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Roorda et al. Vol. 12, No. 8/August 1995/ J. Opt. Soc. Am. A 1647
Geometrical theory to predict eccentric
photorefraction intensity profiles in the human eye
Austin Roorda and Melanie C. W. Campbell
School of Optometry and Department of Physics, University of Waterloo, and Guelph– Waterloo
Program for Graduate Work in Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
W. R. Bobier
School of Optometry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Received March 30, 1994; revised manuscript received September 6, 1994; accepted October 3, 1994
In eccentric photorefraction, light returning from the retina of the eye is photographed by a camera focused on
the eye’s pupil. We use a geometrical model of eccentric photorefraction to generate intensity profiles across
the pupil image. The intensity profiles for three different monochromatic aberration functions induced in
a single eye are predicted and show good agreement with the measured eccentric photorefraction intensity
profiles. A directional reflection from the retina is incorporated into the calculation. Intensity profiles for
symmetric and asymmetric aberrations are generated and measured. The latter profile shows a dependency
on the source position and the meridian. The magnitude of the effect of thresholding on measured pattern
extents is predicted. Monochromatic aberrations in human eyes will cause deviations in the eccentric
photorefraction measurements from traditional crescents caused by defocus and may cause misdiagnoses of
ametropia or anisometropia. Our results suggest that measuring refraction along the vertical meridian is
preferred for screening studies with the eccentric photorefractor.
Key words: eccentric photorefraction, ocular aberrations, retinal reflections.
1. INTRODUCTION
We have set out to develop a simplified geometrical-
optical technique for predicting the paths of rays reflected
from the retina in nonimaging applications such as ec-
centric photorefraction. Using this geometrical tool, we
analyze the effects of ocular aberrations on the intensity
profiles measured across the pupil in photorefraction, the
degree and effect of thresholding in traditional measure-
ments, and the effects on and implications of asymmetric
aberrations for intensity profiles.
The predicted results from the geometrical model are
verified experimentally in a single eye, where differing
aberration functions are induced by the application of
varifocal contact lenses.
A. Background
Eccentric photorefraction is a technique for measuring
the refractive state of the human eye. The technique is
remote from the subject and quick, which makes it par-
ticularly useful in refracting the eyes of children or un-
cooperative subjects for whom traditional methods, such
as retinoscopy, are impractical and difficult to perform.
The eccentric photorefractor employs a small light source,
positioned a fixed distance from the limiting aperture of
the camera, that projects a blurred spot onto the retina
(Fig. 1). The eccentricity refers to the distance of the
light source from the limiting aperture. The camera is
focused on the pupil plane, and light reflected from the
retina appears as an intensity distribution or crescent
in the pupil. The size and the orientation of the cres-
cent are used to determine the refractive state of the
eye.1,2 A set of equations was developed simultaneously
in 1985 by Bobier and Braddick1and by Howland,2who
used a paraxial geometrical theory that gives the re-
fractive state in terms of the crescent width and other
measurement parameters such as eccentricity and camera
distance. Eccentric photorefraction methods that use an
extended light source and measure the slope of the inten-
sity distribution have also been developed.3
Our main concern is in the region near the dead zone,
where, in the paraxial approximation, no light distribu-
tion is expected across the pupil. This dead zone occurs
when the eye is focused on the source. One can narrow
the width of this dead zone (in refractive state) by reduc-
ing the eccentricity of the source. In screening studies
the width of this dead zone is set such that ametropias
greater than a specified value (say, 62 D) will produce
a light distribution. Subjects with ametropias less than
this are expected to have no intensity distribution, that
is, their refractive states are said to lie in the dead zone.
We will show that, in this region, the photorefractor is
very sensitive to aberrations. We previously showed that
aberrations may cause the misclassification of ametropias
and a decrease in instrument sensitivity.4For higher re-
fractive states the defocus is the main contributor to the
retinal blur, so we show that, as the refractive state in-
creases, the crescent assumes a more traditional form,
as is predicted for the aberration-free eye. The impor-
tant factor in the sensitivity of the eccentric photorefrac-
tor to refractive error is the angle subtended between the
source and the limiting aperture, which is approximately
the eccentricity edivided by the camera distance from the
subject. Therefore our settings of 1-mm eccentricity and
0.33-m working distance used throughout this paper are
equivalent to those of an eccentric photorefractor with
0740-3232/95/081647-10$06.00 1995 Optical Society of America
1648 J. Opt. Soc. Am. A/Vol. 12, No. 8 / August 1995 Roorda et al.
Fig. 1. Light from the source forms a blur uv on the surface of
the retina. The light is diffusely reflected from the retina, and
an image of blur uv can be constructed at the far point of the eye.
