4.1.3. Higher-order spherical aberration

Considering the importance of center field correction, higher-order spherical aberration - also called secondary, being result of the second term in conic surface equation - does have some significance in certain types of amateur telescopes with typically more strongly curved optical surfaces. In particular, in most apochromatic refractors, as well as Maksutov-Cassegrain (MCT) and Schmidt-Cassegrain (SCT) telescopes. Since the Schmidt corrector can directly compensate for higher-order terms as well, they are not a factor in the properly made SCT.

Putting Schmidt-type aspheric on one or more surfaces of an apochromat or MCT is not a practical option; they also generate significantly more of higher-order spherical aberration than a comparable SCT. Higher-order spherical in the MCT can be very effectively lowered by moderately aspherizing primary mirror, which allows to relax corrector's radii. However, all-spherical MCT, as well as apochromats, may have significant level of residual higher-order spherical aberration - 5th order transverse (ray), or 6th order on the wavefront. It has to be minimized by balancing it with a similar amount of lower-order spherical aberration and defocus. This is accomplished by adjusting the radii of curvature. Near-optimum reduction of the 6th order aberration P-V wavefront error by balancing is by a factor of ~0.22, with the PV-to-RMS wavefront error ratio also reduced, approximately by a factor 0.78. In all, the RMS error is reduced approximately by a factor of 0.17 (FIG. 23B).

Higher-order spherical aberration is merely a consequence of using (close) approximation of the conic surface, instead of the actual surface, in calculating the aberration. General conic surface is described by an expansion series, determining its sagitta (depth at the center) as z=(d2/2R)+[(1+K)d4/8R3]+[(1+K)2d6/16R5]+..., with d being the surface half-diameter, R the radius of curvature and K the conic constant. The first term alone accurately describes paraboloid (K=-1) which, consequently, doesn't have a higher-order aberration component. The second term describes surface approximations for K≠-1, producing lower-order (3rd order transverse ray, 4th order on the wavefront) spherical aberration for object at infinity (and any object location except that coinciding with the conic surface's geometric focus); the third term in the series adds a correction to that surface approximation, and with it an additional, relatively small amount of the aberration, the already mentioned 5th order transverse, or 6th order on the wavefront.

The order designation is generally based on the sum of exponents in surface half-diameter d and field angle α. Since spherical aberration doesn't change with the field angle, the exponent in α is zero, and the second term in the expansion series for the surface - and thus for the wavefront as well - is of 4th ordert; since the exponent in d for the transverse geometric aberration resulting from the 4th order surface approximation is 3, it is a 3rd order transverse aberration.

Terms beyond third in the expansion series are insignificant in amateur-size telescopes.

FIGURE 23: (A) - Higher-order spherical aberrations of reflecting (shown) or refracting conic surface result from corrections to the lower-order surface approximation. The higher-order term that can be significant in amateur telescopes is following the lower-order term in the expansion series. According to the power of zonal height in its expression, it is referred to either as 5th order transverse ray aberration term, or 6th order on the wavefront (by the same token, longitudinal aberration is 2nd order in its lower-order form, and 4th order in the next higher-form). In the presence of the lower-order aberration, it adds to both, its ray (LA=longitudinal aberration, and TA=transverse aberration, shown for the paraxial focus plane) and wavefront error (W). With the lower-order aberration cancelled, it remains as a residual aberration. (B) - When significant, it needs to be minimized by correcting the surfaces in their lower-order approximation so that they better approximate the higher-aberration term surface profile. This is typically the case in telescopes using strongly curved surfaces, such as meniscus corrector in Maksutov-Cassegrain, or lens objective in apochromatic refractors. It is described as "adding" similar amount of lower-order aberration of opposite sign to that of the higher-order. Since the two forms of deviations are not identical, the aberration cannot be cancelled out, but is significantly reduced, by a factor of ~0.2. (C) - The minimized wavefront deviation form has an additional curl toward the edge. For given P-V error, compared to the lower-order wavefront deviation, it has roughly similar average amplitude from zero mean (dotted line). These extreme deviations, however, affect relatively small area of the wavefront, thus also have smaller effect on the wavefront RMS error. Consequently, for given P-V wavefront error, the RMS wavefront error for balanced 4th and 6th order spherical aberration, given by RMSB=P-VB/√28, is smaller by a factor of ~0.63. Note that this balanced form of spherical aberration is often called simply "higher (5th or 6th) order" spherical, which may be confused with the higher-order spherical directly related to the surface term. For this reason, it is better to use the term "balanced 6th/4th-order spherical" for the former.

For object at infinity, reflecting surface forming a perfect wavefront is the paraboloid. Since surface error doubles in the wavefront for paraxial focus, the lower-order P-V wavefront error with respect to the reference sphere centered at paraxial focus is given by double the second term, or W4=(1+K)(ρd)4/4R3. Likewise, the remaining 6th order P-V wavefront aberration is W6=(1+K)2(ρd)6/8R5, with ρ being the height in pupil normalized to 1 and d the pupil (aperture) radius. At best focus, as mentioned, the P-V wavefront error for the 4th order spherical aberration is smaller by a factor of 0.25. Due to somewhat different form of deviation, best focus error for the 6th order aberration (balanced with defocus) is reduced somewhat less, by a factor of ~0.4; location of the best focus is also shifted from midway between the marginal and paraxial foci to somewhat closer to paraxial focus.

