Laguerre–Gauss type light beams are exact solutions of the Helmholtz equation in the paraxial approximation, expressed in cylindrical coordinates. These beams are characterized by carrying orbital angular momentum (OAM), associated with the azimuthal index l. When l ≠ 0, the phase of the field varies as $e^{i l \varphi}$, producing a helical phase structure and an optical vortex at the beam center, where the intensity vanishes. The 'Beams of Light' simulator allows you to visualize these structures and explore their spatial properties and interference.

In the application, it is possible to plot the cross section (x–y plane) of one or two Laguerre–Gauss beams with azimuthal index $l$ varying from −3 to 3. For two simultaneous beams, the program explicitly displays the interference pattern resulting from the coherent superposition of the fields. In this way, you can investigate how orbital angular momentum, spatial displacements, and phase differences influence the observed intensity distribution.

In paraxial form, the complex electric field of a Laguerre–Gauss mode $ ext{LG}_{p}^{l}$ can be written as:

\[ E_{p,l}(r,\varphi,z) = E_0 \, \frac{W_0}{W(z)} \left( \frac{\sqrt{2}\,r}{W(z)} \right)^{|l|} L_{p}^{|l|}\!\left( \frac{2r^2}{W^2(z)} \right) \exp\!\left( -\frac{r^2}{W^2(z)} \right) \\ imes \exp\!\left( -i k \frac{r^2}{2R(z)} \right) imes \exp\!\left( i l \varphi \right) \exp\!\left( -i (2p+|l|+1)\,\zeta(z) \right) \]

where $L_{p}^{|l|}$ are the associated Laguerre polynomials, $W(z)$ is the beam radius, $R(z)$ is the radius of curvature of the wavefront, and $\zeta(z)$ is the Gouy phase. The radial index $p \ge 0$ determines the number of radial rings in the intensity, while the azimuthal index $l$ defines the helical structure of the phase and the orbital angular momentum per photon, equal to $l\hbar$.

The simulator allows you to directly control the main physical parameters of the beams:

• Center position $ (x_c, y_c) $;
• Beam width $W$;
• Amplitude of the electric field $ E $;
• Modal indices $ l $ and $ p $;
• Radius of curvature $ R $;
• Wave number $ k $;
• Wave vector components in $x$ and $y$, i.e., the $k_x$ and $k_y$ components.

With these controls, it is possible to study transverse displacements, beam tilts, wavefront bending effects, and interference patterns between multiple modes.