Zernike and Seidel AberrationsPhase MOSAIC

Zernike and Seidel Polynomials


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Zernike polynomials

Surfaces can be represented by a linear combination of Zernike polynomials.

zern exp

Z[n] are known as Zernike polynomial coefficients and have units of surface height. Positions are normalized to a unit circle radius so that the polynomials Z(n) are unitless (R = r / Normalization radius.)  Here is a partial listing of the Zernike polynomials used in Phase MOSAIC:

Z(0) = 1

Z(1) = R cos(q )

Z(2) = R sin(q )

Z(3) = 2 R2 - 1

Z(4) = R2 cos(2q )

Z(5) = R2 sin(2q )

Z(6) = (3R2 - 2) R cos(q )

Z(7) = (3R2 - 2) R sin(q )

Z(8) = 6 R4 - 6 R2 + 1

Z(9) = R3 cos(3q )

Z(10) = R3 sin(3q )

Z(11) = (4R2 - 3) R2 cos(2q )

Z(12) = (4R2 - 3) R2 sin(2q )

Z(13)= (10R4 - 12R2 + 3) R cos(q )

Z(14) = (10R4 - 12R2 + 3) R sin(q )

Z(15) = 20R6 - 30R4 + 12R2 - 1

Note: In some Zernike numbering orders, the Zernike polynomial series is truncated at Z(35) and the next purely radial polynomial used in place of Z(36). This is not done in Phase MOSAIC. The following pure radial Zernike is Z(48).

Seidel polynomials

The Seidel polynomials can be represented by linear combinations of Zernike polynomials. This means the magnitude of the Seidel polynomial coefficient can be calculated from Zernike polynomial coefficients. Phase MOSAIC adopts the common convention of using only the first eight Zernike coefficients to calculate Seidel coefficients:

Seidel AberrationMagnitudeAngle
piston mag
tilt mag
tilt angle
astig mag
astig angle
coma mag
coma angle
spherical mag
See Also

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