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13.4. Monochromatic eye aberrations   ▐    13.5. Higher-order eye aberrations
 

13.4.2. 2ND ORDER EYE ABERRATIONS, OFF-AXIS

Due to the imperfections in alignment and form of its optical surfaces, eye aberrations over wider retinal area are often asymmetrical, varying randomly in magnitude from one portion of the field to another. In general, level of aberrations increases progressively with the retinal eccentricity outside the fovea. This is of little importance for eye acuity in normal use. Main function of the outer retina is providing peripheral vision, important for orientation and registering movement; neither is significantly affected by the average level of off-axis eye aberrations. More so considering generally low resolution capability of extrafoveal retinal photoreceptors. 

Outer visual field, however, becomes more important for telescope users. Assuming eyepiece exit pupil located at the eye pupil, the effective visual field projected onto retina is approximated by the eyepiece's apparent field of view (AFOV). With regular eyepieces, it is commonly in the 40-50 diameter range, and up to twice more with wide-field eyepieces. Conventional eyepieces generate strong astigmatism off-axis, even with medium fast systems. Fast Newtonian telescope also add significant amount of off-axis coma. Wavefront subjected to these aberrations ultimately arrives to the eye, where the final reshaping of its form takes place.

Eye-off-axis aberrations become significant at large eye pupil diameters, i.e. at low magnifications. But eyepiece aberrations also follow that pattern and, generally, dominate eye aberrations by a wide margin. Following text will determine more specifically the magnitude of eye's oblique aberrations vs. that in typical amateur telescopes.

The two 2nd order eye aberrations, defocus and primary astigmatism, are largest in magnitude and, therefore, of primary interest here.

Off-axis defocus

Off-axis defocus error is defined as the differential between axial defocus and defocus for a given retinal eccentricity. Thus, off-axis defocus error different in sign than axial defocus implies lesser magnitude of the actual error.

Unlike axial image, off axis defocus is not the largest eye off-axis aberration - that epithet belongs to astigmatism. Farther off-center, retinal shape (contour) - which is generally prolate ellipsoid, but of varying vertex radius, conic and local deviations - is also a factor that can be significant in determining the magnitude of defocus error, and so are the Petzval and astigmatic field curvature of the image created by the eye. Together with optical power of the eye, these are the factors determining the magnitude of off-axis defocus error.


FIGURE 226: Variation of eye aberrations over 40x32 visual field (based on Mathur, Atchison, Charman, 2009, 2010; 5mm pupil diameter, ten young emmetropes aged 20-30y, nine young myopes aged 22-35y and seven older emmetropes aged 50-71y). Central circle approximates the size of foveal area; T, N, S and I stand for temporal, nasal, superior (upper) and inferior visual field. Top row shows off-axis defocus aberration, so called relative peripheral refractive error (RPRE), which is measured relative to the central defocus error set to zero. Field asymmetry is obvious, with emmetropic eyes tending to have different asymmetry pattern than myopes. Bottom raw shows combined field aberration with 2nd order astigmatism, and 3rd (coma, trefoil) and 4th (spherical, secondary astigmatism, quadrafoil) order aberrations. Defocus error significantly alters field maps, with 1D of defocus error corresponding to nearly 0.9 micron RMS wavefront error.

As FIG. 226 illustrates, off-axis defocus error of the eye is significant even in emmetropes (20/20 or better eyesight). Of course, off-axis defocus can be corrected at the eyepiece, just as axial defocus. The problem is that they cannot be corrected at the same time: when center is in focus, outer field is not, and vice versa (younger eyes, with ample range of accommodative power, are likely to be correct field defocus - effectively field curvature - without eyepiece refocusing).

The effect of eye defocus error is always dependent on the error input by the telescope, i.e. field curvature induced by the objective and eyepiece. Mathur et al. indicate that off-axis eye defocus tends to be myopic, with the image field curving away from the retina and toward eye lens. In that case, it generally diminishes curvature of telescopic image concave toward the eye, and adds to the convex one (the latter being is generated by most telescope types; well designed eyepieces have nearly flat field, but eyepiece field curvature specs are usually not published). In either case, off-axis defocus of the eye effectively induces randomly asymmetrical deviations in field curvature, which may be noticeable, and difficult-to-impossible to correct, either by eyepiece refocusing or accommodation. However, the effect is generally small in comparison with the typical magnitude of off-axis eyepiece aberrations; for that reason, it could be more noticeable in slower systems with well corrected eyepieces.

Off-axis astigmatism

Of course, in addition to the common axial (or central) astigmatism, human eye also suffers from off-axis astigmatism rapidly increasing with the distance from fovea (so called retinal eccentricity). Since it results from incident light pencils passing through the cornea and eye lens at an angle, its major component is the regular vertical/horizontal astigmatism (Z5 i.e. J0), with the oblique term Z3 (J45) being near-negligible on the average.


