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◄ 13.3. Eye aberrations   ▐    13.4.2.  2nd order defocus and astigmatism, off-axis ►
 

13.4. MONOCHROMATIC EYE ABERRATIONS

Axial (also central, or foveal) eye aberrations are the primary limitation to its proper functioning both, in daily life and as a part of optical train involving use of a telescope. Off-axis (also oblique, or abaxial) eye aberrations are comparatively unimportant, but will be also addressed, since they can affect quality of extended object images, or extended fields projected onto the retina.

Generally, the magnitude of aberration contribution decreases with increasing Zernike radial order. Second order aberrations account for roughly 90% of total aberration of the average eye. The most significant higher-order terms are 3rd order coma and trefoil, and 4th order spherical aberration, as summed up in the table below (as mentioned before, zero-order piston is not an aberration for image formed by a single aperture, and 1st order tilt/distortion is not a point-image quality aberration; for that reason, they are usually omitted from evaluating  eye aberrations).
 

RADIAL ORDER (n)	ABERRATION	ZERNIKE TERM		ZERNIKE COEFFICIENT		ZERNIKE CIRCLE POLYNOMIAL
		Double-index notation ()	Single-index notation (Zj)	Double-index notation	Single-index notation (zj)
2ND	Oblique primary astigmatism	Z	Z3	z2,-2	z3	ρ2sin2θ
	Defocus	Z	Z4	z2,0	z4	2ρ2-1
	Vertical primary astigmatism	Z	Z5	z2,2	z5	ρ2cos2θ
3RD	Vertical trefoil	Z	Z6	z3,-3	z6	ρ3sin3θ
	Vertical primary coma	Z	Z7	z3,-1	z7	(3ρ3-2ρ)sinθ
	Horizontal primary coma	Z	Z8	z3,1	z8	(3ρ3-2ρ)cosθ
	Oblique trefoil	Z	Z9	z3,3	z9	ρ3cos3θ
4TH (the most significant 4th order aberration)	Primary spherical aberration	Z	Z12	z4,0	z12	6ρ4-6ρ2+1

Following the main division of monochromatic eye aberrations on lower and higher orders, according to the ANSI Zernike terms expansion, the text continues with more detailed description of 2nd order eye aberrations, on axis (foveal) as well as off-axis. Next page addresses higher-order eye aberrations.
 

Axial defocus

In ophthalmology, defocus is second-order aberration, denoted with the Zernike term Z and Zernike (expansion) coefficient z2,0 (short notation may only use its n and m integers, i.e. 2,0). Its j-number, or mode, is 4, which means that the term can also be identified as Z4, and the coefficient as z4. As mentioned, defocus is by far the largest single inherent eye aberration. Luckily, it is also the easiest to correct - a simple positive or negative lens placed in front of eye will do.

Axial eye defocus is caused primarily by the mismatch between optical and/or accommodative power of the eye and the size of vitreous body; other contributing factors can be weakened muscles surrounding eye lens, and/or lack of elasticity of the lens itself. The more the separation between cornea and retina deviates from 23-24mm, the more likely significant axial defocus error; in general, larger eye tends to suffer from myopia (short focus for distant objects, improving with reducing object distance), and smaller eye from hyperopia (long focus for distant objects, worsening with the reduction in object distance). Overgrowth of the vitreous body - either as a whole, or as an elongation along visual axis, or as an outward deviation in the concavity of the rear retinal wall - is much more common than undergrowth, hence myopia is also more frequent than hyperopia.

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. Together with optical power of the eye, these are the factors determining the magnitude of off-axis defocus error, which is addressed in the subsection on off-axis eye aberrations.

Following table is a short summary of defocus-related terms and relations.
 

