Gaussian Beams

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 $latex z$ direction, the intensity profile is given by

$latex \displaystyle I(\rho, z) = I_0 \left( \frac{W_0}{W(z)} \right)^2 e^{-\frac{2 \rho^2}{W^2(z)}} $

where $latex \rho$ is the cylindrical radius, $latex W_0$ is the ‘width’ of the beam waist at $latex z = 0$, and $latex W(z)$ is the ‘width’ of the beam as a function of $latex z$, given by

$latex \displaystyle W(z) = W_0 \left[ 1 + \left(\frac{z}{z_0} \right)^2 \right]^{1/2} $

$latex z_0$ is a parameter of the beam, given by

$latex \displaystyle z_0 = \frac{\pi W_0^2}{\lambda} $

The full width half maximum (FWHM) of the beam intensity is related to the ‘width’ parameter $latex W(z)$ as FWHM(z) = $latex \sqrt{2 \ln{2}} W(z)$.

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 $latex W >> \lambda$.  Once again, Wikipedia has a very nice visualization of the field variation in the beam:

As you can see, the intensity is peaked at the center of the beam, and drops off for larger $latex \rho$ 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 $latex z$ 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 $latex z$ dependence of the beam width $latex W$, which increases with $latex z$.  At the same time, the peak intensity in the center of the beam decreases as $latex z$ increases, as it must to conserve power.  The integral of the intensity over any constant-$latex z$ 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 $latex z = 0$ but increasingly curved as $latex z$ 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.