R aQt i

dr dr

rQr+r(rQr)dr de

Fig. 3.2 External load and shearing forces providing the equilibrium of a plate element in z-direction z = znm = Anmrn cosmO, with m < n, m + n even > 2, (3.6)

where n and m are integers. Given the condition m < n, the series development of such optics modes generates the terms of a triangular matrix. For low-order modes we use the simple suffix denotation nm instead of n, m. With m + n = 4, the three terms coming after the dioptrics are the third aberration modes z40, z31, and z22, i.e. spherical aberration, coma, and astigmatism respectively, also denoted Sphe 3, Coma3, and Astm3.

The following derivative elements d2z

1 d7 1 d2z

r dr r2 dO2

allow one to derive from (3.1a, b, c) the bending and twisting moments as

Mr = [n(n- 1) + v(n -m2)] D(r,O)Anmrn-2cosmO, (3.8a)

Mt = [n-m2 + vn(n - 1)] D(r, O)A„mrn-2cosmO, (3.8b)

Mrt = m (n - 1)(1 - v) D(r,, O) Anmrn-2 sin mO. (3.8c)

Restricting to Vtds of rotational symmetry, which is an interesting case for practical applications, let us consider a rigidity D(r,, O) = D(r) of the following analytical form [14]

where the unknown constant coefficients AO, Ai, and at are only functions of the external loading. Denoting

and substituting the rigidity in Eq. (3.9), the moments are represented by

Mr = [n(n - 1) + v(n - m2)] [AOrn-2lnr + £Airn-2-ai] cosmQ, (3.11a) Mt = [n - m2 + vn(n - 1)] [A0rn-2lnr + £A'rn-2-ai] cosmQ, (3.11b) Mrt = m(n- 1)(1 - v) [AOrn-2lnr + £Airn-2-ai] sinmQ. (3.11c)

After calculation, the radial shearing force, derived from the equilibrium (3.3), is

Qr = -[n -2)(n2 -m2) AOrn-3lnr cosmQ - [n(n - 1) + v(n - m2)] A'0 r"-3 cos mQ

+ [ n(n - 1) + v(n - m2)] at} A'lrn-3-ai cos mQ. (3.12)

and the tangential shearing force derived from (3.4) is

+ £ m [n2 - m2 - (n - 1)(1 - v) a] A' rn-3-ai sin mQ. (3.13)

Finally, the equilibrium equation (3.5) provides the external load applied to the surface of the plate. After calculation, the load is represented by q = (n2 - m2) [ m2 - (n - 2)2 ] AO rn-4ln r cos mQ

- [n(n - 2)(2n - 1 + v) - m2(2n - 3 - v)] AOrn-4cosmQ ( + 1"(" - 1) + v(n - m2)] a2N

- [ n(n - 2)(2n - 1 + v) - m2(2n - 3 - v)] ai cos mQ.

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