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13. THE EYE   ▐    13.4. Monochromatic eye aberrations

13.3. Human eye optical properties

Considering its biological origin - as opposed to precisely crafted optics - it is not surprising that optical aberrations of the human eye are significant. They include both monochromatic and chromatic aberrations. The former include aberrations typical of conic surfaces of revolution, but also irregular wavefront deformations caused by local deformations of eye surfaces. Eye constitutes the last optical element in the objective-eyepiece-eye train, thus its aberrations affect the final visual image just as those of telescope itself.

One exception is defocus error - either common focusing errors such as myopia (short sight) and hypermetropia (or hyperopia, long sight) - by far the most significant single aberration of the average eye. Low axial defocus error, allowing for "normal" acuity, (usually considered to imply either less than 0.5 diopters, or 1 in terms of visual acuity, i.e. 20/20 vision, or better) is called emmetropia. Luckily, eye defocus error doesn't affect retinal image quality when looking through a telescope, due to its correction by focusing via focuser.

FIGURE 220: The most significant factor causing eye defocus error is a mismatch between the size of eye ball and eye lens' power (left). Eye wavefront aberration are commonly measured by reverse imaging, i.e. analyzing reflection of a point source at the retina.

In general, in reverse imaging, wavefront originating at a point-source at the center of fovea would exit the perfect eye perfectly flat, i.e. as a collimated beam; real eyes are always aberrated, and wavefront originating from retinal reflection (which is the common way of assessing wavefront quality) is not flat.

On the other hand, we normally do not notice appreciable eye color errors, due to perception of color blurring due to eye chromatism being mainly filtered out during brain signal processing. Nevertheless, spread of energy on the retina caused by eye chromatism does lower optical quality of the retinal image just as much as with the color separation fully detectable.

Individual deviations in eye aberration level are quite wide. Main causes are individual deviations in size and shape of eye surfaces involved in image formation, and the efficiency of eye compensatory mechanisms. Integral part of the final (perceived) error level here are the spatial and functional properties of retinal photoreceptors. What separates eye from a non-biological detector is that its neural signal processing non-trivial significance in determining the final outcome.

Basic optical properties of human eye, typical deviations from the idealized schematic eye, main determinants of pupil size and accommodative power, are presented on FIG. 221.

FIGURE 221: (A) Eye aberrations are the result of its optical simplicity, combined with imperfect surface shapes and alignment of its optical components. The single largest aberration, defocus, accounting for about 1/2 of the total aberration of the average eye, is the consequence of a limited range of refocusing (accommodation), mainly a function of the eye (crystalline) lens. Eye's accommodative power - defined as the differential between relaxed (infinity) focus and distance of nearest focus (i.e. 1/d in diopters, with d being the distance of nearest focus in meters) - declines with age, from nearly 15 diopters in early childhood to
1-2 diopters after age 50. Second largest axial aberration, astigmatism (on average about 3 times smaller in magnitude than defocus), mainly originates at a deformed front surface of the cornea. Spherical aberration is minimized by aspheric (prolate) shapes of eye surfaces, as well as by decreasing refractive power of the eye lens toward its edges. Axial coma mainly results from tilt and decenter of the eye lens relative to the visual axis. The later also causes axial lateral color, while axial chromatic defocus, in the form of primary spectrum (i.e. shorter wavelengths focusing closer to the eye lens that longer wavelengths) is due to relatively narrow range of refractive/dispersive powers of eye's optical elements. (B) Eye pupil diameter varies both, with illumination level and individually. For given illumination level, variations in pupil size up to a factor of two are not uncommon (top; smoothed out plots for 250 subjects from Richman et al. 2004). Decrease in illumination causes pupil to increase; it nearly doubles vs. size in average daylight conditions and comes close to its maximum diameter already in twilight illumination. Pupil size steadily declines with age, but individual differences are wide (bottom; 91 subjects from Winn et al. 1994 and 70 subjects from Yang et al. 2002, with the latter only qualifying illuminance level as "photopic" and "mesopic'). The plots are for the average pupil size; individual deviations are in the range of about +/-2mm. While older individuals have generally smaller pupil, largest pupils at the old end are still be significantly larger than the smallest pupils at the young end (also, the largest pupils at old age are nearly as large as largest pupils at a young age). Pupil size also changes with object distance, generally being somewhat larger for closer objects.

Most of the refractive power of the eye - about 2/3 - comes from cornea, and in particular from its front surface. This is due to both, strongly curved corneal surface and the refractive index differential being the highest here (1 vs. 1.38). Within the eye, refractive indici vary between  1.33 and 1.41, thus having only secondary effect on the optical power and with it, secondary effect on eye aberrations as well. In general, cornea and eye lens tend to induce errors of opposite sign that partly - and often significantly - offset in the combined wavefront.

