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Optical Engineering Science. Stephen RoltЧитать онлайн книгу.

Optical Engineering Science - Stephen Rolt


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order, n, of 4 and a polar dependence, m, of zero. Similarly, coma has a radial order of 3 and a polar dependence of one. Table 5.2 provides a list of the first 28 Zernike polynomials.

      (5.22)equation

      Unfortunately, a variety of different numbering conventions prevail, leading to significant confusion. This will be explored a little later in this chapter. As a consequence of this, the reader is advised to be cautious in applying any single digit numbering convention to Zernike polynomials. By contrast, the n, m numbering convention used by Born and Wolf is unambiguous and should be used where there is any possibility of confusion.

      5.3.3 Zernike Polynomials and Aberration

      As outlined previously, there is a strong connection between Zernike polynomials and primary aberrations when expressed in terms of wavefront error. Table 5.2 clearly shows the correspondence between the polynomials and the Gauss Seidel aberrations, with the 3rd order Gauss-Seidel aberrations, such as spherical aberration and coma clearly visible.

      The use of defocus to compensate spherical aberration was explored in Chapters 3 and 4. In this instance, for a given amount of fourth order wavefront error, we sought to minimise the rms wavefront error by applying a small amount of defocus.

equation

      Hence, without defocus, adjustment, the raw spherical aberration produced in a system may be expressed as the sum of three Zernike terms, one spherical aberration, one defocus and one piston term. The total aberration for an uncompensated system is simply given by the RSS of the individual terms. However, for a compensated system only the Zernike n = 4, m = 0 term needs be considered. This then gives the following fundamental relationship:

ANSI# N m Nn,m R(ρ) G(ϕ) Name
0 0 0 1 1 1 Piston
1 1 −1 images ρ sin φ Tilt X
2 1 1 images ρ cos φ Tilt Y
3 2 −2 images ρ 2 sin 2φ 45° Astigmatism
4 2 0 images 2ρ2 − 1 1 Defocus
5 2 2 images ρ 2 cos 2φ 90° Astigmatism
6 3 −3 images ρ 3 sin 3φ Trefoil
7 3 −1 images 3ρ3 − 2ρ sin φ Coma Y
8 3 1 images 3ρ3 − 2ρ cos φ Coma X
9 3 3 images ρ 3 cos 3φ
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