Zernike polynomials: Mathematical Expression

Zernike polynomials used in Sensoft - our wavefront sensor software: expressions for the first 7 terms
 
Zernike_convention Zernike polynomials are generally expressed in terms of the normalized radius r of the  pupil,  and  the  azimuthal  angle φ ,  by the following expression :  rn cos (m φ + φ0)  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.   φ0   is the zero-offset : it gives the orientation of the particular aberration with respect to the x-axis.
 
  Aberration Annular Zernike polynomials Seidel polynomials
  Defocus
  (n=2,m=0)
Defcus_annular Defocus_seidel
  Tilt
  (n=1,m=1)
Tilt_annular Tilt_seidel
  Coma
  (n=3,m=1)
Coma_Annular Coma_seidel
  Spherical 3rd order
  (n=4,m=0)
Sa3_Annular Sa3_seidel
  Astigmatism
  (n=2,m=2)
Ast3_Annular ASt_seidel
  Triangular Coma
  (n=3,m=3)
 TComa_annular TComa_seidel
  Quadratic Astigmatism
  (n=4,m=4)
Qast_Annular QAst_seidel