11. MISCELLANEOUS TELESCOPE OPTICS

11.4. WAVEFRONT SPLIT ERROR IN THE AMICI PRISM

In order to restore the proper horizontal orientation to the image, Amici prism uses configuration with its back side split into two surfaces coming together in the plane containing optical axis, and at 45 degrees with respect to it. As a result, converging wavefronts containing this line are split in two, with each portion being reflected to the opposite side, and after reflection on that side merging together in the point image. If the prism is less than perfectly symmetrical, these two parts of the wavefront will have different optical path lengths, with the phase differential producing aberrated diffraction images.

In addition, since a prism acts as a plane parallel plate, inducing longitudinal chromatism, the color foci of the two wavefront portions won't coincide, which can result in noticeable color infidelities. But this effect is generally smaller and less important than the diffraction effect at the best focus.

Images below are OSLO simulations of these diffraction effects, for two simple scenarios: (1) even phase error between the two wavefront portions, caused by one side of the prism being slightly longer, and (2) the error gradually increasing away from the dividing line, as a consequence of one back side being at a slightly different angle. The two parts of the wavefront have a constant path difference. In this case, the part of the wavefront left of the central line is delayed, i.e. having the longer path with respect to the other one. Converging beam has a relative aperture of f/5, with the prism front side 100mm in front of the original focus, and about 10mm wavefront diameter at the splitting line. About 1/9 wave of spherical aberration induced by 50mm in-glass path is present in all simulations.

The in glass differential δ produces optical path differential (n-1) δ, where n is the glass refractive index (in this case the glass is Schott BK7, with n=1.517 for the 546nm wavelength).

The side length error is generally acceptable for δ~λ/4 and smaller (corresponding to little over 1/8 wave of optical path differential). It is still better than diffraction limited for twice as large error, but doubling it again makes it unsuitable for higher magnifications (with the wavefront diameter at the splitting line of 10mm, the width of the field affected in the final image is nearly as much). At δ=1 and the wavefront split in two halves, the resulting diffraction image is split in a double maxima (MTF graphs below show the contrast consequence).

In the second scenario, the path difference, i.e. wavefront error gradually increases away from the split. The prism side angle deviation is 1/4, 1/2 and 1 arc minute (the actual error is somewhat larger, due to the longer path to the opposite side). Since the wavefront becomes folded, resulting aberration has similarities with astigmatism, particularly when the two wavefront portions are comparable in size.

For smaller prism errors, resulting wavefront errors are the smallest for the wavefronts split in two, since they are positioned over the area of lower deviation (that changes with the largest prism error, because large wavefront errors result in a different, less predictable phase combining). MTF graphs on the bottom shows contrast loss for the three patterns with the largest prism error. The simulations suggest that the acceptable prism error of this kind should be below 10 arc seconds. It is possible to correct this kind of errors with phase coatings, but it would require accurate measurement of the prism shape before it can be applied; in other words, it would make it prohibitively expensive.

Another kind of diffraction artifact comes from the middle line where the two double slanted sides meet. If it was near-perfect, with negligible width, there would be no noticeable effect. If it is, instead, say, 0.05mm wide, with the cone width at that location of, say, 10mm, it would be effectively equaling a vane of 1/200 aperture width (e.g. 0.5mm with 100mm aperture). With bright objects, it would cause a thin long spike orthogonal in orientation to the prism's middle line.

11.5. ABERRATIONS OF THE PRISM DIAGONAL

When in a converging light cone, prism diagonal generates aberrations, both chromatic and monochromatic. Since it acts like plane parallel plate, Eq.105.1 applies, with d/L becoming 1/2F, F being the focal ratio (f/D, focal length by aperture diameter). Since d/L becomes a constant for any given system, prism distance, i.e. beam diameter on its front surface becomes non-factor, with the only remaining factors being the focal ratio, in-glass path (thickness) and glass refractive index. Taking for the index n~1.5, gives for the only two possibly significant monochromatic aberrations the P-V wavefront error (mm) as W=T/1380F⁴ (spherical aberration) and W=T/65F³ (coma). Graph below shows how they change as a function of focal ratio (F).

Spherical aberration and coma affect all wavelengths neary equally, which makes them a part of chromatic error as well. Purely chromatic errors are longitudinal chromatism, caused by the change in refraction with the wavelength, and lateral color, which is generally negligible. Picture below illustrates these aberrations on a 32x32m prism (BK7) in f/10 and f/5 cone. The objective is a "perfect lens", so all aberrations come from the prism. Note that different prism types have different in-glass path length: for a given clear opening, Amici prism has it about 60%, and penta prism 3.4 times longer than the standard 90-degree prism.

At f/10, Zernike term for primary spherical aberration (8) is 0.002853, which divided with 5^0.5 gives the RMS wavefront error as 0.001276. The corresponding P-V error is larger by a factor 11.25^0.5, or 1/234 wave (both in units of 546nm wavelength). It, expectedly, agrees with the equation, since no other significant aberrations are present. The term for coma (4) is 0.003984, which divided by 8^0.5 gives the RMS error as 0.00141 (the P-V error is larger by a factor 32^0.5). So, at f/10 coma is somewhat larger than spherical aberration, but both are entirely negligible.

Longitudinal chromatism has a form of reversed primary chromatism, with longer wavelengths focusing shorter than shorter wavelengths (the consequence of the refraction at the front surface being diverging). It is a consequence of image displacement caused by the cone angle narrowing inside the prism (looking at the raytrace side view, it is causing the oblique line sections to become longer). The displacement is given by (1-1/n)T, and the variation in n (δn) with the wavelength produces longitudinal chromatism, given by (T/n²)δn. It remains nominally unchanged with any focal ratio (of course, due to the smaller Airy disc at the faster focal ratio, chromatic error increases correspondingly). At f/5, the error in both, F and C line is over 0.4 wave P-V. It will change relatively little in a fast achromat, unless a very small, but it would introduce noticeable color error in the reflecting systems, or any other fast systems with a very low level of chromatism.

Misaligned prism will induce all-field coma, astigmatism and lateral color. Coma dominates astigmatism at fast focal ratios, while the latter can be larger at ~f/10 and slower. At f/5, 1-degree prism tilt vs. optical axis will induce 0.023 wave RMS of coma, and 0.0068 wave RMS of astigmatism. Since coma changes with the 3rd power of focal ratio, and astigmatism with the 2nd, at f/10 coma drops to 0.0028, and astigmatism to 0.0017 waves RMS. But with the coma changing with the tilt angle, and astigmatism with the square of it, at 2-degree tilt the latter will be slightly larger. Since the magnitude of tilt -induced aberrations can be significant only at fast focal ratios - except at insanely large tilt angles - it is only coma that could be of concern.

Tilt-induced lateral color (prism effect) doesn't change nominally with the focal ratio, but its magnitude vs. Airy disc does, in proportion to it. At f/5 and 1-degree tilt, the mid-field separation of the F and C lines is 0.002mm, or about 30% of the Airy disc diameter. For the field center, the separation shouldn't be larger than half the Airy disc diameter. Since it increases with the tilt anlgle, it should stay below 2 degrees. At f/10, 1-degree tilt will induce only half the error at f/5, i.e. the F and C lines separation will be about 15% of the Airy disc diameter.

11.6 DIFFRACTION EFFECT OF NON-SYMMETRICAL VANES ARRANGEMENT

The standard 4-vane spider is the simplest form of the kind, but rotational stability is not its strongest point. To improve on that, the vanes need to be rearranged, breaking the symmetry of a cross. Simulations below show diffraction effect of two of such arrangements vs. standard 4-vane form.

Due to the different converging angles from the vane sections, the modified spider produces wider, complex spikes of similar length. While the amount of energy transfered out of the Airy disc depends solely on the vane face area, significantly wider spikes could appear either more, or less pronounced, depending on the detector's filtering. The eye could be biased toward wider, fainter spikes, or toward narower, brighter ones (for identical vane area). Which it is, has to be established experimentally; it is quite possible that it could vary individually. A variation of the middle vane arrangement, used on some commercial telescopes lately, have the vanes shifted off only slightly, so that their sides lie on a common strait line splitting the aperture in half (below).

Diffraction images show similar doubled spikes, but with one clearly dominant. Since the spike energy is nearly identical with that of the standard cross arrangement, spikes should be a bit less pronounced visually, with the faint spike likely remaining invisible (note that these patterns are brighter due to 2.5 times lower normalization value for the unit intensity, 0.02).

Doubling the vanes to increase spider rigidity will also alter their diffraction effect. The reason is diffraction interference between the vanes. As the simulation below shows, the spike of a doubled vane is

broken into bright segments and extended vs. single vane pattern. The consequence is slightly more energy transferred to the outer areas, even if the vane area is kept unchanged (graph on the right is effectively magnified by showing 5 times smaller radius on the same frame size). On the doubled area vanes' pattern (right) can be detected presence of secondary side maximas, which would imply less energy in the principal maxima, i.e. main spike. As with the vane configurations above, whether the eye would be more sensitive to a longer, segmented, and slightly less bright spike, only experimental examination will answer.

11.7 RAYTRACING EYEPIECES: REVERSE vs. DIRECT

Eyepiece performance level is commonly determined by reverse raytracing, i.e. in a setup when the eyepiece exit pupil becomes aperture, and collimated light pencils passing through it travel through the the eyepiece in reverse, to form an image in front of the field lens, at the nearly same location where forms the image of the objective. Image formed by raytracing eyepiece in this manner is real image, but it is neither image that a perfect lens would form if placed at the eyepiece exit pupil, nor image of the objective. Rather, it is an image at the location of objective's image that would produce perfectly collimated pencils at the exit pupil end. As such, this image reflects aberrations of the eyepiece in their kind and magnitude, some of them reversed in sign, some not. For example, if reversed raytracing produces field curvature concave toward eyepiece, it implies that such curvature would produce perfectly collimated exit pencils because eyepiece itself generates curvature of opposite sign (it seems illogical since the two curvatures seemingly coincide, but the image space of the eyepiece is behind the eye lens, not in the objectve's image space; hence with a flat objective's image the eyepiece would produce exit pencils becoming converging toward outer field - because the field points in the image of the objective would've been farther away than what it needs to form a collimated beam - i.e. would form a curved best image surface of the opposite sign). On the other hand, if reverse raytracing produces overcorrection, this means that the eyepiece would, with zero spherical aberration from the objective, form exit pencils with rays becoming divergent toward pencil edge (since off-axis points on unaberrated image surface are in this case closer to the eyepiece), focusing farther than rays closer to the center, i.e. also generating overcorrection.

As long as the geometry of pencils passing through the exit pupil is identical, so will be the aberrations generated. But this is strictly valid only for points close to axis. The farther off field point, the more likely that the perfect exit pencils that we are starting with in reverse raytracing won't exactly match those generated by a perfect input from the opposite end, and that will cause different aberration output as well, possibly significant. One particular difference is that reverse raytracing can be done from one fixed pupil location at a time. It is generally insignificant with eyepieces having relatively small exit pupil shift with the change of field angle (so called spherical aberration of the exit pupil, with the pupil generally shifting closer toward eye lens with the increase in field angle), but when it's significant, raytracing from any fixed pupil location will cause gross distortion of the astigmatic field for field zones with different exit pupil location. The only way around it is to raytrace for several different exit pupil locations, and piece up the actual field from that. This problem vanishes in direct raytracing, where every point's cone simply goes to its actual exit pupil.

To make easier to detect the differences, a wide field eyepiece is needed, and for the ease of accessing its optical process it should be simple as possible. The perfect candidate is a modified 1+1+2 Bertele, which is for this purpose designed to produce 80° apparent field of view (AFOV). While not in the league of the (much) more complex designes for this field size, it is still significantly better than other conventional designs. Below is how it raytraces in reverse, and directly, the latter using OSLO "perfect lens" as the objective and at the eye end. Eyepiece focal length is 10mm, and exit pupil diameter is 1mm, hence it processes f/10 beam.

Top half shows reverse raytracing. Surface #1 is the aperture stop, and #9 the image. The section between two marginal cones (6.4mm radius) is the actual image, and the full length of the vertical dashed line (8.4mm radius) is Gaussian image, i.e. image that would've been seen w/o distortion at the entry field angle (in effect, the image seen at that angle from the distance equal to the eyepiece focal length, as illustrated with the dotted lines at left). Column of numbers at right are the heights of the marginal chief ray (central ray) at all surfaces. Astigmatism plot shows reversal of the tangential line (in the plane containing axis and chief ray) toward field edge, preventing further increase in the outer 20%, or so, of the field radius. At 40° off, the P-V wavefront error is 2 waves. Note that due to the exit pupil shift (3.7mm eye relief for 40° to 5.5mm for half-field), tangential line for 3.7mm pupil position slightly magnifies the longitudinaludinal aberration for the upper half of the field; the actual line is a bit flatter over that section. As the wavefront map shows, on its way through the lenses the wavefront acquires vertical elongation, despite it being cut into a horizontal ellipse (0.5x0.383mm) at the aperture stop (OSLO Edu doesn't have pupil-that-tilts-with-field-angle feature). Marginal cone focusing into the final image is noticeably wider than the axial cone, which indicates negative (barrel) distortion. Since wider cone in effect acts as one of the faster f-ratio, it produces smaller diffraction pattern; with the effectively shorter focal length image magnification drops and the entire image shrinks as a result toward field edge (black square around red grid representing the image area is the distortionless form of it, with its diagonal equal to the field diameter). Nominally, distortion nears 25%. Illustration at left shows how a beam of light passing through a lens surface generates astigmatism: the two radii perpendicular in the centre of wavefront footprint are different, creating astigmatic wavefront deformation. It also shows how refracted beam changes its width, increasing vertically when it travels from left to right (shown), or decreasing when traveling in the opposite direction (for clarity, refraction difference between the two rays is neglected - they remain nearly parallel after refraction, as they do when a section of lens surface is small enough to be nearly flat).

