Two-mirrors resonator

We will discuss in details in this part the classical and very common resonator made of two spherical mirrors (radii of curvature R1 and R2, separated by a distance d)

If a Gaussian beam is a mode for such a resonator, then its radius of curvature at the mirrors position matches the radius of curvature of the mirrors. This condition is necessary to insure that the beam returns the same way after bouncing on one mirror.

Let z1 and z2 be the M1 and M2 mirrors positions respectively : we have then R(z1) = -R1 et R(z2) = R2.

We can then make use of this characteristic to determine the mode geometry inside the cavity without using the general ABCD law : we only need to apply the relations we already demonstrated in the “Gaussian spherical wave” paragraph, with the origin located at the waist.

We then write the two conditions on the mirrors :

and

We also have . We then just need to solve this 3 equations / 3 unknowns (z1, z2 and ZR) system to obtain their values as a function of the distance d between the mirrors and their radii of curvature.

If we do the whole calculation (do it as an exercise !), and with (i=1,2), we come to the following relations :

 We can find here again the stability condition for a two-mirrors resonator, namely with strict inequalities this time.

To generalize the stability diagram shown on figure 7 so that it could be applied to Gaussian beams, one then need to exclude the hyperbola itself as well as the gi = 0 axis.

The simplest example is the Fabry Pérot resonator, made of two facing plane mirrors. As the A and D matrix elements are equals to unity, we have g1 = g2 = 1: the stability condition is verified in ray optics, but not for Gaussian beams.

Nevertheless, it is possible to obtain stable laser operation with such a cavity, as some other elements could stabilize the resonator. For example, the amplifying medium itself often acts as a converging lens under pumping (thermal lensing).

A mode could resonate if the field is the same after a round trip inside the cavity. In other words, the phase variation along this round trip has to be equal to a multiple of ;

The phase term for a gaussian spherical wave is (see the corresponding paragraph) where . The first exponential term is simply the phase shift due to propagation, whereas the second one is a specificity of Gaussian beams.

If  is the phase at the abscissa z, we should have :

q is here an integer equal to the number of half-wavelengths over a distance d (nothing to do with complex radius of curvature here !): there is (q-1) nodes et q antinodes in the cavity (NB : for standard cavity lengthes (between one centimeter and one meter), q is a huge number !)

The resonant frequency of TEM00q Gaussian modes in the cavity are consequently (replace k by ) :

This expression could be written differently thanks to

that is (after some calculations) :

A very similar formula was given for a plane wave in the “longitudinal and transverse modes” paragraph : however we have here an additional term . This term is added for each frequency, so that we still have : .

The previous study was performed in the simple case of the two-mirrors linear cavity. For more complex resonators (see figures 17 and 18), we have to calculate the phase variation for each Gaussian beam during a round trip.