Zernike polynomial ($Z_n^m(\rho ,\varphi )$)

The real Zernike polynomials form an orthogonal polynomial basis on the unit Euclidean disk, with radial factor $R_n^m(\rho )$ and angular factor $\cos m\varphi$ or $\sin m\varphi$. Here $0\le m\le n$ and $n-m$ is even. They decompose wavefront errors into modes useful for optical correction. Normalizations differ, so coefficients should specify the chosen convention.

The unnormalized rotationally symmetric mode is $Z_4^0(\rho )=6{\rho }^4-6{\rho }^2+1$. It describes balanced primary spherical aberration. The lower-degree terms make it orthogonal to constant piston and quadratic defocus under the disk area measure: direct integration gives ${\int }_0^1{Z}_4^0\rho d\rho =0$ and ${\int }_0^1{Z}_4^0(2{\rho }^2-1)\rho d\rho =0$.