Karthiksrao said:
gee thanks!
Significant part of the paper deals with Dyadic Green's function (DGF)..
The mathematical parts I can always look up.. But can you tell me what was the purpose of introducing these and in what context these DGFs are used ?
As far as I know, if DGF of a particular medium is known, then you can calculate Electro magnetic fields in that medium. .. Am I right? If so, Is there a standard procedure for finding DGF of a medium ?
I cannot come up with the general definition for a Green's function off the top of my head. In many ways though, a Green's function is an integrating kernel. In electromagnetics, the Green's function is the scalar wave solution that arises from a point source (delta function). If you recall, if you convolve a function with a delta function the result is the evaluation of the function at the location of the delta function. This allows us to use the Green's function as the kernel for the fields excited by a given source.
However, this is a scalar function, electromagnetic fields and waves are vectors. A dyad is another word for a tensor (it may only be three rank tensors but I am not sure of that condition). The dyadic Green's function will relate a vector current source to the resulting excited electric or magnetic vector fields. In essense, the dyad map a vector source to a vector field. The elements of the dyad are of the form:
\hat{x}\hat{x}, \ \hat{x}\hat{y}, \ \dots , \hat{y}\hat{z}, \ \dots
What this means is that the xy entry of the dyadic will map the y component of the source vector (electric or magnetic current) to the x directed component of the resulting field (electric or magnetic). This is why we use a dyadic Green's function.
A large portion of my research is in layered medium and so I have spent a lot of time with the Dyadic Green's Function for Layered Medium (DGLM). This is useful, for example, in a method of moments (MOM) solver. A MOM solver uses the Green's function to relate the source currents with the excited field. The currents are unknown so we use the method of moments to solve for the currents (based upon the known boundary conditions). In this case, the Green's function is central to the solver and the homogeneous Green's function is a dyad. A DGLM is useful because if we have a layered medium, then we do not need to model it as an additional scatterer(s). The layered inhomogeneity will be taken care of by the Green's function and we only need to mesh and solve over the other scatterers. For example, if we have a patch antenna on a FR4 board, this is air, patch, substrate, copper cladding, air. We can model the substrate and copper ground layers in the DGLM and thus the only scatterer we need to mesh and solve over is the patch (which is a surface problem if we model the patch as a PEC, the substrate, if included, would make the problem a volumetric problem and thereby greatly increase the number of unknowns.).
So how to find the dyadic Green's function. Simply you just find the field response from a point source in the medium. There are only a few situations where we can do this analytically (and rarely in closed form). Homogeneous medium are easy and we can do layered medium (planar, cylindrical or spherical). The layered medium Green's function is a Sommerfeld integral and in the exception of only PEC layers does not have an exact closed form solution. If you want to know more about layered medium dyadic Green's function (DGLM) or the homogeneous dyadic Green's function, take a look at Weng Cho Chew's Fields and Waves in Inhomogeneous Media (but there is an error in his derivation of the embedded source, he published a correction in IEEE). I am pretty sure there is an entire book on the dyadic Green's function as it applies to EM that is also published by the IEEE Press but I have not looked at it very much.
Outside of EM, you can find a lot of stuff on deriving the Green's function in any mathematical methods textbook. The only one I know off hand is written by Keener but his is a horrible textbook to learn from (as a reference it is good, but for learning... eh...).
EDIT: Oh, by the way, you may come across the Mie scattering in other applications. For example, Griffith's Quantum Mechanics textbook derives the Mie scattering coefficients and I am sure that you will find it in the acoustic scattering off of a hard sphere as well. The boundary conditions for the EM waves are common for various wave problems and so you may find some Deja vu with the derivation.
