JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
VOLUME 52, NUMBER 9
SEPTEMBER 1962
Light Distribution in the Image Formed by the Living Human Eye*
GERALDWESTHEIMERANDFERGUSW. CAMPBELL
Sc/tool of Optometry, University of California, Berkeley 4, California
Physiological Laboratory, University of Cambridge, Cambridge, England
(Received February 20, 1962)
By photoelectric scanning, the light distribution was determined in the aerial ophthalmoscopic image of
a thin light filament viewed by an observer with an homatropinized eye. Light distributions were obtained
for various pupil sizes and- degrees of defocusing. Measurements were also obtained with bar and grating
objects.
To compute the line-spread function on the fundus, correction was made for the double passage of the
light through the optical system of the eye on the assumption that the spread in angular measure is the same
in both directions. The results may be considered to depict distributions which are possibly broader, but
certainly not narrower, than the real distributions in the retinal image. The line-spread function on the
fundus was determined to have a half-width at half-height of one minute of arc for an eye in best focus with
a 3-mm pupil, and this suggests that the point-spread function has half-width 0.66 min of arc as an upper
estimate.
plane C. Precautions are taken to ensure that the slit is
FOR imagery of objects varying in only one dimension, for example a grating or a straight edge, the
fundamental measure of the quality of an optical system
is the distribution of light across the image of a line
object of infinitesimal width-the line-spread function.
1
parallel
The images of all such objects may be considered as the
photomultiplier tube is positioned behind the slit, and
its output is displayed on an oscilloscopeor pen recorder.
Care is taken with the signal amplification so that a
linear relation exists between light flux through the slit
and recorder deflection. It is necessary to screen stray
light in the apparatus by blackening all surfaces at
which unwanted reflections may occur and by using
summation of the images of all the lines making up the
objects, providing certain conditions of additivity hold.
Direct measurements of the line-spread function of
the human eye were made by Flamant.' Her results
have been subjected
to the aerial image of the filament. A 931 A
light baffles.
to criticism 2 on the basis of com-
A dim fixation mark is provided to permit the subject
parison with experiments on excised mammalian eyes,
but it will be shown that the criticism is not justified.
Yet the long photographic exposure and the multistage
optical system for illuminating the retina employed by
Flamant suggest that a more rapid and optically
simpler approach to determine the line-spread function
1
S
of the human eye would be welcome. The following
experiment was carried out with this aim.
EXPERIMENTAL PROCEDURE
A vertical incandescent tungsten filament S (Fig. 1)
is placed 80 cm in front of the eye. Its width subtends
an angle of 30 sec of arc at the center of the entrance
pupil of the eye, and its vertical extent is at least 20.
The eye is homatropinized and an artificial pupil (A.P.)
carefully centered on the dilated natural pupil.
The image of the filament is formed by the optical
system of the eye on the eye's fundus; an auxiliary lens
L is usually necessary to ensure optical conjugacy of
the filament and the fundus. Light reflected from the
fundus in turn forms an aerial image in C, a plane also
conjugate to the fundus. A beam-splitting pellicle P
serves to separate the directions of the ingoing and outgoing beams. A narrow vertical slit, its width subtending
15 sec of arc at the eye, and its length 10, is placed in
A preliminiary report of this research was read before the Fall
Meeting of the Optical Society of America, Los Angeles, October,
1961 [J. Opt. Soc. Am. 51, 1463 (1961)].
1 F. Flamant, Rev. optique 34, 433 (1955).
2 D. W. DeMott, J. Opt. Soc. Am. 49, 571 (1959).
*
I
C.
A
iIC
* PHOTOCELL
0
.P
SLIT
FIG. 1. Schematic diagram of apparatus. Incandescent filament
S is imaged by auxiliary lens L and the optics of the eye. Light
reflected from the fundus forms an aerial image at C. A narrow
slit and a photomultiplier serve to measure the light distribution
in plane C. S can be moved in the direction indicated. A.P.,
artificial pupil; P, beam-splitting pellicle.
1040
September1962
LIGHT DISTRIBUTION
IN IMAGE FORMED BY LIVING EYE 1041
PUPIL DA.6MM
to maintain steady fixation. The filament is now moved
by a motor-driven device at a constant speed in a plane
normal to the subject's line of sight. In this manner the
image of the filament sweeps at constant
speed across
the area of the fundus which has the slit as its geometrical image. The output of the photomultiplier, since it
is a linear function of the light falling on its cathode, is
then a measure of the light distribution in the aerial
image of the filament's image on the fundus. Displacement of the filament is expressed in minutes of arc
subtended at the center of the entrance pupil of the
eye in object space.
