8.4.3. Two-mirror tilted component telescopes 2
In the course of years, arrangements with three and four mirrors are added, in various combinations of surfaces. While adding more mirrors does not necessarily improve performance significantly, it is generally easier to correct aberrations in a system with more surfaces. Few of those systems can achieve satisfactory performance at focal ratios closer to /10, most of them being still in the /15 to /25 range. More elements, however, means more difficult alignment which, in practical terms, can take away most of the performance enhancement.
Further reduction of aberrations, allowing for faster focal ratios and more compact instruments, can be also achieved using a multiple-lens corrector. The best example of this type of tilted-element telescopes is catadioptric schiefspiegler designed by Ed Jones (Jones' Chief). It uses a pair of tilted PCX and PCV lenses with a single paraboloidal mirror to practically correct the astigmatism - on and off axis - down to /7, and even faster. Remaining aberrations are visually near negligible off-axis coma and lateral color. As the system below shows, the lenses are placed close (minimizes lateral color) in an inverse V configuration open toward the incident pencil of light. Combined lens power is weakly positive, resulting in /7.7 system with /7.9 mirror. For given pair of lenses corrected for longitudinal chromatism, axial astigmatism can always be corrected by adjusting lens' tilt. In general, the front lens is tilted counterclockwise (w/lenses above mirror, as shown), and the rear lens clockwise, roughly 50% more than the front lens. Either lens induces both, astigmatism of opposite, and and coma of the same sign (opposite to those of the mirror, as raytrace output at right indicates), but of a different magnitude and in a different proportion. That allows for perfect axial monochromatic correction, as well as minimized (negligible) off-axis astigmatism, with coma remaining the only potentially significant (depending on relative aperture) monochromatic off-axis aberration. Axial lateral color needs to be minimized by slightly decentering the rear lens. However, due to the chief rays passing through lens sections of different power, off-axis lateral color is quite unbalanced, being negligible on one side of the field, but significantly larger on the opposite side (bottom).
For any given lens' location, there is only a single radii pair (within a narrow envelope) that will also correct for axial coma. If stronger, lens will - when optimally aligned to cancel out axial astigmatism - have residual positive (tail up) axial coma, and negative axial coma if weaker. For instance, if both radii in the given system are 10mm stronger, axial coma will vary depending on the particular tilt arrangement of all four elements (that cancels out axial astigmatism) from 0.05 to 0.1 wave RMS. This means that the radii should be within ±3mm (±3%, or so), in order to keep axial coma acceptable or negligible. However, it only applies if the radius deviation is of the same sign for both lens elements. If, for example, the front lens radius is 3mm longer, and rear lens radius 3mm shorter, axial correction can be still very good (0.011 wave RMS), but off axis astigmatism is nearly 30% larger, and lateral color on the positive field radius more than doubled (on the negative side is already large, hence little affected by this magnitude of change; also, longitudinal color is roughly 50% larger, but the error is still quite small, about 0.05 wave RMS in the F-line). Thus, for staying close to the design field performance, lens radii should stay within ±2%, or so, of the design.
Also mirror tilt and radius need to be close to the design values. Increasing mirror tilt 0.1 results in 0.06 wave RMS of axial coma and about 50% greater off-axis astigmatism. Tilt smaller by the same amount results in nearly 0.04 wave RMS of negative axial coma (off axis astigmatism doesn't change appreciably, but it becomes imbalanced, increasing roughly 50% on one half of the field, while nearly reducing to zero on the other). Mirror radius 2% longer would require about 13.5mm increase in the mirror-to-front-lens separation, with the residual axial coma of little over 0.02 wave RMS requiring in addition about 2/3 of a 10th of a degree mirror tilt reduction, and a slight adjustment of the tilt of either front or rear lens (0.08° less and 0.2° more, respectively). With only lens separation adjustment, axial coma is acceptable (Strehl better than 0.95) up to about 3% mirror radius deviation.
Placing lenses closer to the focal plane does not significantly change best possible correction of monochromatic aberrations, on and off axis, nor correction of longitudinal chromatism. Lateral color, however, worsens approximately in (inverse) proportion to the reduction in the corrector-to-focus distance (shown on the bottom for the corrector 100mm farther away from mirror), due to the stronger lenses required for correction of monochromatic aberrations.
