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‫ חיבור זה או‬,‫ להפיץ באינטרנט‬,‫ לאחסן במאגר מידע‬,‫ לתרגם‬,‫ להדפיס‬,)‫אין להעתיק (במדיה כלשהי‬
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© Technion - Israel Institute of Technology, Elyachar Central Library
STUDIES OF LIGHT SCATTERING FROM THE
HUMAN RETINA
Research Thesis
In Partial Fulfillment of the
Requirements for the Degree of
Master of Science in Physics
Idan Mishlovsky
Submitted to the Senate of the Technion – Israel Institute of Technology
October 2014, Haifa, Elul 5774
© Technion - Israel Institute of Technology, Elyachar Central Library
The Research Thesis Was Done Under Supervision of Dr. Erez N. Ribak of the
Department of Physics.
I deeply thank Dr. Erez Ribak for his candid guidance and support throughout this entire
research.
The Generous Financial help of the Technion – Israel Institute of Technology is gratefully
and deeply acknowledged.
© Technion - Israel Institute of Technology, Elyachar Central Library
Table of Contents
Table of Figures ..................................................................................................................... 3
1
Synopsis ......................................................................................................................... 1
2
Background .................................................................................................................... 3
2.1
The ocular system ................................................................................................... 3
2.2
The Human Retina. ................................................................................................. 5
2.2.1
Müller Cells ..................................................................................................... 6
2.2.2
The Photoreceptors – Rods and Cones ............................................................ 8
2.2.3
The Retinal Pigment Epithelium ................................................................... 11
2.3
2.3.1
Optics of the Eye ........................................................................................... 13
2.3.2
Resolution ...................................................................................................... 14
2.3.3
The Stiles Crawford Effect ............................................................................ 18
2.4
3
Physics of the Ocular System ............................................................................... 13
Background Summary and Motivation ................................................................. 19
The Method .................................................................................................................. 20
3.1
Theoretical Framework ......................................................................................... 20
3.1.1
Wave Equations of Light. .............................................................................. 20
3.1.2
Waveguiding. ................................................................................................. 21
3.1.3
Light Scattering ............................................................................................. 25
3.2
Coherent and Incoherent Addition of Waves ....................................................... 27
3.3
Beam Propagation Method ................................................................................... 28
3.3.1
Fast Fourier Transform Split Step Beam Propagation Method (BPM). ........ 28
3.3.2
Spherical Symmetric FFT Beam Propagation Method. ................................ 31
3.4
The Computational Model .................................................................................... 32
3.4.1
Computational Model Setup .......................................................................... 32
3.4.2
Inward Light Propagation .............................................................................. 35
3.4.3
Backward Light Scattering ............................................................................ 37
© Technion - Israel Institute of Technology, Elyachar Central Library
3.5
4
Methods – Short Summary ................................................................................... 37
Results .......................................................................................................................... 38
4.1
Inward Propagated Intensities at the RPE............................................................. 38
4.2
Inward Control Propagation.................................................................................. 41
4.3
Backscattered Intensities at the Vitreous Gel ....................................................... 41
4.4
Comparison of Results with Empirical Results .................................................... 44
4.5
Results Summary .................................................................................................. 51
5
Summary and Further Studies ...................................................................................... 52
6
Bibliography ................................................................................................................. 54
© Technion - Israel Institute of Technology, Elyachar Central Library
Table of Figures
Figure 1: Schematic Human Eye Structure, light propagates left to right [website1] ........... 3
Figure 2: Retina's cross section Diagram. (Left) layers of tissue in the retina; (Right) five
major types of retinas cells [website2] .................................................................................. 5
Figure 3: Müller cell structure; Image was taken using differential interface contrast
microscopy. Bar size is about 25 µm [Franze et al. 2007] .................................................. 7
Figure 4: Retina tissues illuminated from the left using a thin fiber glass, the dotted line
[Agte et al. 2011] (a) fiber laser core placed parallel to the retina. (b) laser light diverging
in the gel. (c) Fiber laser (dotted line) placed just in-between the Müller cell funnels. (f)
Fiber laser is placed right in-front of the Müller cell funnel. (d, g) Graphs show the profiles
of the laser beam propagating between the Müller cells and inside the Müller cells
respectively. The graph is taken at the inner plexiform layer. (e, h) Graphs show intensity
as shown by the irradiance of the laser beam on a plastic membrane placed right after the
layers of tissues. ..................................................................................................................... 8
Figure 5: Normalized absorbance with respect to wavelength. Each curve is normalized by
its own maximum. ................................................................................................................. 9
Figure 6: Rods and Cones densities vs. angular displacement. (Osterberg 1935)................. 9
Figure 7: En Face imaging of cones at the Fovea, bar is 10µm [Curcio 1990] ................... 10
Figure 8: Retinal photoreceptors; on the left, the parafovea retina is shown. There are
approximately 10 rods to every cone in that region of the retina. The right we see the
peripheral retina; about 8 mm eccentricity from the fovea to the temporal region, bar is
10μm for both images [Curcio 1990]. ................................................................................. 10
Figure 9: Schematic photoreceptors structure. On the left we see the main section of the
photoreceptors, amongst others the outer sections, the inner section, the cell bodies and the
synaptic terminals (bottom illumination). On the right (top illumination), we see four
sketches of the current and voltage response functions [Left – website 4 ; right – Burns
2005] .................................................................................................................................... 11
Figure 10: Schematic summary of RPE function [Strauss 2004] ........................................ 12
Figure 11: RPE spectral reflectance [Delori and Pflibsen 1989]......................................... 13
Figure 12: Human schematic eye [Website 3]..................................................................... 14
Figure 13: Examples of intensity distributions of given optical effects. a - Diffraction
limited image, with the visible Airy Pattern; b – Spherical Aberrations. c – Coma. D –
astigmatism [Lipson et al. 2010] ......................................................................................... 16
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 14: Optical Chromatic Aberrations. Upper drawing describes the axial chromatic
aberration while the lower one the lateral chromatic aberration. ........................................ 17
Figure 15: Relative luminance as a function of pupil position with respect to the entering
beam. Data for measurements taken from the left eye of W.S Stiles [Westheimer, 2008] . 18
Figure 16: Total internal reflection in an optical fiber [website5] ...................................... 22
Figure 17: Left handed shows the step index optical fiber, note that n1>n2; to the right,
cylindrical coordinate system, with z the propagation direction. ........................................ 23
Figure 18: The left side of equation [23] is plotted from the with dotted lines; intersection
defines the supported modes in the optical fiber. ................................................................ 25
Figure 19: Spatial distribution of energy [website 6] .......................................................... 25
Figure 20: Spherical Coordinates [Website 5] .................................................................... 31
Figure 21: Simulated Müller cells funnel layout as appears from the top of the retina ...... 33
Figure 22 – Refractive indices throughout the retina. The first
are where the
Müller cells are positioned; right after them the cones come and create a step function in
the cell’s refractive index. ................................................................................................... 34
Figure 23 – Rods and Cones scattering layer; The cones are colored gray and the rods are
colored white. Scale bar is 6.5µm long. .............................................................................. 35
Figure 24: Incident beam as seen at the Müller cells funnels’ layer ................................... 36
Figure 25: Intensity at the bottom of the retina, right before the impact with the RPE.
Illumination pattern as in Figure 24. The blue color shows several modes clusters about the
Müller cells. ......................................................................................................................... 39
Figure 26: As in Figure 25. The green propagation shows more accentuated modes than the
blue, the cluster inter-border lines are more blurred. .......................................................... 39
Figure 27: As in Figure 25. There are several far red modal structures about the Müller
cells locations. ..................................................................................................................... 40
Figure 28: Simulation of the 550nm Gaussian light beam propagated into the retina, as
seen from the RPE layer, black scale bar is 6.5µm (left). An image taken from forward
scattering of a 532nm laser beam, shot at a retinal tissue (right) , white scale bar
corresponds to 5µm [Agte 2013]. ........................................................................................ 41
Figure 29: Propagation control results for λ=0.6µm; please note that all effects associated
with the presence of the waveguides are eliminated. The entry beam is equal to the exit
beam, and These results repeat for all wavelengths. ........................................................... 41
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 30 – Backscattered intensity at the vitreous gel of a blue monochromatic
wavelength. Four big bundles of scattered light about the Müller cells can be seen, scale
bar is 6.5µm long. ................................................................................................................ 42
Figure 31 – Backscattered intensity at the vitreous gel of a green monochromatic
wavelength. Scale bar is 6.5µm ........................................................................................... 43
Figure 32 - Backscattered intensity at the vitreous gel of a near infrared monochromatic
wavelength, scale bar is 6.5µm long. .................................................................................. 43
Figure 33: Backscattered green light at the vitreous surface, superimposed on the Müller
cell tapestry. Left: a macro image of the entire simulated retina. Right: a zoom unto a
single Müller cell area. Cone and Müller cell missalignment in these two is due to a wide,
, incidence angle. .......................................................................................................... 44
Figure 34 – Normalized mean radial calculation with respect to Müller cell’s center. ....... 45
Figure 35: Total scattered intensity vs. wavelength color. The Blue line represents
simulated total intensity, while the green represents average reflectance of retinal fundus
measurements; circles and crosses represents empirical reflectance measurements from the
nasal (black circle), temporal paraforvea (red ex) and the fovea (blue crosses)
[Berendschot et al 2003]...................................................................................................... 46
Figure 36: Parafovea AOSLO image of the retina, image size is 168 × 122 µm [Cooper et
al., 2011]. ............................................................................................................................. 46
Figure 37. Left: AOSLO image of the retina [Cooper et al. 2011] (Figure 35), using the
same color-map as in my simulation, image size is 168 × 122 µm. Right: simulated
backscattering image of the retina, image size is 66.5 × 66.5 µm....................................... 47
Figure 38. AOSLO image of the cones uppermost layer [Dubra et al. 2011]. .................... 48
Figure 39 – Normalized mean radial for scattered 680nm light incoherent intensity
addition with respect to Müller cell’s center. Addition was done at the lowest level of cone
cells. The red solid line represents the red light simulation results, while the blue solid line
represents the same algorithm applied on Dubra et al. [2011] measurements. All data in
the figure is arithmetically averaged. .................................................................................. 48
Figure 40 – Upper cone cell level intensity incoherent addition; Normalized mean radial
scattered light intensity with respect to Müller cell’s center, 680nm wavelength. The red
solid line represents the red light simulation results, while the blue solid line represents the
same algorithm applied on Dubra et al. [2011 ] measurements. All data in the figure are
arithmetically radially averaged. ......................................................................................... 49
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 41 [Labin et al. 2014] Experimental imaging of the guinea pig retina. (a) The
experimental setup included light injection from a halogen lamp into a confocal
microscope. (b) Red light transmission throughout the retina was reconstructed in 3D. (c)
Light is imaged right above the photoreceptors. Images showing four 48µm x 48µm
squares corresponding for 417nm, 491,nm 577nm and 695nm respectively. ..................... 50
© Technion - Israel Institute of Technology, Elyachar Central Library
1
Synopsis
In vertebrate eyes, the retina is structured from layers of cells organized in a specific and
apparent “reverse” order. The light sensitive photoreceptors are positioned at the back of
the retina, forcing light that is focused on the inverted retina to pass through all retinal layers until it reaches the photoreceptors in order to convert that light to neural signals. Energetically, only a third of the incident light is converted to electrical signals by the photoreceptors. A large fraction of light is absorbed in the outermost layer of the retina, the retinal
pigment epithelium (RPE); however, there remains a portion of light that is scattered back
to the eyeball.
Interlaced throughout the entire multi-layered structure of the retina are cylindrical-like
Müller cells that widen into a conical funnel facing the vitreous humour (the eyeball gel).
Measurements of these cells demonstrate a higher refractive index compared to their vicinity, which implies a wave-guiding phenomenon.
A comprehensive three-dimensional computer model of the retina was constructed based
on measured optical and physical parameters describing both geometrical outline and refractive indices. Since an analytical solution is inapplicable due to the complexity of the
problem, I used the Fast Fourier Transform Split-Step Beam Propagation Method (BPM)
to solve the Helmholz equations of light travelling inward. Light was propagated from the
vitreous humour to the RPE. Next, using Spherical Wave Light Propagation Techniques
while adding a random phase to each scattering point, light was scattered from both types
of the photoreceptor layers and from the RPE simultaneously, and their intensities were
summed incoherently back at the vitreous humour.
Scattering results created an image that indicates strong spatial correlation between the
back-propagated light intensities at the center of the cones at the bottom of the retina, and
the corresponding scattered light intensities at the vitreous humour. The rods being small
and positioned randomly at the photoreceptors layers created the corresponding indirect
scattered image. The results of this analysis demonstrate direct light scattering from the
cones and indirect light scattering from the rods, supporting the hypothesis that Müller
cells guide light and advocating for directionality in light propagation through the retina
for improved visual acuity. As opposed to previous simulations, this resultant light pattern
is corroborated by actual measurements, to achieve a fine qualitative behavior.
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List of Symbols and Abbreviations
Symbol
Meaning
AOSLO Adaptive Optics Scanning
Symbol
Meaning
J
Electrical Current Density Vector
Laser Ophthalmoscope
BPM
Beam Propagation Method
J1
Bessel Function of the First Order
Circ
Circular Aperture Function
k
Temporal Frequency
FFT
Fast Fourier Transform
n
Refraction Index
ILM
Inner Limiting Membrane
N
Natural Integer Number
IPL
Inner Plexiform Layer

