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This also applies to the secondary mirror in two-mirror systems. The difference is that lens correctors have more surfaces - commonly four - each contributing its own aberrations. Consequently, the appropriate aberration calculation is significantly more complex. For that reason, the consideration here will be limited to a general form of aberration relations for a single lens, showing how the three point-image quality aberrations, spherical, coma and astigmatism, as well as field curvature, are affected and inter-related at a sub-aperture lens corrector.

Axial correction

Aberration coefficient for lower-order spherical aberration of sub-corrector lens element is the same as for a single lens element in general:

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). Obviously, only the value of p is different from those for object at infinity.

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. This assumes nearly identical marginal ray height h of the axial cone on the two elements, when the difference in (h1/h2)4 is negligible; otherwise, the coefficient 2 needs to be corrected by (h2/h1)4 before the addition (assuming that h1 is used to calculate peak aberration coefficient). Another option is to calculate peak aberration coefficient for each element, and then do the addition.
 

Off-axis aberrations

For lower-order coma, the aberration coefficient for a single lens element with the stop at the surface is given by

with the P-V wavefront error given by W=2C/3=cαD3/12, with C0=cαd3=cαD3/8 being the peak aberration coefficient for coma with the stop at the surface, α being the lens image point field angle (the RMS wavefront error is ω=W/√32), and p and q, as before, the lens position and shape factor, respectively. A doublet coma coefficient is 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, the chief - or central - ray for off-axis points shifts off the lens center. This shift of incident rays relative to the lens surface changes their optical path lengths relative to each other, changing by that off-axis aberrations (in terms of the wavefront, the stop effectively directs the section of the wavefront entering through it to a different portion of the lens, at somewhat different angle of incidence).

C = C0-4yS0         (98.1)

where C0 and S0 are the peak aberration coefficients for coma and spherical aberration with the stop at the lens, respectively, and

y = Tα/r        (98.2)

the relative pupil coordinate shift caused by the displaced stop, with T being the stop-to-surface separation - numerically positive 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 linear height in the image plane), and r the stop radius.

Aberration coefficient of lower-order astigmatism for a single thin lens with the stop at the surface is given by:

a = -1/2f        (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, which is for the stop at the surface given by A0=aα2d2=aα2D2/4, becomes:

A = A0-2yC0+4yS0          (99.1)

Change in astigmatism causes change in best image field curvature, which is now, also as the peak aberration coefficient, given by:

U = A0-yC0+2yS0         (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. By the virtue of their function, sub-aperture correctors do have inherent (corrective) aberrations, and normally require use of stop-shift relations to calculate their off-axis aberrations.

Aberration coefficients for sub-aperture corrector can be also calculated as a sum of aberration coefficients for each surface, as outlined with field flattener lens example. It is even more time consuming and, as the above relations, only covers lower-order aberrations; that limits their usefulness for assessing aberrations of sub-aperture correctors, which commonly have significant higher-order aberration component.

A form of the sub-aperture corrector with the simplest aberration expressions is a single Schmidt plate in a converging cone, usually at some distance in front of the final focus. The aberration coefficients for lower-order spherical aberration, coma and astigmatism are s=bΔ4/8, c=-bTΔ3/2 and a=-bT2Δ2/2, respectively, where b is the aspheric coefficient, Δ is the relative plate-to-focus separation in units of the system focal length, and T is the stop (exit pupil) separation for the plate. As with the Schmidt camera, the lower-order aspheric coefficient is found from -(b/8)+ss=0, where ss is the aberration coefficient of lower-order spherical aberration for the rest of the system. The stop separation T equals the mirror-plate separation (numerically negative) in a single-mirror system with the stop at the surface. In two-mirror systems, T=E-Δƒ, where E is the exit pupil (i.e. image of the primary formed by the secondary) separation, and ƒ is the system focal length (note that both E and Δƒ are numerically positive, as is the stop-to-plate separation T for Δƒ<E).
 

Chromatic aberration

Chromatic aberration of a doublet corrector can be 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 the 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. The only difference is that the focal lengths of the lens elements in Eq. 42 are replaced by the actual lens powers, given by p=oi/(o-i), where o is the object distance for the element, and i its image separation (object for the these lens element is the image formed by preceding element). The actual lens power figure results from the angle of refraction, as illustrated below.

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

with the required lens separation for achromatism given by:

where ƒ1, ƒ2 and V1, V2 are the front and rear element focal length and Abbe number, respectively. Since in this form the relation apply for object at infinity (collimated incident light at the front lens), for object at a finite distance - most common scenario with sub-aperture corrector, where the object is the image formed by the optical element preceding corrector - f1 is replaced by image separation for the first lens, obtained from the Gaussian lens formula.

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:

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:

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.

◄ 9.1. Designing doublet achromat    ▐    10.1.2. Sub-aperture corrector examples ►

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