LaguerreGauss

Laguerre-Gauss (LG) modes are a family of solutions to the paraxial wave equation in cylindrical coordinates that describe monochromatic, scalar light fields with orbital angular momentum. They are labeled by two integers: the azimuthal index l (topological charge) and the radial index p ≥ 0. The complex field distribution of the LG_p^l mode at a distance z from the source is proportional to:

(w0 / w(z)) (√2 r / w(z))^{|l|} L_p^{|l|}(2 r^2 / w^2(z)) exp(- r^2 / w^2(z)) exp(i l φ) exp(- i k r^2 / (2 R(z))) exp(i (2 p + |l| + 1) ζ(z))

where w(z) = w0 sqrt(1 + (z / z_R)^2), R(z) = z [1 + (z_R / z)^2], z_R = π w0^2 / λ, and ζ(z) = arctan(z / z_R). L_p^{|l|} is the associated Laguerre polynomial.

The phase term exp(i l φ) imparts a helical structure to the wavefront, giving the field an orbital angular momentum of lħ per photon. For l ≠ 0 the intensity typically exhibits a central null with a donut-shaped ring, whose exact shape depends on p and l. LG modes form a complete set for a given mode order and can be generated with spatial light modulators, spiral phase plates, q-plates, or mode-converting optics.

Applications include high-capacity optical communications exploiting the orbital angular momentum degree of freedom, optical trapping and micromanipulation, quantum optics, and precision metrology. LG beams are related to Hermite-Gaussian modes and have become a standard tool in laser optics and photonics research.