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Wave Optics & Transverse Waves

Polarization

Understand the transverse nature of light through polarization. Rotate polarizers and analyzers to verify Malus's law, block glare with sunglasses, and explore wave filtering with rope models.

Polarization Physics Lab ⓘ

Observe light amplitude reduction, Malus's Law curve, and polarization alignment.

Simulation active

Polarization Telemetry

I = I₀ cos²θ
Incident Intensity (I₀)
100 W/m²
Transmitted Intensity (I)
25 W/m²
Transmission %
25 %
Relative Angle (θ)
60.00°
Polarization Direction
Linear (Vertical)

Postulates of Light Polarization

Polarization is the confinement of transverse electromagnetic wave vibrations to a single geometric plane perpendicular to the wave's path of travel. Unpolarized light consists of electric fields vibrating in all possible directions.

  • Polarizer: A grid or sheet that restricts incident unpolarized light to a single axis, reducing total intensity by exactly 50%.
  • Malus's Law: When plane-polarized light is incident on an analyzer, the transmitted intensity varies according to the angle θ between their axes: I = I₀ cos²θ.

Polarized vs Unpolarized

Unpolarized light has electric fields vibrating in all directions perpendicular to the ray direction. Polarized light has electric fields confined to a single vibration plane.

Because sound waves are longitudinal, they do not possess transverse vibration axes and cannot be polarized.

Malus's Law Formula

Transmitted intensity through a rotating analyzer filter is governed by:

Malus's Law

I = I0 × cos2θ

Here, I₀ is the polarized light intensity hitting the analyzer, and θ is the angle difference.

Eliminating Reflected Glare

Light reflecting off flat horizontal surfaces like lakes or wet asphalt is horizontally polarized. Polarized sunglasses block this glare by filtering light through a vertical axis.

Vertical axes block horizontal glare

This provides crystal-clear vision for driving and boating, highlighting Brewster's angle application.

Step-by-Step Solved Problems

Master Malus's Law and filter intensity drops with these step-by-step solutions.

Example 1 Problem Statement

Unpolarized light of intensity 100 W/m² is passed through a polarizing filter, and then through a second polarizing filter (analyzer) whose axis is oriented at 60° to the first. Calculate the final transmitted light intensity.

View Mathematical Solution Steps
  1. Recall the effect of the first polarizer: Unpolarized light passing through a polarizer loses exactly half its intensity. Therefore, the intensity after the first filter is: I1 = I0 / 2 = 100 / 2 = 50 W/m².
  2. Recall Malus's law for the second filter: I = I1 cos²θ.
  3. Substitute values: θ = 60°, cos 60° = 0.5, cos² 60° = (0.5)² = 0.25.
  4. Calculate the final transmitted intensity: I = 50 W/m² × 0.25 = 12.5 W/m².

Final Derived Answer: Final Transmitted Intensity I = 12.5 W/m².

Example 2 Problem Statement

Two polarizing sheets have their transmission axes crossed at 90° so that no light is transmitted. A third polarizing sheet is inserted between them with its transmission axis at 45° to the first. Find the fraction of light transmitted.

View Mathematical Solution Steps
  1. Identify the incident light intensity after the first sheet as I1.
  2. Recall Malus's law for the middle (second) sheet at 45°: I2 = I1 cos²(45°) = I1 × (1/√2)² = 0.5 I1.
  3. Recall Malus's law for the third sheet. