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13.2. Eye aberrationsConsidering its biological origin - as opposed to precisely crafted optics - it is not surprising that optical aberrations of the eye are significant. Common focusing errors such as myopia (short sight) and hypermetropia (long sight) don't affect image quality through a telescope, due to automatic correction at the focuser. Similarly, considerable chromatism produced by the eye is somehow corrected during brain signal processing. No such luck with roughness, astigmatism and spherical aberration, caused by surface deformations and lack of media homogeneity. Even a good average eye - mostly due to inhomogeneous medium and local surface errors - generates enough of the wavefront error to require pupil diameter smaller than 2.5mm for near diffraction limited level (~λ/14 RMS for the 550nm wavelength). The error rapidly increases with the pupil diameter to ~1/3 wave RMS at 5mm. It also plummets with the decrease in pupil diameter, to bellow 1/50 wave RMS at the pupil diameter of 1mm. Most of the refractive power of the eye comes from cornea, and in particular 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. Spherical Spherical aberration of the eye varies widely individually, with the possible average being about 0.25 diopters at 5mm pupil diameter. 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 ~60. The larger diopter, the stronger optical power. The difference between the optical powers of the central and outer zones 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 62. 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 58D 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. Defocus expressed in diopters (D) can, obviously, easily be converted into longitudinal defocus L, from L~DƒE/60. It is directly related to the P-V wavefront error of low-order spherical aberration at best focus, given by Ws=L/64F2, with the focal ratio number F for the eye given by P/ƒE, P being the effective pupil diameter. Since longitudinal defocus due to spherical aberration varies with the square of the aperture diameter, an average defocus is approximated by L~ƒE(P/5)2/240. Hence, the wavefront error of spherical aberration in the eye typically varies between 1/3600 and 1/5.7 wave P-V at 1mm and 5mm pupil, respectively. A note on why the effective focal length ƒE~17mm is used to calculate the wavefront error, and not the actual eye focal length of ~22mm. What matters in the error calculation is the size of longitudinal/transverse aberration relative to the Airy disc diameter. And 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. Why this occurs is due to a media 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 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). Same goes for errors induced by the eye lens: they 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 are reduced in size, but so is the wavelength. Should the eye retina be an over-sampled detector, its conventional linear diffraction cutoff frequency λF would be smaller by a factor ~0.75, due to reduced effective wavelength λ; and so would its nominal angular cutoff frequency λ/D. However, general form of the diffraction resolution limit - λ/D - remains unchanged. Back to using the "air" eye focal length: as the Airy disc formed at the retina is appropriate in size to one produced by the F-number resulting from F=actual eye f.l./1.33P (P being the pupil diameter), not the one given by F=actual eye f.l./P, the appropriate focal length to use as a basis for calculating the defocus error is 22.2mm/1.33=17mm. It gives a proper match of the longitudinal/transverse errors and the Airy disc size. Eye astigmatism Astigmatism in human eye can be axial, which most often originates at the cornea due to a small cylindrical component built into it, or regular off-axis astigmatism. The extent of the former is determined by the severity of deformation. The latter becomes significant only at relatively large incoming angles. Longitudinal off-axis astigmatism in the average eye has little over 7 diopters (D~7) defocus at 60° off-axis, and changes approximately in proportion to α1.6, α being the off-axis angle. With the longitudinal defocus L~DƒE/60, and the P-V wavefront error of astigmatism Wa=L/8F2, an average off-axis astigmatic error ranges from ~39 waves at 60° off-axis to ~2.2 waves at 10° off-axis, for 5mm pupil. Since the 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, the off-axis astigmatism is 25 times smaller than for 5mm, or about 1.6 wave and 1/11 wave at 60° and 10° off-axis, respectively. Note that the RMS error of 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 spherical aberration. Coma Coma appears as an axial aberration in human eye only exceptionally. Off-axis, it is relatively insignificant compared to astigmatism. Eye chromatism The eye suffers from significant chromatism, in the form of secondary spectrum. Chromatic defocus can also be expressed in diopters; for the blue F-line, which focuses ~0.6 diopters shorter than green light (0.6D), and the red C-line that focuses ~0.4 diopters longer (-0.4D), the combined defocus between the two is about 1 diopter (Fig. 126).
FIGURE 126: Secondary spectrum in the human eye focused at infinity (exaggerated for clarity). Unlike spherical aberration, astigmatism, and some other aberrations, there seem to be relatively little individual variation in eye's chromatism. Shorter wavelengths focus shorter than green light, and longer wavelengths longer. The blue focus (F-line) is defocused by about 0.6 diopters, or 1/100 of the eye lens' f.l., while the red (C-line) defocus is about -0.4 diopters, or 1/150 the eye lens' f.l. The brain effectively removes chromatism, as long as all the wavelengths appear as if coming from one point. This is not the case when observing with an achromat, whose red/blue foci are typically closer to the eyepiece field lens than the optimum green focus thus appear to the eye as originating from locations significantly nearer to it . However, even if the brain automatically exerts its natural chromatic correction, the effect is negligible, since its effective selective extension (blue) or reduction (red) of the actual focal lengths for various colors nearly averages out with similarly defocused red/blue wavelengths. Averaged secondary spectrum is, thus, about 0.5D, or 1/120 of the eye lens' focal length. That is some 16 times more than in a doublet achromat, and is quite significant at medium and large pupil diameters: at the 6mm pupil size, the averaged red/blue blur is some 13 times larger than the Airy disc (a 4" ƒ/3 achromat level). The fact that we don't see this chromatic defocus indicates that the brain's corrective refocusing approximates -0.6 diopters in the blue, and 0.4 diopters in the red. Combined aberrations, diffraction The eye also suffers from higher order aberrations, which can be significant, especially at large pupil sizes. Some other errors, like "floating motes", don't affect the wavefront, but can affect quality of observing. And so can a variety of perception and signal processing errors. Eye aberrations can be approximated by an optical system made to its general specifications. There is a number of optical models of the eye. The simplest one is the Emsley model (1946), employing a single front surface of 5.5mm radius of curvature (convex) followed by imaging medium of 1.333 refractive index. The model most closely reproducing eye aberrations is a complex system proposed by Liou and Brennan in 1997, incorporating aspheric surfaces and varying refractive indici. The effect of wavefront aberrations - either those induced by the eye itself, or externally - is adding to the effect of diffraction in a further spread of the point-image energy, as illustrated on FIG. 127. For pupil diameters larger than ~2mm, it is eye aberrations that determine image quality. Diffraction become dominant for pupil diameters smaller than 2mm. In terms of the RMS wavefront error, eye aberrations increase from ~1/80 wave at 1mm pupil, to ~1/4 wave at 5mm (individual variations increase with the pupil size, at 5mm being in the 1/3 to 1/5 wave RMS range).
FIGURE 127: Top left,
optical power (in diopters) of the two main elements of the human eye,
cornea and crystalline lens (number
in
brackets is corresponding f.l. if imaging in air; for the eye as a whole,
assuming a single imaging element with 22.2mm f.l. effective in the
aqueous medium).
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. Eye begins to recognizing image shape, as its
size exceeds ~3 arc minutes. Any detail bellow that size appears
point-like to the eye. Consequently, in order for the eye to discern the
effect of wavefront aberration 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 ƒ/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 ~ƒ/5 paraboloid.
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.
Besides eye aberrations, important factors determining
properties of the image formed by the eye are its spectral response, intensity response, and contrast sensitivity. |
