Flattened Gaussian profile

class fbpic.lpa_utils.laser.FlattenedGaussianLaser(a0, w0, tau, z0, N=6, zf=None, theta_pol=0.0, lambda0=8e-07, cep_phase=0.0, propagation_direction=1)[source]

Define a linearly-polarized laser such that the transverse intensity profile is a flattened Gaussian far from focus, and a distribution with rings in the focal plane. (See Santarsiero et al., J. Modern Optics, 1997)

Increasing the parameter N increases the flatness of the transverse profile far from focus, and increases the number of rings in the focal plane.

More precisely, the expression in the focal plane uses the Laguerre polynomials \(L^0_n\), and reads:

\[ \begin{align}\begin{aligned}E(\boldsymbol{x},t)\propto \exp\left(-\frac{r^2}{(N+1)w_0^2}\right) \sum_{n=0}^N c'_n L^0_n\left(\frac{2\,r^2}{(N+1)w_0^2}\right)\\\mathrm{with} \qquad c'_n = \sum_{m=n}^{N}\frac{1}{2^m}\binom{m}{n}\end{aligned}\end{align} \]

  • For \(N=0\), this is a Gaussian profile: \(E\propto\exp\left(-\frac{r^2}{w_0^2}\right)\).

  • For \(N\rightarrow\infty\), this is a Jinc profile: \(E\propto \frac{J_1(r/w_0)}{r/w_0}\).

The expression far from focus is

\[ \begin{align}\begin{aligned}E(\boldsymbol{x},t)\propto \exp\left(-\frac{(N+1)r^2}{w(z)^2}\right) \sum_{n=0}^N \frac{1}{n!}\left(\frac{(N+1)\,r^2}{w(z)^2}\right)^n\\\mathrm{with} \qquad w(z) = \frac{\lambda_0}{\pi w_0}|z-z_{foc}|\end{aligned}\end{align} \]

  • For \(N=0\), this is a Gaussian profile: \(E\propto\exp\left(-\frac{r^2}{w_(z)^2}\right)\).

  • For \(N\rightarrow\infty\), this is a flat profile: \(E\propto \Theta(w(z)-r)\).

Parameters:

  • a0 (float (dimensionless)) – The peak normalized vector potential at the focal plane.

  • w0 (float (in meter)) – Laser spot size in the focal plane, defined as \(w_0\) in the above formula.

  • tau (float (in second)) – The duration of the laser (in the lab frame)

  • z0 (float (in meter)) – The initial position of the centroid of the laser (in the lab frame)

  • N (int) – Determines the “flatness” of the transverse profile, far from focus (see the above formula). Default: N=6 ; somewhat close to an 8th order supergaussian.

  • zf (float (in meter), optional) – The position of the focal plane (in the lab frame). If zf is not provided, the code assumes that zf=z0, i.e. that the laser pulse is at the focal plane initially.

  • theta_pol (float (in radian), optional) – The angle of polarization with respect to the x axis.

  • lambda0 (float (in meter), optional) – The wavelength of the laser (in the lab frame) Default: 0.8 microns (Ti:Sapph laser).

  • cep_phase (float (in radian), optional) – The Carrier Enveloppe Phase (CEP, i.e. the phase of the laser oscillation, at the position where the laser enveloppe is maximum)

  • propagation_direction (int, optional) – Indicates in which direction the laser propagates. This should be either 1 (laser propagates towards positive z) or -1 (laser propagates towards negative z).