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◄ 11. MISCELLANEOUS OPTICS   ▐    12.2. Eyepiece aberrations I ►

12. TELESCOPE EYEPIECE

12.1. Eyepiece functions and properties

•Focal length   •Field of view   •Exit pupil size   •Eye relief   •Light transmission

The image formed by the telescope objective is real, and can be observed directly. However, as explained in
1.4. Main functions of a telescope, its magnification is limited to ~ƒ/250, ƒ being the objective focal length. Also, since an average eye can't focus properly looking at objects closer than ~25cm, it can grasp only a fraction of light coming from the image formed by the objective (FIG. 6). It is telescope eyepiece that solves both of these problems. It increases telescope magnification by a 250/ƒe factor (ƒe being the eyepiece f.l. in mm), and brings all the light emitted by the object-image to the eye pupil (FIG. 7).

Telescope eyepiece is a complex positive lens system placed between the eye and the image formed by the objective. If the object image formed by the objective is located at the ocular's front focal plane, the eyepiece images every image point at infinity - in other words, it transforms (desirably) spherical wavefronts emerging from the object image's points into flat wavefronts merging at the location of the eyepiece exit pupil (in terms of rays, the eyepiece turns diverging cones emerging from the objective's image into collimated pencils of light, FIG. 209). These wavefronts enter the eye, which transforms them into near-spherical and have them focus onto the retina, creating the final magnified apparent image.

The figure illustrates basic properties of a telescope eyepiece: focal length, field of view (directly determined by its field stop size), exit pupil and eye relief. 

 
FIGURE 209: (A) Standard eyepiece with the object-image formed by the objective at its front focal plane turns the diverging cone from every point source of the object-image into a collimated pencil. Image of the objective formed by the eyepiece (dotted lines) is the exit pupil, where the pencils of collimated light merge, making possible for the eye - with the eye's iris placed approximately at the exit pupil - to receive all the light emitted by the object image (note that the exit pupil is inide the eye, changed by refraction through cornea, hence "exit pupil" shows only its diameter before entering the eye). Since the pencils entering the eye are collimated, the image formed by the eye is projected at infinity. With the objective image magnification given by the image distance ƒƒe/(ƒ-ƒe) to the object distance ƒ+ƒe ratio, the exit pupil diameter is Pex=Dƒƒe/(ƒ2-ƒe2) or, practically, Pex=Dƒe/ƒ=ƒe/F~D/M, M being the telescope magnification, and F the telescope F#. Telescope magnification M=tanε/tanα, ε being the apparent eyepiece image angle, and α the apparent object angle. The distance ER between the eye lens and the exit pupil is called eye relief. For full field edge illumination, the front lens needs to be somewhat larger than object-image in the focal plane (specific value varies with the eyepiece type - i.e. stop to field lens separation - and telescope ƒ-ratio).


(B) Smyth-type eyepiece has a negative front lens - in effect an integrated Barlow lens - separated from the positive lens froup following it. Smyth lens forms a new focus, generally farther away from the objective, also magnifying the image. In order to produce magnification corresponding to the eyepiece designated focal length, the positive lens group has to have focal length larger by the factor of Smyth lens magnification. Main benefit of this configuration - whose origins date back to the 19. century - is the reduction of off axias astigmatism by offsetting the "regular" astigmatism of the positive-power lens group by that of the Smyth lens. Consequently, it has become configuration of choice for the widefield and ultrawidefield eyepieces.
Basic properties of the configuration are: (1) Smyth lens acts a focal tele-extender, magnifying objective's image, (2) in order to produce the designated eyepiece focal length, the positive lens group has to have the actual focal length equal to the designated eyepiece focal length multiplied with the Smyth lens magnification factor, (3) the field stop diameter for the designated eyepiece focal length is determined by the actual field stop, in front of the positive lens group, and equals the actual stop divided by the Smyth lens magnification factor (so, when, for instance, Tele Vue lists the field stop of its "Ethos" 21mm unit as 36.2mm, it is the virtual field stop at the original focus of the objective, obtained by dividing the actual, internal field stop by the Smyth lens' - in this case, the Smyth lens group, that inludes a positive meniscus - magnification).

Assuming perfect eyepiece and eye, image space of the image formed by them will be identical in size and shape to the image in the objective's image space. Below is raytrace of a 200mm f/5 mirror alone, and with eyepiece, one of 17mm focal length - identical to the optical focal length of the human eye - and one with twice longer focal length. For both, eyepiece and eye is used OSLO "perfect lens". The only aberrations are those of the mirror, less than 0.02mm longitudinal astigmatism at 0.25° field radius with the best field curvature radius identical to the focal length, only of opposite sign.

