06 Planar Waveguides 2010

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

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ) ( - ± =
Background image of page 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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/23/2011 for the course EE 474 taught by Professor Lingo during the Spring '11 term at USC.

Page1 / 29

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online