Average accommodation power of the eye decreases with age, from 10-15 diopters at age 20, to 1-2 diopters at age 60 (FIG. 221 bottom left). Accommodation power in diopters can be expressed simply as 1/S, with S being the eye-to-object separation in diopters (i.e. in units of 1 meter); hence accommodation power of 10 diopters implies ability of eye lens to refocus from zero (object at infinity) to 0.1m, and 1 diopter accommodation power implies ability to refocus from infinity to objects 1m away. The smallest distance at which eye can sharply focus is referred to as the least distance of distinct vision or, more simply, near-focus distance.

Diopter (or dioptre, abbreviated by D), expressed as D=1/ƒ, in which ƒ represents the lens power (focal length) in meters, is used as a measure of longitudinal defocus. Since the effective focal length of the eye (the f.l. that the eye would have with air as imaging medium) is ƒE~17mm, or ~0.017m,  its diopter is ~59. The larger diopter, the stronger optical power, i.e. the shorter focal length. The difference between the optical powers of rays from the central and outer zones of eye pupil indicates the amount of defocus. A 2D defocus caused by a different optical power of the outer zone of the eye lens means that the outer zone diopter is 61. In effect, it is by ~1/30 of the proper focal length shorter, as if formed by rays falling onto the eye lens being slightly converging. Negative diopter indicates that the outer zone focuses longer. Thus -2D defocus indicates 57D power for the outer zone of the eye lens, or focal length by 1/30 of the central rays' focal length longer, as if coming - approximately - from a point not at infinity, but located at 30 eye lens' focal lengths - or 2 diopters - in front of the eye.

Formaly, diopter prescription with eye defocus indicates the power of lens needed to correct the aberration. So if eye defocus is, for instance, 3 diopters, making its focal length 62 diopters, it will be corrected by putting a -3D lens (negative, with focal length of -1/3 meter) in front of the aberrated eye. This is due to the combined focal length being approximated with a thin lens equation, as 1/ƒC=(1/ƒE) +(1/ƒG), which for focal lengths in diopters becomes DC=DE+DE, with subscripts C, E and G are standing for "combined", "eye" and "(eye)glasses", respectively.

Defocus expressed in diopters can be converted into longitudinal defocus L, as L~DƒE/60, and from it into P-V wavefront error W=L/8F2 which, with the focal ratio F~17/P, where P is the eye pupil diameter, can be written as W=LP2/2300. Substituting DƒE/60 for L, and taking ƒE~17mm, in terms of defocus error in diopters D it becomes W=DP2/8100. The corresponding RMS wavefront error is smaller by a factor 1/√12, and equals absolute value of the Zernike expansion coefficient. Denoting RMS error with ω, it can be written as ω=|D|P2/28,000. Solving for P and ω=0.000041mm (diffraction limited RMS error for 0.00055mm wavelength) determines the limit for diffraction limited defocus error in diopters as D~1.15/P2. That gives 0.29 diopters defocus limit at 2mm pupil diameter, and 0.13 diopters at 3mm pupil diameter.

It is safe to say that relatively few individuals have natural eyesight with such a low level of defocus error. Looking at a few recent studies, inherent eye defocus errors are quite common, and often quite large. For instance, among 109 "normal" participants aged 21 to 65 years (41y average), defocus, or spherical refractive error was between +6D and -12D (-3D mean, ±3D standard deviation). The average eye here (Porter et al. 2000)is far from diffraction limited even at 1mm pupil diameter. Another study on 100 students of Indiana University School of Optometry, aged 22-35y (26.1y average), had nearly identical overall defocus error, +5.5D to -10D range, with -3.1D mean ±3D SD (Indiana Aberration Study, Thibos et al. 2002). These numbers indicate that eye defocus error does not increase significantly with age. In the Indiana Aberration Study, only about 20% of participants had defocus error less than 1D. Majority had myopia (short sight, numerically negative defocus error), which is representative of the population in general.