The method measures changes in the light reflected
by a thin, long fundal region as the line image traverses
it. Measurements are unaffected by any of the normal,
small fixation eye movements that occur in the few
seconds taken for a traverse. Artifacts due to large eye
movements or blinks are easily recognized.
To assay the effect of defocusing on the light spread,
light distributions are obtained for a range of auxiliary
lenses. The lens yielding the narrowest distribution is
then used in further experiments with black bar and
grating targets seen against a uniformly illuminated
background. For this purpose a uniform broad ribbon
filament is imaged in plane S by a high-quality wideaperture lens, and the grating and bar targets, photographically reproduced on transparent glass plates, are
mounted to move in the same manner as the line filament in the original experiment. It was ascertained that
the diffraction image in the plane S of each point on the
ribbon filament is small compared with the details to
be resolved in the images of the grating or bar; problems
of lack of incoherence of radiation do not, therefore,
arise in any of the experiments described in this paper.
Gratings of 1:1 light/dark ratio with square waveforms
were used within the range of 2.3 min of arc per halfcycle and 7.2 min of arc per half-cycle. The single black
bars range in width from 1 to 20 min of arc.
0*O-
0-5o
-I
OD
-12
-8
4
ARC
a
12
The effective wavelength band of the radiation used
in this experiment is determined by the emission
characteristics of the filament and the sensitivity of the
photomultiplier. An experimental determination of the
joint effect of these two factors was made. Figure 2
shows the output of the photomultiplier tube used
when the tungsten filament under the operating conditions in the experiment was scanned through the wavelength range of a grating spectroscope with nonselective
spectral transmittance.
RESULTS
Measurements
were obtained
in one eye of each of
four subjects in the age range of 21-33 years. None of
the eyes had any appreciable astigmatism.
Figure 3 shows the light distribution in the aerial
CD
w
'>
0
OF
FIG. 3. Light distribution in the aerial images when a line
filament is presented to the homatropinized eye of subject JK
with a 6-mm artificial pupil for a variety of auxiliary lens powers.
Ordinates: relative intensity.
1.0
2
-4
MINUTES
0.5
w
image of the line filament for the homatropinized
.-
of one observer with a 6-mm artificial pupil and for a
wJ
variety
0
0
200
400
600
800
WAVELENGTH IN m
F'1IG.2. Effective wavelength band of the radiation
used in the experiment.
1000
of auxiliary
eye
lens powers. Figure 4 shows a
similar group of findings with a 3-mm artificial pupil.
Displacement in the ordinate indicates linear change in
illuminance at the photocell. In each curve the ordinatehas been adjusted so that the areas under all curves
are equal.
An indication of the light spread in the aerial image
is the half-width at half-height of the distribution, i.e.,
G. WESTHEIMER
1042
AND F. W. CAMPBELL
Vol. 52
half the distance, in angular measure at the center of
the entrance pupil of the eye, between the two points
at which the light intensity has fallen to half the peak
value. This is plotted as a function of auxiliary lens
power in Fig. 5. It may be seen that the light spread is
somewhat less for a 3-mm pupil than for a 6-mm pupil,
even in best focus, and that the image quality deteriorates to a greater extent on defocusing with a 6-mm
pupil than with a 3-mm pupil. These findings are in
accord with the known characteristics of the eye.
Figure 6 shows the output of the photomultiplier
behind the slit when gratings of square waveforms with
various periods were moved at constant speed in plane
HALF- 6
WI DTH
MIN
4
* 3MM
2
o 6MM
DIOPTERS
FIG. 5. Half-widths at half-height of light distributions
in Figs. 3 and 4.
and a red-sensitive photomultiplier, we were able to
obtain light distributions for wavelengths in the vicinity
of 700 myt. Compared with a similar one for the previously described spectral sensitivity, the long wavelength distribution appears to come to a near-zero level
05)
more slowly.
DISCUSSION
-0-5D
The principle of our experiment may be expressed as
follows: The light from the line filament is spread across
the retina, and it is this spread function which we wish
to determine. Since this is not accessible in the intact
eye, we resort to measuring the light spread in the aerial
image of the fundal image. This means that two stages
of optical imagery are produced by the same optical
system; for the second passage the system is acting in
-I-50
-12
4
4
0
-4
MINUTESOf ARC
8
12
FIG. 4. Light distribution in the aerial images when a line
filament is presented to the homatropinized eye of subject JK
with a 3-mm artificial pupil for a variety of auxiliary lens powers.
