4. INTRINSIC TELESCOPE ABERRATIONS

As mentioned, these are the aberrations caused by inherent properties of properly positioned optical elements, thus caused either by aberration limitations of a conic surface, or fabrication error (i.e. surface radius or conic). The only difference vs. these aberration forms caused by miscollimation is that they cannot be eliminated in alignment procedure.

Intrinsic telescope aberrations include the five primary aberrations intrinsic to conical surfaces of revolution - spherical, coma, astigmatism, field curvature and distortion - as well as chromatism and aberrations resulting from fabrication errors.

4.1. Spherical aberration

Spherical aberration - or correction error - is the only form of monochromatic axial aberration produced by rotationally symmetrical surfaces centered and orthogonal in regard to the optical axis. The attribute spherical probably originates in this aberration being inherent to the basic optical surface - spherical - for object at infinity. However, spherical aberration will appear whenever optical surface form doesn't properly match that of the incident wavefront. Thus, it is induced with the change of object distance or, with multi-surface objectives, with deviations in proper spacing. Spherical aberration affects the entire image field, including the very center. For that reason, its correction in a telescope is more important than that of other inherent conic surface aberrations, which affect the outer field.

Spherical aberration in the majority of amateur telescopes - especially more traditional ones, like Newtonian reflector or achromat refractor - is sufficiently accurately presented based on 4th order surface approximation, which includes the first two terms in the conic surface expansion series. Axial aberration associated with this surface approximation is called lower-order, or primary spherical aberration (also, 4th order wavefront, or 3rd order transverse ray aberration). Telescope objectives with strongly curved surfaces - like Maksutov-Cassegrain or doublet apochromatic refractors - generate significant amount of higher-order (6th on the wavefront, or 5th transverse ray) spherical aberration, which requires inclusion of the third term in the series i.e. upgrading, or correcting 4th order surface approximation to the 6th order surface. Very rarely, yet higher order terms also need to be taken into account.

4.1.1. Lower-order spherical aberration

FIG. 20 illustrates under-corrected (negative) form of primary spherical aberration, characteristic of a spherical mirror for object at infinity. Due to the actual wavefront being not spherical, rays projected from it do not meet at the same point; the wavefront becoming more strongly curved toward the edge causes the foci for rays projected from its outer zones to fall progressively closer to the mirror.

FIGURE 20: Spherical aberration of a concave spherical mirror, commonly called under-correction (due to marginal rays falling short of paraxial focus). RIGHT: The actual wavefront W is increasingly more curved toward the edge than the reference sphere SP coinciding with wavefront produced by reflection from the imaginary paraboloid P (for object at infinity) and centered at the paraxial (Gaussian) focus G. 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 M. Best focus B is midway between the marginal and paraxial focus, due to the deviation of the actual wavefront from perfect reference sphere Sb centered at this point being the smallest. Best focus wavefront error peaks at 0.707 aperture radius; it is smaller from the wavefront error at either marginal or paraxial focus by a factor of 0.25. The aberration at paraxial focus is primary spherical aberration, and at 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 with respect to the Gaussian focus. LEFT: Relative deviation of the actual wavefront in regard to perfect reference spheres for the marginal (Sm), paraxial (SP) and best, or diffraction focus (Sb) is constant for any amount of aberration.

With over-corrected (positive) spherical aberration, marginal rays focus farther away than paraxial rays. In either case, geometrical structure of the defocused zone remains identical in regard to the paraxial focus. For the longitudinal aberration normalized to 2 (Lg0≤Λ≤2, i.e. with 0 at paraxial focus, increasing with longitudinal shift to 2 at the marginal focus), the geometric (ray) spot has a constant relative size (vs. size of longitudinal aberration) and structure for a given value of Λ, as illustrated in FIG. 21. 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).

The relative wavefront error - either P-V or RMS - for any point between the two foci, in units of the error at the paraxial or marginal focus, is also constant, as given by:

ŵ = [1+0.9375Λ(Λ-2)]1/2 (6)

It gives the minimum relative aberration of 0.25 for Λ=1, which is the mid point between marginal and paraxial focus, as shown on the graph at left. The error is four times larger at either paraxial or marginal focus. At the location of smallest geometrical blur (circle of least confusion) the normalized error is 0.545, or larger than the error at best focus by a factor 2.18.

FIGURE 21: Defocus caused by spherical aberration, illustrated by selected rays projected from the aberrated wavefront. Axial separation between the foci for paraxial and marginal rays determines longitudinal aberration L (Λ0 when normalized to 2. Note that Λ=P/S, P and S being the peak aberration coefficients for defocus and spherical aberration, respectively. Both, transverse and wavefront aberration vary with the relative defocus Λ within the aberrated focal zone. Transverse blurs, from left, at the paraxial, or Gaussian focus (Λ=0), at the best, or diffraction focus (Λ=1), at the location of the circle of least confusion (Λ=1.5), and at the marginal focus (Λ=2). Pupil semi-diameter is d, and arbitrary paraxial zone height (illustration only) is p. Darker blur coloration roughly indicates increased ray density. Smallest blur radius is determined by the point of intersection of marginal ray and ray originating at the 0.5d zone. The relative blur radius, in units of the paraxial blur, is given by (Λρ/2)-ρ3 with ρ=1 for Λ=0, 1 and 1.5 (paraxial, best focus, and smallest circle location, respectively), and with ρ=1/31/2 for Λ=2 (marginal focus, where the blur radius is of opposite sign to the former three due to being measured, for positive ρ, relative to the converging ray above the axis). That gives the relative blur size as 1, 0.5, 0.25 and 0.385, respectively.

Aberration shown on FIG. 21 is spherical under-correction; the term probably originates from the ray geometry, with the rays from outer zones focusing slightly shorter than paraxial rays. Neither blur size/structure, nor size of wavefront error for given (absolute) value of Λ change for over-correction, where the geometry is symmetrically reversed, with the outer rays focusing longer than paraxial rays.

Note that the parameter Λ is related to peak aberration coefficients for spherical aberration and defocus, S and P, respectively, as Λ=|P/S|. Since, for given focal ratio, the P-V wavefront error of defocus (equal to the peak aberration coefficient of defocus) is double that for spherical aberration (equal to the peak aberration coefficient for spherical aberration) 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).

Thus, in terms of defocus error, spherical aberration is minimized, or balanced, for P=-S, or for spherical aberration at paraxial focus combined with the identical P-V wavefront error of defocus aberration (the minus sign indicates the direction of defocus, which is from paraxial toward marginal focus when the defocus aberration is opposite in sign to spherical aberration).

Follows detailed review of quantifying primary spherical aberration in both, wavefront and ray (geometric) form for reflecting surfaces and lenses.

◄ 3.5.2. Zernike aberration coefficients ▐ 4.1.2. Lower-order spherical: aberration function ►

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