4.1.3. Higher-order spherical aberration

PAGE HIGHLIGHTS
• Origin in conic surface equation   • Balancing secondary with primary spherical   • Longitudinal and transverse
• Pure and balanced secondary spherical

FIGURE 37 TOP: (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 to the zonal height in its expression, it is referred to either as 5th order transverse ray aberration term, or 6th order on the wavefront for the transverse and wavefront aberration form, respectively (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 on a single surface, 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 surface aberration.
(B) - The form of wavefront deviation for pure secondary spherical at the best focus is very similar to that for the primary aberration. Its function is ρ6-0.9ρ2, which for its first derivative 6ρ5-1.8ρ (i.e. 6ρ4-1.8 after reduced by ρ) set to zero gives the value of ρ with the maximum deviation as ρ=0.30.25=0.74. When significant, this higher order residual needs to be minimized by correcting the surfaces in their lower-order approximation so that they better approximate the actual 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. Secondary spherical can be either removed by applying aspheric term on a surface, or minimized - i.e. "balanced" - with primary spherical aberration, keeping surfaces spherical (note that the relevant form in balancing for both aberrations is the one balanced with defocus, i.e. best focus aberration). Balancing can be described as "adding" similar amount of lower-order aberration of opposite sign to that of the higher-order. Specifically, the P-V wavefront error of the balancing primary spherical is 0.885 of the P-V wavefront error of secondary spherical, both at their respective best focus location. Balancing is usually accomplished by an appropriate change in the radius of one or more surfaces. Since the two forms of deviations are only similar, not identical, the aberration cannot be cancelled out, but is significantly reduced, by a factor of ~0.15 in the RMS wavefront error. Graph at left shows minimizing 6th order spherical residual in a wider context, starting with the error at paraxial focus. The first two terms in the relation for the balanced 6th/4th order aberration are a sum of the respective errors at the paraxial focus, i.e WFE at that location. The third, defocus term, represents the shift to the best focus location.
(C) - The minimized wavefront deviation form (expanded in the width for clarity) has an additional curl toward the edge. Compared to primary spherical at best focus, it has greater deviation above 91% of the radius, and below 64%, but the overall deviation is smaller due to its smaller deviation over the 64-91% radius section. Since the P-V/RMS ratio of an aberration is given by a sum of the (absolute) values of its Zernike aberration term for the two values of ρ corresponding to the maximum wavefront deviation from zero mean and unit coefficient (znm=1) value, it is here given by a sum of absolute values of √7(20ρ6-30ρ4+12ρ2-1)z60 for ρ=0 and ρ=1 (as can be seen above, under C), i.e. 2√7. 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, i.e. the RMS error is smaller by a factor of ~0.63 than for primary spherical (and slightly larger for pure secondary spherical) aberration). 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.
BOTTOM: As with spherochromatism in general, only a single wavelength can be fully optimized, or in the case of balancing 6th and 4th order aberration optimally balanced, i.e. fully minimized. Other wavelengths will be either balanced only partially, or will have the two aberrations add up (being similar in form, when adding up, the sum is roughly arithmetic). Below are illustrated stages of partially balanced 6th order spherical for both, longitudinal aberration (LA) and the corresponding wavefront.

The top shows five stages of balancing, starting with pure secondary spherical aberration and adding one quarter of the full primary spherical needed for optimum balancing at the time. Bottom inset starts with the fully balanced form and deducts one quarter of the full amount of 6th order spherical aberration in it, ending with the full balancing 4th order alone (generated by OSLO). Thus, top shows the full amount of balancing higher order with various amounts of primary spherical, and the inset shows the full amount of balancing spherical with various amounts of higher-order spherical. The number columns show the corresponding P-V and RMS wavefront error (WFE) for the P-V error of the fully balanced aberration normalized to unity, for best (B), paraxial (P), marginal (M) and the point of maximum defocus (MD, with respect to paraxial focus, when different from the marginal focus), from top to bottom. Since the two forms are generally similar, but somewhat different, the LA graphs and wavefront shapes for partial balancing are also somewhat different when the 6th order aberration is dominant (top) and when the 4th order is dominant (bottom inset). However for the same numerical proportion of one and the other one, the best focus error and the extent of LA remain at a similar level (comparable mixed aberrations are starting from either full order toward the optimally balanced form, e.g. second from left on top to the forth in the inset). The only significant difference is that the error is the largest at the marginal focus for the former, and at the paraxial focus for the latter. Since the reduction rate at the optimum balance is by a factor of 4.6, 6 and 3.9 for the P-V WFE, RMS WFE and LA extent, the relation between the LA and P-V WFE, given by LA/64F2 (exact for the 4th order and nearly exact for the 6th order), gradually transforms to approx. LA/76F2 at the optimum balance, with the RMS error smaller by a factor of 4.44. The rate of change is somewhat slower up to the level when one of the aberrations is less than a half of the other one, increasing after that. For the points outside the LA range, the P-V WFE error increases roughly by the amount of defocus l from the paraxial or marginal focus, given by W=l/8F2. Note that the specific full magnitude of 6th order aberration here is 0.28 waves P-V, hence with the optimally balanced 6th+4th order of 0.06 wave P-V. Note that OSLO somewhat deviates from the values implied by the theory - 4.3 times smaller P-V and 6.7 times smaller RMS for the minimized vs. pure 6th order spherical - giving somewhat larger reduction in the P-V, but smaller in the RMS error; the difference probably comes from the best focus criterion used by the software.

