The real Zernike polynomials form an orthogonal polynomial basis on the unit Euclidean disk, with radial factor and angular factor or . Here and 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 . 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 and .
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Zernike polynomials are a set of orthogonal polynomials defined over a unit disk, which are commonly used in various fields such as optics, imaging science, and surface metrology. They are particularly useful for describing wavefronts and optical aberrations, as they provide a convenient mathematical framework for representing complex shapes and patterns.