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Research Article  |   January 2011
Absence of an integrated Stiles–Crawford function for coherent light
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Journal of Vision January 2011, Vol.11, 19. doi:10.1167/11.1.19
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      Brian Vohnsen, Diego Rativa; Absence of an integrated Stiles–Crawford function for coherent light. Journal of Vision 2011;11(1):19. doi: 10.1167/11.1.19.

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Abstract

The Stiles–Crawford effect that relates visibility to pupil point is typically expressed by a Gaussian function at any given wavelength of illumination. The pupil location of the maximum and the width of this function refer, respectively, to the pointing and waveguide properties of individual cone photoreceptors. In vision simulations, the function is integrated across the pupil when estimating effective retinal images, but the validity of this approach has still not been unequivocally confirmed. Indeed, aberrations and coherence properties may significantly alter not only the amplitude but also the phase distribution of the light at the retina in a way that differs fundamentally from that of the Maxwellian illumination configuration used when characterizing the effect. Here, we report on an experimental comparison of the traditionally determined Stiles–Crawford function and the equivalent for annular and half-annular apertures using extended highly coherent and incoherent sources. We show that an integrated Stiles–Crawford function is absent for coherent light but remains valid for highly incoherent light at the pupil. The results are supported by numerical evidence for coherent light propagation and are in agreement with a light-coupling understanding of retina photoreceptor waveguides.

Introduction
The Stiles–Crawford effect of its first kind (SCE-I) describes a marked reduction in visibility of a narrow Maxwellian beam of light incident near the pupil rim (Stiles & Crawford, 1933). The effect depends on retinal eccentricity (Westheimer, 1967) and illumination spectrum (Stiles, 1937) and originates in a reduced light-coupling efficiency of obliquely incident light to individual photoreceptors (Snyder & Pask, 1973; Vohnsen, Iglesias, & Artal, 2005). The functional relationship between visibility, η, and pupil point, r, is commonly expressed by a Gaussian function with respect to the point of maximum visibility (typically near the pupil center): 
η = 10 ρ r 2 ,
(1)
where ρ is the characteristic directionality factor (dependent on retinal eccentricity and illumination spectrum). 
It is tempting to apply this characteristic SCE-I function to the case of vision with an unobstructed natural pupil to predict effective retinal images in vision design (Noorby, Piers, Campbell, & van der Morren, 2007). To accomplish this, Equation 1 has typically been integrated across the pupil to define a reduced effective pupil size (Drum, 1975; Enoch, 1958; Enoch & Lakshminarayanan, 2009; Ercoles, Ronchi, & Toraldo di Francia, 1956; Moon & Spencer, 1944; Vohnsen, 2009). Whether this approach is generally valid remains, however, questionable. 
Additivity of the SCE-I was examined by Drum when viewing a Fraunhofer image of a diffuser exposed to quasi-monochromatic light (578-nm wavelength) from a mercury lamp (Drum, 1975) confirming the procedure of integrating Equation 1 across the pupil. Nonetheless, an integrated SCE-I function is likely to depend on both ocular aberrations and source coherence that can alter the amplitude and phase distribution of light at the retina thereby influencing the fraction of waveguided light in the effective retinal image (Vohnsen, 2007). To examine this in more detail, we devised an experiment similar to the one of Drum using different-sized annular and half-annular apertures imaged onto the eye pupil to characterize the SCE-I for extended illumination. A laser source was chosen to ensure not only a high degree of monochromaticity in the illumination (which was also the case in the experiment of Drum, 1975) but also a high spatial and temporal coherence across the pupil image of the aperture. The experiment was then repeated using broadband light from an incoherent source and the measured directionality factors were compared with those found in a separate experiment using a traditional SCE-I Maxwellian illumination. Finally, the configuration was modeled numerically to visualize the mechanisms involved. 
Methods
The experimental setup used with annular and half-annular apertures is shown in Figure 1. In the case of coherent illumination, the beam from a 5-mW HeNe laser (632.8 nm wavelength) is split in two, expanded, and both samples of the beam pass through an attenuating paper diffuser (nominal paper thickness is 104 ± 2 μm) rotating at a speed of Ω = 80 rpm, which is sufficient to remove the visual impact of speckles while scattering the beam into an angular spectrum. It should be stressed, however, that at any instant the incident light distribution is coherent and thus subject to both amplitude and phase addition at the retina. In the case of incoherent illumination, a fiber-guided tungsten–halogen lamp was used together with a broadband red diffusion filter (Roscolux #120 with a 12% transmission peak at 660 nm) instead of the laser source in the setup. The red-colored incoherent light incident on the rotating paper diffuser improved measurement accuracy in comparison to white light that underwent a slight hue change caused by the spectral properties of the beam splitters used. 
Figure 1
 
