◄ 11.4. Schmidt-Newton telescope   ▐    11.5.1. SCT close focusing ►
 

11.5. Schmidt-Cassegrain telescope (SCT)

Among the most popular commercial designs, the Schmidt-Cassegrain telescope (or SCT, FIG. 88) owes its success mostly to the possibility of relatively inexpensive commercial production of a well-performing Schmidt corrector, particularly in combination with two spherical mirrors. The road for the SCT commercial production was paved by Celestron's Tom Johnson who, back in the '60-ties, pioneered the method for corrector's mass-production.

Aberration-wise, there are two significant differences between the SCT and all-reflecting Cassegrain varieties. One is that the SCT can be made free from both, coma and astigmatism, while an all-reflecting arrangement can only correct for one. On the other hand, the Schmidt corrector induces some sphero-chromatism.

The only significant monochromatic aberration introduced by the Schmidt corrector in collimated light is spherical. Off-axis errors are practically induced by the two mirrors alone. The P-V wavefront errors for spherical aberration, coma and astigmatism at best focus for a general SCT system are as follows:

- lower-order spherical aberration:

                                                              Ws=(scr+sM)d4/4                                           (113)

with scr being the corrector spherical aberration coefficient, d the pupil (aperture) radius and the mirror aberration coefficient sM being the sum of the individual mirror spherical aberration coefficients, sM=s1+s2. The individual mirror coefficients for object at infinity are those for a two-mirror system alone, given by:

 s1=(K1+1)/4R13    for the primary, and     s2=-k[K2+(m+1)2/(m-1)2](1-1/m)3/4R13       (113.1)

                                                             =-[K2+(1-2r/k)2]k4/4R23

for the secondary, with K1 being the primary conic, R1 the primary radius of curvature, k the height of marginal ray at the secondary in units of the aperture radius, m the secondary magnification and ρ=R2/R1, the secondary radius of curvature in units of R1. Thus, the system P-V wavefront error at best focus can be written as:

                                Ws= {P-(K1+1)+k[K2+(m+1)2/(m-1)2](1-1/m)3}D/2048F13                       (114)

with P being the needed corrector power to cancel system spherical aberration, and ss={}/4R13 being the system aberration coefficient.

Of course, for zero system spherical aberration, the aberration coefficient for the corrector scr, which is related to the corrector power P as scr=-P/4R13, needs to be nominally equal to the sum of mirror aberration coefficients sM, and of the opposite sign. If the primary is spherical, to cancel its aberration alone, the corrector aberration coefficient needs to be scr=-b/8 (with the aspheric term b=2/R13, it comes to scr=-1/4R13, same as the spherical primary, but of the opposite sign). The corrector with this value of the aspheric term is said to have a unit power P. In such arrangement, for cancelled spherical aberration of the system, secondary conic needs to be K2=-[(m+1)/(m-1)]2, same as in the classical Cassegrain. With both, primary and secondary spherical (K1=K2=0), needed power P of the corrector for zero system spherical aberration is (from Eq. 113.1) P=1-k(m-1)(m+1)2/m3.

For closer objects, spherical aberration coefficients for all three, corrector, primary and secondary change (negligible for the corrector), disturbing the zero-balance for distant objects, and resulting in spherical aberration. The chage of aberration contribution on the two SCT mirrors is similar to that in all reflecting two-mirror system (Eq. 92). Main difference is with SCT systems that focus by moving the primary. Here, the error induced by a relatively small object distance is in part offset by under-correction induced by refocusing, which requires an increase in mirror separation. More specifically, reduced object distance lowers under-correction of the primary, and increase over-correction of the secondary. That makes the system over-corrected; the increased mirror separation needed to bring the focus point to its fixed location diminishes the effective diameter of the secondary, reducing over-correction induced by it, and by that the overall system over-correction as well. This makes a typical SCT better suited for terrestrial observations than a similar system with fixed-mirror focusing. More details on this subject are given on the next page.

- lower-order coma P-V wavefront error for object at infinity is given by:

                                                                          Wc=2csαd3/3                                              (115)

with the system coma aberration coefficient cs being the sum of the individual aberration coefficients for the two mirrors, cs=c1+c2, and α the field angle. The individual coma coefficients for the primary and secondary, respectively, are given by:

  c1=[1-s(K1+1)]/R12     and    c2=k(2s-1)[uK2+(m+1)(1-u)/(m-1)](m-1)2/m2R12       (116)

with s being the corrector-to-primary separation in units of the primary radius of curvature (σ=-s/R1, s being the corrector-to-primary separation, positive in sign), and the extracted common factor

 
                                     u=(m-1)[1+k(2s-1)]/2mk(2s-1) = (m-1)[1+1/k(2s-1)]/2m            (116.1)

which can also be written as u= φ/ρ, φ being the exit pupil separation for the secondary in units of the primary radius of curvature. This gives the needed secondary conic for zero coma as:

                        K2=-{[1-s(K1+1)]m2/ku(2s-1)(m-1)2}-(1-u)(m+1)/(m-1)u             (116.2)

This is so called aplanatic SCT, that recently became commercially available as the "improved Ritchey-Chrétien". While admittedly with less astigmatism than comparable RC, it is - needless to say - still "only" an aplanatic SCT.

Another option for correcting the coma is to keep the mirrors spherical, but move the corrector farther away from the primary. Taking a common ~ƒ/2/10 system with k~0.25 and m~5 gives u=0.4+1.6/(2σ-1), and after substitution c2=0.288σ-0.528 which, after setting c2=-c1 thus 0.288σ-0.528=-(1-σ) (omitting the common denominator R1), gives needed corrector separation as σ=0.663, or about 2/3 of the mirror radius of curvature.