This aerial image is labeled u0v0. Some of the rays from point
v pass through v0and enter the aperture of the camera. These
rays are the shaded bundle of rays in the figure. If a ray from
a particular point in the pupil enters the camera aperture, then
that point on the pupil will appear illuminated. This is how the
crescent is formed. The front view of the pupil illustrates the
crescent that would appear in this case. In this figure the eye is
myopic with no aberrations. One situates the limiting aperture
at the source plane by putting a thin shield across one half of
the lens. The eccentricity is the distance from the light source
to the edge of the shield.
5-mm eccentricity and a 1.67-m working distance (i.e.,
more-traditional settings).
B. Effects of Aberrations
The human eye is subject to more than just defocus errors.
Chromatic aberrations in the eye are wellknown and have
predictable effects on eccentric photorefraction patterns.5
The effects of astigmatism on pattern extents and
orientations have been explored by Wesemann et al.6
Hodgkinson et al.7have developed a physical optical
model to predict the effects of chromatic, spherical, and
astigmatic aberrations on intensity profiles for a knife-
edge photorefractor. Their method, however, was based
on the assumption that the returned light always tends
to a Gaussian form. We show that this is not always
the case.
The theory in this paper is based on our previous study
that developed a geometrical method to predict the maxi-
mum extents of the photorefraction pattern given any gen-
eral aberration in the eye.4This model accounted for
the variability of aberrations among eyes and predicted
that crescent edge measurements and inferred refractive
states can be affected by the presence of aberrations, espe-
cially near the dead zone. We showed that the crescent
may be larger or smaller than that predicted by parax-
ial theory, depending on the type of aberration present.
We extend this framework to predict the intensity pro-
file given any aberration, and we also verify our predic-
tions experimentally. The new analysis is comparable
with that used in the Foucault knife-edge measurement
of aberration adopted by Berny and Slansky8to measure
the wave-front aberration in the eye. Their method, how-
ever, eliminated the first pass by illuminating the retina
in Maxwellian view. This was technically a more dif-
ficult measurement, but it greatly simplified the analy-
sis so that they could measure the wave-front aberration
in the pupil from the knife-edge photographic patterns.
Our intention, on the other hand, is to predict the effects
of particular aberrations on classical double-pass inten-
sity patterns.
C. Thresholding
The intensity distribution at the edges of photorefraction
patterns is not sharp but decreases gradually. This cre-
ates some uncertainty in the measurement of the posi-
tion of crescent edges and creates a dependency of pattern
extent on the particular system (i.e., film sensitivity, cam-
era lens, etc.) and on the person performing the mea-
surement. In general, measured pattern extents are less
than that predicted by the theory.1,2,9 This reduction is
not the same for all refractive states, and it was found
that in some cases experimental pattern extents may ac-
tually exceed theoretical predictions.1,9 This difference,
however, was found by comparisons with the paraxial
theory. Monochromatic aberrations have been shown to
increase the predicted extents of the patterns.4,10 Cur-
rently, thresholding is dealt with by an experimental
calibration for the particular instrument.9To our knowl-
edge, no theoretical work has been published that predicts
the magnitude of the effect of thresholding on measured
refractive states.
D. Asymmetric Aberrations
We use our geometrical model to calculate the intensity
profile along any meridian of the eye. We show that (i)
for rotationally symmetric aberrations, such as spherical
aberration and defocus, measurements are independent of
source position or meridian and (ii) for asymmetric aber-
rations, such as the complex aberrations found in the hu-
man eye,11 primary astigmatism and coma, the pattern
is dependent on the source position and the meridian.
The meridian of the source refers to the orientation of the
line from the source through the center of the limiting
aperture. The position of the source refers to the loca-
tion of the source with respect to the limiting aperture as
viewed by the subject (i.e., nasal, temporal, superior, infe-
rior). We discuss the implications of this dependence for
measurements in which two eyes are photorefracted si-
multaneously. In this analysis we restrict ourselves to
measurements along principal astigmatic and comatic
meridians.
2. GEOMETRICAL-OPTICAL ANALYSIS
A geometrical theory was chosen instead of a physical op-
tical theory to calculate the expected intensity profiles.
This approach is justifiable since the aberrations in the
human eye are generally greater than one wavelength.12
Diffraction effects are predominant for aberrations less
than one quarter of a wavelength, but, for aberrations
that are several times this value, a geometrical approach
is reasonable.13 The geometrical theory has faster calcu-
lation times and can take advantage of the existing theory
for predicting the location of the crescent edges.
