Hermite Gaussian Transverse Profile

Used to define a Hermite-Gaussian transverse laser profile. Hermite-Gaussian modes are a family of solutions to the paraxial wave equation written in cartesian coordinates. The modes are characterised by two transverse indices \(n_x\) and \(n_y\).

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class lasy.profiles.transverse.HermiteGaussianTransverseProfile(w0, n_x, n_y, wavelength=None, z_foc=0)

A high-order Gaussian laser pulse expressed in the Hermite-Gaussian formalism.

More precisely, the transverse envelope (to be used in the CombinedLongitudinalTransverseLaser class) corresponds to:

\[\mathcal{T}(x, y) = \, \sqrt{\frac{2}{\pi}} \sqrt{\frac{1}{2^{n} n! w_0}}\, \sqrt{\frac{1}{2^{n} n! w_0}}\, H_{n_x}\left ( \frac{\sqrt{2} x}{w_0}\right )\, H_{n_y}\left ( \frac{\sqrt{2} y}{w_0}\right )\, \exp\left( -\frac{x^2+y^2}{w_0^2} \right)\]

where \(H_{n}\) is the Hermite polynomial of order \(n\).

Parameters:

w0float (in meter)

The waist of the laser pulse, i.e. \(w_0\) in the above formula.

n_xint (dimensionless)

The order of hermite polynomial in the x direction

n_yint (dimensionless)

The order of hermite polynomial in the y direction

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.