Notes 13 Spring 2005

Notes 13 Spring 2005 - Notes 13 Spring 2005 Plane wave...

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Notes 13 Spring 2005 Plane wave propagation in general directions Previously we’ve described the electric field via (for example). Technically, we could have written jkz 0 y y e E E = r k j 0 y r k j 0 y z z jk 0 y y e E e E e E E r r r r = = = , where we now use a vector formulation for k and write z ˆ z r and z ˆ k k z = = r r where r r is the position vector locating a point in space with respect to an origin. For the given electric field then, we have ( ) z k 0 cos r k r k z = = r r r r (the angle is zero here because in this case the wave vector 1 , k r , is in the same direction as r r ). Usually the subscript on the k is left off if the wave is traveling along a single Cartesian direction, hence where the phase is seen to change with z (along the direction of propagation) at a rate dictated by the phase constant (spatial frequency) k. Therefore, to determine the phase at a specific location, one multiplies the position (z-value, for instance) along the direction of propagation by the phase constant in that direction ( for instance). jkz 0 y r k j 0 y y e E e E E = = r r z k In reference to the figure below we see that the phase constant is given by and the position vector is given by z ˆ k x ˆ k k z x + = r z ˆ z x ˆ x r + = r , so that ( ) ( ) z k x k z ˆ z x ˆ x z ˆ k x ˆ k r k z x z x + = + + = r r . Then the magnitude of in the direction of the position vector k r r r is the projection of k in the direction of r r r : θ = cos k k r r r . Therefore to determine the phase at a specific position denoted by r r we write r k cos k r k r r r r r r r r = θ = (recall the geometric interpretation of the dot product). So we write () z k x k j 0 r k j 0 z x e E e E E + = = r r . θ k r r r r k r x z x k z k r k 1 The wave vector k r is in the direction of wave propagation. It is convenient to decompose it into two orthogonal Cartesian components, i.e., z ˆ k x ˆ k k z x + = r , as we shall soon see. cos k r k r r r r r cos k k r r r r r r = θ θ = = Wave Crests Direction of wave propagation 1
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Notes 13 Spring 2005 2 z 2 x k k 2 k 2 + π = π = λ r 2 z 2 x p k k k v + ω = ω = r x x k 2 π = λ x px k v ω = z z k 2 π = λ z pz k v ω = Note that the phase velocity along x and z directions can be greater than c. This, however does not violate any physical law since the energy (power) only flows in the direction of (the direction of propagation, i.e. the Poynting vector direction). k r III. Plane Wave Reflections at Oblique incidence angles (Throughout this development we will assume non-magnetic materials, i.e., ) 0 2 1 µ = µ = µ Plane of incidence : The plane formed by the two following vectors: 1. The normal unit vector to the boundary surface, n . ˆ
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.

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Notes 13 Spring 2005 - Notes 13 Spring 2005 Plane wave...

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