Longitudinal aberration plot shows entirely negligible spherical aberration, and not entirely negligible axial chromatism. Defocus δ in the blue F line of less than 0.1mm indicates ~0.2 waves P-V wavefront error at f/10 (for 486nm wavelength, from δ/8F²), and four times as much at f/5. Error in the violet g-line is three times larger. lateral color is well controlled accross the field, with the F-to-C separation reaching half the Airy disc diameter at the field edge.

Bottom half shows direct raytrace of the same eyepiece with a "perfect lens" as the 89.4mm f/10 objective (the focal length is determined from the field-end angle of the marginal chief ray in reversed raytracing). Focal length of the "perfect lens" at the image end is set to 10mm, to produce a directly comparable f/10 system. Longitudinal chromatism and spherical aberration are nearly identical to those in reversed raytracing. But that is where the similarity ends. Here, Gaussian image is smaller than the actual, apparent one, as a result of the positive (pincushion) distortion, nominally the inverse of the negative distortion in the reversed raytracing (1.32 vs. 0.76). As a result of positive distortion, with the edge cone visibly more elongated, the Airy disc at 40° is noticeably larger than on axis, unlike the reversed tracing, where it is smaller. Longitudinal astigmatism at 40° is about doubled, but the P-V error is smaller: 1.8 wave. This is mainly the result of the effective f-ratio for that cone being f/13.2, with the transverse astigmatism smaller by a factor 1.75 vs. f/10, and more so vs. effective focal ratio at this field point in the reverse raytracing. Tangential curve now extends farther out, as a result of the higher order astigmatism now being of the same sign as the primary, adding to it instead of taking away as in reverse raytracing. This is caused by the narrower cone of light passing through the eyepiece for the outer field points, which much more affects secondary astigmatism, changing with the 4th power of cone width (as opposed to the 2nd power with primary astigmatism), with the 40° pencil not quite filling out the exit pupil circle outlined by the axial cone, as it does in reverse raytracing. As a result of the different astigmatism plot, field curvature also changes: instead of zero accommodation at the edge, and +1 diopter required for 0.7 field radius, now edge requires somewhat over +1 diopter, and the 0.7 field radius is flat, requiring zero accommodation (small box bottom right, accounting for the effective 13.2mm focal length of the perfect lens at this point). However, it should be noted that if the two astigmatism plots would be corrected for the distortion effect, they would become very similar, despite the difference in the sign of secondary astigmatism.

Unlike the 40° wavefront in the reverse raytracing, elongated vertically, here it's flattened, and noticeably more so. This is caused by refraction at large angles, compressing or expanding wavefront vertically, and shows the true extent of it (the elongation in reverse raytracing is partly offset by the horizontally elliptical wavefront outline determined at the entrance pupil). This asymmetrical astigmatic shape will result in asymmetry of both, ray spot plots and diffraction images along the extent of longitudinal aberration. Here, the tangential line, laying in the sagittal plane, is noticeably thicker, because it's formed by the (shorter) vertical wavefront sections focusing into it (blue on the wavefront map is its delayed area, hence it forms convex surface focusing farther away), as well as about 50% longer, since the wavefront extends that much more horizontally. As a result, best focus is not in the middle between sagittal and tangential focus, but closer to the sagittal line, which is laying in the tangential plane (the one containing axis and the chief ray), as indicated on the astigmatism plot.

Overall, the differences in aberrations magnitude are small, but one needs to keep in mind that some of them do reverse in sign with the change in light direction. For instance, nominal lateral color error is reversed, and roughly doubled, but the actual change is relatively small due to the larger Airy disc (it is similar with astigmatism, with the two plots appearing grossly different, but with the actual P-V error diferential being near negligible).

11.8 POLYCHROMATIC STREHL: PHOTOPIC vs. MESOPIC vs. CCD

Polychromatic Strehl for telescopes with refracting elements is commonly given for photopic (daylight) eye sensitivity. Strictly talking, it is valid only for daytime telescope use, but both sides of the market seem to be neglecting it, or simply are unaware of it. Since the Strehl figure is used as a qualifier of the level of optical quality - 0.80 for so called "diffractin limited", and 0.95 for "sensibly perfect", it does matter to know that it is limited to the sensibility mode used for calculating the Strehl. In general, due to the higher overall sensitivity in the mesopic mode - and particularly toward blue/violet - and the error usually being greater in the blue/violet, the mesopic (twilight level) Strehl will be lower than photopic (broad daylight). It likely worsens somewhat toward scotopic (night conditions) mode, but telescopic eye is most likely to be within the range of mesopic sensitivity. How significant is the difference between photopic and mesopic Strehl depends primarily on the magnitude of chromatic error in the red and blue/violet, with the latter being more significant since, unlike the blue/violet, sensitivity to the red generally declines toward mesopic and scotopic mode. While these eye sensitivity modes are relevant for visual observing, for CCD work it is the chip sensitivity that needs to be used for obtaining the relevant, CCD Strehl. Here, CCD sensitivity is a rough average of the range of sensitivities of different chips.

As illustration of the difference between Strehl values for photopic, mesopic and CCD Strehl, it will be calculated for a highly corrected TOA-like triplet in two slightly different arrangement: one with the standard Ohara crown, S-BSL7, and the other with its low-melting-temperature form, L-BSL-7. Slight difference in dispersion between the two is sufficient to produce larger axial error, particularly in the red and violet, which will show the difference in correction level between two seemingly highly corrected "sensibly perfect" systems, when judged by the photopic Strehl value alone. Strehl values are calculated using 9 wavelengths spanning the visual range, as shown below. Mesopic sensitivity is approximation based on empirical results, somewhat different from the official mesopic sensitivity, which is merely a numerical midway between photopic and scotopic values.

From top down, first shown is a lens using S-BSL7, extremely well corrected for axial chromatism. So much so, that the mesopic Strehl, and even CCD Strehl are only slightly lower. This lens practically has zero chromatism in the violet, a rarity indeed.

Replacing S-BSL7 with L-BSL7 (same prescription, except slightly stronger R1 - 2380mm - to optimize red and blue) roughly doubles axial chromatism, except for the violet, which is now at the same level as the deeper red. Photopic Strehl is still excellent, suggesting there is no noticeable difference in the chromatic correction between the two. However, mesopic Strehl tells different story: this lens is not "sensibly perfect", and its CCD Strehl sinks toward 0.80.

Mesopic Strehl gives different picture for achromats too. According to its photopic Strehl, a 100mm f/12 achromat is slightly better than the "diffraction limited" at its best diffraction focus (0.09mm from the e-line focus toward the red/blue; first column shows Strehl values at the best green focus). But its mesopic Strehl, also at the best diffraction focus, is only 0.63, and its CCD Strehl dives down to 0.45. Knowing that the Strehl number reflects average contrast loss over the range of MTF frequencies (for the mesopic Strehl it is, for instance, 37%), implies that this achromat is nowhere close to "diffraction limited" under average night-time conditions. For that, it needs to be twice as slow, f/24. In conclusion, it is hard to draw a precise line for where a "sensibly perfect" photopic Strehl should be for telescopes used at night, but it seems safe to say that it does need to be significantly better than 0.95; probably close to 0.99.

11.9 PLASTIC ACHROMATS

Optical plastics are widely used for production of small and not so small lenses for all kinds of cameras, glasses and optical devices, but rarely for telescopes, and when used, nearly without exception for those of low quality. Most important optical plastics are acrylics, polycarbonates and polystyrenes, but some other are also viable. Optically, they can be as good as glass, but have several times higher thermal expansion, and a 100-fold higher variation of the refractive index with temperature. Also, they are more prone to static charges, and more difficult for coatings. On the good side, they are ligther, safer, and cheaper. Technological advances resulted in a wider number of optical-grade plastics available, which makes their application for small telescope objectives easier. Follows overview of the performance level of optical plastics - mainly those listed in OSLO Edu catalog - as components of achromatic 100mm f/12 doublets and triplets. In general, they have better color correction, occasionally approaching - even exceeding - the minimum "true apo" requirement of 0.95 Strehl.

Doublets are of the Steinheil type, with the negative element in front, because the "flint" element in most of objectives, polycarbonate, is more resistant to impact and temperature (in general, order of elements does not significantly change the output). Performance level is illustrated with a chromatic Focal shift graph, against that for the standard glass achromat (BK7/F2, black plot). Chromatic focal shift shows the paraxial focus deviation for other wavelengths vs. optimized wavelength (546nm, e-line). In the absence of significant spherochromatism - which is here generally the case - it is a good indicator of the level of longitudinal chromatic correction. The P-V error of defocus can be found from the graph for any wavelength, using P-V=δ/8F², where δ is the focus shift from the e-line focus (0 on the graph) and F is the focal number. Graphs are accompanied with the corresponding photopic Strehl (25 wavelengths, 440-680nm), except for the last two, whose rear element plastics are not listed in OSLO (direct indexing for five wavelength was entered from ATMOS). The Strehl value is for the diffraction focus, which for most of these objectives does not coincide with the e-line focus (amount of defocus is given as z, and can be positive or negative, depending on the plot shape).

All but one plastic lens combination have a higher Strehl than the glass achromat (0.81). Some combinations have near-apo correction in the blue/violet, some in the red, but most important is how well corrected is the 0.5 to 0.6 micron section (approximately). Two doublets have Strehl value exceeding 0.9, as well as two triplets, with one of them qualifying as a "true apo" by the poly-Strehl criterion of 0.95 or better (#8). It wouldn't satisfy the P-V apo criterion having 2.3 wave error in the violet g-line (1/6 wave F-line, 1/5 wave C, and 1/2.5 wave r-line) but due to the very low eye sensitivity to it in the photopic mode, it has little effect on the photopic Strehl. The more appropriate for night time use, mesopic Strehl, would be somewhat inferior to objectives with a similar photopic Strehl, but better violet correction.

There are other plastics available, and more combinations possible (also, the properties of any given plastic can vary somewhat depending on its production process), but these shown here suffice to conclude that optical-grade plastics can be superior to the standard glasses in chromatic correction. Some could even produce the "true apo" level in the range of mid to moderately long focal ratios.

11.10 HOYA FCD1 vs. FCD100, TRIPLET OBJECTIVE

The older generation of extra-low dispersion glassess, with Abbe number around 81, is commonly considered inferior in their performance limit to the latest generation, with Abbe number around 95 (also called super-low dispersion, or SD glasses). However, the larger Abbe number gives one single advantage: with any given mating glass the higher order spherical aberration residual is lower, allowing for somewhat faster lens for a given design limit in the optimized wavelength. But the difference is generally small. Let's illustrate this with Hoya's FCD1 and FCD100 glasses in a 5-inch f/7.5 triplet objective.

Limiting the mating glass to Hoya's catalog, the best match for FCD1 is BCD11, and for FCD100 BSC7 (Hoya's equivalent of Schott BK7). As image below shows, a 5" f/7.5 triplet with FCD1 (top) has photopic polychromatic Strehl rounding off to "sensibly perfect" 0.95 (mesopic value would be somewhat lower, but not by much, considering relatively low errors across well balanced spectrum). The FCD100 triplet (middle) does have better polychromatic Strehl - rounding off to 0.98 - but about half of the differencial comes from the optimized line correction. Since the limit in the optimized e-line for the FCD1 triplet is at the level of 1/15 wave P-V of primary spherical aberration, the actual units with a similar optimized line correction would have no perceptible difference in color correction (granted, any given optimized line correction level would be easier to achieves in the FCD100 triplet, due to its more relaxed inner radii). Note that the FCD1 triplet correction mode minimizes the error in the violet g-line, which is not the best general correction mode. With slightly less of the positive power, the error in violet would increase, but would decrease in the other three wavelengths, with the F and C nearly touching at the edge zone on the OPD graph. In such case, the FCD1 poly-Strehl increases to 0.966 which, considering the unequal error in the optimized line, would imply the same level of chromatic correction.