Scattering Parameter
INL
Inner Nuclear Layer
v
Curl of the Vector v
NA
Numerical Aperture
2v  r 
Laplacian of a Vector Field
ONL
Outer Nuclear Layer

x
Partial Derivation in x
OPL
Outer Plexiform Layer

Dielectric Polarization Coefficient
ROI
Region of Interest
  r , 
Aperture Function
RPE
Retinal Pigment Epithelium
1,2
Field
d
Diameter

Wavelength
E r 
Vector Electric Field
0
Magnetic Permeability
H r 
Vector Magnetic Field

Normalized Frequency
I r 
Light Intensity
 r 
Complex Wave Function
fx , f y
Spatial Frequency
F
Fourier Transform
F 1
Inverse Fourier Transform
Average

2
Commutator
© Technion - Israel Institute of Technology, Elyachar Central Library
2
Background
In the following chapter I will give an overview regarding the biological and optical structure of the ocular system, i.e. the eye, and its interaction with light. First a description of
the biological system in general is presented, with its main components and their optical
roles highlighted. Next the mechanism with which an image is created onto the retina is
explained, with special respect to the physics and optics involved. Finally, a thorough description of the inverted vertebrate retina is given; main optical characteristics of its structure are introduced and examined.
2.1
The ocular system
The human eye is one of the most sophisticated and complex systems in nature, consisting
of various cells and tissues. Geometrically, the eye is best described as an ellipsoid (24mm
long 22mm across) mostly made of water-like liquid, dozens of different cell types, neural
cells. Figure 1 shows a schematic structure of the human eye with its main components,
which are described in the following short survey.
Figure 1: Schematic Human Eye Structure, light propagates left to right [website1]
Sclera: ancient Greek for “hard” can be simply described as the white of the eye [Cassin et
al. 1990]. This tough, opaque tissue serves as the eye’s engulfing protective layer, making
about five sixths of total outer layer of the eye. The sclera is moved by six muscles connected to it, and is attached to the optic nerve at the back of the eye.
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Cornea and Crystalline Lens: These two lenses compose the entire imaging power of the
eye. The cornea takes the remaining one sixth of the outer layer of the eye providing the
eye with fixed and relatively high focusing capability. The crystalline lens is rather symmetric, bi-convex lens, controlled by a dynamic set of muscles, the ciliary body that control both front and back radii of curvature, hence changes its focus length, and the entire
effective focal length of the ocular system. This process, also known as accommodation
uses a sort of biological “close circuit control loop” in order to constantly provide the imaging system with the sharpest image possible. In example, a young human being can
change his focal field of interest from infinity to about 7cm in approximately 350msec, this
phenomenal vision performance is dramatically impaired with age, and the mean minimal
range for a healthy 50 year old rises to ~30cm [Abraham et al. 2005].
Iris and Pupil: a circular tissue comprised of colorful radial strips which give the eye its
color; the iris sits in between the cornea and the crystalline lens and responsible to the diameter of the eye’s pupil, setting the amount of light imaged on the retina. The diameter of
the pupil can range from 3mm to 9mm, allowing adaptability to both low (photopic vision)
and high (scotopic vision) energy of ambient light [Gray 1918].
Vitreous Gel (humour): a clear liquid that fills the volume between the crystalline lens
and the retina; this gel has an index of refraction which is very close to water (i.e.
) and viscosity of about two to four time that of water which gives it its gelatinous
properties [Hecht 2002].
Retina: the neural hub of the visual system, containing more than 70 different types of
cells and tissues. The retina is a layered structure of tissues lining the inner surface of the
eye’s sclera, its outer surface is in contact with the choroid and its inner surface with the
vitreous humour. Geometrically, the retina is about 70% of a whole sphere, 1mm thick
and 22mm in diameter. It contains various kinds of cells and tissues, including among others the visual photoreceptors (the rods and cones), whose main role is to convert the incident imaged light to electrical signals which are than transformed to the visual cortex in
the brain. The lowest layer of the retina is the Retinal Pigment Epithelium (RPE), an absorptive layer which absolves all light that did not interact with the photoreceptors; the
RPE makes sure that no minimal amount of stray light is returned to the interior of the eye.
The next section will provide an in depth description of all the layers, tissues and cells par-
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ticipating in both light propagation and absorption in the retina [Gray 1918, Masland
2001].
2.2
The Human Retina.
As shown in Figure 2, the retina is composed of a wide variety of cells and tissues which
can be divided to ten distinguishable layers of tissue and cells.
Figure 2: Retina's cross section Diagram. (Left) layers of tissue in the retina; (Right) five major types of retinas
cells [website2]
The inner limiting membrane (ILM) is the topmost layer, adjacent to the vitreous humor
and is essentially a tapestry of Muller cells funnels. The layer is a network of nerve fibers,
acting as connecting channels for electrical signals from the following layer; the ganglion
cells, which fire signals through their axons. This fiber network is basically axons connecting the ganglion cells to the optic nerve that collects signals from the entire eyeball, and
leads the electrical signals to the visual cortex section of the brain. Next layer contains the
ganglion cells nuclei and the base of these cells axons. The following layer is the inner
plexiform layer (ILP) that contains synapses between the cells that inhabits the Inner nuclear layer and the discussed ganglion cells. The inner nuclear layer contains nuclei and
body of the bipolar cells that connects horizontally between ganglion cells. The outer plexiform layer (OPL) containing end-foots of the photoreceptors (cones and rods) and inhabits
mainly synapses between the retinas photoreceptors layer and the bipolar cells in the inner
nuclear layer (INL). The outer nuclear layer (ONL) and the external limiting membrane
contain the cell bodies of the rods and the cones – the eye’s photoreceptors. The rods and
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cones are the eyes light sensitive cells, converting light photon to electrical signal and
sending them via the axons, through the optic nerve to the brain. Engulfing the entire
outermost layer of the retina is the retinal pigment epithelium (RPE); this skin like layer of
tissue is noted in Figure 2 as the pigment cell layer, and like its name suggests is fully rich
with pigments that absorbs most of the light, but scatters a portion of the incoming light
back to the eyeball. Out of these ten layers of cells and tissue there are four major cells that
would be thoroughly discussed in the upcoming section; The Müller cells, the RPE, and
the photoreceptors (rods and cones) all share a unique and important part in the process of
light interaction and image processing with the retina, from the moment light is focused on
the retina to the moment signals are being sent back to the brain [Wade 2007].
2.2.1 Müller Cells
2.2.1.1 Morphology of Müller Glia Cells
Glial cells are non-neuronal brain cells that serve as support cells for neurons of the brain.
Generally, there are three types of glial cells in the mammalian retina, one micro-glial cell
type which takes an important role in the general defense system against invading microorganisms and two forms of neuron supporting macro-glial types the astrocytes and Müller
cells. The Astrocytes enter the cells of the developing retina along the optic nerve creating
a tube through which the axons run from the eye to the brain, and play an important role in
the supporting mechanism of the brain eye neural connectivity [Bringmann et al. 2006].
The Müller cells are the principal glial cells of the retina; they form mechanical support
across the thickness of the retina and are adjacent to both the vitreous on the one hand and
the retina pigment epithelium on the other. The Müller cells span the entire length of the
retina, which for adults is normally 130 µm. Müller cells can be characterized generally as
long cylinders, which widen at the inner end-foot to a funnel, and stretch half way into the
retina without radial change [Chao et al. 1997]. Midway through the retina is the nucleus
of the Müller cell, i.e. the soma, which can be seen in as a bulge in-between the IPL and
the OPL of the retina. The lower part of the Müller cell intersects with the photoreceptors,
which in the parafovea sits right in front of cone cells. To name a few, Müller cells main
functions include metabolic support and the nutrition of the neurons [Newman and Reichenbach 1996],mechanical support of the retina, protection against oxidation stress, keeping
potassium and water homeostasis, removing waste to the humour vitreous and many more
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[Robinson and Reichenbach 1995]. Since glia cells serve first as metabolic channels, they
face the center of the eyeball. The photoreceptors, which face the pupil, are thus best
aligned with the glia cells near the centre of the eye.
2.2.1.2 Optical Properties of Müller cells
Measurements of Müller cells using confocal microscopy and measurements of their indices of refraction have shown that compared with other cells in the vertebrate retina Müller
cells have a slightly higher refractive index, which can describe a then unknown phenomenon regarding the retina light guidance [Franze et al. 2007]. Franze and his team have discovered that the indices of refraction of the Müller calls are slightly higher at the cell’s
end-foot, n=1.359 of the Müller cells compared with n=1.335 on the vitreous gel. Moving
along the Müller cell, the cell’s index of refraction itself grows ever so slightly, up to
n=1.409; in the meantime the retina’s index of refraction rises to n=1.358, but throughout
the entire length of the Müller cell, the index relations are
[1]
nMüller _ Cell ( z )  nVicinity  z  ; z .
In a another experiment shown in Figure 4, a single mode fiber laser (a) was positioned in
front of the retinal surface; the divergent light was visualized (b) by submerging the fiber
Figure 3: Müller cell structure; Image was taken using differential interface contrast microscopy.
Bar size is about 25 µm [Franze et al. 2007]
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glass into an agarose gel that caused scattering of the light and allowed light detection
normal to the beam; adjacent to the tissue layer a plastic membrane was positioned to image the spot at the end of the Müller cell. Test results yielded a tight spot when light was
propagated through the Müller cell with spot diameter of about
whereas
the light that propagated between two adjacent Müller cells yielded a wider spot size, with
diameter of about
[Agte et al. 2011], these measurements are for the
close periphery, as you move along the eye, they may increase. These results imply that
Müller cells, apart from their diverse biological important roles, and cardinal metabolic
functionalities, may have an important role with the formation of images on the retina on
the small scale, and with the vision acuity of the entire ocular system on the grand scale
[Labin and Ribak 2010].
Figure 4: Retina tissues illuminated from the left using a thin fiber glass, the dotted line [Agte et al. 2011]
(a) fiber laser core placed parallel to the retina. (b) laser light diverging in the gel. (c) Fiber laser (dotted line)
placed just in-between the Müller cell funnels. (f) Fiber laser is placed right in-front of the Müller cell funnel. (d,
g) Graphs show the profiles of the laser beam propagating between the Müller cells and inside the Müller cells
respectively. The graph is taken at the inner plexiform layer. (e, h) Graphs show intensity as shown by the
irradiance of the laser beam on a plastic membrane placed right after the layers of tissues.
2.2.2 The Photoreceptors – Rods and Cones
The photoreceptors are a particular type of retinal cells in all retinae. These cells convert
photons focused on the retina to an electronic (ionic) signal that is then channeled to the
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brain by the ocular axons. There are three possible types of cone types that can be found in
a vertebrate retina, the L, M and S types, standing for long, short and medium wavelengths, ,which are basically the chromatic variation of cones in the retina, and a single
type of rod cell. The distribution of rods and cones varies with respect to the angular displacement from the fovea, while the fovea can be referred to as the polar point of the system, which inhibits the maximal number of cones.
Figure 5: Normalized absorbance with respect to wavelength. Each curve is normalized by its own maximum.
Figure 6: Rods and Cones densities vs. angular displacement. (Osterberg 1935)
Figure 6 shows explicitly the distribution of the photoreceptors of the human retina; most
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of the cone population is concentrated at the fovea, while the rods are
concentrated mostly at the peripheral parts of the retina.
Figure 7: En Face imaging of cones at the Fovea, bar is 10µm [Curcio 1990]
The average human retina contains 4.6 million cones and 92 million rods with a wide variance of about 15% [Curcio 1990], while at the fovea the density is single flavored with an
absolute dominance for the cones, As can be seen in Figure 7, the spatial concentration of
the retinal photoreceptors at the parafovea and the peripheral retina is relatively more
blended which creates a more complex picture as shown in Figure 8. The parafovea inhibits more cones than the peripheral retina, while the rods remain at the same concentration.
Figure 8: Retinal photoreceptors; on the left, the parafovea retina is shown. There are approximately 10 rods to
every cone in that region of the retina. The right we see the peripheral retina; about 8 mm eccentricity from the
fovea to the temporal region, bar is 10μm for both images [Curcio 1990].
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The photoreceptors are responsible for light transduction to electric signals; in the human
retinae, the cones are sensitive to three basic wavelengths red, green and blue and are being mostly used during daylight, at highly illuminated conditions. The rods are much more
sensitive to low light than the cones, but they can only see variations in black and white
with no contribution to chromatic perception. Incident light is focused on the photoreceptors which receive the incoming photons at their outer segments; photon absorption in the
outer segments of a rod or a cone causes a decrease in the inward currents of the photoreceptors. The outer segment currents in the cones are an order of magnitude faster than
those of the rods (Figure 9), which require a much stronger pulse to activate. These pulses
of electrical currents are filtered through voltage gated conducting membranes in the inner
segments of the photoreceptors, eventually causing voltage spikes from the neurotransmitters located at the synaptic terminals at the outer end of the photoreceptors.
Figure 9: Schematic photoreceptors structure. On the left we see the main section of the photoreceptors, amongst
others the outer sections, the inner section, the cell bodies and the synaptic terminals (bottom illumination). On
the right (top illumination), we see four sketches of the current and voltage response functions [Left – website 4 ;
right – Burns 2005]
2.2.3 The Retinal Pigment Epithelium
The Retinal Pigment Epithelium (RPE) is a monolayer of pigmented cells, forming the last
layer of the retina, and being the last barrier between the retina and blood cells at its back.
As a layer of pigmented cells the retina absorbs light focused on it by the lenses of the eye;
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and maintains nutrient support such as glucose, retinol and fatty acids from the blood and
delivers these nutrients to the photoreceptors. Figure 10 below shows that apart from light
absorption, the RPE also provides constant potassium supply to the Glial cells and removes waste from the photoreceptors [Strauss 2004]. The RPE spectral transfer function is
biologically characterized by the high concentration of melanin in the tissue [Hammer and
Schweitzer 2002], which specifically defines its spectral response to light. Melanin is a
highly complex natural polymer which absorbs light in the deep ultraviolet and the blue,
and as the spectrum moves closer to the middle of the visible and towards the red and the
infrared, it reflects more and more light. The ultraviolet radiation absorption of the melanin
protects the RPE from oxidation effects, and serves as a strong natural antioxidant in the
eye. The RPE absorbs light, and therefore creates a dark screen behind the photoreceptors
and thus enhances image acuity of the eye. Figure 11 shows the above described response
of the RPE to light; it is evident that the RPE absorbs most of the blue light (~98%) and
reflects in comparison a higher percentage of the incident light as the wavelength gets
larger.
Figure 10: Schematic summary of RPE function [Strauss 2004]
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2.3
Physics of the Ocular System
After reviewing the biological aspects of the human eye, including a brief look at the major parts of the eye and a deeper inspection at the retina, let us discuss in the following paragraphs the cardinal physical properties of ocular system. The discussion shall start from a
short optical analysis, a quick flashlight on resolution of the eye and eventually the StilesCrawford Effect.
2.3.1 Optics of the Eye
Optically, the human eye is typical of vertebrate eyes in general, fairly in the middle regarding resolution and focusing range [Charman 1991]. Although the human eye is a relatively a simple optical system, it is capable of a nearly diffraction limited performance
close to its axis in good lighting conditions when the pupil’s aperture is small (~ 3 mm).
The eye also provides a wide field of view of a minimal of 1300 full angle provided a
frontal direction of gaze; The variance in the field of view is strongly dependent in the geometric properties of the given individual [Charman 1995]. The schematic eye optical system, provided in Figure 12, shows on the left a schematic, geometric optic analysis of image formation on the retina. This schematic is based upon the Gullstrand-Emsley schematic eye model (1952). This model contains the cornea, represented as two surfaces, the crystalline lens which is represented as two surfaces as well. The model defines the aperture
Figure 11: RPE spectral reflectance [Delori and Pflibsen 1989]
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stop of the entire system at the eye’s pupil. This model includes neither accommodation
nor chromatic aberrations, which are dealt thoroughly in the Navarro et al. model using
differential focal lengths of the crystalline lens and defining each of the surfaces using aspheric curvatures [Navarro et al. 1985]. Regardless of the depth of the model (i.e. accommodation or aspheric curvatures) the analysis of the eye may be reduced to the analysis
represented in the upper right corner of Figure 12; This drawing represent image formation
of any optical system containing as many constraints as are represented in the discussed
model. Image of a point object at the height h in is determined by two rays, one normal to
the axis at the P’ and one passing through the anterior focal point, and running parallel to
the optical axis at the first principal plane. The nodal points N and N’ define the actual
field of view of the system; for instance, the chief ray, running through the center of the
entrance pupil define a field of view that is smaller compared with the actual field of view,
determined using the nodal points by a factor of 0.82.
Figure 12: Human schematic eye [Website 3]
2.3.2 Resolution
2.3.2.1 Diffraction
For the aberration free model of the eye which is the upper limit of the possible optical
performance for a real eye, the parameters affecting the resolution of the eye are the wavelength of the given incident light and the pupil diameter at the moment. That pupil diameter defines the resolution limit known as the diffraction limit of the system. Therefore for
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the diffraction limit case, the problem reduces to a mere calculation of the diffraction pattern of the optical stop of the system; the pupil’s diffraction pattern. The pupil is, as was
described earlier a circular aperture of the eye, and therefore can be defined by the circ
function:
  r ,   circ  R 
[2]
,
where r and θ are the spherical coordinates at the pupil’s plane and R is the diameter of the
eye at a given moment. The Fraunhofer diffraction framework defines the image at the focal plane, and calculates the intensity of the field to be the absolute value of the Fourier
transform of the mask. Simply the complex wave-function is given by:
  r   
[3]
where
,
,
 ik f x '  f y '
 r  e 
dx ' dy ' ,
x
aperture
y
are the spatial frequencies of the system, and the aperture
is the pupil of the eye. Therefore, the intensity is given by squaring the (complex) amplitude,
[4]
I r    r  .
2
Plugging [2] into[3], one can derive [Lipson et al. 2010] the Airy complex function, given
by
[5]
 2 J1   R  
,
 R 
  ,     R 2 
where  and  are defined by f x   cos  and f y   sin  , in polar coordinates, and J1 is
the Bessel function of the first order. Using [4] and substituting [5] into it and we get the
Airy intensity pattern
 2J  R  
I  r   I0  1
 .
 R 
2
[6]
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2.3.2.2 Optical Aberrations
The paraxial approximation in Gaussian optics assumes that the entire fan of rays does not
stray much from the optical axis. More explicitly, a paraxial ray is such a ray whose angle
with respect to the axis may be expressed using
[7]
 sin   .
The eye’s pupil expands to a full diameter of about 8 mm, while its optical axis length
from entrance to the retina is about 22 mm, hence the full angle defined at about 100; such
an angle cannot be approximated according to the paraxial approximation, and must be
tended with special care to aberration. Expanding the sine of the angle to higher orders we
obtain
[8]
sin     
3 5 7
3!