Schematic (not to scale) of the setup used to measure visibility with annular and half-annular apertures (probe light) whose images are centered at the pupil point of maximum SCE-I visibility where also a brightness-tunable Maxwellian reference is incident (reference light). The large distance of the diffuser from the eye makes the use of a lens in the reference branch prior to the iris unnecessary.
Figure 1
 
Schematic (not to scale) of the setup used to measure visibility with annular and half-annular apertures (probe light) whose images are centered at the pupil point of maximum SCE-I visibility where also a brightness-tunable Maxwellian reference is incident (reference light). The large distance of the diffuser from the eye makes the use of a lens in the reference branch prior to the iris unnecessary.
A sample of the beam (from the coherent laser or from the incoherent lamp) scattered by the diffuser is brought to a focus in Maxwellian illumination at the pupil point of maximum visibility (reference) whereas another sample of the beam passes through an annular (or half-annular) aperture that is imaged into a Maxwellian ring (or half-ring) in the pupil plane (probe). The iris sets the viewing angle seen by the subject and was kept constant with a 1.5° visual field centered at the observer's fovea. The field appears highly uniform because of the diffuser used in the illumination path (i.e., each point on the annular aperture contributes to the entire visual field). The visibility of the probe light for each aperture is compared to the visibility of the reference light with the subject adjusting the brightness of the Maxwellian reference using a variable neutral density filter. To facilitate the comparison, the subject blocks in turn the reference and the probe light with a mechanical shutter. The viewing time of either the probe or reference light was typically a few seconds, as decided by the subject, and could be repeated until equal visibility of the two illumination pathways was established by adjusting the variable filter. At no time was the subject simultaneously exposed to both lights. 
Seven equal-area annular apertures with a central diameter of 1 to 7 mm (in increments of 1 mm) when projected onto the pupil plane were used. In addition, another set of half-annular apertures was prepared and used for comparison. The area of each projected aperture in the pupil plane was 2.50 mm2 (annular) and 1.25 mm2 (half-annular), respectively. Apertures were printed 2.5 times larger on film with the difference in transmission between transparent and opaque regions being superior to 103. The transmission of each aperture was calibrated a priori to reduce sources of error by measuring its transmission in the eye pupil plane through the 2.5 times reduction telescope of the setup. The distance from the diffuser to the eye pupil was 1.4 m for the probe and 1.5 m for the reference. Typical examples of light distributions recorded with a CCD camera in the pupil plane and schematically in the retinal plane (here obtained by means of an f 25-mm achromatic lens instead of the eye) are shown in Figure 2. To highlight the influence of speckles, the diffuser was not rotated while these stationary speckle images were recorded. When rotating, the light distribution appeared uniform to the subject. 
Figure 2
 
CCD images (pixel size = 4.65 μm) obtained with 3-mm diameter (a, b) annular and (c, d) half-annular apertures in the (a, c) pupil plane and (b, d) retinal plane. An f 25-mm achromatic lens was used in place of the eye to record the retina-like images. To highlight the appearance of speckles, the rotation of the diffuser was not activated during the 133-ms exposure time.
Figure 2
 
CCD images (pixel size = 4.65 μm) obtained with 3-mm diameter (a, b) annular and (c, d) half-annular apertures in the (a, c) pupil plane and (b, d) retinal plane. An f 25-mm achromatic lens was used in place of the eye to record the retina-like images. To highlight the appearance of speckles, the rotation of the diffuser was not activated during the 133-ms exposure time.
The subject was presented to each aperture in a random sequence ensuring that a total of 5 measurements were made with each annular and half-annular aperture. Each series of 7 apertures was fully completed before being repeated. The average light power from the annular probe light reaching the eye was 0.5 nW for each aperture. The corresponding retinal illuminance with an annular aperture was approximately 2.2 log Trolands. The right eye of each author (BV: age 40 and DR: age 29) having normal vision was examined with dilated pupil (tropicamide 1%) to avoid clipping of illumination from large apertures. Ethical approval for the research was obtained from the UCD Human Research Ethics Committee. 
Experimental results
Coherent light
The SCE-I was examined for both subjects when viewing through the full and half-annular aperture images at the pupil with the HeNe laser used as source. The results obtained are shown in Figure 3
Figure 3
 
Fovea visibility versus pupil point measured using coherent light for BV and DR: with annular apertures (blue, triangles), with half-annular apertures (red, circles), and with the conventional, separately obtained, SCE-I Maxwellian illumination configuration (black, squares). The symbols indicate the average value and ±1.0 standard deviation for each measurement. Each curve is the best Gaussian fit for the visibility based on Equation 1.
Figure 3
 
