# polarization of the lower mode gaussian beam

In most introductory analysis of Gaussian beam optics, Helmoltz scalar optics is assumed. Hence polarisation is ignored. But I'm not clear what are the possible orientations for the lower transverse mode $T_{00}$ when full vectorial electric and magnetic fields are allowed.

Assuming cylindrical coordinates, where the beam propagates in the axial coordinate z. The intensity of the fundamental mode is axially symmetric; but How does look the electric field orientation for a linearly polarized gaussian beam? is it always aligned toward a fixed direction? it has axial symmetry as well?

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In the center of a Gaussian beam, the field structure is close to that in a plane wave with the same polarization. So the field is not axially symmetric for a linearly polarized Gaussian beam.

Let me note that Gaussian beams are not precise solutions of the free Maxwell equations. For this reason, a few years ago, I derived some precise solutions of the free Maxwell equations that are close to Gaussian beams in some sense (http://arxiv.org/abs/physics/0405091 , eq.22). I considered circular polarization only, but it is not difficult to build solutions with linear polarization starting from the solutions with circular polarization.

EDIT (07/04/2013): I am trying here to answer lurscher's question in the comments. It depends on where you place the surface of the reflecting material. If you place it in the center of the waist of the Gaussian beam, where the structure of the field is close to that in a plane wave, the reflection coefficients will be very close to those for a plane wave, I believe. If the reflecting surface is far from the center of the waist, the structure of the field will be close to that in radiation from a point source (within some angle of divergence). Directions of propagation will span some solid angle, so it seems the only way to eliminate s-polarization completely is to use a surface orthogonal to the axis of the Gaussian beam, however, in this case, reflection will be relatively small. On the other hand, if the Gaussian beam is narrow, the share of s-polarized radiation can be made small with a different choice of the reflecting surface. I don't have time right now to give more details.

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thanks, i assume that far-field properties like beam divergence and beam waist relationships are unchanged when taking polarisation into consideration right? divergence is still $\frac{\lambda}{\pi w_0}$? – lurscher Jun 19 '13 at 0:23
I believe so, but too lazy to check now:-) – akhmeteli Jun 19 '13 at 0:31
@lurscher: So what kind of details would you like to see? – akhmeteli Jun 30 '13 at 4:18
sorry I didn't replied earlier, I forgot about this bounty :-) Some materials have different reflectivities for p-wave and s-wave polarisations, so I want to know, if a Gaussian Beam hits a surface at an angle, which what kind of polarisation I'm dealing with? I was hoping that linear polarisation would stay aligned in a single spatial direction, so a mirror perpendicular to that would maximise reflectivity – lurscher Jul 3 '13 at 23:45
@lurscher: Please see the edit to my answer. – akhmeteli Jul 4 '13 at 18:20

The Gaussian beam is a model of physical beam propagation (not even one that exactly satisfies Maxwell's equations at that, although it does satisfy certain paraxial approximations of the equations). In typical optics use, polarization is neglected, as you realize. However, if you specify a linearly polarized gaussian beam, it would have all the E vectors lined up in the same direction at a snapshot in time, and the intensity would vary in the usual Gaussian shape away from the axis. Most cheap diode lasers actually have this behavior, which you can verify experimentally with a laser pointer and linear polarizing filter. Turn the laser or filter axially and you should be able to visually identify the polarization axis by watching the intensity variation in the transmitted beam. To my knowledge, the polarization can be controlled using various optical devices to achieve circular or radial polarization as well, or more exotic polarization profiles; there is some exciting research in this area.

The exact field solutions for these probably can't be written down in closed form anyway, so we use the Gaussian beam as a model we can handle mathematically.

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