Astro890L6 - Zernike Polynomials The Zernike polynomials...

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Zernike Polynomials The Zernike polynomials were derived in the early 20 th century by Dutch physicist Frits Zernike (famous for the invention of the phase-constrast microscope for which he won the 1953 Nobel Physics Prize). The classic treatment of Zernike polynomials for the description of atmospheric turbulence is Noll, R.J. 1976, J. Opt. Soc. Am., 66, 207. This is the form most often seen in the astronomical literature, but beware that there are many other forms in use, and it can get quite confusing. The Zernike polynomials are a set of orthogonal polynomials defined on a unit disk. They can be expressed in either Cartesian (x,y) or Polar (r, θ ) coordinates, but the polar form is most common: , , 0 1 ( ) 2 cos( ), m 0 1 ( ) 2 sin( ), m 0 1 ( ), m=0 m jeven n m jodd n jn Zn R r m R r m R r θ =+ where () / 2 2 0 (1 )( ) ! ![( ) / 2 ]! [( ) / 2 ]! s nm m ns n s Rr r snm s nm s = −− = +− The values of n and m are always integers, and satisfy mn and even −= . The index j is the mode ordering number, and is a function of n and m.
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Astro890L6 - Zernike Polynomials The Zernike polynomials...

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