Hi Mark,

It sounds right for the Schmidt profile on the front surface, light traveling left to right, the first (corrector radius) term should be numerically positive, and the aberration terms negative. For the rear surface, reversed.

There is no good reason for the (correctly typed in) formulae not to give correct values. If you look at FIG. 168 left, it is a graphic illustration of the parameters in the Schmidt profile equation. The first term is the corrector r.o.c. sagitta, and the aberration terms are effectively the (inverse) wavefront deformation "magnified" by the factor of light slowdown inside glass (so it is not d^4/4R3 s.a. coefficient for air, rather d^4/4(n'-n)R^3 for the first aberration term). The corrector radius term is merely used to flex this glass shape based on the aberrated wavefront, moving the neutral zone along the radius.

From this, it is obvious that needed depth of the corrector is directly related to the P-V wavefront error at chosen focus location, given by

W=xd^4/4R^3,

with x=[1+1.875N^2(-2+2N^2)]

being the relative error for the specific focus (i.e. neutral zone) location in units of the P-V error at paraxial focus, where "d" is the aperture radius, R the mirror r.o.c. and N the relative height of the neutral zone in units of aperture radius.

Maximum profile depth - the one at the neutral zone - is simply W/(n'-n). It gets more complicated with higher-order terms, but since the primary aberration term accounts for nearly all of it, it is a good checkup.

For instance, a 300mm f/3 Schmidt, so d=150, R=-1800, N=sqrt.0.5 (for the minimum spherochromatism and depth), Rc=-(n'-n)R^3/(Nd)^2 - taking the rear surface so the entry media (glass) index for BK7 and e-line is n=1.518 and the exit media index n'=1 - Rc=-269,000, and the first aspheric parameter (often called "coefficient")

A1=1/4(n'-n)R^3=8.28^-11.

These give the profile as z=(1/2Rc)(rd)^2+A1(rd)^4 - and taking the 0.707 relative pupil height of the neutral zone for the value of normalized in-pupil height "r" as the location of maximum depth - z=-0.021+0.0105=-0.0105mm.

The P-V wavefront error at best focus - which applies to this neutral zone position - is given by W=D/2048F^3 which, for D=300mm and F=3 gives W=0.0054mm. Hence, the needed corrector depth at 0.707 zone is z=0.0054/0.518=0.104mm, in good agreement with the formula.

After the primary spherical is corrected, the next higher order term is added to the profile curve. Since it is a 6th power function, it becomes significant only toward the edge. The effect of adding it is illustrated at the graph, grossly exaggerated, since it is usually much smaller than primary spherical. Effectively, it raises the edge, but since glass cannot be added, this means that the inner portion of corrector, from 0.7 zone to the center, need to be evenly deepened as much as it is the value of the second aberration term for r=1 (solid blue vs. dashed red). The deepest zone shifts just slightly below 0.707 radius, but that is insignificant. As with primary spherical, the actual P-V wavefront error is enlarged by a factor of 1/(n'-n) in the glass profile, which means that the paraxial focus error is by a factor (n'-n) smaller than the secondary aberration term itself (for r=1). However, since the amount of secondary spherical at its best focus is 2.6 times smaller than at the paraxial focus - which is the focus of correction for secondary spherical - leaving it uncorrected results in as much smaller secondary spherical, i.e. smaller by a factor of (n'-n)/2.6 than the secondary aberration term.

Vla