Expansion of polarized plane waves into spherical harmonics,

[tecne1982]

expansion of polarized plane waves into spherical harmonics, please help!

Hi all,

I would like to get some guidance in how to expand a polarized (i.e. linear polarization) plane wave into a series of spherical harmonics. I am aware of the formula applying to scalar plane waves (please see attached file). I would appreciate anyone's help.

Thanks

Regards

Tony

 

[Andy Resnick]

The attached .png isn't the expansion of a plane wave (exp(-ikz)), it's the general expansion for a spherical wave (exp(ikr)). Even so, it's possible to make the scalar expansion work with the usual z = r cos (theta) substitution.

In any case, going over to vector functions is considerable more difficult. The trick I learned was to construct a scalar field $\psi$, from which E and B (vector) fields are related by:

E = -r x $\nabla \psi$ and B = i/w $\nabla$ x (r x $\nabla \psi$) for a TE mode and something similar for a TM mode (w is frequency). Becasue Poisson's equation still holds for $\psi$, it's possibleto recast everything in terms of the scalar field, solve that, and then take the derivatives to get E and B.

There is also vector versions of spherical harmonics:

http://en.wikipedia.org/wiki/Vector_spherical_harmonics

But I have never really used those.

 

[tecne1982]

Hi,

Thanks for your reply which I appreciate a lot. Unfortunately I cannot quite understand what you mean and I would like to see a complete solution of the problem (if it exists). Is there any book that you might be aware of, that describes the expansion of polarized plane waves into spherical harmonics?

Thanks and Regards

Tony

 

[Andy Resnick]

Tecne1982,The result is fairly well-known, I just don't have the time to LaTex out the derivation here. A good place to find where this type of result is written out in detail is a book on light scattering: Boren and Huffman is excellent, but there's a lot of others out there- the equations are all over the internets as well.

A good keyword to look for is "multipole exansion".

 

[Antenna Guy]

I would like to see a complete solution of the problem (if it exists).

I think it's obvious Andy meant "multipole expansion", but you might also try "spherical nearfield theory" or "spherical modes".

That said, I doubt you'll find a spherical wave representation of a plane wave. Plane waves don't actually exist - they are the (unidirectional) limit as r approaches infinity of a spherical wave. This is not to say that plane waves are not a useful concept (far from it), but they have their limitations.

Regards,

Bill

 

[Andy Resnick]

Oops- yeah, sorry there.

Antenna Guy, doing a multipole expansion is very useful for solving problems with spherical and cylindrical geometry- scattering of a sphere is the basic example. In performing the expansion, a particular point is defined to be the coordinate origin, and a direction also defined (say, the wavevector). When that occurs, the boundary conditions at a material interface become very easy to express: the fields must be continuous at the surface r=R, for example. Any decent derivation of Mie scattering will present the expansion of a (scalar) plane wave into spherical components.

To be sure, plane waves are as ficticious as spherical waves- point sources don't exist either, and a plane wave is simply a spherical wave located at infinity.

 

[Redbelly98]

The attached .png isn't the expansion of a plane wave (exp(-ikz)), it's the general expansion for a spherical wave (exp(ikr)).

If k and r are vectors, and the OP means their dot-product, then

$$e^{i k \cdot r}$$

would be a plane wave traveling in the direction of k (not necessarily along z).

 

[Antenna Guy]

doing a multipole expansion is very useful for solving problems with spherical and cylindrical geometry- scattering of a sphere is the basic example.

I didn't say plane wave expansions were not useful - just that plane waves are not real.

In performing the expansion, a particular point is defined to be the coordinate origin

That point is typically referred to as "the phase center".

and a direction also defined (say, the wavevector).

Here you are speaking of plane waves - not spherical waves. "k" is a scalar as far as spherical waves are concerned (i.e. the phase term: $e^{-jkr}$).

When that occurs, the boundary conditions at a material interface become very easy to express: the fields must be continuous at the surface r=R, for example.

Since we are speaking of a sphere of radius R, and the currents on that sphere are continuous over the entire surface, one might assume that the fields about that sphere (at some r>R) are continuous as well. This continuity (at constant r) is what the spherical waves represent.

When one considers reciprocity, it becomes evident that it is impossible for the sphere to emit a plane wave in any direction. However, it is reciprocity that makes the plane wave expansion useful.

Any decent derivation of Mie scattering will present the expansion of a (scalar) plane wave into spherical components.

If you find one that expresses a plane wave as a spectrum of spherical waves, let me know.

To be sure, plane waves are as ficticious as spherical waves

Spherical waves are no more ficticious than continuous currents on a sphere. When one considers a radiating object, the k-space is limited by the smallest sphere that completely contains the object. The diameter of that minimum sphere (in terms of wavelengths) defines the k-space. The same is true of either planar or spherical expansions.

Regards,

Bill

 

[Andy Resnick]

<snip>


If you find one that expresses a plane wave as a spectrum of spherical waves, let me know.

<snip>

Bill,

Again, I don't have the time to LaTex out the whole derivation. Jackson, chapter 16, has one. Specifically, 16.128:

exp(ikz cos$\theta$) = $\sum i^{l} (2l+1) j_{l}(kr) P_{l}(cos \theta)$.

Jackson does go further to work out some of the vector expansion (16.131 through 16.139).

Again, any decent light scattering book (Bohren and Huffman, Van DeHulst, etc) will have this spelled out in excruciating detail.

The rest of your postappears to be commentary on an appropriate modeling method. An aperture in a plane screen should be modeled differently that an fluorescent molecule. And still different from a linear, center-fed antenna (Jackson, 16.7).

 

[Antenna Guy]

exp(ikz cos$\theta$) = $\sum i^{l} (2l+1) j_{l}(kr) P_{l}(cos \theta)$.

That looks like a plane wave expansion to me.

Substitute in $\theta=\frac{\pi}{2}$, and let me know what you get.

Regards,

Bill

 

[Andy Resnick]

<snip>

If you find one that expresses a plane wave as a spectrum of spherical waves, let me know.
<snip>

Bill

That looks like a plane wave expansion to me.

Substitute in LaTeX Code: \\theta=\\frac{\\pi}{2} , and let me know what you get.


Bill

I don't understand what you are asking- I gave the plane wave expansion for a plane wave propogating in 'z' (as requested), and you then asked what the expansion looks like if the wavevector is normal to the propogation direction, an unphysical situation for traveling plane waves. Or any propogating mode, for that matter.

I wonder if you are asking what the expansion of exp(ikr), with k in the radial direction, looks like? That problem is the expansion of a spherical wave in terms of plane waves. I guess I just understand where you are going- you are appearing to claim that all of light scattering analysis is invalid.

 

[Antenna Guy]

I don't understand what you are asking

Obviously.

What I was asking for might look something like this:

$$E(r,\theta,\phi)=k\sqrt{2\zeta}\sum_{m=-int(k r_0)}^{int(k r_0)}\sum_{n=|m|,_n>0}^{int(k r_0)} ( a_{mn} m_{mn} (r,\theta,\phi) +b_{mn} n_{mn} (r,\theta,\phi) )$$

where $a_{mn}$ and $b_{mn}$ are coefficients of the expansion, and $m_{mn}$ and $n_{mn}$ are spherical vector expansion functions (which are more involved than I care to tex in).

I don't think it's possible to express a plane wave in this manner.

you are appearing to claim that all of light scattering analysis is invalid.

Not by any stretch of the imagination.

Regards,

Bill

 

[Ben Niehoff]

The expansion of a plane wave in terms of vector spherical harmonics is presented on this page:

http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html

It's also discussed in Jackson.