three-dimensional view of the antenna is shown in Fig. A4.1. The antenna consists of
two patches separated by a foam spacer of thickness d and εr=1. The patches are etched
on a substrate with relative permittivity εr=3.38, loss tangent tanδ=0.0027, and thickness
of 0.81mm. Another substrate of identical specifications is stacked to the fed-patch.
These two substrates for the fed-patch are glued together to result in total thickness of
1.74mm. We assume a 0.12-mm thick air gap between the two glued substrates. The
dimensions of interest are as follows: fed-patch 64mm×68mm, upper patch
72mm×72mm, and the finite ground plane 100mm×100mm. The coaxial probe feed,
modeled according to work reported in [90, 91], is placed at x=10mm and y=4mm from
the lower left hand corner of the lower patch as shown in Fig. A4.2. To reduce the
computational burden, in both the FDTD and the FEM simulations, an infinite ground
plane was considered.
Fig. A4.1: Three-dimensional view of the electromagnetically coupled patch antenna.
130
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
Fig. A4.2: Lower patch of antenna in Fig. A4.1, with indicated coordinates of the feed point.
A4.3 Return Loss
The measured and simulated results of input return loss (S11) are shown in Fig. A4.3 and
Fig. A4.4 for d=22mm and d=18mm. The center frequency of the -10dB return loss in
simulation and experimental results is close to 1.3GHz and varies slightly with the
change in d.
Fig. A4.3: Return loss for the separation between the lower and upper patch (foam spacer) d=18mm.
131
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
Fig. A4.4: Return loss for the separation between the lower and upper patch (foam spacer) d=22mm.
Fig. A4.5 shows the percentage of -10dB bandwidth versus d both for simulation and
measurement. Fig. A4.5 demonstrates that the measured bandwidth of the antenna
linearly increases with d, with the maximum measured bandwidth of 12% for d=30mm.
As it can be observed in Figs A4.3-5, around 1.25GHz and for the foam spacer less than
24mm, which is the focus of this paper, there is a good agreement between the simulated
and the measured results. A discrepancy between the measurement and simulation can be
noted around 1.8GHz, and we suspect that it was caused by several approximations of the
simulation model. First, we assume εr=1 for the foam layer. Second, we neglect the
electromagnetic properties of the glue. Finally, the scattering effects of the finite ground
plane in the constructed antenna (not modeled in the simulations to reduce the
computational burden) also represent a possible source of error. As discussed in [92], by
considering the finite ground plane it is possible to obtain better simulated results around
1.8GHz.
132
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
12
Measurement
HFSS simulation
Persentage of bandwith (%)
11
10
9
8
7
6
5
4
3
14
16
18
20
22
24
26
28
30
d (mm)
Fig. A4.5: Percentage of bandwidth for different values of the separation between the lower and upper
patch (thickness of the foam spacer), d.
A4.4 Axial Ratio
Fig. A4.6 shows the axial ratio computed with the finite-element software (HFSS) for
different values of d. The minimum value of the axial ratio (0.7dB) occurs at d=20mm at
the center frequency of 1.24GHz. Considering the frequency shift between the
measurements and simulations noted in Figs. A4.3 and A4.4, we can anticipate the
minimum axial ratio value in the measurement at approximately 1.3GHz – the center
frequency of the -10dB bandwidth. The maximum 3dB bandwidth of axial ratio (2.2%) is
observed for d=16mm at the center frequency of 1.23GHz. We note that the increase in d
results in a reduction of the axial ratio bandwidth. As can be seen in Fig. A4.6, for d >
22mm, the 3dB bandwidth of axial ratio is negligible.
Fig. A4.7 shows the variation of axial ratio at φ=0°(x-z plane) versus θ at 1.24GHz and
for different values d. Again, it can be observed that, as d increases, the beamwidth of the
3-dB axial ratio decreases. For d>22mm, the beamwidth is zero. At d=20mm the 3-dB
beamwidth of axial ratio is around 90°.