Ordinates: relative intensity.
S. The auxiliary lens which gave the narrowest linespread function was used in this and the succeeding
experiment. Each record also shows the change in
photomultiplier output when there was a blink.
Finally, we present in Fig. 7 measurements of the
change in light intensity as the images of single black
bars against a uniform light background pass across
the fundus strip sampled by the photomultiplier. For
this experiment a 5-mm artificial pupil was used.
An attempt to determine to what extent these measurements vary with wavelength was only partially
successful. Using a broad ribbon filament as a target
(a)
(b)
(c)
FIG. 6. Light disthe
tribution
in
aerial images when
gratings of various
periods are presented
to the eye. Auxiliary
lens giving best focus
and 5-mm artificial
pupil were used.
Each record shows
an artifact due to
blink. From top to
bottom: (a) 2.3, (b)
3.2, (c) 5.2, and (d)
7.2 min of arc per
half-cycle of grating.
September1962
LIGHT DISTRIBUTION
IN IMAGE FORMED BY LIVING EYE 1043
say, the retina. The effects of neglecting this aspect are
most likely to overestimate the width of the spread
function on the retina. Many associated questions of,
for example, the influence of retinal receptor structure
on the spread measured by us, or, inversely, of the
influence of receptor structure on the effective light
energy in each receptor in the region of the spread, or of
the Stiles-Crawford effect, are left for further
investigation.
If each strip of the image I' is imaged in turn in I" in
the same manner as the original object 0 is imaged at
I', the spread at I" is the self-convolution of the spread
function at I'. The mathematical procedure for determining a function f(x) whose self-convolution is F(x) is
to take the Fourier transform of F(x), take its square
root, and then obtain the inverse Fourier transform.
Certain difficulties may arise due to the ambiguity of
the sign of the square root. In our case we are dealing
(a)
(b)
FIG. 7. Dips in light
distribution in aerial images
of fundus as black bars of
various widths pass across
fundus strip sampled by
slit and photomultiplier.
Auxiliary lens giving best
focus and 5-mm artificial
pupil were used. From top
to bottom: records for bars
of width (a) 1.0, (b) 1.4,
(c) 2.4, (d) 3.5, (e) 7.5, and
(f) 21 min of arc, each with
its base line.
(c)
(d)
wo.
3-MM
JK
.
PUPIL
-
(f
.4
the reverse direction. Figure 8 illustrates the two stages,
but for convenience of discussion the second stage has
been "unfolded." In reality a diffuse reflection occurs
at I'.
.
I
The assumptions underlying our procedure for computing the spread at I' from the measured spread at I"
are these:
(1) The optical system of the eye is reversible, i.e.,
the spread in adjusted angular measure going from 0 to
I' is the same as that going from I' to I".
(2) The receiving screen does not contribute a
further spread of the image. Our experiment does not
provide by itself any information about the level of the
fundus at which reflection occurs, nor of what the
special properties of internal light scatter may be of,
o
-:I"=
I'
FIG. 8. Diagram to illustrate the principle of the experiment and
the method used for analysis. I' is the image of line object 0 as
formed by the optical system of the eye and the eye's fundus.
Light reflected from the fundus forms an aerial image I" by reverse
passage through the optical system of eye. The reverse passage is
shown "unfolded."
o
. .. .: .. ..
2
MINUTES
ARC
a
:;--..-......,..
to
12
FIG. 9. Measured light distribution in best-focus aerial image
and computed light distribution on fundus for a thin line object.
Subject JK, 3-mm artificial pupil.
with a symmetrical function in which the use of the
cosine Fourier transform suffices. If the latter decreases
monotonically without reaching zero, the question of
the sign of the square root need not be raised. In out-offocus imagery this is not the case, and the theory in this
simple form is not directly applicable. Allowance may
also be made in the Fourier domain for the finite width
of the filament and of the slit. In our experiment these
two dimensions were so small, compared with the overall width of the spread function, that such a correction
did not appear necessary.
