2318
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
Vohnsen et al.
Guided light and diffraction model of human-eye
photoreceptors
Brian Vohnsen, Ignacio Iglesias, and Pablo Artal
Laboratorio de Óptica, Universidad de Murcia, Campus de Espinardo (Edificio C), 30071 Murcia, Spain
Received November 8, 2004
The photoreceptors of the living human eye are known to exhibit waveguide-characteristic features. This is
evidenced by the Stiles–Crawford effect observed for light incident near the pupil rim, and by the directional
component of light reflected off the retina in the related optical Stiles–Crawford effect. We describe a model for
the coupling of light to/from photoreceptors on the basis of waveguide theory that includes diffraction between
the eye pupil and the photoreceptor apertures, and we show that valuable insight can be gained from a Gaussian approximation to the mode field. We apply this knowledge to a detailed study of the relationship between
the Stiles–Crawford effect and its optical counterpart. © 2005 Optical Society of America
OCIS codes: 170.4470, 330.4060, 330.5310.
1. INTRODUCTION
The human eye has long been known to be sensitive to the
angle of incidence of light on the retina as demonstrated
by the psychophysical Stiles–Crawford effect (SCE), i.e., a
reduced visual sensitivity to light rays that intersect the
eye pupil off axis1 with a resulting impact on visual
performance.2 The directionality also manifests itself in
light reflected off the retina3 in what is often referred to
as the optical Stiles–Crawford effect (OSCE).4 In either
case the responsible mechanism has been identified with
the waveguide nature of the photoreceptors.5–8 In an unaberrated eye the photoreceptors are oriented with their
central axis aimed toward a point near the pupil center9
since this is the configuration where their coupling to incident light is favored the most. Centered at the same pupil point, and superimposed on an essentially uniform
background of scattered light, the OSCE produces a
peaked intensity distribution of directional light returned
from the eye fundus.10–12 Both the SCE and the OSCE are
normally fitted to a Gaussian function at the pupil plane
(for simplicity assumed to be rotationally symmetric although its actual cross section might be slightly
elliptical13):
f共r = 冑x2 + y2兲 = C + A ⫻ 10−␳r = C + A exp共− ␣r2兲,
2
共1兲
where C is a constant background, A / 共A + C兲 determines
the degree of directionality, and the parameters ␳ and ␣
= ␳ / log共e兲 ⬇ 2.30␳ both represent the spread in directionality (a smaller value equals a larger spread). Although
they are similar in origin, there is an important distinction between the intensity distribution measured for the
OSCE and the sensitivity curve measured for the SCE.
The directionality parameter, as measured with both
methods at approximately the same retinal location, differ typically by a factor of 2, with the OSCE distribution
being the narrowest.10,12 With simultaneous scanning of
both the entrance and the exit pupil of the eye, even
higher values for the OSCE directionality parameter
1084-7529/05/112318-11/$15.00
(equaling 3 to 4 times the value for the SCE) have been
found.4,14
It has been suggested that both the SCE and the OSCE
are caused by variations in photoreceptor alignment,13,15
but this contradicts the finding of highly aligned foveal
receptors.16 Moreover, it has been argued that the observed narrowing of the OSCE, as compared with the
broader acceptance curve for the SCE, could be caused by
an energy redistribution among modes in the photoreceptors following reflection.14 Recent studies have favored a
model that explains the directionality through a combined effect of photoreceptor wave-guiding by light-ray
excitation and a scattering mechanism from the simultaneous illumination of many photoreceptors.17,18 Both may
produce a Gaussian peak at the pupil and the combined
effect of two distributions can explain the observed radiation narrowing for the OSCE.
Another concern has been the dependence on wavelength ␭ for both effects.19–21 Experimental studies of the
OSCE21 have verified a 1 / ␭2 dependence in agreement
with the photoreceptor scattering model.17,18 In this case
the contribution from wave-guided light is assumed to be
wavelength independent so that scattering alone produces the dependence observed. Other studies of the SCE
have suggested a more delicate wavelength dependence to
take into account also minor variations produced as the
number of allowed photoreceptor modes change.7
In this paper we propose an alternative model that can
account for both the known deviation between the directionality parameter of the SCE and the OSCE and the observed overall 1 / ␭2 dependence. Instead of considering
that light propagates as rays in the eye to/from photoreceptors as commonly assumed, the model includes diffraction of both the incoming beam and the light emanated by
the photoreceptors. This choice is dictated by the small
size of the photoreceptor apertures and is supported by
both the observation of individual modes6 and current
models of light propagation inside photoreceptors.7,14,22,23
© 2005 Optical Society of America
Vohnsen et al.
Our description is applicable on a single-photoreceptor
level and therefore appears highly suited also for analysis
of directional light in the context of imaging with a confocal scanning laser ophthalmoscope.24
The paper is organized as follows. In Section 2 the
model that accounts for the coupling of light to an individual photoreceptor and the corresponding generation of
a back-reflected directional component is described. The
importance of the model for the SCE is discussed in Section 3, and the relation to the OSCE is discussed in Section 4. In Section 5 the influence of a Maxwellian illumination scheme is addressed for both effects, and in
Section 6 the influence of coherence is discussed. In Section 7 we present our conclusions.
Vol. 22, No. 11 / November 2005 / J. Opt. Soc. Am. A
at which the intensity has dropped to 1 / e2 ⬇ 0.14 of the
on-axis value) is incident on the eye and focused on the
retina to a spot size of
wr =
In the following the model of light coupling to and from
photoreceptors, including diffraction of both the incoming
beam towards the retina and the back-reflected light, is
described in accordance with the notation shown in Fig. 1.
A collimated Gaussian beam of width 2w0 (i.e., the width
Pr
共2兲
␲neyew0
=
冏冕冕
*
␺ r␺ m
dxdy
冏
2
共3兲
.