A. Calculation of the Intensity Profile
The general method followed is one previously developed4
and is based on similar assumptions. The light source is
assumed to be a point. Light from the source enters the
eye and is imaged on the retina as a blurred spot. The
entering light is traced to the eye ray by ray. Each ray
Roorda et al. Vol. 12, No. 8/August 1995/ J. Opt. Soc. Am. A 1649
has an intersection point on the retina determined by the
aberration function in the pupil. The aberration function
is discussed in terms of transverse ftsrdg and longitudinal
fksrdg aberrations. The term tsrdcan be defined as the
ray intercept position on the retina measured from the
paraxial intercept, projected into object space, as a func-
tion of the ray’s position in the pupil. ksrdis defined as
the far-point distance from the eye as a function of the
ray position in the pupil. Both representations define the
same aberration according to the relation
ksrd
l
"12tsrd
r#,(1)
where ksrdis the longitudinal aberration (far-point shift),
tsrdis the transverse aberration, lis the object distance,
and ris the vector ray position in the pupil. The density
of intersecting rays on the retina as a function of ray
position defines the retinal point-spread function.
Each point on the retina is assumed to emit light in all
directions as a diffuse reflector. The directional compo-
nent of the reflection is discussed in Subsection 2.B. A
principal ray can be drawn from each point on the retinal
point-spread function, and it emerges from the center of
the entrance pupil. The density of intersecting principal
rays in the source plane is a scaled version of the reti-
nal point-spread function (Fig. 2). So the retinal point-
spread function is calculated not directly but rather in
terms of the principal rays emerging from it. This calcu-
lation can be performed without knowledge of any of the
optics of the eye other than the aberrations as a function
of position in the entrance pupil. The intersection of the
principal ray in the source plane is calculated by4
xpr srdr?fp1ksrdg
2ksrd,(2)
where xpr srdis the position of intersection of the principal
ray in the source plane with respect to the source (in mil-
limeters), ris the ray position in the pupil (millimeters), p
is the camera distance (meters), and ksrdis the far-point
position as function of ray position in the pupil (meters).
We find the density of the projected retinal point-spread
function by sampling the intersecting principal rays per
unit length in the source plane. The unique feature of
the model is that it is suitable for any type of eye, model
eye or living eye, provided that the aberration function
across the entrance pupil is known.
The diffusely reflected rays emerging from a single
point on the retina will intersect their principal ray out-
side the eye at a far point determined by the radius at
which the ray leaves the pupil (Fig. 3). If there are no
aberrations present, all rays from a single point on the
retina will intersect at a single point along the princi-
pal ray. With aberrations present, the intersection point
will vary according to the aberration function. In this
analysis we assume that the point-spread and the aber-
ration functions are the same in the second pass as in the
first.
We determine the maximum crescent extents by tracing
the rays from the extreme edge of the blur on the retina
and finding which rays enter the limiting aperture of
the camera. The original monochromatic theory of the
effects of aberrations on photorefraction4considered rays
from this point only. Rays from the other points on the
retinal blur will not increase the crescent size but will
contribute to the crescent intensity. We calculate the
crescent intensity profile by first calculating the pattern
extents in the pupil for each point on the retinal blur.
This is illustrated in Fig. 3 for a general aberration. The
edges of the pattern extents for each position on the blur
on the retina are given by4
ysrdxpr 1r?fp1ksrdg
2ksrd,(3)
where ysrdis the position of intersection of the returning
ray and xpr is the position of intersection of the principal
ray as defined in Eq. (2).
The positions in the pupil, r, for which ysrde, the
source eccentricity, are the edges of the crescent from a
particular point on the retina.4In some cases, such as
is shown in Fig. 3, there may be two or more rays from a
single point on the retina that intersect the limiting aper-
ture. This is the source of the split patterns (crescents
appear to originate from either side of the pupil) and the
central bright regions (a pattern in which the peak in-
tensity occurs near the center of the pupil and tails off
toward the margins) that often appear in photorefraction
profiles.14
Each pattern is given an intensity weighting according
to the intensity of the corresponding point on the retinal
blur, derived from Eq. (2). The sum of all the patterns
from each point gives the final intensity profile.
Fig. 2. The principal ray from each point on the retinal blur
emerges from the center of the entrance pupil. Thus the an-
gular density of principal rays will be a scaled distribution
equivalent to the intensity of the retinal point spread.
Fig. 3. The diffusely reflected rays from a single point on the
retina are traced out of the eye. The principal ray is the
heavy dashed line from point U on the retina. This ray trace
illustrates several rays traced out of the eye through the points
labeled 4 to 24 that intersect the principal ray at the corre-
sponding far points. The bundles of rays (shaded) that enter
the limiting aperture of the camera define the regions where
crescents will appear in the pupil. In this case a comatic-type
aberration combined with defocus results in a split crescent
photographed in the pupil plane.
1650 J. Opt. Soc. Am. A/Vol. 12, No. 8 / August 1995 Roorda et al.
We model the change in the refractive state by adding
a constant defocus factor to the aberration function. We
assume that the aberrations are constant with object
vergence.4In this way the crescent intensity profiles can
be calculated for a range of refractive errors.
