Topological insulators and their optical properties

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Homework Statement

Attached as a filed below. Need help for b and c.

Relevant Equations

Also attached as a file below.

I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an angle (which is related to 5.c) since the boundary condition iii) states that the electric field parallel to the surface should be equal in both magnitude and direction over the two regions.

I tried to see if I could use the constitutive relations to my advantage, but I failed to see how since the components D and H make an equation only consisting of E and B hard to write.

Overall I tried to use what I had learned in class, which was using boundary conditions to obtain a relation between waves in different regions. Help would be very much appreciated.

Thanks in advance.

 

[Nate810]

To answer your question, the boundary conditions in this case are as follows: i) The tangential component of the electric field (E⊥) must be continuous across the boundary. ii) The tangential component of the magnetic field (H⊥) must also be continuous across the boundary. iii) The normal component of the electric field (E⋅) must be equal in magnitude and direction on both sides of the boundary. iv) The normal component of the magnetic field (H⋅) must also be equal in magnitude and direction on both sides of the boundary. Using these conditions, we can derive a relation between the incident and the reflected waves at the boundary: the ratio of the amplitudes of the reflected wave to the incident wave is equal to the ratio of the sines of the angles of incidence and reflection. This is known as the law of reflection. You can use the constitutive relations to derive this relation, by substituting the components of the electric and magnetic fields into the wave equations, and then using the boundary conditions to solve for the parameters of the reflected wave.