# Calculation and application of dyadic Green's function

Jeffrey Yang
Hello everyone:

I'm confusing with the construction and application of dyadic green's function. If we are in the ideal resonant system where only certain resonant mode is supported in this space (such as cavity), the Green's function can be constructed by the mode expansion that is:

Gij(r,r') = ff = fi(r)⋅fj(r')

In this formula, fi and fj are the normalized mode.(Here we consider electric field). Bold means vector.

For example, in a numerical simulation, we got the electric field E0(r), and then we normalized it to f(r) = E0(r)/A with normalization number A (which will be the square root of total energy of the system). Then, the dyadic Green's function will be constructed, such

Gxx(r,r') = fx(r)⋅fx(r') = E0x(r)⋅E0x(r')/A2

Now, if we want to calculate the field E(r), induced by a dipole P(r0). Then, the x component of the field will be:

Ex(r) = Gxx(r,r0)Px(r0) + Gxy(r,r0)Py(r0) + Gxz(r,r0)Pz(r0)
=(E0x(r)⋅E0x(r')⋅Px(r0) + E0x(r)⋅E0y(r')⋅Py(r0) + E0x(r)⋅E0z(r')⋅Pz(r0))/A2

These are my understanding, are they correct?