# Calculation and application of dyadic Green's function

• Jeffrey Yang

#### 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?

!

Dear forum member,

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.

Best regards,

## 1. What is a dyadic Green's function?

A dyadic Green's function is a mathematical tool used in electromagnetics and other fields of physics to solve problems involving the interaction of electric or magnetic fields with materials. It describes the response of a material to a localized electric or magnetic source, and can be used to calculate the fields at any point in space.

## 2. How is a dyadic Green's function calculated?

Dyadic Green's functions are typically calculated using integral equations, which represent the response of a material to a point source. These equations are then solved numerically, using techniques such as the method of moments or the finite difference time domain method.

## 3. What are some applications of dyadic Green's functions?

Dyadic Green's functions are used in a variety of applications, including antenna design, radar and satellite communication, and electromagnetic scattering problems. They are also used in the study of plasmonic and metamaterial structures, and in the development of new materials for applications in optics and electronics.

## 4. How does a dyadic Green's function differ from a regular Green's function?

A dyadic Green's function is a generalization of a regular Green's function, which is typically used in scalar fields. Dyadic Green's functions take into account the vector nature of electromagnetic fields, and can be used to solve problems involving polarization and anisotropic materials.

## 5. Are there any limitations to using dyadic Green's functions?

While dyadic Green's functions are a powerful tool for solving electromagnetic problems, they do have some limitations. They are typically only valid for linear, isotropic materials, and may not accurately describe the behavior of highly nonlinear or anisotropic materials. Additionally, they can be computationally expensive to calculate, especially for complex structures and configurations.