B. Incorporation of Directional Reflection Effects
The Stiles–Crawford effect of the first kind refers to the
perceived intensity of light as a function of the ray en-
trance position in the pupil. The function peaks near
the pupil center and drops off parabolically from this
point. Gorrand et al.,15 van Blokland,16 and R¨
ohler and
Schmielau17 have all found that reflections from the
retina have a similar directional feature. This effect
is attributed primarily to the limited acceptance angle
of a single photoreceptor. The photoreceptor has wave-
guiding properties like those of a small optical fiber.
The intensity of the reflection is generally greatest near
the center of the pupil and drops off toward the mar-
gins. One can incorporate this reflection effect into the
crescent intensity profiles by multiplying the intensity
at each point across the pupil by an attenuation factor
that is a function of the radius. A function was selected
that matched the typical reflection function found by the
authors mentioned above15–17 [Eq. (4) below]. The expo-
nential decay term in the function is similar to that used
by Artal,18 but we have added a baseline term. This
function also incorporates a decrease in the directionality
with increasing spot size (or increasing refractive state)
by means of a change in D. A calculation of the change
in directionality with spot size is included in the results.
The form of the attenuation factor chosen was
IsrdI08
<
:
11expf2Ds ryrmax d2g
29
=
;
,(4)
where I0is the uncorrected crescent intensity, Isrdis the
attenuated crescent intensity, Dis a directionality fac-
tor (D0;no directionality; D4;50% attenua-
tion at pupil edges), rmax is the maximum radius of the
pupil (millimeters), and ris the ray position in the pupil
(millimeters).
C. Calculations
The calculations are iterative and were performed on a
computer. The aberration functions were entered into
the program as the variation of the far point as a function
of the pupil position (longitudinal aberration). A two-
dimensional array of the intensity as a function of refrac-
tive state and pupil position was generated and plotted
as a three-dimensional surface chart.
3. EXPERIMENT
An experiment was performed to compare empirical ec-
centric photorefraction patterns with the theoretical pre-
dictions for three ocular aberration functions in the eye.
Effects of aberrations, thresholding, and asymmetric
aberrations were investigated. We used a single eye
to maintain a constancy between the cases, the only dif-
ference between them being the aberration of the optical
elements. Differences in retinal reflectivity, the direc-
tionality of the retinal reflections, and ocular scatter are
ruled out.
The aberrations were induced in the eye by application
in two cases of soft varifocal contact lenses of two types,
PA1 (Bausch & Lomb, nominal power 20.5 D) and PS45
(Nissel, nominal power 20.5 D), respectively; the third
case is for the unaided eye. The lenses were designed
with changing power for each annular zone to provide a
range of foci to aid vision for the presbyopic eye, which
has little or no accommodative facility. The idea is that
one could obtain a long depth of focus by inducing spheri-
cal aberration with the contact lens in place. Hence the
application of these contact lenses is equivalent to induc-
ing monochromatic aberrations in the eye.19 The lenses
have different effects on the aberrations for each person20
and produce values two to three times larger than that
found in an average adult human eye.11
The aberrations were measured along the horizontal
meridian for three cases: with the PA1 lens, with the
PS45 lens, and with the unaided eye, and were used to
predict the expected intensity profiles. Measured eccen-
tric photorefraction intensity profiles were then compared
with the theoretical predictions. The subject was an em-
metrope (,0.25 D) with no astigmatism (,0.25 D), 26
years of age, and without correction had relatively low
aberrations across the horizontal meridian. Visual acu-
ity was measured as 20y15.
We measured the aberrations across a single meridian
in the eye with and without the contact lenses at a fixed
accommodative state, using a previously developed modi-
fied Ivanoff apparatus,11 as the aberrations are known to
vary with the accommodative state. The eye was given
a stimulus for accommodation at 3 D. In the measure-
ment the observer was required to align a target in a
Maxwellian view with a similar target in a normal view.
The normal and Maxwellian views were seen simultane-
ously in a split field. The subject perceives a horizontal
separation of the targets that varies with the horizon-
tal position of the Maxwellian view in the pupil. This
separation is due to the aberration corresponding to the
particular pupil position. By moving the actual position
of the target that appears in the Maxwellian view, the
subject can achieve a perceived alignment of the two tar-
gets. The separation of the targets in real space, after a
subjective alignment, for each position of the Maxwellian
view across the pupil is directly related to the trans-
verse ray aberration. A fifth-order polynomial is fitted
to the transverse aberration to produce a smooth aberra-
tion function:
tsrdA1r31A2r51B1r21B2r41Cr1Dt,(5)
where A1and A2are coefficients of third- and fifth-order
spherical aberration, respectively, B1and B2are coeffi-
cients of third- and fifth-order coma, respectively, Cis
the defocus coefficient, and Dtis the zero-offset term.
