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PolarizationComments

Polarization

PolarizationA property of light is that it can be polarized. Unpolarized light points in all sorts of direction, while polarized light has its E vectors going in the same direction.
Malus' Law

Where I0 is the intensity of unpolarized light, and I is the intensity after polarization:

For polarized light going at angle θ to the polarizer: I = I0 cos2 θ

For unpolarized light, since Since the average value of cos2 θ is ½ then: I = ½ I0

Brewster's Angle (θP)

When a ray travelling through a medium with index of refraction n1 strikes a second medium with n2, then the angle θP is when the reflected ray and refracted ray have an angle of 90° between them. The angle θP is formed between the incident ray and the normal to the surface.

n1 sin θP = n2 sin θ2

n1 sin θP = n2 sin ( 90° - θP )

n1 sin θP = n2 cos θP

tan θP = n2 / n1 (Brewster's Law)

Practice problems

The critical angle for total internal reflection at a boundary between two materials is 40.0°. What is Brewster's angle at this boundary? Give two answers, one for each material.

Critical angle: sin θc = n2 / n1
Brewster's angle for first material: tan θP = n2 / n1
Brewster's angle for second material: tan θP = n1 / n2

sin 40.0° = n2 / n1
n2 / n1 = tan θP for first material
0.64278761 = tan θP
θP = 32.7° for first material

sin 40.0° = n2 / n1
n1 / n2 = tan θP for second material
0.64278761 = 1 / tan θP
1.55572383 = tan θP
θP = 57.3° for second material

Two polarizers are oriented at 50° to one another. Light polarized at an 30° angle to each polarizer passes through both. What is the transmitted intensity?

I = I0 cos2 θ

I0 = Initial intensity
I1 = Intensity after first polarizer
I2 = Intensity after second polarizer
θ1 = 30° is incident light angle to first polarizer
θ2 = 50° since the second polarizer is rotated 50° relative to the first polarizer.

I1 = I0 cos2 θ1 = I0 cos2 30°
I2 = I1 cos2 θ2 = I0 cos2 30° cos2 50°
I2 = I0 0.309881933
I2 / I0 = 0.309881933 = 31% transmitted intensity

Two polarizers are oriented at 70° to each other and plane-polarized light is incident on them. If only 10% of the light gets through both of them, what was the initial polarization direction of the incident light?

I = I0 cos2 θ

I0 = Initial intensity
I1 = Intensity after first polarizer
I2 = Intensity after second polarizer (10% of I0)
θ1 = Incident light angle to first polarizer
θ2 = 70° since the second polarizer is rotated 70° relative to the first polarizer.

I1 = I0 cos2 θ1
I2 = I0 cos2 θ1 cos2 70°
I2 / I0 = 10% = .10
.10 = cos2 θ1 cos2 70°
0.854863217 = cos2 θ1
0.924588134 = cos θ1
θ1 = 22°