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8.4.3. TCT 2
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9. REFRACTING TELESCOPES
► 8.4.4. Offaxis Newtonian
PAGE HIGHLIGHTS Offaxis Newtonian uses an offaxis mirror segment cut out of a larger paraboloid (parent mirror), so that it focuses outside the incoming pencil of light (FIG. 136). The parent mirror has to be paraboloidal; offaxis segment from any other conic would suffer from spherical aberration, as well as from tilt induced center field
FIGURE 136: Offaxis segment of a paraboloid focuses pencil of incoming light out of its path. Reflecting flat directing converging light to a more convenient location is also out of the incoming light path, allowing unobstructed system. The parent mirror is usually ~f/4, which is close to the upper limit of its relative aperture, both, with respect to the economy of production and the amount of aberration leftover in the offaxis segment mirror (usually f/10 to f/12 effective fratio). Segment of any other conic would produce axial aberrations. coma and astigmatism, the farther from paraboloid, the more so (it is obvious for a sphere, whose any offaxis section is, in fact, a smaller sphere tilted with respect to the incoming light). The question is how much of the aberration of the parent mirror  and in what form  is passed onto its offaxis segment. The usual guess is that the main offaxis aberration of an offaxis section is also coma, only somewhat reduced. A glance at the aberrated wavefront of the parent mirror reveals that it is not so (FIG. 137, left). While the wavefront profile for the segment could be determined by recalculating the coma pupil function for the segment area, it is easier and more illustrative to have it extracted from major points of the existing wavefront, assuming  with a degree of approximation  that the wavefront segment is purely astigmatic (that is, neglecting the low coma component). With this assumption, the PV error is determined by the peak aberration of the "parent" mirror wavefront along its axis of aberration, on one hand, and the segment size and position with respect to the wavefront on the other (FIG. 137, right). This is not a strict analysis  rather informal one and with approximate results, but illustrating the basis of transformation from the parentmirror coma into offaxis segment astigmatism.
The above illustration shows that the relative PV wavefront error of coma wc of this (top) half of the wavefront of parent mirror  in units of the peak aberration coefficient C in Eq. 12, with θ set to zero  with respect to the its reference sphere (solid red) over mirror radius d, is a sum of the absolute values of the two peak errors obtained from wc=ρ3(2ρ/3) for ρ=1 (zonal height for the maximum positive deviation), denoted on above illustration as w1, and for ρ=0.47 (zonal height for the maximum negative deviation), denoted as w2. Thus, the sum of the two is w1+w2=0.33+0.21=0.54. It makes 81% of the relative PV coma wavefront error wP (which is also in units of the peak aberration coefficient) over the entire parent mirror wavefront, given by a sum of wc=ρ3(2ρ/3) for ρ=1 and ρ=1, or 0.67, with respect to its reference sphere. For practical reasons, all wavefront deviation values hereafter are given as absolute (i.e. positive). However, with respect to the reference sphere adjusted for the tilt of the wavefront produced by the segment's surface, the relative PV wavefront error at the center of the segment is somewhat smaller. In the horizontal plane, it is approximated  for practical relative (in units of parent mirror's radius d) segment radius ρS values of ~0.3 or greater  by wSh~w'+w", where w' is the relative (vs. parent mirror's) PV wavefront error given by w'=ρ3(2ρ/3) for radius value ρ=(1ρS), and w"=(w1w3)/2, with w3 obtained by substituting ρ=(12ρS), for ρ in ρ3(2ρ/3). The approximate sagitta value at the