When 6th order spherical aberration is minimized by combining with the lower-order form, the resulting aberration - the 6th/4th order spherical - is also balanced by certain amount of defocus needed for the shift from paraxial focus to the location with the highest peak diffraction intensity. In other words, it is balanced by both, the lower-order form and defocus (this balancing should not be confused by balancing of the 6th order spherical alone, which is only done by defocusing). As a result, the aberration function changes from WS6=S6ρ6 for the P-V wavefront error of pure 6th order spherical aberration paraxial focus, where S6 is the peak aberration coefficient for 6th order spherical aberration S6=(1+K)2d6/8R5, to WS(6/4)=S6(0.88ρ6-1.32ρ4+0.528ρ2-0.044) for the peak wavefront error of balanced 6th/4th order spherical aberration. The term in ρ4 is for balancing with lower-order spherical, and the term in ρ2 for balancing with defocus. The constant term quantifies the relative peak aberration for ρ=1 or ρ=0, which are identical in magnitude, but of opposite sign (as illustrated on FIG. 23C). Compared to the relative P-V wavefront error of pure (unbalanced) 6th order spherical (1 for ρ=1), the fully balanced form's P-V error is smaller by a factor of 0.088.

The transverse 5th order spherical aberration for reflecting conic surface, as the difference between height of marginal ray reflected from 3rd and 5th order surface approximation in the paraxial focus plane (FIG. 23A) is given by TA5=3(K+1)(K+3)(ρd)5/8R4. Hence, it is smaller than 3rd order transverse aberration, whose blur radius is TA3=(K+1)(ρd)3/2R2, by a factor of 3(3+K)/64F2. Looking at the wavefront-error-to-transverse-aberration ratio, we see that the one for the lower-order aberration is independent of the conic, with the paraxial focus transverse error being larger than the wavefront error by a factor of 16F.

For the higher-order aberration, the wavefront error and transverse aberration doesn't scale evenly with the conic. This means that the aberration changes its form somewhat with the conic, with the paraxial transverse aberration being greater than the wavefront error approximately (due to the best focus wavefront error varying slightly with respect to the error at paraxial focus) by a factor of 16(K+3)F/3(K+1). Given P-V wavefront error, the 5th order transverse paraxial blur for spherical reflecting surface (K=0) is, as expected due to the similar form of wavefront deformation, nearly identical to the lower-order aberration paraxial blur; the corresponding RMS wavefront error is only slightly lower in the former. Best focus blur, being located closer to the paraxial focus, is somewhat larger for the 5th order aberration.

The transverse blur for given P-V wavefront error of balanced 4th and 6th order spherical aberration is larger than the 3rd order transverse aberration at best focus location, by a factor of ~1.8, mainly due to the strong deviation at the edge of the wavefront. It, however, affects relatively small wavefront area. The RMS wavefront error for the balanced 5th order aberration is actually smaller by a factor of ~0.75, due to relatively large portion of the P-V error being caused by the central deformation, small in area (FIG. 23C). Contrary to the 3rd order aberration, the paraxial blur for balanced 3rd and 5th order aberration tends to be smaller than the best focus blur. This is due to the paraxial and marginal foci nearly coinciding for optimally balanced 5th order aberration (FIG. 23B, bottom; also, FIG. 24 below).

FIGURE 24: Ray spot plot to the left illustrate the difference in transverse aberration between unbalanced and balanced form of 5th order spherical aberration at "diffraction-limited" level (0.80 Strehl) for selected points of defocus. Longitudinal defocus Λ is normalized to 2, with Λ=0 being the location of paraxial focus, and Λ=2 location of the farthest defocus point (it coincides with the marginal focus point for the unbalanced form, and with 0.71 zone focus with the balanced form of aberration).

Blur sizes and structure indicate two distinctly different form of spherical aberration. While the unbalanced form is more similar to a 3rd order spherical aberration in regard to the blur size and structure, as well as, somewhat less, in regard to the corresponding P-V and RMS wavefront errors, balanced 5th/3rd order spherical has significantly larger nominal blur for given RMS wavefront error at best focus: it is as much as eight times larger than the Airy disc (compares to 3.28 and 4.6 times larger blur with lower-order and unbalanced higher-order spherical, respectively). The scarcity of widely scattered rays, however, indicates relatively small wavefront area contributing to ray scatter, and correspondingly small loss of energy from the Airy disc.

Correction of the 5th order spherical aberration is easy in instruments using Schmidt corrector. It is completely eliminated by adding 5th order aberration parameter to the corrector's aspheric curve (appropriately overcorrecting it with the 3rd order aberration alone would significantly reduce the 5th order aberration error; in, say, 200mm ƒ/2 Schmidt camera, from 0.83 wave P-V and 0.227 wave RMS of 6th order spherical to ~0.19 wave P-V and less than 0.05 wave RMS error of balanced aberration).

Higher-order spherical is more of a problem in larger apochromatic refractors and, particularly, in instruments using strongly curved meniscus correctors. As the wavefront error of 6th order aberration increases inversely to the 5th power of surface radius, meniscus corrector can generate enormous amount of higher spherical aberration, that cannot be minimized to an acceptable level. This limits relative apertures of mirrors used in these instruments to ~ƒ/3, or slower, even in moderate to small apertures. An effective - although adding to the expense - solution to this problem is aspherizing the primary, which then requires weaker corrector, with significantly lower higher-order spherical aberration contribution.

◄ 4.1.2. Lower-order spherical: aberration function ▐ 4.2. Coma ►

Home | Comments