Table 3.22 gives the radial polynomials derived from (3.431) up to degree 8. It should be remembered that each radial polynomial with m > 0 has two components depending on p cos ^ and p sin giving the number of linearly independent polynomials of degree < n as 1 (n +1)(n + 2), as indicated above.

Resolving in this way, we derive from Table 3.22 the systematic ordering of radial polynomials given, following Dierickx [3.122], in Table3.23. This comprises the terms in a triangle of Table 3.22 with base extending from n =1 to n =10 and apex at n = m = 5. By analogy with the Characteristic Function of § 3.2, the three lowest terms are the Gaussian effects. Terms with m = 0 and n > 4 are termed spherical aberration; with m =1 and n > 3 coma; with m = 2 and n > 2 astigmatism; m = 3 triangular; m = 4 quadratic; m = 5 five-fold. The classical order numbers are given by (n + m — 1).

From the normalization of (3.430), all functions are unity if p = 1. Integration of the functions confirms the orthogonality condition, that the integral be zero [3.123], is fulfilled. This is physically obvious for all the terms in Table 3.23 with m = 0 since they depend on cos m^> or sin m^>. Setting U = m^>, then J cos m^>d^> = 1 sin U, giving a zero integral as in the case of the basic polynomial p cos Similarly, it is easily shown that the radial polynomials represent a balance of classical terms up to the defined degree which minimizes the rms wavefront value 1 2. Thus the focus term R2 (No. 3) balances

The rms wavefront error is given by r2 — 21 1/2

i.e. the square root of the [ mean of the square of the wavefront aberration minus the square of the mean wavefront aberration 1 (see § 3.10).

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