- lower-order astigmatism P-V wavefront error for object at infinity is:

                                                               Wa=asα2d2                                                (117)

with the system astigmatism aberration coefficient as being the sum of the two individual mirror coefficients for the primary and secondary, as=a1+a2, with:

             a1=[K1s2 + (1-s)2]/R1    and     a2=-k(m-1)(2s-1)2[u2K2+(1-u)2]/mR1           (117.1)

While generally low in the typical compact commercial SCT, astigmatism still requires attention. Relatively small design changes can result in significantly increased astigmatism level. For instance, typical ƒ/2/10 SCT with spherical primary, σ=0.4 and k=0.25 would, in an aplantic arrangement, require secondary conic K2=-0.77 for cancelled coma; its astigmatic P-V error would be W=-0.000155(αd)2. Change of the secondary magnification to m=4, corrector separation σ to 0.375 and relative marginal ray height on the secondary k to 0.3 (for identical back focus distance), would require zero-coma secondary conic K2=-1.176 and would have the P-V wavefront error of astigmatism W=-0.000374(αd)2 - greater by a factor of 2.4 (despite that, best image surface curvature would slightly improve, due to the lower Petzval curvature).

- field curvature; of interest are Petzval field curvature, which is not affected by the corrector and, thus, remains as for any two-mirror system, 1/RP=(2/R2)-(2/R1), and best image surface curvature which is, in the presence of astigmatism, given by 1/Rm=(1/RP)+4as or: 

                 1/Rm=1/RP +4{K1s2 + (1-s)2 - k(m-1)(2s-1)2[u2K2+(1-u)2]}/mR1                (118)

- spherochromatism in the SCT originates at the corrector. Given aperture and F#, it is proportional to the relative power of the Schmidt corrector P. In other words, to the relative value of the aspheric term b needed to cancel spherical aberration of the system. It can be written as:

                                   P= bSCT/bS=1+K1-k[K2+(m+1)2/(m-1)2](m-1)3/m3                             (119)

                                            = 1+K1-[K2+(1-2r/k)2]k4/r3

with bs being the aspheric factor - or the corrector spherical aberration coefficient - needed to correct for spherical aberration of the primary alone. As mentioned, b=2/R13, from the general form of the aspheric coefficient b=2n[(m+1)/(m-1)]2/R13, for the mirror magnification m=0 (object at infinity) and the index of incidence n=1.  Aspheric coefficient cancelling the aberration of a spherical mirror is bs=-b/8=-1/4R13, while  that for an SCT system,  equal to its aberration coefficient scr, is bSCT=-P/4R13, where P is the corrector power, positive in sign.

The P-V wavefront error of spherochromatism for a particular SCT arrangement is obtained by multiplying relative power of its corrector (Eq.  119) with the wavefront error for the unit power corrector (Eq. 106). For the transverse aberration (ray spot diameter), the relative power is to be multiplied with the transverse aberration for the unit power corrector (Eq. 107.1) and the SCT secondary mirror magnification.

In general, the SCT spherochromatism is low. For a typical commercial ~ƒ/2/10 version, with both mirrors spherical, k~0.25, σ~0.4 and m~5, relative corrector power ~0.72, and 0.866 neutral zone placement, the red (C-line) and blue (F-line) geometric blurs are still within the Airy disc. For the 0.707 radius neutral zone placement (FIG. 89), the blurs are doubled, but the wavefront error halves for the lowest chromatism level achievable with the Schmidt corrector.

In regard to misalignment sensitivity, the SCT, expectedly, has elements of both, two-mirror telescope and a Schmidt camera. Mirror misalignment alone results in wavefront errors as given by Eq. 91.1-91.3, while the misaligned corrector will cause the P-V wavefront error of coma given by:

                    Wcd=2pvd3/3R13 = pv/96F13    and    Wct=2pτd3/3R12 = pτD/48F2          (120)

for decenter and tilt, respectively, with v being the decenter, p the corrector relative power, D the aperture diameter and τ the tilt angle in radians. Sensitivity to despace is practically zero. Of course, any combination in misalignment of the two mirrors and corrector is possible.

Since the commercial SCT uses primary mirror focusing, it is interesting to determine how it affects its wavefront quality. For any change ∆ in mirror separation, the final focus shifts axially by -∆m2/[1+∆(m-1)/s], m being the secondary magnification at the optimum separation, and s the secondary mirror to primary focus separation, positive in sign (note that ∆ is negative when reducing the primary-secondary separation). For ∆ up to ~1mm, good approximation of the resulting focus shift is given by -∆m2. Depending on the accessories used, required shift from the optimum focus can be more or less than that. From Eq.82 (after substituting for k' and m', as given for the despace error), for D=200mm, F1=2, k=0.25, m=5, and K1,2=0, every mm of reduction in mirror separation (nearly for every inch of focus extension) induces ~1/23 wave P-V wavefront error of under-correction, and as much of over-correction for widening the separation. Aspherizing the secondary would significantly increase the error.

However, this only applies to the change in mirror separation to accommodate the use of various accessories, while the object of observing remains at a large distance. Focusing on relatively close objects has to deal with optical consequences of the change in shape of the incident wavefront, which is now convex spherical. Follows more detailed consideration of how it affects performance level of a typical commercial SCT.
 

 ◄ 11.4. Schmidt-Newton telescope   ▐    11.5.1. SCT close focusing ►

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