def qdht(r: np.ndarray, f: np.ndarray, order: int = 0) -> Tuple[np.ndarray, np.ndarray]:
"""Perform a quasi-discrete Hankel transform of the function ``f`` (sampled at points
``r``) and return the transformed function and its sample points in :math:`k`-space.
If you requires the transform on a frequency axis (as opposed to the :math:`k`-axis), the
frequency axis :math:`v` can be calculated using :math:`v = \\frac{k}{2\\pi}`.
.. warning::
This method is a convenience wrapper for :meth:`.HankelTransform.qdht`, but incurs a
significant overhead in calculating the :class:`.HankelTransform` object. If you
are performing multiple transforms on the same grid, it will be much quicker to
construct a single :class:`.HankelTransform` object and call
:meth:`.HankelTransform.qdht` multiple times.
:param r: The radial coordinates at which the function is sampled
:type r: :class:`numpy.ndarray`
:param f: The value of the function to be transformed.
:type f: :class:`numpy.ndarray`
:param order: The order of the Hankel Transform to perform. Defaults to 0.
:return: A tuple containing the k coordinates of the transformed function and its values
:rtype: (:class:`numpy.ndarray`, :class:`numpy.ndarray`)
"""
transformer = HankelTransform(order=order, radial_grid=r)
f_transform = transformer.to_transform_r(f)
ht = transformer.qdht(f_transform)
return transformer.kr, ht
def test_round_trip_r_interpolation(radius: np.ndarray, order: int, shape: Callable):
transformer = HankelTransform(order=order, radial_grid=radius)
# the function must be smoothish for interpolation
# to work. Random every point doesn't work
func = shape(radius)
transform_func = transformer.to_transform_r(func)
reconstructed_func = transformer.to_original_r(transform_func)
assert np.allclose(func, reconstructed_func, rtol=1e-4)
def test_1_over_r2_plus_z2_equivalence(a: float):
r = np.linspace(0, 50, 1024)
f = 1 / (r ** 2 + a ** 2)
transformer = HankelTransform(order=0, radial_grid=r)
f_transformer = 1 / (transformer.r**2 + a**2)
assert np.allclose(transformer.to_transform_r(f), f_transformer, rtol=1e-2, atol=1e-6)
kr, one_shot_ht = qdht(r, f)
assert np.allclose(kr, transformer.kr)
standard_ht = transformer.qdht(f_transformer)
assert np.allclose(one_shot_ht, standard_ht, rtol=1e-3, atol=1e-2)
def test_round_trip_with_interpolation(shape: Callable, radius: np.ndarray,
transformer: HankelTransform):
# the function must be smoothish for interpolation
# to work. Random every point doesn't work
func = shape(radius)
func_hr = transformer.to_transform_r(func)
ht = transformer.qdht(func_hr)
reconstructed_hr = transformer.iqdht(ht)
reconstructed = transformer.to_original_r(reconstructed_hr)
assert np.allclose(func, reconstructed, rtol=2e-4)
def test_round_trip_r_interpolation_2d(radius: np.ndarray, order: int,
shape: Callable, axis: int):
transformer = HankelTransform(order=order, radial_grid=radius)
# the function must be smoothish for interpolation
# to work. Random every point doesn't work
dims_amplitude = np.ones(2, np.int)
dims_amplitude[1 - axis] = 10
amplitude = np.random.random(dims_amplitude)
dims_radius = np.ones(2, np.int)
dims_radius[axis] = len(radius)
func = np.reshape(shape(radius), dims_radius) * np.reshape(
amplitude, dims_amplitude)
transform_func = transformer.to_transform_r(func, axis=axis)
reconstructed_func = transformer.to_original_r(transform_func, axis=axis)
assert np.allclose(func, reconstructed_func, rtol=1e-4)
def propagate_using_object(r: np.ndarray, field: np.ndarray) -> np.ndarray:
transformer = HankelTransform(order=0, radial_grid=r)
field_for_transform = transformer.to_transform_r(field) # Resampled field
hankel_transform = transformer.qdht(field_for_transform)
propagated_field = np.zeros((nr, Nz), dtype=complex)
kz = np.sqrt(k0**2 - transformer.kr**2)
for n, z_loop in enumerate(z):
phi_z = kz * z_loop # Propagation phase
hankel_transform_at_z = hankel_transform * np.exp(
1j * phi_z) # Apply propagation
field_at_z_transform_grid = transformer.iqdht(
hankel_transform_at_z) # iQDHT
propagated_field[:, n] = transformer.to_original_r(
field_at_z_transform_grid) # Interpolate output
intensity = np.abs(propagated_field)**2
return intensity
def example():
# Gaussian function
def gauss1d(x, x0, fwhm):
return np.exp(-2 * np.log(2) * ((x - x0) / fwhm) ** 2)
nr = 1024 # Number of sample points
r_max = .05 # Maximum radius (5cm)
dr = r_max / (nr - 1) # Radial spacing
ri = np.arange(0, nr) # Radial pixels
r = ri * dr # Radial positions
beam_radius = 5e-3 # 5mm
propagation_direction = 5000
Nz = 200 # Number of z positions
z_max = .25 # Maximum propagation distance
dz = z_max / (Nz - 1)
z = np.arange(0, Nz) * dz # Propagation axis
ht = HankelTransform(0, radial_grid=r)
k_max = 2 * np.pi * ht.v_max # Maximum K vector
field = gauss1d(r, 0, beam_radius) * np.exp(1j * propagation_direction * r) # Initial field
ht_field = ht.to_transform_r(field) # Resampled field
transform = ht.qdht(ht_field) # Convert from physical field to physical wavevector
propagated_intensity = np.zeros((nr, Nz + 1))
propagated_intensity[:, 0] = np.abs(field) ** 2
for n in range(1, Nz):
z_loop = z[n]
propagation_phase = (np.sqrt(k_max ** 2 - ht.kr ** 2) - k_max) * z_loop # Propagation phase
propagated_transform = transform * np.exp(1j * propagation_phase) # Apply propagation
propagated_ht_field = ht.iqdht(propagated_transform) # iQDHT
propagated_field = _spline(ht.r, propagated_ht_field, r) # Interpolate output
propagated_intensity[:, n] = np.abs(propagated_field) ** 2
return transform, propagated_intensity
# %% # Set up beam parameters Dr = 100e-6 # Beam radius (100um) lambda_ = 488e-9 # wavelength 488nm k0 = 2 * np.pi / lambda_ # Vacuum k vector # %% # Set up a :class:`.HankelTransform` object, telling it the order (``0``) and # the radial grid. H = HankelTransform(order=0, radial_grid=r) # %% # Set up the electric field profile at :math:`z = 0`, and resample onto the correct radial grid # (``transformer.r``) as required for the QDHT. Er = gauss1d(r, 0, Dr) # Initial field ErH = H.to_transform_r(Er) # Resampled field # %% # Perform Hankel Transform # ------------------------ # Convert from physical field to physical wavevector EkrH = H.qdht(ErH) # %% # Propagate the beam - loop # ------------------------- # Do the propagation in a loop over :math:`z` # Pre-allocate an array for field as a function of r and z Erz = np.zeros((nr, Nz), dtype=complex)