Info Fig. 1.22 (Up) Wavefronts converging at the Gaussian-, mean confusion-, and marginal-focus in the presence of primary spherical aberration w = 5Xp4, 5X(p4 — | p2), 5X(p4 — 2p2). (Center) Wavefronts of primary coma w = 4Xp3 cos 0, 4X(p3 — p) cos 6. (Down) Wavefronts of primary astigmatism w = 1.6Xp2cos26, 1.6Xp2(cos26 + 1), 1.6Xp2(cos26 + 2)

Fig. 1.22 (Up) Wavefronts converging at the Gaussian-, mean confusion-, and marginal-focus in the presence of primary spherical aberration w = 5Xp4, 5X(p4 — | p2), 5X(p4 — 2p2). (Center) Wavefronts of primary coma w = 4Xp3 cos 0, 4X(p3 — p) cos 6. (Down) Wavefronts of primary astigmatism w = 1.6Xp2cos26, 1.6Xp2(cos26 + 1), 1.6Xp2(cos26 + 2)

1.8.2 Seidel Aberration Modes - Elastic Deformation Modes

The active optics methods are used to aspherize mirrors and lenses whose surface shape is generated by elasticity and represented by one or several terms zn,m = An,m pn cos mQ of the power series (1.40b).

In this representation, the normalized image height parameter r) - or normalized field angle p - and successive powers in l do not explicitly appear. By regrouping in W the aberration terms of same integers n and m, we may obtain the An,m coefficients. For instance, if a mirror is elastically aspherized for simultaneously correcting a1,3,1 fj p3 cos Q and a3,3,1 f]3p3 cos Q, which represent the first two terms of linear coma, of respective orders KW = 3 and 5 in Table 1.2, the coefficient A31 should be

From the two integers n and m, we may define a zn,m term of (1.40b) as an optical surface mode included in the order KO of optical surface modes,

For simplicity, we will refer to each optical surface mode by abbreviating the same name as that of the aberration wavefront appearing in the lowest order KW of the wavefront function, followed by its order number KO .8

From (1.40b), both optics and elastic modes, which are used hereafter and in the next chapters for active optics, are denoted by the following abbreviated modes,

Cv 1 Sphe3 Sphe5

+ An r cos 9 + A3i r3cos 9 + A5i r5cos 9 + ••• Tilt1 Coma3 Coma5

+ A22 r2 cos 29 + A42 r4 cos 29 + ••• (1.52) Astm3 Astm5

From the Clebsch elasticity theory which concerns constant thickness plates, we will show that the modes belonging to the two lower diagonal lines of this optics triangular matrix can be easily generated and co-added by flexure of a circular plate (cf. Sect. 7.2). Also including the Sphe 3 mode, we will often refer to them as Clebsch-Seidel modes.

1.8.3 Zernike rms Polynomials

For wavefront testing analysis, the rms values of the deviation to a theoretical surface are usually required. From a Zernike circle polynomial, one may build its associated rms polynomial, the Zernike rms polynomials zn,m (Noll ). From the radial components Rn,m{p} defined by (1.40d), zn,m are given by zn,m{p, 9} = kn,mRn,m{p, 9}, kn,m = Vn + 1 X i ^ ^ m 0 (1.53)

The constant kn,m allows one to obtain the rms polynomial zn,m from the ptv polynomial Rn,m (Table 1.3).

8 N.B: We refer to the same abbreviations for the optical surface modes in zOpt as for the wavefront terms in W , but generally their meaning is different. For instance, the aberration wavefront term Astm3 is represented by the two terms f]2p2 and f)2p2cos29 (cf. (1.41) and Table 1.2). Now using the abbreviation of the optical surface modes in (1.52), this term is obtained by the co-addition of Cv 1 and Astm 3 modes. In the context, the words "wavefront aberrations term" and "optical surface mode," or more concisely "aberration term" and "optical mode," allow one to distinguish between them.

Table 1.3 Coefficients kn,m and Zernike polynomials Rn,m defining the rms polynomials zn,m up to the order KO — n+m-1 — 9 (after Hugot )

i

Mode

k

Rn,m{P, 6)

Ab