5!

7!
 ...
Taking higher orders into account, it is evident that a single point is nonexistent in practice, and therefore, monochromatic aberrations must be taken into account. A simple explanation is that the point of coincidence which is defined by the Gaussian optics does not
coincide with the actual point defined by the higher orders of the sinus. Following the definition of the angle given earlier, one can assume a strong correlation between the aperture
of the system and the effect the aberrations make on the image; and indeed, for a given effective focal length, the wider the aperture, the larger the aberrations in the image, with
respect to the higher orders required by the sine. An image without any aberrations is
called diffraction limited image, which defines the best obtainable image with the given
optics. Figure 13 gives several examples of monochromatic aberrated images, with the
Figure 13: Examples of intensity distributions of given optical effects. a - Diffraction limited image, with the visible Airy Pattern; b – Spherical Aberrations. c – Coma. D – astigmatism [Lipson et al. 2010]
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Airy pattern illustrating the diffraction-limited image.
Even in the paraxial approximation, with a monochromatic aberrations-free regime, taking
color into account results in optical aberrations. The eye is a lens-based optical system,
filled with refractive natural materials such as the vitreous gel. Thus, it is important to take
into account the fact that the refractive index ( ) of transparent material is a decreasing
function of the wavelength, with differences between the materials. Chromatic aberrations
arise from the variations in refractive indices due to wavelengths; applying this to the eye,
it is modeled using the refractive index of water, since all the transparent tissues in the eye
are basically made of water and small additions of lipids (fats). The following formula defines the refractive index of water with respect to wavelength [Bennett and Tucker 1975].
[9]
n2  1.7662  1.38 108  2  6120   2  1.41108   4
Taking chromatic aberrations into account, Figure 14 illustrates the effect of chromatic aberrations on lateral and axial imaging. For both cases we have obtained different focal
points for different colors, yielding blurry images; while the collimated beam hitting the
lens ended with axial displacement of the two focal points on the optical axis, lateral
chromatic aberration is the difference in the size of the images of the same object being
imaged.
Figure 14: Optical Chromatic Aberrations. Upper drawing describes the axial chromatic aberration while the
lower one the lateral chromatic aberration.
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2.3.3 The Stiles Crawford Effect
The human eye is directionally sensitive; this statement is one of the most pronounced optical phenomena in the ocular system. Basically, when white light is positioned at the center of the eye’s pupil, and then moved in any direction, there is a significant decrease in the
perceived intensity as a function of the displacement from the pupil’s center. Figure 15
shows the relative decrease in luminance levels with respect to the source location translation in front of the human’s pupil. This phenomenon is known as Stiles-Crawford effect of
the first kind [Snyder and Pask 1973]. Keeping that scenario in mind, consider the same
experiment, only done with a monochromatic source; there is a similar decrease with the
observed light intensity with respect to the light source’s location in front of the pupil.
Figure 15: Relative luminance as a function of pupil position with respect to the entering beam. Data for measurements taken from the left eye of W.S Stiles [Westheimer, 2008]
Moreover, different monochromatic colors yield different intensity response curves, and
present different modal responses, which may be deduced to a color dependent phenomenon, in our proposal, wave-guiding [Snyder and Pask 1973]. Until today, most of the
measurements of the Stiles-Crawford effect were conducted in the fovea.
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2.4
Background Summary and Motivation
The ocular system in general and the retina in particular are fundamental building blocks
of all optical phenomena. Wave-guiding in the retina is still a controversial approach for
describing light propagates into the eye, especially with regard to Müller cells; nevertheless, all research done in the field so far was unable to connect the dots between light
propagation into the retina and light scattering out of the retina, which represents the forefront of lab measurements today. The motivation of this work is to provide a different approach, supporting the hypothesis that light-guidance phenomena occur in the retina, and
that the results obtained with this computational and theoretical study can be corroborated
with other works in the field.
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3
The Method
In the following chapter I will give an overview portraying the methods and physical tools
that were used throughout the work. The theoretical framework will be highlighted, including wave-guiding propagation technique, light scattering models and modal analysis. An
explicit description of the computational model will be given at the end of the chapter, using the knowledge that was shown earlier to clarify used methods.
3.1
Theoretical Framework
3.1.1 Wave Equations of Light.
Let us start with the differential Faraday’s law
 E  
[10]