133
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
18
d=16mm
d=18mm
d=20mm
d=22mm
d=24mm
16
Axial ratio (dB)
14
12
10
8
6
4
2
0
1.2
1.21
1.22
1.23
1.24
1.25
1.26
Frequency (GHz)
1.27
1.28
1.29
1.3
Fig. A4.6: Axial ratio versus frequency for different values of d.
10
d=16mm
d=18mm
d=20mm
d=22mm
d=24mm
9
8
Axial ratio (dB)
7
6
5
4
3
2
1
0
0
10
20
30
θ (deg)
40
Fig. A4.7: Axial ratio at φ=0° versus θ for different values of d at 1.24GHz.
134
50
60
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
A4.5 Modal Analysis
In order to explain the broadband behaviour and the circular polarization of the antenna,
we simulate the antenna using finite-difference time domain (FDTD) method and extract
the current distribution on the patches at different frequencies.
(a)
(b)
135
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
(c)
(d)
Fig. A4.8: Current distribution at 1.3GHz on the patches, illustrating the simultaneous excitation of TM01
and TM10 modes. a) Jx on lower patch b) Jy on lower patch c) Jx on upper patch d) Jy on upper patch.
In order to obtain the current distribution (Jx and Jy) we calculate and save the current
distribution at each time step at the late simulation time to ensure the higher order modes
136
ANALYSIS OF A SINGLY-FED CIRCULARLY POLARIZED ELECTROMAGNETICALLY COUPLED PATCH ANTENNA
have diminished and only dominant modes exist on the patches. We then apply the
Fourier transform on these current distributions to obtain the current distributions in
frequency domain. As depicted in Fig. A4.8, the current distribution at 1.3GHz
corresponds to TM01 and TM10 modes with approximately the same amplitude. The
simultaneous excitation of these two modes matches with the broadband behaviour
obtained at around 1.3GHz. Further, since the modes are spatially orthogonal this
explains the circular polarization when the appropriate phase conditions are met for
certain values of the separation between the upper and lower patch (d).
A4.6 Conclusion
A broadband circularly polarized electromagnetically coupled patch antenna has been
analyzed through simulation and experiment. The measured bandwidth increases linearly
with the separation between the lower and upper antenna patch d and the maximum
bandwidth of 12% around 1.33GHz (L-band) occurs for d=30mm. Through FEM
simulations, we observed that the 3-dB bandwidth of axial ratio degrades as the
separation of the patches d increases. A modal analysis on the current distributions
obtained by FDTD has been performed and the current distribution at 1.3GHz clearly
shows the simultaneous excitation of TM01 and TM10 modes with nearly equal amplitudes
on the lower and upper patches. This can explain the broadband and circular polarization
characteristics of the antenna.
Acknowledgements
This work was funded by Natural Science and Engineering Research Council (NSERC)
of Canada Discovery Grant and by the Le Fonds Québécois de la Recherché sur la Nature
et les Technologies Nouveaux Chercheurs Grant.
137
References
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
Y. Wang, "On cognitive mechanisms of the eyes: the sensor vs. the browser of the
brain," in Proceedings of the Second IEEE International Conference on Cognitive
Informatics (ICCI’03), 2003.
E. N. Marieb and K. Hoehn, Human Anatomy and Physiology, 7 ed.: Benjamin
Cummings, 2007.
L. J. Raj, "How to assess and plan for the management of visually challenged
children in the context of multiple “different-abilities," Community Eye Health
Journal, vol. 20, pp. 91-92, 2007.
N. J. Wade, "Image, eye, and retina," Journal of The Optical Society of America,
vol. 24, pp. 1229-1249, 2007.
W. Lotmar, "Theoretical eye model with aspherics," Journal of the Optical
Society of America, vol. 61, pp. 1522-1529, 1971.
A. C. Guyton, Text book of Medical Physiology, 8 ed.: W.B. Saunders Com, 1991.
J. M. Enoch, "Visualization of waveguide modes in retinal receptors," American
Journal of Ophthalmology, pp. 1107-1118, 1961.
A. W. Snyder and R. Menzel, Photoreceptor Optics: Springer-Verlag 1975.
R. Szema and L. P. Lee, "Biologically inspired optical systems," in Biomimetics:
Biologically Inspired Technologies, Y. Bar-Cohen, Ed.: CRC press: Taylor and
Francis Group, 2006, pp. 291-308.