Here ␺r is the normalized incoming field amplitude at the
retina and ␺m is the normalized mode field amplitude
(both including their phases), and the integration is to be
carried out in the retinal plane.25 Ideally, with only one
photoreceptor the area of normalization and integration
in Eq. (3) extends to infinity. This methodology will be
used in the following derivations. However, for a wellconfined mode and illumination, the area to be considered
needs to be only slightly larger than the actual photoreceptor aperture. This approximation will be used in the
numerical work of Section 5 (see also footnote of Table 2
below). The total power carried forward by the photoreceptor is distributed among a small number of modes,6,7
but when the incident beam is Gaussian and wellmatched to the location and width of the photoreceptor,
the largest coupling will be to the fundamental mode
LP01, given that for a cylindrical waveguide this mode is
nearly Gaussian.26 When this is not the case more of the
incident field may couple to higher-order modes. The actual number of possible modes is found from the V number of the waveguide defined by V = 共␲di / ␭兲冑ni2 − n2s , where
the (assumed) uniform inner segment has refractive index
ni ⬇ 1.353 and width di, and the refractive index of the
surrounding medium is ns ⬇ 1.340 (the values shown are
those used by Snyder and Pask; see Ref. 7 and references
therein). When V ⬍ V0 = 2.405 only the fundamental mode
is allowed which for the selected parameters occurs if di
⬍ 4.09␭. The fraction of light that remains uncoupled is
essentially lost from the photoreceptor but might be absorbed while traversing adjacent receptors.28 For the
present, we assume that the incident beam couples light
only to the fundamental mode represented by a Gaussian
function of width 2wm. In this case the fraction of power
transmitted to the photoreceptor T = Pm / Pr can be found
from Eq. (3) as27
T共u兲 =
Fig. 1. Schematic of the configuration considered for light coupling to and from a photoreceptor. (a) The Gaussian beam incident on the eye is focused at the retina slightly off axis from the
photoreceptor axis and coupled to mode m. (b) The incident
Gaussian beam is displaced in the pupil a distance rp with the
result of intersecting the photoreceptor at an angle ␪. (c) The
back-reflected beam diffracts from the photoreceptor and produces a Gaussian intensity distribution at the eye pupil.
␭feye
in the absence of aberrations. The eye parameters used
are those of the reduced eye, i.e., a focal length feye
= 22.2 mm and a constant index of refraction neye = 1.33.
The focused beam can couple its power Pr at the retina to
the guided modes of an underlying photoreceptor, where
the amount of power Pm coupled to mode m can be expressed via an overlap integral as
Pm
2. MODEL OF LIGHT COUPLING TO AND
FROM PHOTORECEPTORS
2319
冋
2wrwm
2
w2r + wm
册 冋
2
exp
− 2u2
2
w2r + wm
册
,
共4兲
where u is the displacement of the incident beam with respect to the photoreceptor axis [Fig. 1(a)]. Alternatively
the incident beam may intersect the photoreceptor at an
angle ␪ = rp / feye Ⰶ 1, where rp is the shift of the incident
beam in the pupil plane with respect to the central posi-
2320
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
Vohnsen et al.
tion [Fig. 1(b)], with the fraction of transmitted power
given by27
T共␪兲 =
冋
2wrwm
2
w2r + wm
册 冋
2
exp
− 2共␲neyewrwm兲2␪2
2
␭2共w2r + wm
兲
册
.
共5兲
In the case in which the beam is both shifted and tilted
with respect to the photoreceptor axis, the transmission
factor can be found to a first approximation by a combined
action of Eq. (4) and the exponential factor from Eq. (5).
When the width of the incident beam is perfectly matched
to that of the mode, wr = wm = w, these expressions reduce
to
冋 册
w2
共6兲
,
冋冉 冊 册
T共␪兲 = exp −
␲neyew
␭
␭feye
␲neyewm
共9兲
.
In the following sections we apply this photoreceptorto-light coupling model on both the SCE and the OSCE.
The description focuses on cone photoreceptors, as this is
where most experimental data are available, but the principle ideas should be equally applicable to rods albeit the
directionality, and thus the wave-guiding effect, manifests itself less directly for them than it does for
cones.6,31,32
3. STILES–CRAWFORD EFFECT REVISITED
− u2
T共u兲 = exp
wp =
2
␪2 .
共7兲
In the above derivations the refractive index of the inner
photoreceptor segment, which is slightly larger than that
of the reduced eye, enters indirectly through the calculation of the mode width 2wm. In turn, the small reflective
loss 共⬃0.01% 兲 due to the difference in indices has been
omitted as it is clearly negligible when compared with
other factors.
It is known that in order to avoid absorption of visible
light in the outer photoreceptor segment, and thereby enhance the contribution of directional back-reflected light
to the OSCE, bleaching of the photoreceptors is required.
Thus, the directional light originates either from reflection at the outer photoreceptor end face or beyond,12 or
possibly from the refractive index modulation by the numerous photopigment-containing discs in the outer segment itself.29 In any case only a small number of scattering processes can be involved since the directional
component retains a large degree of the incident light polarization (although the exact amount, apart from being
wavelength dependent, is still a topic of debate).10,30 The
scattering will excite one or more back-reflected modes,
but for simplicity we again assume that only the fundamental Gaussian-like mode is efficiently excited. As this
field radiates from the photoreceptor it diffracts in the eye
and ideally produces a Gaussian beam largely centered at
the eye pupil. This beam generated by the photoreceptor’s
light guiding carries forward an intensity distribution
[Fig. 1(c)]
Idir共r兲 = I0TR exp关− 2共r/wp兲2兴,
共8兲
where I0 is a constant (for a given incident power and spot
size), T is the incident coupling efficiency as determined
by Eqs. (4)–(7), and R is the (directional) photoreceptor
reflectivity. It should be stressed that only the fraction of
scattered light coupled back through the photoreceptor is
contained in R. The unguided components are either lost
(absorbed) or contribute to a practically uniform background at the pupil [not included in Eq. (8)].10 The spot
size at the pupil, wp, is given by [cf. Eq. (2)]
The width of the sensitivity curve for the SCE is traditionally obtained by registering changes in apparent
brightness while translating an incident beam in the pupil plane and fitting the results to Eq. (1). The translation
produces a corresponding change in angle of incidence at
the retina and the coupling efficiency is therefore described by T共␪兲. When the beam is well matched to the
fundamental mode of the photoreceptor, the directionality
parameter ␳SCE can be obtained directly from Eq. (7) as
␳SCE = log共e兲
冉 冊
␲neyew
␭feye
2
.
