fringes were produced on the retina directly. Threshold contrasts optical aberrations in the eye. (Received 12 January 1967)

Transcription

1 J. Phy8iol. (1967), 19, pp With 5 text-figure8 Printed in Great Britain VISUAL RESOLUTION WHEN LIGHT ENTERS THE EYE THROUGH DIFFERENT PARTS OF THE PUPIL BY DANIEL G. GREEN From the Department of Ophthalmology, University of Michigan, Ann Arbor, Michigan U.S.A. (Received 12 January 1967) SUMMARY 1. Threshold contrasts for resolution of sinusoidal gratings imaged on to the retina through a decentred 2 mm pupil were measured. 2. No loss in resolution was found when the pupil was decentred parallel to the lines of the gratings. A loss in resolution by a factor of 3 occurred when the pupil was decentred by 3 mm perpendicular to the lines of the gratings. 3. The effects of focus on the threshold contrast for a grating viewed through a centred and decentred pupil were used to show that at least a portion of the loss in resolution is due to optical aberrations. 4. Using a neon-helium gas laser as a coherent light source, interference fringes were produced on the retina directly. Threshold contrasts for resolution of the fringes were determined for different positions of entry of the beams of light through the pupil. When the Stiles-Crawford brightness effect was compensated for, no loss in resolution was found to occur for decentred entry of the beams. 5. It is concluded that the off-axis loss of visual acuity is wholly due to optical aberrations in the eye. 6. The ratios between the threshold contrasts for sinusoidal gratings and for interference fringes are used to calculate the optical transfer functions of the off-axis aberrations of the eye. INTRODUCTION Visual acuity through an artificial pupil placed close to the cornea is maximal when the aperture is over the approximate centre of the natural pupil. As the aperture is moved from the centre to the edge of the pupil, visual acuity decreases in the meridian at right angles to the meridian of displacement (Campbell, 1958). By dilating the pupil and using a grating 37-2

2 584 DANIEL G. GREEN test object, Campbell reported that visual acuity decreased by a factor of 8 when a 1 mm artificial pupil was displaced from the centre of the natural pupil by 4 mm. The effect is not due to simple spherical aberration since it cannot be corrected using spectacle lenses. However, a significant portion of the effect was shown by Campbell & Gregory (196) to be due to higher order optical aberrations. They concluded that only about 2% of the measured loss in acuity might be retinal in origin. It has been suggested that an explanation of the retinal component of the off-axis loss in acuity might involve the optical properties of the foveal cones. Campbell & Gregory (196) point out that the foveal cones are long, thin structures. The outer segments are about 4 It long and about 1 It wide, and the inner segments are about 3,u long and 2 It wide. Therefore, an obliquely incident ray of light could activate one or more additional receptors and this effective spread of light might in turn lead to a reduction in contrast and resolution. Campbell & Green (1965) describe an interferometric technique using a neon-helium gas laser which permits one to measure the resolving properties of the retina independent of the image-forming properties of the eyes' optics. In this paper, the results of applying this method to measure directly the retinal component to the off-axis visual acuity loss are described. In addition, by comparing these interferometric measurements with data obtained using sinusoidal gratings which are imaged on to the retina in the conventional way, it has been possible to estimate the optical transfer function of the off-axis aberrations of the eye. METHODS The methods were essentially those described by Campbell & Green (1965). In summary, the sinusoidal grating test target was generated on a cathode-ray oscilloscope (Tektronix, type 547) having a tube with a green (P 31) phosphor. The vertical axis of the oscilloscope was driven at 15 kc from an oscillator with a triangular output. The horizontal axis was driven by the oscilloscope's own time base at a frame frequency of 5 cls or higher. To form the grating patterns, a sinusoidal wave form was applied through a capacitor to the brightness control grid of the cathode-ray tube. The modulation depth of the grating pattern was varied by changing the magnitude of the voltage reaching the control grid of the cathode-ray tube. It was possible to run the screen at a luminance of up to 1 cd/m2. The contrast of the gratings is proportional to the voltage applied to the grid from zero to about -8 contrast. The subject bit on a bite bar and viewed the oscilloscope from 57 in. (144.8 cm) through a 2 mm artificial pupil placed close to the cornea of his right eye. About 3 min before the start of the experiment, the pupil of this eye had been dilated using a combination of 1 % cyclogyl and 1 % neo-synepherine instilled into the conjunctival sac. The subject adjusted the contrast until he was satisfied that the grating could just be resolved. The position in the natural pupil which optimized high-frequency resolution was taken as the effective centre of the pupil. All displacements of the artificial pupil were made with respect to this centred position.

