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### Parallel to Plane of Incidence

Now the magnetic field is parallel to the surface so and . This time three equations survive, but they cannot all be independent as we have only two unknowns (given Snell's law above for the reflected/refracted waves). We might as well use the simplest possible forms, which are clearly the ones where we've already worked out the geometry, e.g. (as before for ). The two simplest ones are clearly: (E_0 - E_0'')(&thetas#theta;_i) & = & E_0'(&thetas#theta;_r)
&epsi#epsilon;&mu#mu;(E_0 + E_0'') & = & &epsi#epsilon;'&mu#mu;' E_0' (from the second matching equations for both and above).

It is left as a moderately tedious exercise to repeat the reasoning process for these two equations - eliminate either or and solve/simplify for the other, repeat or backsubstitute to obtain the originally eliminated one (or use your own favorite way of algebraically solving simultaneous equations) to obtain: E_0' & = & E_0 2nn'(&thetas#theta;_i) &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) + nn'^2 - n^2^2(&thetas#theta;_i)
E_0'' & = & E_0 &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) - nn'^2 - n^2^2(&thetas#theta;_i) &mu#mu;&mu#mu;' n'^2(&thetas#theta;_i) + nn'^2 - n^2^2(&thetas#theta;_i)

The last result that one should note before moving on is the important case of normal incidence (where and ). Now there should only be perpendicular solutions. Interestingly, either the parallel or perpendicular solutions above simplify with obvious cancellations and tedious eliminations to: E_0' & = & E_0 2n n' + n
E_0'' & = & E_0n' - nn' + n

Note well that the reflected wave changes phase (is negative relative to the incident wave in the plane of scattering) if . This of course makes sense - there are many intuitive reasons to expect a wave to invert its phase when reflecting from a heavier'' medium11.7.

Next: Intensity Up: Dynamics and Reflection/Refraction Previous: Perpendicular to Plane of   Contents
Robert G. Brown 2017-07-11