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Unformatted text preview: Physics 54 Wave Optics 1 Not only is the universe stranger than we think — it is stranger than we can think. — Werner Heisenberg Superposition of Harmonic Waves The essential characteristic of energy transport by waves is that waves obey the superposition principle . This means that two waves in the same spatial region can interfere , rearranging the energy in space in a pattern often quite different from that of either wave alone. Since light propagates as a wave, we must analyze this phenomenon. We begin with a mathematical problem: How do we Fnd the wavefunction for the resulting wave when two waves interfere? We consider two harmonic em waves of the same frequency and wavelength, both moving along the xaxis, but differing in phase by the angle δ . The wavefunctions are the EFelds, which we assume to oscillate in the ydirection, so when we write E we mean the ycomponent of E . We have: E 1 ( x , t ) = E 1 cos( kx − ω t ) E 2 ( x , t ) = E 2 cos( kx − ω t + δ ) The result of superposition of these waves will be another harmonic wave with the same frequency and wavelength, also moving in the xdirection. Thus E = E 1 + E 2 must have the form E ( x , t ) = E cos( kx − ω t + φ ) Our problem is to Fnd the constants E and φ in terms of the amplitudes of the original waves and the phase difference δ . Our main interest is in the intensity of the resulting wave, which is proportional to E 2 , so we are usually less interested in φ . To solve this problem we employ a trick based on a famous theorem: Euler’s theorem e i θ = cos θ + i sin θ Here i is the imaginary unit, with i 2 = − 1 . This remarkable formula says that exponential functions and trigonometric functions are related through the complex numbers. Let us Frst review of some facts about complex numbers. PHY 54 1 Wave Optics 1 Any complex number z can be written in two forms, related by Euler’s theorem: z = x + iy z = re i θ In the Frst form, x and y are real numbers; x is the real part of z [one writes x = Re( z )] while y is the imaginary part of z [written y = Im( z )]. In the other form, r is the amplitude and θ is the phase of z . ¡rom Euler’s theorem we Fnd x = r cos θ , y = r sin θ . One often displays complex numbers graphically by showing the real and imaginary parts in a twodimensional plot, as shown. Each point on the diagram corresponds to a particular complex number....
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This note was uploaded on 10/19/2011 for the course PHYSICS 54L taught by Professor Thomas during the Summer '09 term at Duke.
 Summer '09
 Thomas
 Physics, Energy

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