Other unusually fast TCT arrangements are possible, but they are generally more complex and/or bulky. In addition, they commonly require more strict tolerances in spacing, centering and tilt angles than a comparable axial Newtonian. This is the practical negative of this type of system, since imperfect collimation error can more easily make the difference between high and mediocre - or worse - optical quality in actual use. On the other hand, both the primary and the lens pair in the Jones catadioptric schiefspiegler, for instance, are significantly less sensitive to miscollimation than secondary mirror in the Ritchey-Chrιtien of comparable aperture and -ratio.
Reducing the aperture size allows for larger TCT relative apertures, but there is not much room in that direction, since they are already relatively small. An interesting compact solution is given by Herrig's two-mirror 4-reflection system. As the system below shows, both mirrors are oversized, but their separation is significantly smaller than in the typical TCT, and smaller by nearly a third from Ed Jones' Chief (keeping in mind the Herrig is slower, although still fast by TCT standards). Both mirrors are spherical, the first mildly convex, the second concave. Each does two reflections, with the first mirror being primary/tertiary, and the second secondary/quaternary. Size and positions of these reflections are shown next to each mirror as partly rotated numbered circles. In general, it is the aberrations of the secondary/quaternary that dominate, due to its more strongly curved surface. Off axis aberrations, however, are very low, since the effective tilt of the secondary vs. (reflected) primary axis is low, and the quaternary is comparatively small in size.
Dominant axial aberration is spherical, mostly coming from the secondary/quaternary (which is somewhat offset by that from the primary/secondary). Secondary to it is trefoil, creating triangular pattern within the ray spot. Since spherical aberration changes with the 3rd power of focal ratio, relatively small changes in the power of two mirrors can have appreciable effect on its axial correction level.
Farther off axis, it is astigmatism and coma that dominate (noting that the change in astigmatism, being a sum of the two mirrors' contributions opposite in sign and roughly comparable in magnitude is, for practical field sizes, closer to linear than quadratic vs. field angle). The magnitude of off-axis aberration is also very sensitive to the system focal ratio (i.e. to mirror radii). For instance, making the top system /16 cuts off axis aberrations nearly in half, and to less than half on axis. Similar change, only in the direction of error increase, would result from making the system faster.
The second system (middle) illustrates the mechanics of Herrig's arrangement. Decrease in tilt of the first mirror requires compensatory decrease in the second mirror relative to the first, in order to keep their combined off axis aberrations at a minimum. As a result, tertiary reflection on the first mirror is drawn closer to the first reflection, thus smaller mirror diameter is needed. However, lower part of the first mirror now throws 4th reflection onto the second mirror higher, separating 4th and 2nd reflection wider, thus requiring larger second mirror. The imaging cone is noticeably shorter, but the system focal ratio is only slightly faster, and the level of correction is nearly identical. The mirror radii here are somewhat different, because they were used to correct for the residual axial astigmatism and coma, resulting from the change in tilt of the first mirror. Changing the radius of either mirror alone can correct for the astigmatism, but correcting the coma too requires change in both radii, in a different proportion. Mirror separation is also a bit larger, which helped reduce the positive (tail up) axial coma residual (it also reduces off axis coma to a similar degree).
For a given set of mirrors, however, switching from - in this example - 9° to 8° first mirror tilt requires an increase in mirror separation in order to correct for the residual axial coma, and adjustment of the second mirror tilt to correct for axial astigmatism (bottom; this system is given in a 3-D projection, also showing the individual reflections). The resulting system is only slightly slower, but the imaging cone is somewhat shorter. First mirror is slightly smaller, and the second slightly larger than with 9° first mirror tilt. The second mirror is closer to the incoming beam of light, but there is still room left for another half a degree, or so, of the first mirror tilt reduction. That, however, would not produce appreciable change in the correction level, nor a gain in system compactness.