Fovea visibility versus pupil point measured using coherent light for BV and DR: with annular apertures (blue, triangles), with half-annular apertures (red, circles), and with the conventional, separately obtained, SCE-I Maxwellian illumination configuration (black, squares). The symbols indicate the average value and ±1.0 standard deviation for each measurement. Each curve is the best Gaussian fit for the visibility based on Equation 1.
For comparison, also the outcome of a separate traditional 0.6-mm aperture Maxwellian illumination experiment obtained with a 10-nm band-pass filter at 620-nm wavelength from the tungsten–halogen source is shown (here the average value of four independent measurements at each pupil point was used to fit the SCE-I visibility function). The small horizontal shift and reduced range of the Maxwellian SCE-I curve is due to the different experimental setting used (for simplicity not described here). It has been included to allow comparison with the results for annular and half-annular apertures and to confirm a resemblance to earlier studies using Maxwellian illumination (e.g., Stiles, 1937). 
Incoherent light
The SCE-I was reexamined for both subjects when viewing through the full and half-annular aperture images at the pupil with the broadband red-filtered incoherent light source. The results obtained are shown in Figure 4. For comparison, also the outcome of a separate Maxwellian illumination experiment obtained with the unfiltered white light tungsten–halogen source is shown (here the average value of four independent measurements at each pupil point was used to fit the SCE-I visibility function). 
Figure 4
 
Fovea visibility versus pupil point measured using incoherent light for BV and DR. Other details are the same as in Figure 3.
Figure 4
 
Fovea visibility versus pupil point measured using incoherent light for BV and DR. Other details are the same as in Figure 3.
Measured directionality
The directionality measured with the different coherent (coh) and incoherent (incoh) illuminations and apertures are summarized in Table 1
Table 1
 
Values of the measured directionality factors ρ (in units of 1/mm2) determined from Figures 3 and 4 for subjects BV (top) and DR (bottom). For comparison also the directionality factors obtained with a conventional Maxwellian illumination configuration in a different setup are shown to the right (determined with narrow bandwidth red light and white light, respectively).
Table 1
 
Values of the measured directionality factors ρ (in units of 1/mm2) determined from Figures 3 and 4 for subjects BV (top) and DR (bottom). For comparison also the directionality factors obtained with a conventional Maxwellian illumination configuration in a different setup are shown to the right (determined with narrow bandwidth red light and white light, respectively).
Source Annular Half-annular Maxwellian
BV Coherent 0.008 ± 0.003 0.019 ± 0.002 0.051 ± 0.003
Incoherent 0.064 ± 0.005 0.085 ± 0.005 0.049 ± 0.005
DR Coherent 0.008 ± 0.004 0.026 ± 0.002 0.058 ± 0.003
Incoherent 0.072 ± 0.005 0.083 ± 0.005 0.062 ± 0.005
Numerical analysis
To improve the understanding of the optical mechanisms involved in this study, numerical simulations for the coherent light case simulating each step of the propagation from the diffuser to the retinal plane (assuming a schematic eye f eye = 22.2 mm and n eye = 1.33) have been performed. The optical Fourier Transforms (FTs) of each lens element, from the aperture and diffuser plane to the iris (FT1), from the iris to the pupil plane (FT2), and from the pupil to the retinal plane (FT3), have all been highlighted in Figure 5 and allow calculation of both the field amplitude and phase of the light in each Fourier plane. 
Figure 5
 
Schematic of the configuration used to simulate coherent light propagation in the annular, half-annular, and Maxwellian configurations from the paper diffuser to the retinal plane. Each angular scattering component from the diffuser projects to a different retinal location, but here only the axial component has been highlighted. The included numerical example shows the calculated field amplitude and phase in each of the 4 planes for the particular case of a 5-mm annular pupil. In the amplitude images, the values range from zero (black) to maximum (white), whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale. The small yellow square in (d) and (h) highlights the magnified (zoom-in) regions of the retina shown in Figures 6 and 7.
Figure 5
 
Schematic of the configuration used to simulate coherent light propagation in the annular, half-annular, and Maxwellian configurations from the paper diffuser to the retinal plane. Each angular scattering component from the diffuser projects to a different retinal location, but here only the axial component has been highlighted. The included numerical example shows the calculated field amplitude and phase in each of the 4 planes for the particular case of a 5-mm annular pupil. In the amplitude images, the values range from zero (black) to maximum (white), whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale. The small yellow square in (d) and (h) highlights the magnified (zoom-in) regions of the retina shown in Figures 6 and 7.
A proper light scattering model of the paper diffuser is challenging and beyond the scope of this work, but random-walk approaches can give reasonable estimates (Carlsson et al., 1995). The high uniformity of the paper thickness might favor forward scattering, which would appear as a bright spot in the center of the retinal images. However, uniform light scattering across the entire retina patch defined by the adjustable iris was observed experimentally (see Figure 2) both when full and half-annular apertures were used. 
We have chosen to model the diffuser plane across an area of 80 mm × 80 mm divided into 2048 × 2048 pixels with an individual diffuser pixel size of 39 μm. This corresponds to a 20° scattering angle in the diffuser comparing well to experiments of transmitted light scattering in paper (Chen, Baranoski, & Lin, 2008). The large dimension of the diffuser plane (extending well beyond the aperture) is required to obtain a high resolution of calculated speckles in the retinal plane with a retinal pixel size of 0.33 μm. 
A random phase variation of the diffuser uniformly distributed on 2π has been assumed. For the simulation, the diffuser is held fixed creating a unique speckle pattern in the retina plane. The amplitude and phase distribution of this light is analyzed since at any given instant both may impact the amount of light coupled to the underlying cone photoreceptor waveguides. Oblique retinal incidence will cause a reduced coupling of light and therefore lower predicted visibility (Vohnsen, 2007). Any prevailing wavefront tilt at the retina for a given position of the diffuser would remain also once the diffuser is set into motion smearing out the visual appearance of the intensity speckles. 
Simulated results of the light intensity (amplitude squared) in the pupil and magnified retina planes are shown in Figures 6 and 7 together with phase maps for 3-mm and 5-mm annular, half-annular, and Maxwellian illuminations, respectively. Horizontal sections have been included that highlight variations of intensity and phase across a single line of the images and speckles. 
Figure 6
 