center of the segment along its vertical diameter is wSv~w'w, with w being the wavefront deviation for the edge point on the segment's wavefront horizontal radius, with coordinates ρ and 1ρ. The deviation is given by w=[ρ'3(2ρ'/3)]cosθ, with ρ'=[ρ2+(1ρ)2]1/2 and tanθ=ρ/(1ρ). In order to be expressed in units of the PV wavefront error of the entire comatic wavefront of parent mirror wP, all wavefront deviation values are divided by 0.67. As already mentioned, ρS is the relative segment diameter in units of the parent mirror diameter. The relative segment size ρS is limited by the number of segments cut out from the parent mirror. Usually, it is three or four which, for the theoretical limit given by ρSmax=1/[1+(1/sinβ)], with β=180/N in degrees, N being the number of equalsize segments, gives the maximum segment diameter ranging between 0.4 and 0.45 of the parent mirror diameter. Taking ρS=0.4, centered at ρ=0.6, gives w' by substituting 1ρS=0.6 for ρ in ρ3(2ρ/3) as w'=0.184/0.67=0.28, in units of the PV wavefront error of parent mirror. Likewise, w"=(w1w3)/2, with w1=0.33 and w3=0.125, gives w"=0.1/0.67=0.15. Hence the tiltadjusted PV error for the segment's wavefront at its center in the horizontal plane as wSh~w'+w"=0.43wP, with wP being, as before, the coma PV wavefront error of the parent mirror in units of the peak aberration coefficient C. But this is not the effective PV wavefront error for the segment. It is obvious that somewhat more strongly curved sphere is a better fit to this portion of the wavefront. Hence best focus for this wavefront section shifts somewhat closer (for light coming at the opposite angle, the section of reflected wavefront is weaker, with its best focus shifting somewhat farther; these foci shifts result in a tilted image surface, as explained in more detail a bit later). The actual PV error of the wavefront reflected from the segment results from its unequal crosssectional radii. The PV error equals the maximum sagitta differential, i.e. that between the vertical and horizontal wavefront radius, or wS=wShwSv. With the former approximated by wSh~0.48wc, and the latter obtained from wSv~w'w, with w=[ρ'3(2ρ'/3)]cosθ, with ρ'=[ρS2+(1ρS)2]1/2 and tanθ=ρS/(1ρS), hence w=0.09/0.67=0.13, and wSv~0.280.13=0.15, i.e. wSv~0.15wP, with the effective PV error of the segment's wavefront as wS=wShwSv=0.43wP0.15wP~0.27wP. This is only slightly larger than the segment's wavefront error given by raytrace (wS~0.26wP). It implies that the PV wavefront error of a typical offaxis segment is nearly four times smaller than the PV coma wavefront error of the parent mirror at a given offaxis distance. Applying the 0.26 factor to the coma PV wavefront error expression for the parent mirror (Eq. 70), gives good approximation for the mainly astigmatic PV wavefront error of such mirror segment as: WS ~ αD/180F2 or, alternatively, WS ~ h/180F3 (94) with α being the field angle, h the linear height in the image plane, D the parent mirror aperture diameter and F the parent mirror focal ratio. The raytrace gives, as expected, wavefront deformation form closely resembling that corresponding to best astigmatic focus (FIG. 138). Zernike coefficients below (OSLO, for 100mm /10 offaxis segment from 250mm f/4 parent mirror) indicate that onaxis error ifs practically nonexistent, as long as the mirror is perfectly paraboloidal (K=1). Moderate figure error  in this case K=0.95 conic with 250mm f/4 parent mirror, resulting in 1/5.8 wave PV of primary spherical aberration (best focus)  causes imperfect segment's wavefront as well. The dominant error component is astigmatism, with small amount of coma. The RMS error is 0.032 wave, or a bit over 60% of the onaxis RMS wavefront error of the parent mirror (which is, of course, spherical aberration). This RMS error carries over into the field, so at 0.13° offaxis the segment's wavefront error increases to 0.11 wave RMS.