It is obvious on the LA graph that the FCD1 triplet has significantly higher spherochromatism on the primary spherical level, most of it the result of more strongly curved inner radii. But the sign of higher order spherical residual - after optimally balancing the optimized wavelength - significantly reduces the aberration in the blue/violet, while increasing it only moderately on the red end. As a result, Strehl values for non-optimized wavelengths are generally close to those of the FCD100 triplet.

One other possibility is using moldable glasses. Best match for FCD1 is M-BACD12. If one surface is aspherized, all four inner radii can be equal, and the triplet is nearly as well corrected as the one with FCD100 glass (bottom).

In all, how objective performs still depends more on the combination, than any single glass. It is possible that the older ED glass objective even performs better, although it is generally to expect small to negligible advantage with the higher Abbe# varieties. The difference is more pronounced in the doublets, because the higher order residual increases exponentially with the lens curvature, and doublets require them significantly stronger than triplets.

11.11 MICROSCOPE EYEPIECES IN TELESCOPES

While not common, use of microscope eyepieces in telescopes does happen. How well these eyepieces can be expect to perform? Is a good name on them implying they will be as good as those made for telescopes, or even better? The answers are: "no one can tell", and "no", respectively. There are two main differences between the standard microscope and a telescope with respect to the eyepiece performance: (1) due to the significantly shorter objective-to-image distance - for a standard old-fashioned microscope the main part of it the so called "optical tube length" (OTL), standardized to 160mm - rays entering any given eyepiece field stop have significantly larger divergence, and (2) due to the very small objective, the effective focal ratio is very high (measured as a ratio of objective diameter vs. objective-to-image separation; not to confuse with the microscope numerical aperture, which is measured vs. objective-to-object separation). The former generally increases off axis aberrations, while the latter makes them smaller. In other words, looking only at #1, eyepiece optimized for a microscope would have to be sub-optimized for a telescope with respect to field correction. How much does #2 offset for this?

Since the microscope magnification can also be expressed as a product of the objective and eyepiece magnifications - the former given by OTL/f_o, and the latter by 250/f_e, with f_o and f_e being the objective and eyepiece focal length, respectively - we'll illustrate the divergence vs. focal ratio offset with an average objective of 10mm focal length, and a 20mm focal length Huygenian eyepiece (from the above, they produce 160x250/10x20=200 magnification). Image below shows the optical scheme of a microscope (top) and an actual raytraced system with the given parameters (bottom). The objective and eye lens in the latter are "perfect lens", so neither contributes aberrations (note that the correct magnification for perfect lens 1 should be -16.13, but makes no difference in the ray spot plot).

The eyepiece is upscaled 10mm Huygenian shown under "Individual eyepieces" on eyepiece raytracing page, so its nominal aberrations in a telescope are twice larger than those shown for the 10mm unit. In this microscope setting, the 20mm unit (due to re-orienting its effective focal length is around 22mm) shows entirely negligible aberrations all the way up to its 10mm radius field stop. The effect of the f/86 cone (the paraxial data given below objective is for the objective only) makes the effect of significantly stronger divergence entirely negligible over the strongly curved best image field (-4.5 diopters of accommodation required at the edge). Even over flat field, it dwarfs defocus effect to 1/12 wave P-V at the field edge. Note that the field is given in terms of object height, with 0.618mm corresponding to 25° apparent FOV in the eyepiece, and 3.33° true field of the objective, i.e. angular radius of the object (magnification is not, as with a telescope, related to the angular size of the object in the system, but to its angular size as seen from the standard least distance of distinct vision, 250mm; on the schematic microscope, that angle magnified by the objective is α₀, and the final angular object radius is α).

In all, correction requirements for microscope eyepieces are much lower than those used in telescopes. This applies to both, axial and off-axis correction, and that is the main risk in using microscope eyepieces for telescopes: those performing just fine in a microscope, could become sub-standard in a telescope. Another possible obstacle, not visible in this demonstration, is that microscope eyepieces could be optimized to offset typical aberrations of microscope objectives, while telescope eyepieces are generally designed to produce best possible stand-alone image.

11.12 SCHMIDT vs. HOUGHTON TRIPLET CAMERA vs. STANDARD SCHMIDT

While looking for some modern, "extremely achromatic" camera prescription, a drawing of a triplet catadioptric camera by Bernhard Schmidt caught my eye. It was called an "alternative to the standard Schmidt", and made me curious: just how close it is. Then, in an online PDF file which contained data as close as possible to the prescription ( Journal of Astronomical History and Heritage), there was quite similar triplet camera patented by Houghton some 15-20 years latter (1944, US Pat.#2,350,112). Whether Houghton could know for Schmidt's work is anyone's guess, but doesn't make less interesting finding out how do these two cameras compare, and how close they come to the standard (aspheric plate) Schmidt camera. For the Schmidt triplet, the original handwritten prescription by Schmidt was used, scalled down to 100mm aperture diameter, and optimized by very minor tweaks (there is also a 1934. prototype of the same design which will be mentioned in raytracing analysis). All three cameras are 100mm aperture f/1, to make them directly comparable, and the field radius is 6 degrees. Image below shows raytrace of the downscaled Schmidt 3-lens catadioptric camera. The the outer two lenses are plano-convex and symmetrical with respect to the biconcave mid element. A single glass, probably Schott's old O15 crown (n_d=1.53, v_d=58.99) was used; since it is not listed in OSLO Edu catalogs, the closest found was used (during rescaling the lenses got somewhat squeezed up; increasing the gaps to 6.9mm, needed to clear axial pencil, doesn't appreciably change the output).

Central obstruction size is not given. Image size sets the minimum size at 20% linear, which would practically have to be somewhat larger. Since the effect is near-negligible, both central obstruction and (possible) spider vanes are omitted.

LA graph shows relatively significant higher-order spherical residual on axis. The corresponding wavefront errors for five selected wavelengths are given by the OPD (optical path difference) plot. Best image surface doesn't fall midway between the tangential and sagittal surface due to the presence of odd secondary (Schwarzschild) aberrations (in presence of spherical aberration, the astigmatism plot, originating at the paraxial focus, is shifted away from best focus, but best image surface is vertical when its radius coincides with the one entered in raytrace). While all five wavelengths have a common focus for the 75% zone ray, their best foci - mainly due to spherochromatism - do not coincide, resulting in a nominally significant chromatism. Still, the g-line error is only about three times the error in the optimized wavelength. The ray spot plots indicate relatively insignificant chromatism (Airy disc is a tiny black dot; its e-line diameter is 0.00133mm, or 1/300 of the 0.4mm line). Polychromatic diffraction blur exceeds 0.02mm at 4.1° and 0.04mm at 6° off (the five wavelengths, even sensitivity). Should be mentioned that other than the prescription, there is an actual unit, a prototype of this camera type from 1934. According to the measurements taken by the paper authors, it is nearly identical to the prescription, except that the middle element has slightly weaker radii (perhaps fabrication inaccuracy). When scaled to the comparable 100mm f/1 system (originally 125mm f/1.1) the overall correction is somewhat worse.

Houghton's patented camera differs in that it has two biconvex lenses framing in the biconcave central element. Also, it uses two different glasses. Again, there was no near-exact match listed for a glass quoted for the mid element, but the one used for raytracing is close enough not to make the end result significantly different (the minor optimizing tweaks are probably in better part due to the small differences in glass properties).

While the LA graph looks better on the first sight, due to considerably lower higher-order spherical residual, chromatic correction is significantly suboptimal due to the five wavelengths having a common focus too high, at the 90% zone. It is larger by a factor of 2.6 than what it would be if the common focus was at the 70.7% zone. The ray spot plots indicate more chromatism than in the Schmidt configuration, but diffraction blurring is significantly reduced.

However, when compared to the standard Schmidt below, even the Houghton falls significantly behind.

There is not so much difference in the astigmatism plot - which shows primary and standard secondary astigmatism - but significantly smaller ray spot plots and diffraction images indicate much lower odd secondary aberrations. Nominal chromatism (spherochromatism) is also significantly smaller, but not so relative to the optimized wavelength, which is much better corrected than with the 3-lens correctors. Overall superiority of aspheric plate is undisputable. It can be illustrated with the magnitude of Zernike terms, and encircled energy, both 6° off axis (below).

The standard Schmidt has only three significant terms, primary astigmatism (#4), primary spherical (#8) and secondary astigmatism (#11). The primary astigmatism term indicates 0.63 wave RMS (term divided by √6), corresponding to 3.1 wave P-V, plus 0.43 wave RMS (2.7 wave P-V, term divided with √10) for secondary astigmatism. That is more than ~4.5 waves P-V corresponding to ~0.02mm longitudinal astigmatism on the plot, indicating the presence of lateral astigmatism, of the same form as primary astigmatism, but increasing with the 4th power of field angle, not included in the plot. Similarly, the spherical aberration term indicates 0.53 wave RMS of primary spherical aberration (term divided by √5), much more than what is present on axis. This is due to the presence of lateral spherical aberration, of the same form as the primary, but increasing with the square of field angle.

In the 3-lens Schmidt and Houghton, dominant term is primary astigmatism, followed by primary spherical, secondary astigmatism, primary (#6) and secondary coma (#13). Most of the terms are significantly higher than in the standard Schmidt, and particularly for primary astigmatism. Simiarly to the standard Schmidt, RMS error values indicated by the terms are not in proportion to the graphical output, because it doesn't include odd secondary Schwarzschild aberrations, lateral spherical, astigmatism and coma. The ray spot plot in the paper is elongated vertically probably because it is given for the plotted best astigmatic field not including odd secondary aberrations; the actual best field, according to OSLO, is about 5% stronger (also, spot structure is markedly different than for the one given by OSLO, with the dense part of it for e-line more than twice larger than in OSLO: over 0.05mm vs. 0.025mm; note that system in the paper has 25% larger aperture, but also is somewhat slower, at f/1.13). Polychromatic encircled energy plot (the 5 wavelengths, even sensitivity) shows that the standard Schmidt has about three times smaller 80% energy radius than its 3-lens corrector alternative, with the Houghton midway between.

11.13 FLAT-FIELD QUADRUPLET APO

Flat-field quadruplets can come in various forms. The particular arrangement given here is a contact air-spaced triplet combined with a singlet meniscus at some distance behind such as Ascar 130 PHQ from Sharpstar (its glasses are not published, so that's where similarity ends). The triplet does all corrections, central line and chromatic, but it is slightly modified to compensate for the optical effect of the meniscus (it induces corrective astigmatism and field curvature, but also significant amounts of coma and spherical aberration). The base triplet is NPN 130mm f/7.5 with Hoya's FCD1 and BCD11, given under 11.10 above. After adding the field-flattening meniscus, only the 1st and last triplet radius were changed to correct for coma, and one inner radius to correct for spherical aberration. However, since the meniscus exerts negative power, the focal ratio went from f/7.5 to f/8.6 (image below, top).

Field can be flattened with any meniscus form, but for minimized lateral color a strongly curved meniscus is required. Off axis monochromatic corretion is the best with the astigmatism cancelled, and some slight residual field curvature remaining (flattening field by indroducing a small amount of astigmatism roughly doubled the edge field wavefront error). Meniscus location is pretty flexible; cutting it in half only slightly worsens chromatic correction. However, placing it right after the objective gives rise to a significant higher-order spherical residual, due to correcting for primary spherical requiring significantly larger inequality in radius value between the three equal inner radii and the one correcting for the spherical (trying to correct spherical by bending lenses produces similar result).

One alternate way of correcting field curvature is by placing achromatized meniscus significantly farther from the objective (bottom). The overall chromatic correction is still good, but somewhat less than with the above arrangement. The relative aperture also diminishes, to f/9.7. The singlet meniscus extends the triplet's focal length by roughly 15%, and the achromatized (more widely separaed) closer to 20%. This means that these arrangements need to use triplets capable of achieving good correction at f/6 to f/6.5 in order to produce well corrected f/7 to f/7.5, or so, flat field systems. Note that these are not necessarily the best glass combinations, or separations: they illustrate general system properties (however, with the singlet meniscus, as mentioned, the differences are fairly small).

Does the triplet arrangement - NPN vs PNP - affects the outcame? In general, like with the doublets, where reversing order of positive and negative elements generaly has little effect on chromatic correction, shouldn't be substantial, although it can be significant in some respect. Ascar's 130 PHQ uses PNP triplet which, since the positive element has to be ED glass, means it has two ED glass elements. Similarly to the doublets, placing the negative element in front requires significantly stronger inner radii, because the glass used for it always have significantly stronger index of refraction, requiring stronger radii to compensate for the initial chromatic error by the weaker-index positive glass (image below).

Significantly weaker inner radii of the PNP arrangement (bottom) seem to be producing less spherochromatism and better overall correction. The exception is the violet g-line, which is slightly worse, due to defocus, but the rest of lines are significantly better. About twice smaller displacement of the astigmatic field origin - which is by default at the paraxial focus - indicates as much smaller spherical aberration in the e-line. However, the NPN lens is set to produce the smallest error possible in the g-line, which comes at a price of sub-optimal correction in the F and C. By making the front radius 1-2mm stronger, F and C lines come to their near-optimal correction, and the difference in F/C chromatic correction becomes insignificant, while the g-line error becomes about 25% larger than in the PNP. What remains unchanged is the twice larger minimum error in the optimized wavelength.

