Gaussianbeam

A Gaussian beam is a fundamental solution to the paraxial wave equation that describes the transverse electric field of many laser beams. It is characterized by a near-Gaussian intensity profile in the plane perpendicular to the direction of propagation and often corresponds to the TEM00 mode of a stable laser cavity. Gaussian beams are widely used because they approximate diffraction-limited propagation and can be described with a small set of parameters.

Key parameters describe how the beam evolves as it travels. The beam has a waist radius w0 at the location z = 0, where the beam is smallest. The beam radius at distance z is w(z) = w0 sqrt(1 + (z/zR)^2), where zR = π w0^2 / λ is the Rayleigh range and λ is the wavelength. The radius of curvature of the wavefront is R(z) = z [1 + (zR/z)^2], and the Gouy phase is φ(z) = arctan(z/zR). The complex beam parameter q(z) = z + i zR satisfies 1/q(z) = 1/R(z) − i λ/(π w^2(z)). The field amplitude can be written in a compact form, and the intensity profile is proportional to exp(-2 r^2 / w^2(z)).

Properties include approximate diffraction-limited behavior, a minimum beam divergence θ ≈ λ/(π w0), and a beam quality factor M^2 ≈ 1 for an ideal Gaussian beam. Real beams may deviate due to higher-order modes, astigmatism, or optical aberrations.

Applications span laser machining, optical communication, microscopy, and optical trapping, where the predictable focusing and propagation of Gaussian beams are advantageous.