3 OFF-AXIS VISUAL RESOLUTION 585 The interference fringes were formed directly on the retina using a Perkin-Elmer (type 11) gas laser. The beam from the end of the laser was divided into two portions using a beam-splitting cube. The beam reflected by the beam splitter was reflected a second time from a front surface mirror and directed so as to superimpose with the direct beam on the entrance pupil of a microscope objective. A second source of light from a tungsten filament was matched in colour to the laser light by placing a Wratten no. 29 filter in its beam. The light from this noncoherent source was divided into two parts by the beam splitter and mirror arrangement and combined with the coherent light from the gas laser. The separation of successive maxima in the intensity distribution on the retina can be expressed as a= A/a, (1) where a, is the visual angle between maxima expressed in radians, A is the wave-length of the light, and a is the separation of the images of the coherent source in air. In these experiments, A = 632 8m#t and a was determined by direct measurement. From the above formula it is possible to express the spatial frequency of the fringes on the retina in terms of the separation of the images f = 27-6a c/deg. (2) where a is expressed in millimetres. By placing his eye close to the microscope objective, the observer viewed the interference patterns. A polaroid was placed in the tungsten lamp beam at right angles to the angle of polarization of the laser beam. It was possible to vary the contrast and keep the mean luminance constant by rotating an analysing polaroid which was just behind the objective. The subject's head was fixed by a bite bar. Using a micrometer screw, the bite bar could be displaced horizontally with respect to the beams of light thereby changing the position in the pupil through which the light entered. RESULTS Resolution of sinusoidal gratings. Figure 1 illustrates the results of an experiment in which the subject viewed the grating patterns generated on the oscilloscope through a 2 mm artificial pupil which was decentred both perpendicular to and parallel to the lines of the grating. Contrast sensitivity, the reciprocal of threshold contrast, is plotted vertically on a log. scale. Contrast is defined in the usual way as Imax - Imin/Imax + Imin- Spatial frequency, which is defined as the reciprocal of the angular distance between successive maxima in the sinusoidal intensity distribution, is plotted horizontally on a log. scale. Smooth curves have been drawn through the results. Comparing the open and closed circles it is found, as reported by Campbell (1958), that there is no loss of acuity when the pupil is decentred by 2-5 mm parallel to the lines of the grating. In making measurements with decentred pupils, the Stiles-Crawford brightness effect (Stiles & Crawford, 1933) has been compensated for with neutral density filters. In addition, primary spherical aberrations have been corrected by using in each case the spectacle lens which optimized high-frequency resolution. The other data points are for the 2 mm pupil shifted temporally perpendicular to the lines of the grating. There is a progressive loss in visual acuity as the pupil is shifted toward the periphery. Extrapolating the measurements to the unity contrast line (sensitivity = 1), it is estimated

4 586 DANIEL G. GREEN that there would be a factor of 3 loss in visual acuity when viewing gratings made up of black and white bars for the pupil displaced by 3 mm from the centre of the natural pupil. The loss in resolution is dependent on the frequency of the grating with the contrast sensitivity measurements showing little or no sign of an off-axis loss at low spatial frequencies. 1 c) o x - -C a). 2 1 OQ vn I I I I I I I II I I I I li I I I Spatial frequency (c/deg.) Fig. 1. Threshold contrast as a function of spatial frequency for centred and decentred 2 mm artificial pupils. The natural pupil was dilated and accommodation was paralysed. For each position of entry through the pupil, the Stiles-Crawford effect has been compensated for and the spectacle lens which optimized high-frequency resolution has been placed before the eye. The points are each the average of three measurements. Thresholds obtained when the pupil was centred are shown as filled circles. The open circles were obtained when the pupil was decentred parallel to the vertical bars of the gratings by 2-5 mm. The other measurements are for the pupil decentred perpendicular to the bars of the gratings by 1, 2, and 3 mm. Is this loss in resolution due to optical aberrations or to a retinal directional acuity effect? The experiment illustrated in Fig. 2 has been designed to give a partial answer to this question. A 15 c/deg. grating was viewed through both a centred and a decentred artificial pupil. The effects of changes in focus on the threshold contrast for resolving the grating have

