Rayleighdistance far-field

The Rayleigh length is the distance (from the waist) where the beam area is twice the beam area at the waist (the radius is times bigger). This parameter is useful to define a “collimated” beam : over this length, the beam size is nearly constant (between and ) - see figure 11.

Beam waist

The Gaussian characteristics of the beam are essentially important in the vicinity of the beam waist. Indeed, when z increases, the complex radius of curvature becomes close to R and the wave could be considered spherical.

Rayleigh rangeGaussian beam

w(z) is the dimension of the laser spot (the “radius” if the spot is circular) in the plane perpendicular to the propagation, at a distance z from the origin. Precisely, it is the radius (at 1/e for the amplitude, or 1/e² for the intensity) of the transverse Gaussian profile at the z abscissa.

For a tightly focused laser beam (w0 = 10 µ) and a 1 µm wavelength, we find ZR = 314 µm and a divergence (half-angle) of 1,8 degrees.

Gaussian beamRayleigh rangecalculator

Rayleigh rangecalculator

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Rayleighlength numerical aperture

Let's take again the origin at the waist position w0, corresponding to a plane wave (infinite radius of curvature). We have defined the Rayleigh length : .

If we consider a “big” waist (1 mm), we find ZR = 3,14 m and a divergence (half-angle) of 0,018 degrees. We then obtain a so-called “collimated beam”.

The main basic expressions related to Gaussian beams were mathematically obtained in the previous paragraph. We will now describe their physical signification.

The divergence of a Gaussian beam is inversely proportional to the size of its waist. In the framework of Gaussian optics, “collimating a beam” is the same thing as “having a big waist”.

When z increases, the beam expands in the transverse direction while its amplitude on the z-axis decrease (energy conservation). The profile shape remains Gaussian.

The size of the beam at the origin, w0, is minimal : the beam will diverge from this point (see figure 11). This minimal dimension is called “beam waist” (the waist is the radius of the spot. The diameter is of course given by 2 w0).