.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optics/diffraction/plot_huygens_wavelets.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_optics_diffraction_plot_huygens_wavelets.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_optics_diffraction_plot_huygens_wavelets.py:


Huygens' wavelets: diffractive spreading of a pinhole
=========================================================

Christiaan Huygens' 1690 construction treats every point of an advancing
wavefront as the source of a new spherical secondary wavelet; the envelope
of all those wavelets a moment later reconstructs the propagated
wavefront. :func:`~physicskit.optics.wave.angular_spectrum_propagate`
realizes exactly this modern form of the picture. The scalar field
:math:`U_0(x,y)` right after the aperture is decomposed into its plane-wave
(angular) spectrum by a 2D Fourier transform; each plane-wave component,
labeled by transverse spatial frequency :math:`(f_x,f_y)`, is a secondary
wavelet that advances rigidly along :math:`z` picking up phase
:math:`e^{ik_z z}`; re-synthesizing (inverse Fourier transform) sums the
interference of all these wavelets at the new plane:

.. math::

    U(x,y;z) = \mathcal{F}^{-1}\!\left[
        \mathcal{F}[U_0](f_x,f_y)\; e^{ik_z z}
    \right], \qquad
    k_z = \begin{cases}
        \sqrt{k^2 - k_x^2 - k_y^2} & k_x^2+k_y^2 \le k^2 \quad \text{(propagating)}\\
        i\sqrt{k_x^2+k_y^2 - k^2} & k_x^2+k_y^2 > k^2 \quad \text{(evanescent)}
    \end{cases}

with :math:`k=2\pi/\lambda`, :math:`k_x=2\pi f_x`, :math:`k_y=2\pi f_y`.
Because this sums the exact plane-wave spectrum rather than a paraxial
approximation, it is Huygens' interference picture computed with no
approximation beyond the scalar-wave treatment of light itself. A
circular pinhole of radius ``radius = 0.01`` mm -- a few wavelengths
across at ``wavelength = 0.5e-3`` mm -- makes an especially clean
demonstration, since ray optics alone would predict a sharp-edged shadow
of the aperture at every distance :math:`z`, while Huygens' picture
predicts the diffractive spreading seen below as :math:`z` grows from
0.05 mm to 2.0 mm.

.. GENERATED FROM PYTHON SOURCE LINES 38-44

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.optics.visualizers import plot_diffraction_pattern
    from physicskit.optics.wave import angular_spectrum_propagate, circular_aperture








.. GENERATED FROM PYTHON SOURCE LINES 45-47

A pinhole a few wavelengths across: an array of Huygens point sources
-----------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-53

.. code-block:: Python


    wavelength = 0.5e-3  # mm
    dx = 0.002  # mm per pixel
    radius = 0.01  # mm
    aperture = circular_aperture((256, 256), dx=dx, radius=radius).astype(complex)








.. GENERATED FROM PYTHON SOURCE LINES 54-57

Every open pixel re-radiates a spherical wavelet; angular_spectrum_propagate
sums their interference at each successive plane -- Huygens' principle,
computed exactly rather than approximated.

.. GENERATED FROM PYTHON SOURCE LINES 57-72

.. code-block:: Python


    U_near = angular_spectrum_propagate(aperture, wavelength=wavelength, z=0.05, dx=dx)
    U_far = angular_spectrum_propagate(aperture, wavelength=wavelength, z=2.0, dx=dx)

    fig, axes = plt.subplots(1, 2, figsize=(10, 4.5))
    plot_diffraction_pattern(U_near, dx=dx, ax=axes[0])
    plot_diffraction_pattern(U_far, dx=dx, ax=axes[1])
    axes[0].set_title("z = 0.05 mm\nclose to the aperture's geometric shadow", fontsize=10)
    axes[1].set_title("z = 2.0 mm\ndiffractive spreading well beyond the shadow", fontsize=10)
    fig.tight_layout()

    print(f"aperture radius: {radius} mm, wavelength: {wavelength} mm")
    print("the far-field pattern has spread well past the aperture's geometric")
    print("shadow -- exactly what Huygens' wavelet construction predicts and")
    print("plain ray optics cannot explain.")



.. image-sg:: /api/gallery/optics/diffraction/images/sphx_glr_plot_huygens_wavelets_001.png
   :alt: z = 0.05 mm close to the aperture's geometric shadow, z = 2.0 mm diffractive spreading well beyond the shadow
   :srcset: /api/gallery/optics/diffraction/images/sphx_glr_plot_huygens_wavelets_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    aperture radius: 0.01 mm, wavelength: 0.0005 mm
    the far-field pattern has spread well past the aperture's geometric
    shadow -- exactly what Huygens' wavelet construction predicts and
    plain ray optics cannot explain.





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.100 seconds)


.. _sphx_glr_download_api_gallery_optics_diffraction_plot_huygens_wavelets.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_huygens_wavelets.ipynb <plot_huygens_wavelets.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_huygens_wavelets.py <plot_huygens_wavelets.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_huygens_wavelets.zip <plot_huygens_wavelets.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_