BEAM WAIST AND DIVERGENCE

BEAM WAIST AND DIVERGENCE

Diffraction causes light waves to spread transversely as they propagate, and it is therefore impossible to have a perfectly collimated beam. The spreading of a laser beam is in accord with the predic- tions of diffraction theory. Under ordinary circumstances, the beam spreading can be so small it can go unnoticed. The following for- mulas accurately describe beam spreading, making it easy to see the capabilities and limitations of laser beams. The notation is consis- tent with much of the laser literature, particularly with Siegman’s excellent Lasers (University Science Books).

Even if a Gaussian TEM00 laser-beam wavefront were made perfectly flat at some plane, with all rays there moving in precisely parallel directions, it would acquire curvature and begin spread- ing in accordance with

 

and

 

where z is the distance propagated from the plane where the wave- front is flat, l is the wavelength of light, w0 is the radius of the 1/e2 irradiance contour at the plane where the wavefront is flat, w(z) is the radius of the 1/e2 contour after the wave has propagated a dis- tance z, and R(z) is the wavefront radius of curvature after propa- gating a distance z. R(z) is infinite at z = 0, passes through a minimum at some finite z, and rises again toward infinity as z is further increased, asymptotically approaching the value of z itself.

The plane z = 0 marks the location of a beam waist, or a place where the wavefront is flat, and w0 is called the beam waist radius.

The irradiance distribution of the Gaussian TEM00 beam, namely,

 

where w = w(z) and P is the total power in the beam, is the same at all cross sections of the beam. The invariance of the form of the distribution is a special consequence of the presumed Gaussian distribution at z = 0. Simultaneously, as R(z) asymptotically approaches z for large z, w(z) asymptotically approaches the value

 

where z is presumed to be much larger than pw02/l so that the 1/e2 irradiance contours asymptotically approach a cone of angular radius

 

This value is the far-field angular radius (half-angle divergence) of the Gaussian TEM00 beam. The vertex of the cone lies at the center of the waist (see figure 36.6).

It is important to note that, for a given value of l, variations of beam diameter and divergence with distance z are functions of a sin- gle parameter, w0, the beam waist radius.