.. _sphx_glr_auto_examples_random_media:

Random media
============

Seeded random fields, numerical ensembles, phase functions, and wavelength dependence.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Generate a reproducible collection of identical spheres in a uniform background. Sphere centres are proposed uniformly within the part of the box that keeps each sphere fully inside. Overlapping proposals are rejected; the generator raises an error if it cannot place every requested sphere within its attempt limit. This is sequential placement, rather than an equilibrium hard-sphere model.">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_random_spheres_thumb.png
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  :doc:`/auto_examples/random_media/random_spheres`

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      <div class="sphx-glr-thumbnail-title">Random nonoverlapping spheres</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Compare finite-sample ensemble phase functions for three Gaussian covariance lengths at a vacuum wavelength of 633 nm. The phase function is a probability density per steradian, normalized over solid angle. The corresponding polar-angle density includes the factor 2pisin(theta) and integrates over radians.">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_phase_function_thumb.png
    :alt:

  :doc:`/auto_examples/random_media/phase_function`

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      <div class="sphx-glr-thumbnail-title">Normalized phase function</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Sweep vacuum wavelength at fixed background refractive index, refractive-index fluctuation strength, and Gaussian covariance length. Total scattering and reduced scattering are different observables: forward scattering increases anisotropy and reduces the contribution to mu_s_prime = mu_s * (1 - g).">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_wavelength_dependence_thumb.png
    :alt:

  :doc:`/auto_examples/random_media/wavelength_dependence`

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      <div class="sphx-glr-thumbnail-title">Wavelength dependence of scattering and anisotropy</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Compare two covariance models with the same refractive index standard deviation and the same length parameter. Their definitions differ: Gaussian covariance is sigma_n2  exp(-r2 / (2  ell2)), while exponential covariance is sigma_n2 * exp(-r / ell). Equal parameters therefore do not give identical real-space profiles or scattering spectra. Here, r is the distance between two points in the medium. The covariance describes how their refractive-index fluctuations vary together as that distance increases.">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_covariance_models_thumb.png
    :alt:

  :doc:`/auto_examples/random_media/covariance_models`

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      <div class="sphx-glr-thumbnail-title">Gaussian and exponential spatial covariance</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Explore the full three-dimensional sample for each spatial covariance model. Drag to rotate, scroll to zoom, and hover to inspect the refractive index. Regions with lower refractive index are transparent; regions with higher refractive index are more opaque. Opacity is a display setting, not absorption. Histograms and central slices follow as secondary checks of the same seeded fields.">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_random_medium_thumb.png
    :alt:

  :doc:`/auto_examples/random_media/random_medium`

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      <div class="sphx-glr-thumbnail-title">Visualize seeded three-dimensional random media</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Gaussian probability distributions and spatial covariance are separate choices. These simulations compare Gaussian, exponential and Matérn covariances using the same refractive-index standard deviation and correlation length. The plotted covariance estimate averages products within each realization, then averages independent realizations. No spatial sample mean is removed.">

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  .. image:: /auto_examples/random_media/images/thumb/sphx_glr_theory_random_fields_thumb.png
    :alt:

  :doc:`/auto_examples/random_media/theory_random_fields`

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      <div class="sphx-glr-thumbnail-title">Theory figure: covariance and simulated random fields</div>
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    </div>


.. toctree::
   :hidden:

   /auto_examples/random_media/random_spheres
   /auto_examples/random_media/phase_function
   /auto_examples/random_media/wavelength_dependence
   /auto_examples/random_media/covariance_models
   /auto_examples/random_media/random_medium
   /auto_examples/random_media/theory_random_fields