What is the mathematical expression for electric field vector of unpolarized light?
Try the following construction:
Let E=[Ex(t),Ey(t),Ez(t)] be an electric field vector, where Ei(t) are random functions. Additionally, you must supplies conditions for E, H, D and B vectors which preserve structure of electromagnetic field governed by the Maxwell equations.
First you shojld know that in real life therd is never an unpolarized light. It is used when the polarization measurement is slower than the rate of polarization change.... Then, you can use just scalar wave by using A(t) as a complsx envelope, multiply it by the phasor in time and length exp(-j.(at+kz)), multiplied by transverse shape E(x,y). If you strictly need a unpolarized dsz read option you can add Jones matrix with a and b as the x and y complex amitudes, and associate random phase to one of them with respect t to the other. If this phase is. for example, uniformly distributed between -pi and +pi then you have in practice "unpolarized" light on long term averaging. Stokes vectors were invented exactly for this purpose of partially or none polarized light. Again. only with respect to the speed of polarization measurement...
Consider Stokes vector [S0; S1; S2; S3]. S0 represents the total intensity where as S1, S2, S3 represent various polarization states. For unpolarized light S1 = S2 = S3 = 0. Using the expressions of Stokes parameters in terms of the electric field, mathematical expression for the "time averages of the electric field" of the unpolarized light can be obtained. Please refer to the link, I found it really useful.
https://books.google.co.in/books?id=aWX5jj603PoC&pg=PA276&dq=unpolarized+light+dennis+young%27s+double+slit&hl=en&sa=X&ei=WaCmVKyaLMOjugSX24KACg&ved=0CB0Q6AEwAA#v=onepage&q=unpolarized%20light%20dennis%20young%27s%20double%20slit&f=false
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