# Gaussian Beams (E&M)

Staff Emeritus
Does anyone have any references or pointers to the complete electromagnetic field of a "Gaussian Beam"?

By Gaussian, I mean that for a beam propagating in the z direction, a cross-section of the beam in the x-yplane has

|E| = k exp( -(x^2+y^2)/w^2 )

see for instance
http://en.wikipedia.org/wiki/Gaussian_beam

I gather that real laser beams tend to have this sort of "Gaussian" profile. However, I haven't been able to find out anything with exact formulas for Ex, Ey, and Ez. $\nabla \cdot E = 0$ seems to imply that Ez is not zero.

Observable
Actually,the exact formulas for Ex,Ey,Ez are the same as separable Helmholtz eqations in Cartesian coordinates. The Gaussian packet(beam) form is just a kind of B.C. You can simply superpose plane waves(in coordinate) to get a Gaussian shape. In this sense, kx,ky are generally non-zero,so Ez could also be non-zero. It´s not a problem though, since this is not plane wave at all.

DaTario
There is a book by Yariv..Quantum Electronics.Wiley 1967
I guess you should find good stuff there.

Best Regards,

DaTario

DaTario
Note that one of the reasons for Gaussian to be used is its easy doing Fourier Transform (spectrum)

The full vector representation is disgustingly complicated .

I was only able to find the vector representation in an optics journal, the reference is;

'Analysis of vector Gaussian beam propagation and the validity of paraxial and spherical approximations.' Carl G. Chen, Paul T. Konkola, Juan Ferrera, Ralf K. Heilmann, Mark L. Schattenburg, JOSA A, Volume 19, Issue 2, 404-412, (2002).

You won't be able to access it though unless your institution has a subsciption with the OSA.

On the paper, I noticed that it goes beyond the Paraxial approximation, a vector solution including the paraxial approximation may be less complicated.

Claude.

Staff Emeritus
Claude Bile said:
The full vector representation is disgustingly complicated .

Heh - I was afraid of that, thanks.

I was only able to find the vector representation in an optics journal, the reference is;

'Analysis of vector Gaussian beam propagation and the validity of paraxial and spherical approximations.' Carl G. Chen, Paul T. Konkola, Juan Ferrera, Ralf K. Heilmann, Mark L. Schattenburg, JOSA A, Volume 19, Issue 2, 404-412, (2002).

I don't have access to that paper, unfortunately, but I did find an interesting reference online when I included "Helmholtz" in my keywords

http://www.hep.princeton.edu/~mcdonald/examples/oblate_wave.pdf

I don't think this paper directly answers my question, but it does give me an idea of how messy the problem is :-)