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10.2.4.5. Secondary spectrum reduction      10.2.4.7. Houghton-Cassegrain: designing
 

10.2.4.6. Two-mirror Houghton telescope with plano-symmetrical corrector

As mentioned, a single-glass plano-symmetrical Houghton corrector type, using a pair of plano-convex and plano-concave lenses of equal surface radii, can cancel mirror spherical aberration inducing chromatism considerably lower than comparable symmetrical aplanatic version (FIG. 204). The price to pay in a two-mirror system, however, is introduction of coma somewhat greater than that in a comparable SCT with spherical mirrors (FIG. 204a), only of the opposite sign (which indicates that coma contribution of the corrector and secondary is stronger than that of the primary). Consequently, the coma can be reduced by moving corrector closer to the the primary, rather than farther away, as it is the case with SCT systems.


FIGURE 204:
Chromatism in a Houghton-Cassegrain using single-glass plano-lenses is significantly reduced compared to a single-glass aplanatic corrector (FIG. 201a), but the field coma is is uncorrected and approximately by 1/3 stronger than in a comparable SCT (a). It is possible, however, to reduce and even cancel the coma by moving the corrector closer to the primary, increasing the coma of the primary until it balances out with corrector's coma. This, of course, requires larger minimum secondary mirror. At the minimum secondary size of 0.5D[1] - thus with the corrector at about half the focal length from the primary - and magnification m=2, the system becomes /5 (b), all four curved surfaces (two at the corrector and the two mirrors) have identical radius of curvature and the coma is effectively cancelled (the system a has the minimum secondary size of 0.3D). This works due to the coma of the primary mirror increasing as the stop (corrector) separation decreases, thus cancelling more of the corrector's coma. Obviously, this wouldn't work with the SCT, since the Schmidt corrector has practically zero coma. Therefore, reduction of the coma here requires increase in the stop separation. The coma level in the two systems nearly equalizes when the back focal distance is reduced from 9" at /10 to 5" (for an /8 system); the linear aberration is reduced in the HCT, while nearly unchanged in the SCT (c). However, HCT chromatism in the violet is about four times larger (for 0.707 SCT neutral zone placement), which puts it at the level of a 4" ~/200 achromat. Since the chromatism of a system changes with the fourth power of the primary's relative aperture, a single glass corrector HCT with an /3 primary has the chromatism level of an SCT with an /2 primary. As Eq. 148 implies, the chromatism also changes in proportion to the aperture.            SPEC'S

[1]For the BK7 glass index. Higher refractive index lowers coma of the corrector, thus allowing for greater corrector-to-primary separation and, for n~1.65-1.7, significantly smaller secondary. Higher refractive index also results in increased chromatism, particularly in the violet, making a two-glass corrector desirable option.


FIGURE 205:
Ray spot plot for the chromatism and field coma in the comparable /10 SCT and MCT. The blur size can be poor indicator of the level of chromatism, as it is the case for aberrations in general. While the geometric blur size indicates similar levels of chromatism in both, the /2.5/10 plano-Houghton and /2.5/10 SCT, the actual chromatism measured by the wavefront error* in the violet h-line (405nm) is some four times greater  in the HCT (0.24 wave RMS of the combined secondary spectrum and spherochromatism) than in the SCT (0.06 wave RMS of spherochromatism). The Houghton chromatism could be reduced by the use of a slightly different glass type for the second lens element, as described in the previous section (for instance, by replacing BK7 glass in the rear lens by BK8 and the increase in lens spacing to 15mm, the h-line error reduces to 0.16 wave RMS, and the r-line from 0.07 to below 0.05). The gain is, in general, smaller, since most of the chromatic error here comes from the spherochromatism, not secondary spectrum. In the Maksutov-Cassegrain, chromatism is even lower than in the SCT, but the image quality suffers from higher order spherical aberration, not correctable without adding higher-order surface term. The system shown has little better than 1/4 P-V wavefront error of mostly higher order spherical, resulting in slightly over 1/20 wave RMS error (comparable to 1/6 wave PV of lower-order spherical) in the optimized e-line. It probably can be reduced somewhat by further optimizing, but not significantly. This underlying correction error increases toward non-optimized wavelengths, also combining with the very small defocus (secondary spectrum) element. As a result, the h-line chromatism for the MCT falls between the other two at ~0.1 wave RMS. The combined h-line/r-line RMS error at ~0.15 wave RMS is still somewhat smaller than in the Houghton (~0.18 wave RMS).                    SPEC'S:  SCT MCT

*It should be noted that the energy lost to the Airy disc grows exponentially with the RMS wavefront error.

While the coma in the plano-lens HCT is not as severe as to make it unusable, it does compromise its field quality to a significant degree. In the SCT, it can be corrected either by aspherizing the secondary, or by moving the corrector significantly farther from the primary, as mentioned. The latter is not an option for the HCT, due to its coma originating from the corrector and secondary, thus requiring more (opposite) coma from the primary; in other words, it requires corrector moved significantly closer to the primary. Secondary conic for a coma-corrected HCT needs to be oblate spheroid - not an easy figure to make, especially with smaller convex mirrors. Fortunately, there is another, easier way of correcting the HCT coma, and that is by employing a sub-aperture corrector placed at the front opening of the baffle tube (FIG. 206). This simple doublet fully corrects


FIGURE 206: 200mm diameter /9.75 aplanatic all-spherical Houghton-Cassegrain with a plano-lens full-aperture corrector and integrated sub-aperture corrector, also consisting of a pair of plano-lenses. The sub-aperture corrector is very easy to fabricate - it can be cut out of the full-aperture corrector - and at a favorable location, the least sensitive to misalignment and not interfering with the accessory end of the telescope. It only requires minor adjustment (weakening) in the power of the full-aperture corrector. Ray spot plots show the only remaining system point-source aberration, low astigmatism. Field curvature is weaker than in the single corrector system. Chromatism is significantly reduced in comparison to the single (full-aperture) corrector version (compare FIG. 113a): the h-line (405nm) RMS wavefront error is reduced to 0.08, or 1/3 the error of the single-corrector system, with the r-line (707nm) remaining nearly unchanged  at and practically negligible. All four lens elements are of a single common glass; achromatizing one or both correctors would likely result in a further significant reduction of chromatism which, as the ray spot diagram to the left shows - already surpasses required chromatism level of a true apo refractor (ray spot diagram generated by OpTaliX-LT raytracing software).                                         SPEC'S

the system coma, at the price of somewhat stronger, but still low astigmatism. Stronger astigmatism has a positive effect of lessening the field curvature. Chromatism is also reduced, and the overall performance level is high, even without detailed system optimization (FIG. 207). As with the camera, aberration coefficients of the sub-aperture corrector are different than those given for the full-aperture corrector, due to it being placed in a converging cone of light. Consequently, the position factor p for both lens elements of the sub-corrector has its value changed according to the relation given for Eq. 97. In a two-mirror system, however, it is the image formed by the secondary (without the sub-corrector in place) that is the object for the first lens element of the sub-corrector.


10.2.4.5. Secondary spectrum reduction      10.2.4.7. Houghton-Cassegrain: designing

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