Average actual eye focal length fA is around 23mm; however, since the medium in which the image is formed has refractive index n~1.33, the effective focal length fE used in the aberration calculation is 23/n, or ~17mm. Due to the effective compression of light waves in its denser medium, the Airy disc formed on the retina is smaller than what it would be in air medium. Since the media in which the eye forms images (vitreous humour) is of refractive index ~1.33, diffraction effect is suppressed compared to that of imaging in air-medium. Thus, the Airy disc diameter, given by 2.44λF/n, n being the refractive index of the imaging media, is smaller by a factor of ~0.75.

More specifically, in  a medium slower by a factor 1/n, the wavelength gets "compressed" by the same ratio, and any given linear optical path difference expressed in units of the standard wavelength effectively increases by a factor of n. In other words, the first diffraction minima - as well as all the successive ones - occur at an angular radius smaller by a factor of 1/n. Also, any nominal wavefront deviation results in the phase error greater by the same 1/n ratio (this doesn't mean that a deviant wavefront entering the eye will have its error multiplied by the ~1.33 factor; due to proportionally slower imaging medium, the wavefront formed within it will have its nominal linear error reduced by the same factor, preserving the effective error size unchanged).

Likewise, errors induced by eye lens will be smaller by the same ~0.75 factor compared to those that would be induced should the lens be imaging in air. As a result, the effective wavefront error doesn't change. Physically, diffraction effects - including those induced by aberrations - are reduced in size, but so is the wavelength. As long as eye retina is an over-sampled detector (i.e. with the linear point image larger than photoreceptor by a factor of 2, or more), its conventional linear diffraction cutoff separation λF would be smaller by a factor ~0.75, due to reduced effective wavelength λ; and so would its nominal angular cutoff angle λ/D. However, general form of the diffraction resolution limit - λ/D - remains unchanged.

Note that, as in the usual telescope optics notation, D here is the aperture diameter. In describing eye aberration, the same symbol is used for one of the basic metrics, diopter. For sake of clarity, this section on eye aberrations uses D for "diopter", while eye pupil diameter is denoted by P.

But calculating wavefront error based on longitudinal aberration requires rescaling the actual eye f.l. fA so that it corresponds to the actual diffraction pattern, when formed in air. Since the Airy disc formed on the retina is appropriate in size to one produced by the F-number resulting from F=fA/1.33P, not the one given by F=fA/P, the appropriate focal length to use as the basis for calculating defocus error is 23mm/1.33~17mm. It gives a proper match of the longitudinal/transverse errors and the Airy disc size.

Quantifying eye aberrations is a difficult task and, not surprisingly, research results are not always in good agreement. Main reasons, in addition to much more complex function of an active biological optical system, such as the eye, are usually small sample size, prone to significant individual deviations from the average, as well as different methods of measurement, methodologies and/or degree of measurement accuracy. To make result interpretation more difficult, ophthalmological concepts are often different than those used in physical optics, and its overall scientific integrity is probably lower.

Table bellow summarizes some of the main differences in the terminology, concepts and presentation between ophthalmic and (optical) telescope optics.

























REFRACTIVE ERROR: Commonly defocus error, although any eye-generated ray/wavefront aberration is, in fact, refractive error
LOWER-ORDER ABERRATIONS: Those with Zernike radial order n lower than 3 (tilt, defocus, primary astigmatism)
HIGHER-ORDER ABERRATIONS: Zernike radial order n of 3 or higher (includes primary coma and spherical aberration)
TRANSVERSE CHROMATISM: Lateral chromatic magnification (lateral chromatism in telescope optics)
DEPTH OF FOCUS:  Range of defocus with acceptable image deterioration relative to best focus (not necessarily diffraction-limited)
SPHERE or FOCUS: Defocus (i.e. lens power needed to correct defocus of the eye; some eyeglasses prescription terms, frequently - and inappropriately - are used for naming the aberrations themselves)
PRISM: Wavefront tilt
CYLINDER: Astigmatism
AXIS: Angle between two astigmatic meridians (doesn't have to be 90)

One needs to keep in mind these important points: unlike the standard eye model, an actual eye is:

(1) an active optical system, with adjustable components and aberrations varying in time,
(2) it is not strictly centered system,
(3) it is not a rotationally symmetrical system, and
(4) final perception is the subject of neural processing.

Consequently, aberration forms of comparable magnitude do not necessarily have identical effect as in a passive optical system, such as telescope and, to a small but not insignificant degree, eye aberrations are random.

