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set2solution

# set2solution - ECE 222B Winter 2010 Solution to Homework...

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ECE 222B, Winter 2010 Solution to Homework Set 2 1. Assume that an anisotropic crystal has a permittivity tensor ~ ~ ε = ε xx ˆ x ˆ x + ε yy ˆ y ˆ y + ε zz ˆ z ˆ z where ε xx , ε yy , and ε zz are all real constants > 0. Now assume that we introduce a propagating wave of frequency ω . (a) If the magnetic field of the wave takes the form ~ H = H 0 exp( - j ~ β · ~ r ) , where ~ β = β 1 z cos( θ ) + ˆ x sin( θ )), what is the wave dispersion (re- lation between β 1 and ω ? Is the phase velocity of the wave con- stant? Calculate ~ D and ~ E , and express them in terms of frequency, wavenumber and magnetic field. Are ~ D and ~ E parallel to each other? Under what conditions? Solution: This problem (both parts a and b) contain a rather glaring typo. The equation for ~ H should have read ~ H = H 0 exp( - j ~ β · ~ r y, For part b, the equation for ~ D should have read ~ D = D 0 exp( - j ~ β · ~ r y, As posed, if we take the equation for ~ H to mean any allowable direc- tion, then it will lie in a plane perpendicular to the direction of ~ beta . For any chosen initial polarization, the wave will then break down into two waves, one with a y-directed H field, and the other with a y directed D field. Here an inconsistency occurs, since both ~ beta and ω have been fixed, but cannot be the same for both waves. In the spirit of the originally intended problem, I will give the solution for a y-directed H field, and then handle the y-directed D field in part b. If ~ H = H 0 exp( - j ~ β · ~ r y , then the electric field will lie in the x-z plane (since E and H must be orthogonal). This means that the solution for the dispersion and mode behavior will be independent of ε yy . As a consequence, the mode properties will be identical to those discussed in the handout for the uniaxial case. (b) If the electric flux density of the wave takes the form ~ D = d 0 exp( - j ~ β · ~ r ) , 1

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where ~ β = β 2 z cos( θ ) + ˆ x sin( θ )), what is the wave dispersion (re- lation between β 1 and ω ? Is the phase velocity of the wave con- stant? Calculate ~ H and ~ E , and express them in terms of frequency, wavenumber and magnetic field. Are ~ D and ~ E parallel to each other?
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