Flattened Gaussian Transverse Profile#

Implements a flattened transverse Gaussian profile. This profile can either be defined close to a focus, in a far-field regime or in a collimated beam in the near field.

class lasy.profiles.transverse.FlattenedGaussianTransverseProfile(field_type, w, N, wavelength, z_foc=0)[source]#

Class for the analytic profile of a Flattened-Gaussian laser pulse.

Define a complex transverse profile with a flattened Gaussian intensity distribution far from focus that transforms into 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.

The implementation of this class is based on that from FBPIC <fbpic/fbpic>.

In the focal plane (\(z=0\)), or in the far field, the profile translates to a laser with a transverse electric field:

\[E(x,y,z=0) \propto \exp\left(-\frac{r^2}{(N+1)w_0^2}\right) \sum_{n=0}^N c'_n L^0_n\left(\frac{2\,r^2}{(N+1)w_0^2}\right)\]

with Laguerre polynomials \(L^0_n\) and

\[\qquad c'_n=\sum_{m=n}^{N}\frac{1}{2^m}\binom{m}{n}\]

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 equivalent expression for the collimated beam in the near field which produces this focus is given by:

\[E(x,y) \propto \exp\left(-\frac{(N+1)r^2}{w^2}\right) \sum_{n=0}^N \frac{1}{n!}\left(\frac{(N+1)\,r^2}{w^2}\right)^n\]

with the relationship between the spot sizes of the beams in the far field and in the near field given by:

\[w = \frac{\lambda_0}{\pi w_0}|z_{\mathrm{foc}}|\]

where \(z_{\mathrm{foc}}\) is the distance between the far field and near field planes.

Note that a beam defined using the near field definition would be equivalent to a beam defined with the corresponding parameters far from focus in the far field, but without the parabolic phase arising from being defined far from the focus.
For \(N=0\), the near field profile is a Gaussian profile: \(E\propto\exp\left(-\frac{r^2}{w^2}\right)\).
For \(N\rightarrow\infty\), the near field profile is a flat profile: \(E\propto\Theta(w-r)\).

Parameters:

field_typestring: Options: ‘nearfield’, when the beam is defined far from focus (e.g., right before the focusing optics), or ‘farfield’, when the beam is in the vicinity of the focus.
wfloat (in meter): The waist of the laser pulse. If field_type == 'farfield' then this variable corresponds to \(w_{0}\) in the above far field formula. If field_type == 'nearfield' then this variable corresponds to \(w\) in the above near field formula.
Nint: Determines the “flatness” of the transverse profile, far from focus (see the above formula). Default: N=6 ; somewhat close to an 8th order supergaussian.
wavelengthfloat (in meter): The main laser wavelength \(\lambda_0\) of the laser.
z_focfloat (in meter), optional: Only required if defining the pulse in the far field. Gives the position of the focal plane. (The laser pulse is initialized at z=0.)