(adsbygoogle = window.adsbygoogle || []).push({}); 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]

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# Homework Help: E&M plane wave interference

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