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Rectangular Optical Wavguide - EM Wave

  1. Jul 28, 2007 #1
    Hello All, this is my first post, and while I'm a newb to the forums I'm not a newb to Optics. However, I have a very light background in low level electromagnetics, I'm more of a network design person.

    I'm working on a very very long term personal project to understand optics better and I've come to the point where I would like to understand how to calculate how an EM-Wave propagates down a Rectangular Optical Waveguide. This waveguide is made of one glass material "n=1.47", it has nothing but air surrounding it. "See the attached image"

    If I take the standard wave equation of:
    [tex]{ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u[/tex]

    and I calculate that the speed of light in the medium "with a refractive index or n=1.47" is
    [tex]c = { c_o \over n } = { 1 \over \sqrt{ \mu \varepsilon } } [/tex]
    which equals: 203945578.23

    What's my next step?
    Am I even going in the right direction?
     

    Attached Files:

  2. jcsd
  3. Jul 29, 2007 #2

    olgranpappy

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    both Jackson and Griffiths (and probably every EM book ever written) have sections on waveguides, and in particular on rectangular waveguides like the one you are interested in... I would probably start with Griffiths if I were you:

    Griffiths, "Introduction to Electrodynamics", section 9.5 (Guided Waves), especially section 9.5.2 (TE waves in a rectangular Wave Guide)
     
  4. Jul 29, 2007 #3
    Thanks, I'll start there.
     
  5. Jul 29, 2007 #4
    I got a hold of a copy of: Griffiths, "Introduction to Electrodynamics", section 9.5 (Guided Waves) .. and read through chapter 9...

    I guess my problem is the level of math is above what I'm used to.

    Can anyone explain the standard wave equation?

    [tex]{ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u[/tex]

    What do each part represent, I can't seem to visualize what it stands for.
     
  6. Jul 29, 2007 #5

    olgranpappy

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    It's easier to start off with just one space dimension, in which case we have:

    [tex]
    \frac{1}{c^2}\frac{\partial^2}{\partial t^2}u(x,t)=\frac{\partial^2}{\partial x^2}u(x,t)
    [/tex]

    Now, consider a function [tex]u(x,t)[/tex] that depends on space and time only through the single variable [tex]x-ct[/tex]. That is, for any function of one variable [tex]f(w)[/tex] define
    [tex]
    u(x,t)=f(x-ct)
    [/tex]

    then
    [tex]
    \frac{\partial u}{\partial x}=f'(x-ct)
    [/tex]

    and
    [tex]
    \frac{\partial u}{\partial t}=-cf'(x-ct)
    [/tex]

    thus
    [tex]
    \frac{\partial^2 u}{\partial x^2}=\frac{1}{(-c)^2}\frac{\partial^2 u}{\partial t^2}
    [/tex]
    I.e., u satisfies the wave equation... so how to interpret this form of u?

    well, for example, if the function f has a pronounced peak somewhere, then as time goes on the location of the peak moves along with a velocity c. If at t=0 f has a peak at x=x_0 then at t=t' f has a peak at x_0+ct'. Etc.

    So, to try and answer your questions:

    c is the velocity at which the wave (who's amplitude is described by u(x,t) at time t and position x) travels.

    in the context of EM we usually arrive at wave equations where the roll of u is played by the amplitudes of the fields.
     
    Last edited: Jul 29, 2007
  7. Jul 29, 2007 #6
    Ok, I've got to go back to the basics again:

    Can you explain curl?

    as in how do I do it, I've read many portions of many books that say by taking the curl of Maxwell's equations 2 and 4, we can derive the Wave equation for E. However I don't understand how they get from this: "See the attached"
     

    Attached Files:

  8. Jul 29, 2007 #7

    olgranpappy

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    "Curl" can be thought of as a differential operator that acts on vectors it's also written with the nabla [tex]\nabla[/tex] and cross-product [tex]\times[/tex] symbols as
    [tex]
    curl\vec V=\vec \nabla\times \vec V
    [/tex]
    I.e., curl V is a vector whose cartesian components are given by
    [tex]
    (curl V)_x=\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}
    [/tex]

    and

    [tex]
    (curl V)_y=\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x}
    [/tex]

    and

    [tex]
    (curl V)_z=\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y}
    [/tex]
     
  9. Jul 30, 2007 #8

    olgranpappy

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    ...now, you get to start memorizing properties of the curl. For example, since the curl of a vector is also a vector you can take the curl twice. For example:
    [tex]
    \vec \nabla \times (\vec \nabla \times \vec V)\;.
    [/tex]

    It turns out that that is equal to
    [tex]
    -\nabla^2\vec V+\vec\nabla(\vec \nabla\cdot\vec V)
    [/tex]

    So, for example, it you have [tex]\vec \nabla\cdot \vec V=0[/tex] (like if V was a magnetic field) then you would have
    [tex]
    \nabla\times\nabla\times\vec B=-\nabla^2\vec B\;.
    [/tex]

    Low and behold there is the spatial part of a wave equation!
     
  10. Jul 30, 2007 #9

    olgranpappy

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    ... but from maxwell's equation we also know that (in the absence of sources)
    [tex]
    \nabla\times B=\frac{\partial E}{c\partial t}
    [/tex]

    and [tex]\nabla\times E=\frac{\partial B}{-c\partial t}[/tex]so that
    [tex]
    \nabla\times\nabla\times B=\nabla\times\frac{\partial E}{c\partial t}=-\frac{\partial^2 B}{c^2\partial t^2}
    [/tex]

    Low and behold, the other half of your wave-equation. Thus all together we have:
    [tex]
    -\nabla^2\vec B=\frac{\partial^2}{-c^2\partial t^2}\vec B
    [/tex]

    ...we can look for solutions to maxwell's equation that are travelling waves whose velocity is c.
     
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