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8.3.1. Paul-Baker, flat-field anastigmatic aplanats   ▐    8.4.2. Two-mirror TCT
 

8.4. Off-axis and tilted component telescopes

Axially 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 optics and
off-axis section systems are limited to relatively small, long-focus systems.

8.4.1. 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. 132), but a very long-focus mirror is still required in order to keep aberrations at an acceptable level.

FIGURE 132: Modern version of the Herschelian reflector, with the primary tilted by an angle τ, and a small reflecting flat placed out of the path of incoming light to direct converging cone to the side, for more convenient eyepiece position. In order to keep tilt-induced aberrations low, the primary is of a very small relative aperture.

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=τD/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. 85), 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. 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 , D and F being the mirror focal length, diameter (in inches) and focal ratio, respectively. For mm, ζ=(f-D-100)/(D+50)F. Between 100mm /20 and 150mm /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, needed mirror focal ratio F for any given RMS wavefront error ωa of astigmatism introduced to the field center is given by
F=(D/128
ωaζ224)1/3.

For D=100mm aperture diameter, ζ=0.6 and ωa=λ/14 (λ=0.00055mm), needed focal ratio 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 below 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. 133). 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.

FIGURE 133: Ray spot diagram for a 100mm /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.

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 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).

Herschelian-type telescope can gain significantly in correction level, compactness, aperture size and/or relative aperture if some type of lens corrector is used to minimize tilt-induced aberrations. The usual choice is wedged lens, or wedged meniscus, inserted at an angle to the axis in the optical path in the converging light cone. A single BK7 meniscus with unequal radii (R1=80mm, R2=88.4mm, concave toward mirror, 9.8mm center thickness, front surface at 1450mm from mirror vertex, 3.8mm positive vertical axial decenter), reduces center-field aberration of a 2.3 tilted 200mm diameter /8 sphere to 0.042 wave RMS. Despite some lateral color, it makes for quite useable unobstructed aperture, with better part of its strong field astigmatism offset by astigmatism of the eyepiece. Better results are possible with more sophisticated - and complex - corrective lenses.

The question is, is there a practical benefit from it? What is common to all these compensating lens correctors is that they have very tight tolerances for spacing, tilt angle, surface curvature and center thickness, due to an enormous load of compensatory aberrations (mainly astigmatism). It requires very accurate mounting and adjustment mechanism, and even then it is questionable how close to its optimum such a system can be objectively maintained.

On the other hand, a 200mm /8 mirror in a regular Newtonian arrangement can work with central obstruction in the 0.15 to 0.25 of the aperture diameter range, satisfying requirements from high-contrast planetary to a wider illuminated field deep-sky instrument. Its particular advantage is comparatively low sensitivity to misalignment, thus likely better field performance than what optical data per se indicates. Considering this, a tilted-mirror telescope is an alternative mainly for those who find in it values other than its optical quality and practicality of use.

Catadioptric Herschelian with full aperture Houghton corrector

  A catadioptric variant of Herschelian that offers good overall correction, while making possible to use significantly faster mirror, uses the simplest form of a zero-power Houghton corrector: two full aperture plano (PCX and PCV) lenses of equal radii placed at nearly the focal length separation in front of the mirror. This is a slightly modified design originally proposed by D. Shafer (Telescope Making 41, as presented in "Reflective and catadioptric objectives" chapter by Lloyd Jones, University of Arizona). It is an all-spherical 150mm /8 system.

The two tilted elements are the rear lens and the mirror. Correction of aberrations is very good over flat field, with the diffraction field diameter exceeding 1 degree. Axial chromatism easily passes the "true apo" requirement; Lateral color is well controlled, with all the wavelengths remaining within the Airy disc. The above system is not fully optimized, so nearly all aberrations can be a bit lower, but it wouldn't bring tangible practical gains.

For the system to approach its design correction level, the two tilted elements have to meet very tight tolerances, with respect to the tilt angle. Also all three elements (the diagonal flat is optional, and optically passive) have to be very well centered around optical axis. In other words, this system requires a high quality mechanical assembly - optical tube in general, and lens and mirror cells in particular.

    
8.3.1. Paul-Baker, flat-field anastigmatic aplanats   ▐    8.4.2. Two-mirror TCT

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