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Homework Help: E&M plane wave interference

  1. Dec 18, 2008 #1

    KFC

    User Avatar

    1. The problem statement, all variables and given/known data
    Find out the condition for two-plane-wave interference. The two plane waves are given as

    [tex]
    \vec{E}_1 = E_0\vec{A}\exp(ikz - i\omega t) + c.c.
    [/tex]

    [tex]
    \vec{E}_2 = E_0\vec{B}\exp(ikz - i\omega t + i\phi) + c.c.
    [/tex]

    where [tex]\vec{A}[/tex] and [tex]\vec{B}[/tex] are complex. All other variables are real.


    2. The attempt at a solution
    Here is how I do the problem. I first let

    [tex]\vec{A} = A\exp(i\alpha)[/tex]

    [tex]\vec{B} = B\exp(i\gamma)[/tex]

    and define
    [tex]F = kz - \omega t, \qquad G = kz - \omega t + \phi[/tex]
    so

    [tex]
    \vec{E}_1 = E_0A \left(\exp[i(F + \alpha)] + \exp[-i(F+\alpha)]\right)
    = 2E_0 A\cos(F+\alpha)
    [/tex]

    [tex]
    \vec{E}_2 = E_0B \left(\exp[i(G + \beta)] + \exp[-i(G+\beta)]\right)
    = 2E_0 B\cos(G+\beta)
    [/tex]

    Now, both [tex]\vec{E}_1[/tex] and [tex]\vec{E}_2[/tex] become real. So I can square the total field [tex]\vec{E} = \vec{E}_1 + \vec{E}_2[/tex] directly

    [tex]E^2 = E_1^2 + E_2^2 + 2E_1E_2 =
    4E_0^2 A^2\cos^2(F+\alpha) + 4E^2_0 B^2\cos^2(G+\beta) + 8E_0^2AB\cos(F+\alpha)\cos(G+\beta)
    [/tex]

    But I remember the interference term should only contain the

    [tex]E_0^2AB\cos(\phi+\alpha-\beta)[/tex]
     
  2. jcsd
  3. Dec 19, 2008 #2

    turin

    User Avatar
    Homework Helper

    What do you mean by "the condition for two-plane-wave interference"? They always interfere, regardless of any specific phases and whatnot.

    Are these waves supposed to be going in exactly the same direction? If so, this problem may be quite easy (depending on what "the condition for two-plane-wave interference" means.)
     
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