Cavity diffraction loss

The energy stored cannot increase ad infinitum, so ultimately the power input to the cavity must equal the sum of the power dissipated resistively and lost by other mechanisms diffraction losses, coupling into the circuit etc. The power lost in the steady state is thus the input power Pin. 

If the reflectivity is very high, diffraction losses may become dominant, in particular for cavities with a large separation d of the mirrors. Since the TEMqo mode has the lowest diffraction losses, the incoming laser beam has to be mode-matched by a lens system to excite the fundamental mode of the resonator but not the higher transverse modes. Similar to intracavity absorption, this technique takes advantage of the increased effective absorption length Leff = LUX — R), because the laser pulse traverses the absorbing sample 1/(1 - R) times. 

The experimental setup is shown in Fig. 1.18. The laser pulses are coupled into the resonator by carefully designed mode-matching optics, which ensure that only the TEMoo modes of the cavity are excited. Diffraction losses are minimized by spherical mirrors, which also form the end windows of the absorption cell. If the absorbing species are in a molecular beam inside the cavity, the mirrors form the windows of the vacuum chamber. For a sufficiently short input pulse (Tp < 7r), the output consists of a sequence of pulses with a time separation Tr and with exponentially decreasing intensities, which are detected with a boxcar integrator. For longer pulses (Tp > 7r), these pulses overlap in time and one observes a quasi-continuous exponential decay of the transmitted intensity. Instead of input pulses, the resonator can also be illuminated with cw radiation, which is suddenly switched off at f = 0. 

The so-called concentric resonator (Figure 3.5d), with r - -r2= L, represents the functional opposite of the plane-plane resonator. It is easiest to align, has the lowest diffraction loss and exhibits the smallest mode volume. For example, CW dye lasers incorporate this type of cavity (see Chapter 4.3) because of the short length of the active medium (the dye jet), strong focusing of the pump and resonator beams is necessary to cause efficient stimulated emission and to generate sufficient gain for laser action. However, this type of spherical resonator is not commonly used with any other laser. 

In Sect. 2.1 we have seen that any stationary field configuration in a closed cavity (called a mode) can be composed of plane waves. Because of diffraction, plane waves cannot give stationary fields in open resonators, since the diffraction losses depend on the coordinates (x, y) and increase from the z-axis of the resonator towards its edges. This imphes that the distribution A x,y), which is independent of X and y for a plane wave, will be altered with each round-trip for a wave traveling back and forth between the mirrors of an open resonator until it approaches... 

The effects of non-parallelism of the mirrors in a nominally plane-parallel cavity have been studied by Fox and Li (1963). The diffraction loss is found to increase rapidly for only slight deviations from perfect alignment or from perfect optical quality of the windows and mirrors. 

For instance, in order to maintain the diffraction loss below 1 per cent per pass at 6328 X in a 1 ra long cavity, having an effective aperture of 3 mm diameter, it is necessary to maintain the parallelism of the mirrors to within 1 arc sec. This is an extremely stringent requirement for a mechanical structure of this size. Fortunately cavities using spherical mirrors are between 10 and 100 times less sensitive to mirror misalignment and consequently the plane-parallel resonator is now seldom used. 

It can be demonstrated by simple ray tracing that not all combinations of mirror radii and mirror separations represent stable cavities since a beam, initially launched parallel to the cavity axis, may after several reflections start to diverge. The cavity then has a very high diffraction loss and is said to be unstable. The conditions which must be satisfied for the resonator to be stable and so possess low-loss, high-Q modes can be deduced from equation (12.14), since Wq must, in this case, be positive. We therefore require that... 

We have seen in preceding sections that the diffraction losses in optical cavities can be reduced to negligible values by careful design. However, all practical resonators have finite losses associated with the output transmission at the mirrors and so the Q of the cavity, defined by... 

To relate Aa> to the cavity losses, we consider a collimated beam of radiation of intensity Iq launched inside the cavity parallel to the cavity axis. After making one complete trip the beam returns to its starting point with an intensity CIq-Sj-Iq) where 6 is the fractional round-trip loss of the cavity. This fractional loss will include the diffraction loss discussed in section 12.2, but will usually be dominated by the losses produced by absorption, scattering, and output transmission. Since a round trip in the cavity takes a time At=2L/c, we have... 

The constant S on the left-hand side of (35) contains only parameters of the resonator, i.e., the active length L, and reflectivity R. Other types of losses, like scattering, diffraction, etc., may be accounted for by an effective reflectivity, Ren The value (A) is the minimum fraction of the molecules that must be raised to the first singlet state to reach the threshold of oscillation. One may then calculate the function (A) from the absorption and fluorescence spectra for any concentration m of the dye and value 5 of the cavity. In this way one finds the frequency for the minimum of this function. 

The total energy stored in the resonator is proportional to the geometrical phase shift of the cavity, kd, where d is the mirror separation, whence we may derive a diffraction Q, go = 2Trd/ka. In general, one must also consider electrical losses (which contribute to the unloaded Q), sample absorption and scattering (which contribute to the sample Q, Q ), and resonator coupling (which contributes to the radiation Q, Q. Ql, the loaded Q, of the cavity, may therefore be written as a sum of terms... 

Reaction 14 is an example of a reaction that can occur between active sulfur compounds and the reaction product, Li2S. The possibility of this type of reaction was investigated by heating equimolar amounts of Li2S and arsenic trisulfide in an evacuated quartz ampoule at 385°C and then at 480 °C for a total of 5 hrs. X-ray diffraction showed that little, if any, of the reactants were present and that a compound, possibly LiAsS2, had been formed. It is likely that compounds of this type could stabilize the sulfur and sulfide, which would assist in decreasing the extent of intermediate (S2" and S22") formation by Reactions 9 and 10. However, the ternary compounds must be insoluble in the electrolyte to prevent loss from the electrode cavity. 

---