共10兲
A similar analysis has previously been carried out for the
case of an incident plane wave.7,14,33 Experiments with
green, yellowish, and red light have shown that, some
variation apart, ␳SCE ⬇ 0.05/ mm2 at the fovea,1,4,7,12,13
which may be restated by use of Eq. (10) as w / ␭ ⬇ 1.8 (or
w ⬃ 1.1 ␮m). It should be stressed, however, that these experiments have been performed with simultaneous illumination of many photoreceptors and may therefore differ
from the above-expressed result. We return to this point
in Sections 5 and 6. The wavelength dependence of the
SCE19 reveals an influence that has been related to the
number of excited modes7 and to the spectral properties of
the eye constituents,29,34 but for wavelengths either in the
blue–green range or for utmost red the major trend fits
the 1 / ␭2 dependence shown. To get beyond this tendency
in a satisfactory manner would require a unification of
the waveguide model with absorption in the entire eye,
which lies outside the scope of the present study. However, it should be stressed that only with a sufficiently
dim light source and a dark-adapted eye can the waveguide contribution to the SCE be properly accessed, since
in this case the incident light will be effectively absorbed
within a fraction of the entire outer segment (thereby
eliminating the possible contribution of backscattered
light34 to the sensation of brightness). The observed increase in ␳SCE with distance from the fovea35 (approximately doubled at 2 deg eccentricity and beyond36) may
also be explained by Eq. (10) since w increases with the
waveguide width and thus with foveal distance. Finally,
the fact that rods appear less directionally sensitive31
might be explained by a smaller mode width as compared
with that of cones (due to their smaller size and possibly
smaller index of refraction7).
Vohnsen et al.
Vol. 22, No. 11 / November 2005 / J. Opt. Soc. Am. A
4. OPTICAL STILES–CRAWFORD EFFECT
REVISITED
As the excited mode field propagates in the outer segment
or beyond, scattering excites the backward-propagating
modes that when diffracted combine to produce a Gaussian peak of directional light at the pupil. As the outer segment is very narrow, do ⬃ 1 ␮m, it appears reasonable to
assume that principally the fundamental mode can be efficiently excited by back-scattered light, as also suggested
by simulations (see Table 2),7,14 with the field matched to
the same mode of the inner segment.37 It should be
stressed that this alone may suffice to explain the Gaussian distribution of directional light at the pupil. Actually,
assuming a (uniform) refractive index7 no ⬇ 1.430 and estimating the corresponding V number one finds that for
␭ ⬎ 0.652 ␮m only single-mode propagation is allowed in
the outer photoreceptor segment.
Several mechanisms have been proposed to explain the
observed deviation between the widths of the OSCE and
the corresponding SCE sensitivity curve, and a number of
different but related methods have been used in part to
resolve this controversy.4,8,10–12,14,17,18,20,21 The use of different techniques and illumination wavelengths, however,
makes an exact comparison of the results obtained difficult. The OSCE has typically been analyzed by aligning
the incident beam at or near the pupil center where the
SCE is maximized, and subsequently recording the reflected light intensity versus pupil coordinate as this
configuration carries the strongest imprints of
directionality.10–12,20,21 This differs from the technique
commonly used to analyze the SCE where the incident
beam is translated in the pupil plane (Section 3). Thus,
for the OSCE with a fixed but optimal angle of incidence
we may take T共␪兲 = 1 and from Eqs. (8) and (9) directly obtain
␳OSCE = log共e兲
2
wp2
= 2␳SCE ,
共11兲
in good agreement with experimental studies of both effects when measured on the same subjects.12 A significantly larger value has been found in a similar study
where only a narrow slit defines the exit pupil,20,21 but in
that case the deviation is presumably due to the illumination of a larger retinal area with increased contributions from wider cone photoreceptors. As a numerical example we may take directionality parameters ␳OSCE
⬇ 0.11/ mm2 (at the fovea) and ␳OSCE ⬇ 0.19/ mm2 (at 2 deg
eccentricity) measured at a wavelength ␭ = 0.543 ␮m by
Marcos and Burns (see Ref. 18). When inserting Eq. (9)
into Eq. (11) these values may be converted to corresponding mode widths of wm ⬇ 1.03 ␮m and wm ⬇ 1.35 ␮m, respectively. If the inner segment potentially allows for the
propagation of several modes (i.e., assuming that V Ⰷ V0)
its width may be estimated by di ⬇ V0wm. This relation
has been obtained by fitting a Gaussian distribution to
the exact mode shape (see Ref. 26 in the limit of U01
→ V0. For a waveguide with a given V an optimum fit to
the Gaussian mode should be made numerically, but a
good approximation is obtained by wm ⬇ di / U01). Thus, in
this approximation one finds a diameter di ⬇ 2.5 ␮m for
the inner segment at the fovea in good agreement with
2321
anatomical studies whereas an estimated diameter di
⬇ 3.2 ␮m at the eccentric location is somewhat smaller
than expected 共⬃4.5 ␮m兲.38 The reason for the deviation
at the off-center location may be an increased contribution from higher-order modes like LP11, possibly excited
by intrareceptor scattering of back-reflected light. If this
is the case, the diameter is expected to fall within the
range V0wm ⬍ di ⬍ 1.593 V0wm. It should also be recalled,
however, that there are variations among subjects and
that the parameters used above represent an average of
only three.
