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13.6. Eye chromatism   ▐    13.8. Eye intensity response
 

13.7. COMBINED EYE ABERRATIONS, DIFFRACTION

In any optical system, the effect of wavefront aberrations - either those induced by the eye itself, by the optical train of a telescope, or externally - is adding to the effect of diffraction in further spreading out the energy making up the point-image. Eye is not an exception. At small pupil sizes, eye diffraction is the main determinant of image quality; at large pupil sizes, that role belongs to eye aberrations, and at intermediate pupil sizes, both diffraction and aberrations are significant factors.

Not surprisingly, studies give different indications of the maximum average pupil size at which the eye is still diffraction-limited (0.80 Strehl, or better). The range spans from 1.22mm diameter, or as low as 0.76mm when tilt is excluded (Thibos et al. 2002), to 2.8mm (Howland and Howland, 1977). Prevailing view is that the average eye remains diffraction limited up to about 2mm pupil diameter, after which its exponentially increasing aberrations cause the Strehl to plummet (FIG. 236A). However, the perceived image quality is much less affected than the average nominal Strehl, primarily due to the low effective magnification of the eye (in other words, angular size of either dominant diffraction pattern, or dominant aberrated pattern, remain below detection threshold of the eye).

Size of the geometric aberrated blur (i.e. ray spot size) also varies with the eye model used. There is a number of optical models of the eye, from the simplest one by Emsley (1946), with a single refracting surface on a water-like medium (1.333 refractive index), to a complex system proposed by Liou and Brennan in 1997, incorporating aspheric surfaces and varying refractive indici (the latter is most closely resembling biological eye, and reproducing its aberrations). The plot is based on eye model with physical dimensions of the average eye, constructed to produce its type and level of foveal aberrations. Specs given in table below (surrounding medium for the cornea and lens is water; their dispersive properties are obtained by scaling up those for water approximately corresponding to the change in e-line refractive index).
 

CORNEA

LENS

R1/aspheric

Thickness/n/V

R2/aspheric

R1/aspheric

Thickness/n/V

R2/aspheric

Tilt

Decenter

7.8mm
-0.25*

0.5mm
1.383/1.380/1.376
62.5/62/61.5

6.5mm
0.145**

10.2mm
0*

3.6mm
1.423/1.420/1.416
65.5/65/64.5

-6mm
-3*

2

0.2mm

R=radius of curvature;   n=F/e/C refractive index;   V=F/e/C Abbe number;   *conic;   **toric curvature

As presented previously, both monochromatic and chromatic foveal aberrations are nominally significant at medium to large eye pupils. Their combined and separate effects are best illustrated on MTF graph (FIG. 236B). Same goes for monochromatic aberrations - which are usually the primary concern - and their dependence on the pupil size (FIG. 166C,D). The actual size of diffraction pattern, either nearly aberration-free (at small pupil sizes) or heavily aberrated (at large pupil sizes), does not change much going from small to large pupil sizes - certainly much less than the nominal Strehl, or the RMS wavefront error value (FIG. 236E).

