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12.4. Telescope eyepiece: comparative raytracing       13.3. Eye aberrations


The light passing through the telescope objective and eyepiece is focused onto retina by the optics of the eye, after which it is neurally processed into the visual image. Hence it is the optical media of the eye - cornea, crystalline lens, and aqueous fluid filling the eye - that determine the final shape of the wavefront reaching retinal photoreceptors. Being bio-engineered, eye optics is far from high standards of a quality telescope. However, due to its small aperture and low magnification, significant aberrations generated by it generally do not have much of an effect on the perceived image quality. The two exceptions are eye defocus error - which is effectively corrected by appropriately defocusing the eyepiece, thus inconsequential for the telescope user - and, to a smaller extent, eye astigmatism.

Placing eye at the eyepiece end of a telescope changes its optical parameters. This change is significant enough to justify establishing a specific term for this particular mode of operation: the telescopic eye. The most important change is that its aperture stop is now the eyepiece exit pupil, not the iris. This not only directly determines the level of eye aberrations due to its effective aperture, but also may induce additional aberrations resulting from displaced stop. In addition, unlike the "unarmed" eye, which observes objects directly, the telescopic eye observes diffracted image of these objects. In general, the consequences of it are not significant, but should be addressed nevertheless.

Neural response and processing of light signals in the eye varies significantly with their intensity and wavelength, making it an important subject for the telescope user.

In evaluating the properties of telescopic eye, the starting point is, unavoidably, optical properties of the eye alone

13.1. The human eye: Physical properties, transmittance and acuity

The entire purpose of a visual telescope is to gather light from distant objects and to enlarge the incoming light angles, so that these objects appear brighter, larger and more detailed to the eye. Added benefit of the larger aperture of a telescope is lessening limitations to image quality imposed by diffraction. Since the light passes through both, telescope system and the eye, optical properties of the latter can affect the final image created by the brain.

Being an optical element itself, the eye is, just as the telescope objective, subjected to the effects of diffraction of light and wavefront aberrations. Physical and optical properties of human eye vary individually, often significantly; those presented here are based on experimentally determined averages (FIG. 216).

: Top view of the human right eye cross-section. The light entering the eye first passes through the cornea (refractive index n~1.38), ~0.5mm thin negative meniscus of ~7.8/6.5mm radii. Anterior chamber, between cornea and the eye lens is filled with aqueous humour, watery fluid with n~1.33. The iris is a muscle that forms circular opening - the pupil P, eye aperture - varying from ~2mm to ~8mm in diameter. The eye lens is bi-convex, made of thousands of roughly concentric layers, with refractive index from ~1.41 in the center to ~1.38 at the edges. Change in the lens' shape accomplished by the action of ciliary muscle enables eye accommodation, maintaining focus on the retina for varying object distances. When focused at infinity, the lens is ~3.6mm thick, with front and back radii ~10mm and ~6mm, respectively, and focal length f~24mm. The eye focuses through the vitreous humour, watery fluid (n~1.33) inside the vitreous body, onto the retina, made of the layers of cells and neural wiring connecting photo-receptors to the brain via optic nerve. Small spot on the retina - fovea centralis - is about 4.5° (1.3mm) in diameter, 4-8° off the optical center. Its central ~1/3, is foveola, the most acute vision area of the eye in day-light conditions (as opposed to the outer area, highly sensitive to low-intensity light). Square at right shows simplified scheme of a retinal segment. Light falling onto it first passes through plexiform, the outer retinal layer consisting of a network of neural cells and wiring through which the brain both, controls photoreceptor function and receives light-generated input from them. The two main types of retinal photoreceptors are cones - active in daylight conditions, covering the entire foveola, and most of the outer fovea - and rods, active in low-light conditions, predominant in the wide retinal area surrounding fovea. The last, third retinal layer is choroid, which has a dual function of providing nourishment to the photoreceptors, and absorbing any remaining light, in order to prevent internal reflections.

