8.2 Coupling coefficients for Hermite–Gauss modes

Hermite–Gauss modes are coupled whenever a beam is not matched to a cavity or to a beam segment or if the beam and the segment are misaligned. This coupling is sometimes referred to as ‘scattering into higher-order modes’ because in most cases the laser beam is a considered as a pure TEM00 mode and any mode coupling would transfer power from the fundamental into higher-order modes. However, in general, every mode with non-zero power will transfer energy into other modes whenever mismatch or misalignment occur, and this effect also includes the transfer from higher orders into a low order.

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 q1 is injected into a segment described by the parameter q2. Let the optical axis of the beam be misaligned: the coordinate system of the beam is given by (x, y,z) and the beam travels along the z-axis. The beam segment is parallel to the z’-axis and the coordinate system (x′,y ′,z′) is given by rotating the (x, y,z) system around the y-axis by the misalignment angle γ. The coupling coefficients are defined as

( ) ∑ ( ) unm(q1) exp i(ωt − kz ) = kn,m,n′,m′un′m ′(q2)exp i(ωt − kz ′) , (163 ) n′,m′
where unm(q1) are the Hermite–Gauss modes used to describe the injected beam and un′m′(q2) are the ‘new’ modes that are used to describe the light in the beam segment. Note that including the plane wave phase propagation within the definition of coupling coefficients is very important because it results in coupling coefficients that are independent of the position on the optical axis for which the coupling coefficients are computed.

Using the fact that the Hermite–Gauss modes unm are orthonormal, we can compute the coupling coefficients by the convolution [6Jump To The Next Citation Point]

∫ ∫ ( ′ 2( γ )) ′ ′ ′ ∗ kn,m,n′,m ′ = exp i2kz sin -- dx dy un′m′ exp (ikx sin γ) unm. (164 ) 2
Since the Hermite–Gauss modes can be separated with respect to x and y, the coupling coefficients can also be split into knmn′m′ = knn′kmm ′. 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 γ must not be larger than the diffraction angle. In addition, to obtain correct results with a finite number of modes the beam parameters q1 and q2 must not differ too much.

The convolution given in Equation (164View Equation) can be computed directly using numerical integration. However, this is computationally very expensive. The following is based on the work of Bayer-Helms [6Jump To The Next Citation Point]. Another very good description of coupling coefficients and their derivation can be found in the work of Vinet [55]. In [6] the above projection integral is partly solved and the coupling coefficients are given by simple sums as functions of γ and the mode mismatch parameter K, which are defined by

K = 1(K + iK ), (165 ) 2 0 2
where K0 = (zR − z′)∕z′ R R and K2 = ((z − z0) − (z′ − z′))∕z′ 0 R. This can also be written using q = izR + z − z0, as
i(q − q′)∗ K = ------′--. (166 ) 2ℑ {q }

The coupling coefficients for misalignment and mismatch (but no lateral displacement) can then be written as

′ ′ knn′ = (− 1)n E(x)(n!n′!)1∕2(1 + K0 )n∕2+1∕4(1 + K ∗)−(n+n+1)∕2{Sg − Su }, (167 )
where
[n∑∕2][n′∑∕2](− 1)μ ¯Xn−2μXn′−2μ′ min(∑μ,μ′)(−1)σF¯μ−σFμ′−σ Sg = -(n−2μ)!(n′−2μ′)!-- ------a-------(2 σ)!(μ − σ )!(μ′ − σ)!, μ=0 μ′=0 σ=0 (168 ) [(n−∑ 1)∕2][(n′−∑1)∕2] μ ¯n−2μ−1 n′−2μ′−1min∑(μ,μ′) σ ¯μ−σ μ′−σ Su = (−1(n)−2Xμ−-1)!(n′X−2μ′−1)!- (2(σ−+11))F!(μ−σF)!(μ′−σ)!. μ=0 μ′=0 σ=0
The corresponding formula for kmm ′ can be obtained by replacing the following parameters: n → m, n′ → m ′, X, X¯ → 0 and E (x) → 1 (see below). The notation [n∕2] means
[m ] { m ∕2 if m is even, -- = (169 ) 2 (m − 1)∕2 if m is odd.
The other abbreviations used in the above definition are
¯ ′ ′ √ ------∗ X = (izR − z )sin (γ)∕( 1 + K w0), X = (iz + z′)sin (γ)∕(√1--+-K-∗w ), R 0 F = K ∕(2(1 + K0 )), (170 ) F¯ = K ∗∕2, (x) ( X ¯X ) E = exp − -2- .

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 by

knmm ′n′ = knn ′kmm ′, (171 )
where k ′ nn is given above with
q → qt and (172 ) w0 → wt,0, etc.
and γ → γy is a rotation about the y-axis. The kmm ′ can be obtained with the same formula, with the following substitutions:
n → m, n′ → m ′, q → q , (173 ) s thus w0 → ws,0, etc.
and γ → γx is a rotation about the x-axis.

At each component a matrix of coupling coefficients has to be computed for transmission and reflection; see Figure 54View Image.

View Image

Figure 54: Coupling coefficients for Hermite–Gauss modes: for each optical element and each direction of propagation complex coefficients k for transmission and reflection have to be computed. In this figure k1, k2, k3, k4 each represent a matrix of coefficients knmn′m′ describing the coupling of un,m into un′,m′.

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