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Dielectric constants in different directions -- does this make sense?
So I'm trying to analyze a scenario in which I have a plane wave incident (from, say, some medium with permittivity and permeability [itex]ε_1[/itex] and [itex]μ_1[/itex]), on a plane of a dielectric material that has an anisotropic permittivity: in directions parallel to the plane (of the interface, not the plane of incidence), it is [itex]ε_{para}[/itex] and in the direction perpendicular to the plane, it's [itex]ε_{perp}[/itex].
So I break the problem down into two parts as it is usually done: When the E component of the wave is perpendicular to the plane of incidence (POI), and when it is parallel. Anything else is just a linear combination of these.
The boundary conditions at the interface are:
[itex]\hat{n}\cdot(\vec{D_I} + \vec{D_R} - \vec{D_T}) = 0[/itex] (1)
[itex]\hat{n}\cdot(\vec{B_I} + \vec{B_R} - \vec{B_T}) = 0[/itex] (2)
[itex]\hat{n}\times(\vec{E_I} + \vec{E_R} - \vec{E_T}) = 0[/itex] (3)
[itex]\hat{n}\times(\vec{H_I} + \vec{H_R} - \vec{H_T}) = 0[/itex] (4)
Where [itex]B = \mu H[/itex] and [itex]D = εE[/itex] (they are vectors) and [itex]\hat{n}[/itex] is a unit vector normal to the interface. The angle of incidence is θ and the angle of transmission is [itex]\phi[/itex]
B is related to E through [tex]\vec{B} = \frac{\hat{k} \times \vec E}{v}[/tex]
Where [itex]\hat{k}[/itex] is a unit vector pointing in the direction of the wave's propagation ([itex]\vec k /k[/itex] if you wish, where k is the wave number). v is the velocity of the wave, equal to [tex]v = \frac{1}{\sqrt{εμ}}[/tex]
So this is all very fine and dandy for the normal scenario. But now I'm running into some problems and I would love to know if I'm on the right track.
For the E parallel to the POI, in the boundary conditions, (1) seems to be pretty simple: We just use [itex]ε_{perp}[/itex] because that's the direction of the electric field:
[itex]ε_1 sin(\theta)(E_I + E_R) - ε_{perp}sin(\phi)E_T = 0[/itex]
(2) tells us nothing because B has no components perpendicular to the interface.
(3) is just the tangential E equation: [itex]cos(\theta)(E_I - E_R) - cos(\phi)E_T = 0[/itex]
(4) is the tangential H equation: [itex]\frac{\sqrt{\mu_1\epsilon_1}}{\mu_1}(E_I+E_R) - \frac{\sqrt{\mu_2\epsilon_{para}}}{\mu_2}E_T = 0[/itex]
So, in Jackson, for the normal simple case (not my anisotropic mess), he points out that of the 4 boundary conditions, two are actually equivalent to each other if you use Snell's law.
But here, Snell's law doesn't make immediate sense: [itex]n = \sqrt{\frac{\epsilon\mu}{\epsilon_0\mu_0}}[/itex], but in the anisotropic medium, ε depends on the direction, so it seems like n does too.
So how can I proceed from here? I know from Jackson which two of the boundary condition equations are supposed to be equivalent ((1) and (4)), so I could just used one of them and the other non equivalent one ((3)). But I was trying to prove their equivalence, and besides, I still don't have the angle of transmission.
Anyone have any ideas??
Thanks!
So I'm trying to analyze a scenario in which I have a plane wave incident (from, say, some medium with permittivity and permeability [itex]ε_1[/itex] and [itex]μ_1[/itex]), on a plane of a dielectric material that has an anisotropic permittivity: in directions parallel to the plane (of the interface, not the plane of incidence), it is [itex]ε_{para}[/itex] and in the direction perpendicular to the plane, it's [itex]ε_{perp}[/itex].
So I break the problem down into two parts as it is usually done: When the E component of the wave is perpendicular to the plane of incidence (POI), and when it is parallel. Anything else is just a linear combination of these.
The boundary conditions at the interface are:
[itex]\hat{n}\cdot(\vec{D_I} + \vec{D_R} - \vec{D_T}) = 0[/itex] (1)
[itex]\hat{n}\cdot(\vec{B_I} + \vec{B_R} - \vec{B_T}) = 0[/itex] (2)
[itex]\hat{n}\times(\vec{E_I} + \vec{E_R} - \vec{E_T}) = 0[/itex] (3)
[itex]\hat{n}\times(\vec{H_I} + \vec{H_R} - \vec{H_T}) = 0[/itex] (4)
Where [itex]B = \mu H[/itex] and [itex]D = εE[/itex] (they are vectors) and [itex]\hat{n}[/itex] is a unit vector normal to the interface. The angle of incidence is θ and the angle of transmission is [itex]\phi[/itex]
B is related to E through [tex]\vec{B} = \frac{\hat{k} \times \vec E}{v}[/tex]
Where [itex]\hat{k}[/itex] is a unit vector pointing in the direction of the wave's propagation ([itex]\vec k /k[/itex] if you wish, where k is the wave number). v is the velocity of the wave, equal to [tex]v = \frac{1}{\sqrt{εμ}}[/tex]
So this is all very fine and dandy for the normal scenario. But now I'm running into some problems and I would love to know if I'm on the right track.
For the E parallel to the POI, in the boundary conditions, (1) seems to be pretty simple: We just use [itex]ε_{perp}[/itex] because that's the direction of the electric field:
[itex]ε_1 sin(\theta)(E_I + E_R) - ε_{perp}sin(\phi)E_T = 0[/itex]
(2) tells us nothing because B has no components perpendicular to the interface.
(3) is just the tangential E equation: [itex]cos(\theta)(E_I - E_R) - cos(\phi)E_T = 0[/itex]
(4) is the tangential H equation: [itex]\frac{\sqrt{\mu_1\epsilon_1}}{\mu_1}(E_I+E_R) - \frac{\sqrt{\mu_2\epsilon_{para}}}{\mu_2}E_T = 0[/itex]
So, in Jackson, for the normal simple case (not my anisotropic mess), he points out that of the 4 boundary conditions, two are actually equivalent to each other if you use Snell's law.
But here, Snell's law doesn't make immediate sense: [itex]n = \sqrt{\frac{\epsilon\mu}{\epsilon_0\mu_0}}[/itex], but in the anisotropic medium, ε depends on the direction, so it seems like n does too.
So how can I proceed from here? I know from Jackson which two of the boundary condition equations are supposed to be equivalent ((1) and (4)), so I could just used one of them and the other non equivalent one ((3)). But I was trying to prove their equivalence, and besides, I still don't have the angle of transmission.
Anyone have any ideas??
Thanks!