Longitudinal astigmatism in the eye image is a bit smaller than for mirror alone, probably due to beam manipulation by perfect lens (in order to keep image aberration-free, perfect lens has to widen oblique pencils in the vertical plane in order to: 1 - cancel astigmatism, and 2 - compensate for the longer path of oblique pencil to a flat image plane; hence, pencils are wider by a 1/cos2α factor, α being the pencil-to-axis angle). Image height on the retina remains unchanged with 34mm eyepiece, despite the twice larger image by the objective (assumed is unchanged eyepiece AFOV, hence twice larger field angle in the image by the objective), because the twice larger exit pupil makes the eye twice faster a system, with the image scale cut in half. In the real world, both eye and eyepiece contribute their own aberrations, which affects the final astigmatism and field curvature, but the overall image scale remains unchanged.

                                                                  1. FOCAL LENGTH

The main eyepiece parameter, determining its basic function - magnification - is its focal length ƒe. Normally, it cannot be directly measured as the separation between the lens and image of a distant object, because the lens (i.e. eyepiece) thickness is large compared to the f.l., and the effective entrance pupil is projected into the eyepiece. From Pex=ƒe/F it follows that ƒe=FPex, i.e. eyepiece focal length is given by a product of the telescope focal ratio and exit pupil diameter. That, of course, requires knowledge of telescope's focal length a precise measurement of the eyepiece exit pupil.

An alternative method of establishing ƒe is measuring the image size of a relatively distant object vs. size of its image produced by the eyepiece. From the geometry of image formation (FIG. 7), the proportion between object and image height ho and hi, respectively, equals that between object and image lens separation, So and Si, in the same order, thus

Si = Sohi/ho,

irrespectively of lens thickness. With focal length defined by Eq. 1.4 as ƒ=SiSo/(Si+So), and hi/ho=Si/So, the relation between image separation and focal length is Si/ƒ=(Si+So)/So=(Si/So)+1=1+(hi/ho) and

 ƒ = Si/[1+(hi/ho)] = So/[1+(ho/hi)]

Thus eyepiece focal length can be obtained from the measurements of the object size (ho), image size (hi) and object distance (So). The object's image can be measured either as projected on a piece of paper, or directly as seen at the eye lens. Best object type is a bright, uniform surface with well defined boundaries, such as a well lit window. Size of the image should be kept at the minimum allowing accurate measurement, in order to minimize distortion (which tends to enlarge outer field portion, thus to result in shorter than actual eyepiece focal length if that portion of the field is used). Also, greater object distance - 50 to 100 focal lengths - will minimize the error of (most likely) adding to the object distance, measured from the field lens, a small differential between that point and second principal plane of the eyepiece.

                                                                 2. FIELD OF VIEW

The next important eyepiece parameter is its field of view. Limit to the eyepiece apparent field is set by its field stop, an axially centered opening in front of the field lens, which for focused eyepiece coincides with the objective's image plane. Angular size of the field stop as seen from the center of the entrance pupil (α, FIG. 209) is called true field of view (TFOV), and its angular size as seen through the eyepiece (ε, FIG. 209) is apparent field of view (AFOV) of a telescope (here, both are presented as field radius; often times, the terms are also used for the field diameter).

Illustration above shows the geometry of eyepiece's apparent and true field of view. Since eyepiece effectively allows the eye to observe objective's image from the distance equaling eyepiece's focal length, the lenses can be entirely left out. Eyepiece magnification M is given by a factor of reduction of observing distance vs. naked eye, equaling V/f (which multiplied by magnification of the objective, f'/V, f' being the objective's f.l., gives telescope magnification). With zero rectilinear distortion, condition of which is that any zonal height "h" in the field stop becomes M times larger through the eyepiece, i.e. that the tangent of its magnified angle is M times larger than tangent of the unmagnified angle. The corresponding angular size of the field stop in the eyepiece is the geometric field of view (GFOV). Deviation from this condition causes curving of straight lines in the eyepiece image. With zero angular distortion, every zonal angle in the stop is M times larger in the eyepiece image and, since multiplied angle is larger than the angle corresponding to its multiplied tangent, the apparent stop size with zero angular distortion is always larger than with zero rectilinear distortion, i.e. requires non-zero (positive) rectilinear distortion. Deviation from zero angular distortion causes elongation of shapes, either vertical (positive), or horizontal (negative). Actual apparent stop size can be anywhere from smaller than GFOV (w/negative rectilinear and angular distortion) to larger than zero-angular-distortion angular size (w/positive rectilinear and angular distortion), but usually is close to the zero-angular-distortion size.

