8.1 Coupling of Hermite–Gauss modes

Let us consider two different cavities with different sets of eigenmodes. The first set is characterised by the beam parameter q1 and the second by the parameter q2. A beam with all power in the fundamental mode u00(q1) leaves the first cavity and is injected into the second. Here, two ‘misconfigurations’ are possible:

The above misconfigurations can be used in the context of simple beam segments. We consider the case in which the beam parameter for the input light is specified. Ideally, the ABCD matrices then allow one to trace a beam through the optical system by computing the proper beam parameter for each beam segment. In this case, the basis system of Hermite–Gauss modes is transformed in the same way as the beam, so that the modes are not coupled.

For example, an input beam described by the beam parameter q1 is passed through several optical components, and at each component the beam parameter is transformed according to the respective ABCD matrix. Thus, the electric field in each beam segment is described by Hermite–Gauss modes based on different beam parameters, but the relative power between the Hermite–Gauss modes with different mode numbers remains constant, i.e., a beam in a u00 mode is described as a pure u00 mode throughout the entire system.

In practice, it is usually impossible to compute proper beam parameter for each beam segment as suggested above, especially when the beam passes a certain segment more than once. A simple case that illustrates this point is reflection at a spherical mirror. Let the input beam be described by q1. From Figure 49View Image we know that the proper beam parameter of the reflected beam is

q = -----q1------, (162 ) 2 − 2q1∕RC + 1
with RC being the radius of curvature of the mirror. In general, we get q1 ⁄= q2 and thus two different ‘proper’ beam parameters for the same beam segment. Only one special radius of curvature would result in matched beam parameters (q1 = q2).
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