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8.4.3. TCT 2
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9. REFRACTING TELESCOPES
► 8.4.4. Off-axis Newtonian
PAGE HIGHLIGHTS Off-axis Newtonian uses an off-axis 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; off-axis segment from any other conic would suffer from spherical aberration, as well as from tilt induced center field
FIGURE 136: Off-axis 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 off-axis segment mirror (usually f/10 to f/12 effective f-ratio). 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 off-axis 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 off-axis segment. The usual guess is that the main off-axis aberration of an off-axis 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 P-V 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 parent-mirror coma into off-axis segment astigmatism.
The above illustration shows that the relative P-V 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 P-V 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 P-V 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) P-V wavefront error given by w'=ρ3-(2ρ/3) for radius value ρ=(1-ρS), and w"=(w1-w3)/2, with w3 obtained by substituting ρ=(1-2ρS), 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 1-ρ. The deviation is given by w=[ρ'3-(2ρ'/3)]cosθ, with ρ'=[ρ2+(1-ρ)2]1/2 and tanθ=ρ/(1-ρ). In order to be expressed in units of the P-V 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 equal-size 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 1-ρS=0.6 for ρ in |ρ3-(2ρ/3)| as w'=0.184/0.67=0.28, in units of the P-V wavefront error of parent mirror. Likewise, w"=(w1-w3)/2, with w1=0.33 and w3=0.125, gives w"=0.1/0.67=0.15. Hence the tilt-adjusted P-V 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 P-V wavefront error of the parent mirror in units of the peak aberration coefficient C. But this is not the effective P-V 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 P-V error of the wavefront reflected from the segment results from its unequal cross-sectional radii. The P-V error equals the maximum sagitta differential, i.e. that between the vertical and horizontal wavefront radius, or wS=wSh-wSv. 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+(1-ρS)2]1/2 and tanθ=ρS/(1-ρS), hence w=0.09/0.67=0.13, and wSv~0.28-0.13=0.15, i.e. wSv~0.15wP, with the effective P-V error of the segment's wavefront as wS=wSh-wSv=0.43wP-0.15wP~0.27wP. This is only slightly larger than the segment's wavefront error given by raytrace (wS~0.26wP). It implies that the P-V wavefront error of a typical off-axis segment is nearly four times smaller than the P-V coma wavefront error of the parent mirror at a given off-axis distance. Applying the 0.26 factor to the coma P-V wavefront error expression for the parent mirror (Eq. 70), gives good approximation for the mainly astigmatic P-V 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 off-axis segment from 250mm f/4 parent mirror) indicate that on-axis error ifs practically non-existent, 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 P-V 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 on-axis 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° off-axis 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 P-V 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 P-V - 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 off-axis section mirror (0.4D cut-out from 10" f/4 parent mirror) at 0.16 degrees off-axis (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 RMS-to-PV 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 off-axis 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 P-V wavefront error approximation (Eq. 94) gives the field RMS wavefront error (in units of the wavelength) of a typical paraboloidal off-axis 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 off-axis 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 equal-aperture (to
parent mirror) paraboloid with
F-ratio 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 off-axis 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 f-ratio identical to the parent mirror. Assuming
segment's f-ratio 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 f-number 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 equal-aperture
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 equal-aperture paraboloid with comparable field
size to that in the segment, and F the parent mirror
f-ratio. For ρS~0.4, as before, that gives FA~1.13F.
With the mainly astigmatic blur in
off-axis 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 lower-order
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 off-axis segment from
230mm f/5.2 parent mirror, when
used with 20mm symmetrical Plossl, has astigmatism rising to the
diffraction-limited 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 off-axis segment is its image tilt. Due
to the wavefronts from different incident angles taking on different form of
deviation after reflection from an off-axis 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 parent-mirror-edge-oriented segment radius tilts
(rotates) away from the segment radius.
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.
For pure astigmatism, longitudinal
aberration is given by LA=16WaF2
(note that Wa
is half the P-V 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 off-axis segment cut out of a 10" f/4 parent mirror.
Added significance of the off-axis paraboloid segment configuration is
in it being relatively frequently employed with larger Newtonians using
off-axis 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 off-axis
arrangements is negligible.
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FIGURE
139: 
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 4-5 diopters (0.4-0.5mm 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 eyepiece-alone
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 