Optical system of human eye is usually represented by the schematic eye, which uses average dimensions with idealized, centered and rotationally symmetrical surfaces, in modeling eye imaging and aberrations. As every complex system, eye is described by its entrance and exit pupil and 6 cardinal points: object and image space focal point, first and second principal plane, and first and second nodal point (FIG. 222).

: LEFT - Illustration of the optical scheme of human eye; there is a number of schemes, from those simplest ones, with a single spherical refractive surface, to those with two lens elements with aspheric surfaces and gradient index. The above nominal values are based on the Gullstrand-Emsley schematic eye, which uses a single corneal refractive surface, but adding the second surface has little effect on the optical parameters shown. Eye pupil is the system's aperture stop; its image by the preceding optical element (cornea) forms the system's entrance pupil - which is the apparent boundary to entering rays from object point - at about 3mm behind the anterior cornea, 13% larger than eye pupil. Image of the aperture stop by the optical element following it in the optical train (eye lens) forms system's exit pupil - the apparent boundary to rays from the image point - only slightly smaller than eye pupil (as indicated by the red dotted line representing refracted ray; it is made slightly larger than eye pupil to avoid it being covered by the ray). Eye exit pupil is, for all practical purposes, at the eye pupil (iris). Due to the eye being an unequifocal system, with its object space (anterior) focal length smaller than its image space (posterior) focal length, nodal points are not contained in the principal planes; they are shifted deeper into the eye, straddling the rear surface of eye lens.
- Scheme of image formation by unequifocal system (not to eye's specifications; principal and nodal points are more widely separated, for clarity; also, image plane farther out than posterior focal plane corresponds to accommodated eye). Image of a point-object at the height h in the outer field is determined by two rays originating at the object point, one normal to the 2nd principal plane (normal to the axis at the 2nd principal point P'), and the other passing through through the anterior focal point and turning parallel with the axis at the first principal plane (normal to the axis at the first principal point P). The significance of nodal points is that the ray from object point directed toward 1st nodal point (N) acts as if it follows the axis to the 2nd nodal point (N'), from which it connects to the image point keeping the original ray orientation. In other words, nodal points define the actual incident angle, i.e. actual field of view. Apparent incident angle, and field of view, determined by the ray passing through the center of exit pupil (i.e. intersecting the axis at a point between nodal and principal points) - the chief ray - is smaller than the actual angle and field; for the schematic eye, by a constant factor of about 0.82. RIGHT - BOTTOM: Imaging by human eye often can be sufficiently well modeled with a simplified version of schematic eye, called reduced eye. Shown is Emsley's reduced eye, which assumes a single refractive surface and 60 diopters effective (in air) eye focal length. With aperture stop at the surface, its entrance and exit pupils coincide with it; likewise, its two principal points coincide at the refractive surface (P), and two nodal points coincide at the surface's center of curvature (N). Foveal center is assumed to coincide with the posterior focus, thus optical and visual axis also coincide.

A basic aberration-defining reference line in a general optical system is the chief ray: a ray that passes through the center of aperture stop, determining the reference optical path length against which are measured optical path differences of other rays. For an axial image point, the chief ray coincides with optical axis; for off-axis image points, chief ray is the one passing through the center of the aperture stop. In ophthalmology, on the other hand, it is common to refer to line-of-sight (LOS) as the chief ray equivalent - a concept not compatible with the conventional aberration theory (taking a ray other than one passing though the center of the aperture stop as the reference for optical path difference changes the wavefront aberration form, making symmetrical aberrations asymmetrical).

Optical axis associated with schematic eye is defined same as for any centered system with rotationally symmetrical surfaces, as a line connecting surface vertices. Other relevant axes and lines are illustrated on FIG. 223.

FIGURE 223: Axes and lines associated with the eye. Two of them pass through foveal center: visual axis and line of sight. The former connects fovea with the 2nd nodal point (visual axis in the image plane), and then 1st nodal point with the object point. The latter, also a broken line, connects fovea to the object point through the exit and entrance pupil centers. Pupillary axis is not associated  with schematic eye; by definition, it is a normal to the anterior cornea, directed to the entrance pupil center. This is possible only if the corneal surface is deformed, or eye elements misaligned (otherwise, normal to anterior cornea would nearly point at its center of curvature). Finally, fixation axis connects object point and eye's center of rotation.

