Professor Neil Charman concludes his discussion of wavefront aberration analysis with a few cautionary words. Equations and Tables from Part 1 (optician, July 8, page 24) are referred to in this week's optican, however, to view the equations as in the hard copy of the article, click below
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THE FULL SPHERO-CYLINDRICAL CORRECTION
Having considered the spherical defocus, we can now turn to the problem of deriving the full sphero-cylindrical correction from the total wavefront errors. If we examine the Zernike polynomials in Table 1 we see that
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Since sin2<03B8> = +1 when <03B8> = 45 or 225,
= -1 when <03B8> = 135, 315, with zeros at <03B8> = 0,90,180, 270 degrees, and there is a <03C1>2 dependence, we can see from the same basic sag formula ideas as were used in the spherical defocus case that this polynomial represents a crossed-cylinder wavefront with axes at 45/135. Taking account of the fact that many of the higher-order polynomials (Z4-2, Z6-2 etc) also contain terms in <03C1>2sin2<03B8> and proceeding by analogy with the spherical case by considering the curvatures in the principal meridians, the full, paraxial, crossed-cylinder correction in the 45 and 135 meridians can be found as:
J45 = -(2Ã6.C2-2 - 6Ã10.C4-2 + 12Ã14.C6-2 + still higher order contributions)/rmax2....(5)
Here J45 is the Jackson crossed-cylinder power. Use of only the first, C2-2, term in the bracket would give the least-squares value of J45.
Clearly the other second-order polynomial
Z22 = Ã6<03C1>2cos2<03B8>
is also a crossed cylinder, but its axes are at 90/180, since cos2<03B8> = + 1 when <03B8> = 0, 180, and = -1 when <03B8> = 90, 270. The corresponding paraxial crossed-cylinder power, including the contributions of higher as well as second-order polynomials, is:
J180 = -(2Ã6 C22 - 6Ã10 C42 + 12Ã14 C62 + still higher order contributions)/rmax2 .....(6)
As was shown by several authors,13,14,17 the conventional sphero-cylindrical correction in the form S/C x <03B1>, with a negative cylinder, is then given by:
C = -2Ã(J1802 + J452) .............................(7)
S = M - C/2 .......................................(8)
<03B1> = [tan-1(J45/J180)]/2 ............................(9)
If J180 is zero, the equation for <03B1> gives an indeterminate result. In this case, if J45<0, the <03B1> = 135, and if J45