To compute the amount of coupling the beam must be projected into the base system of the cavity or beam segment it is being injected into. This is always possible, provided that the paraxial approximation holds, because each set of Hermite–Gauss modes, defined by the beam parameter at a position z, forms a complete set. Such a change of the basis system results in a different distribution of light power in the new Hermite–Gauss modes and can be expressed by coupling coefficients that yield the change in the light amplitude and phase with respect to mode number.
Let us assume that a beam described by the beam parameter is injected into a segment described by the parameter . Let the optical axis of the beam be misaligned: the coordinate system of the beam is given by () and the beam travels along the z-axis. The beam segment is parallel to the z’-axis and the coordinate system () is given by rotating the () system around the y-axis by the misalignment angle . The coupling coefficients are defined as
Using the fact that the Hermite–Gauss modes are orthonormal, we can compute the coupling coefficients by the convolution x and y, the coupling coefficients can also be split into . These equations are very useful in the paraxial approximation as the coupling coefficients decrease with large mode numbers. In order to be described as paraxial, the angle must not be larger than the diffraction angle. In addition, to obtain correct results with a finite number of modes the beam parameters and must not differ too much.
The convolution given in Equation (164) can be computed directly using numerical integration. However, this is computationally very expensive. The following is based on the work of Bayer-Helms . Another very good description of coupling coefficients and their derivation can be found in the work of Vinet . In  the above projection integral is partly solved and the coupling coefficients are given by simple sums as functions of and the mode mismatch parameter , which are defined by
The coupling coefficients for misalignment and mismatch (but no lateral displacement) can then be written as
In general, the Gaussian beam parameter might be different for the sagittal and tangential planes and a misalignment can be given for both possible axes (around the y-axis and around the x-axis), in this case the coupling coefficients are given byy-axis. The can be obtained with the same formula, with the following substitutions: x-axis.
At each component a matrix of coupling coefficients has to be computed for transmission and reflection; see Figure 54.
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