*product of the RMS constant and Zernike coefficient's value Other types of parent mirror surface errors will also affect correction level of the segment. Wide zonal deviation, for instance, will induce about as much of astigmatic PV error to segment's mid field. Since it will be nearly offset by astigmatism induced due to entrance beam inclination, best focus would shift somewhat off field center. Local (or overall) roughness of parent mirror, however, would produce near identical roughness PV  and possible proportionally larger roughness RMS  in the segment's wavefront. Note that the RMS errors do not simply add arithmetically; the final wavefront shape results from the aggregate sum of positive and negative deviations at every wavefront point. Statistical error sum is given as the square root of the individual RMS wavefront errors squared. The RMS values indicate the magnitude of aberration contribution of the specific aberration forms to that final wavefront shape. When one aberration form strongly dominates, then its RMS error is a fairly good indicator of the RMS error of actual wavefront.
FIGURE 138: Best focus wavefront deformation of an f/10 offaxis section mirror (0.4D cutout from 10" f/4 parent mirror) at 0.16 degrees offaxis (perfect reference sphere is flat circle). The form of deformation is very similar to that of pure astigmatism, except that one of the tips of the "saddle" does not deviate as much relative to the center point as do the two bottoms (blue). In effect, one side of the wavefront morphs toward cylindrical form, making this form of astigmatism sort of cross between best focus astigmatism on one, and sagittal or tangential astigmatism on the other side. This wavefront form results in slightly lower RMStoPV error ratio than that with "ordinary" astigmatism, as well as in triangular (instead of round) ray spot form at the best focus. The triangular form of pattern deformation probably makes this astigmatism form more similar to coma in appearance. It also "behaves" as coma, changing in proportion to the field radius, not its square, as the "regular" astigmatism does. The RMS/PV ratio for pure astigmatism at the best focus is 1/√24. For somewhat peculiar wavefront form produced by an offaxis segment of a paraboloid, OSLO gives larger RMS error, varying somewhat with the field point orientation, as shown on FIG. 140 (it also varies with ρ' value). The mean RMS/PV ratio here can be rounded off to ~1/√23. Taking this RMS value, and applying it to the PV wavefront error approximation (Eq. 94) gives the field RMS wavefront error (in units of the wavelength) of a typical paraboloidal offaxis segment as: ω ~ h/860F3, (94.1) h being the linear height in the image plane in mm, and F the parent mirror F#. In units of 550nm wavelength, the RMS error is ω' ~ 2.1h/F3, (94.2). Compared to the parent mirror's RMS wavefront error of coma (Eq. 70), quality linear and angular field size in the typical offaxis segment mirror is about 3.2 times larger, over its tilted best image surface. Since the quality linear field size changes in proportion to F3, that would make segment's linear field comparable to that in a paraboloid with the F# greater by nearly a factor of 1.5 than that of the parent mirror. In other words, with an f/4 parent mirror, the segment with the above specs would have linear diffraction limited field comparable to an f/6 paraboloid. And since quality angular field, for given aperture size, changes with F2, its quality angular/field is comparable to that in an equalaperture (to parent mirror) paraboloid with Fratio greater by nearly a factor of 1.8, or f/7.2. Setting ω=0.0745λ in Eq. 94.1 gives diffraction limited linear field radius of an offaxis segment as h~58λF3; in units of 550nm wavelength, it is: hDL ~ fαDL ~ F3/28 (94.3) with the corresponding angular diffraction limited field αDL ~ F2/28ρSD (94.3.1) with ρSD being the aperture diameter of the segment. Comparison with Eq. 70.2 shows that its linear diffraction limited field is more than three times larger than in a paraboloid of identical aperture to that of the segment, and fratio identical to the parent mirror. Assuming segment's fratio larger by a factor of 2.5, this implies that segment's diffraction limited linear field is nearly 5 times smaller than in a paraboloid of identical fnumber and aperture. Since the linear size of comatic blur changes with F3 only, and quality angular field with F2/D, an �/4 paraboloid of the same aperture as the segment  in this case 2.5 times smaller than the parent mirror  has identical linear coma blur as the parent mirror  thus identical linear field  but since its focal length is 2.5 times