FIGURE 236: (A) Generalized scheme of the respective magnitudes of aberration-free (diffraction-limited) and average real (i.e. aberrated) eye, on axis. Geometric blur size and RMS wavefront error have nearly identical rate of change, thus both scales apply to the same plot. With all eye aberrations included, diffraction dominates for pupil diameters smaller than ~2mm, and for larger pupils eye aberrations. Geometric size of point-image blur - as aberrated ray spot blur, or diffraction blur conventionally represented by the Airy disc - is smallest at ~2mm pupil diameter. For larger pupils, it is enlarged due to eye aberrations, and for smaller pupils due to diffraction. With defocus corrected, which applies to the telescopic eye, eye is diffraction limited for pupils diameters smaller than ~3.5mm. Relative magnitude of higher-order aberrations increases with the pupil size, becoming comparable to the defocus-corrected eye (i.e. plagued only by primary astigmatism) at about 7mm pupil diameter.
(B) Foveal eye contrast transfer  for 5.7mm pupil diameter without correction, with corrected 2nd order aberrations, and with corrected monochromatic aberrations (based on Guirao et al. 2002). The latter effectively shows contrast transfer drop due to eye's chromatic aberrations; at this pupil size, eye chromatism (longitudinal and lateral error combined) cause more contrast loss than secondary spectrum in 200mm f/5 achromat (shown in the corresponding relative range of frequencies). Spatial frequency - given by 1/
ν, with the frequency ν determined by the inverse of the wavelength-to-aperture ratio, in radians - is, by default, limited to 60 per degree, i.e. one arc minute, despite the theoretical cutoff frequency being 200 per degree, or 0.3 arc minute (as shown on graph C, this theoretical resolution doesn't materialize, due to the contrast level at these frequencies falling below the minimum required by the eye).
(C) Foveal eye contrast transfer for three different pupil sizes, with 2nd order aberrations - defocus and primary astigmatism - corrected. Since eye defocus error is cancelled out by defocusing the eyepiece, these plots are close to the actual monochromatic (around e-line) contrast transfer for the telescopic eye, with eye astigmatism being relatively insignificant at this error level. The MTF for larger pupil sizes indicates considerably higher cutoff frequencies, but the actual cutoff frequency is similar for all three pupil sizes, being determined by the minimum contrast level required by eye for resolution.
(D) Foveal MTF for the same three pupil sizes with spatial frequency normalized to 1. A comparison with MTF for 24mm Knig eyepiece for identical exit pupil sizes shows that eye aberrations lower contrast more than the eyepiece aberrations for all three pupil sizes. At 2mm exit pupil diameter, the Knig is essentially aberration-free on axis, and nearly so even at 4mm exit pupil (with the diameter of eyepiece exit pupil is given by the ratio of eyepiece f.l. and system's focal ratio, these exit pupils correspond to an
/12 and /6 system, respectively). Since the only axial aberration of an eyepiece is spherical aberration, which increases much faster with the pupil diameter than the astigmatism/coma/spherical/trefoil mix of the eye, its contrast transfer at 6mm exit pupil diameter (i.e. with /4 system) is comparable to that of the eye with 4mm pupil (C and D based mainly on Liang and Williams 1997 and Thibos et al 2000).
(E) Polychromatic and monochromatic diffraction spots for four eye pupil diameters. At 1mm eye pupil diameter, the two patterns are essentially identical, indicating near-zero eye chromatism; at 2mm, the difference is only slight. At 4 and 6mm, polychromatic pattern becomes noticeably fainter, indicating loss of energy due to chromatic error. Patterns are generated by OSLO, with the eye model specified in above table; real eye pattern at larger pupil sizes - as depicted in the dark circle - commonly shows significant random asymmetry, caused by random local deviations of eye surfaces from the rotationally symmetrical conic surface. Change in nominal size of the conspicuous portion of diffraction blur is much less pronounced than its change relative to the Airy disc for corresponding pupil sizes, shown at right. The blur is relatively large with respect to the cones (~2.3μ-10μ from foveola to the outer retina, respectively) and rods (~2.5μ-5μ), hence with the limiting resolution determined entirely by the blur size.

Eye aberrations lower object-to-image contrast transfer, hence the perceived image quality and resolution limit. Actual effect on image quality, however, depends not only on the wavefront error, but also on the retinal image size (magnification). An aberrated image has to be large enough, angularly, to allow the eye to detect the aberration. Average eye begins to recognize image shape, as its size exceeds ~3 arc minutes. Any detail below that size appears point-like to the eye. Consequently, in order for the eye to discern the effect of wavefront aberrations in the retinal image - whether an extended detail or a point-object image - has to be larger than ~3 arc minutes.

That is why the edge-field coma in an /6 paraboloid is so inconspicuous; at 15mm off-axis, with the coma wavefront error at 2.5 waves P-V (0.45 wave RMS), the sagittal blur size produced by a 25mm eyepiece is still only 3.6 arc minutes on the retina. It takes the blur size of ~5 arc minutes in diameter for the average eye to clearly recognize that it is not point-like; in this case, it would take ~/5 paraboloid. Considering that the astigmatism of standard eyepieces usually adds significantly to the blur size that far off axis, 5 arc minutes blur size is probably still borderline between inconspicuous and obvious level of aberrations for the average observer.

The required minimum spot size for shape recognition is greater for less contrasty details, as well as for those that fall in the low-sensitivity range of eye photoreceptor cells, cones and rods.

A glance at FIG. 236E reveals that despite its very high nominal aberrations, the aberrated diffraction blur of the average eye is still significantly smaller than 5 arc minutes for pupils smaller than about 5mm in diameter, thus with relatively small effect on the perceived image quality. The exception is eye defocus error (assumed corrected in the patterns shown), which is typically by far the largest eye aberration. It often very noticeably degrades quality of the visual image; fortunately, it does not affect the telescope user.

In addition to eye aberrations, other important factors determining properties of the image formed by the eye are its spectral response, intensity response, and contrast sensitivity.
 

13.6. Eye chromatism   ▐    13.8. Eye intensity response

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