Light beams

Laguerre–Gauss light beams are exact solutions of the Helmholtz equation in the paraxial approximation, expressed in cylindrical coordinates. These beams carry orbital angular momentum (OAM), associated with the azimuthal index l. When l ≠ 0, the field phase varies as $e^{il\varphi}$, producing a helical structure and an optical vortex at the beam center, where the intensity vanishes. The simulator plots the transverse section (x–y plane) of one or two Laguerre–Gauss beams with azimuthal index $l$ ranging from −3 to 3; when two beams are present, the interference pattern resulting from the coherent superposition of their fields is displayed explicitly. In the paraxial form, the complex electric field of a Laguerre–Gauss mode $\text{LG}_{p}^{l}$ is written as:
$$E_{p,l}(r,\varphi,z) = E_0 \, \frac{W_0}{W(z)} \left( \frac{\sqrt{2}\,r}{W(z)} \right)^{|l|}$$ $$\times \, L_{p}^{|l|}\!\left( \frac{2r^2}{W^2(z)} \right) \exp\!\left( -\frac{r^2}{W^2(z)} \right)$$ $$\times \, \exp\!\left( -i k \frac{r^2}{2R(z)} \right) \exp\!\left( i l \varphi \right)$$ $$\times \, \exp\!\left( -i (2p+|l|+1)\,\zeta(z) \right),$$ where $L_{p}^{|l|}$ are the associated Laguerre polynomials, $W(z)$ is the beam radius, $R(z)$ is the wavefront radius of curvature, and $\zeta(z)$ is the Gouy phase. The radial index $p \ge 0$ determines the number of radial intensity rings, while the azimuthal index $l$ defines the helical phase structure and the orbital angular momentum per photon, equal to $l\hbar$.

Adjust the parameters in the "Beam 1" and "Beam 2" panels—the center position ($x_c$, $y_c$), width ($W$), amplitude ($E$), modal indices ($l$ and $p$), radius of curvature ($R$), and wave-vector components ($k$, $k_x$, $k_y$). In the "Graph" panel, change the plotted region size (Span) and resolution (pixels); in the "Scale" panel, choose the color map. Use the "Examples" menu to load preset configurations.

The orbital angular momentum carried by these beams (distinct from the spin angular momentum associated with polarization) has applications in optical tweezers, high-capacity optical communications, and nanoscale particle manipulation—each photon in an LG mode carries orbital angular momentum l·ħ, an integer multiple of the reduced Planck constant.

Marco P. M. de Souza