Why does the Jones formalism not apply to incoherent light?

Question

I could not really find an answer to this, because basically any literature immediately states that for the description of incoherent polarized light the Stokes formalism should be used, wihtout going further into detail.

Suppose you have a beam of fully polarized white light, say for example by the use of a reasonably good polarizer, and the light further passes a medium or optical component with wavelength dependent changes to the polarization. Finally, the light passes a second polarizer and hits a detector. The conventional description would suggest describing the light as a 4D real Stokes vector, because the light is incoherent and depolarization occurs because of the wavelength-dependent change in polarization.

However, why is it not possible to describe the light with a Jones vector for each frequency component, and at the detector add them up incoherently by computing and adding the intensities? And if it is possible, could someone recommend literature where this is elaborated on in more detail?

For the record, I am aware that the Stokes formalism is considered the better approach, because it generalizes better for other kinds of depolarizing effects and usually requires less computation.

Answer 1

This is the same difference as with wavefunctions $\left|\psi\right>$ vs density operators $\rho=\sum_ip_i\left|\psi_i\right>\left<\psi_i\right|$. When you have a pure state, you can use the simpler formulations, ie wavefunctions or Jones vector. When you have mixtures, density operators or Stokes vectors come into play. Of course, you can model any physical system purely using multiple wavefunctions or Jones vectors and then manually accounting for the fact that it is actually a mixture, but that is just cumbersome. Why would you not want to learn a formalism that accurately captures the fact that you are dealing with mixtures?

But of course, there is always a trade-off. Some computations are a LOT easier in wavefunction form than in density operator form. As such, you may find that decomposing a specific Stokes vector into some combination of Jones vectors may vastly simplify your computation, and even become faster and stabler. However, such decompositions would likely not be unique, and you might find that you still want to periodically compute the Stokes vector just to be sure that you are on the right track, or for absolute comparison, to know if there is a real difference between any pair of quantum states. This is particularly the case if you find that a specific decomposition into Jones vectors is converging slowly; some other decomposition might converge faster.