8.4.3. Off-axis Newtonian

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. 89). 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 89: 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 ~ƒ/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 ƒ/10 to ƒ/12 effective ƒ-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. 90). 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).

FIGURE 90: Along the axis of aberration, comatic wavefront deviation for a full aperture mirror has two lobe-like formations of opposite signs. Deformation of the portion of the wavefront reflected from an off-axis segment - green shaded - is mostly astigmatic, despite it being originating from mainly comatic deformation of the wavefront as a whole. Section of the wavefront reflected from the segment will have the radius along the axis of aberration - imaginary central vertical line on the plot - more strongly curved than in the direction perpendicular to it. The P-V wavefront error of the segment is determined by the P-V wavefront error of the appropriate portion of the comatic wavefront over its effective pupil (aperture). Since this astigmatic wavefront deformation originates from the parent mirror coma, it changes linearly, with the field angle - not with the square of it, as astigmatism does when it originates from the incoming wavefront being tilted with respect to the optical surface (which is the "usual" astigmatism form for off-axis image points). Variation of wavefront profile with the wavefront incoming angle also produces peculiar (somewhat asymmetric) image field.

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. 91). 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.

FIGURE 91: Wavefront profile along the axis of aberration (dotted green) for parent mirror's wavefront over its pupil radius d is the basis for calculating approximate P-V error of astigmatism (as the dominant aberration) in an off-axis section mirror. Perfect reference sphere for this wavefront (solid red) determines relative actual wavefront deviation at any pupil height ρ normalized to 1, according to ρ3-(2ρ/3) (Eq. 12; for the axis of aberration, θ=0). For given segment diameter 2ρ'd, with its upper edge nearly coinciding with that of parent mirror's, the P-V wavefront error with respect to the same reference sphere can be approximated by W=W1+W2. However, it has an obvious tilt element in regard to this reference sphere, which is centered at the best focus for the comatic wavefront as a whole. Best focus of the wavefront segment lies in a slightly different direction; in effect, it is shifted downward with respect to the best focus direction of the parent mirror wavefront. Better reference sphere for the segment portion of the wavefront (dashed red) is the one centered at the segment's diffraction focus. It corrects for tilt error, with the P-V error along the axis of aberration reduced to WcS'~W'+W". It is further reduced to a final error value WcS by fitting in a final best reference sphere - essentially bending the dashed red line until the lowest net deviation from the sphere defined by its vertex-centered rotation with respect to the corresponding portion of the original wavefront is reached. The half-segment of the original wavefront radius shown is in the tangential (vertical) plane, thus determining the tangential astigmatic focus. Since the wavefront is more strongly curved vertically than in the direction perpendicular to it, this section focuses closer. With the wavefront error being mainly astigmatic in its form, best focus lies near midway between the tangential and sagittal foci.

It is evident that the relative peak wavefront error wc (in units of the peak aberration coefficient C in Eq. 12, with θ set to zero) with respect to the coma reference sphere (solid red) over radius d is a sum of the peak errors obtained from wc=ρ3-(2ρ/3) for ρ=1 (zonal height for the maximum positive deviation) and ρ=0.47 (zonal height for the maximum negative deviation), or 0.54. This sum makes 81% of the relative P-V coma wavefront error wc (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.

However, with respect to the reference sphere adjusted for tilt of the wavefront produced by segment's surface, the relative P-V wavefront error wcS' is smaller, approximated - for practical relative (in units of parent mirror radius d) segment radius ρ' values of ~0.3 or greater - by wcS'~|w'|+|w"| (FIG. 91), where w' is the relative (vs. parent mirror's) P-V wavefront error given for ρ=(1-ρ'), and w"=(|w1|-|w3|)/2 is the difference to the approximated relative wavefront error for the segment. Note that w1 and w3 are obtained by substituting ρ=1 and ρ=(1-2ρ'), respectively, for ρ, into into wc=ρ3-(2ρ/3).

Note that all three values, w' and w1 and w3 are, in order to be expressed relative to the P-V wavefront error of the entire comatic wavefront of parent mirror wc, divided by 0.67. As already mentioned, ρ' is the relative segment diameter in units of the parent mirror diameter.

The relative segment size ρ' 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 ρ'max=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 ρ'=0.4, centered at ρ=0.6, thus with w', obtained by substituting 1-ρ'=0.6 for ρ in ρ3-(2ρ/3) and vs. its full-aperture value (for ρ=1), as w'=-0.184/0.67=-0.275, with w"=(|w1|-|w3|)/2=(0.33-0.125)/2 as 0.1/0.67=0.15, gives the tilt-adjusted P-V wavefront error for the segment's wavefront as wcS'~w'+w"=0.425wc, with wc being the coma P-V wavefront error in units of peak aberration coefficient C.

