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10.2.2.5. Schmidt-Cassegrain cameras

As mentioned before, Schmidt-Cassegrain optical arrangement, similarly to Schmidt camera, allows for the 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/f1, s being the mirror separation, and f1 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:

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

- 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 camera types:

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), needed corrector power is somewhat lower than in a commercial Schmidt-Cassegrain telescope with identical primary mirror f-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), which minimizes the geometric blur, 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 vs. 0.866 neutral zone, 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 ~f/3, the larger aperture, the more so.

Neither the 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, and residual coma cancelled by moving corrector slightly farther away, according to σ1=[2-k2(2-k)(3-k)]/2[K1+1-k2(2-k)2]. The only consequence is introduction of a 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, in a 300mm f/3.6 system, is adding about 0.1 wave RMS wavefront error at 1° off-axis, nearly 0.06 wave due to the 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 much smaller wavefront error and a 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 size (i.e. 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 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 k=0.4, the corresponding values are ρ=0.7 and m=2.33, with its chromatism higher by a factor of 6.5 than in a comparable Schmidt camera. With the primary as fast as f/1.5 in the concentric Schmidt, it would be an f/3.5 system, and the h-line blur diameter would be 0.048mm (48 microns) at 200mm aperture diameter.

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 simplest way to flatten the curved image field of the concentric SC camera is by placing a thin, strongly curved meniscus 40-60mm (approximately, varies with the system speed) in front of final focus, facing the secondary with its convex side. The meniscus is weakly negative, and at certain radii strength its Petzval offsets the camera Petzval, reducing it to zero, while the system astigmatism reduces to zero at one specific meniscus-to-focus separation. The only induced aberration that may become significant is lateral color, which can be eliminated either by allowing for somewhat higher crossing (i.e. above 0.707 zone), at a prica of somewhat higher spherochromatism, or by adding to the meniscus a weak PCX lens.

The flat-field aplanatic Schmidt-Cassegrain, an arrangement first designed by Slevogt, is corrected for coma, with the slight residual astigmatism which combines with the mirrors' Petzval curvature to flatten best image field. Both mirrors are spherical. Needed corrector-to-primary separation - with the aperture stop at the corrector - for corrected coma is given by σ1=[2ρ3-(2ρ-k)(2ρ+1-k)k2]/2[ρ3-(2ρ-k)2k2] for fixed ρ value of ~0.96 i.e.

σ1=[1.77-(1.92-k)(2.92-k)k2]/[1.77-2(1.92-k)2k2].

Since the median (best) astigmatic surface for any astigmatic field is flat when the 4th order astigmatism aberration coefficient a is 1/4 of the inverse value of Petzval curvature - and of opposite sign - given for two mirror systems with 1/RP=2(1-ρ)/ρR1, RP being the Petzval curvature and R1, as before, the primary's r.o.c., the astigmatism coefficient of the Slevogt is a=1/4RP=(1-ρ)\2ρR1. However, since astigmatism and the Petzval have different rates of change with the system parameters, not every ρ value will allow for the flat astigmatic field. In fact, it is possible only for ρ~0.96, i.e. for the secondary radius of about 96% of the primary's. There is no simple expression for the needed ρ value in terms of main system parameters, since it involves Petzval curvature in addition to the aberrations, but the context can be graphically illustarated.

Going through the range of secondary magnification, from m=1.5 to m=2.5, for what is about the minimum usable backfocus (BFL) of 0.2 primary's focal lengths - thus the minimum relative (in units of aperture diameter) secondary size is k=(1+BFL)/(m+1)=(1+0.2)/(m+1), and the secondary-to-primary radius of curvature ratio is  ρ=mk/(m-1)=1.2m/(m2-1) - shows opposite in sign 4X astigmatism significantly smaller than the Petzval for the location of corrector canceling coma (σ little over 1), except for the very narrow range of secondary magnification around m~1.8. For the lower magnifications, Petzval and astigmatism are of the same sign, hence astigmatism only worsens the curvature given by the Petzval. For higher magnifications, the Petzval quickly races ahead of the (opposite) astigmatism for the locations of aperture stop canceling coma for those magnifications (σ falling toward 1 and somewhat lower).

At m=1.8, however, the Petzval is 0.89, with the 4X astigmatism for that magnification reaching -0.89 for σ=1.09. The zero-coma stop location for that magnification is σ=1.14, but shifting to that σ value will change the astigmatism - which is bottoming there - only slightly. Thus the system is a flat field aplanat, and the corresponding secondary-to-primary r.o.c. ratio is ρ=1.2m/(m2-1)=0.964.

