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10.2.2.5. Schmidt-Cassegrain cameras

As mentioned before, Schmidt-Cassegrain optical arrangement, similarly to Schmidt camera, allows for correction of all five lower-order aberrations. In addition, axial higher-order spherical aberration is easily controlled with the 5-th order term on the corrector. However, there is a price to pay to accomplish this: it is necessary to either sacrifice compactness, or strongly aspherize the mirrors. Also, since relatively low secondary magnifications are needed, an accessible final image requires relatively large secondary mirror, not suitable for visual observing. For this reason, highly-corrected Schmidt-Cassegrain systems are generally used for astrophotography.

Assuming corrected astigmatism, the flat-field requirement for a two-mirror system is that radii of curvature of the mirrors are identical. Thus, R2=R1, and ρ=R2/R1=1. With the secondary magnification for a general two-mirror system given by m=ρ/(ρ-k), k being, as before, relative height of the marginal ray at the secondary in units of the aperture radius (closely approximated by k~1-s/ƒ1, s being the mirror separation, and ƒ1 the primary's focal length) it is, therefore, for the flat-field Schmidt-Cassegrain given by m=1/(1-k). Since the condition for accessible image is mk>(1-k), this sets the corresponding minimum for the secondary size as k>1/(m+1), with relatively large secondary required for m<2.

With the minimum secondary size for accessible surface k~1/(m+1) combined with secondary magnification m=1/(1-k), determines the minimum k value for accessible image (i.e. η=0, or larger) as kmin=1.5-√1.25, or practically k>0.4. Back focus distance in units of the primary's focal length is, for a general two-mirror system, η=(m+1)k-1, which with m=1/(1-k) becomes η=(3k-k2-1)/(1-k) for the flat-field Schmidt-Cassegrain.

Aperture stop is assumed to be at the corrector, with σ1 being stop(corrector)-to-primary separation in units of the primary mirror radius of curvature.

Note that "accessible" here is somewhat formally based on the final image being distanced from the secondary more than the primary mirror (i.e. with positive back focus η values). In practice, very low positive back focus values may not be sufficient to provide accessible image with small instruments, while even low negative back focus values may be sufficient to make the final image accessible to a camera with large apertures. Thus the actual condition for image accessibility is best determined with a particular design/camera combination.

 The two general configurations of flat-field Schmidt-Cassegrain anastigmatic aplanat are:

(1) non-compact, keeping the mirrors spherical by maintaining stop (corrector) location at or near the primary's center of curvature, and

(2) compact, with the corrector as close as desired, but with strongly aspherized mirrors (oblate ellipsoids) needed to cancel off-axis aberrations. As a result of higher level of spherical aberration induced by mirrors, the latter also requires significantly stronger corrector, resulting in correspondingly higher spherochromatism.

Among the non-compact solutions, the three main variations are: (1) flat-field anastigmatic aplanat, which requires slightly aspherized primary (one of Baker's designs, hereafter referred to as non-compact Baker-Schmidt), (2) concentric Schmidt-Cassegrain, with spherical mirrors but some field curvature remaining, and (3) flat-field aplanat, with spherical mirrors and low residual astigmatism which flattens the field. Either of the last two is sometimes referred to as Slevogt camera; the original design by Karl Slevogt was flat-field aplanat.

Following table lists main parameters of four Schmidt-Cassegrain cameras:

TABLE 13: Main parameters of highly corrected Schmidt-Cassegrain systems in terms of the relative height of marginal ray at the secondary k, given in units of the aperture radius (in effect, the relative minimum secondary size in units of the aperture) and secondary mirror radius of curvature ρ, in units of the primary mirror radius of curvature. Note that all systems except concentric Schmidt have flat image field.

Follows more detailed description of these four systems.
 

