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Let the two input beams have different phase fronts and have orthogonal polarization states (say linearly polarized along x and y direction respectively). If these beams are superimposed, then the final polarization states will modify depending on the phase profiles of both input beams. However, how to determine the phase distribution of final beam ? ( it will have components in both x and y direction)

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If the polarisation states are orthogonal, the two slits will not interact with each other. Simply calculate the one-slit patterns from the two sources. Remember that the phase of the (presumably coherent) light and the polarisation have nothing whatsoever to do with each other -- Temporally coherent classical light in a polarisation-independent medium can be completely represented by an element of $L^2(\mathbb{R}^3) \otimes \mathbb{C}^2$ -- that is, a pair of (non-interacting) square-integrable functions of space e.g. $\mathbf{\Psi}(x) := (\psi_L(x), \psi_R(x))$ in the left-right circular basis.

Supposing you have some appropriate photon propagator $K(x, y)$, it is possible to calculate the final wave-field completely by $\mathbf{\Psi}(x) = \int d^3y K(x; y) \mathbf{\Psi}(y) $ - just propagate the independent components separately.

Your life will however get tricky if

  • There is any object in the beam's path that couples the two polarisations - e.g. an optically active sugar solution
  • The light is temporally incoherent, i.e. the beam shape changes noticeably over time.
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