Update doc

Author: RemiLehe

_modules/fbpic/lpa_utils/laser/laser.html

@@ -200,11 +200,11 @@ <h1>Source code for fbpic.lpa_utils.laser.laser</h1><div class="highlight"><pre>
 
 <span class="sd">    .. math::</span>
 
-<span class="sd">        E(\\boldsymbol{x},t) = a_0\\times E_0\,</span>
-<span class="sd">        \exp\left( -\\frac{r^2}{w_0^2} - \\frac{(z-z_0-ct)^2}{c^2\\tau^2} \\right)</span>
+<span class="sd">        E(\boldsymbol{x},t) = a_0\times E_0\,</span>
+<span class="sd">        \exp\left( -\frac{r^2}{w_0^2} - \frac{(z-z_0-ct)^2}{c^2\tau^2} \right)</span>
 <span class="sd">        \cos[ k_0( z - z_0 - ct ) - \phi_{cep} ]</span>
 
-<span class="sd">    where :math:`k_0 = 2\pi/\\lambda_0` is the wavevector and where</span>
+<span class="sd">    where :math:`k_0 = 2\pi/\lambda_0` is the wavevector and where</span>
 <span class="sd">    :math:`E_0 = m_e c^2 k_0 / q_e` is the field amplitude for :math:`a_0=1`.</span>
 
 <span class="sd">    .. note::</span>
@@ -233,7 +233,7 @@ <h1>Source code for fbpic.lpa_utils.laser.laser</h1><div class="highlight"><pre>
 
 <span class="sd">    ctau: float (in meter)</span>
 <span class="sd">        The duration of the laser (in the lab frame),</span>
-<span class="sd">        defined as :math:`c\\tau` in the above formula.</span>
+<span class="sd">        defined as :math:`c\tau` in the above formula.</span>
 
 <span class="sd">    z0: float (in meter)</span>
 <span class="sd">        The initial position of the centroid of the laser</span>
@@ -249,7 +249,7 @@ <h1>Source code for fbpic.lpa_utils.laser.laser</h1><div class="highlight"><pre>
 
 <span class="sd">    lambda0: float (in meter), optional</span>
 <span class="sd">        The wavelength of the laser (in the lab frame), defined as</span>
-<span class="sd">        :math:`\\lambda_0` in the above formula.</span>
+<span class="sd">        :math:`\lambda_0` in the above formula.</span>
 
 <span class="sd">    cep_phase: float (in radian), optional</span>
 <span class="sd">        The Carrier Enveloppe Phase (CEP), defined as :math:`\phi_{cep}`</span>

_modules/fbpic/lpa_utils/laser/laser_profiles.html

@@ -638,29 +638,29 @@ <h1>Source code for fbpic.particles.particles</h1><div class="highlight"><pre>
 <span class="sd">        vector throughout the simulation.</span>
 
 <span class="sd">        .. math::</span>
-<span class="sd">            \\frac{d\\boldsymbol{s}}{dt} = (\\boldsymbol{\\Omega}_T +</span>
-<span class="sd">             \\boldsymbol{\\Omega}_a) \\times \\boldsymbol{s}</span>
+<span class="sd">            \frac{d\boldsymbol{s}}{dt} = (\boldsymbol{\Omega}_T +</span>
+<span class="sd">             \boldsymbol{\Omega}_a) \times \boldsymbol{s}</span>
 
 <span class="sd">        where</span>
 
 <span class="sd">        .. math::</span>
-<span class="sd">            \\boldsymbol{\\Omega}_T = \\frac{q}{m}\\left(</span>
-<span class="sd">                 \\frac{\\boldsymbol{B}}{\\gamma} -</span>
-<span class="sd">                 \\frac{\\boldsymbol{B}}{1+\\gamma}</span>
-<span class="sd">                 \\times \\frac{\\boldsymbol{E}}{c} \\right)</span>
+<span class="sd">            \boldsymbol{\Omega}_T = \frac{q}{m}\left(</span>
+<span class="sd">                 \frac{\boldsymbol{B}}{\gamma} -</span>
+<span class="sd">                 \frac{\boldsymbol{B}}{1+\gamma}</span>
+<span class="sd">                 \times \frac{\boldsymbol{E}}{c} \right)</span>
 
