Two light waves of the same frequency and fixed phase difference arrive together at a…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Wave propagation and light in media

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

Two light waves of the same frequency and fixed phase difference arrive together at a point:

E_1(t) = E_1\cos\omega t

and

E_2(t) = E_2\cos(\omega t + \delta)

, where

E_1

and

E_2

are their (possibly unequal) amplitudes and

\delta

is the fixed phase difference between them. ((a)) Form the sum

E(t) = E_1(t) + E_2(t)

, square it, and take the time average over one full cycle, using

\langle\cos^2\omega t\rangle = \langle\sin^2\omega t\rangle = \tfrac12

and

\langle\cos\omega t\sin\omega t\rangle = 0

. Show that

\langle E^2\rangle = \tfrac12\left(E_1^2 + E_2^2 + 2E_1E_2\cos\delta\right)

. ((b)) Using

I \propto E_0^2

for a wave of amplitude

E_0

, with the same constant of proportionality for both beams (so

I_1 \propto E_1^2

I_2 \propto E_2^2

), rewrite the result of part (a) as

I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\delta

. ((c)) Check the formula of part (b) for two beams of equal irradiance

I_0

\delta = 0

\delta = \pi/2

and

\delta = \pi