# 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 :  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)  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)  