Add laser model description to docs

docs/source/index.rst

@@ -79,6 +79,7 @@ In case you are already fluent in compiling C++ projects and HPC, running PIC si
 
    models/pic
    models/AOFDTD
+   models/lasers
    models/total_field_scattered_field
    models/shapes
    models/LL_RR

docs/source/models/lasers.rst

@@ -0,0 +1,238 @@
+.. _model-lasers:
+
+Analytic Expressions for the 3D Laser Profiles
+==============================================
+
+.. sectionauthor:: Klaus Steiniger
+
+
+Among others, PIConGPU offers the ``GaussianPulse`` and ``DispersivePulse`` profiles to model complex, Gaussian-like laser pulse profiles in simulations.
+
+The ``GaussianPulse`` profile allows modelling standard Gaussian pulses defined by a Gaussian distribution of laser energy along the longitudinal and transverse directions.
+Furthermore, it allows including higher-order transverse modes via Laguerre-Gaussian modes [Pausch2022]_.
+
+The ``DispersivePulse`` profile allows modelling standard Gaussian pulses which feature dispersions, i.e. frequency specific modifications of the laser's spectral phase, up to third order.
+That is, time delay (TD), angular dispersion (AD), spatial dispersion (AD), group-delay dispersion (GDD), and third-order dispersion (TOD) are self-consistently taken into account in the propagation of these pulses.
+The profile assumes a Gaussian shape for the laser's spectrum.
+The respective in-focus values of these dispersions can be provided as parameters.
+However, the electric field values in time domain are computed from the field's values in frequency domain by a discrete Fourier transform.
+Therefore, it is possible to use any other shape of the laser's spectrum by modifying the profile's source code.
+
+A very concise description of the equations used in the ``GaussianPulse`` and ``DispersivePulse`` profiles are provided in the following.
+Please also refer to the in-code documentation of these and the other profiles in order to obtain more profile-specific information.
+Furthermore, refer to :ref:`Workflows / Adding Laser <usage-workflows-addLaser>` in order to see how to add a laser to a PIConGPU simulation.
+
+
+
+Definitions
+-----------
+Following [Steiniger2024]_, the electric field in frequency-space domain is assumed to be polarized along :math:`x` and defined as
+
+.. math::
+  \hat{\vec E}(\vec r, \Omega) = \hat E_\mathrm{A}(\vec r, \Omega) e^{-\imath \varphi(\vec r, \Omega)}\vec{\mathrm e}_x\,,
+
+where :math:`\Omega=2\pi\nu` is the angular frequency and :math:`\vec r` the position considered, :math:`\hat E_\mathrm{A}` is the spectral amplitude and
+:math:`\varphi=\tfrac{\Omega}{c} \vec{\mathrm e}_\Omega \cdot \vec r`
+the spectral phase of the pulse,
+with :math:`\vec{\mathrm e}_\Omega` being the propagation direction of frequency :math:`\Omega`.
+Dispersions are assumed to occur only in the plane spanned by the direction of pulse propagation :math:`\vec{\mathrm e}_z`, being equal to the direction of propagation of the central frequency :math:`\Omega_0`, and polarization :math:`\vec{\mathrm e}_x`, i.e. :math:`\vec{\mathrm e}_\Omega \cdot \vec{\mathrm e}_y = 0`.
+This implies, that the spectral phase does not vary along the :math:`y`-direction in focus, i.e.
+
+.. math::
+  \varphi |_{z=0} = \varphi(x,\Omega) |_{z=0} = k_x(\Omega) \cdot x\,,
+
+where :math:`k_x = \tfrac{\Omega}{c} \vec{\mathrm e}_\Omega \cdot \vec{\mathrm e}_x = -\tfrac{\Omega}{c}\sin\theta` with :math:`\theta=\theta(\Omega)` being the angle enclosed by the propagation directions of frequency :math:`\Omega` and the central laser frequency :math:`\Omega_0`.