11.14 ED doublets with lanthanum

ED doublets using lanthanum as mating element have become common these days. While they leave something to be desired in the violet end correction, their main advantage is their large Abbe# (dispersion) differential vs. ED glasses, combined with sufficiently small relative partial dispersion (RPD) differential to keep secondary spectrum small to acceptable. Large Abbe# differential is a must for fast ED doublets, since the larger it is, the less strongly curved inner radii required, and less higher-order spherical aberration induced. Some other factors are also potentially significant - like the refractive index ratio, actualy not favoring lanthanum in general - but large enough Abbe# differential would compensate for it too. The problem is that, due to the architecture of RPD diagram, the larger Abbe# differential, the higher on it is lanthanum glass, and the larger RPD differential vs. ED glass, i.e. the larger becomes secondary spectrum (higher RPD is in general offset by sufficently larger Abbe# differential, but the tendency is secondary spectrum increase). Thus the choice of lanthanum is always a compromise between a low higher-order spherical residual, determining the central line correction level, and low secondary spectrum. For the former the Abbe differential needs to be as high as possible, for the latter - about as small as possible. The advantage of high Abbe# ED glasses (~95) is that they can use lanthanums that are lower on RPD diagram, i.e. with a smaller RPD differential, hence smaller secondary spectrum as well. It will be illustrated how much of a difference it makes vs. lower Abbe# ED glasses (~81), starting with the latter.

Raytrace below shows three variation of an ED doublet using Chinese (CDGM) glasses, FK61 ED and lanthanum (these are similar to the Astro-Tech 4" f/7 AT102ED). Doublet of this type has R2 significantly stronger than R3. From the LA (longitudinal aberration) and OPD (optical path difference, i.e. wavefront error) plots it is immediately visible that the soft spot is correction in the violet. Top doublet uses lanthanum with smaller Abbe# differential than the other two, hence has higher secondary spherical residual, and higher minimum error in the central line. The middle doublet has the highest Abbe# differential, and the bottom one is in between the two. Photopic polychromatic Strehl (0.43-0.67 micron, shown boxed) slightly favors the latter (note that the Strehl values are for the location of best e-line focus; due to the presence of secondary spectrum, best poly-Strehl is shifted toward F/C lines - nearly 0.02mm for all three - 0.890, 0.898 and 0.903, top to bottom, respectively). OSLO quotes the price of its lanthanum glass as 5 times the BK7, vs. 3.5 times for the other two, which makes the top combination most likely.

F and C lines are nearly balanced in the second and third combination, at ~0.155 and ~0.125 wave RMS, respectively (0.53 and 0.44 wave P-v of defocus), which puts them at the level of a 100mm f/24 and f/27 achromat, respectively. The top combination has correction somewhat biased toward blue/violet, with 0.074 wave RMS (0.26 wave P-V of defocus) in the F line, and 0.15 (0.52) in the C. If made nearly equal (g-line in that case goes over 1.1 wave P-V), they come at ~0.39 wave P-V of defocus, comparable to f/31 100mm achromat. The middle combination has significantly larger error in the violet, but it has little effect on the photopic polychromatic Strehl, due to the low eye sensitivity to violet in this mode. However, in the more appropriate to night-time observing, mesopic mode, eye sensitivity to violet is significantly larger, and this combination would have more effect on contrast (0.72 vs. 0.76 mesopic poly-Strehl vs. top combination, in part due to the higher sensitivity in the red vs. green/yellow as well) in addition to more violet fringing. This magnitude of violet defocus (comparable to that in a 100mm f/18 achromat) would be visible on bright objects, unless a special lanthanum doped coating, selectively absorbing in violet, is applied (which was likely the case with the APM 140mm f/7 lanthanum doublet).

Using ED glass with higher Abbe# - also called "super-dispersion" (SD) - makes possible to use lanthanums with higher Abbe#, with less of RPD differentiial, i.e. inducing less of secondary spectrum, assuming the ED glass is of similar RPD value (FCD100/FPL53 have somewhat lower RPD than FK61/FCD1, thus the advantage is partly offset). Taking Hoya's FCD100 and two possible Hoya lanthanum matching glasses, show reduced secondary spectrum, and significantly better correction in the violet (these are similar to Astro-Tech 102EDL). As a result, the poly-Strehl is significantly higher than with the lower Abbe# ED glass.

Due to the presence of secondary spectrum, best polychromatic focus is shifted from the best optimized line focus (defocus Z in mm). Despite its lower F/C error (0.07 vs. 0.08 wave RMS, when the two lines are equilized), the top combination has lower polychromatic Strehl, because its lower optimized-line Strehl weighs more in the poly-Strehl value. But its poly-to-optimized-line Strehl ratio shows that it has better chromatic correction (0.962 vs. 0.954).

Interestingly, going somewhat slower while using the lower Abbe# lanthanum glass in order to eliminate higher-order spherical residual is likely to result in a small drop in the chromatic correction, not only due to a bit more of secondary spectrum, but also due to the higher-orderspherical actually reducing error in the blue/violet. For example, taking 125mm f/7.8 AT125EDL configuration with assumed matching lanthanum CDGM's H-LAF50B (equivalent of Ohara S-LAH66, or Hoya's TAF1), with no higher-order spherical residual, produces photopic poly-Strehl at the diffraction polychromatic focus of 0.923 (shown is objective with the positive element in front, but the reverse arrangement produces identical correction).

Even at f/7 there is no higher-order spherical, and chromatic correction is only slightly worse (0.912 poly-Strehl) but it was probably made slower to be geared toward visual observers, since its violet correction leaves something to be desired on CCD level. Visually, its violet g-line (0.436μ) is at the level of a 100mm f/28 achromat (or 60mm f/17), i.e. unintrusive. Its F/C correction is at the level of a 100mm f/33 achromat.

Non-lanthanum alternatives at this fast f-ratios do exist for SD glasses, but only a few. Short flints, like Schott N-KZFS2, paired with FCD100, Ohara FPL55/53 or LZOS OK4 would produce better overall correction, with the poly-Strehl exceeding 0.95. Schott N-ZK7 crown would produce better chromatic correction than lanthanums, but because of the central wavelength limit to 0.96+ due to more of the higher-order spherical residual its poly-Strehl is lower, at ~0.91 (the inner radii of such objective would be also very strongly curved, requiring very tight fabrication and assembly tollerances). The best match for lanthanums is fluorite, which has the highest RPD value, hence the high Abbe differential lanthanums matched with it would produce less of secondary spectrum.

When comparing secondary spectrum in ED doublets with that in achromats, it should be kept in mind that even at near-identical error levels the effect is not the same, because in the former the aberration is a more or less balanced mix of 6th and 4th order spherical, with some amount of defocus, while in the latter it is mainly defocus. They have different forms of intensity distribution, and in so much different effect on contrast. Good indication of this difference is given by their respective MTF plots (below).

Since at these (low to moderate) error levels spherical aberration spreads energy wider (it is not fully apparent at the intensity normalized to 0.1 shown, but would be more visible at lower normalization values, or with logarithmic base), it causes more of a contrast loss at low frequencies, but less at mid frequencies, where is the approximate cutoff for bright low-contrast objects, like planetary surfaces. However, it is again more detrimental at high frequencies (lunar, doubles, globulars). Diffraction simulations for 6th/4th order spherical are based on the actual F-line wavefronts in the ED doublets, and defocus simulations are pure defocus.

11.15 ED doublet with plastic

It was demonstated above that optical plastics can work well replacing glass in an achromat. How well they can work as a matching element to ED glass? Here's what it looks like with some of plastic materials listed in OSLO Edu. Objective is 80mm f/7, alike AT80ED, which is not using lanthanum (and probably couldn't considering its low price), and cheap suitable crowns have too small Abbe# differential to work well at f/7. It is not to imply AT80ED uses plastic mating element, but it is a possibility.

With what appears to be a mix of styrene and acrylic (top), correction in F and C is very good, just over 0.2 wave P-V of defocus. It is comparable to a 100mm f/56 achromat, and satisfies the "tru apo" requirement. At about 1 wave P-V, the red r line is at the level of a 100mm f/22 achromat, while 4.1 wave P-V in the violet g line puts it at the level of a 100mm f/14.5 achromat.

Using carbonate as mating element (bottom) produces markedly better correction in the violet, but worse in the other three lines. With just over 0.8 wave P-V in F and C, it is comparable to a 100mm f/15 achromat, with the red r line at the level of f/12, and violet g line f/15. Other plastics did not produce good correction levels, but it is very likely that better correction than these two shown are possible. Of course, using plastics for triplets widens the possibilities. These two combined, with STYAC in front (reversed order is not as good) produce f/7 system with F/C lines at the level of a 100mm f/12 achromat, and the violet g-line more than 2.5 times better, i.e. at the f/31 level. Still better is FK61/CARBO/STYAC f/7 arrangement, with F/C at the level of a 100mm f/15 achromat, and g-line at the level of f/25. These are unusual modes of correction, illustrating that use of plastics could enhance both correction level and correction choices in lens objectives (note that optically there is no difference between the standard H-FK61 glass and low-softening-temperature, moldable D-FK61, but the latter is nearly twice more expensive).

Plastics have the advantage of being lighter, but their other physical and chemical properties so should be at least close to those of optical glass. As the production technology advances, it will be becoming more viable as a glass substitute.

11.16 Can 52° AFOV fit 32mm 1.25" Plossl?

Different brands of this eyepiece come with anywhere from 44° (Celestron) to 52° (Orion, Meade "Super Plossl", generic brands) apparent fieldof view (AFOV) claimed. Taking that the apsolute limit for the field stop radius is the inner radius of the 1.25" barrel - around 14mm - implying 23.6° zero-distortion angular field (~47° diameter), the limit to the AFOV is imposed by field distortion. In most cases, distortion is positive, enlarging image away from axis, in which case the AFOV is bigger than zero-distortion FOV according to the extent of distortion. For Plossl eyepiece, it is about 10%, implying nearly 52°. Raytrace exercise below tells somewhat different story (it is illustrated using Plossl design, but in general can be applied to any other).

Top design is downscaled Plossl from the Rutten/Venrooij's book. Reverse raytracing shows that the size of optical image is 14.2mm, a bit over 14mm, implying 25.6° (51.2° diameter) as the limit to AFOV. Design below, the upscaled patented Nagler Plossl design, implies 26.5° (53° diameter) limit. The difference comes from different distortion rates: the Nagler has somewhat larger distortion, resulting in a larger AFOV transmitted (note that reverse raytracing gives distortion of opposite sign to the actual distortion, hence zero-distortion - or Gaussian - image is larger than the optical image, showing so called barrel distortion).

However, the numbers come out differently with direct raytracing - at least at first sight. The Nagler Plossl is plugged in with two "perfect lenses": one for the 125mm f/8 objective, and the other one at the eye end with a 17mm f.l. Here, the true field angle just fitting into 14mm barrel radius is 0.80°. With the objective focal length of 1000mm, and the corresponding 31.25x magnification, the zero-distortion field radius is 25°. That is 2.4° more than what the field stop radius vs. eyepiece f.l. implies. The larger field produces higher distortion (in proportion to the 3rd power of field radius), now about 14%, with the coresponding 57° AFOV. Anyway, that would be the usual way of calculating it. However, magnification is not defined as magnification of the angle, rather as magnification of its tangent. So the correct AFOV in this case can be found from: (1) multiplying tan(0.8°) with 32mm f.l. to obtain unmagnified height corresponding to the eyepiece focal length, (2) having that height multiplied with 31.25x magnification, and (3) from arctan of that height divided by 32mm f.l. obtain the corresponding zero-distortion angle of view. In this case, the unmagnified height is 0.447mm, the magnified one is 13.96mm, and the corresponding zero-distortion angle is 23.6° (47.2° in diameter). With 14% positive (pincushion) distortion, it gives 53.8° AFOV. It is less than a degree larger than the AFOV obtained directly from the stop radius vs. eyepiece f.l., i.e. the corresponding angle, multiplied with the distortion ratio. This difference is most likely due to rounding off the nominal distortion numbers.

The size of transmitted AFOV doesn't depend on barrel length, or stop (i.e. image) location within it - the last passing cone is vignetted by over 50%, and the very next one higher is not making it trough at all. Its change with the focal ratio (f/8 shown) is negligible. There is little use of an oversized field lens transmission-wise, but it is desirable in order to avoid light passing near the very edge of a lens. Placing a 1mm wide stop into the barrel would reduce the opening to 12mm radius, with the corresponding zero-distortion field radius reduced to 20.25°, and the corresponding AFOV to 22° (44° AFOV diameter). So, the answer is: it is possible to pack 52° AFOV into 32mm 1.25" barrel eyepiece, but only with no field stop in the barrel.

