Zernike polynomials

Since A x,y) can really be any function bounded by the aperture of the system it is best to use as general description as possible. One such description of this function is to expand A x, y) = A p, 0) about (p, 0) in an infinite polynomial series. One set of polynomials that are frequently used are Zernike polynomials. Thus one can write A(p, 0) = Y.m,n CmnFmn p, 0). 

The lowest order Zemikes are tabulated in Table 1 along with their common optical name. The next figure shows fhe first 12 Zernike polynomials with a pseudo-color graphical representation of their value over a unit circle. In the plots, blue values are low and red values are high. 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

Noll, R., 1976, Zernike Polynomials and atmospheric turbulence, JOSA.A 56, 207... 

The aberration function is decomposed into sums of mathematical functions, making possible the classification of the aberrations. For this decomposition, it has proven convenient to use particular sets of orthogonal polynomials, the most common of which are the Zernike polynomials. With the Zernike polynomials, each polynomial represents a particular type of aberration, having its own characteristic effect on imaging and optical lithography. ... 

The Zernike polynomials are functions of the polar coordinates (r, 0) of positions r and angle 6 in the exit pupil. Because the same point in the exit pupil is specified by 6 and (6 -1-360 deg), it is convenient for the polynomials to be expressed as functions of sin(u0) and cos(u0), where n is an integer." Although the Zernike polynomials comprise an orthogonoal polynomial series with an... 

History. Starting from the ID point statistics of Zernike and Prins [116] J. J. Hermans [128] designs various ID statistics of black and white rods. He applies these models to the SAXS curves of cellulose. Polydispersity of rod lengths is introduced by distribution functions, / , (,r)108. Hermans describes the loss of correlation along the series of rods by a convolution polynomial . One of Hermans lattice statistics is namedparacrystalby Hosemann [5,117]. Hosemann shows that the field of distorted structure is concisely treated by the methods of complex analysis. A controversial subject is Hosemann s extension of ID statistics to 3D [63,131,227,228],... 

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