Frequency Spectrum of Passive Resonators

The fundamental axial modes TEMoo (m = n = 0) have the frequencies v = q + )c ld and the frequency separation of adjacent axial modes is 

Equation (5.49) reveals that the frequency spectrum of the confocal resonator is degenerate because the transverse modes with q = q and m- -n = lp have the same frequency as the axial mode with m=n = Q and q = qi+ P- Between two axial modes there is always another transverse mode with m- -n 1 = odd. The free spectral range of a confocal resonator is therefore 

If the mirror separation d deviates slightly from the radius of the mirror curvature R, the degeneracy is removed. We obtain from (5.34) with p = qn and 0 = d/R 7 1 for a symmetric nonconfocal resonator with two equal mirror radii Ri = R2 = R 

As has been shown in [5.21] the frequency spectrum of a general resonator with unequal mirror curvatures R and Ri can be represented by 

Now we briefly discuss the spectral width Av of the resonator resonances. The problem will be approached in two different ways. 

Equation (5.49) reveals that the frequency spectrum of the confocal resonator is 

The stationary field configurations of open resonators, discussed in the previous sections, have an eigenfrequency spectrum that can be directly derived 

Since the laser resonator is a Fabry-Perot interferometer, the spectral distribution of the transmitted intensity follows the Airy formula (4.64). With an incident intensity Iq, a transmission factor T, and a reflectivity R of each resonator mirror, the intensity stored within the resonator is 

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The second approach for the estimate of the resonance width starts from the quality factor Q of the resonator. With total losses per second, the energy W stored in a mode of a passive resonator decays exponentially according to (5.18). The Fourier transform of (5.18) yields the frequency spectrum of this mode, which gives a Lorentzian (Sect. 3.1) with the halfwidth Ar r = P/2n. With the mean lifetime F = 1/ of a photon in the resonator mode, the frequency width can be written as... 

Assume that a wave with the spectral intensity distribution 7o(y) traverses an interferometer with two mirrors, each having the reflectivity R and transmission factor T (Fig. 5.21). For the passive interferometer we obtain a frequency spectrum of the transmitted intensity according to (4.50). With an amplifying medium inside the resonator, the incident wave experiences the amplification factor (5.58) per round-trip and we obtain, analogous to (4.61) by summation over all interfering amplitudes, the total transmitted intensity... 

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