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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 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. 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. Still, mainly due to the correction 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. For that reason, 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).

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.
 

10.2.4.8. Houghton-Cassegrain comparison       12. THE EYEPIECE

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