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Master wavefront aberrations with four versatile polynomial conventions.
Zernike Polynomials in Sensoft
Accurately represent and analyze phase aberrations with Sensoft's advanced wavefront modeling. Whether you need standard mathematical fitting or need to account for central optical obscurations, our software adapts to your exact measurement needs.
Mathematical Formulation
Zernike polynomials represent a particular phase aberration W at a point P on a circular pupil and they are generally expressed in terms of the normalized radius r of the pupil and the azimuthal angle φ, by the following expression:
General Expression:rⁿ cos(mφ + φ₀)
r : normalized radiusφ : azimuthal angleφ₀ : zero offsetε : annulusW : wavefront aberration
Aberration Balancing & Conventions
- This is the general expression in terms of the Seidel polynomials. The Standard Zernike polynomials, on the other hand, involve aberration balancing, in which aberrations of a lower order are combined with those of the higher order for reducing the wavefront error. For example, the expression for 3rd order spherical aberration also contains a defocus term. φ0 is the zero-offset : it gives the orientation of the particular aberration with respect to the x-axis.
Annular Zernike Polynomials: Finally, the Annular Zernike polynomials also take into account the effect of the annulus ε of the optical element.
Zernike polynomial options in Sensoft
SpotOptics wavefront sensor software SenSoft allow you to select among four polynomials to fit to the data.
01
Annular Zernike polynomials
Annular Zernike polynomials are based on diffraction theory and take into account the presence of the central obscuration in the pupil. They are the most general form of Zernike polynomials. They are normally used in astronomical wavefront sensors (like Puntino) to take into account the obscuration by the secondary mirror of the telescope.
02
Standard Zernike polynomials
Standard Zernike polynomials follow from Annular Zernike polynomials, with the central obscuration factor, and are normalized over the pupil.
03
Fringe Zernike polynomials
Fringe Zernike polynomials are the same as Standard Zernike polynomials without the normalization factors. They are mainly used in the field of interferometry, as suggested by the name.
04
Seidel polynomials
The Seidel polynomials are the extension of Seidel aberrations from geometrical optics.
Zernike polynomial comparison
| Aberration | Standard Zernike | Annular Zernike | Seidel |
|---|---|---|---|
| Tilt (n=1,m=1) | 2r cos(φ + φ₀) | 2r cos(φ + φ₀) / √(1 + ε²) | r cos(φ + φ₀) |
| Defocus (n=2,m=0) | √3 (2r² - 1) | √3 [2r² - (1 + ε²)] / (1 - ε²) | r² |
| Coma (n=3,m=1) | √8 (3r³ - 2r) cos(φ + φ₀) | √8 [3r³ - 2r(1 + ε²)] cos(φ + φ₀) / [(1 - ε²)²] × normalization terms | r³ cos(φ + φ₀) |
| Spherical 3rd order (n=4,m=0) | √5(6r⁴ - 6r² + 1) | √5 [6r⁴ - 6(1 + ε²)r² + (1 + ε² + ε⁴)] / (1 - ε²)² | r⁴ |
| Astigmatism (n=2,m=2) | √6 r² cos(2φ + φ₂) | √6 r² cos(2φ + φ₂) / √(1 + ε² + ε⁴) | r² cos(2φ + φ₂) |
| Triangular Coma (n=3,m=3) | √8 r³ cos(3φ + φ₃) | √8 r³ cos(3φ + φ₃) / √(1 + ε² + ε⁴ + ε⁶) | r³ cos(3φ + φ₃) |
| Quadratic Astigmatism (n=4,m=4) | √10 r⁴ cos(4φ + φ₄) | √10 r⁴ cos(4φ + φ₄) / √(1 + ε² + ε⁴ + ε⁶ + ε⁸) | r⁴ cos(4φ + φ₄) |
Summary
Zernike polynomials provide a mathematical representation of wavefront aberrations over a circular pupil. Sensoft provides multiple polynomial conventions for fitting wavefront sensor data, including Annular Zernike, Standard Zernike, Fringe Zernike and Seidel polynomials.