Retinal arc extends ~32mm through the central meridian. Outer retinal area of relatively low sensitivity to daylight surrounds the yellowish oval spot of ~4mm (nearly 15°) in diameter, centered at ~3.4mm (nearly 12°) from the optical axis of the eye, called macula, which converges toward fovea, the highest daylight sensitivity area. From the outskirts of macula outwards, roughly 20° wide, extends the ring-shaped area of the highest sensitivity to low-intensity light.

Eye light transmittance is relatively high in the 500nm-700nm range (and beyond, into infrared), but falling off quickly toward the blue/violet end of the spectrum (FIG. 217).

FIGURE 217: (A) Range of eye spectral transmittance, based on several small-scale studies. The results indicate wide individual differences, although it could also result from small sample sizes (four to nine individuals in four separate studies) and/or differences in procedures. It is unclear whether eye transmittance - specifically its preference for mid- and longer wavelengths within the visual spectrum - has been factored out from the eye spectral response (sensitivity) curve. If not, it would superficially lower actual sensitivity of the eye in the blue/violet relative to that in the green and red for both, cones and rods. Since the relative change in transmittance over the range of wavelengths doesn't seem to vary significantly with the transmission level, it shouldn't affect individual perception of chromatism. Variations in eye transmittance would mainly affect perceived brightness, with the difference between high and low transmission level being close to one magnitude, roughly evenly across the visible spectrum (possible exception is that some individuals may have the ability to sense wavelengths well below 400nm, and some not).
(B) Equally as important as light transmittance, and the final judge of how much light actually is detected, is retinal absorption. There are individual variations here as well, but a study on 28 subjects should be sufficiently representative of this aspect of light detection (left). The difference between total transmission and absorption is mainly light scatter and absorption by other mediums. As the graph shows, unlike the total transmission, which remains quite high well into the infrared, retinal absorption steadily decreases from its maximum at about 0.51 micron toward longer wavelengths. On the other hand, the drop-off is very similar to that of retinal transmission toward short wavelengths.

Eye photoreceptors cells, cones and rods, form the light-sensing lining of the retina. We need them to sense light, just as we need nerve endings in the skin to sense touch. They range in size from ~2μ to over 10μ, in general becoming larger toward the outer area of the retina, the cones more so than the rods. Dominant retinal photoreceptors at medium-to-small pupil sizes (daylight and indoor light conditions) are the cones, while at large pupil sizes (in low-light conditions) the dominant photoreceptors are the rods. The two differ significantly in, among other properties, their respective resolution limits. Eye resolution level is termed acuity. It varies over the retina, depending on the receptor type and size. It is also a function of the illumination level (FIG. 218).

FIGURE 218: LEFT, approximate average resolution of cones and rods over the retina varies with their size, density, and type of neural connection. Center of the fovea (foveola) is populated exclusively with cones, their density approaching 200,000 per mm2. Outside fovea, cone density quickly decreases, and their size increases, with the cone resolution falling to a fraction of that in foveola. Rods reach their highest density just outside of fovea; there, they are nearly as numerous per mm2 (which directly implies the size of individual receptor) as foveal cones, but their resolution is only a fraction of the cone resolution due to the neural convergence of signals from individual receptors (as opposed to individual processing of the signals from foveal cones). To the RIGHT, resolution in lines per arc minutes as a function of illumination level, for the photopic (bright-light), scotopic (low-light) and mesopic (transitional) eye modes. Maximum rods resolution is somewhat over 5 arc minutes, a fraction of the maximum resolution of the cones. As mentioned, resolution of rods is inferior due to the input from several rods merging before they reach the eye nerve; the purpose is increased sensitivity, at a price of lowered resolution. Cones, on the other side, send individual inputs to the eye nerve, maximizing resolution, while partly sacrificing sensitivity. With the Airy disc for a typical 2mm photopic eye pupil diameter being ~1.6 arc minutes, diffraction resolution  (defined as the FWHM of the PSF, or 0.4 of the Airy disc diameter) is in the 0.6'-0.7' range (in laboratory conditions; not to be confused with the limit in field conditions, which is ~1 arc minute at best, and usually somewhat more).

