We’ve been dealing mostly with plane waves, which have infinitely large wavefronts in the planes perpendicular to their direction of propagation, but of course this is unphysical. It is possible to write a closed-form solution of the Helmholtz wave equation which represents a more realistic finite-width beam. Gaussian beams are one such solution. They derive their name from the intensity profile of the beam, which is Gaussian. For the lowest-order Gaussian beam propagating primarily in the direction, the intensity profile is given by

where is the cylindrical radius, is the ‘width’ of the beam waist at , and is the ‘width’ of the beam as a function of , given by

is a parameter of the beam, given by

The full width half maximum (FWHM) of the beam intensity is related to the ‘width’ parameter as FWHM(z) = .

The validity of the Gaussian beam solution of the Maxwell equations depends on the transverse variation of the field being much slower than the longitudinal variation, or in other words, we must have . Once again, Wikipedia has a very nice visualization of the field variation in the beam: https://en.wikipedia.org/wiki/Gaussian_beam

As you can see, the intensity is peaked at the center of the beam, and drops off for larger with a Gaussian shape. All finite-sized beams are susceptible to spreading; only plane waves do not spread, and that’s because they’re infinitely large. Put another way, we could decompose any finite-sized beam into a sum (integral really) of plane waves, and it would include some plane waves whose wavevectors did not align with the axis. The Gaussian beam actually exhibits minimal spreading compared to any other finite-sized beam solution of Maxwell’s equations of equivalent width. The spreading of the Gaussian beam is described by the dependence of the beam width , which increases with . At the same time, the peak intensity in the center of the beam decreases as increases, as it must to conserve power. The integral of the intensity over any constant- plane is the total power crossing that plane, which must be the same for all of those planes.

It can also be noted in the lovely Wikipedia diagram that the wavefronts (constant phase surfaces) are flat at but increasingly curved as increases. This is consistent with the beam spreading we’ve described.

Many lasers provide light output that can be described using a Gaussian beam. There are also higher-order Gaussian beams with more variation in the transverse plane, but that’s beyond our scope for now.