What units are the slider coefficients?

Amplitudes are in waves (λ) multiplying orthonormal Zernike modes on the pupil. Because those modes are RMS-normalized, a single coefficient is approximately that mode’s RMS wavefront error, not its peak-to-valley value.

How is Strehl ratio defined here?

S is the ratio of the peak aberrated PSF intensity to the peak diffraction-limited PSF from the same circular pupil — the standard intensity Strehl definition.

Why does Strehl differ from the Maréchal estimate?

The Maréchal formula S ≈ exp[−(2πσ)²] assumes small RMS phase error and many modes; a single large Zernike coefficient or high-order mixing breaks the approximation.

Pupil apodization, central obscuration, polychromatic light, vector diffraction, and higher Zernike orders beyond Z₄⁰.

Zernike Wavefront Aberrations

Defocus, astigmatism, coma, spherical Zernike modes on a circular pupil; PSF via FFT, Strehl ratio and Maréchal estimate.

Live graphs

About this model

Optical aberrations are conveniently expanded in Zernike polynomials Z_n^m(ρ,θ) on the unit pupil (ρ ≤ 1). This simulator superposes the lowest-order modes with user coefficients in waves (fractions of λ): defocus Z₂⁰, astigmatism Z₂² (with rotatable axis via Z₂² and Z₂⁻² components), coma Z₃¹ and Z₃⁻¹, and spherical Z₄⁰, using the standard orthonormal forms. The aberrated pupil field is E = P(ρ) exp(i2πW) with uniform amplitude inside the disk; the point-spread function PSF = |ℱ{E}|² is computed with a 128×128 FFT and compared to the diffraction-limited reference. The Strehl ratio is reported as the peak PSF ratio S = max(PSF)/max(PSF₀), and the RMS wavefront error σ (in waves) yields the Maréchal approximation S ≈ exp[−(2πσ)²] for small aberrations. Three panels show the wavefront map, aberrated PSF (optional log display), and reference Airy-like PSF, plus a radial intensity profile.

Who it's for: Undergraduate and graduate optics students studying wavefront sensing, adaptive optics, microscope alignment, and telescope aberration theory.

Key terms

Zernike polynomial
Wavefront aberration
Point spread function
Strehl ratio
Defocus
Astigmatism
Coma
Spherical aberration
Maréchal approximation

How it works

Zernike wavefront aberrations: defocus, astigmatism, coma, and spherical modes on a circular pupil; PSF via FFT and Strehl ratio.

Key equations

W(ρ,θ) = Σ cₙᵐ Zₙᵐ(ρ,θ) · φ = 2πW/λ

Strehl ≈ exp[−(2π σ)²] (σ = RMS in waves, Maréchal)

Frequently asked questions

What units are the slider coefficients?: Amplitudes are in waves (λ) multiplying orthonormal Zernike modes on the pupil. Because those modes are RMS-normalized, a single coefficient is approximately that mode’s RMS wavefront error, not its peak-to-valley value.
How is Strehl ratio defined here?: S is the ratio of the peak aberrated PSF intensity to the peak diffraction-limited PSF from the same circular pupil — the standard intensity Strehl definition.
Why does Strehl differ from the Maréchal estimate?: The Maréchal formula S ≈ exp[−(2πσ)²] assumes small RMS phase error and many modes; a single large Zernike coefficient or high-order mixing breaks the approximation.
What is left out?: Pupil apodization, central obscuration, polychromatic light, vector diffraction, and higher Zernike orders beyond Z₄⁰.

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