# Monochromatic Polarization

What does it mean when we say that the monochromatic wave is un-polarized? Is this kind of wave actually possible? What would the equation of such a wave look like? Since we can consider also that there are no ideal monochromatic waves. How are these un-polarized waves generated?

Ex=βπΈπ(π)βcos(ππ‘βπ½π§ + ππ₯(t))

Ey= βπΈπ(π)βcos(ππ‘βπ½π§ + ππ¦(t))

Ξπ(t)=ππ₯(t)βππ¦(t)

ππ₯(t)=time varying phase offset of x polarized wave and

ππ¦(t)=time varying phase offset of y polarized wave

When these waves are superposed do we get any specific polarization?

Is it Possible to have a phase offset varying with respect to space?

Ξπ(z)=ππ₯(z)βππ¦(z)

• Monochromatic means that the light possesses a single wavenumber $$k = 2\pi/\lambda$$.
• The polarisation vector $$\vec \epsilon$$ describes in which direction the electric field $$\vec E(z,t)$$ oscillates. Unpolarised means that the polarisation changes randomly.

Hence, an unpolarised, monochromatic wave propagating in the $$z$$ direction is a wave described by $$\vec E(z,t) = E_0 \cos(\omega t - kz) \cdot\vec \epsilon(t)$$ where the polarisation vector changes it's direction randomly within the $$(x,y)$$-plane.

Physically the polarisation vector is unable to change its direction randomly within an arbitrary small amount of time. Hence, if we consider a "small enough" time interval $$\Delta t$$ the polarisation will always have a preferred direction, and therefore a polarisation. However, because

1. visible light has an oscillation frequency of the order of $$\nu \approx 5 \cdot 10^{14}Hz$$, and
2. the measurement time $$\Delta t$$ is for most applications larger than $$1\mu s$$,

almost all measurements are done by averaging millions of oscillations. Therefore, if we consider a thermal light source (containing "many" point-like sources) the averaged polarisation is to a good approximation random within the measurement time interval $$\Delta t$$. Therefore, such a source is often considered to possess a random polarisation.

• One could speak about correlation time over which light is polarized. For polarized light you can turn the light on (a laser for example) and if you measure it's polarization at one moment, $t=0$, and at another moment days later you will find that the polarization hasn't changed. However, for an unpolarized field you may find that the polarization changes from one second to the next, or even from one nanosecond to the next. Unpolarized means the correlation time for polarization is very very small. Jun 22, 2020 at 18:41
• If we have Ex and Ey component in the EM wave and we find that the phase difference between the two EM waves is varying with time i.e. Ξπ(t) = ππ₯(t)βππ¦(t). Then can we comment that we have no specific polarization? @jgerber Jul 15, 2020 at 8:53
• If the relative phase of the horizontally ($E_x$) and vertically ($E_y$) polarized components of the light field are varying in time then we would say (assuming the amplitudes for these two modes are fixed) that the ellipticity of the polarization is fluctuating in time. If the variation is slower than the detection speed of our polarization detector then we would say the ellipticity is fluctuating in time. If the variation is faster than our detection speed would say that the ellipticity is random. Jul 15, 2020 at 9:44
• You raise a little bit of an interesting idea here. When people talk about unpolarized light they often mean the amplitudes of the two linear components of light are rapidly oscillating. If the amplitudes are fixed and the phase is randomly varying then if you use a standard polarization analyzer which sorts light based on the power in each of its linear components (like polarized sunglasses or a polarizing beamsplitter) then you would actually see no temporal fluctuation in the two polarizations. you would need a circular polarization analyzer to see that the light has random ellipticity. Jul 15, 2020 at 9:46

Light from a single atom is polarized. The electric field for unpolarized light is given by a sum of the radiation from many uncorrelated individual atoms, so no particular polarization direction can be defined.

A sodium lamp is an example of a (very nearly) monochromatic unpolarized light source. Such light, and incoherent waves in general, must be described with a density matrix.