+
+
+
+In the following, the spectral amplitude is assumed to be separable in a spectrum :math:`\epsilon_\Omega`, as well as transverse envelopes :math:`\epsilon_x` and :math:`\epsilon_y`
+
+.. math::
+  \hat E_{\mathrm A} = \epsilon_\Omega \epsilon_x \epsilon_y\,.
+
+
+.. math::
+  \epsilon_\Omega(\Omega)& = e^{-\frac{\tau_0^2}{4}(\Omega-\Omega_0)^2}\,, &
+    \tau_0& = \tau_\mathrm{FWHM,I} / \sqrt{2 \ln 2} \\
+  \epsilon_x(x)& = e^{-\frac{\left[x-x_0(\Omega)\right]^2}{w_{0,x}^2}}\,, &
+    w_{0,x}& = w_{\mathrm{FWHM,I},x} / \sqrt{2 \ln 2} \\
+  \epsilon_y(y)& = e^{-\frac{y^2}{w_{0,y}^2}}\,, &
+    w_{0,y}& = w_{\mathrm{FWHM,I},y} / \sqrt{2 \ln 2} \\
+
+:math:`x_0=x_0(\Omega)` is the center position of the spatial distribution of frequency :math:`\Omega` along the polarization direction in the focus.
+
+
+3D field in frequency-space domain
+----------------------------------
+
+Prerequisite: Huygens' integral in the Fresnel approximation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+[Siegman1986]_ shows on p. 633 and p. 636 formulas for the propagation of paraxial beams in 3D and 2D, respectively.
+They have an equal kernel
+
+.. math::
+  \hat K_s(s^\prime, z, \Omega) = \sqrt{\frac{\Omega}{2\pi c}}
+    \frac{e^{\imath\frac{\pi}{4}}}{\sqrt{z}}
+    e^{-\imath \frac{\Omega}{2 c z} (s-s^\prime)^{2}}\,,
+
+such that propagation can be computed as
+
+.. math::
+  & \text{(2D)}& \hat E_x(x, z, \Omega)& =
+      e^{-\imath\frac{\Omega}{c}z}
+      \int\limits_{-\infty}^{\infty}
+        \hat K_x(x^\prime, z, \Omega) \hat{E}(x^\prime, z=0, \Omega) \, \mathrm dx^\prime
+    \qquad \clubsuit \label{eq::Fresnel2D} \\
+  & \text{(3D)}& \hat{E}(x, y, z, \Omega)& =
+      e^{-\imath\frac{\Omega}{c}z}
+      \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}
+        \hat K_y(y^\prime, z, \Omega) \hat K_x(x^\prime, z, \Omega)
+        \hat{E}(x^\prime, y^\prime, z=0, \Omega) \, \mathrm dx^\prime \mathrm dy^\prime \\
+  & \text{(3D)}& \hat E(x, y, z, \Omega)& =
+      \epsilon_\Omega e^{-\imath\frac{\Omega}{c}z}
+      \int\limits_{-\infty}^{\infty}
+        \hat K_x(x^\prime, z, \Omega) \epsilon_{x^\prime}
+        \mathrm e^{-\imath \varphi(x^\prime, \Omega)}\, \mathrm dx^\prime
+      \int\limits_{-\infty}^{\infty}
+        \hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime \\
+  & & & = \hat E_x(x, z, \Omega)
+      \int\limits_{-\infty}^{\infty}
+        \hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime\,. \qquad \bigstar \label{eq::Fresnel3Dsep} \\
+  & & & \qquad \text{(provided $\hat E$ is separable and dispersion-free along $y$)}
+
+
+Derivation
+^^^^^^^^^^
+[Steiniger2024]_ computed the 2D :math:`\hat E_x(x,z,\Omega)` part of eq. :math:`\bigstar`, i.e. computed :math:`\clubsuit` and thereby omitted the last integral in :math:`\bigstar`.
