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10.2.4.2. Houghton camera 2   ▐    10.2.4.4. Secondary spectrum reduction
 

10.2.4.3. Houghton-Newtonian

The only difference between Houghton camera and a telescope in Newtonian configuration is in the focal length of the mirror, and position of the corrector. In general, corrector separation in Houghton-Newtonian is somewhat larger, to accommodate for a smaller obstruction by diagonal mirror. The two corrector types most interesting to amateurs are symmetrical aplanatic and plano-symmetrical. The only remaining aberration with the former is mild field curvature (FIG. 198b). The latter retains low residual coma of little or no consequence for general observing, with similar mild field curvature (FIG. 198a). Follows its description in more details.

For n=1.52, the corrector position for zero coma would be at 0.264R (Eq. 145). Moving it to a more practical location in the Newtonian arrangement, say, 0.43R, wouldn't change coma coefficient of the corrector (Eq. 138), but would change that of the mirror. Consequently, sum of the two coefficients is not zero anymore and, for spherical primary, it determines the system coma aberration coefficient as cs=(σ*-σ)/R2, where σ and σ* are the zero-coma and the actual corrector-to-mirror separation, respectively. In this case, cs=0.166/R2, which is 1/6 of the mirror coma aberration coefficient, given by 1/R2. This means that the system will have 1/6 of the mirror coma. For an /4.5 primary, that would result in the coma-free linear field at the level of an /8.2 mirror. It is about half the coma of the comparable Maksutov-Newtonian, and ~1/3 the coma of the Schmidt-Newtonian.

FIGURE 198: Ray spot plots for Houghton-corrector systems in a Newtonian arrangement. (a) Plano-symmetrical corrector type consists of a plano-convex and plano-concave element, thus does not cancel the system coma for every corrector location. However, this has little practical significance. At the location supporting diagonal flat in a regular Newtonian configuration (~0.85f in front of the mirror), some residual coma is visible in the spots, but hardly any in actual observing. (b) Symmetrical aplanatic corrector type, with two radii on four surfaces, also corrects for coma. Coma with plano-symmetrical corrector can be cancelled if the corrector is moved half-way between the mirror and the focal plane (c), which would result in larger central obstruction, still acceptable for photographic purposes. The small circle represents the size of the e-line (446nm) Airy disc.        SPEC'S

Chromatic correction is excellent, as long as the mirror is not significantly faster than ~/4.

10.2.4.4. Two-mirror Houghton-cassegrain telescope - aplanatic

Full-aperture Houghton corrector can also be used as a part of two-mirror catadioptric system (FIG. 199). The only difference vs. single-mirror arrangement on the above is that the aberrations of the secondary are to be added. They are identical to those given for Schmidt-Cassegrain telescope. The Houghton-Cassegrain has the advantage of allowing coma correction in an all-spherical arrangement for any location of the corrector. However, for that it requires putting a curve on at least three out of four surfaces.


FIGURE 199: Symmetrical aplanatic Houghton-Cassegrain (a) has all four corrector surfaces curved, with the two lens elements having identical radii of curvature. It can cancel coma for any corrector location. The plano-symmetrical corrector (b) can only cancel coma for a single corrector location, where the coma of the corrector cancels equal opposite coma of the primary, determined by the stop location. In practice, due to the coma of the plano-symmetrical corrector being typically about 3/4 of that of the primary, only of the opposite sign, and coma of the secondary being typically about 1/2 of the primary's and of the same sign as the corrector's coma, no corrector location can result in cancelled system coma. In general, coma is lower in systems with larger secondary, allowing to place corrector closer to the mirror, but the difference is not significant. Typically, a two mirror HCT with plano-type corrector will have ~1/4 of the primary's coma.

For lower-order spherical aberration of the primary cancelled, the sum of the three aberration coefficients - for the corrector (Eq. 134 in general, Eq. 135 for q1=q2), primary and secondary mirror (Eq. 113) - has to be zero. The coefficient sum sH+s1+s2=sS, respectively, will result in the system P-V wavefront error at the best focus of:

Ws= -ssD4/64            (146)

with D being the aperture diameter. Presence of the glass refractive index n in Eq. 130-131 indicates that spherical aberration can be cancelled only for a single wavelength. The non-optimized wavelengths are affected by spherical aberration, resulting in spherochromatism. In combination with spherical mirror, needed corrector's  aberration coefficient for zero spherical aberration at the optimized wavelength with the refractive index n is s=-n(n-1)(n+1+ι)/4(n+1)(n+ι)(n-1+ι)R3=-1/4R3 for the index differential ι=0 (from Eq.135, with q substituted by Eq. 136). Thus, the aberration coefficient for spherochromatism of the Houghton corrector is given by:

with ι being the index differential ι=n-ni, and R the mirror radius of curvature. Hence, the P-V wavefront error of spherochromatism at the best focus for non-optimized wavelengths (i.e. ι≠0) is:

Ws=s'D4/64           (148)

A good empirical approximation of the spherochromatism coefficient, yielding ~5% greater value, is given by s'~-0.8ι/R3, or s'~ι/103 in terms of the mirror radius of curvature, or focal length, respectively. Substituted in Eq. 148 it gives the P-V wavefront error of spherochromatism at the best focus as:

Ws~ Dι/640F3           (149)

with F being the mirror focal ratio. Negative index differential for shorter wavelengths makes them over-corrected at the best focus, while longer wavelengths are under-corrected. Compared to the P-V wavefront error of spherochromatism at best focus of the Schmidt corrector (Eq. 106), that of the Houghton is greater by a factor of ~1.6 for the Schmidt with the neutral zone at 0.707 the radius, and lower by a factor ~0.8 for the Schmidt with the neutral zone at 0.866 the radius. However, Schmidt corrector with 0.707 radius neutral zone has the advantage of best foci of all the wavelengths coinciding, ensuring virtually zero secondary spectrum. This is not the case with this form of the Houghton corrector (symmetrical aplanat type), which can have significant secondary spectrum. While there is no significant difference in spherochromatism level with other Houghton corrector types, their secondary spectrum is generally significantly smaller.

Also, the comparison is for the lower-order spherochromatism alone; at some point higher-order spherical becomes significant aberration contribution, again, in particular with the symmetrical aplanat type, and needs to be taken into account.

These are comparisons for a single-mirror system, but they remain nearly unchanged for two-mirror systems as well. The only difference is that the power of the corrector - and its chromatism - are somewhat lower, due to the aberration of the primary being partly offset by the secondary.

While spherochromatism does contribute to the level of chromatism of the Houghton corrector, more of a limiting factor can be its secondary spectrum. This is definitely the case with the symmetrical aplanat type, while much less with the plano-symmetrical and asymmetrical type, the later being limited by spherochromatism (FIG. 200). In comparison, Schmidt corrector has secondary spectrum practically cancelled, with the only source of chromatism being spherochromatism, while the Maksutov has nearly cancelled secondary spectrum, but suffers from strong higher order spherical at all wavelengths. 


FIGURE 200:
Longitudinal aberration curves for various Houghton two-mirror systems and a camera. Stronger lens surface curvature needed to correct faster mirrors induce exponentially higher fifth order spherical aberration, increasing defocus for the outer zones, significantly worsening chromatism for the symmetrical aplanatic corrector type (a). It sets the limit of acceptable chromatism for this corrector type at ~/3 mirror focal ratio. Plano-symmetrical Houghton corrector doesn't correct for coma, but suffers significantly less from chromatism, both, in a single- or two-mirror arrangement (b). In comparison, Maksutov corrector produces much tighter longitudinal focusing (secondary spectrum), especially for the outer zones, where the best foci are (e). However, it suffers from unacceptable residual spherical aberration, nearly evenly at all wavelengths. The SCT has by far the best overall correction below /3 (d), approached only by that of optimized (asymmetrical) Houghton corrector (c)SPECS: HCT/SCT/MCT

While not a factor with ordinary Newtonian-style systems, chromatism level of two-mirror Houghton systems sets the limit to how fast the primary can be at ~/3 with the aplanatic single-glass Houghton corrector  (FIG. 201). Reduction in the chromatism can be obtained by allowing for a relatively small amounts of residual coma, allowing for more weakly curved lenses, as illustrated on FIG. 201c.


FIGURE 201
: Aplanatic two-mirror Houghton-Cassegrain with symmetrical corrector type begins to show excessive chromatism as the relative aperture of primary mirror exceeds ~/3. It results from the combined effect of increased secondary spectrum and higher order spherical aberration (FIG. 95). A system with an /2.5 primary (a) has more than doubled chromatic blur diameter compared to a system with /3 primary mirror (c). Reduction in the amount of system chromatism in Houghton-Cassegrain two-mirror systems is possible with some residual coma allowed (d). This level of chromatism is unnoticeable visually, and so is the amount of coma in the system, which is, approximately, at the level of a 200mm /9 paraboloid. However, best performance is achieved with the asymmetrical corrector type, having as little as 0.075 wave RMS error in the violet h-line, which compares  to 1.3 wave RMS with the symmetrical type.                     SPEC'S

Fortunately, Houghton corrector as an optical arrangement offers various possibilities for significant reduction in the level of chromatism. Some that are most appropriate are presented in more details in the following section.
 

10.2.4.2. Houghton camera 2   ▐    10.2.4.4. Secondary spectrum reduction

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