The AFOV/TFOV ratio approximates telescope magnification; however, its approximate accuracy varies with the degree of eyepiece image distortion. Since magnification is defined as apparent image size vs. apparent object size, it is given by the ratio of tangents, or M=tan(AFOV/2)/tan(TFOV/2). With the TFOV being always a small angle, image distortion is negligible, and the actual image angle is practically given by tan(TFOV/2)=T/2ƒ, T being the diameter of eyepiece field stop (being a small angle, it is also closely approximated in degrees by TFOV~180T/2ƒπ).

However, the eyepiece field of view, much larger angularly, suffers from significant distortion. In the telescope eyepiece, it is usually positive distortion, which means that image magnification increases exponentially with the image point height. In effect, the outer image portion is stretched out and seen at a magnification higher than that for the inner image portions (this may and may not be accompanied with spherical aberration of exit pupil). In effect, AFOV inflated by distortion implies the field stop - and true field - larger by the factor of distortion than what it actually is. That is relatively insignificant with small AFOV eyepieces (~10% average in a conventional ~50° eyepiece), but since distortion increases with the third power of the angle, it can be a factor in the wide-field varieties. For instance, a zero-distortion 10mm 60° AFOV eyepiece would have 11.5mm field stop diameter, while one with 10% distortion (i.e. 66° AFOV) would imply ~12.6mm.

The field stop diameter corresponding to the actual (distortionless) angular field is always smaller than the stop implied by the nominal eyepiece AFOV. With the usual magnitude of distortion ranging from up to 5% with 40° AFOV to up to 20% with 80°, it is approximated by T~ƒeAFOV/[58-(AFOV/58)], for AFOV in degrees (plot at left).

The distortionless AFOV is directly related to the field angle of the objective, which is smaller according to telescope magnification (keep in mind that magnification is not angular, but relates to the tangens, so the accurate way of relating the two is by determining the field stop radius from the eyepiece AFOV, divide it with magnification and then findthe corresponding field angle). Figure below shows the limit to zero-distortion AFOV imposed by eyepiece barrel, i.e. without a field stop ring.

3. EXIT PUPIL SIZE

Another important eyepiece-related parameter is the size of its exit pupil. It directly determines image brightness level relative to the object, as well as the level of eye aberrations. The size of eyepiece exit pupil is inversely proportional to telescope magnification. For the relative magnification m, in units of aperture diameter, eyepiece exit pupil is given by 1/m for aperture in mm, and by 25.4/m for aperture in inches. So, for instance, relative magnification of 0.5 per millimeter of aperture (50x for D=100mm, with m=0.5), results in 2mm exit pupil diameter, and so does 12.7x per inch of aperture magnification (m=12.7).

In terms of the telescope focal ratio F=ƒobj /D, exit pupill diameter is given by P=ƒe /F, where ƒe is the eyepiece focal length. Another important number related to the exit pupil diameter is the Airy disc angular diameter, given by 4.6/P(mm) in arc minutes.

At the exit pupil size larger than about 2mm in diameter, eye aberrations begin to dominate diffraction effect, increasing progressively with the pupil size. Thus, telescopic resolution is aberrations-limited for exits pupils larger than ~2mm, and diffraction-limited for smaller pupils.

4. EYE RELIEF

Eyepiece eye relief - the separation between the eye and exit pupil - is mainly important for observing convenience. As with most anything else, too little is as undesirable as too much. Short eye relief, typical of the short f.l. conventional eyepieces, prevents observer from placing eye at the exit pupil, thus effectively reducing apparent field. It also causes eye strain, detrimental to the quality of observing. Too long eye relief, often encountered with long f.l. eyepieces, especially when used with a conventional (not telecentric) Barlow lens, makes it difficult to find exit pupil and keep eye pupil on it. Size of eye relief varies mainly with the eyepiece type and design (FIG. 215 and FIG. 213), although it may vary somewhat within them as well. Image below (under Light transmission shows variation in eye relief of some familiar eyepiece designs.