This is nearly 1/3 larger P-V error than that given by raytrace (wcS~0.32wc), the difference, as mentioned, coming mainly due to the adjustment to the final best reference sphere by raytracing software.

Applying the 0.32 factor to the coma P-V wavefront error expression for the parent mirror (Eq. 70), gives good approximation for mainly astigmatic P-V wavefront error of such mirror segment as:

WcS ~ αD/150F2 or, alternatively, WcS ~ h/150F3 (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. 92).

FIGURE 92: Best focus wavefront deformation of an ƒ/10 off-axis section mirror (0.4D cut-out from 10" ƒ/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 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 best focus is 1/√24. For the 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. 94 (it also varies with ρ' value). The mean RMS/PV value here can be rounded off at ~1/√27. 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/780F3, (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 ω'~2.3h/F3. 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 three times larger, over its tilted best image surface. That would make the segment's off-axis aberration level comparable to that in a paraboloid with the F# greater by a factor of 1.4 than that of the parent mirror. In other words, with an ƒ/4 parent mirror, the segment with the above specs would have field quality - i.e. the size of linear diffraction limited field - before the eyepiece comparable to an ƒ/5.6 paraboloid (linear diffraction limited field only changes with F3; angular diffraction-limited field, however, changes with F3/D, thus the segment has it only 20% larger - from 3/2.5=1.2 - than an equal-aperture ƒ/4 paraboloid).

Since the linear size of comatic blur changes with F2 only, and quality angular field with F/D, an ƒ/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, but 2.5 times greater quality angular field. With the mainly astigmatic blur in off-axis segment being less than half the size of the comatic blur (tangential coma) at the same error level, typical segment has more than 6 times smaller blur size at given field height, and ~20% greater angular field than an equal to it in aperture ƒ/4 paraboloid. In other words, it is at the level of an ƒ/6.9 equal-aperture paraboloid with respect to the size of geometric blur (from F'2/42=3, thus F'/4=√3 and F'=4√3=6.9), and at the level of an ƒ/4.8 equal-aperture paraboloid angularly.

The wavefront error diminishes with the relative segment size, nearly in proportion to (ρ'/0.4)2 or 6.25ρ'2; for ρ'=0.3, the RMS error is smaller by a factor ~0.56, thus ω~h/1393F3 or ω'=1.3h/F3 in units of 550nm wavelength, and the effective F# multiplying factor is ~1.7.

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.

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. 93 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.

FIGURE 93: LEFT: Image tilt formation in an off-axis 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 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 off-axis 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 off-axis 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, flat-to-focuser separation and their respective dimensions. According to raytrace, between a 4-inch ƒ/10 and 6-inch ƒ/6.7, image tilt goes from 4.5º to 5.5º, respectively. Since no appreciable error results from up to ±1º deviation from the exact tilt angle - even somewhat larger - the needed tilt angle can be rounded off to about 5º 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.

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. 94 shows image field of a 4" ƒ/10 off-axis segment cut out of a 10" ƒ/4 parent mirror.

FIGURE 94: Ray spot diagram for a typical small off-axis Newtonian (4" ƒ/10), for 0.3º field diameter. To the left is its image field in ATMOS, over "regular" 1000mm-radius best image surface of parent mirror, The off-axis distortion comes from the image surface of the segment being tilted with respect to the Gaussian image plane. Next to it is a ray spot plot over the tilted (best) image surface, also in Atmos. Evidently, image tilt correction results in significant improvement in image field quality. The size of astigmatic blur is - as usual - deceiving: despite it being smaller than the Airy disc, the P-V wavefront error at 0.15º off-axis is about 0.5 wave (about 1/10 wave RMS, comparable to 1/3 wave P-V error of lower-order spherical aberration), for the 0.62 Strehl. With the error changing nearly in proportion to the off-axis height, it puts segment's diffraction-limited field radius at ~0.11°.
To the right is the same system in OSLO. It gives more detailed view of the same field, with PV/RMS errors along the two perpendicular field diameters. The wavefront error is somewhat smaller than that given by ATMOS, with diffraction limited field radius being over 0.12º (these values are used as the basis for Eq. 94/94.1, since ATMOS' output here may be compromised by it not adjusting the coma wavefront error for tilt correction; it triples the PV/RMS error, and results in much lower Strehl).

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 the 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.

◄ 8.4.2. Two-mirror TCT ▐ 9. REFRACTING TELESCOPES ►

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