# Polarization of Light: Any Direction Possible?

• Niles

#### Niles

Hi all.

Usually we have worked with electromagnetic waves linearly polarized along the z-direction (or x or y for that matter).

Is it possible to have an E.M. wave, which is just polarized along some arbitrary direction r = (x, y, z)T? Or does it necessarily have to be circular or in x, y or z?

Light can only be polarized perpendicular to its direction of motion.
The most general polarization is elliptical. Two limiting cases are planar
(The minor axis of the ellipse is zero.) or circular (The two axes of the ellipse are equal.)
The mathematical description of these states is in textbooks.
The hardest state to describe is unpolarized or partially polarized light.
This requires the use of a "density matrix" or some other averaging technique.

Light can only be polarized perpendicular to its direction of motion.

You are saying something completely wrong and not addressing the question of the OP.

Light DOES NOT need to be polarized perpendicular to its direction of motion, ever heard of TM waves?

To the OP: Light can be polarized in ANY direction if you consider linear polarization. The easiest way to grasp is this the change of coordinate systems. You can choose your coordinate axes however you like, and any -r- vector you choose could be just as well your z-component.

So if you just consider linear polarization it could point anywhere in 3D space.

On top of these there are circular and most generally elliptical polarizations.

Hi all.

Usually we have worked with electromagnetic waves linearly polarized along the z-direction (or x or y for that matter).

Is it possible to have an E.M. wave, which is just polarized along some arbitrary direction r = (x, y, z)T? Or does it necessarily have to be circular or in x, y or z?
The reason you usually work with EM waves linearly polarized along the z direction is that in practice, you choose the z axis so that it points along the polarization.

Think about it: if light could only be polarized along the z axis, how would you figure out which direction in space is the z direction?

You can choose your coordinate axes however you like, and any -r- vector you choose could be just as well your z-component.

But if I find e.g. a dipole moment using polarisiation along r, and I then change my coordinate system so it is polarised along z, then would I get the same dipole moment?

You are saying something completely wrong and not addressing the question of the OP.

Light DOES NOT need to be polarized perpendicular to its direction of motion, ever heard of TM waves?

To the OP: Light can be polarized in ANY direction if you consider linear polarization. The easiest way to grasp is this the change of coordinate systems. You can choose your coordinate axes however you like, and any -r- vector you choose could be just as well your z-component.

So if you just consider linear polarization it could point anywhere in 3D space.

On top of these there are circular and most generally elliptical polarizations.

No he's right. Light is only polarized in the plane perpendicular to the wave vector, it is a transverse wave not a longitudinal wave (for most practical intents and purposes). Both TE and TM polarizations are normal to the propagation vector. What they are transverse to is a matter of convention in terms of your coordinate system, not in terms of the propagation vector.

No he's right. Light is only polarized in the plane perpendicular to the wave vector, it is a transverse wave not a longitudinal wave (for most practical intents and purposes). Both TE and TM polarizations are normal to the propagation vector. What they are transverse to is a matter of convention in terms of your coordinate system, not in terms of the propagation vector.

No. I suggest doing a little research before stating conclusive posts - this is a forum people refer to for common knowledge.
First of all, what do you mean by light? If you are talking about electromagnetic waves; no there are all kinds of polarizations entangled, hybrid waves being the most general.

What you are describing is TEM waves, the simplest, the most pristine solution in FREE space or in transmission line waveguides.

Your descriptions of TE and TM waves are also wrong. It is NOT a matter of convention. For TM waves, the electric field has a component in the direction of propagation. That is E_z component is not zero. E_y and E_x components are transverse components.

For most practical intents and purposes LIGHT is TE or TM because in every little corner there's an optical waveguide and if you look at the most elementary solutions of Maxwell's equation in a rectangular waveguide, you'll see how TEM waves cannot exist in a waveguide.

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Propagating modes of light have the property that the instantaneous direction of the electric field is perpendicular to the direction of propagation, by definition. Non-propagating waves, e.g. the near field, do not have this property.

For plane waves traveling in 'z', the light may be linearly polarized in any orientation within the x-y plane, circularly polarized, elliptically polarized, or randomly polarized. For spherical waves, the polarization varies as one moves around a surface of constant phase, but is always normal to the vector 'r'.

Finally, there are some 'odd' polarization states possible: radial or tangential- these have a singularity at the origin.

Hi all.

Usually we have worked with electromagnetic waves linearly polarized along the z-direction (or x or y for that matter).

Is it possible to have an E.M. wave, which is just polarized along some arbitrary direction r = (x, y, z)T? Or does it necessarily have to be circular or in x, y or z?

The polarization state of a single photon (the 'helicity') is generally circular, but IIRC there are some transition states that admit a linear polarization.

Linear polarization states are the most simple classical description of the EM field, but any general polarization state is admitted as a solution as well (which can always be decomposed into two orthogonal states)

Propagating modes of light have the property that the instantaneous direction of the electric field is perpendicular to the direction of propagation, by definition.