The method for the OSCE can be reconfigured to access
the SCE in a more direct manner by measuring the reflected light distribution at one particular location or integrated over the entire exit pupil as a function of entrance pupil location rp. In either case we directly obtain
␳OSCE = ␳SCE by insertion of Eq. (7) into Eq. (8). This result
agrees best with experiments away from the central fovea
(at 1 and 2 deg eccentricity, respectively in Refs. 8 and
18). Possibly, the deviation at the central fovea is related
to the illumination of an extended 1 deg retinal patch18
with a resulting increased contribution from larger cone
photoreceptors (see Section 6), but it should also be recalled that only limited data are available. Yet another
method used to estimate the degree of directionality has
been to scan simultaneously (in tandem) both the entrance and the exit pupil. Setting rp = r in Eq. (7) and inserting into Eq. (8) the combined measurement yields
␳OSCE = 3␳SCE .
共12兲
This is in good agreement with reported values for both
the foveal and near-foveal region4 although other studies
show an even larger difference (about a factor of 4)14 possibly also due to the illumination of a larger area.
As may be noted from Eq. (11), with wp given by Eq. (9),
the reported 1 / ␭2 dependence of the OSCE21 is a direct
consequence of diffraction within the present model.
Thus, it appears to us that inclusion of scattering from
the combined illumination of many photoreceptors is not
a prerequisite for reproducing both the wavelength dependence and the narrowing of the OSCE directionality
expressed in Eq. (11) as has otherwise been
suggested.17,21 Actually, in the case analyzed here the
scattering model in which the retinal photoreceptor distribution is compared with that of a rough surface17 cannot be expected to hold valid, as only a single or a few
photoreceptors are illuminated at a time by the focused
incident beam. In Section 6 we address in more detail the
case of a larger illuminated area to elucidate the possible
influences that this may have on the analysis. One consequence of the wavelength dependence is that the directionality is easiest seen with a short-wavelength illumination 共␭short兲 since at a longer wavelength 共␭long兲 the
directional peak will be broader by a factor of approximately
共␭long / ␭short兲2
in
good
agreement
with
experiments.17,21 Interestingly, the variation related to
modes7 and spectral properties of the eye34 for the SCE
are not seen clearly in experiments on the OSCE, which
suggests that the latter is not merely a rescaled reproduction of the SCE. A plausible explanation is that a single
reflected mode LP01 dominates the light distribution in
the pupil, as also suggested by other authors.14 It should
2322
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
Vohnsen et al.
be borne in mind, however, that the eye and its lens modifies the spectrum of transmitted light in both the ingoing
and outgoing pathways. This may distort the above wavelength dependence for the SCE and in a comparable manner for the OSCE.
5. MAXWELLIAN ILLUMINATION
CONFIGURATION
Our theoretical model has so far been concerned mainly
with the problem of illuminating the retina with a tightly
focused Gaussian beam as done, e.g., in experiments with
the scanning laser ophthalmoscope.4,24 In contrast, most
experiments on the SCE and the OSCE use a small entrance at the eye pupil from which the light exposes an
extended retinal zone in a Maxwellian illumination system. Thus, at each photoreceptor aperture the incoming
light resembles a plane wave. In a first naïve approach
Eq. (4) and Eq. (5) may be rewritten to address this problem by assuming that wr Ⰷ wm resulting in
冉 冊 冋 册
冉 冊 冋
册
T共u兲 = 2
T共␪兲 = 2
wm
wr
wm
2
wr
2
exp
u2
exp − 2
w2r
␭
2
2
wp2
= ␳OSCE .
Ttotal共␪兲 =
兺
冏冕冕 冉
A exp i
2␲
␭
冊
*
dxdy
neye␪x ␺m
冏
2
,
共13兲
.
共14兲
Reading the spread in directionality of the SCE directly
from Eq. (14), and assuming the OSCE to remain unchanged by the simultaneous illumination of many photoreceptors [cf. Eq. (11)], one finds
␳SCE = log共e兲
M
m=0
,
− 2共␲neyewm兲2␪2
also Section 6). When the illuminated retinal area is
large, photoreceptors of different size may contribute to
the distribution causing some broadening of the radiation
peak.21 Variations in the photoreceptor alignment with
respect to the pupil point9 may slightly broaden the
summed radiation pattern, but when truly random the
distribution will remain rotationally symmetric. When all
contributing receptors are well aligned (as in the normal
eye) the sum is expected to show the same angular dependence as its individual contributions. Thus, it appears to
us that the reason the SCE is approximately twice as
broad as the OSCE (in terms of the spread in directionality) should be found rather through a close examination of
the photoreceptor-to-light coupling. Generally, the incident (quasi-) plane wave may couple to various modes
with the total energy coupled to each photoreceptor given
by
共15兲
More generally, if the beam is focused to an area with
1
wr ⬎ wm any value in the range 2 ␳OSCE 艋 ␳SCE 艋 ␳OSCE may
be found. This immediately raises the question of why experiments with Maxwellian illumination typically differ
from Eq. (15) by the aforementioned factor of 2.
This question is best addressed in two parts by considering the wave-guided light returned to the eye pupil
separately from light entering the eye. For extended illumination each photoreceptor can be considered to produce
a Gaussian beam at the pupil as expressed in Eq. (8) with
the distribution for the OSCE obtained as a sum of these
(including possible interference effects that in the case of
coherent illumination may narrow the distribution17; see
共16兲
where A is the plane-wave amplitude normalized to the
area of integration. The sum includes the number of allowed modes 共M + 1兲 as determined by the V number of
the inner segment.