In addition to the general condition for flat-field anastigmatic systems, R2=R1 (thus ρ=1), the non-compact Baker-Schmidt parameters, as given by Schroeder (Astronomical Optics, p146-147), are:

▪ aperture stop at the corrector,
▪ spherical secondary mirror,
▪ primary mirror conic K1=k2(1-k)2/[4-k2(3-k)2],
▪ relative stop (corrector) separation from the primary, in units of the primary's radius of curvature
σ1=[4-k2(3-k)2]/[4-2k2(2-k)(3-k)], and
▪ corrector's lower-order aspheric parameter A1=b/8(n'-n), where n, n' are the glass and exit medium index of refraction (both for Schmidt surface at the rear of corrector), and the aspheric coefficient
b=2[K1+1-(2k-k2)2]/R13.

With a typical larger secondary (k~0.4), it is somewhat lower than in a commercial Schmidt-Cassegrain telescope with identical primary mirror ƒ/ratio (~0.6 vs. ~0.7); that implies proportionally lower spherochromatism, but the difference is not significant.

In comparison with the Schmidt camera of identical relative aperture, however, its spherochromatism is significantly higher. Aspheric factor for the Schmidt camera is b=2/R3, R being the mirror radius of curvature, equal in effect to a product of the Baker-Schmidt camera's primary radius of curvature R1 and its secondary magnification m. Hence, aspheric coefficient of the Baker-Schmidt is larger by a factor of [K1+1-(2k-k2)2]m3. For given neutral zone placement, that results in as much stronger corrector, hence as much higher spherochromatism.

The lower-order aspheric parameter A1 determines needed corrector radius as Rc=-1/2ΛA1d2, with 0>Λ>2 being the arbitrary focus parameter, and d the corrector aperture radius. The focus parameter determines height of the neutral zone at the corrector as NZ=(Λ/2)1/2 in units of the aperture radius. While designers often opt for NZ=0.866 (for Λ=1.5), with which the geometric chromatic blur is smallest, the wavefront error and, thus, the spherochromatism, are at the minimum for NZ=0.707 (Λ=1). Not only that the chromatic error with the latter is more than cut in half, so is the needed corrector depth, making its fabrication significantly easier.

 Calculation of the higher-order aspheric parameter A2 is more complicated due to the presence of secondary mirror. As in commercial SCT systems, it is given by A2=b'/16(n'-n), with the aspheric coefficient b' for the camera approximated by b'~2σ1/R15. The higher-order error component becomes significant with the primary relative aperture larger than ~ƒ/3, the larger aperture, the more so.

Neither primary's conic K1 nor corrector (stop) separation σ1 change significantly with changes in k (i.e. with changes in mirror separation); primary needs to be aspherized only slightly, to an oblate ellipsoid, and the corrector is approximately at 1.1R1 from the primary. In fact, the primary can be left spherical, with residual coma cancelled by moving corrector slightly farther away from the primary, according to σ1=[2-k2(2-k)(3-k)]/2[K1+1-k2(2-k)2]. The only consequence is introduction of very low astigmatism, with the field remaining practically flat (value of the lower-order aspheric parameter A1 also changes slightly, according to the relation given above, while the  A2 value remains practically unaffected).

Specifically, the consequence of leaving the primary spherical, illustrated on a 300mm f/3.3 system, is near 0.1 wave RMS wavefront error at 1° off-axis, nearly 0.06 wave due to astigmatism, and the rest due to the 12,000mm radius field curvature, placing the flat-field image closer to the sagittal plane (thus resulting in elongated image far from axis). Since both change with the square of field radius, so does the cumulative error. By the photographic blur-size criterion, this error is still negligible, having the blur diameter below 5 microns. However, if significantly wider fields are used, the slightly different flat-field aplanat configuration (described below) is a better option, with nearly twice smaller error and round aberrated image. Note that the error scales with the aperture.

