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') = f⊗f = 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?

Thanks for your help

 

[eljose79]

!
Thank you for your question about the construction and application of dyadic green's function. I can understand why this topic may be confusing, but I am happy to provide some clarification.

First, the dyadic green's function is a mathematical tool used in electromagnetics to solve for the electromagnetic fields in a resonant system, such as a cavity. It is a function that describes the relationship between the electric field at one point and the electric field at another point in space.

In the ideal resonant system, where only certain resonant modes are supported, the dyadic green's function can be constructed using the mode expansion. This means that the green's function can be written as a product of two normalized modes, fi and fj, as you have correctly stated in your forum post. These normalized modes represent the electric field in the system, and the bold notation indicates that they are vector quantities.

To construct the dyadic green's function, the normalized modes can be obtained from a numerical simulation of the electric field in the system, as you have also correctly stated. The normalization number A is the square root of the total energy of the system. This normalization ensures that the dyadic green's function is a physically meaningful quantity.

Now, to calculate the electric field E(r) induced by a dipole P(r0), the dyadic green's function is used. As you have stated, the x component of the field can be written as a sum of three terms, each involving the dyadic green's function and the components of the dipole moment. This is because the dyadic green's function describes the relationship between the electric field at two different points, and in this case, we are interested in the field at point r induced by the dipole at point r0.

Overall, your understanding of the construction and application of dyadic green's function is correct. I hope this helps to clarify any confusion you may have had. If you have any further questions, please feel free to ask.