Laguerre-Gauss profile

class fbpic.lpa_utils.laser.LaguerreGaussLaser(p, m, a0, waist, tau, z0, zf=None, theta_pol=0.0, lambda0=8e-07, cep_phase=0.0, theta0=0.0, propagation_direction=1)[source]

Define a linearly-polarized Laguerre-Gauss laser profile.

Unlike the DonutLikeLaguerreGaussLaser profile, this profile has a phase which is independent of the azimuthal angle \(theta\), and an intensity profile which does depend on \(theta\).

More precisely, the electric field near the focal plane is given by:

\[ \begin{align}\begin{aligned}E(\boldsymbol{x},t) = a_0\times E_0 \, f(r, \theta) \, \exp\left( -\frac{r^2}{w_0^2} - \frac{(z-z_0-ct)^2}{c^2\tau^2} \right) \cos[ k_0( z - z_0 - ct ) - \phi_{cep} ]\\\mathrm{with} \qquad f(r, \theta) = \sqrt{\frac{p!(2-\delta_{m,0})}{(m+p)!}} \left( \frac{\sqrt{2}r}{w_0} \right)^m L^m_p\left( \frac{2 r^2}{w_0^2} \right) \cos[ m(\theta - \theta_0)]\end{aligned}\end{align} \]

where \(L^m_p\) is a Laguerre polynomial, \(k_0 = 2\pi/\lambda_0\) is the wavevector and where \(E_0 = m_e c^2 k_0 / q_e\).

(For more info, see Siegman, Lasers (1986), Chapter 16: Wave optics and Gaussian beams)

Note

The additional terms that arise far from the focal plane (Gouy phase, wavefront curvature, …) are not included in the above formula for simplicity, but are of course taken into account by the code, when initializing the laser pulse away from the focal plane.

Warning

The above formula depends on a parameter \(m\) (see documentation below). In order to be properly resolved by the simulation, a Laguerre-Gauss profile with a given \(m\) requires the azimuthal modes from \(0\) to \(m+1\). (i.e. the number of required azimuthal modes is Nm=m+2)

The non-linear plasma response for this profile (e.g. wakefield driven by the ponderomotive force) may require even more azimuthal modes.

Parameters:

  • p (int (positive)) – The order of the Laguerre polynomial. (Increasing p increases the number of “rings” in the radial intensity profile of the laser.)

  • m (int (positive)) – The azimuthal order of the pulse. (In the transverse plane, the field of the pulse varies as \(\cos[m(\theta-\theta_0)]\).)

  • a0 (float (dimensionless)) – The amplitude of the pulse, defined so that the total energy of the pulse is the same as that of a Gaussian pulse with the same \(a_0\), \(w_0\) and \(\tau\). (i.e. The energy of the pulse is independent of p and m.)

  • waist (float (in meter)) – Laser waist at the focal plane, defined as \(w_0\) in the above formula.

  • tau (float (in second)) – The duration of the laser (in the lab frame), defined as \(\tau\) in the above formula.

  • z0 (float (in meter)) – The initial position of the centroid of the laser (in the lab frame), defined as \(z_0\) in the above formula.

  • 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), defined as \(\lambda_0\) in the above formula. Default: 0.8 microns (Ti:Sapph laser).

  • cep_phase (float (in radian), optional) – The Carrier Enveloppe Phase (CEP), defined as \(\phi_{cep}\) in the above formula (i.e. the phase of the laser oscillation, at the position where the laser enveloppe is maximum)

  • theta0 (float (in radian), optional) – The azimuthal position of (one of) the maxima of intensity, in the transverse plane. (In the transverse plane, the field of the pulse varies as \(\cos[m(\theta-\theta_0)]\).)

  • 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).