Zernike polynomials: expressions for the first 7 terms

The Zernike polynomials are generally expressed in terms of the normalized radius r of the pupil,  and the azimuthal angle f, by the following expression:  .  This is the general expression in terms of the Seidel Zernike polynomials. The classical Zernike polynomials, on the other hand, involve aberration balancing, in which aberrations of the lower order are combined with those of the higher order for reducing the wavefront error. They also take into account if the optical element has an annulus e. For example, the expression for 3rd order spherical aberration also contains a defocus term. is the zero-offset.

NAME	n	m	Classical Annular Zernike Polynomials	Seidel polynomials
Tilt	1	1
Defocus	2	0
Coma	3	1
3rd order Spherical Aberration	4	0
5th order Spherical aberration	6	0
Astigmatism	2	2
Triangular coma	3	3
Quadratic Astigmatism	4	4