11.17 80° AFOV Plossl

What happens when the apparent field of a standard eyepiece, such as Plossl, extends well beyond its usual 50 degrees? If, for instance, the same Nagler Plossl from above is to expand to 80° AFOV? Image below shows reverse raytrace of this design with the angle of divergence from the aperture stop - in effect the exit pupil of the eyepiece - equal to 40° (top). Obviously, the field lens had to be enlarged in order to accept the wider field, but the astigmatic field looks relatively good, with the astigmatism magnitude remaining nearly unchanged over the last 30% of field radius. There is some field curvature, but quite acceptable: edge of field best focus is less than 2mm away from the field center focus. For 32mm f.l. eyepiece - from 32²/1000 - it translates to less than +2 diopters of accommodation (infinity to over 0.5m distance). Distortion of about -30% means that in the actual use the zero-distortion field - the one determined by the eyepiece field stop - would be 56-57°.

However, coma - which increases with the 3rd power of aperture (i.e. cone width) vs. astigmatism increasing with the 2nd power - becomes obvious in the outer field. Also, lateral chromatism is unacceptably large in the mid 50%, or so, of the field radius. Coma can be diminished by flattening R6 while strengthening R1 which, with the change of glasses, also lowers astigmatism over most of the field, as well as lateral color error (bottom). But longitudinal chromatism is significantly larger, and field curvature is significntly more demanding: field edge requires nearly +4.5 accommodation (infinity to ~0.24m distance).

Taking a compromise with some more astigmatism but flatter field gives what is shown below (top). Note that in reverse raytracing Gaussian image height is that of the apparent image (dashed line) and the actual, zero-distortion image, determined by the converging marginal cone, is the actual, "aberrated" image. Similarly, diverging cones entering field lens are unequal in width - a consequence of the all field pencils passing through aperture stop (i.e. eyepiece eit pupil) being by default of equal width. The wider marginal converging cone indicates lower magnification (by forming smaller Airy disc, i.e. image scale), resulting in the negative (barrel) distortion. Note that the half-diagonal of the square representing zero-distortion (Gaussian) image equals the Gaussian image height (radius) in the image plane.

But to find out how it is actually working in a telescope, it is necessary to raytrace directly (bottom). The eyepiece is now working with the field produced by a 100/1000mm (f/10) "perfect lens", and another perfect lens is used on the opposite end to form the image. Field angle needed for identical edge point height in front of the field lens is 1.09°, or 19.08mm. The resulting astigmatic field is now significantly changed, with strong higher-order astigmatism dominating the peripheral field. It is a consequence of different ray geometry, with the diverging cones coming from the objective being of the same width at the field lens, while the marginal cone pencil exiting the eyepiece is, for that reason, significantly more narrow than the axial pencil (this is causing it reaching the retina as a narrower converging cone, forming larger Airy disc i.e. generating positive image distortion).

Different ray geometry also results in uncorrected lateral chromatism as well. Higher order astigmatism is lowered by weakening R4 which, with corresponding change in the glasses, produces a better performing design, as it would look like in the actual use (bottom; weakening R4 increased the focal length to 33.4mm, and to compensate for that the field angle is increased to 1.14°). This particular modality has astigmatism minimized at the field edge, but with minor changes in the radius it can be increased at the edge while decreased in the inner field. With the nominal distortion of +40%, the actual (zero-distortion) field is 57°. The eyepiece would be usable at mid to slow focal ratios, but it would have nearly double distortion of more complex designs with Smyth lens (even Erfle-type eyepieces have generally lower distortion over wider a field than the standard Plossl's).

11.18 Protruding focuser: diffraction effect

It is not uncommon with Newtonian reflectors that bottom end of the focuser protrudes into incoming light. Here, the effect is given for a 50mm wide focuser tube protruding 25mm into the axial pencil of light falling onto a 200m diameter mirror. By area, it is just below 4%, comparable in that respect to 0.20D (20%) linear central obstruction.

While the central obstruction effect is considered generally negligible, the effect of protruding focuser may not be. Obstruction by focuser tube makes central diffraction maxima elongated in the direction of focuser. The PSF shows that the perpendicular maxima radius is slightly smaller than that of a perfect aperture, but te elongated radius is about 10% longer. The consequence is a contrast drop incerasing with orientation change, maxing out for the orientation along the elongated radius, where the effect is similar to that of 5% reduction in aperture diameter. Obviously, the same area of protrusion will cause less of obscuration, and less of an effect with a larger mirror, and more with the smaller one.

11.19 Diffraction effect of mirror edge clips

Mirror clips are pretty common in Newtonian reflectors, and can be found in other instruments using mirrors to collect light. The relative mirror area they are obscuring is generally small, but speculations run wild about their diffraction effect, particularly spikes that they might be causing. This simulation by OSLO Edu shows no spikes to worry about as far as mirror clips are concerned - and little else to worry about altogether.

A 200mm diameter mirror with three 12x6mm clips (as the area over the mirror surface) which block less than 0.7% of incoming light, and the effect is commensurate. The only visible change in the PSF, barely noticeable at intensity normalized to 0.01 (top, middle) are sections of uneven intensity in the 3rd bright ring. Normalizing down to 0.001 brings down similar structure in the subsequent bright rings, but the MTF shows that the overall effect is negligible. Enlarging clips area to 1.5% (bottom, comparable in area to 0.125D linear central obstruction) still shows no sign of spikes in the bright pattern area (note that it is now doubled, 0.1mm square side vs. 0.05mm above). Normalizing intensity down to 0.001 - which means that all points brighter than 0.1% of the central intensity are white - brings out faint outer areas where there is a hexagonal (primary) spike structure developing, but the intensity of these spikes is too low for them to become visible in general observing. The MTF is still not showing appreciable contrast loss, so the conclusion is that mirror clips do not produce neither visible diffraction artifacts, nor noticeable loss of contrast.

11.20 Two apochromatic objectives from "Telescope Optics" by Rutten and Venrooij

As an illustration of difference in chromatic correction between objective using ordinary glasses and those using special glasses, an ordinary (BK7/F3) 200mm f/15 achromat is compared to two designs by Klaas Companar: one using Schott FK51 extra-dispersion glass with KZFSN2 short flint, and the other using fluorite with Schott Lak9 lanthanum glass (p56-59). They are both significantly faster, with the ED doublet at f/10, and the fluorite doublet at f/8; the former has 4.6 times lower secondary spectrum, and the latter only half as much of it, or over 9 times lower than the achromat. The ray spot plots given suggest that the two objectives with special glasses are much better corrected. If the difference in focal ratios, i.e. Airy disc size is taken into account, the advantage of special glasses is significantly lessened, but still substantial. However, do the spot sizes tell all the story?

If we look at the polychromatic Strehl, the difference becomes unimpressive. At the best diffraction focus, the achromat scores 0.747, ED doublet 0.826 and fluorite 0.813 (430-670nm, photopic sensitivity). Because of its poor correction in the violet, the achromat falls farther behind for mesopic sensitivity (closer to night-time observing): 0.547 vs. 0.763 and 0.757, respectively. Converted to the corresponding P-V wavefront error of primary spherical aberration, it relates as a bit more than 0.4 vs. a bit less than 0.3 waves. And taken to the standard, photopic sensitivity, it becomes even less impressive (for the special glass) 0.28 vs. 0.23-0.24, in the same order. We should state that blur size comparison is strictly valid only for a single type of aberration (here we have defocus with the achromat and spherical+defocus with the other two), but the main reason for disappointing performance of the special glasses is suboptimal design. Raytrace below shows the two objectives as given in the book (top two) and near their optimum (bottom two; prescriptions are for the optimized objectives). Note that OPD (optical path difference, or wavefront deviation) scale - given in units of wavelength on vertical axis - is different for the two objectives, because of the larger violet error of the fluorite objective.

For some reason, designer wanted to bring together marginal foci of the blue F and red C line. Not only that the error at this focus location is four times the error at the 0.71 zone, this common focus is farther from the green focus, which makes the error in F and C even larger. The only benefit is better correction in the violet, but if it comes at a price of worsening correction in all other non-optimized wavelengths, it has to result in inferior performance. Bringing the F and C 0.71 zones together (bottom two), minimizes error not only in those two wavelengths, but also in most of the others. The result is photopic Strehl of 0.955 for Apoklaas and 0.927 for fluorite (value under Z is defocus needed from the e-line focus to the best diffraction focus). Note that the Strehl is in a smaller part better because of the minimized error in the optimized wavelength as well.

11.21 Secondary spectrum corrector

Not long ago, refractors using ED glasses for secondary spectrum correction were very expensive. One solution to the problem was offered by Valery Deryuzhin, in the form of subaperture corrector, which he named Chromacor. Its prescription was never published, but Roger Ceragioli tried to find out what can be done with such corrector, and results are pretty good, at least over limited field (note that Deryuzhin stated it still falls behind his Chromacor correction wise, especially with respect to lateral color error).

Top raytrace shows the 6" f/7.9 achromat alone (it was probably intended to be 150mm f/8, but the difference is negligible). Bottom raytrace shows correction level with the corrector added (like Chromacor, it consists of 5 cemented lenses). Correction is at the "true apo" level in the violet g-line (0.85 Strehl), but falling short of it in the F-line (0.66 Strehl) and, more so, in the red. But even the red end is several times better corrected than in the achromat. More specifically, the red C-line, as indicated by the OPD graph, is less than 1/2 wave P-V, or at the level of a 100mm f/27 achromat (with nearly balanced C and F lines). F-line, at 1/3 wave P-V is at the level of a 100mm f/36 achromat. Central line correction is even better with then without corrector (note that the ray spot plot size is not in agreement with other values, and should be more than twice larger - probably some kind of a glitch; the other spot sizes, including the central line's for the achromat alone, appear to be in line with the P-V wavefront error values).

Diffraction images bottom left show lateral color error, limiting quality field radius to about 0.05°, when the blue Airy disc starts visibly separating from the green-yellow Airy disc; at that point the polychromatic Strehl (photopic) is around 0.80 - in the absence of any other aberration (note that diffraction images are enlarged vs. ray spot plots, roughly by a factor of 2; also, color code for them is different than for ray spot plots).

11.22 Cemented doublet as a focal reducer

In general, the benefits of a faster system are wider fields achievable and gain in "speed" (CCD/photography). Dedicated reducers can also flatten image field, and/or correct aberrations inherent to the system. Here, it will be examined how well can a common cemented objective corrected for infinity function as a focal reducer. Let's start with a 100mm f/12 achromat. As image below shows, the maximum visual field that fits in a 2" barrel is a bit over 2 degrees in diameter. Placing a 600mm f.l. cemented doublet about 300mm in front of its focal plane turns it into an f/8 system.

Now, the entire 2-degree field can fit in a 1.25" barrel, or if 2" barrel eyepieces are used, it can expand up to 3.2°. The effect on chromatic correction is nearly negligible, as OPD graphs show, but the reducer adds nearly 1/11 wave of undercorrection. Best field curvature, however, is significantly more relaxed - 1800mm vs. 430mm - with the reducer lens (ray spot plots are for the best field, for e, F and C lines only). The reducer is fairly insensitive with respect to its placement point: as much as several mm deviations will weakly affect the reduction ratio, with no appreciable effect on the correction level. This particular cemented objective uses Schott SF5 flint, because it allows for near-optimal correction with BK7, but any well made cemented objective should have similar effect: reduced focal ratio with the chromatic correction level of the original system. In this case, a 100mm f/8 achromat with chromatism of an f/12 system.

With Cassegrain-like systems, there is no gain in field size from using reducer lens, since it is limited by the baffle tube rear opening and in most cases can be fully accepted by 2-inch barrel. Only large systems, with baffle tube opening wider than 2" could gain significant extra field from the use of a reducer lens. Otherwise, the gain limits to possibly be able to pack the useable field into 1.25" barrel.

Just recently, someone on a Russian forum wanted to know if he could expand usable field of his Santel Maksutov-Cassegrain with a reducer lens, so that he could fit the entire Pleyades into the view. Below is an exercise with what should be close to the Santel MK91, a 230mm f/13.5 system. Placing the same type of cemented doublet objective inside its baffle tube reduces (expands is more appropriate word) its focal ratio to f/9, packing up the original near maximal 0.8° field into 1.25" barrel. The original field with ~40% edge illumination - the minimum acceptable visually - is 1.1°, assuming 46mm rear baffle tube opening, same as Celestron 9.25. With the reducer, field diameter with nearly identical edge illumination is 1.2°, for the nearly 10% gain.

Main effects of the reducer lens are somewhat worsened g-line correction (still safely within the "true apo" requirement), and 2.5 stronger astigmatism for given angular field (2.5 times 2.25 for given linear field). Stronger astigmatism, however, makes best image surface significantly more relaxed: 2000mm vs. 500mm w/o reducer lens. Diffraction image of a star at 0.6° off axis is approx. 0.05mm, which for a 50mm eyepiece, needed for near 6mm eit pupil, translates into 3.4 arc minutes - borderline between appearing as a point source and not to the average eye (from 0.05x57.3x60/50). With the Nagler 31mm and a comparable linear field size, it would increase to 5.5 arc minutes - soft, but acceptable for field edge.