P. A. Liebman, "In situ microspectrophotometric studies on the pigments of single
retinal rods," Biophysical Jornal, vol. 2, pp. 162-178, 1962.
M. Subramanian, "Medical Encyclopedia," American Accreditation HealthCare
Commission 2008.
A. Fein and E. Z. Szuts, Photoreceptors: Their Role in Vision: Cambridge
University Press, 1982.
"Photoreception," Institute Of Neurosciences and Biophysics (INB), 2004.
M. Subramanian, "Eye," in Medical Encyclopedia, A. A. H. C. 2008, Ed.
G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods,
Quantitative Data and Formulae: John Wiley & Sons, 1982.
J. E. Dowling, The Retina: An Approachable Part of the Brain: Harvard
University Press, Cambridge, MA, 1987.
P. N. Prasad, Introduction to Biophotonics Wiley-Interscience, 2003.
138
References
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
C. Pask and A. Stacey, "Optical properties of retinal photoreceptors and the
campbell effect," Vision Res., vol. 38, pp. 953-961, 1998.
J. M. Enoch, "Optical properties of retinal receptors," Journal of Optical Society
of America, vol. 53, Jan 1963.
G. t. D. Francia, "Retina cones as dielectric antennas," Journal of Optical Society
of America, vol. 39, April 1949.
A. W. Snyder and C. Pask, "The Stiles-Crawford effect- explanation and
consequences," Vision Res., vol. 13, pp. 1115-1137, 1973.
B. R. Horowitz, "Theoretical considerations of the retinal receptor as a
waveguide," in Photoreceptor Optics, J. M. Enoch and F. L. Tobey, Eds.: BerlinHeidelberg-New York: Springer, 1981, pp. 219-300.
W. Fischer and R. Rohler, "The absorption of light in an idealized photoreceptor
on the basis of waveguide theory-II: the semi-infinite cylinder," Vision res., vol.
14, pp. 1115-1125, 1974.
A. Stockman and L. T. Sharpe, "Length (µm) of outer segments of human
photoreceptors," in Color and Vision Research: University College London,
Institute of Ophthalmology.
K. S. Yee, "Numerical solution of initial boundary value problems involving
maxwell’s equations in isotropic media," IEEE Trans. on Antennas and
Propagation, vol. 14, 1966.
A. Taflove and S. C. Hagness, Computational Electrodynamics: The FiniteDifference Time-Domain Method: Artech House, 2005.
K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for
Electromagnetics Boca Raton: CRC Press, 1993.
K. L. Slrlagerl and J. B. Sclineiderz, "A selective survey of the finite-difference
time-domain literature," IEEE Antenna s and Propagation Magazine, vol. 37,
1995.
A. Taflove, "A perspective on the 40-year history of FDTD computational
electrodynamics," Applied Computational Electromagnetic Society, vol. 22, pp. 121, 2007.
N. P. Chavannes, Local Mesh Refinement Algorithms for Enhanced Modeling
Capabilities in the FDTD Method vol. 161: Hartung-Gorre, 2002.
G. Mur, "Absorbing boundary conditions for the finite-difference approximation
of the time-domain electromagnetic-field equations," IEEE Trans. on
Electromagetic Compatibility, vol. 23, pp. 377-382, 1981.
J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic
waves," Journal of Computational Physics vol. 114, pp. 185-200, 1994.
Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, "A perfectly matched
anisotropic absorber for use as an absorbing boundary condition," IEEE Trans.
On Antennas and Propagation, vol. 43, 1995.
J. A. Roden and S. D. Gedney, "Convolutional PML (CPML): an efficient FDTD
implementation of the CFS-PML for arbitrary media," Microwave and Optical
Technology Letters, vol. 27, pp. 334-339, 2000.
D. S. Katz, E. T. Thiele, and A. Taflove, "Validation and extension to three
dimensions of the berenger PML absorbing boundary condition for FD-TD
meshes," IEEE Microwave and Guided Wave Letterrs, vol. 4, pp. 268-270, 1994.
