physics.fresnel_diffract

Title: Fresnel Diffraction for Coherent and Monochromatic

Wave Fields

Fresnel Diffraction describes the behavior of a wave field as it moves through free space or interacts with an object under the small-angle approximation. It is particularly useful for near field diffraction.

The following algorithm is an adaptation of the ‘transfer function’ based approach contained in the reference. It is critically sampled when: pixel_size = wavelength * prop_dist / side_length

Or equivalently: pixel_size = sqrt(wavelength * prop_dist / pixel_num)

Under and oversampling occur when the left-hand side is less than or greater than the right-hand side, respectively.

This code is adapted and modified from: Computational Fourier Optics: A MATLAB Tutorial by David Voelz

Functions

_fresnel_diffract_1d(→ numpy.ndarray)	Fresnel Diffraction of 1D Wave Fields.
_fresnel_diffract_2d(→ numpy.ndarray)	Fresnel Diffraction of 2D Wave Fields.
fresnel_diffract(→ numpy.ndarray)	Fresnel Diffraction of 1D or 2D Wave Fields.

Module Contents

physics.fresnel_diffract._fresnel_diffract_1d(wavefunc_0: numpy.ndarray, pixel_size: float, wavelength: float, prop_dist: float) → numpy.ndarray

Fresnel Diffraction of 1D Wave Fields. This private function is called by ‘fresnel_diffract’ to handle the fresnel diffraction of 1D wave fields specifically. Args:

wavefunc0 (np.ndarray): The initial 1D wave field at the unpropagated plane. pixel_size (float): The physical size of a pixel (or data point) at the unpropagated plane. wavelength (float): The wavelength of the wave field. prop_dist (float): The desired propagation distance.

Returns:

np.ndarray: The 1D wave field at the propagated plane.

Examples:

>>> import numpy as np
>>> res = _fresnel_diffract_1d(np.ones(64), 1, 1, 1)
>>> res.shape
(64,)

# Conservation of energy >>> import numpy as np >>> wf0 = np.ones(64) >>> wfz = _fresnel_diffract_1d(wf0, 1, 1, 1) >>> bool(np.isclose(np.sum(abs(wf0)**2), np.sum(abs(wfz)**2))) True

# Test that propagation distance of 0 returns the contact image >>> import numpy as np >>> x = np.linspace(-32, 32, 1) >>> wf0 = np.where(abs(x)<=8, 1, 0) >>> wfz = _fresnel_diffract_1d(wf0, 1, 1, 0) >>> np.allclose(wf0, wfz) True

physics.fresnel_diffract._fresnel_diffract_2d(wavefunc_0: numpy.ndarray, pixel_size: float, wavelength: float, prop_dist: float) → numpy.ndarray

Fresnel Diffraction of 2D Wave Fields. This private function is called by ‘fresnel_diffract’ to handle the fresnel diffraction of 2D wave fields specifically. Args:

wavefunc_0 (np.ndarray): The initial 2D wave field at the unpropagated plane. pixel_size (float): The physical size of a pixel (or data point) at the unpropagated plane. wavelength (float): The wavelength of the wave field. prop_dist (float): The desired propagation distance.

Returns:

np.ndarray: The 2D wave field at the propagated plane.

Examples:

>>> import numpy as np
>>> res = _fresnel_diffract_2d(np.ones((64, 64)), 1, 1, 1)
>>> res.shape
(64, 64)
>>> import numpy as np
>>> wf0 = np.ones((64, 64))
>>> wfz = _fresnel_diffract_2d(wf0, 1, 1, 1)
>>> bool(np.isclose(np.sum(abs(wf0)**2), np.sum(abs(wfz)**2)))
True

# Test that propagation distance of 0 returns the contact image >>> import numpy as np >>> x = np.linspace(-32, 32, 1) >>> X1, X2 = np.meshgrid(x, x) >>> wf0 = np.where(abs(X1)<=8, 1, 0) * np.where(abs(X2)<=8, 1, 0) >>> wfz = _fresnel_diffract_2d(wf0, 1, 1, 0) >>> np.allclose(wf0, wfz) True

physics.fresnel_diffract.fresnel_diffract(wavefunc_0: numpy.ndarray, pixel_size: float, wavelength: float, prop_dist: float) → numpy.ndarray

Fresnel Diffraction of 1D or 2D Wave Fields.

This function calculates the Fresnel diffraction of a given wave field, suitable for near-field diffraction. The wave field is assumed to be coherent and monochromatic.

Args:

wavefunc0 (np.ndarray): The initial wave field at the unpropagated plane. pixel_size (float): The physical size of a pixel (or data point) at the pixel_size (float): The physical size of a pixel (or data point) at the unpropagated plane. wavelength (float): The wavelength of the wave field. prop_dist (float): The desired propagation distance.

Raises:

ValueError: If the input wave field is not 1D or 2D.

Returns:

np.ndarray: The wave field at the propagated plane.

Examples:

>>> import numpy as np
>>> res = fresnel_diffract(np.ones(64), 1, 1, 1)
>>> res.shape
(64,)
>>> import numpy as np
>>> res = fresnel_diffract(np.ones((64, 64)), 1, 1, 1)
>>> res.shape
(64, 64)
>>> import numpy as np
>>> res = fresnel_diffract(np.ones((4, 4, 4)), 1, 1, 1)
Traceback (most recent call last):
    ...
ValueError: Expected a 1D or 2D wavefield, but got (4, 4, 4)

# Test that conservation of energy is obeyed >>> import numpy as np >>> wf0 = np.ones(64) >>> wfz = fresnel_diffract(wf0, 1, 1, 1) >>> bool(np.isclose(np.sum(abs(wf0)**2), np.sum(abs(wfz)**2))) True >>> import numpy as np >>> wf0 = np.ones((64, 64)) >>> wfz = fresnel_diffract(wf0, 1, 1, 1) >>> bool(np.isclose(np.sum(abs(wf0)**2), np.sum(abs(wfz)**2))) True

# Test that propagation distance of 0 returns the contact image >>> import numpy as np >>> x = np.linspace(-32, 32, 1) >>> wf0 = np.where(abs(x)<=8, 1, 0) >>> wfz = fresnel_diffract(wf0, 1, 1, 0) >>> np.allclose(wf0, wfz) True