Questions: Linear Polarization: Production and Analysis Methods

Question 1

Linearly polarized light with intensity I₀ passes through a polarizer whose transmission axis is oriented at 60° to the polarization direction. What is the transmitted intensity?

• A. 0.75 I₀
• B. 0.25 I₀
• C. 0.50 I₀
• D. 0 — only perpendicular orientations transmit light

Answer: (B)

Malus's law: I = I₀cos²θ. At θ = 60°, cos²60° = (0.5)² = 0.25, so 25% of I₀ is transmitted. Option A (0.75) is cos²30°, a common error from using the complementary angle. Option C (0.5) is often guessed because 60° is 'most of the way' to 90°, but the cosine-squared relationship falls off faster than intuition suggests.

Question 2

Two polarizing sheets are crossed (transmission axes 90° apart), blocking all light. A third sheet is inserted between them at 45° to each. What fraction of the intensity emerging from the first polarizer is transmitted through the entire stack?

• A. 0% — an obstruction cannot restore blocked transmission
• B. 50% — the intermediate polarizer halves the intensity once
• C. 25% — Malus's law applied twice: cos²45° × cos²45° = 0.25
• D. 75% — the intermediate polarizer rotates most of the polarization toward the final axis

Answer: (C)

Each stage applies Malus's law independently. After the first polarizer the beam is polarized at 0°. The middle polarizer at 45° transmits cos²45° = 0.5 of what reaches it. The final polarizer at 90° transmits cos²45° = 0.5 of that. Product: 0.5 × 0.5 = 0.25. Option A captures the intuitive but incorrect argument — that an extra absorber can only reduce transmission. The key insight is that the intermediate polarizer *changes the polarization direction*, making the angle between polarization and the final axis 45° instead of 90°, enabling partial transmission.

Question 3

When unpolarized light reflects from a glass surface at Brewster's angle, the reflected beam is completely polarized with its electric field parallel to the surface (perpendicular to the plane of incidence).

• T. True
• F. False

Answer: True

At Brewster's angle θ_B = arctan(n₂/n₁), the reflected and refracted rays are 90° apart. Under this condition, the p-polarized component (field in the plane of incidence) is not reflected at all, leaving the reflected beam entirely s-polarized (field perpendicular to the plane of incidence, i.e., parallel to the surface). This is exactly why polarized sunglasses oriented with a vertical transmission axis block horizontally polarized glare from roads and water.

Question 4

Adding a third polarizer between two crossed polarizers usually reduces the total transmitted intensity compared to having primarily the two crossed polarizers.

• T. True
• F. False

Answer: False

Two perfectly crossed polarizers transmit 0% — they already block everything. Inserting a third polarizer at 45° between them allows 25% of the post-first-polarizer intensity to pass through. Going from 0% to 25% is an increase, not a decrease. This apparent paradox — adding an absorbing element increases transmission — arises because the intermediate polarizer changes the polarization direction of the light reaching the final polarizer, reducing the angle from 90° to 45°.

Question 5

Why does inserting a polarizer at 45° between two crossed polarizers increase the transmitted intensity, even though the inserted polarizer itself absorbs light?

Model answer: The two crossed polarizers block all light because the angle between them is 90° and cos²90° = 0. The inserted polarizer does not simply 'add' transmission — it changes the problem. It projects the linearly polarized light from the first polarizer onto the 45° direction, producing a less intense beam polarized at 45°. Now the angle between this new polarization direction and the final polarizer's axis is only 45°, not 90°. Applying Malus's law at each step: cos²45° × cos²45° = 0.25, giving non-zero transmission. Without the intermediate, the direct 90° crossing guarantees zero regardless of intensity.

The core insight is that Malus's law is a projection relationship — it describes how much of the polarization vector aligns with the transmission axis. The crossed-polarizer system fails because the full 90° angle makes the projection zero. The intermediate polarizer resets the polarization direction to an intermediate angle, enabling partial projection at each subsequent stage.