The polynomials that are almost universally applied to fringe analysis are those devised by Fritz Zernike in 1934; the first 45 have been implemented in the Fringe Master software.
International Standards for the measurement and interpretation of interferograms are currently being drafted (ISO 14999 and 10110). They are likely to recommend that Zernike polynomials be used for the numerical analysis of optical wavefronts.
The orthogonal nature of the polynomials means that the individual terms describing tilt, spherical power, astigmatism etc. can be identified, measured and subtracted from the numerical description of the wavefront. Any residual aberrations can be separately identified and used to quantify the quality of the laser beam.
A phase surface can be described by a distribution denoting the distance W of the surface of constant phase from some reference surface (usually a sphere). If the wavefront is continuous and sufficiently smooth, it can be represented by a two-dimensional function of the radial and azimuth cylindrical co-ordinates W(r,q). A suitable function, when the aperture of the beam is circular and has unity radius, is a linear combination of Zernike polynomials Zr so that:
W(r,q) = SAr.Zr
The first term A1.Z1 is the "piston" or constant term since Z1 =1. The terms with r = 2 and 3, for example, are the tilt terms about the x and y axes where Z2 =.r.Sinq and Z3 = r.Cosq. The coefficients Ar are the product of a normalising term and a constant representing the contribution of the rth Zernike polynomial to the total wavefront aberration.