Zernike (paos.classes.zernike)

class Zernike(N, rho, phi, ordering='ansi', normalize=False)

Bases: object

Generates Zernike polynomials

Parameters:

• N (integer) – Number of polynomials to generate in a sequence following the defined ‘ordering’

• rho (array like) – the radial coordinate normalised to the interval [0, 1]

• phi (array like) – Azimuthal coordinate in radians. Has same shape as rho.

• ordering (string) –

  The ordering of the Zernike polynomials. Can either be:

  • ANSI (ordering=’ansi’, this is the default);

  • Noll (ordering=’noll’). Used in Zemax as “Zernike Standard Coefficients”, R. Noll, “Zernike polynomials and atmospheric turbulence”, J. Opt. Soc. Am., Vol. 66, No. 3, p207 (1976);

  • Fringe (ordering=’fringe’), AKA the “Fringe” or “University of Arizona” notation;

  • Standard (ordering=’standard’). Used in CodeV, Born and Wolf, Principles of Optics (Pergamon Press, New York, 1989).

• normalize (bool) – Set to True generates ortho-normal polynomials. Set to False generates orthogonal polynomials as described in Laksminarayan & Fleck, Journal of Modern Optics (2011). The radial polynomial is estimated using the Jacobi polynomial expression as in their Equation in Equation 14.

Returns:

out – An instance of Zernike.

Return type:

masked array

Example

>>> import numpy as np
>>> from matplotlib import pyplot as plt
>>> x = np.linspace(-1.0, 1.0, 1024)
>>> xx, yy = np.meshgrid(x, x)
>>> rho = np.sqrt(xx**2 + yy**2)
>>> phi = np.arctan2(yy, xx)
>>> zernike = Zernike(36, rho, phi, ordering='noll', normalize=True)
>>> zer = zernike() # zer contains a list of polynomials, noll-ordered

>>> # Plot the defocus zernike polynomial
>>> plt.imshow(zer[3])
>>> plt.show()

>>> # Plot the defocus zernike polynomial
>>> plt.imshow(zernike(3))
>>> plt.show()

Note

In the example, the polar angle is counted counter-clockwise positive from the x axis. To have a polar angle that is clockwise positive from the y axis (as in figure 2 of Laksminarayan & Fleck, Journal of Modern Optics (2011)) use

>>> phi = 0.5*np.pi - np.arctan2(yy, xx)

static j2mn(N, ordering)

Convert index j into azimuthal number, m, and radial number, n for the first N Zernikes

Parameters:

• N (integer) – Number of polynomials (starting from Piston)

• ordering (string) – can take values ‘ansi’, ‘standard’, ‘noll’, ‘fringe’

Returns:

m, n

Return type:

array

static mn2j(m, n, ordering)

Convert radial and azimuthal numbers, respectively n and m, into index j

cov()

Computes the covariance matrix M defined as

>>> M[i, j] = np.mean(Z[i, ...]*Z[j, ...])

When a pupil is defined as \(\Phi = \sum c[k] Z[k, ...]\), the pupil RMS can be calculated as

>>> RMS = np.sqrt( np.dot(c, np.dot(M, c)) )

This works also on a non-circular pupil, provided that the polynomials are masked over the pupil.

Returns:

M – the covariance matrix

Return type:

array

class PolyOrthoNorm(N, rho, phi, ordering='ansi', normalize=False, mask=False)

Bases: Zernike

Generates polynomials that are ortho-normal on the mask provided. This is done by applying the Gram-Smidth orthonormalization process, implemented following Section 3.4 of https://doi.org/10.1117/3.927341. Therefore, the relevant input are the type of Zernike polinomial to be used (ordering and normalization), and the mask defining the pupil.

U[m] = sum_n M[m, n] Z[n]

where Z[m] are Zernike polynomials over a circular domain containing the pupil. If Phi is the field on the pupil, using linear algebra notation:

Phi = b.T U = b.T M Z = c.T Z

And the coefficients of the Zernike expansion are

c.T = M.T c

Parameters:

• N (integer) – Number of polynomials to generate in a sequence following the defined ‘ordering’

• rho (array like) – the radial coordinate normalised to the interval [0, 1]. If rho is a numpy masked array, the mask is used in the orthonormalisation process

• phi (array like) – Azimuthal coordinate in radians. Has same shape as rho.

• ordering (string) –

  The ordering of the Zernike polynomials. Can either be:

  • ANSI (ordering=’ansi’, this is the default);

  • Noll (ordering=’noll’). Used in Zemax as “Zernike Standard Coefficients”, R. Noll, “Zernike polynomials and atmospheric turbulence”, J. Opt. Soc. Am., Vol. 66, No. 3, p207 (1976);

  • Fringe (ordering=’fringe’), AKA the “Fringe” or “University of Arizona” notation;

  • Standard (ordering=’standard’). Used in CodeV, Born and Wolf, Principles of Optics (Pergamon Press, New York, 1989).

• normalize (bool) –

  The normalisation of Zernike polinomials. Set to True generates ortho-normal polynomials. Set to False generates orthogonal polynomials as described in Laksminarayan & Fleck, Journal of Modern Optics (2011). The radial polynomial is estimated using the Jacobi polynomial expression as in their Equation in Equation 14.

• mask (bool array like) – The mask defining the pupil following masked array convention. Pixel within the pupil are masked False. Defaults to False.

Returns:

out – An instance of Zernike.

Return type:

masked array

Example

>>> import numpy as np
>>> from matplotlib import pyplot as plt
>>> x = np.linspace(-1.0, 1.0, 1024)
>>> xx, yy = np.meshgrid(x, x)
>>> mask = xx**2 + (yy/0.5)**2 > 1.0
>>> rho = np.ma.MaskedArray(data=np.sqrt(xx**2 + yy**2), mask=mask, fill_value=0.0)
>>> phi = np.arctan2(yy, xx)
>>> poly = PolyOrthoNorm(36, rho, phi, ordering='noll', normalize=True)
>>> U = poly() # zer contains a list of polynomials, noll-ordered

>>> # Plot the Power polynomial
>>> plt.imshow(U[4])
>>> plt.show()

Note

In the example, the polar angle is counted counter-clockwise positive from the x axis. To have a polar angle that is clockwise positive from the y axis (as in figure 2 of Laksminarayan & Fleck, Journal of Modern Optics (2011)) use

>>> phi = 0.5*np.pi - np.arctan2(yy, xx)

toZernike(coeff)

If Phi is the field on the pupil, using linear algebra notation:

Phi = coeff.T U = c.T Z

This function returns the Zerkinke coefficients c

Parameters:

coeff (vector 1D) – coefficients of the field in the orthonormal base

Returns:

out – coefficiens of the field in the Zernike expansion

Return type:

vector 1D