The data from the above experiment were converted to
longitudinal aberration [Eq. (1)] and input into the com-
puter program for calculating the expected eccentric pho-
torefraction intensity profiles as a function of refractive
state; we used the same settings (i.e., working distance,
pupil size, and eccentricity) as in the actual measure-
ments (described below). The computer program was
Roorda et al. Vol. 12, No. 8/August 1995/ J. Opt. Soc. Am. A 1651
written in BASIC and required less than 1 min on a 486 PC
to generate a profile at a single refractive state. For
the single profiles, rays were traced through the eye at
0.1-mm intervals across the entrance pupil. For the
three dimensional plots rays were traced at 0.3-mm in-
tervals and the refractive state was stepped in 0.3-D
intervals. A flow chart of the computer program is in-
cluded as Appendix A.
An eccentric photorefraction experiment was performed
on the same eye fitted in turn with each of the contact
lenses and with the unaided eye. The camera distance
was 0.33 m to permit high magnification of the pupil.
The eccentricity of the source was 1 mm temporal to make
the instrument sensitive to small aberrations near zero
defocus. We simulated a range of refractive states by
placing corrective lenses in front of the eye. The opposite
eye fixated on a target at 0.33 m (23 D accommodative
state). We assumed that there was no change in aber-
rations with vergence, that the additional aberrations in-
duced by the corrective lenses were small compared with
those of the eye, and that the accommodative state of
the eye was fixed. The light source was limited to a fi-
nite bandwidth in the red (620 – 720 nm). This permit-
ted good reflectivity from the retina and little blurring
because of chromatic aberration. The directionality in
red light is expected to be less variable because in the
bleached or the unbleached state the red-light reflection
directionality is virtually unchanged.16 The images were
captured on a CCD array, and the crescent edges and the
intensity profiles were measured with an image-analysis
software package.
To investigate the thresholding effects we used an un-
biased observer to measure the crescent extents, not-
ing traditional patterns and the split and central bright
patterns. The measurement technique was simply a sub-
jective measurement of the edges of the photorefraction
crescents as would usually be performed. These mea-
sured values were compared with the theoretical predic-
tions corrected for different thresholds.
To investigate the effects of asymmetric aberrations,
two eccentric photorefraction images were taken with the
light source on either side (nasal and temporal) of the
horizontal meridian with the PA1 contact lens in place.
The images were taken with the opposite eye focused on
a light source at 0.33 m.
4. RESULTS
A polynomial [Eq. (5)] was fitted to the aberration data for
the PA1 lens, the PS45 lens, and no contact lens (Fig. 4).
The PA1 induced a positive spherical aberration in the
eye with some decentration. The PS45 induced predom-
inantly negative spherical aberration. The aberrations
of the eye measured without any lens were quite small,
even with respect to a typical human subject.11
A. Intensity Profiles
The experimental results for the eye focused on the source
are shown in Fig. 5. For each pupil image the experi-
mental intensity profile is shown, along with the predicted
experimental profile. Several steps were made in per-
forming this fit. First, the profile was predicted from the
measured aberrations as outlined in Subsection 2.A; then
the directionality factor Dfrom Eq. (4) was adjusted for
a best fit. Fits with no directionality are shown for com-
parison. The background intensity was assumed to be
Fig. 4. Transverse aberration data for the three cases measured
on a single subject (AR, right eye) at 3 D accommodative state.
The data and the curve fit are shown for the PA1 lens to illustrate
the accuracy of the fit. The three cases are unaided eye (low
aberrations), PA1 lens (asymmetric aberration or decentered pos-
itive spherical aberration), and PS45 lens (symmetric, negative
spherical aberration).
Fig. 5. Eccentric photorefraction images at 23 D paraxial re-
fractive state (focused at the source) obtained by use of a temporal
source, 1-mm eccentricity, and a 0.33-m working distance. The
images are of the full pupil. The bright spot in the center
of the pupil is the first Purkinje image and can be ignored.
The experimental profiles (solid curves) are taken directly from
the horizontal meridian of the images. The background scatter
has been subtracted from each profile. The computer-generated
profiles are the expected intensity profiles for the same refractive
state. Long-dashed curves are the profiles with directionality,
and short-dashed curves represent calculations before direction-
ality was incorporated. (a) Unaided eye: no crescent observed,
only background scatter. (b) PA1 lens: diffuse crescent pat-
tern observed extending throughout the pupil with peak intensity
close to the center. (c) PS45 lens: crescent with high intensity
formed in the nasal margin that drops off rapidly across the
pupil. The normalization of the experimental maximum in (c)
determines all other intensity levels.