Concentric Schmidt-Cassegrain is a deviation from the flat-field concept, in that its image surface is more than negligibly curved. With both mirrors spherical, coma and astigmatism are corrected, the latter implying that the two radii of curvature, for the primary and secondary mirror, are not identical. In order to cancel off-axis aberrations, aperture stop (which is placed at the corrector) needs to be at the center of curvature of primary mirror (σ1=1), with mirror radii and secondary location satisfying relation 2ρ=1+k, (as before, ρ=R2/R1). This also determines secondary magnification as m=(1+k)/(1-k). Image surface is accessible for k>1/3.

Since astigmatism is zero, the image curvature equals Petzval's, given by RP=R2R1/2(R1-R2). All three surfaces - that of the final image, primary and secondary mirror - are concentric, with the center of curvature at the vertex of aperture stop (corrector). Hence, the amount of image curvature is mainly determined by the physical length of the instrument and, with accessible image surface, somewhat exceeds primary's radius of curvature. Specifically, with the back focus length in units of primary's focal length η=(3k-1)/(1-k), image curvature in units of the primary's focal length is given by 2+η=(1+k)/(1-k), numerically identical to secondary magnification.

Since the image curvature is also identical to that in a comparable Schmidt camera, the only advantages of the concentric arrangement are accessible image and system length reduced by a factor (1-k)/(k+1). Also, due to more strongly curved secondary, the arrangement allows for somewhat smaller minimum secondary required for accessible image; substituting secondary magnification given above into k=1/(m+1) for zero back focus value results in kmin=0.33 (note that smaller secondary comes at a price of smaller relative aperture for given primary).

On the other hand, spherochromatism is, similarly to the compact Baker-Schmidt camera, significantly higher than in the standard Schmidt. With aspheric factor for the system given by b=2[1-(k2/ρ3)]/R13 - or, in terms of k, b=2{1-[8k2/(1+k)3]}/R13 - corrector power and sphero-chromatism in the concentric Schmidt-Cassegrain are larger than in the Schmidt camera by a factor of [1-(k2/ρ3)]m3. For identical secondary magnification as in the above Baker-Schmidt example, m=1.67 and the concentric Schmidt parameters would be k=0.25 (thus inaccessible image) and ρ=0.625. Needed corrector power and resulting chromatism in this configuration are 30% higher than in the Baker-Schmidt, and 3.5 times higher than in the Schmidt camera.

For identical minimum secondary size k=0.4 in both, non-compact Baker and concentric Schmidt, corresponding values are ρ=0.7 and m=2.33 for the latter, with its chromatism higher by a factor of 6.5 than in comparable Schmidt camera. With the primary as fast as ƒ/1.5 in the concentric Schmidt, it would be an ƒ/3.5 system, and the h-line blur diameter would be 0.048mm at 200mm aperture diameter. With identical system ƒ/ratio and secondary size, comparable Baker-Schmidt would require an ƒ/2.1 primary and, mainly due to lower secondary magnification (m=1.67), would sport 0.016mm h-line blur diameter, nearly 2.5 times smaller than the concentric Schmidt.

The lower-order aspheric parameter for the concentric Schmidt is A1=b/8(n'-n), with needed corrector radius Rc=-1/2ΛA1d2. and the higher-order parameter A2=b'/16(n'-n), with b'~2/R15.

The flat-field aplanatic Schmidt-Cassegrain, an arrangement first designed by Slevogt, is corrected for coma, with slight residual astigmatism which combines with the mirrors' Petzval curvature to flatten image field. Both mirrors are spherical. Needed corrector-to-primary separation - with the aperture stop at the corrector -  is given by σ1=[2ρ3-k2(2ρ-k)(2ρ+1-k)]/[ρ3-k2(2ρ-k)2]. Choice of ρ determines the level of residual astigmatism, with the aberration coefficient given by a=m[(1/ρ)-1]/4ƒ, ƒ being the system focal length. The transverse aberration, as best focus blur diameter, TA=mα2[(1/ρ)-1]D/4, D being the aperture diameter, and the P-V wavefront error WA=aα2d2, d being the aperture radius. For given minimum secondary k (in units of the aperture), the choice of ρ also determines secondary magnification as m=ρ/(ρ-k). 

