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Zernike polynomials: Mathematical Expression
Zernike polynomials used in Sensoft  our wavefront sensor software: expressions for the first 7 terms  
Zernike polynomials are generally expressed in terms of the normalized radius r of the pupil, and the azimuthal angle φ , by the following expression : r^{n} cos
(m φ + φ_{0}) This is the general expression in terms of the Seidel polynomials. The Annular Zernike polynomials, on the other hand, involve aberration balancing, in which aberrations of a lower order are combined with those of the higher order for reducing the wavefront error. For example, the expression for 3rd order spherical aberration also contains a defocus term. They also take into account the effect of the annulus e of the optical element. φ_{0} is
the zerooffset : it gives the orientation of the particular aberration with respect to the xaxis.


Aberration  Annular Zernike polynomials  Seidel polynomials 
Defocus (n=2,m=0) 

Tilt (n=1,m=1) 

Coma (n=3,m=1) 

Spherical 3rd
order (n=4,m=0) 

Astigmatism (n=2,m=2) 

Triangular
Coma (n=3,m=3) 

Quadratic
Astigmatism (n=4,m=4) 