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8.4. Off-axis and tilted elements telescopesAxially symmetric reflective telescopes
have disadvantage of the smaller mirror being in the light path,
causing additional diffraction effect degrading image quality. In order to avoid it, one or more mirrors
either have to be tilted, or made as an off-axis segment of a
larger system. Mirror tilt induces severe coma and astigmatism,
hard to control, except at relatively small apertures. Off-axis systems,
on the other side, have better control of aberrations, but are limited
in size by production difficulties and/or price. Consequently, both, tilted elements and
Herschelian reflector The simplest unobstructed reflecting system is so-called Herschelian reflector, used by the great German/British astronomer of the late 18th and early 19th century, Sir William Herschel. In order to prevent additional light loss on an extra mirror surface of - back then - very low reflectivity, Herschel tilted the primary enough to bring the focus out of incoming light, with the eyepiece mounted on the side of an oversized tube. While it eliminates central obstruction effect and the light loss, mirror tilt results in significant image deterioration. It is possible that Herschel partly corrected for it by tilting the eyepiece, but the design still suffers from aberrations, as well as air turbulence caused by warmth off the observers head, placed next to the path of the incoming air. The more recent variant, with the side flat mirror directing image plane away from the tube eliminates that problem (FIG. 67), but a very long-focus mirror is still required in order to keep aberrations at an acceptable level.
Small long-focus mirrors can be left spherical, since their spherical aberration is negligible. However, the tilt-induced center-field astigmatism and coma still can cause unacceptable image deterioration. The mirror tilt angle τ will result in the P-V wavefront error of astigmatism Wa=τ2D/8F (from Eq.18), and the coma P-V wavefront error of Wc=tD/48F2 (from Eq.12-15.1), both at their respective diffraction foci. If ζ is the relative distance in units of the primary focal length at which the ray reflected from the mirror center breaks out of the path of incoming axial pencil (FIG. 56), then the tilt angle τ=1/4ζF in radians, and the two can be written as: Wa=D/128ζ2F3 and Wc=D/192ζF3. (93) D being the aperture diameter and F the focal ratio number. Setting the minimum flat separation at 1" between the incoming axial pencil and the flat center point (giving ~1" usable field diameter), the relative distance ζ is given by ζ=(f-D-4)/(D+2)F with ƒ, D and F being the mirror focal length, diameter (in inches) and focal ratio number, respectively. For mm, ζ=(f-D-100)/(D+50)F. Between 100mm f/20 and 150mm f/25, ζ varies from 0.6 to 0.7, respectively. Since ζ<1, Eq. 93 indicates that the astigmatism is dominant, with the P-V wavefront error larger by a factor of 1.5/ζ than that for the coma. To make them comparable, the two P-V errors need to be expressed as RMS, which are smaller by a factor of √24 and √32 for astigmatism and coma, respectively. Then, the needed mirror focal ratio number F for any given RMS wavefront error ωa of astigmatism introduced to the field center is given by F=(D/128ωaζ2√24)1/3. For D=100mm aperture diameter, ζ=0.6 and ωa=λ/14 (λ=0.00055mm), needed focal ratio number F=22.3. For these values of D, ζ and F, the coma RMS wavefront error is λ/40. Assuming the two mostly unrelated, the combined RMS wavefront error approximation, from the square root of the sum of errors squared, comes to ~λ/13.2. Still slightly bellow the 0.80 Strehl standard (λ/13.4) in the field center, but it does exceed this level in the best portion of the field (FIG. 68). Since the tilt angle τ=1.2°, at 0.2° off-center in the direction of mirror tilt, the actual incoming pencil angle is 1°, reducing the astigmatism wavefront error by a factor of 0.7 and coma by a factor 0.83, for the combined error of ~λ/18.5 wave RMS, and corresponding 0.89 Strehl.
The wavefront error of a tilted concave mirror can also be expressed in terms of the mirror tilt τ in degrees. For the aperture D in mm, the astigmatism RMS wavefront error in units of the 550nm wavelength is given by wa~Dτ2/71.4F, and that for the coma by wc~Dτ/8.6F2. For D in inches, wa~Dτ2/2.8F, and wc~3Dτ/F2 (expressions are slightly rounded, but accurate to within a couple of percent). Multi-mirror tilted element telescopes By replacing the flat with a toroid, in the same configuration, or with a second curved mirror directing the light back toward primary, needed mirror separation can be reduced roughly two two three times, for a similar level of correction. Field asymmetry is also reduced. Best known systems of this kind are the variants of Yolo and Schiefspiegler. The former originate from S.A. Leonard. It uses two concave mirrors, one of them toroidal (a sphere deformed into a toroid in a specially designed cell). In general, it achieves better - and exceptionally good - performance than the Schief (Schiefspiegler for short), and the systems can be considerably faster. On the other hand, it is also more complex. The Schief is created by Anton Cutter, and originally uses a pair of spherical mirrors, the secondary being convex. The aberration here is present in the field center, and has to be kept acceptably low by limiting the system to quite small relative apertures, ~ƒ/20 and smaller. In the course of years, arrangements with three and four mirrors are added, in various combinations of surfaces. A few of those systems can achieve satisfactory performance at focal ratios closer to ƒ/10, but most of them are in the ƒ/15 to ƒ/25 range. Further reduction of aberrations can be achieved with some sort of the correcting lens element. Reducing the aperture allows for larger relative apertures, but there is not much room in that direction. An interesting compact solution is given by Herrig's two-mirror 4-reflection system. And a design that surpasses all other tilted mirror telescopes in regard to image quality is the Stevick-Paul three-mirror system (FIG. 69). It is the only tilted element system anastigmatic aplant in the ƒ/10-ƒ/12 relative aperture range, with only a mild field curvature remaining.
Despite some of these systems being very well corrected, obstruction-free and relatively insensitive to miscollimation, they never became really popular, even in the small-aperture range. Telescopes of this kind are usually built by amateur enthusiasts. Among the reasons are probably their odd appearance, relatively complicated element positioning and bulkiness of the tube assembly. A system somewhat less affected by these drawbacks is the off-axis Newtonian (other off-axis configurations are possible, but even less price competitive).
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FIGURE
68: Ray spot diagram for
a 100mm ƒ/22.3 Herschelian with the mirror tilt
τ=0.021
radians (1.2°). The circle represents the Airy disc diameter. The
field is aberrated asymmetrically, due to the wavefronts coming at
the mirror from the direction of the tilt finding it inclined at a
smaller angle than wavefronts coming from the radially opposite
direction. The aberration diminishes going from the field center in
the direction of mirror tilt (which is toward the location of the
flat mirror ). The size of
aberration is fairly sensitive to changes in the mirror
F-number. Neglecting the change in ζ
as relatively insignificant compared to change in the ratio number F,
from Eq.93, to a first approximation the wavefront
error for both, coma and astigmatism changes in inverse proportion to the cube of F- number.
Thus,
10% slower mirror
would have the aberrations lower by a factor of ~0.7. On the other hand,
relatively small 10% gain in shortening the focal length would come
at the price of both aberrations increased by about a third. As with
all tilted-mirror systems, the image field is also tilted, although
with the effect being negligible due to usually very low tilt
angles. The field center aberrations are comparable to the effect of
33% central obstruction.