Laguerre Gaussian Transverse Profile#

Used to define a Laguerre-Gaussian transverse laser profile. Laguerre-Gaussian modes are a family of solutions to the paraxial wave equation written in cylindrical coordinates. The modes are characterised by a radial index \(p\) and an azimuthal index \(m\).

The modes can have azimuthally varying fields (for \(m > 0\)) but any single mode will always have an azimuthally invariant intensity profile.

class lasy.profiles.transverse.LaguerreGaussianTransverseProfile(w0, p, m, wavelength=None, z_foc=0)[source]#

High-order Gaussian laser pulse expressed in the Laguerre-Gaussian formalism.

Class for an analytic profile. More precisely, at focus (z_foc=0), the transverse envelope (to be used in the CombinedLongitudinalTransverseLaser class) corresponds to:

\[\mathcal{T}(x, y) = r^{|m|}e^{-im\theta} \, L_p^{|m|}\left( \frac{2 r^2 }{w_0^2}\right )\, \exp\left( -\frac{r^2}{w_0^2} \right)\]

where \(x = r \cos{\theta}\), \(y = r \sin{\theta}\), \(L_p^{|m|}\) is the Generalised Laguerre polynomial of radial order \(p\) and azimuthal order \(|m|\)

Parameters:

w0float (in meter): The waist of the laser pulse, i.e. \(w_0\) in the above formula.
pint (dimensionless): The radial order of Generalized Laguerre polynomial
mint (dimensionless): Defines the phase rotation, i.e. \(m\) in the above formula.
wavelengthfloat (in meter), optional: The main laser wavelength \(\lambda_0\) of the laser. (Only needed if z_foc is different than 0.)
z_focfloat (in meter), optional: Position of the focal plane. (The laser pulse is initialized at z=0.)

Warning

In order to initialize the pulse out of focus, you can either:

Use a non-zero z_foc
Use z_foc=0 (i.e. initialize the pulse at focus) and then call laser.propagate(-z_foc)

Both methods are in principle equivalent, but note that the first method uses the paraxial approximation, while the second method does not make this approximation.