1652 J. Opt. Soc. Am. A/Vol. 12, No. 8 / August 1995 Roorda et al.
Fig. 6. Experimental versus theoretical predictions for 13D
(left) and 29 D (right) refractive states. These refractive states
represent 6 D defocus in either direction from the camera placed
0.33 m from the eye. Solid curves, experimental profiles;
long-dashed curves, predicted profiles with directionality;
short-dashed curves, predicted profiles without directionality.
The plots show that for the three cases the intensity profiles
take a similar form. The defocus term is dominant, and the
other aberrations have a small effect. (a) Unaided eye, (b) PA1
lens, (c) PS45 lens.
due to scatter and was subtracted for each experimental
profile. We did this by subtracting a constant value that
was the lowest intensity across the photorefraction inten-
sity profile for the eye with no contact lens at the 23D
nominal refractive state. The same baseline value was
subtracted from all three profiles. As the intensity scales
for both the theoretical and the experimental profiles were
arbitrary, the profiles for both were scaled for a best fit,
normalized, and superimposed. The same scaling factors
were applied for the three cases. A comparison with the
experimental results, measured from Fig. 5, demonstrates
the agreement of the predicted results of the geometrical-
optical model with the empirical measurements for low
refractive states. For the higher refractive states the de-
focus term dominates, and the crescent takes a more-
traditional shape. The same procedure was performed
in matching the theoretical with the experimental pro-
files. A comparison of the experimental and theoretical
profiles for refractive errors of 6 D myopic and 6 D hy-
peropic with respect to the camera position at 0.33 m is
shown in Fig. 6. A good fit of the predicted to the exper-
imental profiles was obtained across the entire range of
refractive states.
It was found that, as the refractive state increased, the
directionality of the retinal reflection for best fit to the
data decreased. We calculated the directionality for sev-
eral of the PA1 and the PS45 results by comparing the
theoretical and the experimental profiles and adjusting
the factor Dfor a best match. The results of the change
in directionality as a function of paraxial refractive state
are shown in Fig. 7. A directionality factor was not cal-
culated for every condition at each refractive state because
the intensity distribution did not always cover enough of
the meridian to provide sufficient data for fitting. The
calculated profiles for a range of refractive states for the
three cases have been generated incorporating the chang-
ing directionality and are shown in Fig. 8.
B. Thresholding
Traditional crescent-width measurements are often
smaller than the theoretical results because the theoreti-
Fig. 7. The directionality of the reflection is determined by the
ratio of the intensity at the center of the pupil to the attenuation
at the margins [see Eq. (4)]. A higher ratio indicates more
directionality. The directionality was deduced from the experi-
mental results and was highest for low refractive errors. The
decrease in directionality for high refractive states may indicate
some photoreceptor disarray during measurement over large spot
sizes on the retina.
Fig. 8. Computer-generated intensity profiles for a range of
refractive states obtained by use of a temporal source, 1-mm
eccentricity, and a 0.33-m working distance. The profile for
each refractive state has been adjusted for the directionality
of the reflection shown in Fig. 9. (a) Unaided eye: with low
aberrations, the crescents take a traditional form. There is a
dead zone around where the eye is focused on the camera, and
crescent intensity increases are symmetric for the hyperopic and
myopic regions, except that they originate in opposite margins of
the pupil. (b) PA1 lens: the presence of aberration alters the
symmetry of the intensity profiles, and the dead zone does not
appear. (c) PS45 lens: again the presence of aberration alters
the symmetry, and no dead zone occurs.
Roorda et al. Vol. 12, No. 8/August 1995/ J. Opt. Soc. Am. A 1653
Fig. 9. The baseline has been raised for the three cases. The
surface plot represents the theoretical calculations. The pre-
dicted edges occur where the surface plots intersect the base-
line, given by the solid curves. The points represent empirical
measurements of the crescent edges. By raising the baseline,
one obtains a good agreement of the experimental measurements
with the predicted edges. We obtained the best correction by
increasing the baseline by roughly 20% of the maximum intensity
as shown.
cal edge does not have sufficient intensity to be measured
by the camera. This is a common effect in any photore-
fraction technique and is referred to as thresholding.1,2,9,21
By raising the baseline of any of the intensity pro-
files shown in Fig. 8, one can obtain a threshold-corrected
crescent edge position. An unbiased observer performed
crescent-edge measurements for all the experimental re-
sults. The measured results originally had poor agree-
ment with the edges predicted by the theory and were
less than the prediction in most cases. When the base-
lines for the three plots in Fig. 8 were raised, a relatively
good fit to the measured edges was obtained, indicating
that the previous differences had been due to threshold-
ing. We achieved the best agreement by raising the base-
line ,20%. The measured crescent edges are plotted on
the threshold-corrected profiles in Fig. 9. Split crescents
and central bright regions were observed and are shown
in Figs. 9(b) and 9(c).