Numerical results (3-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) as well as cross sections of the corresponding retina intensity and phase distributions for: (a–d) annular; (e–h) half-annular; and (i–l) Maxwellian illuminations. The pupil center is marked with a yellow cross. Selected retina image speckles are highlighted with dashed yellow circles. Horizontal retina cross sections were made along the intensity (blue) and phase (red) lines indicated with small arrowheads. Intensity images have been optimized in contrast, whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale.
Figure 6
 
Numerical results (3-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) as well as cross sections of the corresponding retina intensity and phase distributions for: (a–d) annular; (e–h) half-annular; and (i–l) Maxwellian illuminations. The pupil center is marked with a yellow cross. Selected retina image speckles are highlighted with dashed yellow circles. Horizontal retina cross sections were made along the intensity (blue) and phase (red) lines indicated with small arrowheads. Intensity images have been optimized in contrast, whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale.
Figure 7
 
Numerical results (5-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) and cross sections of the retina intensity and phase distributions. Other details are the same as in Figure 6.
Figure 7
 
Numerical results (5-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) and cross sections of the retina intensity and phase distributions. Other details are the same as in Figure 6.
Discussion
The fovea directionality factor ρ found from a separate standard Maxwellian SCE-I experiment for the two subjects was 0.051/mm2 (BV) and 0.058/mm2 (DR), respectively, using narrowband 620-nm light. Perfect additivity of the SCE-I would imply that the same directionality factor (within experimental error) should be obtained with either annular or half-annular apertures, supporting integration of the visibility function from Equation 1 across the entire pupil when calculating effective retinal images (Enoch & Lakshminarayanan, 2009). In the present study with coherent light (as shown in Table 1), this was found not to be the case. 
The cone photoreceptor distribution of the retina resembles a bundle of single or low-order multimode optical fibers (Enoch, 1963). As demonstrated by the SCE-I, these fibers are sensitive to the angle of incidence of light (expressed by the wavefront) and presumably also to the width of the incident beam that should be matched to the allowed modes to maximize light coupling (Vohnsen et al., 2005). This has been exploited for retinal imaging using scanning laser ophthalmoscopes (Delint, Berendschot, & van Norren, 1997; Vohnsen, Iglesias, & Artal, 2004). Refractive index differences are small and therefore the waveguide picture is little influenced by backscattering or non-guided light that may reach the retinal pigment epithelium without triggering the visual system. 
When the incident light is coherent, the field adds in amplitude as well as in phase at the retina. For symmetrical annular pupil illumination, the resulting wavefront is incident along the axes of the cone photoreceptors and the coupling to waveguided light is maximized (Vohnsen, 2007). For incoherent light, rapid phase fluctuations prohibits interference thereby causing oblique incidence on the retina. Thus, the fields do not add in phase and a reduced coupling of light to the cones results. Both of these cases are shown highly idealized in Figure 8 using for simplicity a 2-dimensional representation of two wave components incident on the retina (corresponding to a single angular scattering component from the diffuser). 
Figure 8
 
Illustration of the relation between light impinging on the retina from opposing sides of the annular illumination and the predicted retina light distribution (intensity and wavefront) for the (a) coherent and (b) incoherent cases. For coherent light, a phase shift Δϕ of one of the contributing wavefronts (wf1) will cause a transverse shift (indicated by a blue arrow) in the location of the bright maxima, but the total planar wavefront (wf1 + wf2) remains unaltered being given by the mean propagation direction for the contributing waves. In turn, a phase shift for one of the incoherent contributions will shift the corresponding wavefront (wf1) but will neither affect the uniform total intensity nor the two non-interacting inclined wavefronts at the retina.
Figure 8
 