 <span class="sd">        and</span>
 
 <span class="sd">        .. math::</span>
-<span class="sd">            \\boldsymbol{\\Omega}_a = a_e \\frac{q}{m}\\left(</span>
-<span class="sd">                 \\boldsymbol{B} -</span>
-<span class="sd">                 \\frac{\\gamma}{1+\\gamma}\\boldsymbol{\\beta}</span>
-<span class="sd">                 (\\boldsymbol{\\beta}\\cdot\\boldsymbol{B}) -</span>
-<span class="sd">                 \\boldsymbol{\\beta} \\times \\frac{\\boldsymbol{E}}{c} \\right)</span>
+<span class="sd">            \boldsymbol{\Omega}_a = a_e \frac{q}{m}\left(</span>
+<span class="sd">                 \boldsymbol{B} -</span>
+<span class="sd">                 \frac{\gamma}{1+\gamma}\boldsymbol{\beta}</span>
+<span class="sd">                 (\boldsymbol{\beta}\cdot\boldsymbol{B}) -</span>
+<span class="sd">                 \boldsymbol{\beta} \times \frac{\boldsymbol{E}}{c} \right)</span>
 
 <span class="sd">        Here, :math:`a_e` is the anomalous magnetic moment of the particle,</span>
-<span class="sd">        :math:`\\gamma` is the Lorentz factor of the particle,</span>
-<span class="sd">        :math:`\\boldsymbol{\\beta}=\\boldsymbol{v}/c` is the normalised velocity</span>
+<span class="sd">        :math:`\gamma` is the Lorentz factor of the particle,</span>
+<span class="sd">        :math:`\boldsymbol{\beta}=\boldsymbol{v}/c` is the normalised velocity</span>
 
 <span class="sd">        The implementation of the push algorithm is detailed in</span>
 <span class="sd">        https://arxiv.org/abs/2303.16966.</span>

_modules/fbpic/particles/particles.html

@@ -240,10 +240,10 @@ <h2>Compact function for a Gaussian pulse<a class="headerlink" href="#compact-fu
 <p>More precisely, the electric field <strong>near the focal plane</strong>
 is given by:</p>
 <div class="math notranslate nohighlight">
-\[\begin{split}E(\\boldsymbol{x},t) = a_0\\times E_0\,
-\exp\left( -\\frac{r^2}{w_0^2} - \\frac{(z-z_0-ct)^2}{c^2\\tau^2} \\right)
-\cos[ k_0( z - z_0 - ct ) - \phi_{cep} ]\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(k_0 = 2\pi/\\lambda_0\)</span> is the wavevector and where
+\[E(\boldsymbol{x},t) = a_0\times E_0\,
+\exp\left( -\frac{r^2}{w_0^2} - \frac{(z-z_0-ct)^2}{c^2\tau^2} \right)
+\cos[ k_0( z - z_0 - ct ) - \phi_{cep} ]\]</div>
+<p>where <span class="math notranslate nohighlight">\(k_0 = 2\pi/\lambda_0\)</span> is the wavevector and where
 <span class="math notranslate nohighlight">\(E_0 = m_e c^2 k_0 / q_e\)</span> is the field amplitude for <span class="math notranslate nohighlight">\(a_0=1\)</span>.</p>
 <div class="admonition note">
 <p class="admonition-title">Note</p>
@@ -264,15 +264,15 @@ <h2>Compact function for a Gaussian pulse<a class="headerlink" href="#compact-fu
 <li><p><strong>w0</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em>) – Laser waist at the focal plane, defined as <span class="math notranslate nohighlight">\(w_0\)</span> in the
 above formula.</p></li>
 <li><p><strong>ctau</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em>) – The duration of the laser (in the lab frame),
-defined as <span class="math notranslate nohighlight">\(c\\tau\)</span> in the above formula.</p></li>
+defined as <span class="math notranslate nohighlight">\(c\tau\)</span> in the above formula.</p></li>
 <li><p><strong>z0</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em>) – The initial position of the centroid of the laser
 (in the lab frame), defined as <span class="math notranslate nohighlight">\(z_0\)</span> in the above formula.</p></li>
 <li><p><strong>zf</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em><em>, </em><em>optional</em>) – The position of the focal plane (in the lab frame).
 If <code class="docutils literal notranslate"><span class="pre">zf</span></code> is not provided, the code assumes that <code class="docutils literal notranslate"><span class="pre">zf=z0</span></code>, i.e.
 that the laser pulse is at the focal plane initially.</p></li>
 <li><p><strong>theta_pol</strong> (<em>float</em><em> (</em><em>in radian</em><em>)</em><em>, </em><em>optional</em>) – The angle of polarization with respect to the x axis.</p></li>
 <li><p><strong>lambda0</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em><em>, </em><em>optional</em>) – The wavelength of the laser (in the lab frame), defined as
-<span class="math notranslate nohighlight">\(\\lambda_0\)</span> in the above formula.</p></li>
+<span class="math notranslate nohighlight">\(\lambda_0\)</span> in the above formula.</p></li>
 <li><p><strong>cep_phase</strong> (<em>float</em><em> (</em><em>in radian</em><em>)</em><em>, </em><em>optional</em>) – The Carrier Enveloppe Phase (CEP), defined as <span class="math notranslate nohighlight">\(\phi_{cep}\)</span>
 in the above formula (i.e. the phase of the laser
 oscillation, at the position where the laser enveloppe is maximum)</p></li>