FIGURE 38: Characteristics of pure secondary (5th order transverse, 6th order wavefront) spherical aberration (top) and its minimized form when balanced with primary spherical (bottom), the latter being the one usually referred to as 3th order (transverse), or 4th order (wavefront) spherical aberration. Wavefront errors are normalized to 1 at the pure secondary spherical aberration (PSSA) error at paraxial focus (actual error 0.4 wave P-V), and the balanced longitudinal aberration (LA) graph is scaled 2.5 times the arbitrary unit on the PSSA LA graph (note that the location of paraxial focus remains nearly unchanged after balancing, but longitudinal defocus shifts to the opposite side of it). 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 0.72 zone focus with the balanced form of aberration). Note that normalizing LA to 2 for pure and balanced secondary spherical is only for the convenience of making it comparable to primary spherical. For given LA, defocus P-V wavefront error is 3 times larger than the s.a. P-V wavefront error at paraxial focus for pure spherical (thus actual Λ=3), and 1.33 times for the balanced form (Λ=1.33).
PSSA wavefront data show overall similarity with primary spherical, main exception being that the error at the paraxial focus is somewhat smaller, and at the marginal focus somewhat larger relative to the best focus error, than with primary spherical (note that the P-V WF error ratio at the paraxial vs. best focus for pure secondary spherical is 2.33, not 2.56 as implied by OSLO, probably due to less than perfect accuracy, or different criterion in determining the best focus; related to it, it is to expect that the RMS error reduction at the best focus is somewhat greater than implied and, indeed, it is about 3.6 times smaller, rather than nearly 3.5, as implied by raytrace). This extends to the size of corresponding geometrical blurs. The smaller error at paraxial focus, relative to that at the best focus location, means that less of defocus error is needed to balance PSSA itself (analogous to balancing primary spherical with defocus, not to confuse with balancing PSSA with primary spherical and defocus in order to have it minimized). However, since both are proportionally smaller, the aberration function for best focus location is proportional to ρ6-ρ2 (it gives the smallest P-V error, with the shape similar to primary spherical, with zero deviation at the center and the edge, with the maximum deviation at ~0.7 zone; however, the smallest RMS error is achieved with slightly less defocus, ρ6-0.9ρ2, and edge deviation of nearly 30% of the maximum peak deviation at 0.74 zone, opposite in sign). The PSSA LA is nearly identical to that for primary spherical for given P-V wavefront error at the best focus. Since the secondary aberration function ρ6-0.9ρ2 generates larger maximum deviation than the primary function ρ4-ρ2, the latter is multiplied by a 1.5 factor of 1.5, i.e. 1.5(ρ4-ρ2) to bring its P-V to the needed proportion vs. secondary aberration P-V to result in the minimum RMS error (specifically, the primary aberration P-V wavefront error is smaller by a factor of 0.885). Thus the wavefront aberration function for balanced secondary spherical aberration is

ρ6-0.9ρ2-1.5(ρ4-ρ2)=ρ6-1.5ρ4+0.6ρ2.

As the plot implies, its maximum is for ρ=1. Setting first function derivative, 6ρ5-6ρ3+1.2ρ, to zero does not give direct solution for the values of ρ for which the tangent of the function is vertical (points of inflection), but solving it by dividing the derivative by 6ρ and reducing it to quadratic function by dividing it with ρ2 gives the solutions as ρ2=0.5±0.051/2, i.e,
ρ1=(0.5-0.051/2)1/2=0.526 and ρ2=(0.5+0.051/2)1/2=0.851.
However, the balanced 5th/3rd (i.e. 4th/6th) order spherical is a distinctly different form of spherical aberration. It is already indicated by the shape of its LA graph, where the paraxial and marginal foci nearly coincide. The wavefront error is nearly six times smaller at the best vs. paraxial focus, the latter having twice the error of the marginal focus. As the numbers show, fully minimizing pure higher order spherical aberration by balancing it with primary spherical, results in 4.6 times smaller P-V, and six times smaller RMS wavefront error. The ray spots are much more even in size, but the best focus spot is significantly larger for given RMS wavefront error: with this specific error (0.0077 wave RMS) best focus spot diameter with primary spherical would have been less than 1/3 of the Airy disc's, which makes it nearly three times smaller. The reason is that the dense central portion of the balanced aberration blur, where most of the rays are (bottom corner, right), is of similar size in both aberration forms, and how wide out are spread remaining 20%, or so, of the rays matters little. The significance of uneven rate of aberration change toward the opposite sides of best focus, indicates that the conventional wave of star testing at nearly equal amounts of defocus either side from best focus will be less reliable than with primary spherical aberration.

While the primary and pure secondary spherical aberration at best focus location have very similar effect with respect to image quality, balanced secondary spherical differs more than negligibly even at relatively small error levels. Plots at left show OSLO output for 0.0745 wave RMS wavefront error of these three forms of spherical aberration. Due to its double bright ring, balanced secondary spherical encircles about as much energy at 3λF radius as the other two do at 2λF. It also has somewhat more energy spread out in the wider area beyond this point. The corresponding Strehl is also somewhat lower, at 0.76 vs. 0.80. The contrast transfer (shown only for primary and balanced secondary, for clarity) is similar, on the average, except the frequencies close to the resolution threshold, which is lower for the balanced form, the consequence of its steep edge wavefront curl being similar to that of turned edge.