System above assumes moving focusing tube, capable of handling the shorter back focal length due to the presence of a reducer lens. More often than not, Maksutov-Cassegrain systems come with mirror focusing instead. In that case, by reducing mirror separation back focal length is extended to the eyepiece field stop, with the eyepiece shoulder remaining at a given, permanent location. The efect of extention is illustrated on a 5" f/12 Maksutov-Cassegrain telescope with aspherised primary (the one at right, with a regular D/10 meniscus thickness). Raytrace below shows that the back focus extension has relatively small effect (before extension -top, after - bottom).

In the prescription, boxed numbers show changed values (that matter) due to the extention. It caused focal ratio to change from f/9.5 to f/8.7, adding some more astigmatism which made the best field curvature yet less strongly curved. Longitudinal astigmatism is over 0.8mm vs. less than 0.5mm in the original arrangement, but since the wavefront error changes in inverse proportion to the focal ratio number, astigmatism is over three times larger. Still, the 0.7° diffraction spot size of roughly 0.06mm implies off axis performance nearly as good as with the Santel above (w/reducer). Field illumination and angular size remain very similar to those without reducer lens, only the field is scaled down linearly and fits into 1.25" barrel. There would be little use of trying to expand this field with 2" eyepieces, since field illumination rapidly drops below acceptable.

11.23 Did Maksutov miscalculate with his first instrument?

The very first actual instrument built according to Maksutov's prescription during the WW2 was a small 100mm aperture Maksutov-Gregorian, intended for educational purposes in elementary schools and for public at large. The original prescription exists in Maksutov papers, and it raytraces as shown below. It's LA graph is leaning to the left, indicating suboptimal correction, and the shear magnitude of aberration is already unaceptably large. Color correction is also suboptimal, since the blue and violet (more so) need to have their inner zones focusing longer, and their outer zones focusing shorter than those of the longer wavelengths. As if it isn't bad enough, the image forms just in front of the primary's surface, which severely limits its accessibility. Is it possible that Maksutov was so sloppy with the first embodiment of his new telescope?

As for the image location, the telescope was supposed to be very simple (read: cheap as possible), so the stick-in eyepiece was a part of achieving that goal. Only longer focal length eyepieces were usable, limiting the highest magnification to about 40x. That was probably why it didn't matter that the central line error was nearly 0.4 waves P-V, or 0.13 wave RMS (balanced 6th/4th order spherical, within annulus with 40% linear central obstruction, comparable to 0.44 waves of primary spherical w/o central obstruction). Note that raytrace measures the P-V error from the non-existing center, hence its figure is significantly larger than the actual P-V error of the wavefront within the annulus.

Also, taking a closer look at the central line error reveals that the correction level determined by Maksutov's prescription is the one with the highest Strehl, and with only slightly larger 80% energy radius than the best possible. Reason for this is that the wavefront error is large enough to cause the Strehl to peak not for the wavefront with the lowest RMS error - when a line connecting paraxial and edge zone focus on the LA graph is nearly vertical - but at a slightly different focus location. The lowest nominal P-V wavefront error is, as the OPD graph shows, an artifact of the measurement being done from the origin, where no actual wavefront is present due to obscuration by central obstruction. The actual wavefront in the annulus is significantly larger for this focus point than for any of the three others.

Some time later, higher-order aspheric was applied to the primary in order to reduce the excessive spherical aberration of the original instrument. Reportedly, it allowed use of 200x magnification (presumably, image was made accessible to shorter focal length eyepieces). No specifics of the aspheric applied were given, but if we assume that it was a complete correction (even the 8th order term is not negligible), resulting rytrace is shown at the bottom, left. The sub-optimal color correction becomes more obvious, and it cannot be improved without changing corrector lens itself. For given meniscus thickness, the radii would need to be somewhat shorter (small box bottom right), but it would leave the image hopelessly inaccessible. By making meniscus nearly 2mm thicker, the color error can be minimized with the image a few mm closer. The reduction is much greater than what the respective ray spot plots indicate, because the defocus error is replaced by spherical aberration error; for given error magnitude, defocus spot is 4.5 times smaller than that for primary spherical aberration, and more so for the higher-orders. Wavefront error is 6-7 times smaller in the red and blue, and about four times in the violet. That said, the original correction was at the level of a 100mm f/25 achromat, with that error nearly cut in half by putting higher-order aspheric on the primary. That would take it close to the "true apo" minimum. Considering that, it is more likely that the sub-optimal color correction was also a result of cutting corners than design error, i.e. settling with somewhat thinner meniscus than what is needed for the best correction.

11.24 Triplet vs. Petzval

It is known that Petzval arrangement makes possible to flatten image field, and to achieve high level of correction, both monochromatic and chromatic aberrations. While the flat field advantage is undisputed, how does it compare to a triplet objective correction-wise? To try to answer that, will raytrace two 140mm f/7 apochromatic arrangements. To make them fully comparable, they both use the same two glasses. As shown below, the triplet comfortably passes the "true apo" criterions, with the polychromatic Strehl (430-670nm, photopic) a bit short of 0.97. Note that this combination can't be made into objective with two pairs of identical radii (suitable for oiling) without aspherizing; since that wouldn't appreciably effect correction level - and to keep the two fully comparable - it is left all spherical, same as the Petzval. Looking below, we see that the Petzval arrangement has better correction in the central line, as well as tighter colors, resulting in the 0.98+ Strehl. Also, field astigmatism is more than twice lower: while at 1° off axis the triplet has diffraction pattern expanded into 0.015mm astigmatic blur, in the Petzval the central maxima remains nearly intact.

Petzval's advantage becomes more obvious with faster focal ratios. If, for instance, we go down to f/5.3, its central line correction is practically unchanged (one advantage of the Petzval is that it allows for the higher order residual to be practically eliminated w/o aspherizing). There is more chromatism, but even at this fast focal ratio and at that aperture it is flirting with the "true apo". And the triplet - which can simply be scaled down with minor tweaks, as opposed to the Petzval which needs some more substantial changes (radii 327.6/172.2/172.2/926.8/-172.2/-174.02/-694.4, axial thickness 9/2.3/14.8/546/14/0.7/7) - is now behind in the central line correction, and even more in its chromatic correction. Summing it up with the poly-Strehl, it comes to 0.87 vs. 0.93 for the Petzval. Linear astigmatism didn't change, but since the wavefront error is inversely proportional to the square of focal ratio, it is about 75% larger in both. With the triplet, it means as much larger astigmatic blur (~0.026mm), and in the Petzval it just reaches the level when the diffraction image at 1° off starts turning into a small cross. If downscaled to 100mm aperture, the Petzval comfortably passes the "true apo" test.

11.25 Maksutov 1m-class refractor challenge

In his book "Astronomical Optics", Maksutov gives prescription for 1m f/9.8 Maksutov-Cassegrain that would be both better corrected and (much) more compact than refractors of that aperture size - its tube would be less than 2m long (2nd ed. 1979, p352). However, with an f/2 primary such a large meniscus would generate very large higher-order spherical aberration residual, which would, even after minimizing it by balancing with the lower order, render it unusable. Hence, Maksutov writes, either the front meniscus surface, or primary mirror, have to be aspherized in order to have spherical aberration corrected. He states that a slightly deeper than 1 micron aspheric on the primary would do the job, but there is no specific data for aspherization. Just how good such telescope would be? Shown below is the system as given in the book, all spherical (top), with aspheric on the primary to do away with spherical aberration (middle), and with aspheric on the primary needed to correct spherical aberration and minimize coma (bottom). Central obstruction is taken to be 30% by diameter, which is about the practical minimum for this configuration.

As raytrace shows, the all-spherical arrangement is hopelessly crippled by spherical aberration. Removing it requires use of three successive aspheric terms: 4th, 6th and 8th order, for primary, secondary and tertiary spherical aberration (taking out the 8th term would leave in as much as 2.5 waves P-V of tertiary spherical, which could be minimized by balancing it with secondary spherical, but would still remain roughly at the 1/2 wave P-V level). The first term is same as conic, with the corresponding value given from K=8(d⁴)A4, where d is the aperture semidiameter and A4 the 4th order coefficient (in this case, K=-0.119, with the minus sign coming from the positive A4 value for mirror surface oriented to left, i.e. with the negative sagitta value). The significance of this term (i.e. conic) is that it induces primary coma: in this case, positive coma offsetting the native negative primary coma of the all-spherical arrangement. However, this conic value offsets less than half of the original coma, leaving the field quality with much to be desired. For fully minimizing coma, more than twice stronger conic is needed, with the excess, or rather, lack of primary spherical aberration compensated for by changing corrector radii (making them more relaxed).

But, according to the book, what Maksutov had in mind is correcting for spherical aberration alone. That would put visual "diffraction-limited" (0.80 Strehl) field radius at 2.9mm, or 0.017°, with the diffraction blur at 0.25° off roughly 0.12mm (nerly 2 arc seconds) in diameter. The value of higher-order terms, given by A_idⁱ, where i is the order, and d the aperture semidiameter, gives the needed edge depth of the aspheric as δ=0.014875-0.006625-0.0008984=0.007352mm, or 7.35 microns. That is much more than 1 micron stated by Maksutov, which indicates that he had in mind aspherization for the best focus location, not the paraxial focus, as OSLO is set to do. Knowing that the best focus error is four times smaller for primary, about 2.5 times for secondary, and nearly two times for tertiary spherical, the needed maximum depth comes to 0.62 microns, approximately at the 0.72 zone. Neglecting tertiary spherical, the depth is 1.07 microns, nearly identical to Maksutov's figure. Apparently, Maksutov went with the first two terms only, and if the third term is included the needed depth is almost cut in half. Since the maximum depth is at the ~0.72 zone, not at the edge as with the paraxial focus terms, the corresponding volume of glass to remove is significantly smaller than what the depth figures alone indicate.

Since including the third term not only improves axial correction, but also makes the aspheric shallower, it is the best option. It is therefore shown for both, system corrected for spherical aberration only (middle), and the one corrected for spherical aberration and coma (bottom). A4 term implies that the conic needed for minimizing coma, by balancing its lower order form, affected by mirror conic, with its nearly constant higher order form, is K=-0.23425. The gain in field correction is significant: "diffraction-limited" field radius increases to 0.105° (17.8mm), and the diffraction blur at 0.25° is about 2.5 times smaller, with the 80% encircled energy radius smaller as much as four times (0.02 vs. 0.08mm, or 0.42 vs. 1.68 arc seconds). Spherochromatism is significntly reduced in the last arrangement (0.905 poly-Strehl vs. 0.838), due to its more relaxed meniscus radii. Astigmatism as also reduced, by some 20%, to ~0.85 wave P-V. The price to pay for these is significantly deeper aspheric - 5.3 microns - due to the larger A4 (i.e. conic) and smaller A6 and A8 terms (higher-order aberrations are reduced because of more relaxed corrector radii). If the back focal length is kept nearly unchanged, the system becomes somewhat faster, at f/9.3. Bottomm left illustrates needed modification of the sphere by the paraxial focus (OSLO) aspheric terms. The A6 and A8 terms are exaggerated vs. A4 - although in the previous arrangement A6 was about half as large as A4 - and the latter is exaggerated vs. starting sphere, for clarity. As mentioned, aspheric based on the best focus correction would have its maximum depth at about 0.72 zone, with no glass to remove at the center and the edge; depth of such aspheric would have been a small fraction of the depth shown in this illustration.

The system proposed by Maksutov has two big advantages vs. similar aperture refractor: color correction, and compactness. For comparison, the Yerkes 1-m refractor is 19m long, with its F/C blur measuring nearly 0.7mm (7.5 arc seconds) in diameter. The aspheric required for the Maksutov is not excessive for the last arrangement, and is quite low for the middle one, so it can be regarded as a better overall alternative. Even with the coma left in, the 1m Maksutov at 0.25° has 80% EE radius 0.0799mm, vs. 0.0749mm on axis for the Yerkes refractor (photopic sensitivity), hence having better correction for the smaller fields.

11.26 - Spectacle glass' telescope

My first telescope was made of a pair of identical spectacle lenses - the pre-cut, round shape, placed at half the focal length separation which - as a source I was using stated - was minimizing chromatism (in fact, chromatism is larger than that of the single lens, but it is lower than chromatism of a single lens with focal length equaling that of the two lenses combined). Those were probably +1 diopter (1m f.l.), round lenses, about 50mm in diameter, stopped down to 25mm, or so, clear opening with a diaphragm. I did see more with, than without it, although it was certainly more colorful too. These are simple to make and inexpensive telescopes, but how good they really are? To find out, will raytrace such a system.