api_reference/lpa_utilities/laser.html

@@ -104,19 +104,19 @@ <h1>Donut-like Laguerre-Gauss profile<a class="headerlink" href="#donut-like-lag
 <dd><p>Define a linearly-polarized donut-like Laguerre-Gauss laser profile.</p>
 <p>Unlike the <a class="reference internal" href="laguerre.html#fbpic.lpa_utils.laser.LaguerreGaussLaser" title="fbpic.lpa_utils.laser.LaguerreGaussLaser"><code class="xref any py py-class docutils literal notranslate"><span class="pre">LaguerreGaussLaser</span></code></a> profile, this
 profile has a phase which depends on the azimuthal angle
-<span class="math notranslate nohighlight">\(\\theta\)</span> (cork-screw pattern), and an intensity profile which
-is independent on <span class="math notranslate nohighlight">\(\\theta\)</span> (donut-like).</p>
+<span class="math notranslate nohighlight">\(\theta\)</span> (cork-screw pattern), and an intensity profile which
+is independent on <span class="math notranslate nohighlight">\(\theta\)</span> (donut-like).</p>
 <p>More precisely, the electric field <strong>near the focal plane</strong>
 is given by:</p>
 <div class="math notranslate nohighlight">
-\[ \begin{align}\begin{aligned}\begin{split}E(\\boldsymbol{x},t) = a_0\\times E_0 \, f(r) \,
-\exp\left( -\\frac{r^2}{w_0^2} - \\frac{(z-z_0-ct)^2}{c^2\\tau^2}
-\\right) \cos[ k_0( z - z_0 - ct ) - m\\theta - \phi_{cep} ]\end{split}\\\begin{split}\mathrm{with} \qquad f(r) =
-\sqrt{\\frac{p!}{(|m|+p)!}}
-\\left( \\frac{\sqrt{2}r}{w_0} \\right)^{|m|}
-L^{|m|}_p\\left( \\frac{2 r^2}{w_0^2} \\right)\end{split}\end{aligned}\end{align} \]</div>
+\[ \begin{align}\begin{aligned}E(\boldsymbol{x},t) = a_0\times E_0 \, f(r) \,
+\exp\left( -\frac{r^2}{w_0^2} - \frac{(z-z_0-ct)^2}{c^2\tau^2}
+\right) \cos[ k_0( z - z_0 - ct ) - m\theta - \phi_{cep} ]\\\mathrm{with} \qquad f(r) =
+\sqrt{\frac{p!}{(|m|+p)!}}
+\left( \frac{\sqrt{2}r}{w_0} \right)^{|m|}
+L^{|m|}_p\left( \frac{2 r^2}{w_0^2} \right)\end{aligned}\end{align} \]</div>
 <p>where <span class="math notranslate nohighlight">\(L^m_p\)</span> is a Laguerre polynomial,
-<span class="math notranslate nohighlight">\(k_0 = 2\pi/\\lambda_0\)</span> is the wavevector and where
+<span class="math notranslate nohighlight">\(k_0 = 2\pi/\lambda_0\)</span> is the wavevector and where
 <span class="math notranslate nohighlight">\(E_0 = m_e c^2 k_0 / q_e\)</span>.</p>
 <p>(For more info, see
 <a class="reference external" href="https://www.osapublishing.org/books/bookshelf/lasers.cfm">Siegman, Lasers (1986)</a>,
@@ -142,23 +142,23 @@ <h1>Donut-like Laguerre-Gauss profile<a class="headerlink" href="#donut-like-lag
 <li><p><strong>p</strong> (<em>int</em>) – The order of the Laguerre polynomial. (Increasing <code class="docutils literal notranslate"><span class="pre">p</span></code> increases
 the number of “rings” in the radial intensity profile of the laser.)</p></li>
 <li><p><strong>m</strong> (<em>int</em><em> (</em><em>positive</em><em> or </em><em>negative</em><em>)</em>) – The azimuthal order of the pulse. The laser phase in a given
