Zernike polynomials: orthogonal polynomials on the unit disk. Z_n^m(ρ,θ) = R_n^m(ρ)·{cos(mθ), sin(mθ)}. Named modes: piston(Z₀⁰), tilt(Z₁¹), defocus(Z₂⁰), astigmatism(Z₂²), coma(Z₃¹), spherical aberration(Z₄⁰).
How to Use
Enter n, m, ρ, θ:
- Value: Z_n^m(ρ,θ)
- Radial: R_n^m(ρ)
- Named: Aberration type
Optics Applications
Any wavefront on a circular aperture can be expanded: W(ρ,θ) = Σ a_{nm}Z_n^m(ρ,θ). Coefficients a_{nm} quantify aberrations. Adaptive optics corrects by applying opposite Zernike modes via deformable mirrors.
Named Aberrations
- Z₀⁰: Piston (constant phase)
- Z₁±¹: Tilt (beam direction)
- Z₂⁰: Defocus
- Z₂±²: Astigmatism
- Z₃±¹: Coma
- Z₄⁰: Spherical aberration
Step-by-Step Instructions
- 1Enter n and m.
- 2Enter ρ and θ.
- 3Compute Z_n^m.
- 4See aberration name.
- 5View mode table.