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Add laser model description to docs
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‎docs/source/index.rst‎

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

‎docs/source/models/lasers.rst‎

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.. _model-lasers:
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Analytic Expressions for the 3D Laser Profiles
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==============================================
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.. sectionauthor:: Klaus Steiniger
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A very concise description of the equations used in the ``GaussianPulse`` and ``DispersivePulse`` profiles are provided in the following.
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Please also refer to the in-code documentation of these and the other profiles in order to obtain more profile-specific information.
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Definitions
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-----------
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Following [Steiniger2024]_, the electric field in frequency-space domain is assumed to be polarized along :math:`x` and defined as
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.. math::
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\hat{\vec E}(\vec r, \Omega) = \hat E_\mathrm{A}(\vec r, \Omega) e^{-\imath \varphi(\vec r, \Omega)}\vec{\mathrm e}_x\,,
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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
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:math:`\varphi=\tfrac{\Omega}{c} \vec{\mathrm e}_\Omega \cdot \vec r`
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the spectral phase of the pulse,
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with :math:`\vec{\mathrm e}_\Omega` being the propagation direction of frequency :math:`\Omega`.
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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}_z`, i.e. :math:`\vec{\mathrm e}_\Omega \cdot \vec{\mathrm e}_y = 0`.
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This implies, that the spectral phase does not vary along the :math:`y`-direction in focus, i.e.
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.. math::
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\varphi |_{z=0} = \varphi(x,\Omega) |_{z=0} = k_x(\Omega) \cdot x\,,
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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`.
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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`
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.. math::
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\hat E_{\mathrm A} = \epsilon_\Omega \epsilon_x \epsilon_y\,.
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.. math::
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\epsilon_\Omega(\Omega)& = e^{-\frac{\tau_0^2}{4}(\Omega-\Omega_0)^2}\,, &
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\tau_0& = \tau_\mathrm{FWHM,I} / \sqrt{2 \ln 2} \\
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\epsilon_x(x)& = e^{-\frac{\left[x-x_0(\Omega)\right]^2}{w_{0,x}^2}}\,, &
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w_{0,x}& = w_{\mathrm{FWHM,I},x} / \sqrt{2 \ln 2} \\
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\epsilon_y(y)& = e^{-\frac{y^2}{w_{0,y}^2}}\,, &
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w_{0,y}& = w_{\mathrm{FWHM,I},y} / \sqrt{2 \ln 2} \\
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: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.
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3D field in frequency-space domain
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----------------------------------
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Prerequisite: Huygens' integral in the Fresnel approximation
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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[Siegman1986]_ shows on p. 633 and p. 636 formulas for the propagation of paraxial beams in 3D and 2D, respectively.
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They have an equal kernel
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.. math::
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\hat K_s(s^\prime, z, \Omega) = \sqrt{\frac{\Omega}{2\pi c}}
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\frac{e^{\imath\frac{\pi}{4}}}{\sqrt{z}}
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e^{-\imath \frac{\Omega}{2 c z} (s-s^\prime)^{2}}\,,
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such that propagation can be computed as
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.. math::
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& \text{(2D)}& \hat E_x(x, z, \Omega)& =
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e^{-\imath\frac{\Omega}{c}z}
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\int\limits_{-\infty}^{\infty}
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\hat K_x(x^\prime, z, \Omega) \hat{E}(x^\prime, z=0, \Omega) \, \mathrm dx^\prime
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\qquad \clubsuit \label{eq::Fresnel2D} \\
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& \text{(3D)}& \hat{E}(x, y, z, \Omega)& =
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e^{-\imath\frac{\Omega}{c}z}
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\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}
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\hat K_y(y^\prime, z, \Omega) \hat K_x(x^\prime, z, \Omega)
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\hat{E}(x^\prime, y^\prime, z=0, \Omega) \, \mathrm dx^\prime \mathrm dy^\prime \\
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& \text{(3D)}& \hat E(x, y, z, \Omega)& =
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\epsilon_\Omega e^{-\imath\frac{\Omega}{c}z}
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\int\limits_{-\infty}^{\infty}
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\hat K_x(x^\prime, z, \Omega) \epsilon_{x^\prime}
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\mathrm e^{-\imath \varphi(x^\prime, \Omega)}\, \mathrm dx^\prime
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\int\limits_{-\infty}^{\infty}
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\hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime \\
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& & & = \hat E_x(x, z, \Omega)
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\int\limits_{-\infty}^{\infty}
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\hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime\,. \qquad \bigstar \label{eq::Fresnel3Dsep} \\
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& & & \qquad \text{(provided $\hat E$ is separable and dispersion-free along $y$)}
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Derivation
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^^^^^^^^^^
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[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`.
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The result of this last integral can be read of the known solution for the 2D part, being provided in eq. (6) of [Steiniger2024]_.
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.. math::
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\hat E_x(x, z, \Omega) = &\
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\epsilon_\Omega
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\left[ 1 + \frac{z^2}{z_\mathrm{R,x}^2} \right]^{-1/4}
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e^{-\left[
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x - \left( x_0 - \frac{c}{\Omega_0 w_{0,x}}\alpha z \right)
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\right]^2
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\left[ \frac{1}{w_x(z)^2} + \imath \frac{\Omega}{2c} R_x^{-1}(z) \right]
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}
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\\ &\ \times
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e^{-\imath \frac{\Omega}{c}z}
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e^{\imath \alpha \frac{x}{w_{0,x}} }
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e^{\imath \frac{\alpha^2}{4}\frac{z}{z_{\mathrm R,x}} }