The threshold correction was also tested on previous
data published by Bobier.9We raised the baseline of the
intensity profiles across the range of refractive states to
investigate whether a level could be found that matched
the experimental data on a human eye. It was found that
a good fit was obtained if the threshold level was raised
to 30% of the maximum intensity at the edges. The re-
sults are shown in Fig. 10. The degree of thresholding
depends on the apparatus used and on the person per-
forming the measurement. A higher-sensitivity system
should require a smaller threshold correction. This ac-
counts for the different threshold levels required for Fig. 9
for a video system and Fig. 10 for a photographic system.
C. Asymmetric Aberrations
The effects of primary coma were investigated as an ex-
ample of asymmetric aberrations. The theoretical re-
sults for a schematic eye with pure primary coma are
illustrated in Fig. 11. In this calculation we simulate a
source that is on either side (nasal and temporal) of the
horizontal meridian.
We used a second PA1 lens in a followup experiment
to investigate the effects of asymmetric aberrations on
intensity profiles. Figures 12(a) and 12(b) represent the
predicted (with directionality) and experimental profiles
for the source on the temporal and nasal sides of the hori-
zontal meridian. In both cases the eye was focused on
the source. For the aberration-free eye no intensity dis-
tribution is expected for either position of the source. If
the aberrations were symmetrical the intensity distribu-
tions for the temporal and the nasal sources would be
symmetrical. Because of the asymmetry of the aberra-
tions induced by the PA1 lens, a considerable difference
was observed between the two results.
Fig. 10. Threshold-corrected crescent edges for a typical eccen-
tric photorefraction measurement. Camera distance, 1.5 m; ec-
centricity, 25 mm; pupil size, 8 mm. The solid curves represent
the expected crescent edges for a perfect detection system. The
dashed curves represent the expected edges if the detection of
the edge occurs at 30% of the maximum crescent intensity. The
data from a previous study by Bobier9(cyclopleged human eye,
photographic photorefraction, using 400 ASA color film) fit the
threshold-corrected curve.
Fig. 11. Dependency of the intensity profiles on the source
position for an eye with asymmetric aberrations. In this case
the eye is modeled with primary coma along the horizontal
meridian. When the source is temporal (solid curve) the result
is a split crescent. For the nasal source (dashed curve) a central
bright region is observed.
1654 J. Opt. Soc. Am. A/Vol. 12, No. 8 / August 1995 Roorda et al.
Fig. 12. Two profiles from photorefraction images taken with
the PA1 lens with (a) a temporal source and (b) a nasal source.
Solid curves, experimental profiles; dashed curves, predicted
profiles (with directionality). The peak in the center of the
experimental profiles is the Purkinje image and can be ignored.
The difference in the two profiles from the nasal and temporal
sources indicates the effects of the asymmetric aberration in-
duced by the PA1 lens.
5. DISCUSSION
This investigation shows that the geometrical model can
be used to predict the general pattern of crescent size and
intensity distributions for eccentric photorefraction, given
any specific monochromatic aberration distribution in
the eye.
Our results verify our expectations that monochromatic
aberrations may have considerable effects on photorefrac-
tion intensity profiles,4especially for low eccentricities
and long working distances. Photorefraction pattern ex-
tents can be larger or smaller than that predicted by the
paraxial theory, but more important, perhaps, is that a
wide range of intensity patterns can be seen, especially
near the dead zone. Instead of the traditional crescent
reflex, one may see split reflexes or central bright spots
in the photorefracted pupil. We have shown that these
patterns can be explained by the presence of monochro-
matic aberrations. For aberrations that are rotationally
symmetric the pattern will be the same for the source in
any meridian. On the other hand, for asymmetric aber-
rations, such as coma, the photorefracted pattern is highly
dependent on the position of the source.
The main effect of aberrations on the intensity profiles
occurs for paraxial refractive states when the eye is fo-
cused near the source position. In the paraxial model a
dead zone occurred for low refractive states for which no
intensity distribution appeared in the pupil. Thus, when
intensity patterns are observed in this range, they can
be attributed to the aberrations alone. Conversely, in a
clinical situation, patterns that appear because of aberra-
tions may be mistaken as being due to refractive errors.
As the refractive state increases, defocus becomes domi-
nant, and the photorefractive profiles take a more typi-
cal form.