The area of highest cone acuity coincides with the area of their highest density and smallest individual size - foveola. Area of the highest rode acuity is just outside the macula, in the ring roughly centered at the fovea, some 10° to 15° in radius.

Highest acuity level doesn't coincide with the highest image quality, in terms of contrast level. For the naked eye, retinal images are of the highest quality at a pupil diameter of ~2mm (which means in bright-light conditions with the cones dominant), when the combined effect of aberrations and diffraction is at its lowest. In regard to point-image resolution, it is better at ~4mm pupil size (in dim light conditions), with the cones still sufficiently active, diffraction disc is half the size of the disc at 2mm pupil, and the aberration level of ~0.15 wave RMS still doesn't significantly affect the size of central diffraction disc, thus neither resolution of near-equal intensity point sources. However, for most other detail forms, resolution is inferior to that at 2mm pupil size.

Retinal resolution shown at left applies to the naked eye, which forms its own point-source diffraction image. For the telescopic eye, there is no point sources, since it images (through the eyepiece) the Airy disc formed by the objective. Hence, given aberration level of the objective, it is the level of eye aberrations that determines image quality which, in general, favors smaller eyepiece exit pupil (this, in turn, favors smaller apertures, with smaller exit pupil for given nominal magnification). However, this effect is, after a certain level, outweighed by the negative effects of higher magnifications.


At the telescope end, eye plays a dual role; it is an optical element and a photo-detector. Eye optics preceding retinal photoreceptors determines quality of the image formed at the receptors, while size and photo-sensitivity of the receptors, combined with the modes of neural processing of their input, determine perceived quality of his image.

Optical part of the eye, consisting of the cornea, eye lens and aqueous fluid, is a simple 2-element system. Expectedly, it generates significant aberrations, on- and off-axis. Eye off-axis aberrations are generally irrelevant, since fixation onto selected object by eye movements brings that object onto the visual axis, with its image falling onto fovea. Image quality in the outer field quickly decreases, but it is inconsequential since the field of high acuity is narrow, and both increasing size and pooled circuitry of the outer retinal receptors farther from central retina - as opposed to individual circuitry of foveal receptors - actually set acuity limit for the outer field, rather than its inherent aberration level.

On-axis eye aberration, on the other hand, have disproportionally small effect on perceived image quality. That is mainly the consequence of low effective magnification (i.e. small size of the Airy disc on retina) of the image formed by eye. However, the magnitude of eye aberrations - in particular, defocus and (central) astigmatism - is commonly large enough to noticeably degrade image quality in daily life. Luckily, telescope users are given two important breaks: eye defocus error is effectively corrected by the offsetting error in focusing the eyepiece, and effective eye pupil (determined by the exit pupil of eyepiece) at higher magnifications is small, exponentially lowering eye aberration level.

Placed behind telescope eyepiece, eye is looking at the image formed by objective that is magnified by the eyepiece. The corresponding optical scheme is different than for the eye looking at an object directly (FIG. 219).

: Simplified scheme of the retinal image formation through telescope eyepiece, with
the back focal plane of the objective nearly coinciding with the front focal plane of the eyepiece. Geometrically, light from every point A in the image formed by the objective is transformed into collimated pencil by the eyepiece. These pencils converge into the eyepiece exit pupil, located at the cornea, and are focused by the eye onto the retina (A). With the iris larger than the exit pupil, as is normally the case, the effective aperture of the eye equals eyepiece exit pupil, and the aperture stop is approximately at the cornea; the effective focal length slightly increases, due to the angle of convergence being slightly smaller than the corresponding tangents (i.e. marginal ray height ratio), nearly as it would with the iris reduced to the size of eyepiece exit pupil. This simplified imaging geometry illustrates that angular size α of the Airy disc subtended at the sky appears magnified by a factor fO/E for eye observing its image from distance E without eyepiece (fO being the focal length of objective), and by a factor fO/fEP observed through eyepiece of focal length fEP (it effectively places focal plane at the distance equal to its focal length from the eye). Angular size of the Airy disc of telescopic eye equals that of its object - the image of Airy disc formed by the objective, subtended (magnified) in the eyepiece - given by Mα, where M=f O/ f EP is the telescope magnification (B).