Not in a confined waveguide, and not necessarily in general.

Another reference:

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)#Analysis

I am willing to post an exact analysis, if you are still in doubt.

Fair enough- I amend my comment to read "Propagating unconfined modes of light have the property that the instantaneous direction of the electric field is perpendicular to the direction of propagation, by definition." And add confined waves (TE and TM modes) to my list of 'oddball' polarization states.

No. I suggest doing a little research before stating conclusive posts - this is a forum people refer to for common knowledge.
First of all, what do you mean by light? If you are talking about electromagnetic waves; no there are all kinds of polarizations entangled, hybrid waves being the most general.

What you are describing is TEM waves, the simplest, the most pristine solution in FREE space or in transmission line waveguides.

Your descriptions of TE and TM waves are also wrong. It is NOT a matter of convention. For TM waves, the electric field has a component in the direction of propagation. That is E_z component is not zero. E_y and E_x components are transverse components.

For most practical intents and purposes LIGHT is TE or TM because in every little corner there's an optical waveguide and if you look at the most elementary solutions of Maxwell's equation in a rectangular waveguide, you'll see how TEM waves cannot exist in a waveguide.

That's why I said for most practical intents and purposes. Zenneck waves arise in a very specific set of circumstances and are surface waves. A Zenneck wave is a mode that arises when you place a dipole antenna on top of a conducting ground, but it is evanescent off of the ground. Plasma waves are longitudinal but are a mixture of physical pressure waves which are manifested in the varience in the charge densities and electromagnetic waves. Again these are rather esoteric conditions, certainly something beyond what the OP is asking about.

Again, though TM and TE are often specified with respect to the coordinate system and not the direction of propagation. We often choose the z-axis for simplicity but realize that when we solve for a TEZ mode, the direction of propagation is not assumed to be in the z direction. In layered medium, we often talk about the TE and TM modes, but we situate the transverse plane to be the plane of homogeniety. The normal direction is the dimension of inhomogeneity. So when we say we have a scattering of a TM wave, we are only saying that the magnetic field lies in the plane of homegeneity, but for a traveling wave the TM wave does not have a component of its magnetic or electric field in the direction of propagation because we did not arrange the transverse direction in any relation to it.

When we solve scattering problems that are invariant in one dimension, we discuss modes that are transverse to the dimension of invariance. So for any 2D problem, we generally choose the z-axis as the dimension of invariance and talk of the TE and TM modes. This does not assume that there is propagation in the z direction.

There are circumstances where we do use TE and TM in terms of propagation. Like in a waveguide. However, the actual waves in most propagating modes are not TE or TM in terms of the actual wave vector (I can't think of any that don't fall into this off hand, even the Zenneck wave is an evanescent mode, just like the TEM mode in a waveguide. Maybe the 01 or 10 case could fall into this, I don't have time to actually analyze all the modes). The waves are still truly TEM in terms of their direction of propagation. They are TE/TM in terms of the guided wave direction. With a waveguide, we really only care about the propagation of the wave along the desired direction. In this case, it's obvious to see that we can't have a TEM mode due to boundary conditions, but the TE and TM modes propagate by reflecting continuously off the walls. They do not propagate in the z direction, but at an angle to it. http://www.amanogawa.com/archive/docs/EM15.pdf

So I'll stand by what I said earlier, I should know since my area of research is Computational Electromagnetics. Yes there are situations where you can have longitudinal waves, but for just about every application that a person will come across in the real world, light is strictly transverse. The choice of TE and TM is purely convention because the basis of it is the ability to decompose a general EM wave into to parts described by an Ez and Hz. It is useful to use this decomposition in problems where we care about properties in a single direction, like 1D inhomogeniety, 2D problems or waveguides. It's a little confusing with waveguides because for most purposes, we only care about the propagation of the wave in the z-direction, so often we ignore the fact that the wave's vector is actually still a mix of x,y and z components.

EDIT: If you'd like examples displayed in textbooks, take a look at Chew's "Fields and Waves in Inhomogeneous Media" for the use of TE/TM in layered medium. Balanis' electromagnetic theory text has numerous examples of TE/TM for 2D problems, like the scattering from an infinite cylinder. The above linked page, Amanogawa.com's lecture notes, were done by a former advisor, Prof. Ravaioli at UIUC. The site's actually really useful, I've even used the applets on the job a few times for some quick calculations.

As for the wikipedia entries, they probably should be clarified to specify that the direction of propagation is not the direction of the propagation of the actual wave but of the waveguide. I mean, look at the rectangular waveguide I linked to. At TE01 the direction of propagation of the wave has both an x and z component while the polarization of the electric field is only in the x-y plane, you cannot get the direction of propagation to align with the direction of guided propagation (that would be a TEM mode).

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