Nowadays, spectacle lenses are made of synthetic materials, such as CR39 plastic, acrylic or polycarbonate. CR39 and acrylic have properties similar to the common crown glass, while polycarbonate has significantly different dispersion and generally produces more chromatism. Image below shows raytracing results for a +1D acrylic lens stopped down to 25mm. That produces a 25mm f/41 singlet lens telescope objective. Longitudinal chromatism, as expected, is in the form of primary chromatism, i.e. with shorter wavelenghts focusing shorter, and longer wavelength longer than the optimized, central wavelength. With these particular parameters, defocus wavefront error is 1.25 wave P-V in the blue F-line, and 1.06 wave in the red C-line. Average for the two, 1.15 waves, nominally corresponds to a 100mm f/10.4 achromat, but the actual chromatism for any given F/C error level is significantly greater with primary spectrum. That is because the defocus error changes with the square of wavelength differential with secondary spectrum, and is closer to changing linerly with primary spectrum. It is reflected in the value of polychromatic Strehl (9 wavelengths, 430-670nm, photopic), which is 0.80 for the achromat (at its best diffraction focus, 0.07mm from the e-line focus toward F/C lines focus), and only 0.51 for the spectacle glass objective, if made of acrylic (nearly identical for CR39 or crown glass). But a lens made of polycarbonate would have poly-Strehl as low as 0.30. Poly-Strehl of 0.51 is at the level of a 100mm f/5 achromat, which imlplies that for a given F/C defocus error the magnitude of chromatism, expressed by focal ratio, is twice larger with primary than with secondary spectrum.

Due to its small relative aperture, monochromatic aberrations of this objective are negligibly small (note that the positive meniscus lens shape induces 2-3 times more of each, spherical aberration and coma, but they remain negligible). Best image curvature has radius of -400mm, concave toward objective, causing nearly 0.1mm best focus shift at 0.5° field angle vs. field center. That corresponds to 1 diopter of accommodation for 10mm f.l. (flat field) eyepiece, and 4 diopters for 20mm f.l. (one diopter correspond to accommodation from infinity to 1m distance, four diopters infinity to 4m distance; with the best field curving away from the eyepiece, field points are farter away than the central point, and the pencils exiting the eyepiece are slightly converging, requiring eye lens to relax for proper focusing). Since 10mm unit produces 40x magnification, which is at the maximum usable level for this objective, field curvature is not noticeable, even with the unnatural, positive accommodation.

For the final image, created by eye, this objective needs eyepiece. Keeping it all rudimentary, as it is likely to be, the eyepiece can be a single positive lens. Below is the final image with a 50mm f.l. planoconvex lens as an eyepiece (20x magnification). Best astigmatic image radius is -5mm, with the focus for 0.5° field point forming 1mm closer to the eyepiece than the center point focus. With 50mm f.l. eyepiece, one diopter of accommodation equals 2.5mm, which means that the required accommodation here is 2.5 diopters, practically same as flat field (zero accommodation) even with the negative accommodation. Ray spot plots show that the eyepiece lens induces significant lateral color error, about 0.05mm F-to-C split at 0.5°. Boxed ray spot plots show the outcome if the objective is repleced with perfect lens (PL): longitudinal chromatism becomes negligible, but lateral chromatism remains. MTF graph shows contrast transfer for axial point, 0.35° and 0.5° field point (for best astigmatic field). Practically all degradation on axis comes from longitudinal chromatism (objective), while additional degradation off axis comes from both lateral color error and astigmatism of the eyepiece. Toward higher frequencies, there is some improvement in contrast transfer off vs. on axis, a consequence of lateral color error becoming less of a factor due to the wider color separation (note that lateral color error effect is at its maximum in orientation perpendicular to that of MTF lines, falling to zero for orientation coinciding with it; in other words, nearly all additional deterioration in contrast transfer for off axis points in tangential plane is due to lateral color error, with the sagittal plane affected mainly by eyepiece astigmatism alone).

The actual image is, of course, subject to eye aberrations, which are neglected here. Eyepiece lens orientation does matter, in that turning the flat side in front increases lateral color error by about 25%, and also makes field curvature a bit stronger. Using biconvex (equiconvex) lens slightly increases lateral color, but makes field curvature more than twice more relaxed, due to significantly lower astigmatism generated vs. single twice more strongly curved lens surface (0.1 vs. 0.24 wave RMS at 0.5°).

11.27 - Tilted Houghton vs. Ed Jones' medial

One possible solution for correcting aberrations of the Herschelian telescope is using full-aperture Houghton corrector with a tilted element, compensating for aberrations of the tilted mirror. Another, described by Ed Jones, retains only a single lens in front, with the reflecting mirror replaced with Mangin mirror, and a small singlet at some distance in front of the final image. Both systems use small folding mirror, flat in the Herschelian configuration, convex in the Ed Jones'. Raytrace below shows how the two compare, both correction wise, and with respect to overall configuration, in a 150mm f/8 system.

The level of correction is similar, except that the Houghton-Herschelian has nearly 6° image tilt, causing more than 1mm defocus at the 0.5° field point. Corresponding to nearly 11mm field radius, it would be just out of the field of the standard 20mm 52° AFOV eyepiece. Since one diopter of defocus for this focal length is 0.4mm, it would require 2.5 diopters accommodation, i.e. infinity to 0.4m distance. It is within the accommodation reach of most eyes. However, since the field top and bottom require opposite in sign accommodation, only one field side can be focused at in any given moment, and switching from one to another requires from the eye to go from +2.5 to -2.5 diopters, making accommodation significantly more difficult. With a 10mm f.l. eyepiece and 0.1mm for one diopter of defocus, the accommodation requirements are doubled. Also, tilted image results in the light cones entering the eyepiece asymmetrically, generating additional monochromatic and (lateral) color aberrations.

The Houghton-Herschel has some residual spherical aberration, which can be removed by putting -0.2 conic on the mirror, but the gain is rather cosmetic (top right). The Jones' medial can be designed practically free of astigmatism and coma, with trefoil as the dominant, yet insignificant aberration (astigmatism/coma showing at the field bottom is a tendency that could be minimized with final optimization). The nominal tilt of the image surface is relative to the rear field lens surface; as the magnified image surface detail shows, image plane is practically perpendicular to the optical axis. Another advantage of the medial is its compactness: it is less than 2/3 as long as the Herschelian.

11.28 - Erfle eyepiece w/o and with Smyth lens

The standard Erfle eyepiece has five elements in three groups (2+1+2). Usually made to allow around 65° apparent field, it was the best known wide-field eyepiece before new generation of (ultra)wide-field eyepieces, utilizing Smyth lens, entered the field with Albert Nagler. Its outer field definition went from pretty good with systems slower than ~f/10, acceptable at f/10 to f/7, and increasingly blurry with faster systems. Could the standard Erfle configuration be significantly improved by adding to it a separated negative element(s)? To get some answers, will start with the Erfle eyepiece given by Rutten and Venrooij, which is optimized to the extent that still leaves it close enough to the traditional in its overall performance level. Here, it is scaled down to 10mm f.l. and raytraced with an f/5 system for 66° AFOV (image below, top). Longitudinal color is practically non existent, lateral color is well controlled, but astigmatism explodes toward outer field. Required eye accommodation for best focus is 4.3 diopters (infinity to 1m/4.3), generally acceptable, but of little importance practically, because astigmatism dwarfs defocus error due to field curvature. By relatively small changes in the lens radii, with the heavy flint change to keep lateral color minimized, it can be improved to a flat field design with somewhat less astigmatism but significantly stronger distortion, mainly due to the stronger 1st and 5th radius (second from top). It is about as much as can be done in this configuration.

Placing a negative element in front of the field lens increases the focal length of the combined unit (it can be visualised following axial light pencil entering from left, exiting the positive group to focus and diverge into the negative front element - Smyth lens - which causes it to diverge more strongly after it, hence shortening the focal length). To make it directly comparable, the unit is scaled up to 10mm f.l.; also, due to the changed ray geometry, the glasses needed to be significantly different to keep lateral color minimized (3rd from top). The Smyth lens is quite weak, so that the intermediate image almost coincide with the telesope image, but the effect on the astigmatic field is significant. Here, the astigmatic field curvature has opposite sign vs. starting Rutten-Venrooij design; in fact, for a better part of the field radius - up to about 0.7 zone - it is Petzval curvature, since astigmatism is practically non-existent. Edge blurs are roughly comparable, but the overall field correction is significantly better than in the traditional Erfle. Required edge accommodation is also comparable, but of opposite sign. Similar trade off between field curvature and outer field definition can be accomplished without adding Smyth lens, but would require more of a field curvature, and it would make more difficult to correct residual coma. It is logical, not only because the negative element contributes some offsetting aberrations, but also because more surfaces generally make correction easier.

Use of Smyth lens allows for significant changes in the positive lens group configuration. As an example, design with somewhat stronger front negative element, similar field curvature as the previous design, with less of edge astigmatism, but more astigmatism in the inner field (bottom). Field properties can be changed toward more curvature and less astigmatism, or vice versa, by manipulating two radii, #3 and #11. Diffraction patterns in the box show the field edge with the given radii values for stronger field curvature and nearly cancelled astigmatism (top); needed accommodation is about 8 diopters. Ray spot plots below them show field edge with the field nearly flat and more astigmatism; they are still significantly smaller than with the stand-alone flat-field Erfle.

Above designs were reverse-raytraced, since it is simpler. With direct raytracing possibly giving somewhat different output (the larger field of view, the more so), the last and first design are also raytraced directly. Since this age ancient of SYNOPSYS doesn't offer the "perfect lens" convenience of OSLO Edu, as a perfect objective is used an f/5 Eisenberg-Pearson system, and as a perfect eye a mirror with r.o.c. R=20mm placed 20mm from the exit pupil (effectively aperture stop placed at its center of curvature), producing 10mm f.l. hence reproducing the f/5 relative aperture. The mirror induces no aberrations except field curvature (R/2), and by entering this field curvature into raytrace it is effectively neutralized.

Direct raytracing of the last design shows somewhat more field curvature, and less astigmatism, hence no significant differences (keep in mind that the direction of light forming the final image is opposite to that with reverse raytracing, thus the orientation of the astigmatic field, i.e. actual field curvature is of opposite sign). Most of curvature added is probably due to the image having lower distorsion, hence the slightly smaller linear image is larger than that in the reverse raytraced unit when distortion effect is taken out.

Focusing on the edge mysteriously changes the form of astigmatic field. It is probably a consequence of the very large angle involved: due to it, refocusing results in a change of the field point height. That by itself shouldn't change the astigmatic field, but there could be a lateral shift of the converging beam too small to be detected at this scale (bottom). Below left, ray geometry for the axial, 0.7 zone and edge beams, shows that the 0.7 zone beam doesn't reflect back in the same direction, due to its exit pupil location - determined by the point of intersection of its central ray and central ray of the axial beam - being closer to the mirror (magnified in box at right). Magnified focus area for the edge beam (top box, focus on axis) shows that the edge pupil location is slightly farther away from the mirror's center of curvature, causing a slight lateral asymmetry vs. incoming central ray. Still, change in the astigmatic field still seems to be a glitch. Defocus does not alter ray geometry, only the pattern of cross section, hance the longitudinal astigmatism remains unchanged.

Direct raytrace of the Rutten-Venrooij Erfle has the astigmatic field change even more, at least in terms of longitudinal aberration. In terms of the field edge blur size, the change is nearly negligible. While this seems absurd, it results primarily from the difference in generating higher-order astigmatism. It is relatively low in reverse raytracing (top), but yet lower in the direct raytrace (bottom). In the former, it is large enough to cause a small wrinkle at the bottom of the wavefront (indicated by an extra contour line on the wavefront map shown), throwing the bottom edge rays significantly farther out. However, since the corresponding wavefront area is near negligible, so is its effect on the ray spot plot.

Another tricky part - for raytracing software - is distortion. While both SYNOPSYS and OSLO Edu give zero (practically) aberration coefficient for distortion for aperture stop at the center of curvature of a concave spherical mirror (2mm aperture diameter, 200mm mirror f.l.), they at the same time give the corresponding distortion plot showing -16% distortion for the 33-degree beam. There is no change in focal length for this beam, it hits the mirror perfectly collimated, just as the axial beam, the only deformation is the reduction in its vertical diameter at the aperture stop, due to its steep incident angle (SYNOPSYS has the option of rotating aperture stop with incident angle, but it isn't working as intended). It results in the effective focal ratio varying with the pupil angle, from the highest in horizontal plane, to the lowest in the vertical. Consequently, diffraction image is enlarged vertically, which effectvely should produce positive distortion, but raytrace apparently bases its distortion plot on the reduction in the vertical beam diameter - in proportion to the cosine of incoming angle - interpreting it as (geometric) image reduction. Since similar beam deformations are common in eyepieces (the obviously wider 33-degree exit pencils in the direct raytracing setup are a consequence of vertical beam expansion due to projection on oblique surfaces within the eyepiece), it makes questionable the accuracy of distortion assessment.