Yet another group of students, this time from Murcia University, Spain (59 students, 20-30y old, 24y average) had significantly lower defocus error at +1.3D mean ±1.7D SD (Castejón-Mochón et al. 2002). These numbers, on the other hand, indicate that there may still be substantial random variability in groups of this size.

Between about 3D average defocus error in the first two, and 1.3D in the third study, a broad population average may be at 2-2.5D level. With the corresponding RMS wavefront error ω=DP2/16√3 in microns, the uncorrected average eye would be diffraction limited defocus-wise (for 0.55μ wavelength, thus ω=0.55/√180) roughly at a pupil diameter P of 0.7mm.

However, regardless of the magnitude of defocus error in the eye, it does not affect the telescope observer: s/he corrects for it by defocusing the eyepiece by an offsetting amount.

It is another focusing-related eye function that can have the effect on perceived image quality in a telescope - specifically, quality of the outer field. It is the eye accommodation power.

The image field produced by telescope objective is usually curved, not seldom quite strongly. Depending on the the sign of curvature, it causes outer field image points to be progressively closer, or farther away from the eyepiece. This, in turn, causes these light pencils to exit the eyepiece slightly diverging (when closer) or converging, i.e. appearing as coming from a finite distance. In order to bring these pencils into focus, eye has to accommodate, primarily by changing the shape of eye lens. The eyepiece itself can produce strong image curvature, which can make curvature of the final best image surface either stronger or weaker than that of the objective's image.

Neglecting effect of the eyepiece - whose specific image curvature is commonly unknown - the power of accommodation required due to the curvature of image formed by the objective can approximated using basic paraxial relation (Eq. 1.4). Substituting ƒ+(h2/2R) for object distance, with ƒ being the eyepiece focal length, h being the point height in image space and R the radius of image curvature, gives the apparent point distance p as:

 p=2R(ƒ/h)2+ƒ.

With ƒ and h in mm, the corresponding accommodation power in diopters is 1000/p. For, say, image curvature radius R=100mm, eyepiece f.l. ƒ=10mm and point height h=5mm (which is at a field edge of the standard 10mm 50° AFOV eyepiece), p=810mm and needed power of accommodation is 1.2 diopters. This level of accommodation is achievable for most people, and quite easy for younger individuals. However, with significantly wider field, or significantly stronger field curvature, or with the eyepiece significantly adding to the final image curvature, older individuals may not be able to see the outer field clearly without adjusting the focuser.

The specific form of image curvature, i.e. whether it is concave or convex toward the eye, is an additional factor that influences how demanding is accommodation. It is harder for the eye to focus a converging beam - such as one coming from off-axis point farther away than the center (i.e. from outer portion of image surface convex toward the eye - since it is not a natural geometry, requiring eye lens to weaken its surface curvatures beyond those needed for focusing near-collimated beam.

The next largest axial aberration of the average eye is primary astigmatism. It is mainly the combination of eye's defocus and astigmatism that is mainly responsible for often grotesquely deformed, stretched out diffraction patterns produced by the eye at larger pupil sizes. As mentioned, eye defocus error is inconsequential for visual observing with a telescope, which acts as a corrective aid. Unfortunately, this automatic correction does not extend to eye astigmatism.

Axial astigmatism

Like defocus, primary astigmatism in ophthalmology is second-order aberration, denoted with the Zernike term Z and Zernike (expansion) coefficient z2,2 for vertical astigmatism (denoted as J0 or J180 when expressed as prescription cylinder power in diopters), and Z for oblique astigmatism, with Zernike (expansion) coefficient z2,-2, denoted by J45 as the corresponding corrective prescription power in diopters. The j-number, or mode, for the former is 3, and for the latter 5.

Astigmatism in human eye can be axial, which most often originates at the cornea, prone to relatively weak cylindrical deformations, or regular off-axis astigmatism. The extent of the former is determined by the severity of corneal deformation, while the latter, in general, becomes progressively larger with the increase in off-axis angle (i.e. with the increase in retinal eccentricity).