Optical resonators and Gaussian beams

Multimode beams propagation and M² factor.

The spatial extension of any given mode is always bigger than the fundamental mode one. We can then define a M coefficient, greater than one, such as :

where  are the waists of the observed beam and the fundamental beam, respectively. By injecting this equation in the complex radius of curvature definition, we obtain :

All the previously demonstrated formulae are then the same, but with instead of everywhere. For M²=1, we obviously find the relationships obtained for the fundamental Gaussian mode.

The« M² factor» is a kind of measurement of the “degradation” of the beam quality compared to the fundamental Gaussian beam, taken as reference.

More precisely, M² could be experimentally defined by the following sentence : “for a given waist, the measured divergence of the studied beam is M² times bigger than the divergence of the fundamental Gaussian beam” or :

where is the divergence of a TEM00 mode with the same waist as the observed beam.

Practically, high order Hermite or Laguerre beams are rarely observed. It is however frequent to deal with “single-lobe” beams with a quasi circular shape and Gaussian-like energy distribution : the M² factor characterize in this case “how far” from a TEM00 you are : it is a measure of the difference between a real beam and the theoretical limit given by the diffraction.

From an experimental point of view, the principle of the measurement is as follows : one had to measure the divergence of the beam together with the waist w0, and then compare the result to   : the ratio will give M².

Technically, you have to focus the studied beam with a converging lens, and then to measure the size w of the beam for different positions along the z-axis with any suitable method (camera imaging, measure of the energy percentage passing through an iris...). You will finally obtain a curve similar to the one depicted on the figure 11 : you can fit this curve with the formula that gives w(z) (with the M² factor of course set as the free parameter),

The beams produced by He-Ne lasers or low power diode pumped solid state lasers are usually diffraction limited (M²=1.1 or less). For high power lasers (for example flash-pumped Nd:YAGs), the transverse structure is often heterogeneous and the M² factor easily reaches values between 2 and 10. The beams could also sometimes suffer from astigmatism, so that the M² factor is not the same in the x and y directions. Finally, for non-Gaussian beams (for example beams from high power laser diodes) we can have M² factors values above 50, even if the physical signification of this widely used parameter has to be discussed in this case...

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