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== Multilayer Green’s Functions ==

The Green’s functions are the solutions of boundary value problems when they are excited by an elementary source. This is usually assumed to be an infinitesimally small vectorial point source. In order for Green’s functions to be computationally useful, they must have analytical closed forms like a mathematical expression, or one should be able to compute them using a recursive process. It turns out that only very few boundary value problems have closed-form Green’s functions. Planar layered structures with laterally infinite extents are one of those few cases, which can be represented by recursive dyadic Green's functions.

In general, a structure may support both electric ('''J''') and magnetic ('''M''') currents. The total electric ('''E''') and magnetic ('''H''') fields can be expressed in terms of the electric and magnetic currents in the following way:

:<math>H = H^{inc} + \iiint\limits_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') \, dv'</math>

where '''G_EJ''', '''G_EM''', '''G_HJ''', '''GH_M''' are the dyadic Green’s functions for the electric and magnetic currents due to electric and magnetic current source, respectively, and '''Eⁱ''' and '''Hⁱ''' are the incident or impressed electric and magnetic fields, respectively. In these equations, '''r''' is the position vector of the observation point and '''r'''' is the position vector of the source point. V is the volume that contains all the sources and the volume integration is performed with respect to the primed coordinates. The incident or impressed fields provide the excitation of the structure. They may come from an incident plane wave or a gap source on a microstrip line, a short dipole, etc. The complexity of the Green’s functions depends on what is considered as the background structure. If you remove all the unknown currents from the structure, you are left with the background structure.