Illustration of the relation between light impinging on the retina from opposing sides of the annular illumination and the predicted retina light distribution (intensity and wavefront) for the (a) coherent and (b) incoherent cases. For coherent light, a phase shift Δϕ of one of the contributing wavefronts (wf1) will cause a transverse shift (indicated by a blue arrow) in the location of the bright maxima, but the total planar wavefront (wf1 + wf2) remains unaltered being given by the mean propagation direction for the contributing waves. In turn, a phase shift for one of the incoherent contributions will shift the corresponding wavefront (wf1) but will neither affect the uniform total intensity nor the two non-interacting inclined wavefronts at the retina.
Coherent in-phase addition from the entire annular pupil would ideally result in a single centrally located bright spot with quasi-planar wavefront, the so-called fiducial plane waves (Linfoot & Wolf, 1956), near the focal point. Thus, the light would be normally incident on the retina, similar to the bright speckles in Figures 6b and 7b, where it would couple axially to the underlying photoreceptor cones. For the case shown in Figure 8 generalized to 3 dimensions, this situation is caused by constructive interference only near the on-axis intensity maximum. This corresponds to viewing of a distant point object (Fraunhofer image), but it would be blurred both by ocular aberrations and by the divergence of the contributing light at the pupil that would spread out the image. As a result, some remnant wavefront slope of the light at the retina is likely to be present that could be reduced by correcting the ocular aberrations. 
In the coherent case with phase randomization caused by the inserted diffuser, different bright intensity speckles appear at retina locations where the light interferes constructively. This may be seen as a random phase shift of the contributing rays (see Figure 8) deviating the intensity maxima to other retinal locations. Defocus (if present) may alter the angle of incidence of the incoming wave components thereby changing slightly the size of the bright speckles, but the predicted wavefront for annular apertures will still be axially incident since defocus is a rotationally symmetric aberration. In turn, a defocused coherent point-spread function is predicted to be attenuated away from its geometrical image center (Vohnsen, 2007) and this may play a role for extended Fraunhofer images (Drum, 1975). The fact that the diffuser rotates to smoothen out the visual appearance of speckles has no implication on the wavefront, but it will randomly displace the bright speckles so that each cone when time-averaged is exposed to the average of the incident intensity. 
For the incoherent case, light from different parts of the pupil will not add with any phase relation at the retina, and thus, on average, the effect would be identical to the case of oblique incidence as used when characterizing the SCE-I with traditional Maxwellian illumination. This holds true both for a stationary and for a rotating diffuser. 
In the following, the experimental results obtained for the coherent and incoherent cases in Figures 3 and 4 are discussed, in relation to the numerical analysis summarized in Figures 6 and 7, to examine further the validity of the additivity hypothesis for the SCE-I. 
Coherent light
For coherent illumination, phase variations caused by the diffuser spreads the light at the retina into a circular patch set by the iris as shown in Figure 2. Within this area, bright speckles appear at locations of constructive interference. If the illumination configuration is symmetrical (such as for the annular apertures), essentially no wavefront slope is present at the locations of bright speckles. This can be appreciated in the phase maps and cross sections shown in Figures 6 and 7. Across the encircled speckles, the wavefront slope (and the corresponding phase gradient) is negligible and thus it causes no attenuation in the coupling of light to the underlying photoreceptors (Vohnsen, 2007). For an asymmetrical configuration (half-annular aperture and Maxwellian illumination), a wavefront slope is introduced that increases with pupil asymmetry and is present across individual speckles as shown also in Figures 6 and 7. The larger wavefront slope with increased eccentricity can be appreciated comparing the half-annular aperture with Maxwellian illumination and when comparing the 3-mm to 5-mm cases. 
Since the average size of the speckles is inversely proportional to the diameter of the pupil aperture, a very large aperture may produce speckles that are comparable in size to the individual cone photoreceptor diameters. A further reduction in relative size (increasing either the pupil aperture or exposing the larger parafovea cones) will eventually lead to under-matching of the photoreceptor waveguides and a predicted reduction in light coupling and visibility. 
For annular apertures, the experimental observation in both subject's eyes with coherent light confirms a negligible small directionality of ρ ≅ 0.008/mm2 in the results of Figure 3. This small positive but non-zero value may be due to uncertainties as well as remnant minor intensity variations across the annular apertures that break the symmetry of the illumination. A possible reduction in visibility due to the small speckle size for 7-mm apertures is also possible though not statistically relevant according to the experimental results. For both subjects, enhanced visibility was observed with the 5-mm annular pupil (2.5-mm pupil point) that may be a systematic variation or photoreceptor mode matching of the incident speckles but to understand it further would require examination of more subjects. 
For half-annular apertures, the integrated light reaching the retina will correspond to an oblique wavefront producing a situation somewhere between the traditional Maxwellian SCE-I from a single point on the aperture and the full annular aperture where the wavefront is incident along the photoreceptor axes as described above. 
The effective eccentricity for a thin half-annular aperture of diameter d may be estimated by its center-of-mass location r c, which is located at the pupil point r c = d/π. As a result, the directionality factor for coherent light with a half-annular aperture
ρ ˜
may be expressed in terms of the directionality factor ρ of the traditional Maxwellian-determined SCE-I such that 
ρ ~ = ( 2 / π ) 2 ρ .
(2)
 