5. LIGHT TRANSMISSION

The two factors determining light transmission in the eyepiece is reflection off its lens surfaces, and in-glass absorption. Nothing can be done about the latter, which leaves reflection as the only concern, and not so much with respect to light transmission itself, but more for the negative effect of reflected light on image quality. Aside a few specialty eyepieces employing only a few refractive surfaces, with either air-glass or glass-glass boundaries, modern eyepiece designs have anywhere from several to a dozen or more such surfaces. It makes application of antireflective coatings even more important than with refracting objectives or subaperture correctors. More so considering large incident angles at their lens elements, increasing reflection loses toward lens edge. Four random designs show that incident angles for the edge cone central ray go up to 40° with the standard 50-deg AFOV eyepiece, and up to 50°, or so, with 80-deg AFOV eyepieces (below). All four are 10mm f.l. units, with the Nagler's positive lens group having 23mm f.l., indicating the Smyth lens magnification factor of 2.3. For each surface, given are the angle between z-axis (optical axis) and ray exiting surface (Yang) and angle of ray incidence onto surface (Iang), in degrees.

With today's ultrawides edge incident angles can be even larger (with each surface, the top angle is vs. optical axis, and the angle below it is vs. normal to the surface). This creates the basis needed to relate meaningfully to the reflectance vs. incidence graph (below). It shows how reflection on the lens surface depends on the incident angle (uncoated glass, unpolarized light; ni is the index of incidence, and nt index of transmitting medium). Plots at right are a magnified area of the plot at left, showing that reflections off glass-glass surfaces (3, 4) can be significant, more so considering that they cannot be suppressed with coatings.

The four plots outline reflection range on air-glass and glass-glass surfaces (form of Fresnel equations - square of a differential - makes order, air-glass vs. glass-air, and glass1-glass2 vs. glass2-glass1 inconsequential with respect to the reflectance level). The denser glass, i.e. the higher refractive index, the higher reflectance: for normal incidence air-glass (or the other way around) reflectance is 4% for glass refractive index n=1.5 (1), and more than twice as much for n=1.8 (2). While reflections off air-glass surfaces (1,2) can be significantly reduced with antireflective coatings, reflections off glass-glass surfaces (3,4) remain and can be significant: over 2% at 45° incidence with n=1.5 and n=1.8 (3). Also, a single layer antireflective coating, like MgF2, while relatively stable with respect to the angle of incidence (left), still leaves too much light bouncing between surfaces, to end up as a stray light on the retina. More sophisticated, multi-layer coatings, on the other hand, can reduce reflection to a fraction of a percent, but are generally more sensitive to the incident angle increase. Specifics vary with the coating, but large incident angles may not only cause significant shift in the optimal wavelength, but also significant - and asymmetrical - increase in reflectance over certain parts of the spectrum. It is illustrated by a coating reflectance plot from Thorlabs, showing that at 45° incidence optimum wavelength shifts to ultraviolet, while reflectance increases toward green and red, with disproportiontelly high reflectance in the red. In general, transmission loss due to reflectance is a from about 1.4% with a single-layer (MgF2) coating, to a fraction of a percent per surface with multilayer coatings, likely to increase toward field edge.

For near-normal incidence, reflectivity depends on the relative strength of refractive index of the coating vs. subtrate. For 1/4 wave thick layer of MgF2 (refractive index 1.337 at 546nm), reflectance varies from less than 0.8% with n=1.9 subtrate, to 2.5% with n=1.4 (below, left).

The differnce can be significant, as shown at right for MgF2 layer on BK7 (1) and N-LASF31 (2). Reflectance can be further reduced with multi-layer coatings: with only two layers reflectivity around central wavelength can be reduced close to zero (3). With three layers a more uniform reflectivity over wider spectral range (4). The top and bottom layer are 1/4 wave thick, with 1/2 wave thick mid layer (based on data from Lens Design by Milton Laikin)

Absorption is about 4% per inch of in-glass path, decreasing toward field edge, although partly offset by the negative - usually dense - elements becoming thicker toward the edge. For the four eyepieces shown above (10mm f.l. units), axial glass thickness is from below 9mm (Plossl, Abbe being a bit thicker, around 10mm), to 16mm (Erfle) and 36mm (Nagler1, only the positive group), hence with approximately 1.4%, 2.5% and 5.8% absorption loss, respectively.

As mentioned before, eyepiece light transmission is primarily important because of the light reflected of air-glass and glass-glass surfaces. Part of it is making to the retina, negatively affecting image quality. Internal reflections off eyepiece lens' surfaces can be dominantly diverging toward the eye, dominantly converging, or a mix of the two. Converging reflections can create ghost images, but what is the least desirable - and amounts to a poor design - is when reflections off different surfaces are of the same kind, reinforcing each other. That may, of course, be to some extent dictated by the need to minimize the aberrations, but should be kept as low as possible.
 

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