5 OFF-AXIS VISUAL RESOLUTION 587 been determined. Straight lines have been drawn through the negative and positive lens-power portions of the results. It should be noted that if the procedure of Green & Campbell (1965) is followed and the in-focus lens power is defined as the intersection of the lines, then there is a + 75 D x ~~~~~~~ / x x 2 - x~~~~~~~ X -2 / Lens power (D) Fig. 2. Effect of positive and negative defocus on contrast sensitivity measured with a 2 mm pupil at a spatial frequency of 15 c/deg. The crosses are for the pupil centred and the circles for the pupil decentred temporally by 2 mm. Each point is the average of three measurements. change in focus between the centre and the periphery of the pupil. If the off-axis loss in acuity was solely a retinal effect, then defocus, which simply reduces the contrast of the sinusoidal patterns formed on the retina, should be compensated for by increasing the contrast of the grating object by a constant ratio independent of whether the pupil is centred or not. One would expect the straight lines drawn through the threshold contrasts of Fig. 2 to be parallel. This is by no means so, since the slopes of the straight portions of the curves depend on the position of the artificial pupil with respect to the natural pupil, and we are led to the inference that at least a portion of the effect is optical. Resolution of interferencefringes. The threshold contrast for just resolving the interference fringes has been measured for entry through different parts of the pupil. The beams were displaced horizontally in the pupil plane. In each experiment, a neutral density filter was placed in front of the viewing objective so as to compensate for the subject's Stiles-Crawford

6 588 DANIEL G. GREEN brightness effect. Figure 3 shows the graph of the Stiles-Crawford effect for my own eye whichwas used in determining the densities of the compensating filters. The interference fringes are formed by two beams of equal intensity which pass through different parts of the pupil and overlap on the retina, -5 i -1 M l l l l l Nasal Temporal Point of entry (mm) Fig. 3. The luminance efficiency of a bundle of rays entering the pupil of the subject at various points. The points are the average of two measurements. The empirical equation used by Stiles & Crawford (1933), n/no = 1 -O(T-*4)1 is shown as a smooth line in the figure. forming fringes. Because of the Stiles-Crawford effect, the two beams do not have equal effectivity in stimulating the receptors. This latter effect can be compensated for by treating the Stiles-Crawford effect as if it were an apodization (Metcalf, 1965) in the plane of the pupil, that is to say, as if the transmission through the pupil varied with radius. It is well known that the contrast (C) of interference fringes formed by coherent beams of unequal intensity is given by the formula C = 21(Il -1), (3) '1 + '2 where Ii and 12 are the intensities of the beams (Born & Wolf, 1964). Figure 4 shows the threshold contrast as a function of the position of entry through the pupil. The measured thresholds have been corrected using the above formula and the Stiles-Crawford data shown in Fig. 3. Thresholds have been determined at three spatial frequencies. These frequencies correspond to a separation of.5, 11, and 1*7 mm between the interfering beams in air. Correction of the measured contrasts using the above formula

7 OFF-AXIS VISUAL RESOLUTION 589 is significant only for the high-frequency data. Even in the 1-7 mm case, the correction amounts to only a 2 % reduction in the threshold contrast measured when the beams pass through the extreme edge of the pupil. These data show that there is no off-axis visual acuity loss for the resolution of the interference fringes. It is therefore concluded that the off-axis loss in resolution shown in Fig. 1 is wholly due to optical aberrations. In fact, it 2 = 1 n~ 4' o 14 c/deg o 5 *,I,, L A ^ 5, 3 c/deg * 2 - X- X X X 47 c/deg -o 1_ Q I i I I I I I Nasal Temporal Point of entry (mm) Fig. 4. Dependency of threshold contrast for resolving interference fringes on the position of entry in the pupil of the interfering beams. The position of the beams is defined by a point midway between the two beams. The point of entry of the beams was decentred perpendicular to the interference fringes. Correction for the Stiles-Crawford effect has been made by placing neutral filters in the beams and using equation (3) to calculate the contrast of the fringes on the retina. Measurements have been made at three spatial frequencies:, 14 c/deg.; A, 3 c/deg.; and x, 47 c/deg. The points are each the average of three determinations. is possible to demonstrate that the loss in resolution using ordinary incoherent light simply by sufficiently restricting the width of the entering beam and correcting for focus and brightness changes through different parts of the pupil. When a vertical grating was viewed through a 3 mm pin hole pupil, visual acuity was found to change by a factor of not more than 1-5 when the pin hole was displaced horizontally from the centre to the edge of the dilated natural pupil. Transfer function of the off-axis aberrations. Using the technique of Campbell & Green (1965), it is possible to determine the optical transfer functions of the off-axis aberrations. This determination rests on the assumption that the threshold contrast for a sinusoidal fringe pattern

