4. INTRINSIC TELESCOPE ABERRATIONS

 

FIGURE 33: Spherical aberration of a concave spherical mirror, commonly called under-correction (due to marginal rays falling short of paraxial focus). LEFT: After flat incident wavefront reflects off spherical surface, it is increasingly more curved toward the edge than the reference sphere centered at paraxial focus, which would coincide with wavefront produced by reflection from the imaginary paraboloid P (for object at infinity) of identical vertex radius. As a result, its outer rays focus progressively closer to the mirror: while central rays meet at the paraxial focus, the edge rays meet at the marginal focus. Best focus is midway between the marginal and paraxial focus, due to the deviation of the actual wavefront from perfect reference sphere centered at this point being the smallest. Best focus wavefront error peaks at 0.707 aperture radius. It is measured with respect to reference sphere centered at the best focus, which would have been generated by a paraboloid of slightly smaller vertex radius than that of the spherical surface. The best focus P-V wavefront error is smaller from the error at either marginal or paraxial focus by a factor of 0.25. The aberration at the paraxial focus is primary spherical aberration, and at the best focus location it is balanced primary spherical aberration (it is balanced - or minimized - with defocus aberration). Ray geometry determines the longitudinal (L) and transverse (T) aberration, shown at the Gaussian focus. RIGHT: The relative wavefront deviation from their respective reference spheres is constant for the marginal, paraxial, best, or diffraction focus and the minimum geometric blur focus for any amount of the aberration. Taking paraxial (Gaussian) focus as the reference point, the wavefront error W at any point of the longitudinal aberration L can be expressed as a sum of the spherical aberration P-V WFE at paraxial focus WP and defocus P-V WFE WD with respect to the reference sphere centered at the Gaussian focus, which gives to it the opposite sign. Thus W=WP+WD, with WP being constant and WD ranging from zero at the paraxial focus to WD=-2WP at the marginal focus (for given longitudinal aberration the defocus error is double the s.a. error). Since for a given longitudinal aberration the defocus P-V WFE is twice that of spherical aberration, and the P-V WFE for both equals the peak aberration coefficient, with WP=Sρ4 and WD=Pρ2, S and P being the peak aberration coefficients for primary spherical and defocus, respectively, and ρ being the zonal height for the aperture radius normalized to 1, the relative coefficients can be expressed in terms of the relative defocus value alone. Taking S for unit, with P ranging from 0 to 2, we can write the relative aberration in units of the s.a. error at paraxial focus as

W/S=(ρ4-xρ2), and the actual WFE of spherical aberration as W=(ρ4-xρ2)S,

with x ranging from 0 at the paraxial to 2 at the marginal focus (note that the sign of actual wavefront deviation for ρ4 is by the sign of (ρ4-xρ2) and that of S; with the latter for the specific case shown at left - undercorrection - being numerically negative). The right side of the second equation is the general form of aberration function for primary spherical aberration, giving the actual error at any point in the pupil For x=0, thus W=Sρ4, the maximum deviation, or P-V WFE, is for ρ=1 (also for x<0, and x≥2). For 2≤x≤1, the P-V WFE is given directly by the deviation at the point of deflection (i.e. point of WFE plot reversal, tangent to which is parallel to the reference sphere line). Value of ρ for this point is obtained by setting first derivative of the aberration function - f'(x)=nxn-1 for the functions of f(x)=xn type, and for f'(x)=Σf'(x) with a function that is a sum of more than one exponential term of xn type - to zero.
For instance, for best focus location (x=1) the maximum P-V wavefront error for f(ρ)=ρ4-ρ2 and
f'(ρ)=4ρ3-2ρ=2ρ(2ρ2-1)=0 is for ρ=0.50.5=0.707. For 0<x<1, the WFE plot crosses the reference sphere line and the P-V error is given as a sum of the absolute values for the deviation at the point of plot reversal, and deviation for ρ=1.

If normalized to unit error at the best focus location, which may be more convenient in a simplified context, the relative wavefront error ŵ along the longitudinal aberration length normalized to 2, Λ, is:

ŵ = [16+15Λ(Λ-2)]1/2          (6.1)

giving ŵ=1 for Λ=1 (best focus location), ŵ=4 for Λ=0 (paraxial focus) and Λ=2 (marginal focus) and ŵ=2.18 for Λ=1.5 (smallest ray spot). So, a 6-inch f/8.16 sphere, having 1/4 wave P-V of primary spherical aberration at the best focus, has 1 wave P-V at either paraxial or marginal focus, and 0.54 wave P-V at the location of the smallest ray spot.

Note that the parameter Λ is related to the peak aberration coefficients for spherical aberration and defocus, S and P, respectively, as Λ=-|P/S|, with the defocus coefficient always of opposite sign. Since, for given focal ratio, the P-V wavefront error of defocus (equal to the peak aberration coefficient of defocus) from paraxial to marginal focus is double the P-V error of spherical aberration (equal to the peak aberration coefficient for spherical aberration) at either paraxial or marginal focus for identical longitudinal error, the absolute value of Λ ranges from the minimum 0, at paraxial focus, to 2 at the marginal focus (the sign of aberration coefficient is negative for undercorrection and positive for overcorrection, while can be of either sign for defocus, depending on the direction).

While the above relations hold for any level of spherical aberration with respect to the wavefront error, the corresponding PSF peak, being determined by the phase sum, shifts away from the mid focus, more as the aberration exceeds 0.625 waves P-V (FIG. 36B).

For the longitudinal aberration L normalized to 2 (0≤Λ≤2, i.e. with 0 at paraxial focus, increasing with longitudinal shift to 2 at the marginal focus), the geometric (ray) spot increases steadily from mid focus toward paraxial focus, while initially decreasing and then resuming increase toward marginal focus, as illustrated in FIG. 35. Location of the smallest geometric blur does not coincide with the location of lowest wavefront error.

In units of the paraxial blur diameter (Λ=0), the blur size is 0.385 for Λ=2 (marginal focus), 0.25 for Λ=1.5 (circle of least confusion) and 0.5 for Λ=1 (diffraction focus). In general, for 0≤Λ≤1.5 the relative blur diameter is given by (2-Λ)/2; for 1.5≤Λ≤2, it is closely approximated by (Λ-0.5)/4 (the approximation is exact for Λ=1.5, wit the error increasing to a -2.6% maximum at Λ=2).