Rayleigh Length

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

The Rayleigh length (or Rayleigh range) of a laser beam is the distance from the beam waist (in the propagation direction) where the beam radius is increased by a factor of the square root of 2. For a circular beam, this means that the mode area is doubled at that point.

Older literature often uses the confocal parameter ($b$), which is two times the Rayleigh length.

Rayleigh Length of Gaussian Beam

Typically, the Rayleigh length is considered for Gaussian beams; it is determined by the Gaussian (($1/e^2$)) beam waist radius ($w_0$) and the wavelength ($\lambda$):

where ($\lambda$) is the wavelength in the medium, i.e., the vacuum wavelength divided by the refractive index ($n$) of the material.

Rayleigh Length of Higher-order Beam

There are higher-order modes such as Hermite–Gaussian modes. These all have the same Rayleigh length as the corresponding Gaussian mode (i.e., the one with the same Gaussian mode waist radius parameter ($w_0$)). The reason is that their amplitude and intensity profiles all scale with the Gaussian beam radius ($w(z)$), only with a constant (order-dependent) prefactor. The same holds for Laguerre–Gaussian modes.

Note that the Gaussian mode waist radius parameter ($w_0$) is not the same as the effective waist beam radius of a higher-order mode; the latter is generally larger. For a given effective waist beam radius, a higher-order mode has a smaller Rayleigh length.

Effective Rayleigh Length

An effective Rayleigh length can be defined for an arbitrary beam profile as the propagation length after the focus where the beam radius increases by the factor ($\sqrt 2$).

If the M² factor, characterizing the beam quality, is known in addition to the beam waist radius ($w_0$) according to the D4σ criterion, the effective Rayleigh length can be calculated as

For the above example of a Hermite–Gaussian beam, ($w_0^2$) and ($M^2$) are increased by the same factor, so that the Rayleigh length remains the same as for the corresponding Gaussian beam. (Note that the above formula needs to be applied with ($w_0$) according to the D4σ criterion, not with ($w_0$) of the corresponding Gaussian beam.)

Optimization of Beam Focusing

The (effective) Rayleigh length is a convenient quantity for calculations in the context of focused laser beams. Essentially, it determines the depth of focus.

When focusing a laser beam into a crystal, e.g. a pump beam into a nonlinear crystal for frequency doubling, it is often advisable to focus such that the Rayleigh length is of the order of the crystal length. One could achieve even higher optical intensities in the crystal with stronger focusing, but in that case only over a shorter length because of the strong beam divergence. Similarly, for end pumping off a laser crystal and will usually not want to make the (effective) Rayleigh length substantially shorter than the crystal length.

The article on laser beams contains a paragraph titled “Limitations for the Focusing of Laser Beams”, where the effective Rayleigh length is also discussed.

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