Final answer:
When two coherent waves with different amplitudes and a non-zero phase difference combine, they produce an elliptically polarized wave due to the vector sum of the waves being complex and tracing out an ellipse over time.
Step-by-step explanation:
When two coherent waves with different amplitudes A and B, and a non-zero phase difference φ, combine, the result is not a simple linearly polarized wave, but rather an elliptically polarized wave. To show this, consider the superposition principle, which states that the resultant wave is the vector sum of the individual waves. If the two waves have amplitude A and B (A ≠ B) and phase difference φ, and assuming they have the same wave number and angular frequency, the amplitude of the resultant wave is given by a more complex expression that accounts for the phase difference.
As stated, any phase difference other than 0 or 180° would result in a composite wave that has a phase shift. The two initial waves can be represented as x(t) = A × cos(ωt) and y(t) = B × cos(ωt + φ), where ω is the angular frequency of the waves and t is time. If the phase difference φ is anything other than an integer multiple of π2π, the waves will not cancel each other out or perfectly align. As they interact, their superposition will trace out an ellipse over time, which is characteristic of elliptical polarization. The shape of the ellipse will depend on the values of A, B, and φ.
Elliptical polarization can be visualized by plotting the component vectors of the electric fields (or other wave characteristics) in two orthogonal directions. The relation between intensity I and the amplitude E squared, as described by Malus's law, indicates that the observed intensity through a polarizing filter will also vary based on the angle of polarization, further confirming the non-linear relationship and the elliptical nature of the polarized wave.