0 H ,
t


where
⃗
Vector electric field, namely (
).
⃗ – Vector magnetic field, namely (
).
Applying curl operator on both sides of equation [10] yields
 E  
[11]

0  H .
t


Using Ampere’s law which is specified below, and taking the case that no current sources
are in the given region of interest, we get
[12]
 H  J 
where we have substituted ⃗
 
 E
D
,
  H 
t
t
⃗ , which defines the electric field with respect to the die-
lectric polarization in the matter. Substituting [12] into the right-handed side of [11] we
obtain
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[13]
    

   E
0   H   0 
t
t  t

n2  2
2
  0 0 n 2 2 E   2 2
1

c t
t
  0 0
n
2



0


2
E
0
t 2
 
,
E
c
where c is the speed of light in vacuum and n is the index of refraction. Next, we are assuming a harmonic solution to [13] i.e. (
)
(
)
and substituting
[11] into it we may derive


    E     E   2 E   2 E  
0
[13]
  E  x, y, z  e
2
 it
n2  2
E
c 2 t 2
n2 2
  c2   E  x, y, z  e it  
n2 2
 E r   2  E r   0 .
c
2
This last equation is also known as the Helmholtz equation and it describes the electric
field component of Maxwell’s representation of light in a source-free space [Saleh et al.
1991, Yariv and Yeh 2007]. Helmholtz equation will be the basic input for both our chosen
light propagation method, as would be described in depth in chapter 3.1, and for the basic
analysis of simple light wave-guiding phenomenon.
3.1.2 Waveguiding.
3.1.2.1 Geometrical Optics Approach
Optical waveguides consist of a core, in which the light is confined, and a cladding which
is defined by the surroundings of the core. As a general rule, for a wave guiding phenomenon to occur, the refractive index of the core should be slightly higher than the refractive
index of the cladding, or more explicitly, following the notation from Figure 16,
.
A critical condition for beam confinement to the core is taken from total internal reflection,
which requires that the angle of incident light follows
[14]
n 
n1 sin c  n2  c  sin 1  2  .
 n1 
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Figure 16: Total internal reflection in an optical fiber [website5]
The maximal incident angle for the fiber to guide rays in the core following the paraxial
estimation is thus [Okamoto, 2006]
[15]
max  sin 1

n12  n2 2
.
Since the difference between the refractive indices is relatively small, [15] can be estimated by
[16]
max
n12  n2 2 .
This maximal angle is also called the numerical aperture of the system, or NA of the
waveguide.
The simple ray optics picture of light wave guidance does not explain the entire phenomenon [Saleh et. al, 1991]. Although all guided waves in the optical waveguide must obey the
total internal reflection necessary condition, not all angles in the range will be guided
through the fiber’s core. Since electromagnetic waves are associated with the optical propagation through the fiber, a discrete set of angles, temporal and spatial frequencies can be
supported in the core, which would be the guided modes of the waveguide.
3.1.2.2 Guided Modes
To better understand the phenomenon of wave-guiding, let us examine a step index optical
fiber, depicted in Figure 17. It consists of two different refractive indices, which comply
with the earlier mentioned rule of thumb i.e. n1 > n2, constant for all z.
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Figure 17: Left handed shows the step index optical fiber, note that n1>n2; to the right, cylindrical coordinate
system, with z the propagation direction.
To treat this optical fiber, first let us rewrite the Helmholtz equation [13] with cylindrical
coordinates; the z axis is the direction of propagation, therefore for the electrical field
 2 Ez 1 Ez 1 
 2 Ez

 2
Ez  2  k 2 Ez  0
2
2

   
z
[17]
  x2  y 2
.
 y
 
  tan 1  
x
Using separation of variables, we get a product of three different and independent functions of each of the coordinates for the electrical field z component:
[18]
Ez   ,  , z   F       Z  z  .
Mixing it all and we get a harmonic eigenvalue equation for each of the coordinates,
[19]
d2
Z  z    2Z  z   0
2
dz
d2
    m 2     0
2
d
,
d 2 F    1 dF     2
m2 
2


k



F   0

2 
d2
 d



where
are coefficients, real numbers, which need to be found. The solutions for the
two homogeneous equations for Z and
are complex exponents, scaled by the parameters
respectively. The solutions for the F equation are either the Bessel or the modified Bessel functions [Spiegel, 1968], or more explicitly,
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Z  z   exp  i  z 
    exp  im 
[20]
.
F     Bessel Functions
Substituting these solutions into Eqs. [19], we may obtain the solution for the electric and
the magnetic field,
[21]
 AJ m  n12 k0 2   2    eim ei z ;   a



Ez  
2
2 2
im i  z
CK m    n2 k0    e e ;   a
 BJ m  n12 k0 2   2    eim ei z ;   a



Hz  
2
2 2
im i  z
 DK m    n2 k0    e e ;   a
where A,B,C,D are coefficients, and
,
are the Bessel and modified Bessel functions
respectively, the solution outside the fiber’s core is described by the modified Bessel function, which exponentially diminishes. The other components can be computed in terms of
the electric and the magnetic field [Yariv, 2007]. The modes which can be guided in this
optical fiber can be calculated by requiring continuous parallel field at the interfaces between the core and the cladding, or more explicitly,
 J m '  pa  K m '  pa    J m '  pa  n2 2 K m '  pa   m2  1
1   1 n2 2 1 

 2
[22] 

  2  2  2  2  2 2  ,
n1 q 
 pJ m  pa  qK m  qa    pJ m  pa  n1 qK m  qa   a  p q  p
where
p 2  n12 k02   2 ; q 2   2  n22 k02 and a is the core size.
This eigenvalue equation describes all the guided modes supported by the fiber, so for that
purpose let us define an important parameter, the normalized frequency,
[23]
  k0 a n12  n2 2 .
Substituting the normalized frequency and taking m = 0, we get a graphical solution for the
guided modes equation. Each of the intersection in Figure 18 specifies a certain mode, for
a certain wavelength, at a certain fiber setup, and therefore for each of the
solution, there
is a different spatial distribution of energy and different polarization. Spatial distribution of
the optical fiber modes is shown in Figure 19.
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Figure 18: The left side of equation [23] is plotted from the with dotted lines; intersection defines the supported
modes in the optical fiber.
Figure 19: Spatial distribution of energy [website 6]
3.1.3 Light Scattering
Light scattering on a biological tissue from a physical point of view can be treated as a
composition of light scattering from different sized particles; hence a correlation between
the appropriate scattering model and the scattering particle size must be taken into account.
Another factor that must be considered is refractive indices of the scattering particles. Thus
the nature of the scattering behavior would arise strictly from these two parameters. The
refractive index of biological tissues averages about 1.4; it varies between 1.33 for water
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based tissues up to 1.5 for non-water based tissues, such as proteins, lipids and other heavy
molecules [Bolin et al. 1989]. Biological tissues, and the retina not being apart from that
generalization, are non-homogeneous volumes comprised of different surfaces such as
membranes and nucleus, each measures a different index of refraction [Franze, 2007,
Brunsting and Mullaney, 1974]. Since the scattering particles in the retina are living tissues, each with its own variety of refractive indices, then each single scattering event can
be described by the cosine of the scattering angle, receiving its own independent scattering
phase function, or more explicitly [Agte, 2013]
[24]
p  s, s '   p  cos  ,
where the probability p for scattering between direction s into direction s’ is defined by the
probability to be in the cosine of the incident scattering angle . Apart from phase differences there are also intensity differences at light scattering which varies with the size of
the scattering particle; considering the relation between the typical size of the scattering
particle and the wavelength of the incident beam, we may derive a governing parameter to
describe scattering phenomena [Hulst, 1981]:
[25]

2 a

,
where a is the typical size of the scattering particle and λ is the scattered wavelength.
There are three regimes that govern light scattering from different sized particles that can
be derived from that parameter the Rayleigh scattering, the Mie scattering and geometrical
scattering. Geometrical scattering occurs when
; in that regime the wavelength is
much smaller than the typical size of the system, and scattering can be treated by simple
geometrical optics tools such the likes of mirrors. The second extreme regime is when
; in that case, the system is much smaller than the wavelength, and therefore light
must have a profound interaction within the scattering process, therefore phenomena such
as Rayleigh scattering and quantum optics processes may appear. In the mid sections of the
parameter, we may find Mie scattering. In that regime where the scattering particles are
of the same size as the wavelengths of the scattered light, we observe higher light intensity
at smaller scattering angles than at larger angles. In general, from a macroscopic point of
view, for the biological tissue case in the retina, we may take into account that the scattered light will follow the basic rule that an arbitrary phase will be attached to each of the
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electric fields of the scattered light and that for smaller angles we may observe higher light
intensities than for larger angles.
3.2
Coherent and Incoherent Addition of Waves
Propagated light into the retina is considered to be light emitted from a single source at the
bottom of the vitreous gel, as was imaged by the two lenses at the entrance of the eye
[Vohnsen, 2005], and therefore since it is a single source for the retina it should be related
as a coherent light source In contrast, light scattered from either the RPE or the photoreceptors, which inherits an arbitrary phase once scattered from the surface, should be related to as an incoherent source. Let us consider a simple system of two wave sources with
fields
and
. We know that for coherent waves, the resulting intensity shall be the
square of the summed fields, or more explicitly,
I  1  2  .
2
[26]
Plug-
Now, let us assume without loss of generality that
ging these into [26] and we get that for coherent wave sources, the total intensity is
[27] I  1  2    A cos1  B cos2   A2 cos2 1  B2 cos2 2  2 AB cos1 cos2 .
2
2
On the one hand, the average of a square of the cosine is
regardless of the wave sources
nature, whereas on the other hand, the average of the cosines product, taking into account
that the phases are random which is the incoherent case, and we get that:
[28]
2 AB cos 1 cos  2  AB cos 1   2   cos 1   2   
,
1
1
AB cos 1   2   AB cos 1   2   AB  AB  0
2
2
meaning that the total intensity for two incoherent wave sources should be summed as
[29]
Iincoherent 1 , 2   1  2
2
2
,
whereas the total intensity for two coherent wave sources should be summed up as
[30]
I coherent 1  2   1  2  .
2
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These two approaches [Saleh et al. 1991] lead to two different intensity addition approaches taken in the simulation model, one is coherent intensity addition, following the single
source at the entrance to the retinal system and the other is incoherent intensity addition,
following the random behavior of scattered light from both the RPE and the photoreceptors.
3.3
Beam Propagation Method
The beam propagation method is the most powerful technique for investigating linear and
non-linear light propagation phenomena in axially varying waveguides. The beam propagation method will assume weak light guidance, meaning that the difference between the
waveguide and its environment is relatively small, and that the electric field is varying
slowly with respect to the optical axis. In this section we will investigate both beam propagation of axially coupled waveguides and beam propagation of light from a spherically
symmetric source.
3.3.1 Fast Fourier Transform Split Step Beam Propagation Method (BPM).
Light propagation in space is expressed by the Helmholtz equation, as was shown in Section ‎3.1.1, Equation [13]. Let us write that equation again for the electric field:
[30]
2 E  r   k 2 n2  r  
2 E  r  2 E  r  2 E  r  2 2


 k n r   0 .
x 2
y 2
z 2
We are interested in this stage of the analysis in light propagation into the eye; therefore let
us examine the following inward propagating beam,
E  r     r  e kn0 z .
[30]
Since there is a different behavior for the transverse components of the electric field and its
lateral components, we are inclined to substitute
and [30] into [30],
2 E  r   k 2n2  r   0
[30]
.
 E 2

