# Unpolarized light vs. randomly rotating polarized light?

I am confused with physical picture about unpolarized light.

Is unpolarized light very fast rotating polarized light? or co-existing state of two orthogonal polarization? (or something else?)

If there is a linear polarizer which rotates very very fast and randomly (the polarizer in imagine), the output light is same to unpolarized light? I don't think so but I am not sure.

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or, instead of linear polarizer, a Faraday rotator with magnetic field whose amplitude is randomly chnaged can be considered, I think.

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Does randomly rotating polarized light mean linear polarized light but randomly direction of amplitude? – qfzklm Aug 22 '13 at 9:03
yes. axis of linear polarizer is randomly rotating – Jae-Hwang Jung Aug 22 '13 at 9:13

Unpolarized light can be thought of as a superposition of wave trains of a finite duration of order $0<\tau<\infty$, each of which has a certain pure polarization, which may be elliptical, with a random direction. The phases of the pulses and their start and end times are also random.

What this means in practice is that any unpolarized light source has a coherence time $\tau$. If you look at the polarization with higher temporal resolution than this, you will see a pure polarization (per spectral component! If the light source is not monochromatic the picture is more complicated). If you measure with a lower resolution, the randomly rotating polarization will average out and you will observe no polarization effects.

To put things in scale, the coherence length ($=c\tau$) of sunlight is about $0.6\,\mu\text m$ (doi). In practice this means that any polarization-dependent interferometry must involve path differences shorter than that, or you will be seeing the (lack of) interference between two different pulse trains with random relative polarizations and phases.

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I am confused by your notion of polarization “per spectral component”. How do you define the spectral components? As I see it, you will have to arbitrarily choose a bandwidth, and then their coherence time will be the inverse of this arbitrary bandwidth. Also, your coherence length for sunlight seems to me grossly exaggerated. You probably mean the coherence length of the light coming out of you favorite spectral-component-defining filter. – Edgar Bonet Nov 29 '14 at 10:24
You're right - there was a typo in the sunlight coherence length; I've linked to the source. This is for a point source at 6000K, so it will hold for non-monochromatic sunlight after a pinhole. By a spectral component, I do mean a pure monochromatic component, as mathematically defined. As you point out, and I say in the post, for a non-monochromatic source the picture is more complicated, regardless of the coherence times of each component. I don't think, however, that an in-depth treatment of that is really what the OP was asking for. – Emilio Pisanty Nov 29 '14 at 13:41
If you mean truly monochromatic, then you should remove the sentence in parentheses: any truly monochromatic light source has an infinite coherence time and is fully polarized. Your description of unpolarized light is fine, but you seem to imply that it is only valid for monochromatic light, while the exact opposite is true. BTW, thanks for the ref.: I was just wondering what the self-coherence function of black-body radiation looked like, and you gave me the answer. – Edgar Bonet Nov 29 '14 at 16:00
I don't really think this can be simplified much at the level of the OP's question. I would be interested in your take on how blackbody radiation looks like on timescales shorter than its coherence time (as defined in the link). Perhaps you could post it as an answer here? – Emilio Pisanty Nov 29 '14 at 17:01
No need to simplify anything: just remove the misleading statement in parentheses and you answer is just fine. My take on BB radiation would be like in my answer: an E vector furiously jerking around. It could be visualized by running Gaussian white noise through a digital filter shaped like the square root of the BB spectrum. Do this for both components of E. I would not take this picture too seriously though: we are probably not in the many-photons-per-period limit required to have a well defined classical field. – Edgar Bonet Nov 29 '14 at 17:20

The picture you have about unpolarized light is correct, I think, but I would try to avoid the idea of "rotating fast", because it gives an idea of continuity, that I think is what you try to avoid in the concept of unpolarized light.

So, in essence unpolarized light is modelled by short wave trains of some arbitrary pure polarization; this is because if you interfere this light with itself, the interference pattern will blur at some point, that correspond to the average length of these trains.

I never thought about the idea of getting unpolarized light from purely polarized light, but, I think what you propose could work in theory. Now, if you see a real Faraday rotator, I don't think it can do the job.

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I will give you my personal mental image of unpolarized light, maybe it will help.

In a given point in space, the E field is a vector lying in the plane perpendicular to the propagation. In this plane, if you put the tail of the vector at the origin, then the tip of the vector is a point jerking in a random fashion around the origin. The important thing is that it is random, not periodic, as purely monochromatic light cannot be unpolarized.

If the light is narrow-band, the movement will look kind of periodic (and thus polarized) over short time scales. You would then be able to define an “instantaneous polarization”. But this polarization will slowly change over the time scale corresponding to the bandwidth. You cannot assume anything about the instantaneous polarization: it could be linear, elliptical or circular. I would assume though that it changes continuously, unless the spectrum of the light is quite heavy-tailed: discontinuities in the time domain always make heavy tails in the frequency domain.

If it is white light, then the tip of the vector is just jerking randomly, with a hardly discernible frequency corresponding to the middle of the band. Maybe more a time scale than an actual frequency. It would be very hard to identify an instantaneous polarization, because such polarization would be changing practically in the same time scale.

You could describe both situations as the superposition of two fields with perpendicular polarizations: the combined polarization can be computed from the amplitudes and phases of the components. But since those amplitudes and phases have a finite coherence time, then your polarization is always changing.

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