For the directionality factors found with the Maxwellian illumination, one finds
ρ ~
≅ (2/π)20.051 ≅ 0.0207/mm2 (BV) and
ρ ~
≅ (2/π)20.058 ≅ 0.0235/mm2 (DR) in good agreement with the measured directionality values for the half-annular apertures given in Table 1
The center-of-mass argument used to estimate the effective directionality factor for the half-annular aperture may also be applied to the other cases considered. The center of mass for a full annular aperture is located at its geometrical center coinciding with the SCE-I pupil peak location (and thus no wavefront slope at the retina), whereas the center of mass for a conventional Maxwellian illumination coincides with the small pupil point location (causing oblique incidence at the retina if displaced from the SCE-I peak location). The predicted ratio between retina wavefront tilt of the half-annular aperture and Maxwellian illumination is r c/r = 2/π ≅ 0.64. If comparing the numerically determined wavefront tilt in Figure 6 cross section (h) with (l), a ratio of approximately 3.5/5.0 ≅ 0.7 is obtained in good agreement with the center-of-mass prediction. 
It is important to note that unless the pupil incidence is asymmetrical there will be no wavefront slope of the field at the retina. Although the diffuser introduces random phase variations on the light, its propagation direction into the eye pupil is not random but restricted by the chosen aperture. At each pupil point, the wavevectors of the contributing light are confined to a cone of illumination whose angle is set by the adjustable iris. Asymmetries at the pupil give rise to a wavefront slope at the retina and a reduced light coupling to the photoreceptors (Vohnsen, 2007). 
Incoherent light
For highly incoherent illumination, i.e., ideally where light at each point of the pupil is incoherent from that of neighboring points, the light would not interfere in the retinal plane and the predicted outcome with annular and half-annular apertures becomes identical to that of the standard Maxwellian SCE-I configuration. If so, it would be correct to integrate Equation 1 across the pupil to determine its impact on vision. The results shown in Figure 4 are in fair agreement with this view. However, the annular, and in particular, the half-annular apertures reveal a larger directionality factor than for the traditional Maxwellian configuration as apparent in Table 1. For incoherent light, any displacement of the annular or half-annular pupil image from its location of symmetry at the peak of the SCE-I visibility function would lead to an uneven weighting of the integrated SCE-I function. In such a case, the part of the illumination nearest to the peak would impact more on the distribution and thus a higher directionality factor could be obtained. A numerical integration of Equation 1 suggests a 22% increase of the directionality factor for a 0.5-mm displacement of a full annular aperture from the SCE-I peak location (estimated on a 2-mm pupil) and a 48% increase for a half-annular aperture displaced accordingly (estimated on a 6-mm pupil). This may well suffice to explain the deviations observed for the case of incoherent light. However, also the fact that white light was used for the traditional Maxwellian illumination study may explain the slightly different directionality factor observed (Stiles & Crawford, 1933). 
The coherent case implies an absence of the SCE-I for any annular pupil (and thus also for the entire pupil) whereas the incoherent case implies that the SCE-I function can be added along the ring illumination and thus integrated across the entire pupil. 
Conclusions
The visibility versus pupil point dependence for highly coherent and incoherent lights, respectively, has been measured using extended sources imaged onto the eye pupil. The resulting directionality factors were compared with those obtained using conventional Maxwellian illumination. For coherent light with annular pupils, it was found that an integrated SCE-I function is absent thereby confirming the visual importance of wavefront slope and phase for light at the retina. 
A numerical analysis that highlights the wavefront slope caused by asymmetrical incidence of light at the pupil using either half-annular or a shifted Maxwellian illumination has been performed. Bright retina image speckles appear at locations of constructive interference carrying a wavefront slope induced by the asymmetrical pupil incidence. The corresponding phase variation attenuates the light coupling to the underlying photoreceptors and therefore diminishes the visibility. For a symmetrical (annular or circular) pupil, no wavefront slope is present across the image speckles. Therefore, they are not attenuated in the visual response and the integrated SCE-I function is effectively cancelled (ρ ≅ 0). This is important for visual optics tests when laser light is used since a traditional SCE-I directionality for the entire pupil may then be completely absent. In turn, for highly incoherent illumination, an integrated SCE-I function has been found to remain valid within the accuracy of the experiment. 
The phase distribution of the light incident on the retina is also important for interferometric measurements with interference gratings produced at the retina (Green, 1967; MacLeod, Williams, & Makous, 1992; McMahon & MacLeod, 2001) where asymmetries in the illumination configuration will mimic the visibility variation of the SCE-I (Vohnsen, 2007). For example, if two coherent point sources are simultaneously located in the pupil plane, the effective angle of incidence at the retina will be determined by the location of their common center of mass. If the sources are displaced away from the peak location of the SCE-I (whether parallel or perpendicular to the observed interference fringes), the impact on visibility remains unaltered as also observed experimentally (McMahon & MacLeod, 2001). For larger pupils, aberrations can alter the situation, and in general, both the amplitude and phase variation of the light reaching the retina should be analyzed. 
It should be stressed that in the experimental procedure followed (see Figure 1) individual points of the diffuser are projected onto the pupil plane producing a Maxwellian ring of illumination that, in the case of coherent light, will interfere and cause speckles in the retina plane. This approach is different from the Fraunhofer-projected images used in the earlier study of SCE-I additivity (Drum, 1975). We believe that our configuration comes closer to the original Maxwellian illumination SCE-I experiment whereas Drum's approach is closer to a normal viewing situation but with an obstructed pupil center. The reported additivity of the SCE-I in the former study may relate to the lower degree of source coherence and to the fact that the subject did not view directly the extended source but rather an extended image of the source projected without spatial filtering (and therefore with low coherence) onto a diffuser. This re-scattering from the diffuser could lower additionally the coherence of the light. Drum's conclusion for coherent light appears startling, as it seems to imply a failure of the superposition principle in the linear addition of electromagnetic fields. In any case, to study this with higher precision across an unobstructed natural pupil could benefit from the use of adaptive optics where also the phase scrambling and propagation perturbance caused by ocular aberrations could be significantly reduced. 
In certain real viewing situations, some degree of coherence may be present, which would suggest a situation situated somewhere between the two extreme cases studied here with a plausible reduction in the influence of the SCE-I. 
Acknowledgments
This research was supported by Science Foundation Ireland (Grants 07/SK/B1239a and 08/IN.1/B2053) and Enterprise Ireland (PC/2008/0125). We wish to thank Prof. C. O'Brien and Mr. A. F. Simonsen for valuable help during the experiments, the Editor Prof. D. I. A. MacLeod and reviewers for stimulating comments on the manuscript, and Dr. B. Drum for our enlightening conversation during the ARVO 2010 Annual Meeting. 
Commercial relationships: none. 
Corresponding author: Brian Vohnsen. 
Email: brian.vohnsen@ucd.ie. 
Address: School of Physics, UCD, Dublin 4, Ireland. 
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Figure 1
 