8 59 DANIEL G. GREEN formed by interference is the same as that for a similar grating imaged on to the retina by the optics. Since there is no decrease in contrast threshold for the interference fringes with eccentric position of entry through the pupil, to calculate the optical transfer function it is only necessary to take the ratio between the contrast sensitivities measured in Fig. 1 and the 1. 'O8 - o2~ ~~~~. I,, Normalized spatial frequency Fig. 5. Transfer functions of the off-axis optics. Spatial frequency has been normalized by the diffraction limited cut-off frequency for a 2 mm pupil using A = 54 m,u, the peak in luminance of the oscilloscope phosphor. The dasheddotted line indicates the transfer function for a diffraction limited optical system. The dotted line is the transfer function of an optical system with small amounts of third and fifth order spherical aberrations (from Fig. 6-6 (a) of O'Neill, 1963). The dashed line is the transfer function of an optical system with small amounts of comatic aberration (from Fig. 34 of Barakat & Houston, 1966). The circles are from the measurements with a centred 2 mm pupil; the squares, for 1 mm of decentring of the pupil; the triangles, for 2 mm of decentring. The smooth curves have been drawn by eye through the calculated points. contrast sensitivities for interference fringes plotted in Fig. 5 of Campbell & Green (1965). The results of this calculation are plotted in Fig. 5. The transfer functions have been determined for a centred 2 mm pupil and for the pupil decentred by 1 and 2 mm. The shape of these transfer functions is similar to the transfer functions calculated from diffraction theory for optical systems having higher order aberrations (Barakat & Houston, 1966).

9 OFF-AXIS VISUAL RESOLUTION 591 DISCUSSION When Campbell (1958) discovered that the off-axis visual acuity loss could not be corrected with spectacle lenses, he apparently thought that the foveal cones might exhibit a directional acuity effect similar to the Stiles-Crawford effect. By measuring acuity using Fraunhofer diffraction images, Campbell & Gregory (196) established that the loss in acuity was due mainly to aberrations in the eye. There remained, however, a portion of the effect which could still be due to directional acuity of the photoreceptors. The basis of retinal directional acuity was the idea that, since foveal cones are long and thin, if only a portion of an obliquely incident ray of light was absorbed in the receptor then a single ray could stimulate one or more adjacent receptors. Several authors have proposed theoretical explanations of the Stiles-Crawford effect which incorporate the fact that the refractive index of the receptors is higher than the surrounding tissue. This being the case, the cone-shaped receptors would tend to funnel light rays which are incident parallel to the axis of the receptor and to leak increasing amounts of light as the angle of incidence of the rays is increased (Wright & Nelson, 1936; O'Brien, 1946). If one assumes that the light which leaks out of the receptor can stimulate other receptors, then a loss of visual acuity for obliquely incident light is to be expected. Dunnewold (1964) has used measurements of the off-axis acuity loss to estimate the amount of light leaking out of the receptors. Using improved techniques, we now have good evidence to suggest that the loss of acuity for obliquely incident rays is not due to directional acuity of the receptors but is due to aberrations in the dioptric system of the eye. It is not clear whether this means (a) that the portion of an oblique ray which does not bleach photo pigments is not leaked out of the receptors or (b) that light is leaked out of the receptor but, for some reason, it is unable to excite adjacent receptors. What is the nature of the aberrations which cause the loss in resolution for off-axis imagery? It is well known that the eye has significant amounts of spherical aberration (Koomen, Tousey & Scolnik, 1949; Ivanoff, 1956). The transfer function measured for a 2 mm centred pupil fits that of an optical system having small amounts of third and fifth order spherical aberrations (see Fig. 5). A lens system with spherical aberration usually exhibits the aberration known as coma for imagery off of the optical axis. By examining the distortions of the shadow fringes formed on the retina when a fine Ronchi grating is placed close to the pupil and a distant point source is viewed through a + 5D lens, it is possible to observe the aberrations in one's own eye. Various aberrations cause characteristic types of

