Betatron radiation
===================

This plugin allows to calculate on-fly the synchrotron radiation assuming a 
simplified description valid for the relativistic particles and the strong-wiggler regime. 
These assumptions can be formulated as the conditions on the Lorentz factor of the emitting 
particle :math:`\gamma_p\gg 1`, and the angular excursion of it's oscillations is 
:math:`\sigma_{\theta}\gg 1/\gamma_p`. This is a typical case for the laser-plasma betatron 
or non-linear Compton source.

For the assumptions above, radiation emitted by an electron during time :math:`\Delta t` can be described by the dimensionless spectral density distribution as:

.. math::
	\frac{\partial E_{rad} }{ \partial \hbar \omega_{ph}} = \frac{P_{rad}\; \Delta t  }{ \hbar\omega_c }  \; S\left(\frac{\omega}{\omega_c}\right) \,,

where the instantaneous power follows from the relativistic Larmor formula: 

.. math::

	P_{rad} = \frac{e^2 \gamma^6}{6 \pi\epsilon_0 c} \; [|\dot{\boldsymbol \beta}|^2 - (\boldsymbol \beta\times\dot{\boldsymbol \beta})^2] \,.

The spectral profile is approximated with the standard function of :math:`\xi=\omega/\omega_c`:

.. math::

	S(x) = \frac{9\sqrt{3}}{8\pi} \; x\; \int_x^{\infty} K_{5/3}(\xi)\; d\xi'\,,

where critical frequency is calculated from with `curvature radius along trajectory <https://en.wikipedia.org/wiki/Radius_of_curvature#Derivation>`_: 

.. math::

	\omega_c = \frac{3}{2}\;\gamma^3\; \cfrac{c }{ \rho} = \frac{3}{2}\;\gamma^3 \frac{\sqrt{ \left|\dot{\boldsymbol \beta}\right|^2 - \left(\boldsymbol \beta \cdot  \dot{\boldsymbol\beta} \right)^2 / |\boldsymbol \beta|^2} } {|\boldsymbol \beta|^2} \,,

and motion quantities are calculated from the motion equations with fields:

.. math::

	\boldsymbol\beta = \mathbf{u} /  \gamma  \,,\;\;  \partial_t \mathbf{u} = - \frac{e}{m c} ( \boldsymbol E + \boldsymbol\beta \times c \mathbf{B} )\,, 

	\partial_t \gamma = - \frac{e}{m c} \; \boldsymbol\beta \cdot \mathbf{E} \,,\;\;   \partial_t \boldsymbol\beta = \cfrac{1}{\gamma} \;(\partial_t \mathbf{u} - \boldsymbol\beta \; \partial_t \gamma)\,.

Radiation is emitted in the direction along electron propagation 
:math:`\boldsymbol u/|\boldsymbol u|` into the angular profile approximated as 
:math:`\propto (1-\beta\cos\theta)^{-5}`. The field is calculated in the discrete 
3D spectral-angular domain :math:`(\theta_x, \theta_y, \hbar\omega)`, and the particles are deposited 
using fast bilinear interpolations, so to account for the changing angular shapes and emission 
directions we add to emission direction a random angular spread 
:math:`\delta\theta \sim \mathcal  N(2^{-3/2} \gamma^{-1})`, and with time averages to 
the proper profile.

Optionally the plugin can deduce the photon momentum from the emitting electron in order to 
simulate the classical radiation reaction.

.. warning::
    Currently this plugin only works in the lab-frame simulations.

Plugin is activated for the chosen particle specie (electron) with the following method:

.. automethod:: fbpic.particles.Particles.activate_synchrotron

The calculated radiation data in the dimensionless units 
(:math:`\partial^2 E_{rad} / \partial \hbar \omega_{ph} \partial \theta`) can be added as a 
standard openPMD diagnostics:

.. autoclass:: fbpic.openpmd_diag.SynchrotronRadiationDiagnostic