Since the error of astigmatism, according to the aberration coefficient, is nearly in proportion to (1/ρ)-1, keeping astigmatism low requires ρ close to 1. The long secondary radius of curvature also keeps secondary magnification relatively low, setting the limit to a minimum secondary size needed to produce accessible image. In general, ρ should be greater than 0.9, but it is not a strict limit. With ρ=0.9 and secondary magnification m=1.8 at k=0.4, the P-V wavefront error of astigmatism at 1° off-axis in 200mm ƒ/3.6 camera is 0.00021mm, or 0.38 waves of 550nm wavelength (0.078 wave RMS, just below 0.80 Strehl). The corresponding blur diameter is still as small as 3 microns (0.003mm).

Choice of ρ and k doesn't influence significantly field flatness of the system. With the best (median) image curvature given by

the field remains flat for any practical ρ and k values.

Needed lower-order aspheric parameter of the corrector is A1=b/8(n'-n), with b=2[1-k2(2ρ-k)2/ρ3]/R13. It makes the corrector stronger than in a comparable Schmidt camera by a factor of [1-k2(2ρ-k)2/ρ3]m3. For average value of k=0.4, with ρ=0.95, and m=1.73, that gives spherochromatism greater than in a comparable Schmidt camera by a factor of 3.

Needed corrector radius of curvature is Rc=-1/2ΛA1d2, and higher-order aspheric parameter for the corrector is A2=b'/16(n'-n), with the aspheric coefficient b' approximated by b'~2σ1/R15.

Compact Baker-Schmidt

Finally, the compact Baker-Schmidt design illustrates requirements and performance of an all-corrected Schmidt-Cassegrain system with the stop (i.e. corrector) moved significantly closer to the primary - in this case, at the distance of primary's focal length (thus σ1=0.5). As a result, off-axis aberrations of the fast primary mirror are very significant, particularly coma. Cancelling these aberrations requires both mirror aspherized. With given general requirements for the flat-field anastigmatic aplanat - ρ=1, m=1/(1-k) - needed conic constants for the primary and secondary are K1=1+2k, and K2=-1+2(1+k)/k2.

It is immediately apparent that both mirrors need to be strongly aspherized for practically any value of k. The fact that needed conic is for both an oblate ellipsoid doesn't make fabrication any easier. At near-minimum secondary size for formally accessible image, with k=0.4, required mirror conics are K1=1.8, and K2=16.5. Strongly aspherized primary nearly triples the amount of under-correction of corresponding sphere with the secondary offsetting only smaller portion of it. Compensating that much of spherical aberration requires significantly stronger corrector, as reflected in the corrector's required lower-order aspheric coefficient b=2[1+K1-k2(2-k)2-k4K2]/R13, and corresponding aspheric parameter A1=b/8(n'-n).

Strongly aspherized primary also generates enormous amount of higher-order spherical aberration, again, only in small part offset by the opposite aberration on the secondary. As a result, needed higher-order aspheric parameter A2=b'/16(n'-n) is also significantly higher than previous systems, with the required aspheric coefficient approximated by b'~20/R15.

Compared to a Schmidt camera of identical focal ratio, compact Baker-Schmidt has spherochromatism higher by a factor of [1+K1-k2(2-k)2-k4K2]m3. For the above arrangement with k=0.4 and m=1.67, that factor is 9.2, significantly higher than in the Schmidt-Cassegrain cameras above. Combined with its extraordinary fabrication difficulty, it makes the compact Baker-Schmidt camera less favorable option performance-wise, despite its size advantage. If system compactness is desired, Houghton cameras, either as a single- or two mirror flat-field system, are better overall choice.

Following illustrates physical outline and performance of the above systems.

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