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11.14. Houghton-Cassegrain telescope: designingIn designing a two-mirror Houghton, the most important additional effect is that of the secondary mirror. It induces both spherical aberration and coma of the opposite sign to that of the primary. In effect, it changes the amount of aberration to correct. The new corrector requirement in this respect is obtainable by correcting the system aberration coefficient for spherical aberration and coma for the effect of the secondary. For the lower-order spherical, the secondary to primary mirror aberration contribution is given by s*=s2/s1=-k(m-1)(m+1)2/m3 (153) while for the lower-order coma, the secondary contribution relative to that of the primary mirror is: c*=c2/c1=(m+1)(m-1){2mk(2σ-1)-(m-1)[1+k(2σ-1)]}/2(1-σ)m3 =(m2-1)[(2σ-1)(m+1)k-(m-1)]/2(1-σ)m3 (154) where, as before, k is the relative height of the marginal ray at the secondary in units of the primary semi-diameter, m is the secondary magnification and σ is the stop (corrector) to primary separation in units of the primary radius of curvature (this means that 2σ-1 is numerically negative for σ<0.5, which is usually the case with compact systems). Both, s* and c* are negative due to the two contributions being of the opposite sign. The two ratios give the effective amount of aberrations to be cancelled by the corrector. Thus, for spherical aberration, the aberration coefficient in the form (1+s*)/4R3 replaces 1/4R3 for a single mirror, and for the coma it is now (1+c*)(1-σ)/R2 instead of (1-σ)/R2, R being the primary radius of curvature. Both coefficients are for the spherical surface; for aspheric surfaces, appropriate values for s* and c* can be obtained from Eq. 116. With these substitutions, the zero spherical aberration shape factor is qs=n(n-1)(1+s*)ƒ3/(n+1)R3, and for the coma qc=2n(n-1)(1-σ)(1+c*)ƒ2/(n+1)R2. The lens element focal length (absolute value) for the corrector cancelling spherical aberration and coma is then closely approximated by: ƒ~2(1-σ)(1+c*)R/(1+s*) (155) while the appropriate q factor (when obtained from the q factor for zero spherical aberration) is: q=n(n-1)(1+s*)ƒ3/(n+1)R3 (156) Again, s* and c* are numerically negative. Needed lens radii are then obtained from Eq. 142-143, or from the following summary. A small secondary mirror location adjustment and slight correction of one of the two curves are usually sufficient to bring a system obtained through this procedure to a near-optimum level. Note that this calculation is for lower-order spherical aberration; as the primary relative aperture exceeds ~ƒ/2.5, higher-order spherical becomes significant with the standard, aplanatic Houghton corrector, and will result in a discrepancy between results obtained for the lower-order aberration alone and the actual system error. Two-mirror Houghton system can also be arranged with a pair of plano-convex and plano-concave lenses with identical curvatures (as illustrated on Fig. 108, right). The needed radius (absolute value) is, from Eq. 142, closely approximated by: R1,3~(n-1)ƒ1,3/(1+s*)1/3 =(n-1)[(n+1)/n(n-1)(1+s*)]1/3|R| (157) Since for a corrector consisting of plano-lenses q equals 1 for both elements, and the needed lens element focal length is approximated by ƒ~[(n+1)/n(n-1)(1+s*)]1/3 R, its coma coefficient is, from Eq. 138, given by: ccr~(1+s*)[(n+1)/n(n-1)(1+s*)]1/3/2R2 (158) Evidently, the s* parameter affects both, spherical aberration and coma of the corrector. For accessing its coarse effects, its form given by Eq.153 can be simplified to a single parameter. Assuming back focal distance being nearly equal to the aperture diameter, the secondary magnification is m~(1-k+1/F)/k and, assuming corrector location nearly coinciding with that of the secondary, σ~(1-k)/2. After the substitution, the parameter is approximated by: s*~(a2-2a)[2-(1-k2)a]/(1+k)(a-1)3 (159) with a=(1+1/F)/k, and F being the primary mirror focal ratio number. For given primary focal ratio number F, radius R and the corrector refractive n, there is only a single variable left to access the approximate coma level for various values of k (as before, the height of the marginal ray at the secondary in units of the aperture radius). With the primary mirror coma coefficient given by cp=(1-σ)/R2, now the system coma aberration coefficient can be approximated by cs~ccr+(1+s*)cp, and the resulting P-V wavefront error of coma is: W~csαD3/12 (160) In general, the plano-symmetrical two-mirror Houghton system has coma comparable to that of an all-spherical (mirrors) SCT, which also can be cancelled or reduced by aspherising the secondary (oblate ellipsoid), or by reducing stop separation. Houghton-Cassegrain coma, as already mentioned, can be cancelled for any corrector location with the symmetrical aplanatic corrector type, as well as asymmetrical aplanatic varieties. It is only a matter of generating exactly as much of the coma of opposite sign by the corrector to cancel that of the two mirrors (which is identical to that in a Schmidt-Cassegrain, given with Eq. 116). Same goes for astigmatism and field curvature, with the Eq. 117-118 practically applicable to the Houghton-Cassegrain as well. Misalignment sensitivity of the Houghton-Cassegrain secondary is given by the general two-mirror system relations (Eq. 91.1-91.2). For the corrector itself, raytrace suggests sensitivity to decenter similar to that of the Schmidt corrector, and sensitivity to tilt about tenfold greater (similar to the tilt sensitivity of a full-aperture meniscus corrector). Sensitivity to despace is practically non-existent for all three corrector types. Sensitivity to despace between the two lens elements of the Houghton corrector is very forgiving, and so is the lens thickness tolerance. |