Minimizing backfocus also minimizes the secondary size, but 0.2f₁1 BFL may be to short for smaller cameras. Increasing it to 0.3f₁1 slightly changes all three, the Petzval, astigmatism and zero-coma stop plots (dashed). Looking for a match between the Petzval and astigmatism leads to m=1.88 for which the Petzval is about unchanged, 0.89, and the opposite astigmatism around the stop location correcting for coma (σ=1.16) is -0.9 (inset at bottom left). The -0.01 differential means that the best field will have -0.02/R1 inverse of curvature (1/RC) which is, considering that R1 is numerically negative, of positive sign, i.e. convex toward secondary. For R1=-1120 (280mm f/2 primary for the systems with which all calculations were double checked) this comes to RC=56,000mm. The effect of both, slightly more astigmatism and curvature is negligible; for D=280mm systems with f/2 primary the RMS error at 1° off axis was 0.048λ with BFL=0.2f₁1 and 0.046λ with BFL=0.3f₁1 (for λ=546nm).

Again, ρ=1.3m/(m2-1)=0.964. Further extending BFL adds more of both, astigmatism and field curvature, but at a near-negligible rate. Going for as much as 0.44f₁1 BFL results in 0.057λ wave flat-field RMS wavefront error at 1° off axis (0.0523λ on the best RC~30,000mm image surface). Secondary magnification m=2, and the minimum secondary size k=0.48, only moderately larger than for the minimum BFL~0.2f₁, which is k=0.43. Corrector separation is slightly larger, at σ=1.18. With BFL=0.44 and m=1.88, we still have ρ=1.44m/(m2-1)=0.96.

Extending BFL by strengthening the secondary r.o.c., i.e. by reducing ρ value below 0.96, would quickly generate much larger error. For instance, changing ρ from 0.964 to 0.9 in the first arrangement, which would extend BFL only to 0.24f₁1, would result in 0.107λ RMS WFE at 1° off axis flat field, and 0.034λ on the best curved surface (RC=-7300mm). While the astigmatism is somewhat reduced, its combination with the (stronger) Petzval makes the system no longer flat-field. Small deviations from ρ=0.96, however, won't appreciably affect image quality. With ρ=0.94, for example, the Petzval is RP=ρR1/2(r-1)=-8770mm, with its inverse value multiplied by R1 equaling 0.128, and the fourfold astigmatism coefficient, also multiplied by R1, is probably around -0.085, giving the median curvature (0.128-0.085)/R1=-26,000mm. While not strictly flat, the defocus error on the best image surface is still only 1/10 wave P-V at 1° off-axis.

With ρ=0.9, however, the Petzval value is already 0.22 (about -5,100 curvature radius), and all that -0.066 fourfold astigmatism can do is to soften the best image curvature to -7,300mm.

As noted, these are all systems using f/2 primary. With slower primary and ρ, m and k unchanged (that increases nominal BFL in proportion to the F-number), angular astigmatism changes inversely to F1 (primary's F-number), which means that on the linear field it changes inversely to the square of it.

With the astigmatism aberration coefficient for flat astigmatic field a=1/4RP=(1-ρ)\2ρR1=m[(1/ρ)-1]/4f, f being the system focal length, the transverse aberration, as the 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. With ρ~0.96, a~m/100f, and TA~Dmα2/100. These are, however, only close approximations, since we've seen that the equality between the Petzval and fourfold astigmatism coefficient, 1/RP=-4a, is strictly valid only for the arrangement with m~1.8. 

With ρ=0.964 and secondary magnification m=1.8, it gives the P-V wavefront error of primary astigmatism 1° off-axis in 280mm f/3.6 camera as 0.000107mm, or 0.194 waves P-V with 550nm wavelength (0.04 wave RMS). The corresponding blur diameter is still as small as 1.5 microns (0.0015mm). It is about 15% less than the raytrace output for the actual system, the difference of 1/167 wave RMS probably due to the presence of other non-zero lower and higher order aberrations.

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 ρ=0.96 and the average values of k=0.45 and m=1.9, that gives spherochromatism greater than in a comparable Schmidt camera by a factor of 3.5. The reason is obvious: the corrector needs to correct spherical aberration of the primary that is nearly twice faster than the system, hence with nearly eight times more spherical aberration, with about half of it offset by the secondary's.

Needed corrector radius of curvature is Rc=-1/2ΛA1d2, and the higher-order aspheric parameter for the corrector is A2=b'/16(n'-n), with the aspheric coefficient b' approximated by b'~2σ1/R15. Alternatively, since the value of k is nearly constant, with its 6th order spherical, of opposite sign to that of the primary, well over half that of the primary, 6th order coefficient also can be, from Eq. 104.1, approximated as 1/3 of that for the primary alone, or A₂=1/8(n'-n)R₁⁵.

THE 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 of ρ=1 and m=1/(1-k), the 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 any practical value of k. The fact that the needed conic is for both an oblate ellipsoid doesn't make fabrication any easier. At near-minimum secondary size for an 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).

The 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, the needed higher-order aspheric parameter A2=b'/16(n'-n) is also significantly higher than in previous systems, with the required aspheric coefficient approximated by b'~20/R15.

Compared to the 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 arrangement with k=0.42 and m=1.72, that factor is 8.6, 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 length advantage. If system compactness is desired, Busack astrograph, or even Houghton cameras, either as a single- or two mirror flat-field system, are better overall choice.

Following graphs give physical outline and performance of the above systems.

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