# 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?

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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?

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(x 1,t) = sin(k x 1 + constant) E(x 2,t) = sin(k x 2 + constant)

E(x 1,t) = sin(k x 1 + constant) E(x 2,t) = sin(k x 2 + constant) Phase of wave at x 1 = k x 1 + constant Phase of wave at x 2 = k x 2 + constant

E(x 1,t) = sin(k x 1 + constant) E(x 2,t) = sin(k x 2 + constant) Phase of wave at x 1 = k x 1 + constant Phase of wave at x 2 = k x 2 + constant Phase difference = k x 2 - k x 1 = k ( x 2 – x 1 ) = k L

E(x 1,t) = sin(k x 1 + constant) E(x 2,t) = sin(k x 2 + constant) Phase of wave at x 1 = k x 1 + constant Phase of wave at x 2 = k x 2 + constant Phase difference = k x 2 - k x 1 = k ( x 2 – x 1 ) = k L k = 2 B / 8, so that phase difference = 2 B 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 2 B 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 2 B L / (wavelength of light in material) What is the wavelength of light in the material?

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material 8 0 f 0 = c 8 m f m = v

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material 8 0 f 0 = c 8 m f m = v ( 8 0 f 0 ) / ( 8 m f m ) = c / v = n

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material 8 0 f 0 = c 8 m f m = v ( 8 0 f 0 ) / ( 8 m f m ) = c / v = n f 0 = f m

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material 8 0 f 0 = c 8 m f m = v ( 8 0 f 0 ) / ( 8 m f m ) = c / v = n f 0 = f m Therefore, 8 0 / 8 m = n

8 0 = wavelength of light in vacuum 8 m = wavelength of light in material 8 0 f 0 = c 8 m f m = v ( 8 0 f 0 ) / ( 8 m f m ) = c / v = n f 0 = f m Therefore, 8 0 / 8 m = n Or, 8 m = 8 0 / n

The phase has changed by 2 B L / 8 m

= 2 B L / ( 8 0 / n) = 2 B n L / 8 0

The phase has changed by 2 B L / 8 m = 2 B L / ( 8 0 / n) = 2 B n L / 8 0 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 2 B L / 8 m = 2 B L / ( 8 0 / n) = 2 B n L / 8 0 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) = E 0 exp[i(kx- j t + N )] At a fixed time this is E = A e ikx where A = E 0 exp[i(- j t + N )] The wave amplitude at x 1 is The wave amplitude at x 2 is 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[2 B i n L / 8 0 ]

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