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I have seen a unpolarized wave defined in a number of places (e.g. here) as a wave such that: \begin{align} E_x&=E_0 \cos(kz-\omega t) \\ E_y&=E_0 \cos(kz-\omega t+\varphi) \end{align} Where $\varphi=\varphi(t)$ is a random function in time.
My question is why do we not have $\varphi=\varphi(x,y,z,t)$ with it been a random function in time and space?

(This question follows from discussion in the comments of: Introducing a phase, what changes?)

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  • $\begingroup$ Who says it isn't? There's a trivial relation between time and space in the direction of propagation, but the temporal / longitudinal coherence in this dimension is different from spatial coherence in the transverse direction. (This is hopefully a good starting point.) $\endgroup$ – Emilio Pisanty May 12 '16 at 16:34
  • $\begingroup$ @EmilioPisanty Sorry I am not sure I fully understand your comment, 'Who says it isn't?'... The linked paper, and several other sources. $\endgroup$ – Quantum spaghettification May 12 '16 at 16:42
  • $\begingroup$ To clarify: have you considered the possibility that the model you quote is sufficient to model unpolarized light, but that it is not necessary to impose that simplicity? Other authors can (and do) include a random spatial variation in the phase; Kanseri and Kandpal use that model because it is the minimal model with the behaviour they want (unpolarized light) without undue complications. If you're treating spatial coherence, on the other hand, the variation of $\phi$ with transverse position is obviously critical. $\endgroup$ – Emilio Pisanty May 12 '16 at 19:09
  • $\begingroup$ @EmilioPisanty Yes I have, and I agree with what you have said. But for the assumption that $\phi$ is a function of $t$ only seems to simplistic and I cannot see what assumptions can be made for this to hold? $\endgroup$ – Quantum spaghettification May 12 '16 at 19:13
  • $\begingroup$ As you already noted, along the propagation direction temporal and spatial coherences are interchangeable, so that one can always be whittled down to time. Along the transverse direction, you can assume a flat phase if your source has enough spatial coherence over your range of interest. Whether it does or doesn't depends on the situation - for more details, see a good textbook on optical coherence. (That would make a nice recom question, btw.) $\endgroup$ – Emilio Pisanty May 12 '16 at 19:29
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I'll try to build an answer based on the above comments. There are two things to consider :

  • Temporal coherence, which has nothing to do with polarization. One simple way to modelize it is the following : imagine you have an incoherent pointlike source that emits EM radiation at some frequency $\omega$. This source can be a thermal one, or spectral for instance. Whatever the mechanism of emission is, you can imagine that there exists collisions between atoms, or spontaneous emission that will introduce a random-varying phase $\phi$ which value jumps arbitrarly between $0$ and $2 \pi$ on a typical timescale $\tau$. If your field propagates along $z$, neglecting polarization effects, you can write your wave as such

\begin{align} E_x&=E_0 \cos(kz-\omega t+\varphi(t-z/c)) \\ E_y&=E_0 \cos(kz-\omega t+\varphi(t-z/c)) \end{align}

because the phase at point $z$ in space corresponds to the phase induced at the moment of emission which is $t-z/c$, accounting for propagation delay. If you use this wave in an interferometer, you can observe interference as long as the difference in time to propagate through the two different paths is less than $\tau$. Otherwise, the phase difference will have randomly jump and it destroys the interference pattern because of measure averaging : the typical acquisition time of any light detector is many orders of magnitude longer that the period of visible light (which is of order $10^{-14}$ s).

  • Polarization, which like temporal coherence can be modeled in many different ways. The model the authors seem to have chosen is to say that there exists some random phase difference between the $x$ and $y$ components of the field. surely this phase depends on space as well as time (because your wave is propagating) but they neglect this effect because all they need for the sake of the argument is a phase randomly varying in time between the $x$ and $y$ components.
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