139
References
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
J. Yamauchi, M. Nibe, and H. Nakano, "Scalar FD-TD method for circularly
symmetric waveguides," Optical and Quantum Electronics, vol. 29, pp. 451-460,
1997.
W. P. Huang, S. T. Chu, A. Goss, and S. K. Chaudhuri, "A scalar finite-difference
time-domain approach to guided-wave optics," IEEE Photonics Technology
Letters, vol. 3, 1991.
N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "A scalar finite-difference timedomain method with cylindrical perfectly matched layers: application to guided
and leaky modes of optical waveguides," IEEE Journal of Quantum Electronics,
vol. 39, pp. 487-492, 2003.
W. P. Huang, S. T. Chu, and S. K. Chaudhuri, "A semivectorial finite-difference
time-domain method," IEEE Photonics Technology Letters, vol. 3, pp. 803-806,
1991.
Y. Chen, R. Mittra, and P. Harms, "Finite-difference time-domain algorithm for
solving Maxwell’s equations in rotationally symmetric geometries," IEEE
Transactions On Microwave Theory And Techniques, vol. 44, pp. 832-839, 1996.
W. Yu, D. Arakaki, and R. Mittra, "On the solution of a class of large body
problems with full or partial circular symmetry by using the finite-difference
time-domain (FDTD) method," IEEE Transactions On Antennas And
Propagation, vol. 48, pp. 1810-1817, 2000.
R. Holland, "Threde: a free-field EMP coupling and scattering code," IEEE
Transactions on Nuclear Science, vol. 24, pp. 2416-2421, 1977.
D. E. Merewether, R. Fisher, and F. W. Smith, "On implementing a numeric
huygen's source scheme in a finite difference program to illuminate scattering
bodies," IEEE Transactions on Nuclear Science, vol. 27, pp. 1829-1833, 1980.
K. Umashankar and Allen Taflove, "A novel method to analyze electromagnetic
scattering of complex objects," IEEE Transactions On Electromagnetic
Compatibility, vol. 24, pp. 397-405, 1982.
M. E. Watts and R. E. Diaz, "Perfect plane-wave injection into finite FDTD
domain through teleportation of fields," Electromagnetics, vol. 23, pp. 187-201,
2003.
C. Guiffaut and K. Mahdjoubi, "A perfect wideband plane wave injector for
FDTD method," in IEEE Antennas and Propagation Society International
Symposium, 2000.
S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, and J. Joseph T.
Walsh, "Exact solution of Maxwell’s equations for optical interactions with a
macroscopic random medium," Optical Letters, vol. 29, pp. 1393-1395, 2003.
S. H. Tseng, J. H. Greene, A. Taflove, D. Maitland, V. Backman, and J. Joseph T.
Walsh, "Exact solution of Maxwell’s equations for optical interactions with a
macroscopic random medium: addendum," Optics Letters, vol. 31, pp. 56-57,
2005.
Q. Liu and G. Zaho, "Advances in PSTD Techniques," in Computational
Electrodynamics:The Finite-Difference Time-Domain Method, Third ed, A.
Taflove and S. C. Hagness, Eds.: Artech House, 2005.
R. J. Meuth, "GPUs surpass computers at repetitive calculations," IEEE
Potentials, 2007.
140
References
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
E. J. Kelmelis, J. P. Durbano, J. R. Humphrey, F. E. Ortiz, and P. Curt, "Modeling
and simulation of nanoscale devices with a desktop supercomputer," in
Nanomodeling II San Diego, California, USA, 2006.
D. Miklivcic, N. Pavselj, and F. X. Hart, "Electric Properties of Tissues," in Wiley
Encyclopedia of Biomedical Engineering, 2006.
C. Gabriel, S. Gabriel, and E. Corthout, "The dielectric properties of biological
tissues: I. Literature survey," Phys. Med. Biol. , vol. 41, pp. 2231-2249, 1996.
G. L. Carlo, M. Supley, S. E. Hersemann, and P. Thibodeau, "Where does the
energy go? Microwave absorption in biological objects on the microscopic and
molecular scales," in Wireless Phones and Health, A. R. Sheppard, Ed.: Springer
US, 2002, pp. 165-175.
R. Lubbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, "A
frequency-dependent finite-difference time-domain formulation for dispersive
materials," IEEE Transactions On Electromagnetic Compatibility, vol. 32, pp.
222-227, 1990.
J. L. Young, A. Kittichartphayak, and Y. M. Kwok, "On the dispersion errors
related to (FD)2TD type schemes," IEEE Transactions On Microwave Theory And
Techniques, vol. 43, August 1995.
D. F. Kelley and R. J. Lubbers, "Piecewise linear recursive convolution for
dispersive media using FDTD," IEEE Transactions On Antennas and
Propagation, vol. 44, pp. 792-797, 1996.
J. W. Schuster and R. J. Luebbers, "An accurate FDTD algorithm for dispersive
media using a piecewise constant recursive convolution technique," in IEEE
Antennas and Propagation Society International Symposium: IEEE, 1998.
T. Kashiwa and I. Fukai, "A treatment by FDTD method of dispersive
charectristics associated with electronic polarization," Microwave Optics
Techology Letter, vol. 13, pp. 203-205, 1990.
R. M. Joseph, S. C. Hagness, and A. Taflove, "Direct time integration of
Maxwell's equations in linear dispersive media with absorption for scattering and
propagation of femtosecond electromagnetic pulses," Optics Letters, vol. 16, pp.
1412-1414, September 15 1991.
M. Okoniewski, M. Mrozowski, and M. A. Stuchly, "Simple treatment of multiterm dispersion in FDTD," IEEE Microwave and Guided Wave Letters, vol. 7,
1997.
Y. Takayama and W. Klaus, "Reinterpretation of the auxiliary differential
equation method for FDTD," IEEE Microwave and Wireless Components Letters,
vol. 12, pp. 102-104, March 2002.
A. Pozo, F. Perez-Ocon, and J. R. Jimenez, "FDTD analysis of the light
propagation in the cones of the human retina: an approach to the Stiles–Crawford
effect of the first kind," Journal Of Optics A: Pure And Applied Optics, vol. 7, pp.
357–363, 12 July 2005.
M. J. Piket-May and A. Taflove, "Electrodynamics of visible-light interactions
with the vertebrate retinal rod," Optics letters, vol. 18, April 15 1993.
J. Yamauchi, Propagating Beam Analysis of Optical Waveguides: Research
Studies Press LTD., 2003.
141
References
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
T. Scharf, Polirized Light in Liquid Crystals and Polymers: John Wiley and Sons,
2007.
J. M. Enoch and F. L. T. Jr., Vertebrate Photoreceptor Optics: Springer-Verlag
Berlin, 1981.
A. Hajiaboli and M. Popovic, "Human retinal photoreceptors: electrodynamic
model of optical micro-filters," IEEE Journal of Selected Topics in Quantum
Electronics, vol. 14, pp. 126-130, Jan./Feb. 2008.
A. Hajiaboli and M. Popovic, "FDTD analysis of light propagation in the human
photoreceptor cells," IEEE Transaction On Magnetics, vol. 44, pp. 1430-1433,
2008.
P. J. DeLint, T. T. J. M. Berendschot, J. v. d. Kraats, and D. v. Norren, "Slow
optical changes in human photoreceptors induced by light," Investigative
Ophthalmology & Visual Science, vol. 41, pp. 282-289, 2000.
F. L. Harosi, "Microspectrophotometry and optical phenomena: Birefringence,
dichroism and anamalous dispersion," in Vertebrate Photoreceptor Optics, J. M.
Enoch and J. F. L. Tobey, Eds.: Springer-Verlag, 1981, pp. 337-399.
P. A. Liebman, W. S. Jagger, M. W. Kaplan, and F. G. Bargoot, "Membrane
structure changes in rod outer segments associated with rhodopsin bleaching,"
Nature, vol. 251, pp. 31-36, 1974.
D. G. Stavenga and H. H. V. Barneveld, "On dispersion in visual photoreceptors,"
Vision Res., vol. 15, pp. 1091-1095, 1975.
