Diffraction Losses of Open Resonators

The diffraction losses of a resonator depend on its Fresnel number Ap = a /dX (Sect. 5.2.1) and also on the field distribution A x,y,z = Aidjl) at the mirror. 

The fundamental mode, where the field energy is concentrated near the resonator axis, has the lowest diffraction losses, while the higher transverse modes, where the field amplitude has larger values toward the mirror edges, exhibit large diffraction losses. Using (5.31) with z = djl and (5.33) the Fresnel number A/p = j dX) can be expressed as 

The confocal resonator with the smallest spot sizes at a given mirror separation d according to (5.39a), (5.39b) also has the lowest diffraction losses per round-trip, which can be approximated for circular mirrors and Fresnel numbers A p 1 by [291] 

7T jrwj2 JT confocal TEMqo mode area on the mirror 

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These open resonators are, in principle, the same as the Fabry-Perot interferometers discussed in Chap. 4 we shall see that several relations derived in Sect. 4.2 apply here. However, there is an essential difference with regard to the geometrical dimensions. While in a common FPI the distance between both mirrors is small compared with their diameter, the relation is generally reversed for laser resonators. The mirror diameter 2a is small compared with the mirror separation d. This implies that diffraction losses of the wave, which... 

The mode configurations of open resonators can be obtained by an iterative procedure using the Kirchhoff-Fresnel diffraction theory [5.17]. Concerning the diffraction losses, the resonator with two plane square mirrors can be replaced by the equivalent arrangement of apertures with size 2a) and a distance d between successive apertures (Fig. 5.7). When an incident plane wave is traveling into the -direction, its amplitude distribution is successively altered by diffraction, from a constant amplitude to the final stationary distribution An(x,y). The spatial distribution An(x,y) in the plane of the nth aperture is determined by the distribution An- (x, y) across the previous aperture. 

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... 

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