+The result of this last integral can be read off from the known solution for the 2D part, being provided in eq. (6) of [Steiniger2024]_.
+
+.. math::
+  \hat E_x(x, z, \Omega) = &\
+    \epsilon_\Omega
+    \left[ 1 + \frac{z^2}{z_{\mathrm{R},x}^2} \right]^{-1/4}
+    e^{-\left[
+        x - \left( x_0 - \frac{c}{\Omega_0 w_{0,x}}\alpha z \right)
+      \right]^2
+      \left[ \frac{1}{w_x(z)^2} + \imath \frac{\Omega}{2c} R_x^{-1}(z) \right]
+    }
+    \\ &\ \times
+    e^{-\imath \frac{\Omega}{c}z}
+    e^{\imath \alpha \frac{x}{w_{0,x}} }
+    e^{\imath \frac{\alpha^2}{4}\frac{z}{z_{\mathrm R,x}} }
+    e^{\imath \frac{1}{2} \arctan\frac{z}{z_{\mathrm R,x}} }
+    e^{-\imath \frac{1}{2}GDD_\mathrm{foc}(\Omega-\Omega_0)^2}
+    e^{-\imath \frac{1}{6}TOD_\mathrm{foc}(\Omega-\Omega_0)^3}\\
+  z_{\mathrm R,s} = &\ \frac{\Omega_0 w_{0,s}^2}{2c} \\
+  w_s(z) = &\ w_{0,s} \sqrt{1 + \frac{z^2}{z_{\mathrm R,s}^2}} \\
+  R_s^{-1}(z) = &\ \frac{z}{z^2 + z_{\mathrm R,s}^2} \\
+  \alpha(\Omega) = &\
+    \frac{w_0}{c}\left[
+      \Omega_0 \theta^\prime (\Omega-\Omega_0)
+      + \frac{1}{2}\left( 2 \theta^\prime + \Omega_0 \theta^{\prime\prime} \right) (\Omega-\Omega_0)^2
+      \right. \\
+    &\ \qquad \left. + \frac{1}{6}\left(
+        3\theta^{\prime\prime} + \Omega_0\theta^{\prime\prime\prime} - \Omega_0 {\theta^\prime}^3
+      \right) (\Omega-\Omega_0)^3
+    \right]
+
+Note, :math:`k_x \approx - \alpha / w_{0,x}`.
+
+The sought result for :math:`\int \hat K_y \epsilon_{y^\prime}\,\mathrm dy^\prime` is obtained from the 2D solution :math:`\hat E_x(x,z,\Omega)` by dropping :math:`\epsilon_\Omega` and :math:`\mathrm e^{-\imath(\Omega/c)z}`, plus letting :math:`\alpha \rightarrow 0` and :math:`x_0 \rightarrow 0`.
+Hence,
+
+.. math::
+  \hat E_y(y,z,\Omega)& := \int\limits_{-\infty}^{\infty}
+        \hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime \\
+  & = \sqrt{\frac{\Omega}{2\pi c}} \frac{e^{\imath\frac{\pi}{4}}}{\sqrt{z}}
+      \int\limits_{-\infty}^{\infty} e^{-\frac{{y^\prime}^2}{w_{0,y^\prime}^2}}
+        e^{-\imath \frac{\Omega}{2 c z} (y-y^\prime)^{2}} \,\mathrm dy^\prime \\
+  & = \left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
+    \mathrm e^{-y^2\left[\frac{1}{w_y(z)^2} + \imath \frac{\Omega}{2c}R_y^{-1}(z)\right]}
+    \mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}\,.
+
+
+Result
+^^^^^^
+According to the definitions above
+
+.. math::
+  \hat E(x,y,z,\Omega) = \hat E_x \cdot \hat E_y\,.
+
+
+3D field in time-space domain
+-----------------------------
+Derivation
+^^^^^^^^^^
+For the Fourier transform to time-space domain, again the results of [Steiniger2024]_ can be reused since the modifications due to the presence of :math:`\hat E_y` are easy to incorporate.