Visual axis is defined as the line connecting foveal center to the 2nd nodal point. Being parallel to the line connecting the corresponding object point and 1st nodal point, this line defines the actual visual field angle (this is the prevailing definition of visual axis; some authors have a different view, defining visual axis as one connecting fovea with the center of entrance pupil). Due to foveal eccentricity (4°-8° toward temporal, and about 2° toward inferior retina), this line is inclined to the optical axis; temporal inclination angle is usually denoted by α. A term related to visual axis is achromatic axis, defined by a ray passing through eye's exit pupil and nodal point onto the foveal center with zero lateral color error; according to the Indiana University School of Optometry, real eyes data show that mean angular differential between this axis and visual axis gravitates toward zero, suggesting that the two nearly coincide in the average eye (near-zero lateral color is only a statistical figure, indicating that positive and negative lateral color errors nearly offset each other in the sum of individual errors; according to the same source, average foveal lateral color significantly differs from zero).

Line of sight (LOS) is a broken line connecting object point to the foveal center through the centers of the entrance and exit pupils. As mentioned, it is commonly - and incorrectly - regarded as the chief ray equivalent. In fact, it is not the rays that pass through the centers of entrance and exit pupils that constitute the chief ray: it is the ray that passes through the center of the aperture stop (thus representing the actual central ray of the entering pencil) and appears as if passing through the center of the entrance pupil (from the object point side) and through the center of the exit pupil (from the image point side).

Pupilary axis is defined as a normal to the front corneal surface (hence passing it w/o refraction) directed at the center of the entrance pupil. It is sometimes, too, referred to as the chief-ray-equivalent, which is incorrect, since chief ray passes through the center of the aperture stop. Such axis in schematic eye would always point at the corneal center of curvature. In real eyes, however, this angle (determined by measuring the angle of light reflected from anterior cornea at which it projects back at the entrance pupil center), in general, indicates corneal deformation and/or misalignment in the optical path, the larger its deviation from the visual axis - usually denoted by κ - the more so (this does not necessarily result in significantly higher level of eye aberrations; for instance, larger kappa-angle is associated with higher level of eye's compensatory coma, with both cornea and eye lens generating more of the aberration, but off opposite signs, tending to minimize the total aberration). Another angle associated with pupillary axis is that between it and LOS, usually denoted by λ (some authors use κ, which may be confusing).

Since the main purpose of exploring eye aberrations here is to gain insight into their effect on the telescope image quality, i.e. the interaction between axial and off-axis aberrations of the telescope (including eyepiece) vs. those of the eye, it is important to note that, due to the active nature of eye function, this interaction does not not produce a simple balance of axial and off-axis aberrations of the the two. Specifically, off-axis aberration of the eye are very significant, but their effect on off-axis image quality in a telescope is normally small to negligible (FIG. 224).

FIGURE 224: Reflex eye movements at the eyepiece bring image of selected field object onto the retinal spot with highest acuity - the fovea. This, in turn, requires bringing selected object onto the visual axis, connecting selected object and fovea. If the selected object is an object located in the outer eyepiece field, it effectively becomes a nearly axial object for the eye. In other words, it is primarily foveal aberrations of the eye that interact with telescope aberrations, regardless of the field location of point observed. When optical axes of the eye (right eye, top view, assuming eyepiece exit pupil nearly coinciding with eye pupil) coincides with that of the eyepiece (A), observer is looking at a point somewhat off the field center (the angle is approximated by the discrepancy between eye's optical and visual axis, usually 4-8°). In order to bring field center into foveal area, eye has to rotate clockwise around center of rotation, which nearly coincides with the center of vitreous body, until the visual axis nearly coincides with eyepiece optical axis. The off-axis point H, farther out in the field, is imaged by the eye onto the outer retinal field; it is present in the field of view, but very distorted due to the combined effect of low-resolution photoreceptors in the outer retina, off-axis aberrations of the eyepiece, and off-axis aberrations of the eye. Turning attention to this point causes reflex eye movement - counterclockwise rotation by angle a - bringing it into foveal area; since eye rotation causes eye pupil displacement vs. eyepiece exit pupil, rotation is accompanied with a slight head displacement (s), keeping the two pupils nearly coinciding (B).

Despite eye off-axis aberrations having little or no influence on the quality of telescopic image, they will be also addressed; not only for the completeness of information but, more importantly, as an indicator of possible magnitude of axial eye aberrations caused by excessive deformation and/or misalignment of eye's optical surfaces. One of possible misalignment forms is decenter or despace (or both) of eyepiece pupil with respect to eye pupil; it can induce significant axial aberrations, primarily coma and lateral color.

Eye aberrations will be addressed in four main sections:

(1) monochromatic aberrations, axial 2nd order and off-axis 2nd order,
(2) higher-order monochromatic aberrations,
(3) chromatic aberrations, and
(4) combined eye aberrations.

13. THE EYE   ▐    13.4. Monochromatic eye aberrations

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