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10.2.4.8. Houghton-Cassegrain comparison       12. THE EYEPIECE

11. MISCELLANEOUS TELESCOPE OPTICS

11.1 FOCAL EXTENDERS AND REDUCERS

Nearly as commonly used as eyepieces are the telescope accessories for extending, or for compressing the effective focal length - focal extender (also: Barlow, tele-extender), and focal reducer (telecompressor) lens. The former are used for adding more magnification options with a given set of eyepieces; also, extending/narrowing of the converging light cones improves eyepiece performance (not long ago, added benefit of a Barlow lens was extending tight eye relief of the conventional short-focus eyepieces, but that is less important with new generations of eyepieces with longer eye relief). Focal reducer lens, on the other side, can also serve the purpose of obtaining more magnification options, but is mainly interesting to those who want to make their systems "faster", particularly for astrophotography. For that reason, it is commonly made to acts as a field flattener as well.

The two main parameters of either extender or reducer are its focal length and the inside separation from the original focus. In general, the larger either one, the larger the effect. Scheme below shows Barlow lens extending the original cone and, by the same factor M, multiplying the focal length and image magnification, i.e. L/L0=M. In the thin lens approximation, if the extended cone was the original one, and the lens was positive, twice stronger, the new focus would form where the dashed lines meet, with magnification.

Optically, the effect of either extender or reducer on the focal length, expressed as a magnification factor, is given with the same equation - it is only the sign of their focal length that produces magnification greater (extender), or smaller (reducer) than one. Graph below shows how system magnification changes with the focal length of the extender/reducer lens, with both lens-to-new-focus separation L and the lens focal length ƒ in units of the lens-to-original-focus separation L0. With the decrease in the relative focal length, extender lens' magnification asymptotically approaches infinity, and reducer lens' zero. Raytrace examples below illustrate some diverging-beam extenders.

All are paired with a perfect 1000mm f.l. lens, so all aberrations are produced by the Barlows.

Conventional Barlow lens

Conventional Barlow lens is a cemented doublet achromat, such as one given by Rutten/Venrooij (1), and have moderate lengths and ray divergence, as long as their magnification factor doesn't significantly exceed 2. Fancier glasses produce better performance (2, a Russian 2x extender), but it is, in this case, paid for with noticeably stronger divergence. Divergence is, expectedly, even stronger with TAL's 5x double doublet Barlow (3, as given by Klevtsov in "New serial telescopes and accessories" 2014); note that in the box above is raytrace of a single doublet of this Barlow with 2x magnification at ƒ/8.

Finally, the "shorty" Barlow (4) unavoidably also has strong divergence and, for given glasses, slightly inferior performance. As the plot shows, it induces typical for tele-extenders overcorrection, which is in this case not quite negligible at ƒ/5. In order to reduce it significantly in this particular design type, either flat field or off axis correction has to be further compromised (the original design from the Smith/Ceragioli/Berry has less than 1/10 wave P-V of overcorrection and near-perfect off axis correction, but also a strong 100mm field curvature radius (note that the scale differs from one example to another; 1 and 3, and 2 and 4, are fairly comparable, while the former two are roughly 2-3 times larger vs. the other two than what it appears on the picture.

Telecentric Barlow lens

More recent development in both, focal extenders and reducers arena are the telecentric types. Unlike their conventional counterparts, they produce near-zero divergence exit beams. The advantage of it is that the added element doesn't affect - generally negatively - performance of telescope eyepieces, which are by default designed for near-telecentric (i.e. parallel with optical axis) entrance beams. For creating telecentric exit beams, a two lenses, or group of lenses opposite in their power sign, and with a wider separation, are needed. Two examples of telecentric Barlow below are, as before, with a perfect 1000mm f.l. lens, hence all aberrations come from the Barlow.

The first example is flat-field at ƒ/5, but developing some field curvature at ƒ/8. The other one, more compact, has nearly constant, strong field curvature (over 6 diopters, or approximately infinity-to-8 inch accommodation). However, even with zero accommodation, it is still comparable to the longer design (the reason is the very small relative aperture, below ƒ/22, hence fairly insensitive to defocus). In general, higher magnification requires longer units.

Focal reducers

The simplest form of the focal reducer is a small achromat, usually cemented, corrected for infinity. Below is shown the effect of such random lens with a perfect 1000mm f.l. ƒ/10 perfect lens (top), a 100mm ƒ/10 doublet achromat (bottom left), and a 200mm ƒ/10 standard SCT. While its performance with a perfect lens is acceptable, it doesn't produce appreciable improvement with the SCT, as its original spots in the box show (the flat field SCT-alone spot is roughly 20% larger). Achromat's astigmatism actually enlarges the wavefront error, but what matters in the outer field is the angular size.

