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3.5.3. Zernike expansion schemes       4.1.2. Lower-order spherical: aberration function

4. INTRINSIC TELESCOPE ABERRATIONS

As mentioned, these are the aberrations caused by the inherent properties of properly positioned optical elements, thus caused either by the aberration limitations of a conic surface, or fabrication error (i.e. error in surface radius or conic). They are identical in form to the corresponding aberration caused by miscollimation, but they cannot be eliminated in the 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 wavefront aberrations resulting from fabrication errors.

Specific notation for the five conic surface aberrations is based on a power series expansion of the aberration function that sums up all the geometric deviations in point imaging for Gaussian image point, in which a general aberration term is given as 2l+mwnmdnh(2l+m)cosmθ, where w, d, h and θ are the aberration coefficient, pupil radius, point height in the image space and pupil angle, respectively, and l, m, and n are the power terms from the function. The latter are associated with specific aberrations as shown in table below.

ABERRATION l n m 2l+m Term Order
2l+m+n
Primary
(Siedel)
SPHERICAL 0 4 0 0 0w40d4 4
COMA 0 3 1 1 1w31d3hcosθ 4
ASTIGMATISM 0 2 2 2 2w22d2h2cos2θ 4
FIELD CURVATURE (DEFOCUS) 1 2 0 2 2lw20d2h2 4
DISTORTION (TILT) 1 1 1 3 3w11dh3cosθ 4
Secondary SPHERICAL 0 6 0 0 0w60d6 6
COMA 0 5 1 1 1w51d5hcosθ 6
ASTIGMATISM 0 4 2 2 2w42d4h2cos2θ 6
ARROWS (TREFOIL) 0 3 3 3 3w33d3h3cos3θ 6

TABLE 4: Wavefront aberrations: notation for selected aberrations.

Wavefront aberration forms with 2l+m+n=4 are called 4th order or primary aberrations, those with 2l+m+n=6 are 6th order, or secondary, those with 2l+m+n=8 are 8th order or tertiary, and so on.

The entire term constitutes the peak aberration coefficient, equaling the value of the peak (coma, tilt), or peak-to-valley (spherical, astigmatism, field curvature) wavefront error at the Gaussian image point.  For primary spherical, it is paraxial focus, where the P-V wavefront error changes with (ρd)4, ρ being the height in the pupil normalized to 1. Thus, for unit radius, it is proportional to ρ4, which is its aberration function. For coma, the P-V wavefront error changes with (ρd)3, i.e. in proportion to ρ3, and so on. Strictly talking, these are Siedel aberrations, but best (diffraction) focus is shifted away from the Gaussian image point, as given in Table 3.

Sum in the last column is the order of the aberration term. Fourth order wavefront aberrations are called primary, and sixth order aberrations are called secondary aberrations (since this sum is smaller by one for transverse ray aberrations, primary aberrations are also called third order, and secondary aberrations fifth order).

For practical purposes, the front subscript is often omitted, with the aberration coefficient identified only as wnm. The subscript is sometimes entirely omitted, for simplicity, as it is in this text, where aberration coefficients for primary spherical aberration, coma, astigmatism and field curvature are denoted by s, c, a, p and g, respectively, with the corresponding peak aberration coefficients S, C, A, P and G. Aberration coefficients for secondary aberrations are more complex and are not specified, but secondary aberrations are described and illustrated for the three point-image quality aberrations, spherical, coma and astigmatism.

This nominal notation can also be used to identify the aberration with Zernike coefficients.

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 the 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 (primary) spherical aberration

FIG. 33 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 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=4-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 - '(x)=nxn-1 for the functions of (x)=xn type, and for '(x)=Σ'(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
)42 and
'
)=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.

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.

The relative wavefront error - either P-V or RMS - for any point between the two foci - the paraxial and marginal - in units of the error at the paraxial or marginal focus, is 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 the best focus by a factor 2.18.

FIGURE 34: Wavefront error of primary spherical aberration normalized to 1 at the paraxial and marginal focus, for the range of longitudinal aberration (LA) normalized to 2. The error is symmetrical with respect to the mid point between paraxial and marginal focus, with the rate of change becoming nearly linear for the portion of defocus range outside its central 1/4 (as the plot indicates, linear rate of change extends beyond the defocus range). The rate of wavefront error increase relative to the error at the mid point is approximated with xwmin~8Δ, where x is the ratio of increase, wmin the error at mid point, and Δ is the longitudinal separation from the mid point in units of the LA range normalized to 1, for simplicity. That gives x=2 for Δ=1/4 , or wavefront error doubled at 1/4 of the LA range from the mid point  (correct x value 2.18), and x=4 for Δ=1/2, or fourfold larger wavefront error  at 1/2 of the LA range from the mid point (correct x value 4).

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 /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).

FIGURE 35: 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. The 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 ρ/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.

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).

Aberration shown on FIG. 35 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.

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

3.5.3. Zernike expansion schemes       4.1.2. Lower-order spherical: aberration function

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