.. _sphx_glr_api_gallery_optics_gaussian_beams:

Gaussian beams
--------------

Complex-beam-parameter (Gaussian) wave optics: Kogelnik and Li's ABCD
transformation of the complex beam parameter through a thin lens,
Siegman's :math:`M^2` beam-quality factor for real, non-diffraction-limited
beams, and the doughnut intensity and helical phase of an Allen et al.
Laguerre-Gaussian orbital-angular-momentum vortex.


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


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

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="After Maiman&#x27;s laser it became clear that real laser beams are neither idealized geometric rays nor infinite plane waves, but Gaussian beams. Kogelnik and Li&#x27;s 1966 paper showed that a Gaussian beam is completely characterized by one complex beam parameter q(z), and, crucially, that q transforms through any sequence of lenses, mirrors, or free-space sections by the same ABCD ray-transfer matrix already used for geometric rays -- unifying ray optics and Gaussian-beam wave optics under one formalism. propagate_q implements q&#x27; = (Aq+B)/(Cq+D), and q_to_beam_params recovers the physical beam radius w(z) at any plane.">

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  .. image:: /api/gallery/optics/gaussian_beams/images/thumb/sphx_glr_plot_gaussian_beam_through_lens_thumb.png
    :alt:

  :doc:`/api/gallery/optics/gaussian_beams/plot_gaussian_beam_through_lens`

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      <div class="sphx-glr-thumbnail-title">Kogelnik and Li: focusing a Gaussian beam through a thin lens</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Real laser beams are never perfectly diffraction-limited fundamental Gaussian modes: multimode content, aberrations, and gain-medium imperfections all make them diverge faster than an ideal beam of the same waist. Anthony Siegman showed that essentially every real laser beam&#x27;s second-moment width still propagates by the same functional form as an ideal Gaussian beam,">

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  .. image:: /api/gallery/optics/gaussian_beams/images/thumb/sphx_glr_plot_m2_beam_quality_thumb.png
    :alt:

  :doc:`/api/gallery/optics/gaussian_beams/plot_m2_beam_quality`

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      <div class="sphx-glr-thumbnail-title">Siegman's M^2 beam-quality factor</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Les Allen and coworkers showed that Laguerre-Gaussian light modes carrying a helical (spiral) phase front e^{il\phi} possess a well-defined orbital angular momentum of l\hbar per photon -- an intrinsic, mechanical property of light entirely separate from the spin angular momentum carried by polarization. The discovery opened the field of structured light and optical vortices. laguerre_gaussian_mode constructs the \mathrm{LG}_p^l mode amplitude">

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  .. image:: /api/gallery/optics/gaussian_beams/images/thumb/sphx_glr_plot_orbital_angular_momentum_thumb.png
    :alt:

  :doc:`/api/gallery/optics/gaussian_beams/plot_orbital_angular_momentum`

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      <div class="sphx-glr-thumbnail-title">Allen et al.: orbital angular momentum of a Laguerre-Gaussian vortex</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/optics/gaussian_beams/plot_gaussian_beam_through_lens
   /api/gallery/optics/gaussian_beams/plot_m2_beam_quality
   /api/gallery/optics/gaussian_beams/plot_orbital_angular_momentum