We have calculated the solutions Ulm of the mode equation in two different inner segments and in the outer segment at two different wavelengths as shown in Table 1
(using the photoreceptor parameters of the previous
sections).26 The corresponding coupling efficiency for each
mode at different angles of incidence is shown in Table 2.
Both tables show numerically obtained values for the exact LP modes without the Gaussian approximation used
hitherto for the fundamental mode. In the following, the
results are discussed with emphasis on the visible wavelength. It should be noted, however, that for the OSCE a
near-IR wavelength might in some ways be preferable
since less light gets absorbed in the outer segment
(thereby not requiring prebleaching). In turn, the visual
sensitivity is very small at the near-IR wavelength making the SCE harder to access, but it simplifies the task of
modeling since fewer modes need to be considered.
As the angle of incidence is increased, less light couples
to the fundamental mode of the inner segment but becomes partially redistributed among higher-order modes.
Table 1. Solutions Ulm of the Mode Equation (Ref. 26) for Different Modes LPlm at a Wavelength
of 0.543 ␮m (Upper Half) and 0.785 ␮m (Lower Half)
a
Segment
V
LP01
LP02
LP11
LP21
di = 2.5 ␮m
di = 5.0 ␮m
do = 1.0 ␮m
2.7063
5.4126
2.8888
1.7154
2.0216
1.7511
a
2.5914
3.2014
2.6747
4.2547
di = 2.5 ␮m
di = 5.0 ␮m
do = 1.0 ␮m
1.8720
3.7440
1.9982
1.4824
1.8781
1.5276
Where no value is given the mode is not allowed for the chosen set of parameters.
4.5272
2.9382
Vohnsen et al.
Vol. 22, No. 11 / November 2005 / J. Opt. Soc. Am. A
2323
Table 2. Coupling Efficiency of a Plane Wave to Modes LPlm in the Inner and Outer Photoreceptor
Segments Calculated at Incident Angles of 0, 5, and 10 deg (in Ascending Order) for a
Wavelength of 0.543 ␮m and 0.785 ␮m.a
Wavelength
0.543 ␮m
Segment
LP01
di = 2.5 ␮m
0.851
0.367
0.010
0.691
0.030
0.000
0.836
0.737
0.498
di = 5.0 ␮m
do = 1.0 ␮m
0.785 ␮m
LP02
LP11
0.168
0.108
0.000
0.000
0.241
0.109
0.000
0.137
0.001
0.000
0.063
0.189
LP21
LP01
LP11
0.000
0.138
0.000
0.924
0.600
0.133
0.775
0.169
0.001
0.913
0.854
0.697
0.000
0.238
0.009
a
Obtained from Eq. 共16兲 by integration within a circular area limited by a radius of 0.61d, where d is the respective photoreceptor diameter, as suggested by the diameterto-spacing ratio of 0.82.14,38
The tapered photoreceptor section may redistribute (and
reradiate) some of this energy to the modes available in
the outer segment. It should be recalled, however, that
the SCE is usually studied in unbleached conditions and
most light will therefore be detected near the outer segment entrance (assuming a uniform photopigment density) presumably also affected by modes beyond cutoff.
Note that even if the outer segment does allow for propagation of the LP11 mode along its entire length (i.e., do
⬎ 1.53␭), it will not be efficiently excited by backscattered light at small angles (see Table 2). When many
modes are allowed in the inner segment, the angle at
which each can be efficiently excited can be estimated as
␪lm ⬇
Ulm␭
␲ n id i
.
共17兲
This expression has been derived from a ray–optical mode
consideration in combination with the Ulm solutions of the
waveguide mode equation (its applicability can be appreciated from the individual mode-coupling dependencies
shown in Fig. 2).26 For the inner segment with a diameter
of 2.5 ␮m it is found from Table 1 [using the values of U01
and U11 in Eq. (17)] that 共␪11 / ␪01兲2 ⬇ 2.28 thereby indicating an approximate doubling of the directionality parameter when the mode LP11 is included in the excitation.
However, it is also clear that the expression is mainly apt
to consider higher-order modes. Thus, to examine the
situation in more detail for typical cone sizes, we have calculated the angular dependence of the coupling efficiency,
as shown in Fig. 2 for both inner segments. For the
smaller segment the sum of excited modes (as well as the
fundamental mode) fits accurately a Gaussian distribution. The fitting polynomials may be converted to the form
of Eq. (1) using ␪ = rp / feye, in which case the corresponding
SCE directionality parameters can be found as ␳SCE,01
= 0.102/ mm2 and ␳SCE,sum = 0.048/ mm2. Thus, the simul-
Fig. 2. Calculated coupling efficiency T共␪兲 (thick solid curve) at
␭ = 0.543 ␮m for inner segments (a) 2.5 ␮m and (b) 5.0 ␮m including the contributions from individual modes: LP01 (thin solid
curve), LP11 (dashed curve), LP21 (dashed–dotted curve), and
LP02 (dotted curve). Best-fitted Gaussian distributions versus
angle (in radians): (a) T01共␪兲 = 0.8605 exp关−2␪2 / 0.13182兴, Tsum共␪兲
= 0.8764 exp关−2␪2 / 0.19182兴, (b) T01共␪兲 = 0.6980 exp关−2␪2 / 0.07342兴,
Tsum共␪兲 = 0.7831 exp关−2␪2 / 0.14162兴, shown as a series of crosses
共LP01兲 and circles (sum), respectively.
2324
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
taneous excitation of both modes has lowered the directionality parameter to a similar value as found experimentally. As a consequence of the good fit, Eq. (10) may
still be applied provided that the directionality parameter
used is that of the sum, viz., ␳SCE = ␳SCE,sum or alternatively that the effective mode width is estimated as wm
⬇ di / U11. When more modes are excited the situation becomes more complex as seen in Fig. 2(b) and the Gaussian
fit to the sum becomes less accurate. Nonetheless, it reveals directionality parameters for the large inner segment of ␳SCE,01 = 0.327/ mm2 and ␳SCE,sum = 0.088/ mm2, respectively. Thus, the fit for the sum is in good agreement
with the experimentally found value at the parafoveal
retina.35,36 For light exiting the eye, it may be noted that
the directionality parameter is identical with that of the
fundamental mode for the SCE, viz., ␳OSCE = ␳SCE,01 provided that the back-reflected light distribution is dominated by the LP01 mode as discussed above for the smaller
segment and as suggested by other authors.14 Under this
condition, the width of photoreceptors contributing to the
OSCE may still be estimated as in Section 4, i.e., with the
inner segment diameter estimated by di ⬇ V0wm and with
the fundamental mode width obtained by means of Eq.
(11). It may be noted that the observed increase of the ratio ␳OSCE / ␳SCE ⬃ ␳SCE,01 / ␳SCE,sum with foveal distance is in
qualitative agreement with observations (see Fig. 7 in
Ref. 12). The nontrivial behavior observed with a scanning system for ␳OSCE as a function of foveal distance, increasing to an eccentricity of ⬃2° beyond which it
smoothly decays,4 may also be explained in the context of
the present model. At and near the fovea the situation resembles that of the Maxwellian illumination configuration with simultaneous excitation of more modes as described above, but at larger eccentricities the focused
beam produces a spot at the retina comparable with (Section 4) or smaller than the typical cone photoreceptor resulting in a decrease of the directionality parameter [i.e.,
Eq. (5) with wr ⬍ wm].
The angular dependence of the coupling efficiency may
be examined in a similar manner at other wavelengths,
and specific results obtained with near-IR are shown in
Fig. 3. Here it should be noted that the inner segment
with a 2.5 ␮m diameter allows propagation only of the
fundamental mode. Thus, in this case Gaussian fitting
leads to ␳SCE,01 = ␳SCE,sum = 0.053/ mm2, whereas for the
inner segment with a 5.0 ␮m diameter it can be found
that ␳SCE,01 = 0.176/ mm2 and ␳SCE,sum = 0.087/ mm2.
With regard to the wavelength dependence of the directionality parameter, it may be observed that the ratio
␳OSCE 共 0.785 ␮m 兲 / ␳OSCE 共 0.543 ␮m 兲 ⬃␳SCE,01共0.785 ␮m兲 /
␳SCE,01共0.543 ␮m兲 equals ⬃0.52 for the smaller and ⬃0.54
for the larger inner segment, and is therefore close to the
expected value of 共0.543/ 0.785兲2 ⬇ 0.48. The small deviation from a 1 / ␭2 dependence is due to the quality of fitting
as well as a minor wavelength dependence of the exact
mode width. The same wavelength relationship is not revealed from the fitting of the SCE 共␳SCE ⬃ ␳SCE,sum兲 because of the changed number of allowed modes. Here it
appears only within a given wavelength range when the
number of allowed modes is constant.7 This indicates that
the OSCE might represent the better way of accessing
photoreceptor data when the actual number of allowed
Vohnsen et al.
Fig. 3. Calculated coupling efficiency T共␪兲 (thick solid curve) at
␭ = 0.785 ␮m for inner segments (a) 2.5 ␮m and (b) 5.0 ␮m including the contributions from individual modes: LP01 (thin solid
curve) and LP11 (dashed curve). Best-fitted Gaussian distributions versus angle (in radians): (a) T01共␪兲 = Tsum共␪兲 = 0.9310
⫻exp关−2␪2 / 0.18322兴,
(b)
T01共␪兲 = 0.7833 exp关−2␪2 / 0.10012兴,
Tsum共␪兲 = 0.8062 exp关−2␪2 / 0.14252兴, shown as a series of crosses
共LP01兲 and circles (sum), respectively.
modes is not known a priori (which is normally the case).
A central concern with regard to the modeling of the
SCE and the OSCE is how sensitive the estimations are
to specific model details and to uncertainties in the implemented parameters. On the one hand, a strong sensitivity
holds promise of accurate measurements of physical parameters noninvasively, but on the other it may complicate a direct validation of the model. One issue is the actual step-index fiber model chosen on the basis of the fact
that modes (including those of higher order) observed
near the outer segment termination of in vitro human
photoreceptors resemble strongly the modes used in the
present model.6 In turn, experiments on certain fish photoreceptors have rather suggested a graded-index
model.39 Actually, in the present context a similar feature
would not be of major concern since the Gaussian mode
approximation is exact for the fundamental mode in a
parabolic index distribution (and highly similar for other
index distributions).26
Another issue is the reported wavelength dependence
of the SCE for a single photoreceptor where modeling
seems to produce too abrupt changes (apart from the overall 1 / ␭2 factor) as the number of allowed modes change.7
Vohnsen et al.
Presumably these features are not prominent experimentally due to an averaging and smoothing effect when illuminating an extended retinal patch (containing slightly
different photoreceptors). Actually, with simultaneous illumination of several photoreceptors, as in standard experiments on the SCE, the larger ones will collect a larger
fraction of the incident light and may therefore have
stronger influence on the visual response. A proper
weighting of contribution photoreceptors in Maxwellian
illumination systems may therefore lead to even better
numerical agreement between simulated and histological
data. Moreover, because of its small length the inner segment cannot be expected to have an abrupt mode cutoff in
transmission. Thus, even beyond cutoff some light may
still reach the outer segment and contribute to the sensed
level of brightness.