smaller, it has as 2.5 times greater quality angular field. Since the quality angular (and linear) field of the segment is 3.2 larger than in the parent mirror, it implies that focal ratio of the equalaperture paraboloid with comparable quality angular field is only slightly larger, by a factor of (3.2/2.5)0.5, or about 1.13F. This can be confirmed by setting Eq. 94.3.1 equal to Eq.70.2.1 (with the focal ratio in the former being that of a paraboloid comparable at the segment's aperture level, and focal ratio in the latter being that of parent mirror) which, solved for the comparable focal ratio of a paraboloid equal in aperture to the segment, gives: FA ~ 1.8Fρ0.5 (94.3.1) where FA is the focal ratio of equalaperture paraboloid with comparable field size to that in the segment, and F the parent mirror fratio. For ρS~0.4, as before, that gives FA~1.13F. With the mainly astigmatic blur in offaxis segment being nearly three times smaller than the comatic blur (tangential coma) at the same error level, typical segment has about 9 times smaller blur size at a given field height compared to its parent mirror. The wavefront error diminishes with the relative segment size, nearly in proportion to (ρS/0.4)2 or 6.25ρS2; for ρS=0.3, the RMS error is smaller by a factor ~0.56, thus ω~h/1500F3 or ω'~1.2h/F3 in units of 550nm wavelength, and the effective F# multiplying factors for its quality linear and angular field are nearly 1.8 and 2.4, respectively. Since this wavefront error is, by its form, predominantly lowerorder astigmatism of the sign opposite to that in most eyepieces, it is likely to be further reduced  and quality field expanded correspondingly  when used for visual observing. According to raytrace, 92mm f/13 offaxis segment from 230mm f/5.2 parent mirror, when used with 20mm symmetrical Plossl, has astigmatism rising to the diffractionlimited level (0.0745 wave RMS) at 0.135 degrees (2.8mm) off axis. The eyepiece alone has 0.1 wave RMS of astigmatism for the corresponding eyepiece field angle (~8°), which means that segment's astigmatism partly cancelling out that of the eyepiece resulted in about 15% wider linear diffraction limited field (it also shows that even at these slow focal ratios eyepiece astigmatism still dominates in the outer field). Assuming this effect has similar magnitude with other conventional eyepieces, and at other eyepiece focal lengths (eyepiece astigmatism is proportional to its focal length, which means that the offset with the segment's astigmatism will be generally larger in focal lengths below 20 mm, and vice versa, so 20mm f.l. could represent a rough average), diffraction limited linear field of the segment in combination with these types of eyepieces would be better approximated by hDL~F3/24, with F, as in Eq. 94.3 being the parent mirror's focal ratio. Another interesting property of an offaxis segment is its image tilt. Due to the wavefronts from different incident angles taking on different form of deviation after reflection from an offaxis segment  each form of deviation being a different portion of the original parent mirror comatic wavefront as a whole  they will not focus in the plane orthogonal to the line projected from segment's center to its center focus. Instead, they will form tilted astigmatic image surfaces, with best (median) image surface being at an angle to the straight line connecting center of the segment and field center (FIG. 139 left). In the setup w/o flat, image on the same side of segment's central axis as the parentmirroredgeoriented segment radius tilts (rotates) away from the segment radius. FIGURE 139: LEFT: Image tilt formation in an offaxis section mirror. Curvatures of the reflected wavefronts vary with the angle of incidence, each wavefront being different section of the parent mirror's comatic wavefront (W). In the tangential (vertical) plane, the top image field point is formed by a less curved section of the parent wavefront, while the opposite image field point is formed by a more curved section. These wavefronts are also astigmatic, forming sagittal (S), tangential (T) and best, or median (M) field surface. The image tilt angle t is between the median image surface and focal plane (FP). Due to its origin in the comatic wavefront, this astigmatism changes linearly, and all three image surfaces are nearly flat. Its another odd property is that it diminishes to zero toward the perpendicular (sagittal) field orientation, changing the sign on the opposite field side. Combined with the eyepiece astigmatism (which is of the same sign across the field), this gives best field definition in one direction, worst in the direction opposite to it, and