By means of this theory, computation of that spread
function which, when convoluted with itself, yields the
narrowest light distributions in Figs. 3 and 4 was carried
out with digital computers (EDSAC II of the Mathematical Laboratory, Cambridge University and the
IBM 704 of the Computing Center, University of
California, Berkeley). Figure 9 shows on the same graph
G. WESTHEIMER
1044
AND F. W. CAMPBELL
'Vol. 52
10o0-
6-MM
-
PUPIL
80
J.K.
----
.......
*
601
A.S.
FLAMANT
JK. 3MM
PS.
.
401
.6
301
INT
0
.4
2d
I-
z
15
w
.2
0~~~~~
10
8.c
2
6
MIN
a.;
8
6.0O
FIG. 10. Computed light distributions in the fundal image
thin line object for three eyes in best focus with a 6-mm artif
pupil.
0~*
one side of the best-focus aerial-image light distribut
with a 3-mm artificial pupil for subject JK and
computed spread on the fundus. In Fig. 10 are shc
computed light distributions on the fundus for t1
eyes in best focus with a 6-mm pupil.
The processed data in Fig. 9 are replotted in Fig.
with logarithmic coordinates for illuminance.
comparison, Flamant's data' are also shown. It is s
that a straight line fits the points. It yields, as Flam:ant
had also found, the equation
f(x)=
1
e . 7Ix
where f (x) is the illuminance at a distance x, in minu .tes
of arc, from the center of the distribution. Approximate
average values for the illuminance in the best-fo,cus
image formed by the living human eye of a lc)ng
luminous filament of infinitesimal width may, therefc)re,
be taken to be no more than the following:
Distance from the
center of line image
in minutes of arc
Relative
illuminance
0
1
3.2
6.5
1.0
0.5
0.1
0.01
0
\
4*0o
As discussed earlier, our experimental procedure is more
likely to exaggerate than to minimize the width of the
line-spread function.
Equation (1) cannot be regarded as a good description
of the line-spread function near x=0, because it has a
cusp there. Actually, the line-spread function may be
taken to be concave downwards at x= 0.
Of more general interest than the line-spread function
is the point-spread function, i.e., the light distribution,
3X0t
2-0t
0
I.St
l
*
-o
0
*c}
I I
2
|
w
~
~
~ ~ ~~~
-
4
MINUTES ARC
.
~~~~~~I
6
FIG. 11. Line-spread function for JK best-focus 3-mm pupil
data, plotted on logarithmic coordinates for illuminance. For
comparison, Flamant's data are also shown. Line is from an
empirical equation giving a good description of the spread function
exceptat x =0.
regarded here to be radially symmetrical, in the image
of a point source. This promises to be exceedingly
difficult to measure experimentally. On the other hand,
its computation from the line-spread function involves
the solution of the integral equation
f(x) = 2
f (r) (r2 - x2)-irdr,
(2)
where f(x) is the line-spread function in terms of x, the
distance from its center, and f(r) is the point-spread
function in terms of r, the radial distance from its
center. So far, this equation has not been solved for the
empirical data of Figs. 9 and 10. The problem of the
relationship between the point- and line-spread functions has, however, been studied theoretically by R. C.
Jones' for a number of analytical functions bearing a
close resemblance to our data. From his curves it would
appear that radially symmetrical point-spread functions, which when integrated in the manner of Eq. (2)
R. Clark Jones, Photo. Sci. Eng. 2, 198 (1958).
1
September1962
LIGHT DISTRIBUTION
IN IMAGE FORMED BY LIVING EYE 1045
give line-spread functions similar to those in our
Figs. 9 and 10, have half-widths equal to about of
those of the corresponding line-spread functions. Thus
for an eye with 3-mm pupil and in best focus, the pointspread function would have half-width at half-height
of about 0.66 min of arc in object space. Airy's disk in
such an eye would have half-width of about 0.32 min of
arc. The size of Airy's disk, however, varies inversely
with pupil diameter; our results with a 6-mm pupil
incidentally illustrate the well-known progressively
diminishing significance of diffraction as a limit to the
eye's performance as the pupil diameter is enlarged.
ACKNOWLEDGMENTS
We gratefully acknowledge the collaboration of
W. J. Kilgour who, assisted by a Durham Studentship
awarded by King's College, Cambridge, participated
in several stages of this study. H. F. P. Swinnerton-Dyer
prepared the program for EDSAC II. This research was
aided by Grant B-3154 from the National Institutes of
Health, U. S. Public Health Service, and by a contract
between the Office of Naval Research and the
University of California. Apparatus grants from the
Royal Society and the W. H. Ross Foundation are also
acknowledged.