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Changing the Phase of a Light Wave A light wave travels a distance L through a material of refractive index n. By how much has its phase changed? A light wave travels a distance L in vacuum. By how much has its phase changed? A light wave travels a distance L in vacuum. By how much has its phase changed? How does the amplitude depend on distance? A light wave travels a distance L in vacuum. By how much has its phase changed? How does the amplitude depend on distance? At a fixed time, E(x,t) = sin(kx + constant) E(x1,t) = sin(k x1 + constant) E(x2,t) = sin(k x2 + constant) E(x1,t) = sin(k x1 + constant) E(x2,t) = sin(k x2 + constant) Phase of wave at x1 = k x1 + constant Phase of wave at x2 = k x2 + constant E(x1,t) = sin(k x1 + constant) E(x2,t) = sin(k x2 + constant) Phase of wave at x1 = k x1 + constant Phase of wave at x2 = k x2 + constant Phase difference = k x2 - k x1 = k ( x2 – x1) = k L E(x1,t) = sin(k x1 + constant) E(x2,t) = sin(k x2 + constant) Phase of wave at x1 = k x1 + constant Phase of wave at x2 = k x2 + constant Phase difference = k x2 - k x1 = k ( x2 – x1) = k L k = 2B/8, so that phase difference = 2B L/ 8 Coming back to our original problem, we can say that the phase change the light undergoes in traveling a distance L through the material is 2B L / (wavelength of light in material) Coming back to our original problem, we can say that the phase change the light undergoes in traveling a distance L through the material is 2B L / (wavelength of light in material) What is the wavelength of light in the material? 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 f 0 = c 8m f m = v 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 f 0 = c 8m f m = v (80 f 0) / (8m f m ) = c / v = n 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 f 0 = c 8m f m = v (80 f 0) / (8m f m ) = c / v = n f0= fm 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 f 0 = c 8m f m = v (80 f 0) / (8m f m ) = c / v = n f0= fm Therefore, 80 / 8m = n 80 = wavelength of light in vacuum 8m = wavelength of light in material 80 f 0 = c 8m f m = v (80 f 0) / (8m f m ) = c / v = n f0= fm Therefore, 80 / 8m = n Or, 8m = 80 / n The phase has changed by 2B L / 8m The phase has changed by 2B L / 8m = 2B L / (80 / n) = 2B n L / 80 The phase has changed by 2B L / 8m = 2B L / (80 / n) = 2B n L / 80 In traveling a distance L in the material, the wave changes its phase by the same amount that it would have changed if it had traveled a distance n L in vacuum. The phase has changed by 2B L / 8m = 2B L / (80 / n) = 2B n L / 80 In traveling a distance L in the material, the wave changes its phase by the same amount that it would have changed if it had traveled a distance n L in vacuum. n L is defined as the optical path length. How do we represent a phase change mathematically? In free space, the amplitude function is E (x,t) = E0 exp[i(kx-jt + N )] At a fixed time this is E = A eikx where A = E0 exp[i(-jt + N )] The wave amplitude at x1 is The wave amplitude at x2 is ik x1 L ikx2 E2 Ae Ae E1 Aeikx1 E2 Aeikx2 Aeikx1 eikL E1 eikL If E is the complex amplitude at the entry-face of the material, the complex amplitude at the exit face is E exp[i(phase change)] = E exp[2B i n L / 80 ]