def V(self, n, m, r, f, L_max=35, ncpus=-1):
r"""Evaluate :math:`V_n^m(r, f)`.
Compute
.. math::
V_n^m(r, f) = \epsilon_m \exp(if) \sum_{l=1}^{L_max}(-2if)^{l - 1}
\sum_{j=0}^{(n - |m|)/2} v_{l,j}
(1/l(2\pi r)^l)J_{|m| + l + 2j}(2\pi r).
See Eq. (2.48) in [B2008]_.
Parameters
----------
- `n`: `numpy` vector of integers for the radial orders `n`
- `m`: `numpy` vector of integers for the azimuthal frequencies `m`
- `r`: `numpy` vector of doubles for the radial coordinate :math:`r`,
which is normalised to the diffraction unit `wavelength/NA`
- `f`: `numpy` vector of complex numbers for the defocus parameter,
see [J2002]_, [B2008]_, and [H2010]_
- `L_max`: optional `int` for the truncation order of the series,
see [J2002]_, [B2008]_, and [H2010]_. `L_max <= 0` uses the
default value of `35`.
- `ncpus` : optional `int` for the number of threads. `-1` chooses
all available cpus
Returns
-------
- `vnm`: `numpy` array of shape `(r.size, f.size, n.size)` for
:math:`V_n^m(r, f)`
References
----------
.. [J2002] A. J. E. M. Janssen, "Extended Nijboer–Zernike approach
for the computation of optical point-spread functions," J. Opt.
Soc. Am. A 19, 849–857 (2002). `doi
<http://dx.doi.org/10.1364/JOSAA.19.000849>`__.
.. [B2008] J. Braat, S. van Haver, A. Janssen, P. Dirksen, Chapter 6
Assessment of optical systems by means of point-spread functions,
In: E. Wolf, Editor(s), Progress in Optics, Elsevier, 2008, Volume
51, Pages 349-468, ISSN 0079-6638, ISBN 9780444532114. `doi
<http://dx.doi.org/10.1016/S0079-6638(07)51006-1>`__.
.. [H2010] S. van Haver, The Extended Nijboer-Zernike Diffraction
Theory and its Applications (Ph.D. thesis, Delft University of
Technology, The Netherlands, 2010). `doi
<http://resolver.tudelft.nl/uuid:8d96ba75-24da-4e31-a750-1bc348155061>`__.
"""
n = self._numpify(n, np.int)
m = self._numpify(m, np.int)
r = self._numpify(r, np.float)
f = self._numpify(f, np.complex)
Vnm = vnmpocnp(r, f, n, m, L_max=L_max, ncpus=ncpus)
if Vnm.size == 1:
return complex(Vnm[0, 0, 0])
else:
return Vnm
def U(self, beta, r, phi, f):
r"""Evaluate the complex point spread function at a point.
The complex point-spread function is
.. math::
U(r, \phi, f) = 2 \sum_{n, m} \beta_{n}^{m}
\sqrt{n + 1} i^{m} V_n^m(r, f) \exp(im\phi).
See Eq. (4) in [A2015]_.
Parameters
----------
- `beta`: `list` of the complex Zernike coefficients
:math:`\beta_n^m`
- `r`: float for the radial coordinate :math:`r`, normalised to
the diffraction unit `wavelength/NA`
- `phi`: float for the azimuthal coordinate :math:`\phi`
- `f`: complex-valued defocus parameter, see [J2002]_, [B2008]_, and
[H2010]_
Returns
-------
- `U`: complex value of :math:`U(r, \phi, f)`
References
----------
.. [A2015] Jacopo Antonello and Michel Verhaegen, "Modal-based phase
retrieval for adaptive optics," J. Opt. Soc. Am. A 32, 1160-1170
(2015). `url <http://dx.doi.org/10.1364/JOSAA.32.001160>`__.
.. [J2002] A. J. E. M. Janssen, "Extended Nijboer–Zernike approach
for the computation of optical point-spread functions," J. Opt.
Soc. Am. A 19, 849–857 (2002). `doi
<http://dx.doi.org/10.1364/JOSAA.19.000849>`__.
.. [B2008] J. Braat, S. van Haver, A. Janssen, P. Dirksen, Chapter 6
Assessment of optical systems by means of point-spread functions,
In: E. Wolf, Editor(s), Progress in Optics, Elsevier, 2008, Volume
51, Pages 349-468, ISSN 0079-6638, ISBN 9780444532114. `doi
<http://dx.doi.org/10.1016/S0079-6638(07)51006-1>`__.
.. [H2010] S. van Haver, The Extended Nijboer-Zernike Diffraction
Theory and its Applications (Ph.D. thesis, Delft University of
Technology, The Netherlands, 2010). `doi
<http://resolver.tudelft.nl/uuid:8d96ba75-24da-4e31-a750-1bc348155061>`__.
