Two light waves of slightly different frequencies are present at the same time in the same medium. Will they couple or interfere?
The waves propagate independently in an optically perfect and uniform medium. In such conditions, all the waves will interfere and never will be coupled. Similarly, the modes in a uniform waveguide propagate their energy independently and can only interfere.
The waves (and modes) become coupled if there is a coupling mechanism in an optical system. Some non homogeneity, perturbation, modification of opto-geometrical parameters of the system will possibly result in coupling between propagating waves (modes). Such coupling results in an energy exchange between different waves and pure interference phenomena are no more capable to describe the optical system behavior.
If the two light waves are in the same mode, they will interfere. Interference is simple addition of the electric fields when the intensity is the square of the field. However if there is a nonlinear term that exchanges energy between the waves, it becomes coupling. A typical intensity response is I = I0 + I1 + 2 sqrt(I0)*I1 instead of I = I0 + I1 + 2 sqrt(I0)*sqrt(I1).
Can two light waves with orthogonal polarization couple to each other in a non-linear medium?
Yes provided the axis of the non-linear medium is between the polarization axes of the light beams. The idea is that there will be a component of each light beam along the direction of the axis of the non-linear component.
It means that to prevent the coupling, I have to make sure that the axis of the medium is not along or between the polarization axis of the two waves.
Yes, sort of. You could orient the crystal perpendicular to both the axes of the polarization.
What are the chances of two orthogonally polarized low intensity waves coupling inside a SMPM fiber ring resonator?
If the intensity is low enough, you wont have too much cross coupling any way
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