Schematic (not to scale) of the setup used to measure visibility with annular and half-annular apertures (probe light) whose images are centered at the pupil point of maximum SCE-I visibility where also a brightness-tunable Maxwellian reference is incident (reference light). The large distance of the diffuser from the eye makes the use of a lens in the reference branch prior to the iris unnecessary.
Figure 1
 
Schematic (not to scale) of the setup used to measure visibility with annular and half-annular apertures (probe light) whose images are centered at the pupil point of maximum SCE-I visibility where also a brightness-tunable Maxwellian reference is incident (reference light). The large distance of the diffuser from the eye makes the use of a lens in the reference branch prior to the iris unnecessary.
Figure 2
 
CCD images (pixel size = 4.65 μm) obtained with 3-mm diameter (a, b) annular and (c, d) half-annular apertures in the (a, c) pupil plane and (b, d) retinal plane. An f 25-mm achromatic lens was used in place of the eye to record the retina-like images. To highlight the appearance of speckles, the rotation of the diffuser was not activated during the 133-ms exposure time.
Figure 2
 
CCD images (pixel size = 4.65 μm) obtained with 3-mm diameter (a, b) annular and (c, d) half-annular apertures in the (a, c) pupil plane and (b, d) retinal plane. An f 25-mm achromatic lens was used in place of the eye to record the retina-like images. To highlight the appearance of speckles, the rotation of the diffuser was not activated during the 133-ms exposure time.
Figure 3
 
Fovea visibility versus pupil point measured using coherent light for BV and DR: with annular apertures (blue, triangles), with half-annular apertures (red, circles), and with the conventional, separately obtained, SCE-I Maxwellian illumination configuration (black, squares). The symbols indicate the average value and ±1.0 standard deviation for each measurement. Each curve is the best Gaussian fit for the visibility based on Equation 1.
Figure 3
 
Fovea visibility versus pupil point measured using coherent light for BV and DR: with annular apertures (blue, triangles), with half-annular apertures (red, circles), and with the conventional, separately obtained, SCE-I Maxwellian illumination configuration (black, squares). The symbols indicate the average value and ±1.0 standard deviation for each measurement. Each curve is the best Gaussian fit for the visibility based on Equation 1.
Figure 4
 
Fovea visibility versus pupil point measured using incoherent light for BV and DR. Other details are the same as in Figure 3.
Figure 4
 
Fovea visibility versus pupil point measured using incoherent light for BV and DR. Other details are the same as in Figure 3.
Figure 5
 
