Electromagnetic scattering on a sphere

[Karthiksrao]

My specialization is Mechanical engineering. So I'm dumb to many of the electromagnetic concepts. But right now, I have to analyze this problem.

This is the statement "The incident plane wave as well as the scattering field is expanded into radiating spherical vector wave functions. The internal field is expanded into regular spherical vector wave functions. By enforcing the boundary condition on the spherical surface, the expansion coefficients of the scattered field can be computed "

Can somebody explain in lay man terms what this means ?

Any help will be much appreciated.

Thanks,
Karthik

 

[Andy Resnick]

That sounds like a Mie scattering problem.

The solution, in words, is "the total field is equal to the incident plus scattered field". Since the scatterer is a sphere, the most logical system of coordinates to solve Maxwell's equations are in spherical coordinates.

So, a plane wave (the incident field) can be decomposed into spherical waves, and the scattered field (the field internal to the sphere and the field external to the field) is also written in terms of spherical coordinates.

http://en.wikipedia.org/wiki/Mie_scattering
http://diogenes.iwt.uni-bremen.de/vt/laser/papers/RAE-LT1873-1976-Mie-1908-translation.pdf
http://www.ess.uci.edu/~cmclinden/link/xx/node19.html

 

[Karthiksrao]

Thanks for your reply Andy.

Why is the term 'expanded' used ? and what are expansion coefficients ?

Thanks

 

[Andy Resnick]

In mechanical engineering, have you used the terms 'normal modes'? In other words, generic motion can sometimes be decomposed into a sum of simple modes of motion (i.e. bending modes, torsional modes, etc).

It's the same concept here- a plane wave, which is very simple to write down in (x,y,z) coordinates, can also be written down in terms of (r,$$\theta$$,$$\phi$$) coordinates. If you like, the plane wave is a sum of monopole, dipole, quadrupole, octopole, etc. modes.

Again, the reason for doing that is because the geometry of the problem (the boundary conditions) are very simple to write in spherical coordinates- that's the key reason for complexifying the plane wave.

 

[Karthiksrao]

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 ?

 

[Born2bwire]

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.

 

[Andy Resnick]

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 ?

Can you provide a reference to the paper? I can't tell you why the authors did what they did otherwise...

Green's functions are important to understand, so it's worthwhile taking some time and thinking about them.

 

[Karthiksrao]

Andy:

I actually meant why the DGFs were introduced in general.. i.e what was the motivation to introduce these functions and what are they being used for..

I think born2bwire has tried to address that. Some parts went above my head considering my background. But its fine.. Once I spend some time on it, I could start understanding them.

One small thing: Why aren't all waves described in vector form ? EM fields and waves, as born2bwire said, are vectors.. Is a wave propagating in water or on a string, a scalar function ? If so, what factor distinguishes these waves from EM waves which makes the latter, a vector ? Don't both these waves have directions ?

The paper I am referring to is:
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRBMDO000077000007075125000001&idtype=cvips&prog=normal

He derives the DGF in section 4 of the paper (page 3)

 

[Born2bwire]

Andy:

I actually meant why the DGFs were introduced in general.. i.e what was the motivation to introduce these functions and what are they being used for..

I think born2bwire has tried to address that. Some parts went above my head considering my background. But its fine.. Once I spend some time on it, I could start understanding them.

One small thing: Why aren't all waves described in vector form ? EM fields and waves, as born2bwire said, are vectors.. Is a wave propagating in water or on a string, a scalar function ? If so, what factor distinguishes these waves from EM waves which makes the latter, a vector ? Don't both these waves have directions ?

The paper I am referring to is:
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRBMDO000077000007075125000001&idtype=cvips&prog=normal

He derives the DGF in section 4 of the paper (page 3)

Ahhh... fluctuation dissipation theorem, one of my colleagues is doing work on that. She gave a talk to us about it but I cannot remember any details but I recall she worked out of the Statistical Mechanics Part 2 text by Landau and Lipgarbagez, you can get more information from that book.

EM waves have vectors only because the wave is made of orthogonal electric and magnetic fields. These fields are vector fields. The waes of say water or acoustic waves are not vectors because they are displacement waves. The wave propagates by displacing a volume of water or by a pressure front. In these cases, they are scalar waves. The direction and propagation of a traveling wave is described by a factor usually along the lines of (or in a superposition of terms like)

$$e^{i(\mathbf{k}\cdot\mathbf{r} -\omega t)}$$

The vector k here is the propagation vector. The direction of k denotes the direction of propagation, the magnitude of k denotes the velocity of propagation (not directly as the magnitude k is the wave number). The vector r is the positional vector denoting the point in space that we are evaluating the wave at. This exponential can be combined into sinusoidal forms, which would be the familiar equation for a wave on a string

$$\sin (kx-\omega t)$$

For the paper, definitely look at Chew's text to find some more about the vector spherical wave expansion that they do. The derivation of the dyadic has some subtleties to it, like where that delta function term comes from in Equation (18). Chew's text is fairly dense though but it can at least be a starting point. The Mie series that we mentioned earlier is not derived this way. The Mie series does not use currents to calculate the scattered field. Instead, it just uses the boundary conditions and the method of mode matching (I think that may be the correct term). The plane wave is decomposed into the superposition of spherical waves. The coefficients for the scattered waves in the spherical modes are found by applying the boundary condition of the PEC sphere to the incoming spherical waves.


Ha! I just looked at the references of the paper and he does refer to Landau and Lipgarbagez and to Weng Cho Chew (reference 21 and 18 respectively are the books I am talking about). Collin and Y. T. Lo are also familiar to me, they are bright guys but I have not looked at Collin's book.

 

[Andy Resnick]

Andy:

I actually meant why the DGFs were introduced in general.. i.e what was the motivation to introduce these functions and what are they being used for..

I think born2bwire has tried to address that. Some parts went above my head considering my background. But its fine.. Once I spend some time on it, I could start understanding them.

One small thing: Why aren't all waves described in vector form ? EM fields and waves, as born2bwire said, are vectors.. Is a wave propagating in water or on a string, a scalar function ? If so, what factor distinguishes these waves from EM waves which makes the latter, a vector ? Don't both these waves have directions ?

The paper I am referring to is:
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRBMDO000077000007075125000001&idtype=cvips&prog=normal

He derives the DGF in section 4 of the paper (page 3)

there's a lot here...

Ok, Green's functions first- why have them? At one level, they represent 'elementary' solutions to a differential equation- if I can continue to use the vibration analogy, if you try and analyze a system response to an extended exciter, one way to simplify the analysis is to consider the extended exciter as a (spatial) distribution of point sources, and add together the solutions for each point source- that's equation 11.

Next- vector fields. The reason all waves are not always described in vector form (including scalar representations of electromagnetism) is that the mathematics is considerably simplified. Honestly, that's it. The justification here is that since the solutions are to be valid in the near field, they cannot make the usual simplifications that lead to a reasonable approximation by scalar waves. It may be helpful to compare equations 13-15 with the scalar versions to get a feel for the subsequent manipulations.

Does that help?