E
  2 E  r  eikn0 z   2  ikn0
 k 2 n0 2 E  e ikn0 z  k 2 n2 E eikn0 z  0
z
 z

Light is propagating essentially along the optical axis, therefore we would like to express
the propagation along that axis; thus, let us manipulate this last expression to a first order
derivative with respect to the optical axis, z. First let us examine the transverse dimension
28
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of the light beam with respect to the light’s wavelength. Using geometrical considerations
[Vohnsen et al., 2005] we know that the spot size that the crystalline lens images at the
retina is correlated with the incident light beam width at the pupil using the relation
r 
[31]
where
is the relevant wavelength,
 f eye
 neye0
,
is the eye’s focal length,
the eye’s mean re-
is the incident’s beam width. For an aberration-less system estima-
fractive index and
tion which is fairly sufficient in our case, the expected beam width for green light is about
r 
700nm  22mm
~ 20 m . Therefore, the ratio between the light beam’s width and the
 1.33  2mm
wavelength is about one to fifty. Under this ratio, we may follow the slowly varying amplitude approximation; the light’s electric field amplitude would change fairly slowly with
respect to the optical axis, or more explicitly
2 E
z 2
[32]
2k0
E
.
z
Let us substitute [32] into [30] and take into account the slowly varying amplitude approximation, and we obtain
E 2
E
 ikn0
 k 2n0 2 E  k 2n 2 E  0
2
z
z
E
 2 E  r   ikn0
 k 2  n0 2  n 2  E  0
.
z
ik
i
E
   n 2  n0 2  E 
 2 E
n0
kn0
z
 2 E  r  
[33]
Now, let us look at the refractive indices more explicitly; we know that in the ocular system the differences between the refractive indices of the Müller cells and the environment
are relatively small [Franze et al 2007]. Let us examine the refraction indices of equation
[33] more explicitly,
[34]
n
2
 n2 n0 
n n 
 n0 2   2n0 
   2n0   0   n0  n  n0  ,
n
2 2 
 2n0 2  n0 1
29
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where we have used the assumption that the differences between the core and the cladding
refraction indices are small.
Substituting [34] into [33] and we get
E
i
 ik  n  n0  E 
 2 E
z
kn0
[35]
,
E
  A  B E
z
where we have defined
(
and
).
The solution to this differential equation with respect to the z axis for small steps of h is
the electric field scaled by the exponential of the coefficients defined before. We get
E  z  h, r   e A B h E  z, r  .
[36]
The A and B coefficients are operators that apply to the entire field, and therefore cannot
be simply split into two separate exponents. We use the Baker-Hausdorff theorem for noncommutative operators [Lax et al. 1981, Okamoto (2006)]
1
1


e Ae B  exp  A  B   A, B    A  B,  A, B   ... ,
2
12


[37]
where [
]
is the commutator of the A and B operators, and substituting it
into equation [36] to the first order results in
E  z  h, r   e Ahe Bh E  z, r  .
[38]
Rearranging the right-handed side of equation [38] using Fourier transform and inverse
Fourier transform of the same expression, and time shifting accordingly [Spiegel, 1968],
we obtain the following expression for a finite step of size h beam propagation of the electric field in the medium:
[39]
where
E  z  h, r   F 1F e Ahe Bh E  z, r   F 1 e f F e Bh E  z , r   ,
A
is the Fourier transform of the A operator, and F,
is the Fourier and inverse
Fourier transform operators. This last expression enables us to use the ease of fast Fourier
30
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transform in any computational software, and propagate the electric beam each split step at
a time. This method was indeed used to propagate a well-defined light beam into the retina
and to calculate the light intensities at the outermost layers; in order to scatter light out of
the retina, we had to use a slightly different approach, and that is the Spherical Symmetric
FFT split step BPM.
3.3.2 Spherical Symmetric FFT Beam Propagation Method.
To scatter the light from the retina, let us develop a method for light propagation in a
spherical symmetric system. Let us start again with the Helmholtz equation, i.e. Equation
[13], or more explicitly
[40]
2E  r  2E  r  2E  r 


 k 2n 2  r   0 .
2
2
2
x
y
z
Since the problem is expressed in spherical coordinates system, see Figure 20, let us define
Figure 20: Spherical Coordinates [Website 5]
each spot in the 3d space using three coordinates, r – the radius, the range from the origin;
- the azimuthal change in angle; and
- the elevation angular change from the zenith.
Rewriting [40] in spherical coordinates we obtain [Rubio et al. 1999]
[41]
E  r 
1 2
 E  r   2ikn0
 2k 2  n 2  n02  E  r   0 .
2
r
z
Following the same method as in the split step BPM, we shall use both the slowly varying
amplitude approximation and the small variation in refractive indices, and we derive
31
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E
i 2

E  r   ik  n  n0  E  r  .
r 2kn0 r 2
[42]
D
C
In this method the axial movement is about the radius, and therefore the transversal directions are the azimuthal and the elevation as defined earlier. Following the same path as before, we are inclined to define the solution to this equation using Baker-Hausdorff identity
[37], plugging it all into equation [42] and we arrive at
E  r   r ,  ,    e  C  D  E  r   eC e D E  r  .
[43]
Taking again the inverse Fourier transform of the Fourier transform of the right handed
expression, and time shifting again as we did before, and we obtain the spherical symmetric BPM equation for light scattering off the retina
[44]




E  r  r , ,    F 1 F eC e D E  r , ,    F 1 e f F e D E  r , ,   ,
where F and
C
are the Fourier and Inverse Fourier transform operators respectively and
is the Fourier transform of the operator C. This last expression defines a method to
propagate spherically symmetric waves through random slices of media; the retina, taken
from the view of a pin-point on the RPE or the photoreceptors is comprised of random
media, and therefore, this method could be used to scatter light from the retina.
3.4
The Computational Model
The computational model follows several guidelines laid in the past [Labin and Ribak,
2010] but takes a novel approach regarding light backscattering from different layers in the
retina. Inward propagation takes the earlier described approach of split step beam propagation method, propagating the beam of light each slice at a time. At the RPE and the different photoreceptors layers, the light is multiplied by an arbitrary phase, and scattered back
to the vitreous gel following a point source propagation mechanism. The following section
shall rigorously describe the inputs, the assumptions and the outputs from each section of
the computational model.
3.4.1 Computational Model Setup
A three dimensional simulation model was created, comprised of 512 x 512 matrix sized
tissue slices, summing up to 1300 slices in total. The pitch step size of the model is cubic
and measures
, so the total dimensions of the simulated retinal tissue are 66.56 ×
32
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66.56 × 169 µm. The Müller cells were modeled using an image of the cell [Chao et al.
1997] that was digitized, binarized and used as a baseline throughout the entire simulation
phases. 45 Müller cells were multiplied from the same digitized baseline, forming a parallel puzzle throughout the entire structure; Figure 21 portrays the innermost layer of the
cells as seen from the eyeball’s vitreous gel.
Figure 21: Simulated Müller cells funnel layout as appears from the top of the retina
The refractive indices of the simulation model followed measurements of both the inside
of the Müller cells and outside of the Müller cells [Franze et al. 2007]. The refractive indices of the photoreceptors followed measurements of their width, and coupling efficiency of
the cones and the rods [Vohnsen, 2005]. On these measurements, I superimposed scattering, to create a perturbed version of the refractive indices. Figure 22 depicts the refractive
index throughout the retina; it is evident that the refractive indices of the guiding cells are
slightly higher than those of their neighborhood. The fluctuation in the surroundings refractive index specifies inhomogeneous change of surfaces and textures of the encircling
tissue around the guiding cells [Vohnsen 2005, Franze 2007].
33
[Labin et. al 2014], and that created a large room for the rods to be modeled into; the rods
were simulated using a randomly generated tapestry to fill the gaps between the cones,
creating a different image of the retina each time the simulation was ran. A typical layout
of the rods and cones at the photoreceptors layer is shown in Figure 23; the single circles at
the middle of each black space are the cones, whereas the other circles are the rods, filling
the entire space in between the Müller cells, and providing a different, randomized rod location each time.
Refractive Indices
1.42
Environment
In The cell
1.41
1.4
Refraction Index []
© Technion - Israel Institute of Technology, Elyachar Central Library
Müller cells’ outermost tip was attached to a matching tapestry of cones right underneath it
1.39
1.38
1.37
1.36
1.35
1.34
1.33
0
20
40
60
80
x[m]
100
120
140
160
Figure 22 – Refractive indices throughout the retina. The first
are where the Müller cells are positioned;
right after them the cones come and create a step function in the cell’s refractive index.
34
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 23 – Rods and Cones scattering layer; The cones are colored gray and the rods are colored white. Scale bar
is 6.5µm long.
3.4.2 Inward Light Propagation
The input intensity pattern to the retina is the diffraction pattern of the pupil imaged on the
retina’s inner limiting membrane, the retina’s first tissue. The pupil being a complete circle
creates an Airy pattern on the surface of the retina; examining equation [6] more thoroughly, if we take a single dimension cross section at the main axis of the Airy pattern, and ignore the ripples far from zero, we may rewrite the equation as an decaying exponential
[45]
 q 
I  q   I 0 exp  2  ,
 2s 
where q is the distance from zero and s is the width of the main peak of the Airy pattern
[Born and Wolf, 1959]. The intensity distribution shown in Figure 24 is slanted from the
center of the matrix due to the nature of the investigated section of the retina.
35
1
500
0.9
450
0.8
400
0.13[m]
© Technion - Israel Institute of Technology, Elyachar Central Library
Incident Beam
350
0.7
300
0.6
250
0.5
200
0.4
150
0.3
100
0.2
50
0.1
100
200
300
0.13 [m]
400
500
0
Figure 24: Incident beam as seen at the Müller cells funnels’ layer
As was stated in previous sections, the parafovea is set off the ocular system’s optical axis,
hence the simulation had to take into account that slant of the incident’s light trajectory,
and thus the peak of the incident’s light beam was located that way; this slanting in the
angle of incidence allows the beams that arrive at the reitna at wide angles to fall at the
same photoreceptors as the narrow beams.
Light was propagated using the methods described earlier, and the intensity was measured
at three different locations; at the RPE, at the cones and at the rods layer. At these layers,
the intensity was calculated across the eye’s full aperture, assuming a coherent
monochrimatic light source. An explicit expression of the mean intensities is
[46]
I i  i ,1  i ,2  ...  i ,n
2
,
where Ii is the intensity at either the RPE or the photoreceptors layers, and
tric field amplitude at the
incidence angle.
36
is the elec-
© Technion - Israel Institute of Technology, Elyachar Central Library
3.4.3 Backward Light Scattering
Intensity at the RPE and the photoreceptors was back scattered towards the vitreous gel
from two different layers. Light intensity at either of the scattering layer was divided into
N × N squares, N was taken as 4, which resulted in 0.52μ
wide scattering sources.
Since the scattering surfaces are in-homogenous in both refractive index and outer geometry, each of these phase-less amplitudes measured at either the photoreceptors or the RPE
was multiplied by an arbitrary phase, emulating the incoherent nature of light scattering
[Gotzinger, 2008]. Each of these fields was then propagated back from the scattering surface towards the vitreous gel, using the spherical symmetric propagation methods, where
we have taken the N × N squares as the center of the sphere. At the vitreous layer, the N2
calculated electromagnetic fields that had been back-propagated were added incoherently
to generate the scattered intensity at the vitreous layer. The total intensity at the vitreous
layer, for each wavelength, was
N
[47]
N
2
2
I       PR        RPE     ,