+In fact, when applying the transformation :math:`\Omega \rightarrow \tfrac{1}{\tau_0}\Omega^\prime + \Omega_0`, as is done in the reference,
+
+.. math::
+  \hat E_y(y,z,\frac{1}{\tau_0}\Omega^\prime + \Omega_0) =
+    \left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
+    \mathrm e^{-\frac{y^2}{w_y(z)^2}} \mathrm e^{-\imath \Omega_0 \frac{y^2 R^{-1}_y(z)}{2c}}
+    \mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}
+    \mathrm e^{-\imath \frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0}\Omega^\prime}
+
+only the last exponential depends on frequency :math:`\Omega^\prime` and needs to be taken into account in the Fourier transform.
+This last exponential simply contributes a term to :math:`\gamma_4`, which is located between eqs. (13) and (14) in [Steiniger2024]_, in the form of
+
+.. math::
+  \gamma_4\; +\negthickspace= -\frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0}\,.
+
+
+Result
+^^^^^^
+In total, the 3D time-space domain field :math:`E(x,y,z,t)` of the ``DispersivePulse`` is obtained by taking the existing 2D solution :math:`E(x,z,t)`, cf. eq. (14) in [Steiniger2024]_, and applying
+
+.. math::
+  \gamma_4^\prime& = \gamma_4 - \frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0} \\
+  E(x,y,z,t)& = E(x,z,t,) |_{\gamma_4 \rightarrow \gamma_4^\prime} \cdot
+    \left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
+    \mathrm e^{-\frac{y^2}{w_y(z)^2}} \mathrm e^{-\imath \Omega_0 \frac{y^2 R^{-1}_y(z)}{2c}}
+    \mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}\,.
+
+
+Special case without dispersion: The ``GaussianPulse``
+-----------------------------------------------------------
+
+In that case
+
+.. math::
+  \gamma_1& = 1 \\
+  \gamma_2& = 0 \\
+  \gamma_3& = 0 \\
+  \gamma_4^\prime& = \left[
+      t - \frac{z}{c} - \frac{1}{2c}\left(x^2 R_x^{-1}(z)+y^2 R_y^{-1}(z)\right)
+    \right]\frac{1}{\tau_0}
+
+
+.. math::
+    E(x,y,z,t) = &\
+        \frac{1}{\tau_0\sqrt{\pi}}
+        \left( 1 + \frac{z^2}{z_{\mathrm R,x}^2} \right)^{-1/4}
+        \left( 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right)^{-1/4} \\
+    &\ \times
+        e^{\imath \Omega_0 \gamma_4^\prime \tau_0}
+        e^{\imath \frac{1}{2} \left(\arctan\frac{z}{z_{\mathrm R,x}} + \arctan\frac{z}{z_{\mathrm R,y}}\right)}
+        e^{-\left(\frac{x^2}{w_x^2} + \frac{y^2}{w_y^2}\right)}
+        e^{-{\gamma_4^\prime}^2}\,.
+
+
+References
+----------
+.. [Pausch2022]
+        R. Pausch et al.
+        *Modeling Beyond Gaussian Laser Pulses in Particle-in-Cell Simulations - The Impact of Higher Order Laser Modes*
+        2022 IEEE Advanced Accelerator Concepts Workshop (AAC), Long Island, NY, USA (2022).
+        https://doi.org/10.1109/AAC55212.2022.10822876
+
+.. [Siegman1986]
+        Siegman, Anthony E.
+        *Lasers*,
+        University science books (1986).
+
+.. [Steiniger2024]
+        K. Steiniger et al.
+        *Distortions in focusing laser pulses due to spatio-temporal couplings: an analytic description*,
+        High Power Laser Science and Engineering 12 (2024).
+        https://doi.org/10.1017/hpl.2023.96