In the achromat, it significantly weakens field curvature, at a price of more astigmatism, mixed with some coma, in the outer field (in the box are the e-line spots for achromats best image field).

Performance improves with dedicated achromatized lens pair, either cemented/contact or separated. An example of the former is given by Rutten and Venrooij, as a reducer/flattener for aplanatic (coma-free) SCT. It is shown below also with a perfect ƒ/10 1000mm f.l. lens, with which it does not produce flat field, since its astigmatism/field curvature needs to offset those of the SCT. As ray spot plots and diffraction images (polychromatic, for the wavelengths shown) show, gain over uncorrected flat-field performance is relatively small (in the box are shown flat-field and best image spots for the edge point w/o reducer).

Three more examples include a simple reducer/flattener/coma corrector for the standard SCT (top), roughly similar in form reducer/flattener for an apo doublet, and a random 3-lens reducer with a 100mm ƒ/10 (1000mm f.l.) perfect lens. The SCT reducer produces off axis spots larger than the R&V cemented doublet, but its actual performance is significantly better. It is because a better part of its ray spot are widely scattered rays, due to significant proportion of higher-order aberrations curling up relatively small areas of the wavefront, as opposed to the compact astigmatic spots of the achromat (e.g. for given wavefront error, ray spot plot for primary spherical aberration is nearly 6 times larger than for the primary astigmatism spot). Better indicator of performance are the diffraction images, comparable in scale (important factor is that the air-spaced doublet, unlike the cemented one, also corrects for coma). Performance level of this reducer/corrector probably doesn't fall far behind some simpler commercial units, which perform acceptably up to about 1/3 of a degree field radius. More complex units use more lenses, usually 3 to 4, in any arrangement (e.g. Meade's 0.63x reducer consists of two cemented doublets, and its 0.33x reducer of three singlets), with the main difference being field definition beyond this circle.

The difference in flat-field performance is quite obvious in the case of the 80mm ƒ/8 fluorite doublet (middle). The reducer is telecentric, and unintended extra bonus was correcting the violet end. Finally,the 3-singlet reducer produces near-perfect 2-degree field with a perfect lens. Yet, its performance with systems having significant astigmatism/field curvature is uncertain.

Next, an illustration of performance level of the common f/6.3 SCT focal reducer/corrector. It is similar with both, Meade and Celestron, consisting of two cemented doublets. The configuration can entirely correct for coma, but lenses add some astigmatism of the same sign as the mirror Petzval curvature, thus the field cannot be flattened; in this example, the best field curvature is actually twice as strong (-150mm vs. -300mm radius). Also, entirely correcting for coma seems to be coming at a price of less than satisfactory (below 0.80 Strehl) spherical aberration correction. Still, mainly due to the correcting most of coma, flat field performance is significantly improved, and visual field is free of visible aberrations.

Reducer does add some spherical aberration, lowering axial Strehl in the central line to 0.7 (undercorrection). Actual units probably have somewhat better overall correction, but it can't be much better due to above mentioned limitations of the configuration. Main limitation is that the lenses add astigmatism of the same sign as the Petzval, which increases field curvature as a price of coma correction. It is likely that lenses do not entirely correct for coma, but make it negligible, as this illustration shows (the actual reducer location is somewhat closer to the focal plane, but it doesn't change significantly performance limitations; central line correction, however, does go above 0.80 Strehl). The illustration shows the imaging mode. For visual use, converging beam needs to be extended to reach the eyepiece. This induces undercorrection in the reducer, mainly offset by overcorrection induced by the mirrors.

Finally, one more SCT reducer/corrector configuration, a 3-singlet arrangement used by Meade for its f/3.3 reducer. Shown is f/4.4 reducer using two common glasses, which fully corrects for coma while, similarly to the previous example, makes field curvature somewhat stronger. However, as the image is smaller, the curvature matters less.

Field curvature effect becomes significant only close to the field edge. This reducer would be primarily intended for photography, so its best curved field performance is irrelevant, but the simulations at the bottom illustrate modest effect of the quite strong field curvature (R=-144mm) on flat field performance (which would be still lower with the 0.33 reduction ratio). This reducer also induces spherical aberration (undercorrection) which is reduced if it is placed closer to the focal plane. That, however, tends to increase astigmatism, and make full correction of coma more difficult. As with the previous example, it is easier to make surface flatter with some residual coma left in, since the same surfaces that induce correcting (opposite) coma also induce astigmatism of the "wrong" sign. But, as this example illustrates, good performance is possible even with a strong field curvature. Actual units, being computer optimized, probably deliver still better performance.