Direct raytracing shows somewhat more coma, and less astigmatism, but the differences are relatively small. Edge pencils are noticeably wider than axial in the vertical plane, as a result of projection on oblique surfaces. If sufficiently extended, it would become visible that they are mildly diverging, i.e. having longer focal length than axial beam after focusing, hence creating larger image scale - positive distortion - of the outer field. Direct raytracing gives significantly more spherical aberration, but still well within acceptable, at the level of 1/8 wave P-V.

11.29 Sub f/2 Schmidt vs. Busack camera

Schmidt camera is a long time standard for fast camera systems. In its basic form it consists only from the corrector plate and a spherical mirror. Since it produces strongly curved image field, a singlet lens flattener right in front of the final image can be added, at a price of somewhat compromised field quality. Its main advantage is exceptional correction in optimized wavelength across wide fields. However, at large focal ratios - roughly f/2 and larger - its spherochromatism becomes significant. Here, it will be compared to another highly corrected system, Busack-style Hamiltonian, consisting from a front singlet lens, Mangin mirror and field corrector. While the Schmidt plate uses higher-order aspherics, the Busack is all-spherical system, and in this respect easier to fabricate. Another drawback of the Schmidt camera is its relatively long tube, about twice the focal length; Busack-style camera is twice shorter. The systems considered here are 300mm in aperure diameter, and slightly faster than f/1.9. Although for cameras is relevant wider spectral range, the usual g-r range will be sufficiently illustrative. The Schmidt has somewhat lower limit to minimum central obstruction, but it will be assumed it is 1/3 of aperture diameter for both; it is not likely it would be significantly smaller in the Schmidt.

Starting with the Schmidt, the basic system produces practically perfect central line correction over the 4° field with -555mm surface curvature. However, due to spherochromatism, the polychromatic Strehl for the five wavelengths is only 0.6 (even sensitivity). Color correction is also unbalanced, with the error in the blue-violet significantly larger than in the red. Somewhat better balance would be achieved by inducing low amount of undercorrection (~1/10 wave P-V), but the effect is minor, and will be neglected.

On the other hand, Busack doesn't have quite as good central line correction - although the difference is inconsequential for the camera system of this size - but it has significantly better color correction, with 0.81 polychromatic Strehl. Its distortion is more than twice higher, although still below 0.2%. Central maxima at 2° off is well defined, but the ring structure is less distinct than in the Schmidt.

Looking at the energy distribution plot, the Busack has smaller 80% energy circle. Contrast transfer for the Schmidt is practically identical for all points across the field, somewhat inferior to the Busack's central field transfer, but better than its peripheral field transfer. In the Schmidt with a flattener added (4mm thick plano-convex lens with 190mm surface radius facing the mirror, 548.7mm apart), both energy distribution and contrast transfer are significantly worse than in the Busack. With an integrated flattener (same lens, but the corrector changed to compensate for the errors induced by the flattener), degradation is less pronounced, but overall it is still behind the Busack system.

In all, the two systems are roughly comparable in performance when the Schmidt uses its best, curved surface. Schmidt with integrated flattener is somewhat behind, and with the flattener added it is inferior to the Busack. As a final note, spherical surfaces of the Busack don't imply it is easy to fabricate and assemble; tolerances are still very tight.

11.30 Astro-Physics CCDT67 Telecompressor

This simple telecompressor consists of a cemented positive doublet with 305mm focal length. According to AP, it is originally developed for their 10" f/14.6 Maksutov_Cassegrain, but it is then oriented toward general use. It is not recommended for systems significantly faster than f/9. With a clear opening of 45mm it provides full illumination for KAF-8300 chip (18x13.5mm, 22.5mm diagonal), or alike. Its designated compression factor is 0.67, but it can be varied simply by changing the lens position vs. original focal plane. As for its other effects, it is only stated that it doesn't flatten field, and does not induce coma. Prescription is not published, but there is a drawing online with its outer radii and center thickness, probably measured up by a user. Starting with those values, it is not hard to come to a doublet that would produce nearly flat, coma-free field, with good chromatic correction - as it can be assumed are requirements for such accessory. Below is raytrace of such a lens. Its focal length is 301mm, quite close to the stated 305mm for the actual design. While a number of glass pairs can be employed, best correction level requires low-index front element (generally crown) and high-index rear element (generally heavy flint). With only three radii to play with, such a doublet cannot be fully corrected for both, coma and astigmatism. Since it has inherently non-zero Petzval curvature, it requires some astigmatism of opposite sign for flat field, and it retains some residual negative coma as well (top). Consideration is limited to Schott glasses, but it shouldn't have significant effect on doublet's correction level. The objective is a 200mm f/10 OSLO "perfect lens", hence all aberrations come from the telescompressor.

For this focal length, 11.7mm radius (half of the chip diagonal) corresponds to 0.5° field angle. The lens induces low undercorrection (less than 1/8 wave P-V, from LA/64F²), with coma and astigmatism at 0.5° (flat field) corner-acceptable (astigmatism alone, with LA=0.21mm, is 0.21/8F² or 1 wave P-V for 550nm wavelength; it is in agreement with the Zernike term value, 0.524, indicating 0.524/sq.rt.(6)=0.214 RMS wavefront error, i.e. 0.214*sq.rt.(24)=1.05 P-V wavefront error). Coma alone, according to its Zernike term of 0.317, is at the level of 0.317/sq.rt.(8)=0.112 wave RMS, or 0.112*sq.rt.(32)=0.63 wave P-V. The error diminishes with slower systems (keeping compression ratio unchanged), as illustrated with the boxed ray spot plots for a 133mm f/15 perfect lens). Allowing for a small change in the outer radii values (190,-210,-433mm) corrects for the slight field curvature, reducing the combined astigmatism/coma error at 0.5° off by 15%, with the chromatism only slightly worsened. By a further small reduction in astigmatism, resulting in a mild negative field curvature, the combined error is further reduced by about 5%, but at a price of somewhat more chromatism (190,-216,-400, with SF4 replaced by SF6). This seems to be about the peak correction level for this type of a doublet. As mentioned, relatively low astigmatism is necessery to flatten field; such doublet can be made astigmatism-free, but at a price of larger coma and some field curvature (bottom left). Similarly, if it is made coma-free, astigmatism and field curvature become quite strong (bottom left).

While the actual unit could be little better, we'll use the one based on measured radii to evaluate its performance on some specific telescope systems. Starting with the Schmidt-Cassegrain, a 200mm f/2/10 system, will need the lens at 338mm from the secondary (i.e. 129mm in front of the original focal plane) to produce 0.67 compression. With fixed mirrors, it will form focus at 101mm behind the rear lens' surface, or 48mm in front of the original focus (top).

If the final focus is to be brought back to its original location by moving primary mirror, the telecopressor lens needs to be at 356mm from the secondary. With respect to the system alone, CCDT67 induces low undercorrection (only with fixed mirrors; with moving primary it is nearly offset by the resulting overcorrection) and astigmatism somewhat weakening field curvature, with longitudinal chromatism actually reduced. Number below the telecompressor is the radius of the converging light at the front lens; it is smaller than its clear opening with fixed mirrors, but somewhat larger with moving mirrors, as shown magnified at the bottom.

With an aplanatic SCT, showing the moving mirror case, induced overcorrection is significantly larger at 0.07 wave RMS, due to the undercorrection generated by moving primary adding up to that of the telecompressor lens.

In a 200mm f/3/8 Ritchey-Chretien (similar to the GSO 200mm f/8 RC), CCDT67 astigmatism nearly offsets that of the telescope, leaving it with somewhat less field curvature. Induced overcorrection is negligible, but coma becomes more visible at this relatively fast focal ratio. The error at 0.5° is still 25% lower than in the stand-alone system. Shown is only scenario with fixed mirrors, since it is more likely. The original focal plane is only about 5mm behind, so bringing the final focus to that location by moving primary would produce only a small effect.

11.31 Paracorr-like corrector on a sub-f/3 mirror

Reading Mike Lockwood's 2009 article about using TV Paracorr on his 14.5" f/2.55 mirror, I wondered what the Paracorr correction level with such a fast mirror looks like. Tele Vue states it works well down to f/3, and according to Mike it still worked very well at f/2.55. So, in order to find out what is actually happening past f/3, I raytraced a couple of designs similar to the Paracorr in their basic arrangement. One is from Smith/Ceragioli/Berry, and the other could go as a poor man's Paracorr, using only two cheap glasses. They are raytraced with f/4.5 and f/2.7 300mm in diameter paraboloids.

Starting with the S/C/B design, raytrace at f/4.5 (for an f/5 effective system, with the corrector acting as x1.15 focal extender) shows that it has low residual coma, approximately at the level of an f/8.3 paraboloid. Chromatic correction easily meets the "true apo" requirements, astigmatism is low (0.035mm longitudinal error at 0.5° implies 0.32 wave P-V, or 0.065 wave RMS avefront error) and spherical aberration is negligible (0.045mm longitudinal aberration implies 0.045/64F²=0.0000283 P-V wavefront error, or little over 1/20 wave for 550nm wavelength).

With f/2.7 mirror (for an f/3 system), if maintaining the same back focus, correction looks a lot worse, mainly due to the increase in spherical aberration to 0.63 wave P-V. Also, longitudinal chromatism is not near-optimally balanced anymore, due to the blue/violet end shifting away, and the red closer to the optimized wavelength. Luckily, spherical aberration is easily corrected by pulling corrector little more out: 5mm farther from the mirror brings error down to 1/10 wave P-V (another 0.4mm takes it to its minimum, only ~1/100 wave P-V larger than with f/4.5 mirror). It also reduces chromatic effect, roughly by a half. On the negative side, it makes both, coma and astigmatism larger, but the effect is near negligible (larger astigmatism induces very mild field curvature). Despite it, the bright core of the diffraction image is well defined.

The poor man's Paracorr has mild field curvture and sub-optimal longitudinal chromatism. It would benefit from final optimization, but the gain would be small; it is good enough for the purpose. With f/4.5 mirror (for an effective f/5.5 system, i.e. acting as x1.22 focal extender), it comes close correction wise to the S/C/B design (top). Coma at 0.5° is somewhat more visible, because the linear field is 10% wider, and because of small defocus due to the mild field curvature; it is at the level of an f/8.5 paraboloid.

With f/2.7 mirror it also appears much worse, if placed so that the back focal length remains identical (which is where coma and astigmatism correction remains nearly unchanged). Spherical aberration is by nearly a third lower than in the S/C/B, due to it resulting from the increase in both, primary and secondary spherical, and the S/C/B has them of the same sign. Spherical aberration is reduced to 1/10 wave P-V by pulling corrector 3.4mm out (which is about the minimum with this design, due to its more strongly curved surfaces, hence more of higher-order spherical). Similarly as with the S/C/B, it also partly offsets worsened longitudinal chromatism. However, due to the initial chromatism here being of different sign, the field is effectively flattened. Diffraction image is somewhat more smeared than with the S/C/B, but the bright core is mainly preserved.

Taking a closer look at the wavefront 0.5° off axis, reveals that it is mainly astigmatism and coma causing off axis image degradation. With the S/C/B design and f/2.7 mirror, corrector separation 725mm, the 0.5° focus in the plane of axial focus contains significant Zernike defocus term, indicating that the field is less than perfectly flat (below, left).

Minimizing defocus term by shifting focus slightly toward the mirror leaves primary coma and astigmatism as the main error contributor (right; the sign on the astigmatism plot is opposite to that on the system drawing, since the former by default assigns minus to the direction toward objective, and plus away from it). All terms outside the first 15 are entirely negligible, and even those appearing not negligible, like secondary coma and primary spherical aberration, have little effect because the error adds as the square root of the RMS errors squared (the RMS error is implied by the absolute term value divided by a factor varying with the aberration: 2, sqrt(3), sqrt(6), sqrt(8), sqrt(5), sgrt(8), sqrt(10), sqrt(12) and sqrt(7) for tilt, defocus, primary astigmatism, primary coma, primary spherical, trefoil, secondary astigmatism, coma and spherical, respectively). The cumulative RMS error is 0.586 wave in the plane of central focus, and 0.466 wave with defocus minimized.

Similarly, the poor man's Paracorr has astigmatism and coma as main error contributors, with primary coma somewhat lower, and secondary somewhat higher than the S/C/B. Best field is somewhat less curved, but the best 0.5° image is still found slightly closer to the mirror than the central focus (below).

Minimizing defocus reduces the total RMS error from 0.518 to 0.445. The latter corresponds to 2.5 waves P-V of pure coma, which entirely transforms diffraction image into comatic blur 0.063mm long. However, as a mix of multiple aberrations, diffraction image has no clear resemblance to coma aberration, tending to reflect its dominant component, astigmatism. Note that central obstruction is omitted to leave wavefront maps intact; Mike Lockwood's f/2.55 Newtonian had it at 31% linear, so the effect can be neglected in this context.