Since each surface tilt is relative to the previous surface, the effective image tilt vs. optical axis is a sum of all tilt values. It is 5.97°, 5.34° and 5° from top to bottom, respectively. It should be noted that since OSLO (at least in the free edition) does not allow return to a previous surface, this system has to be constructed by bringing the two reflections on each mirror - each shown as a separate surface on the system drawing - to overlap in their cross sections as closely as it can be visually perceived. Since they are spheres of the same radius, when their central cross sections coincide, they are a part of the same, larger sphere of the same radius. While they are not perfectly in it, the error is generally negligible. Related to this, miscollimation sensitivity for tilt error is about three times larger with the second than first mirror; 0.1° tilt error with the former induces about 0.9 wave P-V (0.2 wave RMS) of mostly astigmatism, nearly evenly across the field (slight residual coma shifts into field center, with the coma-free spot moves off center approximately by the tilt error doubled). As RMS wavefront error, it is slightly more than sensitivity to misalignment of a 200 /5 paraboloid. Decenter of the second mirror results in about 0.025 wave RMS of mostly astigmatism per 1mm, also nearly evenly across the field.
At the end, two more complex TCT systems that, while not two-mirror TCTs, deserve mention for different reasons: one for its unique potential to combine with well corrected small telescopes to expand their effective aperture, and the other for its practically perfect correction.
An arrangement well known and used in the professional astronomy is afocal off-axis arrangement with two confocal paraboloids. Hereafter it will be referred to as off-axis Mersenne. The two paraboloids are a beam reducer, with the secondary off-axis section being smaller than the primary. But for a telescope placed in the collimated beam produced by the secondary, it is practically an aperture expander. While most any type of telescope objective could be used for the imaging element, unobstructed one is needed for an overall unobstructed system.
System below uses such aperture expander to turn a 67mm apo doublet into 200mm aperture. For clarity, very fast paraboloids are used (/1.25 for the full aperture; for amateurs, the limit to fabrication is about /2.5). The reduced collimated beam is free from spherical aberration, coma and astigmatism regardless of the focal ratio of paraboloids. No image tilt, and the only aberration is Petzval field curvature by the secondary (which is only partly offset by that of the primary).
The price to pay for the expanded aperture is lower field quality with respect to the imaging telescope alone, due to the larger off axis angles of rays coming off secondary. Still, correction is good, (much) better than with the obstructed Mersenne, due to the lower off-axis angle magnification factor on this, weaker secondary. Field remains nearly identical if a conventional apo triplet is used, but gets significantly better - as it does with the obstructed Merssenne - if the imager is a Petzval apo. In this example the effective aperture of the doublet is 67mm, with the approximate (from the lens-to-focus separation) /10.7 focal ratio. The Petzval (and the entire system with it) is about /8.5.
For more realistic paraboloids' ratios (/2.5-3 parent mirrors), a small very-long-focus achromat can also be used. Inset bottom right shows field of such a system using 50mm /24 achromat to create, in effect a 200mm /24 (the Airy disc and spots are reduced by a factor of 2, except for the F-e-C spot top left). The field is smaller angularly for given linear size, but correction doesn't worsen significantly to the very edge. The farther off spectral lines are still bloated, but they matter little for visual observing (photopic polychromatic Strehl is about 0.96, in the "true apo" area).
And a design that surpasses all other tilted mirror telescopes in regard to image quality is the Stevick-Paul three-mirror system (FIG. 135), the off-axis cousin of the Paul-Baker telescope invented by Dave Stevick. It is the only TCT anastigmatic aplanat in the /10-/12 relative aperture range, with only a mild field curvature remaining.
Stevick-Paul 3-mirror TCT bases its exceptional performance
on the freedom from off-axis aberrations of a sphere with the stop at
the center of curvature. The convex secondary sphere (2) is confocal with
the paraboloidal primary (1), producing collimated beam and acting as an
aperture stop for the concave spherical tertiary (3). Having identical radii of
curvature, secondary and tertiary cancel each other's spherical
aberration. A small flat (4) makes the final image accessible. The ray
spot plot is for an 8" /11.7
system (based on design from WinSpot, graphics generated by
OSLO). Image quality is practically perfect across the best
image surface, which has 2250mm radius (since astigmatism is
practically non existent - tangential and sagittal surfaces are
slightly separated and nearly concentric - and the two mirrors
with identical radii cancel each other's Petzval out, image
curvature results from the primary's Petzval curvature radius
(equal to the primary's focal length, concave toward eyepiece),
and is well approximated by it (the curvature will cause
asymmetric field on the two opposite sides in the direction of
tube axis, if the image tilt - in this case 8.97°
with the end toward tube
front lower - is not compensated for. On flat field, the system
is better than diffraction limited as far as 21mm off axis. As
plot bottom left shows, distortion is also very low.
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).