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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 e-m waves of the same frequency and wavelength, both moving along the x-axis, but differing in phase by the angle δ . The wavefunctions are the E-Felds, which we assume to oscillate in the y-direction, so when we write E we mean the y-component 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 x-direction. 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 two-dimensional plot, as shown. Each point on the diagram corresponds to a particular complex number....
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