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8.4.1. Off-axis Newtonian 2 Coincidentally or not, it is close to what the raytrace (Atmos) comes up with for the astigmatic P-V wavefront error of an off-axis section mirror at best focus. I am somewhat reserved as to the raytrace underlying parameters, since it doesn't adjust comatic wavefront error for the tilt (as a result, it gives three times the wavefront error at best focus and grossly underestimates the Strehl). There is no apparent correction for the tilt of an off-axis segment wavefront either (the focus location is adjustable only longitudinally), but there might be a mechanism that does enable correct assessment of its properties. The raytrace gives an expected wavefront deformation form, closely resembling that at the best astigmatic focus (FIG. 73).
FIGURE 73: Best focus wavefront deformation of an ƒ/10 off-axis section mirror (0.4D cut-out from a 10" ƒ/4 parent mirror) at 0.16 degrees off-axis. The form of deformation is very similar to that of pure astigmatism, except that the two tips of the "saddle" are of unequal height. This results in somewhat lower RMS-to-PV error ratio than that with "ordinary" astigmatism (which is closer to the RMS/PV ratio of the comatic wavefront), and in the triangular (instead of round) ray spot shape. Also, due to one tip of the saddle being less curved, longitudinal aberration with the segment astigmatism for given P-V wavefront error is somewhat smaller than in the case of pure astigmatism. The triangular form of pattern deformation probably makes this astigmatism form more similar to coma in appearance. The RMS/PV ratio for pure astigmatism at best focus is 1/√24, some 10% greater than the one given by the raytrace for the off-axis section wavefront deformation, which is ~1/√29. Taking the smaller RMS, and applying it to the P-V wavefront error (Eq. 94) gives the field RMS wavefront error (in units of the wavelength) of a typical paraboloidal off-axis segment as: ω~2.7h/F3, (94.1) h being the height in the image plane in mm, and F the parent mirror F#. Compared to the parent mirror RMS wavefront error of coma (Eq. 73), quality linear field size in the typical off-axis segment is by a factor ~2.5 greater than in the parent mirror, over best image surface. That would make it comparable to a paraboloid of the F# greater by a factor of ~1.35. The error diminishes with the relative segment size; for Ω=0.3, the RMS error ω~1.6h/F3, and the effective F# multiplying factor is ~1.6. Another interesting property of an off-axis segment is its image tilt. Due to the wavefronts of 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. 74).
FIGURE 74: 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. 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 determine the image tilt angle υ as υ~23/F in degrees, with F being the parent mirror F#. The raytrace indicates it somewhat smaller: υ=17/F or, for the segment mirror focal ratio number F*, υ~42/F*, probably the consequence of one of the two tangential wavefront tips being lower, resulting in smaller longitudinal aberration. 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. 75 shows image field of a 4" ƒ/10 off-axis segment cut out of a 10" ƒ/4 parent mirror.
Added significance of the off-axis
paraboloidal 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. ◄ 8.4.1. Off-axis Newtonian ▐ NEXT: 9. REFRACTING TELESCOPES ► |
FIGURE
75: Ray
spot diagram for a typical small off-axis Newtonian (4" ƒ/10), for 0.3 degree
field diameter. To the left is flat-field image coinciding with
the Gaussian image plane, with off-axis distortion
coming from the image surface being tilted with respect
to the Gaussian image plane. To the right is
a ray spot plot along the tilted image surface. 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 is nearly 1/2 wave (about 1/10 wave RMS,
comparable to 1/3 wave P-V error of lower-order
spherical aberration), for the 0.62 Strehl. It puts
the mirror's
diffraction-limited field radius at ~0.11°.