Since the object image formed by the objective is subjected to its diffraction, every object point in it is replaced by a diffraction pattern formed by the objective. For 0.55μ wavelength, the true angular size of the Airy disc formed by the objective is 4.6/D in arc minutes, for aperture diameter D in mm. The same relation applies to the final image formed in the eye, corrected by the factor of magnification. The corresponding apparent angular Airy disc size is 4.6F/fEP. This is larger than Airy disc of the objective 4.6/D by a factor of f/f EP - equaling telescope magnification - where f is the telescope focal length.

As long as the exit pupil is smaller than eye pupil, there is no additional diffraction effect for the image formed on the retina. It is because the exit pupil wavefront, as small as it is, is still a part of the continuous optical train, starting with the telescope aperture stop (just like the Gregorian secondary, placed after the intermediate focus, doesn't induce additional diffraction, only magnifies). If, however, exit pupil becomes larger than eye pupil, the physical diffraction pattern on the retina becomes larger correspondingly to the change in f-ratio, which with the eye focal length unchanged, results in as much larger angular size. While it is still the apparent size, with the image scale unchanged it has the effect of that much larger true angular Airy disc. It also results in a light loss, proportional to (D'/D)2 for unobstructed, and [(D'-o)/(D-o)]2 for obstructed apertures (D, D' and o being the original aperture, the effective aperture, and the central obstruction, all normalized for D=1). The relative obstruction size for the latter also increases, but the effect is negligible due to very low magnifications.

Since exit pupil formed by the eyepiece is a real image of the objective, it is as such present in the eye pupil area. Duality of the exit pupil as being "made" of two different wave sources coming from the objective is hinted at on FIG.209: waves from each point of the objective focusing into image points merely pass through the exit pupil as collimated pencils after being refracted by the eyepiece, while each point of the objective fills the field stop with light that actually focuses onto the exit pupil and forms real image of the objective (for the central point this light is outlined above axis within the angle α). With the image field seen through the eyepiece projected at infinity, this real image is suspended in the view, but usually not detectable. Physiologically, image field is produced from beams focused through the exit pupil onto retinal receptors, but there is also light fallout on them from the exit pupil as an object on its own. Image below illustrates location of exit pupil and its dual nature.

Pupil is formed as a final image in front of the eye pupil, after refraction through cornea (note that in the presence of obstruction each pencil has a dark core, but it is not detectable in the point image on the retina, no matter how big the relative obstruction). There is no focused light from the exit pupil on the retina, so there is no real, nor virtual image on it (thus somewhat smaller than the axial pencil exiting the eyepiece). Parallel pencils of light pass through it, but any random point A on it, formed by the light originating on the corresponding point in the aperture, emits diverging cone of light toward the eye lens. It is way too close to be imaged on the retina, but light close to the chief - i.e. middle - ray, being nearly parallel, does get processed by the lens, bent toward retina and at least partly focused on it. This is a small fraction of the total light emited by the point, making this ghost image very low in intensity. However, in the presence of central obstruction, it can become visible if the background is relatively bright (as if in daylight observing, or observing bright objects). This intermediate image does contribute light energy to the receptors, projecting onto the objective's image. The bright portion of it makes it brigher, and the annulus area - when present - not, or in effect makes it darker; that can create a ghost image of the annulus suspended in front of the real image of the observed object. Larger relative size of the obstruction in eye pupil (e.g. with exit pupil larger than eye pupil) make the obstruction shadow appear larger, but the main factor is pupil brightness (which is also dependant on magnification).

12.4. Telescope eyepiece: comparative raytracing       13.3. Eye aberrations

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