Gaussian Beam - Mathematical Form

Mathematical Form

The Gaussian beam is a transverse electromagnetic (TEM) mode. A mathematical expression for its complex electric field amplitude can be found by solving the paraxial Helmholtz equation, yielding

where

is the radial distance from the center axis of the beam,

is the axial distance from the beam's narrowest point (the "waist"),

is the imaginary unit (for which ),

is the wave number (in radians per meter),

,

is the radius at which the field amplitude and intensity drop to 1/e and 1/e2 of their axial values, respectively,

is the waist size,

is the radius of curvature of the beam's wavefronts, and

is the Gouy phase shift, an extra contribution to the phase that is seen in Gaussian beams.

Additionally, the field has a time dependence factor that has been suppressed in the above expression.

The corresponding time-averaged intensity (or irradiance) distribution is

where is the intensity at the center of the beam at its waist. The constant is the characteristic impedance of the medium in which the beam is propagating. For free space, .