N. W. Roberts, H. F. Gleeson, S. E. Temple, T. J. Haimberger, and C. W.
Hawryshyn, "Differences in the optical properties of vertebrate photoreceptor
classes leading to axial polarization sensitivity," Optical Society of America, vol.
21, pp. 335-345, 2004.
A. Hajiaboli and M. Popovic, "Spectral analysis of retinal cone outer-segment," in
International Conference On the Computation of Electromagnetic Fields
(COMPUMAG) Aachen, Germany, 2007.
S. Janz, P. Cheben, A. Delâge, B. Lamontagne, M. J. Picard, D. X. Xu, K. P. Yap,
and W. N. Ye, "Microphotonics: current challenges and applications," in
Frontiers in Planar Lightwave Circuit Technology, Design, Simulation, and
Fabrication, J. Č. S. Janz, and S. Tanev, Ed.: Nato Science Series II Mathematics,
Physics and Chemistry, 2006, pp. 1-38.
A. W. Snyder and P. A. V. Hall, "Unification of electromagnetic effects in human
retinal receptors with three pigment colour vision," Nature, vol. 223, August
1969.
E. J. Warrant and D.-E. Nilsson, "Absorption of white light in photoreceptors,"
Vision Res., vol. 38, pp. 195-207, 1998.
J. M. Enoch and G. A. Fry, "Characteristics of a model retinal receptor studied at
microwave frequencies," Journal of The Optical Society of America, vol. 48, pp.
899-911, 1958.
J. J. Sheppard, Human Color Vision: A Critical Study of the Experimental
Foundation. New York: American Elsevier, 1968.
T. W. Hertel and G. S. Smith, "On the convergence of common FDTD feed
models for antennas," IEEE Transaction on Antenna and Propagation, vol. 51,
pp. 1771-1779, 2003.
142
References
[82]
[83]
[84]
[85]
[86]
[87]
[88]
[89]
[90]
[91]
[92]
S. D. Gedney, "An anisotropic perfectly matched layer-absorbing medium for the
truncation of FDTD lattices," IEEE Transaction on Antenna and Propagation,
vol. 44, pp. 1630-1630, 1996.
J. G. Maloney and G. S. Smith, "The efficient modeling of thin material sheets in
the FDTD method," IEEE Transaction on Antenna and Propagation, vol. 40, pp.
323-330, 1992.
K. R. Umashankar, A. Taflove, and B. Beker, "Calculation and experimental
validation of induced currents on coupled wires in an arbitrary shaped cavity,"
IEEE Transaction on Antenna and Propagation, vol. 35, 1987.
G. Kumar and K. P. Ray, Broadband Microstrip Antennas: Artech House Inc.,
2003.
A. Sabban, "A new broadband stacked two layer microstrip antenna," in IEEE
Antennas and Propagation Society (APS) International Symposium, 1983, pp. 6366.
K. L. Chung and A. S. Mohan, "A systematic design method to obtain broadband
characteristics for singly-fed electromagnetically coupled patch antennas for
circular polarization," IEEE Transaction on Antenna and Propagation, vol. 51,
pp. 3239-3248, 2003.
K. T. V. Reddy and G. Kumar, "Stacked microstrip antennas for broadband
circular polarization," in IEEE Antennas and Propagation Society (APS)
International Symposium, 2001, pp. 420-423.
R. Q. Lee, T. Talty, and K. F. Lee, "Circular polarization characteristics of
parasitic microstrip antennas," in IEEE Antennas and Propagation Society (APS)
International Symposium, 1991, pp. 310-313.
A. Hajiaboli and M. Popovic, "FDTD subcell modeling of the inner conductor of
the coaxial feed: accuracy and convergence analysis," IEEE Transaction on
Magnetics, vol. 43, pp. 1361-1364, 2007.
A. Hajiaboli and M. Popovic, "Comparison of three FDTD modeling techniques
for coaxial feed," in IEEE Antennas and Propagation Society (APS) International
Symposium, 2006, pp. 3432-3435.
"Singly- fed Circularly Polarized Electromagnetically Coupled Patch Antenna,"
FEKO website, knowledge base, application notes, 2007.
143