gbeampro

gbeampro is a small Python package for designing gaussian laser beam propagation and transformations (focusing and collimation, refraction at crystal interface, etc., for instance) according to the ABCD law¹.

Install

pip install gbeampro

Example: Beam focusing into a slab of crystal

This is an example of how to compute the waist diameter inside the crystal and to find the confocal parameter² (twice the Rayleigh range):

$$ 2 z_0 = \frac{2\pi w_0^2 n}{\lambda} $$

At first, import a fundamental (TEM00) Gaussian beam class from gbeampro.beambase.

import ndispers as nd
import numpy as np
import matplotlib.pyplot as plt
from gbeampro.beambase import GaussBeam

Secondly, make an instance of GaussBeam object by giving the five parameters:

1. wl_um : Wavelength of the laser beam (unit: µm)
2. n : Refractive index of the medium in which the beam starts to propagate
3. z_mm : z coordinate of the wavefront. (unit: mm) Wavevector of the gaussian beam is assumed to be along to z axis.
4. R_mm : Wavefront curvature radius. (unit: mm)
5. w_mm : Beam radius, defined as the 1/e**2 intensity half width. (unit: mm)

We will set a beam, named b1, whose waist is located at z=0 and beam radius is 1.5 mm at z=0, as follows.

# setup a GaussBeam instance
b1 = GaussBeam(wl_um=1.064, n=1.0, z_mm=0, R_mm=np.infty, w_mm=1.5)

By entering the beam instance b1, we can check the beam parameters at the present state.

b1

Output:

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=0.00000, R_mm=inf, w_mm=1.50000)
      q : 0.00000e+00 -6.64341e+03i
      theta : 2.257878e-01 mrad

Here a complex q-parameter q and beam divergence (half) angle theta are also printed.

Next, let's put a thin lens (focal length, f=150 mm) at z=0 by using a method,

b1.thinlens(150)

Then the state of b1 has been changed and the transformed beam parameters are shown:

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=0.00000, R_mm=-1.50000e+02, w_mm=1.50000)
      q : -1.49924e+02 -3.38509e+00i
      theta : 2.257878e-01 mrad

Let the beam propagate in the air (n=1.0) upto the entrance face of a crystal,

b1.propagate(150 - 20*0.5 + 5)

(Here + 5 distance is trying to bring the wasit location at the center of the crystal.)

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=145.00000, R_mm=-7.25091e+00, w_mm=0.05977)
      q : -4.92357e+00 -3.38509e+00i
      theta : 5.666828e+00 mrad

The beam enters a slab of LBO crystal (a biaxial crystal). Here we need to compute the refractive index n2 of the crystal.

Xtal = nd.media.crystals.LBO_Newlight_xy() # principal dielectric plane: xy
n2 = Xtal.n(1.064, 0, 149, pol='o') # for ordinary ray of wavelength 1.064 µm and temperature 149 degC.
b1.interface(n2)

(Note that this propagation method can not be applied for extraordinary rays in ansiotropic crystals.)

    GaussBeam(wl_um=1.06400, n=1.604333, z_mm=145.00000, R_mm=-1.16329e+01, w_mm=0.05977)
      q : -7.89905e+00 -5.43082e+00i
      theta : 3.532224e+00 mrad

Let the beam propagate inside the crystal upto its end faceof the crystal, whose length is 20 mm.

b1.propagate(20)

    GaussBeam(wl_um=1.06400, n=1.604333, z_mm=165.00000, R_mm=1.45383e+01, w_mm=0.08270)
      q : 1.21010e+01 -5.43082e+00i
      theta : 2.552784e+00 mrad

At the end face of the crystal, the beam goes out into the air (with n=1.0).

b1.interface(1.0)

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=165.00000, R_mm=9.06187e+00, w_mm=0.08270)
      q : 7.54267e+00 -3.38509e+00i
      theta : 4.095503e+00 mrad

We want the diverging beam to be collimated. So let the beam propagate in the air upto the second lens (f=200 mm).

b1.propagate(200 - 20*0.5 + 5)

(Here + 5 distance is trying to collimate the beam as much as possible after the second lens.)

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=360.00000, R_mm=2.02599e+02, w_mm=2.02623)
      q : 2.02543e+02 -3.38509e+00i
      theta : 1.671490e-01 mrad

The second thin lens collimates the beam.

b1.thinlens(200)

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=360.00000, R_mm=-1.55891e+04, w_mm=2.02623)
      q : -5.87433e+03 -7.55432e+03i
      theta : 1.671490e-01 mrad

Finally, let the beam propagate in air for an illustration purpose.

b1.propagate(100)

    GaussBeam(wl_um=1.06400, n=1.000000, z_mm=460.00000, R_mm=-1.56573e+04, w_mm=2.01330)
      q : -5.77433e+03 -7.55432e+03i
      theta : 1.682224e-01 mrad

Let's plot the beam trajectory in terms of its four parameters:

fig, (ax1, ax2, ax3, ax4) = plt.subplots(4, 1, sharex=True, figsize=(8, 16), facecolor="white")
b1.plot_n(ax1)
b1.plot_w(ax2)
b1.plot_R(ax3)
b1.plot_theta(ax4)

We want to find the beam waist inside the crystal. So use search_BeamWaists method. It searches all the sign-flip points of wavefront curvature (R), at which the beam becomes its waist.

b1.search_BeamWaists()

    Beam waists in z range [0.000, 460.000000] mm
    --------------------------------------------------
    No.0:
      Waist location, z  : 0.0000 mm
      Refractive index, n  : 1.0000
      Waist spot diamter, 2*w0 : 3000.0 µm
      Confocal parameter (2*Rayleigh range) : 13286.81 mm
      2.84 * Confocal parameter : 37734.54 mm
    No.1:
      Waist location, z  : 152.9000 mm
      Refractive index, n  : 1.6043
      Waist spot diamter, 2*w0 : 67.7 µm
      Confocal parameter (2*Rayleigh range) : 10.86 mm
      2.84 * Confocal parameter : 30.85 mm
    No.2:
      Waist location, z  : 360.0000 mm
      Refractive index, n  : 1.0000
      Waist spot diamter, 2*w0 : 4052.5 µm
      Confocal parameter (2*Rayleigh range) : 24244.54 mm
      2.84 * Confocal parameter : 68854.49 mm

We can see the focusing condition inside the crystal at the above result No.1.

Footnotes

1. Kogelnik, Herwig. "Imaging of optical modes—resonators with internal lenses." Bell System Technical Journal 44.3 (1965): 455-494. ↩

2. Yariv, Amnon. Quantum electronics, Third edition. John Wiley & Sons, 1989.) ↩