Some differences are expected to occur between theo-
retical and experimental results for several reasons. Ac-
commodative lag for the 3 D stimulus will cause some
uncertainty of the precise paraxial refractive state for
each measurement. The extended source, multiple layer
reflections22 from the retina, and additional widening of
the blur on the retina resulting from light scattered at the
fundus23 will all tend to smooth out the intensity profiles
across the pupil. These effects are particularly visible
in the disparity between experimental and theoretical re-
sults in Fig. 6(c), where the experimental profile appears
smoother than predicted. These effects, in general, do
not appear large, given the good agreement between the-
ory and experiment, which indicates that the light in the
retina scatters at such wide angles that it can be sim-
ply modeled as a diffuse reflection from a single layer.
A wide-angled broadening of a point source of light may
occur as a result of multiple scatter in the fundus,24 and
this reflection could be modeled as a very wide Gaussian
base added to, but not convolved with, the retinal point
spread. This wide-angle scatter could be used to predict
the constant background observed in our measurements.
We expect that the aberrations in the eye are similar
for the left and right eyes of the human with respect
to the nasal, temporal, superior, and inferior positions.11
In an application of eccentric photorefraction in which
two eyes are measured simultaneously with the source
in the horizontal meridian, the source will be nasal for
one eye and temporal for the other. On the other hand,
if the source is vertical, then the source symmetry will
be the same for both eyes, either superior or inferior.
Therefore, if the aberrations present are asymmetric and
the same for both eyes, the measured photorefraction
patterns in both eyes will differ for a horizontal source
and will be similar for a vertical source when the eyes are
measured simultaneously. A measured difference when
the horizontal source is used might be misdiagnosed as
anisometropia when, in fact, it is due only to asymmetry
of the measurement.
Over the past decade screening of children’s refractive
errors has been conducted by photographic studies25–32
such as the Br ¨
uckner test, based on the principle of eccen-
tric photorefraction. In these studies the child is usually
focused on the camera, where the effects of the aberra-
tions are greatest. Little is known about the degree of
monochromatic aberrations present in childrens’ eyes, so
one should not ignore the fact that high aberrations may
exist. Most of these techniques measure both eyes simul-
taneously in the horizontal meridian. Caution must be
taken, then, to ensure that such an instrument does not
just detect an asymmetric aberration in the eyes.
To avoid these problems one should subtend a larger
angle between the source and the aperture (by increasing
eccentricity or decreasing the working distance). This,
of course, will increase the minimum refractive state that
can be detected by the instrument and decrease the test’s
sensitivity. One should also ensure that simultaneous
measurements of two eyes be made along the vertical
direction on the assumption that the aberrations are the
same in both eyes.
It should be noted that aberrations in the eye, while
they are important in instrumentation, such as pho-
torefraction and ophthalmoscopy, have not been shown
to be severely debilitating to the function of the eye.
Aberrations are generally greatest at the edges of a
Roorda et al. Vol. 12, No. 8/August 1995/ J. Opt. Soc. Am. A 1655
large pupil, where the eye tends to compensate for the
blur through the Stiles– Crawford effect or a smaller
pupil size.
As the refractive state increased, the directionality
needed for prediction of the experimental results de-
creased (Fig. 7). Thus the directionality of the retinal re-
flections decreases with increasing spot size on the retina.
As the existence of directionality is due to photorecep-
tor optics, the loss in directionality is probably due to
photoreceptor disarray over larger areas on the retina.
Some differences between the directionality with the PA1
and PS45 lenses can be seen in Fig. 7. These differences
could not be explained by differences in the spot size re-
sulting from the specific aberrations induced by the con-
tact lenses. Therefore the differences are probably due
to some variability in the experimental results. In gen-
eral, however, the trend of decreasing directionality with
increasing retinal spot size is found.
The degree of thresholding is dependent on the shape
of the intensity profile. If we assume that an edge will
be detected at a specific intensity, then one can make
the prediction of the threshold-corrected crescent edges
by simply increasing the baseline of the intensity plots.
The detection of an edge could also be attributed to the
rate of change of intensity of the profile, but this effect
has not been investigated.
With increasing computational speeds, these iterative
geometrical modeling techniques can be used to generate
reasonably accurate predictions of intensity distributions
for a number of different situations. The model is flexi-
ble enough that calculations can be made for many differ-
ent configurations of the eccentric photorefractor and the
influences of monochromatic aberrations in these situ-
ations, including extended sources, small camera aper-
tures (retinoscopy simulations), and multiple layers
contributing to reflections from the retina. We are ex-
ploring these applications at present.
APPENDIX A: FLOW CHART
FOR COMPUTER MODELING OF
PHOTOREFRACTION
*This is the retinal blur projected into object space.
ACKNOWLEDGMENTS
Thanks to Shannon Cerniuk, Linda Voisin, Peiliang
Zheng, and Chengwan Lu for their assistance in the
experiments. Support from Hughes Leitz Optical
Technologies and the Natural Sciences and Engineer-
ing Research Council (NSERC) Canada, is gratefully
acknowledged.
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