|
telescopeѲptics.net
.......................................................................................... CONTENTS
10.1. Sub-aperture corrector: aberrationsUnlike telescope objectives and most full-aperture corrector arrangements, sub-aperture correctors are normally positioned in a converging light cone, with significant stop (exit pupil) displacement from the front surface. This results in changes of the ray and wavefront geometry and a subsequent change in aberrations - in particular off-axis and image-space aberration - and their expressions. While spherical aberration of sub-aperture corrector elements is not specifically affected, presence of spherical aberration is a factor affecting the size of off-axis aberrations. Thus, it will be included in this overview. Aberration coefficient for lower-order spherical aberration of sub-corrector lens element is the same as for a single lens element in general: s = - [n3 + (n+2)q2 + (3n+2)(n-1)2p2 + 4(n2-1)pq]/32n(n-1)2ƒ3 (97) with n being the refractive index, q=(R2+R1)/(R2-R1) the lens shape factor, and p=(2ƒ/o)-1=1-2ƒ/i the lens position factor (ƒ is the lens focal length, o the object distance and i the image-lens separation). The peak aberration coefficient, equal to the P-V wavefront error at paraxial focus, is S=W=sD4/16, D being the aperture diameter (the wavefront error at best focus is smaller by a factor of 0.25, or W=sD4/64). For the front lens, the object is the image formed by the preceding mirror element (or elements), while the object for the rear lens is image formed by the front lens. The combined error varies with the lens separation, which directly determines relative aperture of the rear lens, as well as its object separation. The doublet spherical aberration coefficient is a combined value of the two elements' coefficients sd=s1+s2, with the final system coefficient being ss=sm+sd, with sm being the mirror aberration coefficient. For lower-order coma, the aberration coefficient for a single lens element with the stop at the surface is given by c=-[(2n+1)p+(n+1)q/(n-1)]/4nƒ2 (98) with the P-V wavefront error given by W=2C/3=cαD3/12, with C being the peak aberration coefficient for coma, and α being the lens image point field angle (the RMS wavefront error is ω=W/√32). The doublet coma coefficient is also a sum of its two elements' coefficients, with the final system coefficient being a sum of the coefficients for the corrector and mirror elements. When the stop is displaced from the front lens, as it is always the case with sub-aperture correctors, the chief ray for off-axis points travels off the center of the front lens, resulting in change in off-axis aberrations. The stop for the front lens element is at the exit pupil formed by the preceding optical element in the system - usually a mirror (primary in a Newtonian, when the exit pupil coincides with the mirror, or secondary mirror in a two-mirror system), or a lens objective, which also nearly coincides with the exit pupil for the front corrector lens. For the rear corrector lens, the exit pupil is the image of the front lens' exit pupil it forms. The coma peak aberration coefficient takes the form: C = C0-4pS0 (98.1) where C0 and S0 are the peak aberration coefficients for coma and spherical aberration with the stop at the lens, respectively, and p=Tα/r (98.2) the relative pupil coordinate shift caused by the displaced stop, with T being the stop separation - numerically negative when the stop is to the left from the lens - α=iα0/(i+T) the new chief ray angle, with i being the lens to image separation, and α0=h/i the chief ray angle with the stop at the lens (with h being the height in the image plane), and r the stop radius. The latter equals mirror radius in a conventional Newtonian (the mirror being coinciding with both aperture stop and the exit pupil), and radius of the image of the primary formed by the secondary - which is the exit pupil for the corrector lens - in a two-mirror system. When the front lens f.l. is much smaller than its stop separation, the image of the stop is not significantly displaced from its focal point, in which case the change in off-axis aberrations of the rear lens also is not significant with respect to those with the stop at the front lens. Aberration coefficient of lower-order astigmatism for a single thin lens with the stop at the surface is given by: a=-1/2ƒ (99) with the P-V wavefront error W=aα2D2/4. Doublet coefficient is a sum of its element's coefficients, and the system coefficient is a sum of the coefficient for the corrector and mirror elements. Similarly to coma, when the stop is displaced from the lens, the peak aberration coefficient becomes: A = A0-2pC0+4pS0 (99.1) Change in astigmatism causes change in field curvature, which is now given by: U = A0-pC0+2pS0 (99.2) Relations for displaced stop - called stop shift relations - apply to optical elements in general, including mirror and lens objectives. They show that change in the stop position doesn't affect spherical aberration, but that uncorrected spherical aberration does affect both coma and astigmatism, with the latter also being affected by uncorrected coma. On the other hand, aplanatic systems (corrected for spherical aberration and coma) are unaffected by stop position. Chromatic aberration of a doublet corrector takes a form of the longitudinal (secondary spectrum) and lateral chromatism. As it is the case for a doublet in general, only one of the two forms can be cancelled, but the overall chromatic error can be greatly reduced by a proper choice of elements' properties (both longitudinal and lateral chromatism cancelled require each of the two elements achromatized). For a contact, or near-contact doublet corrector, achromatizing requirements are identical to those for a doublet objective. For sub-aperture corrector with separated elements, with the separation t defined as that between the back surfaces of the front and rear lens, the combined focal length is given by 1/ƒ = 1/ƒ1 + 1/ƒ2 - t/ƒ1ƒ2 or ƒ = ƒ1ƒ2/(ƒ1+ƒ2-t) (100) with the required lens separation for achromatism given by: t=(ƒ1V1+ƒ2V2)/(V1+V2) (100.1) where ƒ1, ƒ2 and V1, V2 are the front and rear element focal length and Abbe number, respectively. Achromatizing is defined as bringing two widely separate wavelengths to a common focus, thus fulfilling this condition will achromatize for selected wavelengths 1 and 2, provided that the V values used for Eq. 100.1 are those for which the refractive index is the mean value of the respective lens indici: nm=(n1+n2)/2. Consequently, the V number for each lens is V=(nm-1)/(n1-n2). The remaining longitudinal color error is reduced to the displacement of the first and second principal point of the doublet (even if consisting of a pair of thin lenses, a doublet system has properties of a thick lens), given by: ∆l=-ƒt[(1/V1ƒ1)+(M2/V2ƒ2)] (100.2) with M being the transverse corrector image magnification. Longitudinal color error ∆l is zero for M2=-V2ƒ2/V1ƒ1. With the separation t satisfying Eq. 100.1, lateral color error reduces to: ∆h=(tM/V2ƒ2)-∆l/L (100.3) with L being the exit pupil to (final) image separation, approximated by L~-(ƒ-ic), ic being the corrector-to-image separation. Needed lens separation for a single-glass sub-aperture doublet corrector (V1=V2=V) is, from Eq. 100.1, given by t=(ƒ1+ƒ2)/2, with its longitudinal chromatic error reduced to ∆l=-(ƒ2+ƒ1M2)/V and the lateral error reduced to ∆h=(ƒ1M/V)-∆l/L. For zero longitudinal chromatism, the lens elements' focal lengths relate as ƒ2=-ƒ1(1-t/ƒ1)2, due to the back focal distance becoming independent of the wavelength. Obviously, it implies two lenses of opposite powers, with the negative lens stronger for t>0. However, focal length variation ∆l is not zero, resulting in non-zero lateral chromatism. In general, longitudinal and lateral chromatism in a separated doublet cannot be simultaneously cancelled, unless both lenses are achromatic themselves.
|