Zernike polynomials

Most optical systems have circular pupils, and the wavefront error across such a pupil is conveniently expanded in Zernike polynomials, $Z_n^m(\rho,\theta)$, products of a radial polynomial of degree $n$ in the normalized radius $\rho$ and a factor $\cos m\theta$ or $\sin m\theta$. The low orders correspond to the familiar aberrations: $Z_0^0$ is piston, $Z_1^{\pm1}$ are tilts, $Z_2^0$ is defocus, $Z_2^{\pm2}$ astigmatism, $Z_3^{\pm1}$ coma, and $Z_4^0$ primary spherical aberration. Each higher term is balanced against the lower ones; primary spherical aberration in Zernike form already includes the amount of defocus that minimizes its rms, which is why the Zernike term is the natural description for a system that is focused for best image quality.

The property that makes them useful is orthogonality over the unit disk. With the common orthonormal normalization (Noll's), the coefficient of each term is its rms contribution, and the total rms wavefront error is the root-sum-square of the coefficients, excluding piston. Fitting a measured wavefront therefore gives a list of independent numbers whose squares add. For example, 0.05 waves rms of defocus and 0.04 waves rms of astigmatism combine to 0.064 waves rms, and the extended Maréchal approximation, $S \approx \exp[-(2\pi\sigma)^2]$ with $\sigma$ in waves, then gives a Strehl ratio of 0.85. Because tilt and defocus merely shift the image, they are usually removed before quoting the image-degrading residual.

Two indexing conventions are in wide use, Noll's single index $j$ and the ANSI/OSA standard (used in ophthalmology), and they differ in order and in sign conventions for the sine and cosine terms, so a coefficient list must state which it uses. Interferometers and Shack-Hartmann sensors report measured wavefronts as Zernike coefficients, lens design programs specify tolerances with them, and adaptive optics systems often control the deformable mirror in a Zernike basis, correcting the low orders that carry most of the variance in atmospheric turbulence. For non-circular or obscured pupils, as in telescopes with a central obstruction, the polynomials lose their orthogonality and modified sets are used.

References: R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 9; ANSI Z80.28, Methods of reporting optical aberrations of eyes.