With respect to the influence of parameter uncertainties (refractive indices, waveguide diameters) on the predictions of the model, these can perhaps best be appreciated through the V number of the inner segment. Since
any such deviation would produce a corresponding change
in V, the spectral regions where a given number of modes
would be allowed can shift and thereby influence the SCE
directionality parameter (standard experiments would
average such details among a given number of illuminated photoreceptors).7 Thus, only if some of these variables are accurately known by other means does the SCE
appear to be suitable for probing for additional information. In turn, the present model suggests that the OSCE
may be used to directly access the photoreceptor diameter
through the relationship di ⬀ 共␳OSCE兲0.5. Such knowledge
may subsequently be used as input to further studies
with the SCE. It should be emphasized, that the relation
between the OSCE directionality parameter and photoreceptor diameter found in the present study differs from
that of the aforementioned retinal scattering model in
which it was rather related to the average photoreceptor
spacing.17,18 However, since both the inner segment diameter and average cone spacing increase with foveal distance, this may complicate a direct experimental verification in favor of either model.
6. COHERENT VERSUS INCOHERENT
ILLUMINATION AND THE IMPORTANCE OF
PHASE
For simplicity of discussion no direct reference to the
state of coherence of the illumination has yet been made
(although monochromatic coherent light has implicitly
been assumed). Historically, the first experiments on the
SCE were performed with incoherent sources whereas
later experiments on both the SCE and the OSCE have
often made use of highly coherent laser light. It is thus
fair to ask whether this may have any significant influence on either effect and if this can be estimated within
the present model. In the following the possible implications of coherence are discussed first for the SCE and subsequently for the OSCE.
For standard experiments on the SCE no such difference ought to be observable since the angle of incidence
on the retina is the same in both cases and only the trans-
Vol. 22, No. 11 / November 2005 / J. Opt. Soc. Am. A
2325
mitted power [see Eq. (16)] matters for the visual response at any given moment. This is also in agreement
with experimental findings.40 However, if simultaneous illumination from different parts of the pupil is permitted
this need no longer be the case since only a coherent field
may interfere in the retinal plane. In this context the possible additivity of the SCE (i.e., whether different parts of
the illumination will combine to reinforce the SCE or
whether they may interact to either partially or completely nullify the effect) is of central concern. In one such
study, indeed a reduced contribution (although slightly
less than expected) from light near the rim of a large pupil was found with incoherent light as predicted by the
SCE with additivity.41 Such knowledge has been applied
to the design of pupil filters that neutralize the SCE.2
Another study whose object was a direct comparison
with standard experiments on the SCE made use of a sectioned annular pupil aperture (with mean radius rp) so
that all light within the eye was largely confined within a
cone shell having the retina at its apex.42 With the field
added coherently and in phase from the entire annular
aperture, the resulting wavefront at the retina should
show no tilt and therefore no SCE. The field at the retina
can be expressed via a Fourier transform of the pupil
field, i.e., ␺r共x , y兲 ⬀ F兵Pa共x , y兲exp关i␸r共x , y兲兴其 = F兵Pa共x , y兲其
丢 F兵exp关i␸r共x , y兲兴其. Here Pa denotes the magnitude of the
field transmitted by the annular aperture and ␸r the corresponding phase distribution that may be taken to include also ocular aberrations. The symbol 丢 refers to a
convolution. Clearly, if ␸r is not constant on the entire aperture any phase distribution on the retina may result
(different from that of F兵Pa共x , y兲其 alone). Thus, both ocular
aberrations and phase variations in the incident light
may have a detrimental effect and hamper the interpretation of the experimental outcome. In the referenced
study an incoherent source was used in conjunction with
a diffuser (introducing phase variations) and it was concluded that the SCE was additive. However, in our opinion the experimental evidences referred to do not suffice
to reject the present hypothesis of amplitude field addition on which Eq. (3) is based. Rather, a conclusive test
should be performed with a highly coherent source and a
well-controlled wavefront at the entire pupil plane. With
an incoherent source, interference in the retinal plane
would cancel as a result of random phase fluctuations in
the illumination and the expected result would be that of
the standard SCE.
To examine the possible influence on the OSCE of illuminating a large retinal patch (containing numerous photoreceptors) with either coherent or incoherent light, the
variation in photoreceptor length and consequently the
phase of the reflected wave-guided light should be
considered.17 Actually, if no phase differences are produced for coherent light one should expect only a tiny
bright spot near the center of the pupil (toward which all
photoreceptors are oriented9), since in this case the light
reradiated from the retina would add coherently to resemble a convergent spherical wavefront focused at the
pupil. Since this is not what is seen experimentally, the
phase of the reflected light must play an important role.
To describe its effect, the pupil field may be written as a
sum of Gaussian beams originating at each photoreceptor
2326
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
Vohnsen et al.
j with different phase shifts ␸j. The intensity produced at
the pupil can thus be written as
冋
Idir共x,y兲 = I0TR exp −
冏兺
2共x2 + y2兲
wp2
册
N
⫻
j=1
exp关i共kx,jx + ky,jy + ␸j兲兴
冏
2
,
共18兲
where kxj and kyj represent the wave-vector components
in the pupil plane for light originating at each contributing photoreceptor. With only one photoreceptor the expression reduces to that of Eq. (8). For simplicity it has
been assumed that all N photoreceptors radiate equally
brightly and have the same mode width (diameter). If this
were not the case, the Gaussian weighting factor would be
slightly different for each and should therefore act within
the summation sign to produce a different weighting of
each plane-wave component. Clearly, this could slightly
modify the observed directionality parameter as already
mentioned. With random phase shifts ␸j distributed uniformly (when wrapped) on the interval 关0 ; 2␲关, the summation term of Eq. (18) approaches the well-known random walk at any pupil point 共x , y兲. Thus, for coherent
light the intensity at each pupil point may take on any
value from zero to the maximum allowed (due to constructive interference), giving rise to a speckle pattern. It
should be noted, however, that the speckle pattern contained in the summation is in itself not spatially confined
[see Fig. 4(b)]. The only spatial confinement is due to the
Gaussian weighting that is the same as in Eq. (8). Other
authors have argued for a normal distribution of the
phase terms in order to achieve a spatial confinement,17
but in the present model it is simply due to diffraction.
