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Wavefront aberrations at a single optical surface with the stop at the surface, either reflecting or refracting (so called general surface), for the three primary aberrations affecting point-image quality, can be expressed with these three generalized aberration coefficients:

s = -(NJ2+Q)/8            (f)

for spherical aberration, with the peak-to-valley wavefront error at paraxial focus Ws=sd4

c = NJ/2             (g)

for coma, with the peak-to-valley wavefront error at Gaussian image point Wc=2cαd3cosθ, for θ=0, and

a = N/2             (i)

for astigmatism, with the peak-to-valley wavefront error at Gaussian image point Wa=aα2d2cos2θ, for θ=0,
where


 

n, n' being the refractive indici of the incident and refractive/reflecting media, and L, L' the object and image separation from the surface (the latter found using Gaussian approximation), respectively, R being the surface radius of curvature, and K being the surface conic (for coma and astigmatism, valid for the aperture stop at the surface).

Note that the P-V wavefront deviations are those for classical primary aberrations, at paraxial focus; d is the aperture radius, α the field angle in radians and θ the pupil angle (describes the form of rotationally asymmetrical aberrations). By setting θ to zero for coma and astigmatism, the relations give the maximum, P-V error, along the axis of aberration.

If aspheric coefficient b is present, it is added to the sum in the brackets in (f).

Relations for c and a imply that, with the aperture stop at the surface, coma and astigmatism are independent of the surface conic.

Likewise, transverse aberrations, also in the paraxial image space, are given by:

TS = -(NJ2+Q)d3L'/2n' for spherical aberration, as the radius of paraxial blur,
TC = 3NJd2L'α/2n' for coma, as the length of coma blur (tangential coma), and
TA = -NdL'α2/n' for astigmatism, as the diameter of the smallest blur

Transverse ray aberration can also be expressed in terms of the aberration coefficients, as:

for spherical aberration, coma and astigmatism, respectively.

The combined aberration for two or more surfaces is given as a sum of (wavefront) aberration coefficients at each surface, as s=s1+s2, c=c1+c2 and a=a1+a2, assuming equal aperture radius. In practice, telescope systems regularly handle converging - occasionally also diverging - cones, resulting in unequal marginal ray height at individual surfaces. Since the final size of aberration always depends on the size of aperture - expressed as peak aberration coefficients S=sd4, C=cαd3 and A=a(αd)2 for spherical aberration, coma and astigmatism, respectively - either representing the peak-to-valley wavefront error (for spherical aberration and astigmatism), or directly related to it (one half of the P-V wavefront error for coma) - combining aberration coefficients requires factoring in the radius difference. It can be done either directly, by multiplying aberration coefficient with the appropriate radius term before combining aberration contribution from different surfaces into a system sum, or at the level of aberration coefficient, by correcting it for the radius term difference.

For instance, system coefficient for spherical aberration for a two-mirror system with the marginal ray heights D1 and D2 on the primary and secondary, respectively, is given by ss=s1+(D2/D1)4s2, with s1 and s2 being the individual aberration coefficients for the primary and secondary (the D2/D1 ratio, representing the relative aperture of the secondary mirror - specifically, its so called "minimum size" - in units of the primary aperture, is used in calculating two-mirror system aberrations as the k parameter).
 

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