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How to Calculate Zernike Polynomials

Zernike polynomials are an orthogonal set of functions that can be used to represent the wavefront error of an optical system. They particularly come in handy for situations with circular apertures, covering the majority of optical systems. There are many formulations of the Zernike polynomials, and they all do the same job. The most useful formulations are orthonormal when the value for each coefficient represents the contribution of that term to wavefront error.

Instructions

- 1
  Select an order for the Zernike polynomial of interest. The order is represented by two integers, n and m, where m can only be as large as n. The choice is entirely up to you, although values of n and m higher than about 4 are only important in very special situations.
  
  As an example, you could start with: n=3, m=1.
- 2
  Calculate the normalization coefficient, N(n,m). The normalization coefficient is given by
  
  sqrt (2(n+1) / (1 + delta(m,0)); where delta(m,0) is 1 when m=0, and zero everywhere else.
  
  For the example: N(3,1) = sqrt (2(3+1) / (1 + 0)) = sqrt (8).
- 3
  
  When Zernike came up with his polynomials all calculations had to be done by hand --- with modern computers it is child's play.
  
  Calculate the radial portion of the Zernike polynomial. The radial portion is given by
  
  R(n,m,rho) = Sum (from s=0 to s=(n-m)/2) of {[(-1)^s x (n-s)! / (s! ((n+m)/2 - s)! ((n-m)/2 - s)!)] x rho^(n-2s)}.
  
  For the example, this becomes:
  
  Sum (from s = 0 to s = 1) of
  
  {[(-1)^s x (n-s)! / (s! ((n+m)/2 - s)! ((n-m)/2 - s)!)] x rho^(n-2s)}
  
  which equals
  
  {[3!/((2! 1!)] x rho^3 + [(-1)(2!)/1!] x rho}
  
  which equals
  
  (3rho^3 - 2rho).
- 4
  Calculate the angular portion of the Zernike polynomial. This is given by cos(m x theta).
  
  For the example, this is simply cos(theta).
- 5
  Multiply all the separate portions of the polynomial together. This is N(n,m) x R(n,m,rho) x cos(m x theta).
  
  For the example: N(3,1) x R(3,1,rho) x cos(theta) = sqrt(8) x (3rho^3 - 2rho) x cos(theta). This example happens to correspond to an optical aberration called coma.

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