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06 Planar Waveguides 2010

# 06 Planar Waveguides 2010 - Planar Slab Waveguides We have...

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Planar Slab Waveguides We have previously shown that the solutions to Maxwell’s equations in a homogeneous medium is a plane wave. We now want to consider the situations where we have a spatially varying index of refractions that will confine the light. The simplest structure is shown below. The slabs of index n f , n s , and n c are assumed to extend to infinity in the y and z directions. When n c n s , this is called an asymmetric waveguide. Assume the direction of propagation is in the z direction, n s > n c , and that x = 0 at n f /n c interface. There are two possible electric field polarizations to consider: transverse electric (TE) and transverse magnetic (TM). The wave will propagate in the film by being internally reflected from the interfaces. The TE wave has no electric field component along the z axis n c n f n s n f > n s , n c x z y h n c n f n s x H k E o H k E Transverse Transverse Electric Magnetic x z y

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Transverse Electric Fields in a Waveguide E||y with frequency ϖ o and vacuum wave vector k o = ϖ o /c. The solution to this problem is developed by solving for solutions to the wave equation in each medium and then matching the solutions at the boundaries. s n c n f n i n y i y E n k E or , , where 0 2 2 0 2 = = + E y (x,z) is not a function of y because the layers extend to infinity in the y direction. Furthermore, since the structure extends to infinity in the z direction then, 0 ) ( ) ( ) , ( 2 2 2 2 2 = - + = y i o y z j y y E n k x E e x E z x E β where β is the propagation constant. The solutions to this equation depend on the relative magnitudes of β and k o n i . If β > k o n i , the solution will have the form; a exponentially decaying function. x n k o y i o e E x E 2 2 2 ) ( - ± =
Transverse Electric Fields in a Waveguide (cont.) If β < k o n i , then the form of E y (x) is: an oscillatory function. For β > k o n i , we define an attenuation coefficient, γ , where and For β < k o n i , we define a transverse wave vector , κ , where and β,κ and k are related as x n k j o y i o e E x E 2 2 2 ) ( β - ± = x o y i o e E x E n k γ ± = - = ) ( 2 2 2 x j o y i o e E x E n k κ ± = - = ) ( 2 2 2 k = k o n i β κ k 2 = β 2 + κ 2 Longitudinal Transverse wave vector wave vector

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Transverse Electric Fields in a Waveguide (cont.) The longitudinal wave vector, β , is used to identify mode. It is the eigenvalue of the mode. 1. If β < k o n c the wave has the form for all layers and the wave is not confined. 2. If β > k o n c but β < k o n s , the wave is internally reflected at n f /n c boundary and the wave decays exponentially in n c . x n k j o y i o e E x E 2 2 2 ) ( β - ± = x n k j o s f y i o e E x E 2 2 2 ) ( , - ± = x n k o c y i o e E x E 2 2 2 ) ( - ± = 1. If β > k o n s , but β < k o n f the wave is internally reflected at n f /n c and n f /n s boundaries and the wave decays exponentially in n c and n s .
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06 Planar Waveguides 2010 - Planar Slab Waveguides We have...

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