FIGURE 227
: LEFT: At large incident angles, wavefront transforms from circular to strongly ellipsoidal (top); also, due to significant refraction angles of the chief ray at the cornea and eye lens, apparent field of view affectively shrinks (bottom)  MIDDLE: Most of research data indicates that off-axis astigmatism in the average eye tends to be larger on the nasal side of retina. Range of recorded magnitudes varies, as illustrated by two line plots: solid line, representing average of 20 emmetropic eyes (Gustaffson et al. 2001), and dotted-dashed line, approximating averages from two older studies with nearly identical results (Rempt, 1971, and Millodot, 1981, with 726 and 62 eyes, respectively). Shaded area is the envelope containing individual measurements for 20 emmetropic subjects (Umsbo et al. 2000); its results are very similar to those in Gustaffson et al.  In the latter, central astigmatism is not zero, but it is very low due to participant selection (w/less than 0.5D defocus and astigmatism error). Astigmatism here initially diminishes going outside fovea and well into the macula, dropping to zero at about 8
and 14 temporally and nasally, respectively, due to the offset of axial eye astigmatism with that caused by oblique incident pencils. The plot shows noticeable asymmetry in magnitudes of temporal and nasal astigmatism, with the former being greater (common with most othe studies). Averaged off-axis astigmatism for most other studies falls between the two line plots. Possible reasons for discrepancies include measurement techniques and accuracy, random variations within samples, deviations within local and ethnic groups, and generally increasing level of eye aberrations. Results may change with the methodological approach, such as a common problem of assessing strongly ellipsoidal (vs. circular) wavefronts farther off axis with Zernike expansion terms (there are ellipse-specific terms, but for some reason the common practice is to either stretch the minor axis, or to truncate the major axis, in order to obtain circular wavefront). Off axis astigmatism also changes with accommodation level (i.e. object distance), not only in magnitude (generally increasing with the level of accommodation), but also in the form, with the field asymmetry possibly shifting from temporal to nasal dominance (Lee, 2010). Also, some studies set priorities other than the exact magnitude of the aberration (note that the original plots for Gustaffson et al, Remp, and Millodot, are oriented with the tip down, increasing into positive diopters, due to astigmatism being measured along horizontal axis; since its magnitude doesn't change with the meridian, only the sign, all plots are presented at the same side of horizontal axis for ease of comparison - Gustaffson et al. is also mirror-reversed, since originally it shows astigmatism vs. exterior field, where nasal field projects onto temporal retina, and temporal field onto nasal retina). RIGHT: Schematics of the three basic form of off axis astigmatism in human eye. With all three, center field is nearly optimally focused onto the retina, but depending (mainly) on field curvature developing with astigmatism going further off axis, the astigmatic field can become increasingly defocused, with best astigmatic focus falling either behind retinal surface (A), in front of it (C), or remaining close to it (B). Form of astigmatism in actual eyes is commonly asymmetrical , being more or less strong and/or defocused on one than on the opposite side of retina, either relative to the vertical (nasal and temporal retina), or horizontal meridian (inferior - lower - and superior retina). Center field often is not well focused onto the retina, due to a central defocus error, which in turn affects the magnitude of off-axis aberration by adding to it defocus error.

Taking apparent average between the two line plots on FIG. 227, middle, longitudinal aberration at 60 off-axis is about 5 diopters (D~5) on the nasal side, and 8 diopters on temporal side. It changes approximately in proportion to α2, α being the off-axis angle. With the longitudinal aberration given by L~DE/59, D being the aberration in diopters, and P-V wavefront error of astigmatism Wa=L/8F2~DP2/8000 (after substituting E/P for F and taking E~17mm) or Wa~DP2/4.4 in units of 0.55μ wavelength, the average off-axis astigmatic P-V wavefront error for 5mm pupil (P=5) ranges from about 45 waves at 60 off-axis to somewhat over 1 wave at 10, on the temporal side, and nearly 40% less on the nasal side of the retina.

Note that the P-V error for given longitudinal aberration L is identical for astigmatism and defocus. However, since the P-V/RMS wavefront error ratio is 24 and 12 for the former and latter, respectively, the RMS error for given longitudinal aberration (and thus the absolute value of the corresponding Zernike coefficient as well) will be larger for defocus by a factor of 2.

Since longitudinal astigmatism is a function of focal length, and doesn't change with the pupil size, the wavefront error of astigmatism changes with the square of the pupil diameter (due to changes in the transverse aberration and the Airy disc size). So, for 1mm pupil, eye off-axis astigmatism is about 25 times smaller than for 5mm.

Note that the RMS error of balanced primary astigmatism is smaller than the P-V error by a 241/2 factor, thus any given P-V error of astigmatism compares to 1/3 smaller P-V error of primary spherical aberration.

As mentioned before, this very strong off-axis astigmatism of the eye is effectively excluded when observing with a telescope - just as it is unimportant for eye acuity in everyday's life - due to the reflex eye movement that brings the image of a selected object onto the retina. Hence, eye astigmatism interacting with that of the eyepiece is primarily axial eye astigmatism. Being of positive power, they both generate same type of astigmatism, with more power in the tangential plane - the one containing chief ray and optical axis - than in the perpendicular to it sagittal plane. As a consequence, tangential surface in this plane is closer to the eyepiece, and likely to be concave toward it (the sign of curvature depends on the relationship between the strengths of astigmatism and Petzval surface). In short, the two generally add up but, since eye astigmatism and field curvature can be highly asymmetric, eye contribution - and thus the combined magnitude - is likely to vary across the field.

This is probably unimportant farther off-axis with conventional eyepieces, which still generate much more astigmatism than the eye. Only close to the midfield, where eyepiece astigmatism becomes comparable to axial astigmatism of the eye, the latter can significantly change the magnitude of the former - with the increase being more likely - significantly affecting field quality. This may have more of an effect with Nagler-type eyepieces, well corrected for astigmatism. Depending on whether the two add or partly offset - which is mainly determined by which astigmatism component (positive or negative) dominates - the combined field can be anywhere from sensibly perfect to somewhat compromised.

As for the axial astigmatism of the eye, if present, it will not be affected by the eyepiece, which does not generate axial astigmatism. In other words, axial astigmatism of the eye will be the only astigmatism present in the field center. That, of course, unless axial astigmatism is generated by astigmatic surfaces possibly present in a telescope's optical train. Depending on their sign, the two can either add up, or partly offset.


13.4. Monochromatic eye aberrations   ▐    13.5. Higher-order eye aberrations

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