11.2. DILWORTH RELAY TELESCOPE

Don Dilworth's two mirror-relay telescope uses lenses to transfer an internal focus out to an accessible location. It could also be considered a two-mirror system with sub-aperture lens corrector(s), but the relay property makes these systems different from the rest. Unlike other two-mirror relay systems - notable example being Robert Sigler's design - which can have very good axial correction, but much left to be desired field wise (Sigler's 6-inch ƒ/7 system has coma close to that of an ƒ/4.5 paraboloid, and a horrendous field curvature of -44mm), Dilworth's design achieves both. It has an extraordinary monochromatic axial correction - practically zero aberration - weakly curved field, field aberrations lower than comparable aplanatic Cassegrain (Ritchey-Chretien), nearly 0.4 waves p-v of longitudinal chromatism in each, C and F line (comparable to a 100mm ƒ/30 achromat) and no detectable lateral color.

Additional positives include relatively small central obstruction, fast focal ratio, and generous back focus. The negative is more complex alignment, and collimation sensitivity, due to the three widely separated lenses. However, with the relatively slow primary, it should not be significantly out of the ordinary.


11.3. SOLAR TELESCOPE

Majority of the telescopes in use are those made for general astronomy. However, a telescope for general purpose may be limited in its ability to serve for some special purposes, such as observing outside of the visible range (infrared, radio), or observing particular astronomical object with special properties, such as the Sun. Among various specialized instruments for solar observations (coronagraph, spectroheliograph, etc.), probably the most interesting for an amateur is a telescope specialized for use of the H-α (hydrogen alpha) filter. Blocking the rest of abundant solar radiation makes possible observing of a variety of solar features, otherwise less pronounced or invisible (prominences, filaments, solar eruptions, etc.). 

Solar H-α etalon telescope

The H-α solar telescope can either use H-α filter placed in front of the objective, or H-a etalon placed inside telescope, combined with a blocking filter in front of the objective (for astrophotography of emission nebulae, such filter can be mounted close to the image w/o use of blocking filter, but otherwise it is avoided due to the heat-related risk). For the optimum performance, such filter requires near-collimated light, hence a telescope with H-α etalon located behind the objective needs special arrangement providing a collimated section within the light path. It can be created in a simple arrangement of three singlet lenses, two positive and one negative, as shown below.

The advantage of the etalon arrangement is that the filter can be manipulated in order to increase, or modify performance. For instance, double etalon will further narrow the passband; tilting the etalon slightly shifts the passband, allowing optimizing the passband to the detail of observation, and so on.

This simple arrangement cancels all aberrations except field curvature and some residual astigmatism (chromatism, of course, is not corrected, but it is of no consequence operating at a single spectral line). Despite the best field being strongly curved, the 0.7-degree field is still well within diffraction limited even at the edge, due to the small linear field extent.

Width of the collimated section is a function of the front-to-mid lens separation: the smaller separation, the smaller width, and vice versa. The flat-field correction somewhat improves with the smaller separation, but not significantly. For any given separation, the width of collimated section can be also widened by using stronger glass for the mid element. It also improves field correction but, again, only by 10-15%, or so.

The ethalon configuration can be used with an achromat as well. The focal length of the negative front lens needs to be equal to its separation from the original focal plane, and the positive rear lens needs to be slightly weaker (depending on their separation). Best configuration here is with the two lenses facing each other with their curved side. The aberrations induced are a small amount of overcorrection, which actually improves correction in the red, and field curvature. As an example, placing a negative plano-concave lens lens (f=-291mm) at 800mm from the objective in a 150mm f/8 achromat, with the plano-convex lens (f=304mm) 70mm behind it, induces slightly over 1/10 wave P-V in the green e-line, with the error in the red r-line reduced to 1/30 wave. No appreciable effect on chromatism and coma, but the best field curvature goes from -460mm to -270mm.


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 will 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.

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 the aberrations come from the prism.

At f/10, Zernike term for primary spherical aberration (8) is 0.002853, which divided with 50.5 gives the RMS wavefront error as 0.001276. The corresponding P-V error is larger by a factor 11.250.5, 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 80.5 gives the RMS error as 0.00141 (the P-V error is larger by a factor 320.5). 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.

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).

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/fo, and the latter by 250/fe, with fo and fe 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 (nd=1.53, vd=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# differentia, 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# 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). 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.  

10.2.4.8. Houghton-Cassegrain comparison       12. THE EYEPIECE

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