33-4 Polarization#

Prompts

What is the difference between polarized and unpolarized light? What is the plane of oscillation?
How does a polarizing sheet work? Which electric field components are transmitted and which are absorbed?
State the one-half rule and the cosine-squared rule. When does each apply?
What does it mean for two polarizing sheets to be crossed? What fraction of light is transmitted?
Unpolarized light passes through three polarizing sheets with axes at 0°, 45°, and 90°. What fraction emerges? Work through sheet by sheet.

Lecture Notes#

Overview#

Polarized light: electric field oscillates in a single plane (plane of oscillation).
Unpolarized light: electric field direction is random in the plane perpendicular to propagation (e.g., sunlight, bulbs).
A polarizing sheet transmits the component parallel to its polarizing direction and absorbs the perpendicular component.
One-half rule: unpolarized → \(I = I_0/2\). Cosine-squared rule: polarized → \(I = I_0 \cos^2\theta\).

Polarized vs unpolarized light#

Polarized (plane-polarized): All \(\vec{E}\) vectors lie in one plane—the plane of oscillation. Represented head-on by a double arrow along the oscillation direction.

Unpolarized: \(\vec{E}\) is perpendicular to propagation but its direction changes randomly. Can be viewed as a superposition of two perpendicular polarized waves of equal intensity (e.g., \(y\) and \(z\) components).

Type	Electric field	Source examples
Polarized	Single plane of oscillation	TV/radio antennas, lasers
Unpolarized	Random direction in plane	Sun, incandescent bulbs

Polarizing sheet#

A polarizing sheet (e.g., Polaroid) has a polarizing direction (axis):

Parallel component of \(\vec{E}\) → transmitted
Perpendicular component → absorbed

The emerging light is polarized parallel to the sheet’s polarizing direction.

Intensity: one-half rule and cosine-squared rule#

One-half rule (light reaching the sheet is unpolarized):

(241)#\[ I = \frac{1}{2} I_0 \]

Reason: Unpolarized light = two perpendicular components of equal intensity; the sheet absorbs one, transmits the other.

Cosine-squared rule (light reaching the sheet is already polarized):

(242)#\[ I = I_0 \cos^2\theta \]

where \(\theta\) is the angle between the incident polarization and the sheet’s polarizing direction. Transmitted component: \(E_{\parallel} = E \cos\theta\); intensity \(\propto E^2\).

\(\theta\)	\(I/I_0\)
0° or 180°	1 (maximum)
90°	0 (blocked)
45°	1/2

Caution

Use the one-half rule only when the light is unpolarized; use the cosine-squared rule only when it is already polarized.

Polarizer and analyzer; crossed sheets#

Polarizer: first sheet that converts unpolarized light to polarized.
Analyzer: second sheet that analyzes (or further attenuates) the polarization.

Crossed sheets: polarizing directions perpendicular (e.g., vertical and horizontal). No light passes the second sheet—\(\theta = 90°\) gives \(\cos^2 90° = 0\).

Parallel sheets: same polarizing direction. All light passed by the first is passed by the second.

Multiple sheets#

For a chain of polarizing sheets, work sheet by sheet:

Apply one-half rule or cosine-squared rule as appropriate.
The light emerging from each sheet is polarized parallel to that sheet’s direction.
Use that polarization as the input for the next sheet.

Poll: Can unpolarized light pass through three polarizers?

You have three polarizers: one polarizing along the \(y\)-axis, one along the \(x\)-axis, and one along an axis 45° to both. Can initially unpolarized light pass through all three?

(A) Yes, definitely (no matter the order)
(B) No, definitely not (no matter the order)
(C) Maybe—depends on the order of the polarizers

Poll: Maximum intensity—unpolarized through three polarizers

Same three polarizers (axes at 0°, 45°, and 90°). If the initial intensity of unpolarized light is \(I_0\), what is the maximum possible final intensity after passing through all three?

(A) \(0.125\,I_0\)
(B) \(0.250\,I_0\)
(C) \(0.500\,I_0\)
(D) None of these

Poll: Maximum intensity—polarized through three polarizers

Same three polarizers. If the initial light is polarized along the \(y\)-direction with intensity \(I_0\), what is the maximum possible final intensity after passing through all three?

(A) \(0.125\,I_0\)
(B) \(0.250\,I_0\)
(C) \(0.500\,I_0\)
(D) None of these

Example: three sheets

Unpolarized light of intensity \(I_0\) passes through three sheets with axes at 0°, 60°, and 90° (relative to vertical).

Sheet 1 (0°): unpolarized → \(I_1 = I_0/2\), polarized vertically.
Sheet 2 (60°): \(\theta = 60°\) → \(I_2 = I_1 \cos^2 60° = (I_0/2)(1/4) = I_0/8\), polarized at 60°.
Sheet 3 (90°): \(\theta = 30°\) (angle between 60° and 90°) → \(I_3 = I_2 \cos^2 30° = (I_0/8)(3/4) = 3I_0/32\).

Final fraction: \(3/32\).

Other means of polarization#

Light can be polarized by:

Reflection (Brewster’s angle)—see 33-7.
Scattering: e.g., sunlight scattered by air molecules—sky light is partially polarized. Bees and Vikings used such polarization for navigation.

Summary#

Polarized: \(\vec{E}\) in one plane; unpolarized: random direction in plane.
Polarizing sheet: transmits parallel \(\vec{E}\), absorbs perpendicular.
One-half rule: unpolarized → \(I = I_0/2\).
Cosine-squared rule: polarized → \(I = I_0 \cos^2\theta\).
Crossed sheets: \(\theta = 90°\) → no transmission.
Multiple sheets: work through one by one; output polarization = sheet’s direction.