intermediate in between (this would be occurring without any image tilt, but the two effects can combine). Image tilt, if not adjusted for, can significantly degrade offaxis performance of this type of mirror, with the exception of the image field portion near to the plane of tilt. RIGHT: Exaggerated collimation scheme of an offaxis segment mirror w/flat: adjusting focuser to the best image surface requires both tilting focuser toward primary by an angle nearly equal to the image tilt angle, and shifting it away from mirror to bring focuser axis back to the field center. This results in the flat moving from the center of the focuser tube toward mirror. Its final appearance in the focuser opening is determined by the tilt angle, flattofocuser separation and their respective dimensions. According to raytrace, between a 4inch f/10 and 6inch f/6.7, image tilt goes from 4.5° to 5.5°, respectively. Since no appreciable error results from up to ~1° deviation from the exact tilt angle  even somewhat larger  the needed tilt angle can be rounded off to about 5° for commercially available telescopes of this type. Needed focuser shift is given by h(tanτ), where h is the focus height above the bottom of the focuser base, and τ is the tilt angle. Obviously, since image tilt and optical axis are not perpendicular, adjusting eyepiece for image tilt will inevitably make it tilted with respect to optical axis. Raytrace indicates that it is still better overall (below, bottom right; since the image plane is vertical, or nearly so  because there is no tilted elements in the system  zero tilt eyepiece is aligned with it). The difference is minor  a little gain in the field bottom half astigmatism and nearly flat top field half is paid for by the bottom half of the field coming farther out. Added bonus is nearly balanced distortion, indicated by the mismatch between the image surface (representing Gaussian image) and the tip of converging marginal beam, but lateral color is somewhat larger. The eyepiece is 10mm f.l. Koenig, which means that one diopter of accommodation (infinity to 1m object distance) corresponds to 0.1mm on the horizonta scale of the astigmatic plot. That puts outer field beyond accommodation capability of the eye  in fact, if we set as the limit 45 diopters (0.40.5mm on the horizontal scale) for the field edge, usable field w/o refocusing is reduced by a factor of about 0.4 (litlle more top half, little less the bottom). That would make any eyepiece adjustment at this f.l. level inconsequential. However, since diopter of accommodation scales with the square of focal length, longer f.l. eyepiece would likely be able to use more of the field fitting in the field stop. The eyepiecealone spots (bottom left) indicate that it contributes most of the edge astigmatism and lateral color, but not in the inner field. Considering that edge distortion nears 30% (eyepice alone close to 25%), and that 0.25° true field radius at x100 magnification indicates 25° distortionless apparent field, the eyepiece AFOV is closer to 65°. More appropriate ~50° AFOV would eliminate significant part of the field aberrations (see Kutter schiefspiegler). Also, since diopter of accommodation changes with the square of eyepiece f.l. (f^{2}/1000), and field diameter for given apparent field with the focal length, longer focal length eyepieces will have lower accommodation requirements (example: StevickPaul telescope). For pure astigmatism, longitudinal aberration is given by LA=16WaF2 (note that Wa is half the PV error, and F is the effective F#) which, for the above "average" segment, would approximate the image tilt angle υ as υ~23/F in degrees, with F being the parent mirror F#. The raytrace indicates it is somewhat smaller: υ~18/F or, for the segment mirror focal ratio F*, υ~46/F*, probably the consequence of somewhat different wavefront properties vs. that of pure astigmatism. The image tilt causes asymmetric image distortion in the plane orthogonal to the line connecting the field center with the center of the primary. The ray spot diagram on FIG. 140 shows image field of a 4" f/10 offaxis segment cut out of a 10" f/4 parent mirror.
Added significance of the offaxis paraboloid segment configuration is in it being relatively frequently employed with larger Newtonians using offaxis masks. The only difference is in the position of the aperture stop, which is in the "mask arrangement" displaced from mirror surface to the mask. However, since the main aberration comes from the coma of the main mirror, and it is for a paraboloid independent of the stop position, the difference in the size of aberration between these two offaxis arrangements is negligible.
