10.1.2. Sub-aperture corrector examples (2)

EXAMPLE 1: Field flattener lens - Field curvature is rarely, if ever, affecting visual observing, due to the natural ability of the eye to accommodate (refocus). In photographic use, however, field curvature induces defocus error, which can be significant. Good example is the Schmidt camera, which is exclusively a photographic instrument. Its only remaining primary aberration is strong field curvature. This leaves two choices: either use of a curved detector, or sacrificing much of the exquisite field quality by using flat detector. Similar problem exist in two-mirror telescopes, particularly full-aperture catadioptrics employing fast mirrors and, to much smaller extent, in fast Newtonians.

The simplest way to correct for field curvature is by placing a single thin lens just in front of the final focus. With proper choice of parameters, it flattens the field while inducing very low aberrations, except at very large relative apertures. Assuming no appreciable astigmatism induced by the lens itself - valid for telescopes, in general, but not for the Schmidt camera - the field is flattened when lens' Petzval curvature is equal, and opposite in sign, to the median field curvature of a telescope. This determines needed lens shape as plano-concave (negative) for for Cassegrain-like telescopes, and plano-convex (positive) for the Gregorian.

By default, field flattener lens faces image with its flat side, and the sign of its curved surface is determined by the sign of image curvature. Needed radius of curvature of the lens is:

with n being the lens refractive, and Rm the telescope median (best) image radius.

For the Schmidt camera, field flattener lens is positive, with the radius R=[1-(1/n)]RM/2, with RM being the mirror radius of curvature (since it has no astigmatism in the Schmidt arrangement, its best surface coincides with its Petzval surface, which equals RM/2).

For the Newtonian, field flattener lens is of negative power, with the radius R=[1-(1/n)]Rm/2 (according to the sign convention, with the best field concave toward mirror, Rm is numerically positive in the Newtonian), and so is for the refractor. For the latter, median field curvature varies from one to another type; for a typical doublet achromat median field curvature is approximately -ƒ/3, ƒ being the refractor focal length, and the field flattener's radius R~-[1-(1/n)]ƒ/3 (of negative power).

In general, lens aberrations are low if it remains very close to the original focus. But even then, they can be significant with wide angular fields and/or fast focal ratios. Due to the lens thickness being significant relative to object and image separation, the thin lens equations are not appropriate. Instead, aberrations are calculated for each lens surface, similarly to the approach described with the generalized aberration coefficients. Only in this case, with the aperture stop displaced from the front lens surface, the relations need to account for this factor.

Numerical value of the stop separation T is determined by system configuration: in the Schmidt camera, for the front lens (flattener's) surface it is the distance from the corrector (where the mirror re-images the stop) to the surface, numerically positive, and for the second lens surface it is given by T2=α'ch1, h1 being the chief ray height on the second surface (illustrated below; Schmidt flattener thickness exaggerated to show ray paths). In the Newtonian, T1 it is the distance from the mirror to the lens surface, numerically negative (ASM on the illustration below, as opposed to ASS, with the aperture stop at the surface and T1=0), and in two-mirror systems, it is the distance from the image of the aperture stop (primary) formed by the secondary (exit pupil of the system, ExP on the illustration) to the surface, numerically negative.

Likewise, the chief ray angle αc is determined by system configuration: in the Schmidt camera, the chief ray coming through the center of the corrector is reflected back to the same point, therefore converging toward the front lens surface at the same incident angle α. Since it is opening counterclockwise from the axis, it is numerically negative, and so is the projected normal to the surface angle δ at the axis. Since the angles are small enough to be expressed in radians, it determines the chief ray angle α'c after refraction at the front lens surface as α'c=δ-(δ-α)/n, with all the angles in radians, n being the glass refractive. In a two-mirror system, the lens field flattener is of negative power, with both α'cs (chief ray angle after reflection from the secondary, appearing to be coming from the image of the primary formed by the secondary) and the surface normal angle δ numerically negative, the chief ray angle after the front lens surface α'c=δ-(δ-α)/n.

Since the rear surface of the flattener is flat, α'c=nαc. The angle μ1=1/2nF for the marginal ray of axial cone determines location of the image formed by the first lens surface as L1'=h0/μ0 from the front lens surface, h0 being the height of marginal ray at the surface, given as h0=l/2F, l being the front lens surface to original focus separation, and F the focal ratio.

This outlines the general procedure for calculating aberrations of a singlet field flattener lens.

With spherical lens surface, Q=0 and the aberration coefficients for spherical aberration, coma and astigmatism, from Eq. (k)-(m), are given by 

s=-NJ2/8,  c=NJYh/2  and  a=-N(Yh)2/2, respectively, with

N=n2[(1/n'L')-(1/nL)],     J=[(1/L)-(1/R)],     Y=[(1/T)-(1/R)]

and the height of incident point on the surface h=(Tα), where n, n' are the refractive indici of the incident and refractive/reflecting media, L, L' the surface-to-object and image separation, respectively, R being the lens surface radius of curvature, T being the surface-to-stop separation (numerically positive) and α the chief ray angle in radians. Note that the refractive indici are numerically negative, which requires appropriate adjustment to some of the general relations (for instance, needed lens surface curvature to flatten the field is R=(1-1/nl)Rm/2 with the index n numerically positive, and R=(1+1/nl)Rm/2 with the index negative).