10 592 DANIEL G. GREEN fringe patterns (Ronchi, 1964). Using a 2 mm centred pupil, the author observed in his own eye the characteristic curvature of the lines of the fringe pattern which is found in optical systems having small amounts of positive spherical aberration. When the pupil was decentred distortions characteristic of comatic aberrations were observed. The transfer function of an optical system having a small amount of comatic aberration is plotted in Fig. 5 for the situation in which the bars of the grating object are perpendicular to the comatic flare. There is rather good agreement between the theoretical transfer function for a system with coma and the measured contrast ratio for the pupil decentred by 1 mm. Thus, comatic aberration probably contributes significantly to cause the loss in resolution shown in Fig. 1. Enoch (1957, 1959) has suggested that there is a relation between amblyopia and the Stiles-Crawford effect. He argues that if the orientation of the retinal receptors is disturbed, then the Stiles-Crawford maximum will not be directed at the centre of the pupil and for various reasons, such as a loss of brightness and greater sensitivity to scattered light, there might be a concomitant loss of visual acuity. The publication by Campbell of a possible directional acuity effect gave support to the idea that receptor tilt might be a significant cause of amblyopia. These ideas probably will have to be modified in light of these new findings which suggest that there is no loss in acuity other than that due to a loss in brightness when the rays of light forming the image on the retina are oblique with respect to the retinal photoreceptors. What may be of some interest to those concerned with the nature of amblyopia is the fact that there exist optical aberrations which cannot be corrected with the ordinary spheres and cylinders. Whether or not any clinical cases of amblyopia are due to higher order aberrations remains to be determined. The possibility does remain that such aberrations could be corrected with suitable special lenses. This investigation was supported in part by National Institutes of Health Grant FR My thanks are due to the Radar and Optics Division of The Institute of Science and Technology which was kind enough to loan me one of their lasers. It is my pleasure to acknowledge the constructive advice of Dr M. Alpern and Dr F. W. Campbell. REFERENCES BARAKAT, R. & HOUSTON, A. (1966). The aberrations of non-rotationally symmetric systems and their diffraction effects. Optica Acta 13, 1-3. BORN, M. & WOLF, E. (1964). Principles of OptiCs, 2nd edn., p. 55. Pergamon Press. CAMPBELL, F. W. (1958). A retinal acuity direction effect. J. Physiol. 143, P. CAMPBELL, F. W. & GREEN, D. G. (1965). Optical and retinal factors affecting visual resolution. J. Physiol. 181, CAMPBELL, F. W. & GREGORY, A. H. (196). The spatial resolving power of the human retina with oblique incidence. J. opt. Soc. Am. 5, 831.

11 OFF-AXIS VISUAL RESOLUTION 593 DUNNEWOLD, C. J. W. (1964). On the Campbell and Stiles-Crawford effects and their clinical importance. Institute for Perception R VO- TNO, Soesterberg, the Netherlands. ENOCH, J. M. (1957). Amblyopia and the Stiles-Crawford effect. Am. J. Optom. 34, ENOCH, J. M. (1959). Further studies on the relationship between amblyopia and the Stiles-Crawford effect. Am. J. Optom. 36, GREEN, D. G. & CAMPBELL, F. W. (1965). Effect of focus on the visual response to a sinusoidally modulated spatial stimulus. J. opt. Soc. Am. 55, IVANOFF, A. (1956). About the spherical aberration of the eye. J. opt. Soc. Am. 46, KOOMEN, M., TOUSEY, R. & SCOLNIK, R. (1949). The spherical aberration of the eye. J. opt. Soc. Am. 39, METCALF, H. (1965). Stiles-Crawford apodization. J. opt. Soc. Am. 55, O'BRIEN, B. (1946). A theory of the Stiles-Crawford effect. J. opt. Soc. Am. 36, O'NEILL, E. L. (1963). Introduction to Statistical Optics, p. 98. Addison:Wesley Publishing Co. RONCHI, V. (1964). Forty years of history of a grating interferometer. Appl. Opt. 4, STILES, W. S. & CRAWFORD, B. H. (1933). The luminous efficiency of rays entering the pupil at different points. Proc. R. Soc. B 112, WRIGHT, W. D. & NELSON, J. H. (1936). The relation between the apparent intensity of a beam of light and the angle at which it strikes the retina. Proc. phys. Soc. 48,