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e^{\imath \frac{1}{2} \arctan\frac{z}{z_{\mathrm R,x}} }
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e^{-\imath \frac{1}{2}GDD_\mathrm{foc}(\Omega-\Omega_0)^2}
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e^{-\imath \frac{1}{6}TOD_\mathrm{foc}(\Omega-\Omega_0)^3}\\
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z_{\mathrm R,s} = &\ \frac{\Omega_0 w_{0,s}^2}{2c} \\
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w_s(z) = &\ w_{0,s} \sqrt{1 + \frac{z^2}{z_{\mathrm R,s}^2}} \\
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R_s^{-1}(z) = &\ \frac{z}{z^2 + z_{\mathrm R,s}^2} \\
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\alpha(\Omega) = &\
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\frac{w_0}{c}\left[
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\Omega_0 \theta^\prime (\Omega-\Omega_0)
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+ \frac{1}{2}\left( 2 \theta^\prime + \Omega_0 \theta^{\prime\prime} \right) (\Omega-\Omega_0)^2
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\right. \\
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&\ \qquad \left. + \frac{1}{6}\left(
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3\theta^{\prime\prime} + \Omega_0\theta^{\prime\prime\prime} - \Omega_0 {\theta^\prime}^3
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\right) (\Omega-\Omega_0)^3
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\right]
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Note, :math:`k_x \approx - \alpha / w_{0,x}`.
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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`.
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Hence,
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.. math::
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\hat E_y(y,z,\Omega)& := \int\limits_{-\infty}^{\infty}
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\hat K_y(y^\prime, z, \Omega) \epsilon_{y^\prime} \,\mathrm dy^\prime \\
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& = \sqrt{\frac{\Omega}{2\pi c}} \frac{e^{\imath\frac{\pi}{4}}}{\sqrt{z}}
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\int\limits_{-\infty}^{\infty} e^{-\frac{{y^\prime}^2}{w_{0,y^\prime}^2}}
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e^{-\imath \frac{\Omega}{2 c z} (y-y^\prime)^{2}} \,\mathrm dy^\prime \\
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& = \left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
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\mathrm e^{-y^2\left[\frac{1}{w_y(z)^2} + \imath \frac{\Omega}{2c}R_y^{-1}(z)\right]}
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\mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}\,.
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Result
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^^^^^^
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According to the definitions above
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.. math::
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\hat E(x,y,z,\Omega) = \hat E_x \cdot \hat E_y\,.
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3D field in time-space domain
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-----------------------------
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Derivation
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^^^^^^^^^^
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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.
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In fact, when applying the transformation :math:`\Omega \rightarrow \tfrac{1}{\tau_0}\Omega^\prime + \Omega_0`, as is done in the reference,
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.. math::
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\hat E_y(y,z,\frac{1}{\tau_0}\Omega^\prime + \Omega_0) =
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\left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
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\mathrm e^{-\frac{y^2}{w_y(z)^2}} \mathrm e^{-\imath \Omega_0 \frac{y^2 R^{-1}_y(z)}{2c}}
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\mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}
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\mathrm e^{-\imath \frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0}\Omega^\prime}
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only the last exponential depends on frequency :math:`\Omega^\prime` and needs to be taken into account in the Fourier transform.
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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
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.. math::
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\gamma_4\; +\negthickspace= -\frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0}\,.
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Result
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^^^^^^
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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
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.. math::
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\gamma_4^\prime& = \gamma_4 - \frac{y^2 R_y^{-1}(z)}{2c}\frac{1}{\tau_0} \\
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E(x,y,z,t)& = E(x,z,t,) |_{\gamma_4 \rightarrow \gamma_4^\prime} \cdot
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\left[ 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right]^{-1/4}
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\mathrm e^{-\frac{y^2}{w_y(z)^2}} \mathrm e^{-\imath \Omega_0 \frac{y^2 R^{-1}_y(z)}{2c}}
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\mathrm e^{\imath\frac{1}{2}\arctan\frac{z}{z_{\mathrm R,y}}}\,.
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Special case without dispersion: The ``GaussianPulse``
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-----------------------------------------------------------
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In that case
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.. math::
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\gamma_1& = 1 \\
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\gamma_2& = 0 \\
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\gamma_3& = 0 \\
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\gamma_4^\prime& = \left[
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t - \frac{z}{c} - \frac{1}{2c}\left(x^2 R_x^{-1}(z)+y^2 R_y^{-1}(z)\right)
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\right]\frac{1}{\tau_0}
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.. math::
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E(x,y,z,t) = &\
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\frac{1}{\tau_0\sqrt{\pi}}
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\left( 1 + \frac{z^2}{z_{\mathrm R,x}^2} \right)^{-1/4}
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\left( 1 + \frac{z^2}{z_{\mathrm R,y}^2} \right)^{-1/4} \\
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&\ \times
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e^{\imath \Omega_0 \gamma_4^\prime \tau_0}
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e^{\imath \frac{1}{2} \left(\arctan\frac{z}{z_{\mathrm R,x}} + \arctan\frac{z}{z_{\mathrm R,y}}\right)}
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e^{-\left(\frac{x^2}{w_x^2} + \frac{y^2}{w_y^2}\right)}
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e^{-{\gamma_4^\prime}^2}\,.
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References
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----------
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.. [Steiniger2024]
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K. Steiniger et al.
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*Distortions in focusing laser pulses due to spatio-temporal couplings: an analytic description*,
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High Power Laser Science and Engineering 12 (2024).
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https://doi.org/10.1017/hpl.2023.96
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.. [Siegman1986]
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Siegman, Anthony E.
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*Lasers*,
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University science books (1986).
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