.. _sphx_glr_api_gallery_fields_electrodynamics:

Electrodynamics
------------------

Finite-difference time-domain (FDTD) solutions of Maxwell's equations on a
Yee grid, in 1D and 2D, plus a graded-conductivity absorbing boundary in
the spirit of Berenger's Perfectly Matched Layer. Also includes animated
demos of a Hertzian oscillating dipole antenna, a wave crossing a
dielectric slab, and a standing TM cavity mode.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Kane Yee&#x27;s staggered-grid update handles a wave crossing a material interface with no change to its core leapfrog rule -- only the permittivity map eps_r fed into it changes: the TMz-mode Maxwell curl equations solved at every grid cell,">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_dielectric_slab_propagation_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_dielectric_slab_propagation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A wave crossing a dielectric slab</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Twenty-two years after Maxwell&#x27;s equations predicted electromagnetic radiation, Heinrich Hertz built an oscillating dipole antenna -- an induction coil driving a spark gap -- and detected the resulting waves meters away, confirming they travel at the speed of light and can be reflected, refracted, and polarized just like ordinary light. On the 2D Yee grid used here, the out-of-plane field E_z and the in-plane fields H_x, H_y obey the source-free TMz-mode reduction of Maxwell&#x27;s curl equations,">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_dipole_radiation_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_dipole_radiation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A Hertzian dipole antenna radiating on the Yee grid</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="fdtd_2d_tmz and fdtd_2d_tmz_evolve integrate the same TMz-mode Maxwell curl equations as the other FDTD examples in this gallery, but already enforce Ez=0 on all four grid edges every step -- exactly a perfectly-conducting (PEC) cavity wall, no absorbing boundary needed. A rectangular PEC cavity of size L_x\times L_y supports discrete standing-wave modes">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_em_cavity_modes_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_em_cavity_modes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A standing TM cavity mode oscillating in place</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="James Clerk Maxwell unified electricity, magnetism, and optics into four coupled equations. In one dimension, source-free, they reduce to a coupled pair for the transverse electric and magnetic field components,">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_maxwell_speed_of_light_thumb.png
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_maxwell_speed_of_light`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Measuring the speed of light from Maxwell's equations</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Jean-Pierre Berenger solved a problem that had limited FDTD simulations since Yee&#x27;s original 1966 paper: a finite grid needs some boundary condition, and a simple wall reflects outgoing waves right back into the simulation. His Perfectly Matched Layer (PML) is a graded absorbing medium whose wave impedance matches the interior exactly, so outgoing waves are absorbed rather than reflected. pml_conductivity_profile implements a simplified (non-split-field) approximation to this idea: it grades an ordinary electric conductivity \sigma(x) smoothly up from zero over the outermost cells at each edge of the grid, so E_z there obeys the lossy telegraph-style update of a conducting medium rather than the lossless Ampere&#x27;s law,">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_pml_absorbing_boundary_thumb.png
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_pml_absorbing_boundary`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Berenger's Perfectly Matched Layer vs. a hard wall</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Kane Yee&#x27;s 1966 idea was deceptively simple: stagger the electric and magnetic field components on interleaved spatial and temporal grids (a &quot;Yee cell&quot;), so each field&#x27;s curl is naturally centered on the other&#x27;s location. fdtd_1d and fdtd_2d_tmz are direct implementations of this staggered-grid leapfrog scheme for the TMz-mode Maxwell curl equations,">

.. only:: html

  .. image:: /api/gallery/fields/electrodynamics/images/thumb/sphx_glr_plot_yee_fdtd_point_source_thumb.png
    :alt:

  :doc:`/api/gallery/fields/electrodynamics/plot_yee_fdtd_point_source`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A point source radiating on the Yee grid</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/fields/electrodynamics/plot_dielectric_slab_propagation
   /api/gallery/fields/electrodynamics/plot_dipole_radiation
   /api/gallery/fields/electrodynamics/plot_em_cavity_modes
   /api/gallery/fields/electrodynamics/plot_maxwell_speed_of_light
   /api/gallery/fields/electrodynamics/plot_pml_absorbing_boundary
   /api/gallery/fields/electrodynamics/plot_yee_fdtd_point_source