Optical resonators and Gaussian beams

Longitudinal and transverse modes

We can define the open resonator modes by using the expressions found for closed resonators : in the case of a parallelepipedic cavity (the sides of the box have a length a, b and c respectively), the resonant frequencies are given by :

where c is the speed of light in vacuum and m,n , q are integers..

In the case of an open resonator, d>>(a,b) and if we take a=b to simplify the formula , we obtain :

or after a Taylor expansion

This expression gives the TEMmnqmodes frequencies.

  • The frequencies of the longitudinal modes (TEM00q.) are (see figure 2).:

    (this type of mode are also sometimes called “spectral modes”)

Figure 2 : longitudinal and transverse modes in a resonant cavity.
Figure 2 : longitudinal and transverse modes in a resonant cavity. [zoom...]Info

The spectral interval between two longitudinal modes is consequently , that is 1.5 Ghz for L=10 cm fréquentiel de 1,5 Ghz.

A laser with a single well-defined frequency (corresponding to a given value of q) is a “single-longitudinal- mode laser” : only one longitudinal mode could oscillate, and the laser consequently exhibits a high spectral purity (and then an important coherence length)

  • The transverse mode are the TEMmnq modes with m and/or n non equal to zero (and generally inferior to 10, because the main goal of an open resonator is to keep the number of oscillating modes small.)

In a “single-transverse-mode laser”, only the TEM00q modes oscillate.

The spectral interval between two transverse modes (n and q fixed) is :

The spectral repartition of the longitudinal and the first transverse modes is given in the figure 2.

Those results are correct in the plane-wave approximation. We will see later that the expressions have to be modified in the case of Gaussian beams.

Example : Orders of magnitude and numerical examples.
Figure 3 : Spectral repartition of the longitudinal modes for a given laser.
Figure 3 : Spectral repartition of the longitudinal modes for a given laser. [zoom...]Info

What is the spectral width of a (slightly) multimode laser ? And what about a single-mode one ?

Let L be the length of a given optical cavity. The gap between two consecutive modes is c/(2L), that is 1 Ghz if L=15 cm. If we assume that 5 modes are allowed to oscillate (see figure 3), we obtain a spectral width of 5 Ghz (or 17 pm in terms of wavelength). This is gap is too small to be detected by classical spectrometers, and the laser appears to be monochromatic (even it is not strictly single-mode)

For some applications (metrology...), very narrow laser spectra are needed : it is then possible to force the single-mode behaviour (for example by lowering the losses for only one of the modes) The spectral bandwidth is then the natural width of a single laser line, which depends of the nature of the laser medium (gas, solid...) : the order of magnitude could vary from a few Hz to several MHz.

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