 m n1 
n
m 1
where for each scattered and propagated
or
fields, which are the electric fields
from the photoreceptors or the RPE respectively, were summed incoherently at the vitreous layer, at the appropriate wavelength.
3.5
Methods – Short Summary
In the last chapter we have discussed the methods and physical tools used to describe light
propagation into and scattering out of the retina; the following chapter shall thoroughly
depict the calculated results at the RPE and the vitreous gel followed by a discussion of the
implications of such a comprehensive model and intensive calculations.
37
© Technion - Israel Institute of Technology, Elyachar Central Library
4
Results
Following the methods described in the previous chapter, I will now introduce and analyze
both the propagation and scattering results. The results shall be depicted starting from the
propagation results onto the RPE, followed by the scattering results as seen from the ILM
layer at the vitreous gel.
4.1
Inward Propagated Intensities at the RPE
Light propagation into the retina was conducted using an Airy pattern distribution as an
input at the Müller cells base. The Airy pattern distribution, created due to the circular nature of the eye’s pupil is then propagated into the retina and forms an image on the uppermost layer of the RPE. The simulation was conducted in the visible spectrum moving between 400nm to 700nm. A taste of the results at the RPE layer, right at the impact, can be
seen in Figure 25 through Figure 27, for three colors. In all colors the photoreceptors can
be seen with only a slight difference in their intensities. On the one hand, the blue color
consists of fewer hot spots, and therefore less evident modes, which can be traced to the
nature of the diameter and the refraction indices differences to maintain that color propagation. That tendency for light guidance can be expressed in a clean way following the
waveguide characteristic frequency, or more explicitly [Franze et al 2007]
[47]
where d is the diameter and