For incoherent light no interference is observable and the
summation in Eq. (18) can be replaced by unity. Experimental results have shown some difference between the
coherent10–12,14,17,18 and the incoherent case20,21 which
has been attributed to the difference in size of the illuminated areas. However, the available data are still very
few, and have been obtained under different experimental
conditions, making further experiments highly desirable.
In Fig. 4 we have simulated the development of an illuminated extended retinal patch and the gradual approximation of averaged pupil intensity distributions
(each having a random phase at each photoreceptor) to
the distribution for incoherent light. An alternative to averaging numerous similar pupil images is the calculation
of an averaged radial distribution within a single image,
although this would require an accurate determination of
the pupil center. The retinal curvature with all receptors
pointed toward a single pupil point,9 and the fact that
wave-guided light contributes to the intensity distribution at the pupil means that a coherent retinal scattering
model for a corrugated surface17 (where each local facet
points in an arbitrary direction) does not seem entirely
satisfactory in our view. In turn, the present description
gives both directionality parameters and wavelength dependence within the scope of a single model. In this context, it should be stressed that diffraction is a necessity to
explain how light is radiated from tiny structures. This
may also explain why scattering models give the same
wavelength dependence, as they too are based on the diffraction of propagating wave fields.
7. CONCLUSIONS
Fig. 4. Simulated OSCE at an illumination wavelength of
0.543 ␮m of (a) 1° uniformly illuminated retinal patch with 1221
randomly located contributing cones (i.e., a density of
20,000 cones/ mm2) and fundamental mode width wm = 1.35 ␮m
(for simplicity partial cone overlap has not been prohibited in the
model). The coherent plane-wave contribution [i.e., the unweighted sum of Eq. (18)] is shown in (b) at the pupil plane 关6
⫻ 6 mm2兴. Corresponding intensity distributions for a 6 mm pupil are shown for coherent light with (c) 1, (d), 10, (e) 100 averaged frames and compared with the case of (f) incoherent light.
In this paper we have extended the common waveguide
theory of photoreceptors7,14 to formulate a theory of both
the SCE and the OSCE. This has been done analytically
within a Gaussian mode approximation and numerically
for the LP modes. Our description differs from others by
including diffraction for the propagation between eye pupil and photoreceptor aperture for both incoming and
back-reflected light. Despite the small size of photoreceptors, the diffraction in relation to waveguide coupling has
to our knowledge not been considered previously. The
present description is valid for describing illumination of
a single photoreceptor as well as an extended retinal
patch containing numerous photoreceptors. The Gaussian
approximation has simplified the description and allowed
us to derive valuable analytical expressions for the direc-
Vohnsen et al.
tionality parameters used in the description of both the
SCE and the OSCE. We showed explicitly that the commonly observed difference of a factor of 2 for the directionality parameter, as observed with the two techniques in
Maxwellian illumination systems, can be explained by the
difference in coupling of the incident and back-reflected
light. Whereas the incident light may couple to various
modes, the fundamental mode governs the back-reflected
light as suggested by the OSCE. This observation is in
agreement with previous predictions by others.14 The
present model also reproduces the overall wavelength dependence of the directionality parameter,19–21 here identified as a diffraction feature of the waveguided light for the
OSCE, and the larger value for the directionality parameter reported when scanning simultaneously both entrance and exit pupil.4,14
More experiments, at different wavelengths and with
both coherent and incoherent light, are still needed in order to address directly the SCE and the OSCE and
thereby evaluate the validity of the present model. Eventually, such studies may permit in vivo deduction of photoreceptor parameters such as mode width and inner segment diameter at different retinal locations. Finally,
experiments may also serve to elucidate the influence of
other parameters, such as wavefront aberrations and rod
contributions, as well as the influence of a background illumination [C ⫽ 0 in Eq. (1)], which can affect the
observed directionality parameters as well as the quality of cone-mosaic imaging with high-resolution
ophthalmoscopy.24,43
ACKNOWLEDGMENTS
The authors gratefully acknowledge financial support
from the Ministerio de Educación y Ciencia, Spain (grant
BFM2001-0391 and FIS2004-02153) and Ramón y Cajal
research contract RYC2002-006337. The authors are also
grateful for a number of valuable suggestions received
from two reviewers and JOSA A Editor Stephen A. Burns.
Corresponding author Brian Vohnsen’s e-mail address
is vohnsen@um.es.
Vol. 22, No. 11 / November 2005 / J. Opt. Soc. Am. A
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
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2.
3.
4.
5.
6.
7.
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Pm
Pr
26.
2327
=
冕冕
冏冕冕
*
␺ r␺ m
dxdy
兩␺r兩 dxdy
2
冕冕
冏
2
.
兩␺m兩 dxdy
2
Just as before this expression gives the percentage of incident power coupled to the guided mode m. The total
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2328
J. Opt. Soc. Am. A / Vol. 22, No. 11 / November 2005
LP01 is rotationally symmetric and can be represented by
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2
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Ulm
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32.
Jl+1共Ulm兲
Jl共Ulm兲
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= 冑V2 − Ulm
2
Kl+1共冑V2 − Ulm
兲
2
Kl共冑V2 − Ulm
兲
numerically for the unknown Ulm, where Jl is the Bessel
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