Changing the Phase of a Light Wave
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?
Mar 31, 2015
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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 = 2/, so that phase difference = 2 L/
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 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 L / (wavelength of light in material)
What is the wavelength of light in the material?
0 = wavelength of light in vacuum
m = wavelength of light in material
0 = wavelength of light in vacuum
m = wavelength of light in material
0 f 0 = c m f m = v
0 = wavelength of light in vacuum
m = wavelength of light in material
0 f 0 = c m f m = v
(0 f 0) / (m f m ) = c / v = n
0 = wavelength of light in vacuum
m = wavelength of light in material
0 f 0 = c m f m = v
(0 f 0) / (m f m ) = c / v = n
f 0 = f m
0 = wavelength of light in vacuum
m = wavelength of light in material
0 f 0 = c m f m = v
(0 f 0) / (m f m ) = c / v = n
f 0 = f m
Therefore, 0 / m = n
0 = wavelength of light in vacuum
m = wavelength of light in material
0 f 0 = c m f m = v
(0 f 0) / (m f m ) = c / v = n
f 0 = f m
Therefore, 0 / m = n
Or, m = 0 / n
The phase has changed by 2 L / m
The phase has changed by 2 L / m
= 2 L / (0 / n) = 2 n L / 0
The phase has changed by 2 L / m
= 2 L / (0 / n) = 2 n L / 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 L / m
= 2 L / (0 / n) = 2 n L / 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) = E0 exp[i(kx-t + )]
At a fixed time this is E = A eikx where A = E0 exp[i(-t + )]
The wave amplitude at x1 is
The wave amplitude at x2 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[2i n L / 0 ]
11
ikxE Ae2
2ikxE Ae
12 12 1
ik x Likx ikx ikL ikLE Ae Ae Ae e E e
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