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# Beam Divergence

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Definition: a measure for how fast a laser beam expands far from its focus

German: Strahldivergenz

The beam divergence of a laser beam is a measure for how fast the beam expands far from the beam waist, i.e., in the so-called far field. A low beam divergence can be important for applications such as pointing or free-space optical communications. Beams with very small divergence, i.e., with approximately constant beam radius over significant propagation distances, are called collimated beams.

Figure 1: The half-angle divergence of a Gaussian laser beam is defined via the asymptotic variation of the beam radius (blue) along the beam direction. Note, however, that the divergence angle in the figure appears much larger than it actually is, since the scaling of the x and y axes is different.

Some amount of divergence is unavoidable due to the general nature of waves (assuming that the light propagates in a homogeneous medium, not e.g. in a waveguide). That amount is larger for tightly focused beams. If a beam has a substantially larger beam divergence than physically possibly, it is said to have a poor beam quality. More details are given below after defining what divergence means quantitatively.

## Quantitative Definitions of Beam Divergence

Different quantitative definitions are used in the literature:

• According to the most common definition, the beam divergence is the derivative of the beam radius with respect to the axial position in the far field, i.e., at a distance from the beam waist which is much larger than the Rayleigh length. This definition yields a divergence half-angle (in units of radians), and further depends on the definition of the beam radius. For Gaussian beams, the beam radius is usually defined via the point with 1/e2 times the maximum intensity. For non-Gaussian profiles, an integral formula can be used, as discussed in the article on beam radius.
• Sometimes, full angles are used instead, resulting in twice as high values.
• Instead of referring to directions with 1/e2 times the maximum intensity, as is done for the Gaussian beam radius, a full width at half-maximum (FWHM) divergence angle can be used. This is common e.g. in data sheets of laser diodes and light-emitting diodes. For Gaussian beams, this kind of full beam divergence angle is 1.18 times the half-angle divergence defined via the Gaussian beam radius (1/e2 radius).

As an example, an FWHM beam divergence angle of 30° may be specified for the fast axis of a small edge-emitting laser diode. This corresponds to a 25.4° = 0.44 rad 1/e2 half-angle divergence, and it becomes apparent that for collimating such a beam without truncating it one would require a lens with a fairly high numerical aperture of e.g. 0.6. Highly divergent (or convergent) beams also require carefully designed optics to avoid beam quality degradation by spherical aberrations.

## Divergence of Gaussian Beams and Beams with Poor Beam Quality

For a diffraction-limited Gaussian beam, the 1/e2 beam divergence half-angle is λ / (πw0), where λ is the wavelength (in the medium) and w0 the beam radius at the beam waist. This equation is based on the paraxial approximation, and is thus valid only for beams with moderately strong divergence.

A higher beam divergence for a given beam radius, i.e., a higher beam parameter product, is related to an inferior beam quality, which essentially means a lower potential for focusing the beam to a very small spot. If the beam quality is characterized with a certain M2 factor, the divergence half-angle is

As an example, a 1064-nm beam from a Nd:YAG laser with perfect beam quality (M2 = 1) and a beam radius of 1 mm in the focus has a half-angle divergence of only 0.34 mrad = 0.019°.

## Beam Quality Calculations

 Center wavelength: M2 factor: calc Beam parameter product: calc Beam waist radius: calc Divergence half-angle: calc

After you have modified some values, click a "calc" button to recalculate the field left of it.

## Spatial Fourier Transforms

It can be helpful to use the spatial Fourier transform of the complex electric field of a laser beam as a function of the transverse coordinates (Fourier optics). Effectively this means that the beam is considered as a superposition of plane waves, and the Fourier transform indicates the amplitudes and phases of all plane-wave components. For propagation in free space, only the phase values change.

The width, measured e.g. as the root-mean-squared (r.m.s.) width, of the spatial Fourier transform can be directly related to the beam divergence. This means that the beam divergence (and in fact the full beam propagation) can be calculated from the transverse complex amplitude profile of the beam at any one position along the beam axis, assuming that the beam propagates in a homogeneous medium (e.g. in air).

## Measurement of Beam Divergence

For the measurement of beam divergence, one usually measures the beam caustic, i.e., the beam radius at different positions, using e.g. a beam profiler.

It is also possible to derive the beam divergence from the complex amplitude profile of the beam in a single plane, as described above. Such data can be obtained e.g. with a Shack–Hartmann wavefront sensor.

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