GaussianBeams

Gaussian beams are approximate solutions to the paraxial wave equation that model the propagation of laser beams with nearly Gaussian transverse intensity profiles. They describe how beam width and phase evolve along the propagation axis and form the fundamental mode (TEM00) of many laser resonators. The concept provides a simple, analytic framework for predicting how a beam focuses, diverges, and interacts with optical elements.

Key parameters and relations: the beam waist w0 is the minimum spot size at the focus (z = 0). The wavelength is λ. The Rayleigh range is zR = π w0^2 / λ. The beam radius as a function of z is w(z) = w0 sqrt(1 + (z/zR)^2). The radius of curvature of the wavefront is R(z) = z [1 + (zR/z)^2]. The complex beam parameter is q(z) = z + i zR, with 1/q(z) = 1/R(z) − i λ/(π w(z)^2). The Gouy phase φG(z) = arctan(z/zR) describes a longitudinal phase shift.

Intensity profile and focusing: the on-axis intensity scales as I(r,z) = I0 [w0/w(z)]^2 exp[−2 r^2 / w(z)^2], showing how the beam widens with z. A common compact description is that a Gaussian beam remains Gaussian under free propagation, with its width determined by w(z) and its phase by φG(z).

Extensions and limits: Gaussian beams are solutions under the paraxial, monochromatic, and scalar approximations. Higher-order modes include Hermite-Gaussian and Laguerre-Gaussian families. Real beams may exhibit polarization effects and aberrations, or depart from ideal Gaussianity.

Applications include laser optics, focusing and coupling into fibers, optical trapping, microscopy, and communications. The model provides a practical baseline for beam shaping, propagation analysis, and optical system design.