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Zernike polynomials: Mathematical Expression
| Zernike polynomials used in Sensoft - our wavefront sensor software: expressions for the first 7 terms | ||
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Zernike polynomials are generally expressed in terms of the normalized radius r of the pupil, and the azimuthal angle φ , by the following expression : rn cos
(m φ + φ0) This is the general expression in terms of the Seidel polynomials. The Annular 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. They also take into account the effect of the annulus e of the optical element. φ0 is
the zero-offset : it gives the orientation of the particular aberration with respect to the x-axis.
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| Aberration | Annular Zernike polynomials | Seidel polynomials |
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Defocus (n=2,m=0) |
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Tilt (n=1,m=1) |
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Coma (n=3,m=1) |
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Spherical 3rd
order (n=4,m=0) |
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Astigmatism (n=2,m=2) |
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Triangular
Coma (n=3,m=3) |
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Quadratic
Astigmatism (n=4,m=4) |
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