d 2
n 1  n22 ,

is the propagated wavelength.
On the other hand, both the green and the red contain an evident larger number of guided
modes, where at the green the phenomenon is slightly stronger, with more accentuated
modes.
38
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 25: Intensity at the bottom of the retina, right before the impact with the RPE. Illumination pattern as in
Figure 24. The blue color shows several modes clusters about the Müller cells.
Figure 26: As in Figure 25. The green propagation shows more accentuated modes than the blue, the cluster
inter-border lines are more blurred.
39
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 27: As in Figure 25. There are several far red modal structures about the Müller cells locations.
A comparison of the propagation results with experimental work [Agte et al. 2011] shows
several wide ranges of similarities with the research acquired results. Figure 28 depicts the
two side by side; on the right hand the image was taken using a 532nm LED laser diode
for illuminating a mammalian retinal tissue. That forwardly scattered light was collected
using an imaging lens into a CCD camera forming an image on the camera’s sensor. On
the left hand side simulation results show light propagation of a 550 nm Gaussian light
source which emulates the incident Airy pattern on the Retina, and are collected coherently
to an intensity distribution right before the RPE. Amongst other evident similarities, the
phenomenon of accentuated modes at the vicinity of the Müller cells can be seen on both
sides bellow, as well as different responses of the photoreceptors to the incident light.
40
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 28: Simulation of the 550nm Gaussian light beam propagated into the retina, as seen from the RPE layer,
black scale bar is 6.5µm (left). An image taken from forward scattering of a 532nm laser beam, shot at a retinal
tissue (right) , white scale bar corresponds to 5µm [Agte 2013].
4.2
Inward Control Propagation
In order to test the propagation’s tool validity, I conducted a control calculation for a
smooth retina for all colors, but only the 0.6µm wavelength is presented, without limitation
of generality. As the waveguides were eliminated, the light distribution came out smooth
and not patchy (Figure 29).
o
o
Entry intensity.  =0 .  = 0.6m
Exit intensity.  =0 .  =0.6m
500
500
450
450
400
400
350
350
300
300
250
250
200
200
150
150
100
100
50
50
100
200
300
0.2
0.4
0.6
400
0.8
500
100
1
0.2
200
0.4
300
400
0.6
500
0.8
Figure 29: Propagation control results for λ=0.6µm; please note that all effects associated with the presence of the
waveguides are eliminated. The entry beam is equal to the exit beam, and These results repeat for all wavelengths.
4.3
Backscattered Intensities at the Vitreous Gel
Backscattering from the RPE and the photoreceptors layer was calculated in order to try
and imitate high resolution images acquired in ophthalmic imaging of the retina. Each of
the scattering layers was divided to N×N squares, and was added its own arbitrary phase,
41
© Technion - Israel Institute of Technology, Elyachar Central Library
and scattered back towards the ILM and into the vitreous gel. All the fields at the vitreous
border were added incoherently to construct the scattered intensity image. Following the
same colors shown in the previous section, the scattered intensities of blue, green and red
are shown in Figure 30 through Figure 32. The color-bars levels are normalized to the
maximal pixel value calculated, over the entire spectrum; therefore all pixel values are limited to
, and these levels scale the intensities of the scattered light. An examination
of the scattered light images show that the scattered light created bundles about the Müller
cells’ locations. This phenomenon is repeated throughout all examined wavelengths and
may be corroborated with the waveguiding phenomenon which occurs in the retina.
Backscattered Intensity at the Vitreous WL: 0.4 [m]
1
500
0.9
450
0.8
400
350
0.7
300
0.6
250
0.5
200
0.4
150
0.3
100
0.2
50
0.1
100
200
300
400
500
0
Figure 30 – Backscattered intensity at the vitreous gel of a blue monochromatic wavelength. Four big bundles of
scattered light about the Müller cells can be seen, scale bar is 6.5µm long.
42
© Technion - Israel Institute of Technology, Elyachar Central Library
Backscattered Intensity at the Vitreous WL: 0.55 [m]
1
500
0.9
450
0.8
400
350
0.7
300
0.6
250
0.5
200
0.4
150
0.3
100
0.2
50
0.1
100
200
300
400
500
0
Figure 31 – Backscattered intensity at the vitreous gel of a green monochromatic wavelength. Scale bar is 6.5µm
Backscattered Intensity at the Vitreous WL: 0.7 [m]
1
500
0.9
450
0.8
400
350
0.7
300
0.6
250
0.5
200
0.4
150
0.3
100
0.2
50
0.1
100
200
300
400
500
0
Figure 32 - Backscattered intensity at the vitreous gel of a near infrared monochromatic wavelength, scale bar is
43
© Technion - Israel Institute of Technology, Elyachar Central Library
6.5µm long.
To further analyze the nature of the scattered light, let us super-impose the scattered light
results with the Müller cells array that the light was propagated into. Figure 33 describes
the green scattered light at the vitreous layer, superimposed with the Müller cells tapestry
array on the left-handed side of the figure, with the same image of the green color, only
zoomed onto the centermost Müller cell of the pattern. The apparent silhouettes of the rod
photoreceptors show in the right image, whilst a peak of the scattered light intensity can be
Figure 33: Backscattered green light at the vitreous surface, superimposed on the Müller cell tapestry. Left: a
macro image of the entire simulated retina. Right: a zoom unto a single Müller cell area. Cone and Müller cell
missalignment in these two is due to a wide,
, incidence angle.
seen at the center of the Müller cell. The small misalignment between the Müller cell’s
center and the intensity peak stems from a slightly tilted incident beam. These phenomena
would be corroborated with both observations and measurements.
4.4
Comparison of Results with Empirical Results
First, let us employ a thorough analysis which takes all the scattered color intensity densities, and superimposies them with the Müller cells Pattern. These images averages radially
with respect to the Müller cells’ central axis, and the mean radial chromatic normalized
results are drawn in Figure 34. A strong correlation between the incident wavelength and
the width intensity of backscattered light can also be seen. Notice the dark rings, which
represents themselves as a local minima about 1µm at the central lobe, which can also be
seen in experiments [Dubra et al. 2011].
44
Normalized mean Radial Scatterd Light Intensity []
© Technion - Israel Institute of Technology, Elyachar Central Library
Backscattering MeanRadial about a Müller Cell center
1
0.4 [m]
0.55 [m]
0.7 [m]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
2
3
4
5
6
Distance from Müller Cell Center [m]
7
8
9
10
Figure 34 – Normalized mean radial calculation with respect to Müller cell’s center.
The backscattered intensities at the vitreous surface were also spatially integrated over the
entire surface of the gel. The results are expected to imitate, via mere calculation, the red
eye phenomenon as seen in many a layman photographs. By merely summing over each
wavelength’s scattered intensity, and normalizing over the maximal sum of the scattered
intensity, I get
[48]
I tot    
I
i, j
scattered
  , i, j 
max  I tot 
.
Figure 35 exhibits the total intensity of the scattered light at the vitreous from both the
RPE and the photoreceptors versus the initially propagated wavelength. A first glimpse at
the graphics shows higher intensity at the red wavelength, as was expected.
45
0.9
Normalized Total Intensity [a.u]
© Technion - Israel Institute of Technology, Elyachar Central Library
Normalized Total Scattered Intensity vs. Color
1
0.8
Simulated total intensity
Nasal Reflectance Measurements
Perfovea Reflectance Measurements
Fovea Reflectance Measurement
Average Measurements
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.4
0.45
0.5
0.55
0.6
0.65
Wavelength [m]
Figure 35: Total scattered intensity vs. wavelength color. The Blue line represents simulated total intensity, while
the green represents average reflectance of retinal fundus measurements; circles and crosses represents empirical
reflectance measurements from the nasal (black circle), temporal paraforvea (red ex) and the fovea (blue crosses)
[Berendschot et al 2003]
Cooper et al. and Dubra et al. imaged the retina [Dubra et al. 2011] and investigated the
temporal change of reflection in rods [Cooper et al. 2011]. In their research, in vivo monochromatic images of the retina were taken utilizing an adaptive optics scanning laser ophthalmoscope (AOSLO) [Dubra and Sulai 2011]. In order to compare these studies with our
obtained results, we have taken one of the acquired images, Figure 36, and represented the
same data in our color-map; a representation of this comparison is shown in Figure 37.
Figure 36: Parafovea AOSLO image of the retina, image size is 168 × 122 µm [Cooper et al., 2011].
A quick comparison of these images shows high qualitative corroboration between the
simulation results and the AOSLO measurements; the image to the left is an integration of
46
© Technion - Israel Institute of Technology, Elyachar Central Library
line scanning of the retina, whereas the simulated results to the right shows backscattering
image of a single spot of a monochromatic source on the retina.
50
100
150
200
250
50
100
150
200
250
300
350
Figure 37. Left: AOSLO image of the retina [Cooper et al. 2011] (Figure 35), using the same color-map as in my
simulation, image size is 168 × 122 µm. Right: simulated backscattering image of the retina, image size is 66.5 ×
66.5 µm.
To complete the comparison between the computer simulation and measurements, a second image, given in Figure 38 of AOSLO measurement imaging a cone and its neighboring rods, were also radially averaged, as in Figure 34. In order to construct a complete
comparison between the two methods, five cones and rod region of interests (ROI) were
chosen in this setup, to illustrate the mean value of the phenomenon. These ROIs were
compared with a special set of the simulation, imitating the setup to the utmost level. All
the controlled parameters of the simulation were set to be the same as in Dubra’s experiment; the wavelength propagated into and scattered out of the retina was selected to be exactly
, following the same wavelength being used in the experiment. Moreo-
ver, since the nature of AOSLO imaging allows the experimenters to obtain images at a
specific depth, a special care was taken to follow that parameter as well. The obtained results have undergone a similar process; five ROIs were chosen, to represent faithfully the
mean behavior of the special set of the simulation.
47
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 38. AOSLO image of the cones uppermost layer [Dubra et al. 2011].
These five ROIs at both of the setups were radially and then arithmetically averaged; before plotting, each of the averaging results was normalized to the highest peak, which was
expected to be at the center of the Müller cell, namely at zero radius. Figure 39 exhibits the
comparison between the AOSLO measurements results and the simulation results; the blue
line exhibits the average of five mean radial measurements taken in Dubra’s experiments.
Qualitatively, the width of the Müller cell’s main tunnel is repeated in both cases. On the
one hand, Dubra’s experiments exhibit a relatively dark ring about
center, while our simulation shows a weaker minimum at about
from the cells’
from the cells’ cen-
ter.
Figure 39 – Normalized mean radial for scattered 680nm light incoherent intensity addition with respect to
Müller cell’s center. Addition was done at the lowest level of cone cells. The red solid line represents the
red light simulation results, while the blue solid line represents the same algorithm applied on Dubra et al.
[2011] measurements. All data in the figure is arithmetically averaged.
48
of the simulation was made, summing the scattered intensities incoherently at a slightly
higher level than in the first iteration, namely about 7µm above the previous run. Furthermore, cones diameters were changed to test whether the actual measurements were taken
close to the fovea than the initial simulation [Felberner et al. 1072??]. Figure 40 illustrates
the differences in the normalized radially averages between that specific setup of the simulation and Dubra’s experiment; note that the minima of the simulated data slightly moved
to the right, but it is filled with more numeric noise; meaning the limit of the simulation
resolution has been reached in that specific setup.
Simulation Incoherent Addition vs. AOSLO Average Over 5 Cells Measurments | WL = 680 nm
1
Normalized Radially Averaged Intensity [a.u]
© Technion - Israel Institute of Technology, Elyachar Central Library
In order to further investigate the differences between these two graphs, a second iteration
Simulation average over normalized mean radial
Dubra measurements average over normalized mean radial
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.5
1
1.5
2
2.5
3
3.5
Distance from Cone Cell Radius [m]
Figure 40 – Upper cone cell level intensity incoherent addition; Normalized mean radial scattered light intensity
with respect to Müller cell’s center, 680nm wavelength. The red solid line represents the red light simulation results, while the blue solid line represents the same algorithm applied on Dubra et al. [2011 ] measurements. All
data in the figure are arithmetically radially averaged.
Differences between experimental results and both of the simulation iterations consist of
two main issues which are the dark rings surrounding the cones and Müller cells’ centers
and the numerical noise that joins the second iteration result. As was discussed earlier, the
simulation to imitate Dubra et al. results was run providing great and special care to setup
parameters, including the system’s angle of acceptance, the propagated and scattered
wavelength and the depth of the calculated intensity. In the first iteration, shown in Figure
39, a great qualitative match between the simulation results and the experimental measurements can be seen; the dark rings of the scattered light at the bottom of the photoreceptors layer can be clearly resolved, although there is a slight difference in their radius. This
49
© Technion - Israel Institute of Technology, Elyachar Central Library
misalignment can stem from the differences between the depth taken while calculating the
scattered intensity in the simulation and the actual focal length used while taking the images at the AOSLO system setup. The Dubra team focuses their AOSLO imager on the photoreceptors layer; the cone photoreceptors have a specific conic geometry, therefore, an
incentive to change the layer of incoherent intensity addition was pursued, and that is to
prove that hypothesis. In the second iteration, Figure 40, the match between measurements
and simulation results is even qualitatively better, although the dark rings surrounding the
Müller cells’ center are relatively weak, covered with noise. These two simulation iterations, which can be corroborated with actual in vivo measurements, add a convincing
