Enable initialisation and estimation of optional spatio-temperal coupling parameters to gaussian profile

[huixingjian]

PR Description

This PR, synergized with PR #182, introduces the initialization and estimation of first-order spatio-temporal coupling parameters in the GaussianLongitudinalProfile class. Four additional parameters have been added to the constructor:

• Group-delay dispersion phi2 in $s^2$ parameterized by: $\phi^{(2)} = \frac{dt}{d\omega}$,
• Spatio-chirp zeta in $m \cdot s$ parameterized by: $\zeta = \frac{dx_0}{d\omega}$(Here $x_0$ is the beam center position),
• Angular dispersion beta in $s$ parameterized by: $\beta = \frac{d\theta_0}{d\omega}$( Here $\theta_0$ is the propagation angle of this component.)
• Finally stc_theta is the direction of transverse spatio-coupling in xoy plane

Issue addressed

With the introduction of spatio-temporal coupling (STC), the LongitudinalProfile class can leverage additional information from the transverse profile. Along with the aforementioned parameters, the waist and focus length are also incorporated into the constructor. The transverse points x and y have been set to default values of 0 in the evaluate member function.

In the laser utilities, a new function, get_STC, has been added to evaluate the STC parameters using the following formulas, with the laser envelope expressed as $a = a_0 e^{i\theta}$:

• The temporal chirp is calculated through
  $\Phi^{(2)} = \frac{4\phi^{(2)}}{4(\phi^{(2)})^2+\tau^4}$
  Here $\tau$ is duration in s, and phi2 refers to group-delay dispersion $\phi^{(2)} $, and $\Phi^{(2)}$ can be calculated by $\frac{\partial^2 \theta }{\partial t ^2}$.
• Similarly, the spatial chirp is tested through:
  $\nu = \frac{4\zeta }{w_0^2\tau^2+4\zeta^2}$
  Here $L_0$ and $w_0$ are the laser duration and laser waist respectively. $\zeta = \frac{dr_0}{d\omega}$, where $r_0$ is the transverse beam centroid ($\theta$ is the stc_thetamentioned above).
• Finally, the angular chirp term is tested through:
  $\beta = \frac{p}{ k_0}|_{\phi^{(2)}=0,\zeta=0}$
  with $p$ refering to the pulse front tilt $p = \frac{dt}{dr}$

Benefits

First order STC is enabled on both the initialisation and evaluation sides.

Test

The initialisation and the utils test of zeta and phi2 are tested through CI test test_STC.py

Important note

Currently the angular dispersion diagnostics of beta using get_STC only returns correct value when there's no spatio-temporal chirp, namely zeta and phi2 are 0.