Schematic of the configuration used to simulate coherent light propagation in the annular, half-annular, and Maxwellian configurations from the paper diffuser to the retinal plane. Each angular scattering component from the diffuser projects to a different retinal location, but here only the axial component has been highlighted. The included numerical example shows the calculated field amplitude and phase in each of the 4 planes for the particular case of a 5-mm annular pupil. In the amplitude images, the values range from zero (black) to maximum (white), whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale. The small yellow square in (d) and (h) highlights the magnified (zoom-in) regions of the retina shown in Figures 6 and 7.
Figure 5
 
Schematic of the configuration used to simulate coherent light propagation in the annular, half-annular, and Maxwellian configurations from the paper diffuser to the retinal plane. Each angular scattering component from the diffuser projects to a different retinal location, but here only the axial component has been highlighted. The included numerical example shows the calculated field amplitude and phase in each of the 4 planes for the particular case of a 5-mm annular pupil. In the amplitude images, the values range from zero (black) to maximum (white), whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale. The small yellow square in (d) and (h) highlights the magnified (zoom-in) regions of the retina shown in Figures 6 and 7.
Figure 6
 
Numerical results (3-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) as well as cross sections of the corresponding retina intensity and phase distributions for: (a–d) annular; (e–h) half-annular; and (i–l) Maxwellian illuminations. The pupil center is marked with a yellow cross. Selected retina image speckles are highlighted with dashed yellow circles. Horizontal retina cross sections were made along the intensity (blue) and phase (red) lines indicated with small arrowheads. Intensity images have been optimized in contrast, whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale.
Figure 6
 
Numerical results (3-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) as well as cross sections of the corresponding retina intensity and phase distributions for: (a–d) annular; (e–h) half-annular; and (i–l) Maxwellian illuminations. The pupil center is marked with a yellow cross. Selected retina image speckles are highlighted with dashed yellow circles. Horizontal retina cross sections were made along the intensity (blue) and phase (red) lines indicated with small arrowheads. Intensity images have been optimized in contrast, whereas phase images (wrapped on 2π) have been mapped on a −π (black) to +π (white) scale.
Figure 7
 
Numerical results (5-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) and cross sections of the retina intensity and phase distributions. Other details are the same as in Figure 6.
Figure 7
 
Numerical results (5-mm aperture) for the pupil light intensity (10 mm × 10 mm) and magnified zoom-ins (33 μm × 33 μm) and cross sections of the retina intensity and phase distributions. Other details are the same as in Figure 6.
Figure 8
 
Illustration of the relation between light impinging on the retina from opposing sides of the annular illumination and the predicted retina light distribution (intensity and wavefront) for the (a) coherent and (b) incoherent cases. For coherent light, a phase shift Δϕ of one of the contributing wavefronts (wf1) will cause a transverse shift (indicated by a blue arrow) in the location of the bright maxima, but the total planar wavefront (wf1 + wf2) remains unaltered being given by the mean propagation direction for the contributing waves. In turn, a phase shift for one of the incoherent contributions will shift the corresponding wavefront (wf1) but will neither affect the uniform total intensity nor the two non-interacting inclined wavefronts at the retina.
Figure 8
 
Illustration of the relation between light impinging on the retina from opposing sides of the annular illumination and the predicted retina light distribution (intensity and wavefront) for the (a) coherent and (b) incoherent cases. For coherent light, a phase shift Δϕ of one of the contributing wavefronts (wf1) will cause a transverse shift (indicated by a blue arrow) in the location of the bright maxima, but the total planar wavefront (wf1 + wf2) remains unaltered being given by the mean propagation direction for the contributing waves. In turn, a phase shift for one of the incoherent contributions will shift the corresponding wavefront (wf1) but will neither affect the uniform total intensity nor the two non-interacting inclined wavefronts at the retina.
Table 1
 
Values of the measured directionality factors ρ (in units of 1/mm2) determined from Figures 3 and 4 for subjects BV (top) and DR (bottom). For comparison also the directionality factors obtained with a conventional Maxwellian illumination configuration in a different setup are shown to the right (determined with narrow bandwidth red light and white light, respectively).
Table 1
 
Values of the measured directionality factors ρ (in units of 1/mm2) determined from Figures 3 and 4 for subjects BV (top) and DR (bottom). For comparison also the directionality factors obtained with a conventional Maxwellian illumination configuration in a different setup are shown to the right (determined with narrow bandwidth red light and white light, respectively).
Source Annular Half-annular Maxwellian
BV Coherent 0.008 ± 0.003 0.019 ± 0.002 0.051 ± 0.003
Incoherent 0.064 ± 0.005 0.085 ± 0.005 0.049 ± 0.005
DR Coherent 0.008 ± 0.004 0.026 ± 0.002 0.058 ± 0.003
Incoherent 0.072 ± 0.005 0.083 ± 0.005 0.062 ± 0.005
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