clause to our main hypothesis.
Labin et al. used confocal microscopy to recreate a 3D reconstruction of light propagation
through Müller cell tapestry in guinea pigs. Propagating light through
of retinal
tissue and scanning the entire spectral range, they were able to reproduce an in vitro image
of the layered structure of the retina, and more importantly the wave guiding phenomenon
of light propagation into the retina. Figure 41 provides a clear view of their setup and qualitative results. Panel (b) shows slices of the retinal tissue where the vertical strips of light
can be corroborated with the light guidance phenomenon that occurs there. Our results as
were introduced throughout this chapter can be corroborated with theirs, in particular with
the modal effects depicted in panel (c) of the figure.
Figure 41 [Labin et al. 2014] Experimental imaging of the guinea pig retina. (a) The experimental setup included
light injection from a halogen lamp into a confocal microscope. (b) Red light transmission throughout the retina
was reconstructed in 3D. (c) Light is imaged right above the photoreceptors. Images showing four 48µm x 48µm
squares corresponding for 417nm, 491,nm 577nm and 695nm respectively.
50
© Technion - Israel Institute of Technology, Elyachar Central Library
Figure 30 through Figure 32 show the same qualitative behavior as Labin et al. The blue
color propagation shows several modes clustering about the Müller cells. Moving on to the
green monochromatic propagation, the number of modes rises, as well as their peak intensities. The confocal microscopy measures intensity profiles as a function of depth; therefore, these results provide an in-depth insight into our simulated propagation results. The
good fit between the simulation results and the confocal microscopy measurement
strengthens the model’s results.
4.5
Results Summary
In the last chapter I have shown qualitative and quantitative corroboration with Dubra et
al. [2011], Cooper et al. [2011] and Labin et al. [2014] experiments. The Dubra and
Cooper experiments provide the scattering results, whereas the Labin experiment provides
the propagation results confirmation. Moreover, simulation results have shown deep
agreement with the red eye phenomenon; the images that were created at the ILM indicate
a strong spatial correlation between the scattered intensities and the Müller and cone photoreceptors cells location in the retina’s mosaic. The rods on the other hand, being small
and randomly distributed in the retina, created the corresponding indirect scattering image
at the vitreous gel layer. These results, suggesting strong directed scattering from the cones
and indirect scattering from the rods, support the hypothesis that Müller cells guide light in
the vertebrate retinae.
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© Technion - Israel Institute of Technology, Elyachar Central Library
5
Summary and Further Studies
The main purpose of this research is to examine light guidance in the human retina. In order to pursue this question I have formulated three main research questions which examine
the Müller cells as waveguides, the retinal scattering pattern, and ultimately the compatibility with the hypothesis of optical light guidance in the retina.
This work presents calculations and simulations that indicate high spatial directionality of
scattered light from the retina; therefore, a comprehensive and detailed computer model to
describe the light scattering phenomenon from the retina was created. As an input to the
simulation’s kernel, physical properties of all the elements participating in the simulation
were taken from previous experiments and measurements. Light propagation into the retina was conducted using Fast Fourier Transform Beam Propagation Method, such in a way
that way no pre-assumptions were assumed on light densities at the simulation input. Light
densities at the RPE and the various photoreceptors layers were created, these light densities were then taken as an input to be scattered out of the retina, using spherical symmetric
propagation methods, onto the ILM layer; there light intensities were incoherently
summed, and created the retinal scattering pattern.
The scattering patterns, which were generated for each color separately, indicated strong
spatial correlation between the back scattered intensities of the cones at the ILM and the
cones at the bottom of the retina. Moreover, the back scattering images were superimposed with the tapestry of Müller cells that were used in that specific simulation run and
these results showed even greater correspondence. To further strengthen this notion, the
rods patterns that were created in the scattering images, showed none of these properties,
namely were merely indirect at their behavior, providing a random distribution. Next, in
order to corroborate simulation results with careful measurements done in the field, a
comprehensive comparison study of the results was introduced. In that study a highly qualitative match between simulation and measurements behaviors was shown; exhibiting
about the same radius of high intensities at the center of the light’s distribution and silhouettes of dark rings surrounding them.
All these indications, including verification measurement results; high spatial correlation
between the cones and the Müller cells’ tapestry on the one hand and the scattering images
at the ILM layer on the other hand. All this, coupled with the random, indirect behavior of
the scattering images created by the rods photoreceptors, can be summed up logically to
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© Technion - Israel Institute of Technology, Elyachar Central Library
advocate for directionality in light scattering patterns throughout the retina. My main hypothesis is that it all stems from the presence of the Müller cells.
Further studies of this subject are required in order to quantify the sensitivity of the proposed model with the location of the photoreceptors under the Müller cells; my work suggested a comprehensive model describing light propagation unto a well stacked structure
of the retina, and a second look might provide a deeper understanding in that area.
Another area that was taken into consideration during the work was computation time limitation, which resulted in an educated selection of the scattering layers; this selection indeed
helped limiting the time required for each simulation, but may have overlooked physical
phenomena that may ignite further research. A study that finds a more efficient method to
calculate the scattering processes, and then takes into account multi-layered scattering
from the different layers of the retina, may help to quantify the qualitatively achieved results.
AOSLO measurements, and their comparison with the simulation results, are a great step
in the right direction; the dark rings effect (the local minima) of the cone was indeed reproduced qualitatively in the simulation, but in order to obtain a robust model of retina
scattering mechanisms, the kernel of the simulation should be implemented in an environment that takes all the experimental system into consideration, and reproduces the AOSLO
measurements computationally.
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6
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Websites:
[website1] http://webvision.med.utah.edu/imageswv/Sagschem.jpeg
[website2] http://www.vetmed.vt.edu/education/Curriculum/VM8054/EYE/RETINA.HTM
[website3] http://www.telescope-optics.net/images/eye_optics2.PNG
[website4] http://what-when-how.com/wp-content/uploads/2012/04/tmp15F56_thumb4.jpg
[website5]
http://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/3D_Spherical.svg/240px3D_Spherical.svg.png
[Website6] http://spie.org/
59
‫עיקרו של מחקר זה הינו דיון בסוגיות של פיזור האור ברשתית בדגש על בחינת תפקידם של תאי מילר‬
‫(תאי גליה של העין) בהולכת גלים‪ .‬מערכות הראייה של בעלי החוליות ככלל ובני האדם בפרט‪ ,‬ידועות‬
‫ומרתקות בשל מורכבותן הרבה‪ .‬ככלל‪ ,‬מבנה העין הוא בקירוב טוב דומה למבנה כדורי‪ ,‬בעל מפתח‬
‫יחיד בחזית העין ‪ -‬האישון‪ ,‬דרכו חודר האור‪ .‬לאישון תכונה ייחודית בעין‪ ,‬באמצעות שרירי הקשתית‬
‫ביכו לתו לשנות את המפתח האפקטיבי שלו בין שלושה מ"מ לתשעה מ"מ‪ ,‬על מנת לווסת את כמות‬
‫האור הנכנסת אל גלגל העין ולהכניס יותר אור בלילה ופחות אור במהלך היום‪ .‬האור הנכנס דרך‬
‫האישון‪ ,‬מרוכז על ידי עדשות העין הממוקמות על הציר האופטי בדרך אל הרשתית‪ .‬עדשות העין הינן‬
‫מערך של שתי עדשות‪ ,‬האחת מכונה קרנית העין‪ ,‬והיא בעלת העוצמה האופטית החזקה יותר‪ ,‬אולם‬
‫נבצרת ממנה היכולת לשלוט על אורך המוקד שלה‪ .‬העדשה השנייה הינה עדשת העין‪ ,‬שבאמצעות‬
‫שליטה על רדיוס העקמומיות שלה על ידי פעילות שרירים‪ ,‬ניתן לווסת את אורך המוקד האפקטיבי של‬
‫המבנה הא ופטי כולו‪ .‬עדשות אלו יוצרות דמות על גבי רשתית העין‪ ,‬הבנויה משכבות שכבות של תאים‬
‫המאורגנות במבנה ייחודי; בכללן שלוש שכבות של תאי עצב אשר ביניהן שתי שכבות של סינפסיות‪.‬‬
‫אותה הדמות הנוצרת על הרשתית עוברת דרך סבך רב של תאים‪ ,‬המאופיינים במבנה שממבט ראשון‬
‫נראה כמ נוגד להגיון‪ .‬הרשתית בנויה כך שהשכבה הראשונה הינה שכבה של תאי הגנגליונים אשר דרכם‬
‫שולחים תאי העצב בעין את האות החשמלי דרך האקסונים אל המוח ואילו בשכבות פנימיות יותר‬
‫מצויים קולטני האור (הפוטורצפטורים)‪ .‬קולטני האור הינם אותם תאים הרגישים לאור ותפקידם‬
‫להמיר את הקרינה האלקטרומגנטית לאותות עצביים‪ .‬המדוכים‪ ,‬הידועים גם כתאי החרוט‪ ,‬אחראיים‬
‫על חדות הראייה והאבחנה בין הצבעים השונים‪ ,‬ומחולקים לצבעי האדום‪ ,‬הירוק והכחול‪ .‬תאים אלו‬
‫מרוכזים באופן רב במכרז הגומה של הרשתית‪ ,‬וריכוזם יורד באופן דרסטי ככל שמתרחקים מהגומה‬
‫אל עבר ירכתי העין‪ .‬הקנים הינם תאים בעלי מבנה גלילי‪ ,‬והם אחראים על הראייה בתנאי תאורה‬
‫לקויים‪ .‬הקנים רגישים אך ורק לעוצמת האור ולא לצבע‪ ,‬וצפיפותם גדולה בירכתי העין‪ ,‬ואפסית כמעט‬
‫באזור הגומה‪ .‬לאור העובדה כי תאי קולטני האור מצויים בשכבות הפנימיות של הרשתית‪ ,‬מובן כי יש‬
‫מספר לא מבוטל של שכבות תאים שעל האור לעבור בדרכו אל הקולטנים הללו‪ .‬מבחינה אנרגטית‪ ,‬רק‬
‫שליש מהאור הנכנס אל העין מומר לכדי אותות עצביים על ידי הקולטנים; חלק ניכר מהאור נבלע על‬
‫ידי השכבה החיצונית של הרשתית‪ ,‬האפיתליום של פיגמנט הרשתית ( ‪RPE - Retinal Pigment‬‬
‫‪ .) Epithelium‬אולם‪ ,‬קיימת מידה מסוימת של אור אשר מפוזרת בחזרה אל תוך גלגל העין‪ ,‬דבר שעלול‬
‫לפגום בכושר ההפרדה של העין‪.‬‬
‫כאמור‪ ,‬אצל בעלי החוליות‪ ,‬הרשתית מורכבת ממבנה רב שכבתי דרכו שזורים תאי הגליה של העין‪,‬‬
‫תאי מילר (‪ .)Müller or glia cells‬תאים אלו ייחודיים לעין ‪ ,‬ומאופיינים על ידי מבנה דמוי משפך מן‬
‫הצד האחד הקרוב אל הגוף הזגוגי (‪ )vitreous humour‬ואישון העין ועל ידי מבנה גלילי מן הצד האחר‪,‬‬
‫הקרוב אל קולטני האור‪ .‬מגוון רחב של ניסויים ומדידות שבוצעו על תאים אלו בשנים האחרונות גילו‬
‫עובדה מפתיעה‪ ,‬כי מקדם השבירה של תאי מילר שונה ואף גבוה במעט מממקדם השבירה של הסביבה‬
‫בה הם מצויים בעין; דבר המעיד על נטייה לתופעת הולכת גלים‪.‬‬
‫‪I‬‬
‫‪© Technion - Israel Institute of Technology, Elyachar Central Library‬‬
‫תקציר‬
‫הרשתית‪ ,‬להכניס לתוכו אור בצבעים שונים ובזוויות כניסה שונות‪ ,‬ולבחון את פיזורו בתוך גלגל העין‪.‬‬
‫מודל זה נבנה בקפידה רבה והתבסס על מדידות אופטיות וביולוגיות של אלמנטים שונים בתוך העין‬
‫הכוללים ממדים גאומטריים‪ ,‬מקדמי שבירה ופרמטרי פיזור שונים‪ .‬לאור ריבוי הפרמטרים‪ ,‬האילוצים‬
‫הגאומטריים והאופטיים‪ ,‬פתרון אנליטי לבעיות מן הסוג הזה נתפס כבלתי אפשרי‪ ,‬ועל כן יש צורך‬
‫להשתמש בשיטות קירוב שונות לפתרון משוואות הלמהולץ המתארות את התקדמות האור לתוך‬
‫הרשתית‪ .‬באמצעות שיטה נומרית ייעודית לקידום אלומות אור בתווך אנאיזוטרופי באמצעות התמרת‬
‫פורייה מהירה (‪ )Fast Fourier Transform Split Step Beam Propagation Method‬קודמו אלומות‬
‫האור פנימה אל הרשתית מקצה הנוזל הזגוגי אל שכבת האפיתליום של פיגמנט הרשתית‪ .‬פרופיל האור‬
‫המקודם לתוך הרשתית חושב בנקודות הפיזור הקריטיות שלו בתאי הקולטנים ובשכבת האפיתליום‬
‫של פיגמנט הרשתית ושימש כמקור לפיזור האור אל תוך גלגל העין‪ .‬פרופילי העוצמות הללו‪ ,‬הן‬
‫משכבות האפיתליום של פיגמנט הרשתית והן מקולטני האור‪ ,‬חולקו לרכביים בדידים קטנים דיים‪ ,‬כך‬
‫שניתן להתייחס אליהם כאל בעלי סימטריה כדורית יחסית לשאר מבנה הרשתית‪ .‬מכאן‪ ,‬באמצעות‬
‫שימוש בשיטות קידום אור המבוססות על סימטריה כדורית זו‪ ,‬פוזר האור משני הסוגים של קולטני‬
‫האור ומהאפיתליום של פיגמנט הרשתית בו‪-‬זמנית‪ ,‬ונסכם תוך כדי מתן תשומת לב מרובה לחוסר‬
‫הקוהרנטיות הזמנית של מקורות הפיזור הללו‪ .‬התוצר של החישובים המורכבים הללו הינו פרופיל‬
‫פיזור של האור בתחתית הנוזל הזגוגי‪ ,‬כלומר המקור לפיזורי האור המוחזר בתוך גלגל העין‪.‬‬
‫האור שקודם אל שכבות הרשתית הפנימיות ובכללן אל קולטני האור והאפיתליום של פיגמנט הרשתית‬
‫פוזר איפה בחזרה אל גלגל העין; בחינה ראשונית של התוצאות בסביבת תאי מילר בבסיס הרשתית‬
‫אשר נמצאים בצמוד לנוזל הזגוגי גלתה תופעה מעניינת ומבטיחה‪ :‬האור שפוזר בחזרה אל הרשתית‬
‫יצר ריכוזים גדולים סביבות התאים הללו‪ .‬תופעת הריכוזים הללו חזרה על עצמה בעקביות לאורך כלל‬
‫אורכי הגל שנבחנו לאורך המחקר‪ ,‬והיא יוצרת תמונה המעידה על הקורלציה המרחבית החזקה בין‬
‫האור המפוזר מתוך הרשתית לבין האור שקודם פנימה בכל הקשור למיקומי תאי המילר והמדוכים‬
‫במרחב‪ .‬כל זאת ועוד‪ ,‬בבחינה מעמיקה יותר של תמונות הפיזור שנוצרו בנוזל הזגוגי ניתן להבחין‬
‫באופן חד בכיווניות רבה בכל האמור לקידום אלומות האור במרחב פנימה אל שכבות האפיתליום של‬
‫פיגמנט הרשתית מחד גיסא‪ ,‬ולפיזור מהשכבות השונות מאידך גיסא‪ .‬במבט נוסף על פרופיל העוצמה‬
‫באזור בסיס תאי המילר‪ ,‬הקנים‪ ,‬בעודם קטנים יחסית למדוכים וממוקמים באופן אקראי מסביב‬
‫למדוכים בשכבת הקולטנים‪ ,‬יצרו תמונות פיזור עם כיווניות אקראית‪ ,‬לא מוגדרת‪ ,‬כמצופה מפיזור‬
‫אור ממקור מעין זה‪ .‬ניתוח התוצאות הללו‪ ,‬שהניב פיזור כיווני בעבור המדוכים ופיזור לא כיווני‬
‫ואקראי בעבור הקנים תומך בהשערת הולכת הגלים באמצעות תאי מילר‪ ,‬ומעיד באופן חד על כיווניות‬
‫בקידום האור ברשתית כמקור לחדות הראיה‪.‬‬
‫המחקר על אודות סוגיות פיזור האור מהרשתית אל גלגל העין מהווה נדבך נוסף וחשוב ביותר בהוכחת‬
‫ההשערה כי תאי מילר מוליכים גלים ברשתית; מדובר בפעם הראשונה בה ניתן להשוות את תוצאות‬
‫הסימולציה והחישובים עם תוצאות של מחקרים אמפיריים אחרים בתחום‪ ,‬ובעיקר עם תוצאות של‬
‫מדידות מהמחקרים המובילים‪ .‬כבר היום‪ ,‬באמצעות מיקרוסקופ קונפוקלי ניתן למדוד באופן מדויק‬
‫ביותר את התקדמות אלומת אור הליזר לאורך הרשתית ולשחזר באמצעות טכניקות עיבוד תמונה‬
‫מתקדמות את אופיין אלומות האור לאורך רקמות אלו בתלת ממד‪ .‬נוסף על כן‪ ,‬לייזר סורק‬
‫‪II‬‬
‫‪© Technion - Israel Institute of Technology, Elyachar Central Library‬‬
‫על מנת לבחון את התופעה מבחינה תאורטית וחישובית עלה הצורך לפתח מודל תלת ממדי של‬
‫השוו אה של התוצאות שהתקבלו מן הסימולציה עם התוצאות שנמדדו על ידי שתי הטכנולוגיות‬
‫המתקדמות הללו מחזקת את העובדה שתאי מילר הינם מוליכי גלים ברשתית‪.‬‬
‫‪III‬‬
‫‪© Technion - Israel Institute of Technology, Elyachar Central Library‬‬
‫אופתלמסקופי מתעד בעין חיה את קרקעית הרשתית‪ ,‬ומעניק מבט חד אל תוך נבכי העין האנושית‪.‬‬
‫ברצוני להביע את תודתי הכנה לד"ר ארז ריבק‪ ,‬על הנחייתו המסורה‪ ,‬בלעדיה מחקר זה לא היה רואה‬
‫אור יום‪.‬‬
‫למוסד הטכניון – מכון טכנולוגי לישראל‪ ,‬ולפקולטה לפיסיקה‪ ,‬מודה אני על המשאבים שהועמדו עבורי‬
‫ועבור מחקרי‪.‬‬
‫אני מודה לגב' אתי ממן ולגב' יהודית גרינברג על סיוען בפן המנהלי‪.‬‬
‫למשפחתי היקרה‪ ,‬הורי‪ ,‬אשתי ובנותיי‪ ,‬תודה על הסבלנות והתמיכה לאורך כל הדרך‪.‬‬
‫‪© Technion - Israel Institute of Technology, Elyachar Central Library‬‬
‫עבודה זו נעשתה בהדרכתו של ד"ר ארז ריבק מהפקולטה לפיסיקה בטכניון‪.‬‬
© Technion - Israel Institute of Technology, Elyachar Central Library
‫ מאיה ושרון האהובות‬,‫לטניה‬
‫ברשתית האנושית‬
‫חיבור על מחקר‬
‫לשם מילוי חלקי של הדרישות לקבלת התואר‬
‫מגיסטר למדעים בפיסיקה‬
‫עידן מישלובסקי‬
‫הוגש לסנט הטכניון – מכון טכנולוגי לישראל‬
‫אלול תשע"ד‪ ,‬חיפה‪ ,‬אוקטובר ‪4102‬‬
‫‪© Technion - Israel Institute of Technology, Elyachar Central Library‬‬
‫תהליכי פיזור אור‬