-transverse plane varies as <span class="math notranslate nohighlight">\(m \\theta\)</span>.</p></li>
+transverse plane varies as <span class="math notranslate nohighlight">\(m \theta\)</span>.</p></li>
 <li><p><strong>a0</strong> (<em>float</em><em> (</em><em>dimensionless</em><em>)</em>) – The amplitude of the pulse, defined so that the total
 energy of the pulse is the same as that of a Gaussian pulse
-with the same <span class="math notranslate nohighlight">\(a_0\)</span>, <span class="math notranslate nohighlight">\(w_0\)</span> and <span class="math notranslate nohighlight">\(\\tau\)</span>.
+with the same <span class="math notranslate nohighlight">\(a_0\)</span>, <span class="math notranslate nohighlight">\(w_0\)</span> and <span class="math notranslate nohighlight">\(\tau\)</span>.
 (i.e. The energy of the pulse is independent of <code class="docutils literal notranslate"><span class="pre">p</span></code> and <code class="docutils literal notranslate"><span class="pre">m</span></code>.)</p></li>
 <li><p><strong>waist</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em>) – Laser waist at the focal plane, defined as <span class="math notranslate nohighlight">\(w_0\)</span> in the
 above formula.</p></li>
 <li><p><strong>tau</strong> (<em>float</em><em> (</em><em>in second</em><em>)</em>) – The duration of the laser (in the lab frame),
-defined as <span class="math notranslate nohighlight">\(\\tau\)</span> in the above formula.</p></li>
+defined as <span class="math notranslate nohighlight">\(\tau\)</span> in the above formula.</p></li>
 <li><p><strong>z0</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em>) – The initial position of the centroid of the laser
 (in the lab frame), defined as <span class="math notranslate nohighlight">\(z_0\)</span> in the above formula.</p></li>
 <li><p><strong>zf</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em><em>, </em><em>optional</em>) – The position of the focal plane (in the lab frame).
 If <code class="docutils literal notranslate"><span class="pre">zf</span></code> is not provided, the code assumes that <code class="docutils literal notranslate"><span class="pre">zf=z0</span></code>, i.e.
 that the laser pulse is at the focal plane initially.</p></li>
 <li><p><strong>theta_pol</strong> (<em>float</em><em> (</em><em>in radian</em><em>)</em><em>, </em><em>optional</em>) – The angle of polarization with respect to the x axis.</p></li>
 <li><p><strong>lambda0</strong> (<em>float</em><em> (</em><em>in meter</em><em>)</em><em>, </em><em>optional</em>) – The wavelength of the laser (in the lab frame), defined as
-<span class="math notranslate nohighlight">\(\\lambda_0\)</span> in the above formula.
+<span class="math notranslate nohighlight">\(\lambda_0\)</span> in the above formula.
 Default: 0.8 microns (Ti:Sapph laser).</p></li>
 <li><p><strong>cep_phase</strong> (<em>float</em><em> (</em><em>in radian</em><em>)</em><em>, </em><em>optional</em>) – The Carrier Enveloppe Phase (CEP), defined as <span class="math notranslate nohighlight">\(\phi_{cep}\)</span>
 in the above formula (i.e. the phase of the laser