Physics Cup – TalTech 2021 – Pr. 5, Hint 1.

According to the problem statement, the waves are coherent, and hence, you need to add the amplitudes (and not intensities). However, there is a problem: what happens with the phase of the wave when it is jumping to the other fibre? The answer given below is the consequence of the energy conservation law. Suppose a wave A propagating along the first fibre is split at the junction into a straight-going wave A' and a junction-crossing wave A''; denote the phase shift between A'' and A' as \Delta \phi_A. Similarly, we introduce a wave B propagating along the second fibre, and its straight-going and junction-crossing components B' and B'', together with the phase shift \Delta \phi_B. Energy conservation law dictates that \Delta \phi_A+\Delta \phi_B=\pi (showing this is a good exercise!). As an example, consider normal incidence of plane waves at the interface of two transparent media: if the wave A comes from optically denser medium, \Delta \phi_A=0 (there are no phase shifts involved); then the wave B must be coming from optically less dense medium in which case the reflected wave obtains a phase shift \pi so that \Delta \phi_B=\pi, nicely yielding \Delta \phi_A+\Delta \phi_B=\pi.

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