7.2 Transverse electromagnetic modes

In general, any solution u (x, y,z) of the paraxial wave equation, Equation (113View Equation), can be employed to represent the transverse properties of a scalar electric field representing a beam-like electro-magnetic wave. Especially useful in this respect are special families or sets of functions that are solutions of the paraxial wave equation. When such a set of functions is complete and countable, it’s called a set of transverse electromagnetic modes (TEM). For instance, the set of Hermite–Gauss modes are exact solutions of the paraxial wave equation. These modes are represented by an infinite, countable and complete set of functions. The term complete means they can be understood as a base system of the function space defined by all solutions of the paraxial wave equation. In other words, we can describe any solution of the paraxial wave equation ′ u by a linear superposition of Hermite–Gauss modes:
∑ u′(x,y,z) = ajnm unm (x, y,z), (114 ) n,m
which in turn allows us to describe any laser beam using a sum of these modes:
∑ ∑ E (t,x, y,z) = ajnm unm (x,y,z ) exp(i(ωjt − kjz)). (115 ) j n,m
The Hermite–Gauss modes as given in this document (see Section 7.5) are orthonormal so that
∫ ∫ { ′ ′} dxdy unmu ∗n′m ′ = δnn′δmm ′ = 1 if n = n and m = m . (116 ) 0 otherwise
This means that, in the function space defined by the paraxial wave equation, the Hermite–Gauss functions can be understood as a complete set of unit-length basis vectors. This fact can be utilised for the computation of coupling factors. Furthermore, the power of a beam, as given by Equation (108View Equation), being detected on a single-element photodetector (provided that the area of the detector is large with respect to the beam) can be computed as
∗ ∑ ∗ EE = anma nm, (117 ) n,m
or for a beam with several frequency components (compare with Equation (76View Equation)) as
EE ∗ = ∑ ∑ ∑ a a∗ with {i,j | i,j ∈ {0,...,N } ∧ ω = ω }. (118 ) inm jnm i j n,m i j

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