"""
U = complex(0.0)
r = self._numpify(r, np.float)
f = self._numpify(f, np.complex)
assert (r.size == 1)
assert (f.size == 1)
n, m = np.array([0]), np.array([0])
for i in range(self.czern.nk):
n[0], m[0] = self.czern.ntab[i], self.czern.mtab[i]
cf = self.czern.coefnorm[i]
U += complex(beta[i] * cf * ((1j)**m[0]) *
vnmpocnp(r, f, n, m)[0][0][0] *
cmath.exp(1j * m[0] * phi))
return 2.0 * U
def test_vnmpocnp(self):
data = h5py.File('refVnmpo.h5', 'r')
ref = data['ref'].value
r = data['r'].value
f = data['f'].value
n = data['n'].value
m = data['m'].value
data.close()
out = vnmpocnp(r, f, n, m)
self.assertTrue(out.shape[0] == r.size)
self.assertTrue(out.shape[1] == r.size)
self.assertTrue(out.shape[2] == n.size)
err = norm((out - ref).ravel())
self.assertTrue(err < self.max_enorm)
def test_vnmpocnp(self):
data = h5py.File('refVnmpo.h5', 'r')
ref = data['ref'].value
r = data['r'].value
f = data['f'].value
n = data['n'].value
m = data['m'].value
data.close()
out = vnmpocnp(r, f, n, m)
self.assertTrue(out.shape[0] == r.size)
self.assertTrue(out.shape[1] == r.size)
self.assertTrue(out.shape[2] == n.size)
err = norm((out - ref).ravel())
self.assertTrue(err < self.max_enorm)
def make_pol_grid(self, r_sp=None, ph_sp=None, f_sp=None, min_r=1e-9):
r"""Make a polar grid for the complex point-spread function.
The complex point-spread function is
.. math::
U(r, \phi, f) = 2 \sum_{n, m} \beta_{n}^{m}
\sqrt{n + 1} i^{m} V_n^m(r, f) \exp(im\phi).
See Eq. (4) in [A2015]_.
Parameters
----------
- `r_sp`: `numpy` array of the radial coordinate :math:`r`
- `ph_sp`: `numpy` array of the azimuthal coordinate :math:`\phi`
- `f_sp`: `numpy` array of complex numbers for the defocus parameter,
see [J2002]_, [B2008]_, [H2010]_
- `min_r`: float, optional threshold to avoid division by zero in
the image plane
References
----------
.. [A2015] Jacopo Antonello and Michel Verhaegen, "Modal-based phase
retrieval for adaptive optics," J. Opt. Soc. Am. A 32, 1160-1170
(2015). `url <http://dx.doi.org/10.1364/JOSAA.32.001160>`__.
.. [J2002] A. J. E. M. Janssen, "Extended Nijboer–Zernike approach
for the computation of optical point-spread functions," J. Opt.
Soc. Am. A 19, 849–857 (2002). `doi
<http://dx.doi.org/10.1364/JOSAA.19.000849>`__.
.. [B2008] J. Braat, S. van Haver, A. Janssen, P. Dirksen, Chapter 6
Assessment of optical systems by means of point-spread functions,
In: E. Wolf, Editor(s), Progress in Optics, Elsevier, 2008, Volume
51, Pages 349-468, ISSN 0079-6638, ISBN 9780444532114. `doi
<http://dx.doi.org/10.1016/S0079-6638(07)51006-1>`__.
.. [H2010] S. van Haver, The Extended Nijboer-Zernike Diffraction
Theory and its Applications (Ph.D. thesis, Delft University of
Technology, The Netherlands, 2010). `doi
<http://resolver.tudelft.nl/uuid:8d96ba75-24da-4e31-a750-1bc348155061>`__.
"""
r_sp = self._numpify(r_sp, np.float).ravel(order='F')
ph_sp = self._numpify(ph_sp, np.float).ravel(order='F')
f_sp = self._numpify(f_sp, np.complex).ravel(order='F')
# remove nans due to r = 0.0
r_sp = self._trim_r(r_sp, min_r)
Ugrid = np.zeros((r_sp.size, ph_sp.size, f_sp.size, self.czern.nk),
order='F',
dtype=np.complex)
Vnm = vnmpocnp(r_sp, f_sp, self.czern.ntab, self.czern.mtab)
Cnm = np.zeros((ph_sp.size, self.czern.nk),
order='F',
dtype=np.complex)
for k in range(self.czern.nk):
m = self.czern.mtab[k]
Cnm[:, k] = 2.0 * self.czern.coefnorm[k] * (
(1j)**m) * np.exp(1j * m * ph_sp)
for k in range(self.czern.nk):
for f in range(f_sp.size):
v = Vnm[:, f, k].reshape((Vnm.shape[0], 1))
c = Cnm[:, k].reshape((1, Cnm.shape[0]))
Ugrid[:, :, f, k] = np.kron(c, v)
assert (np.all(np.isfinite(Ugrid)))
self.Ugrid = Ugrid
self.Vnm = Vnm
self.Cnm = Cnm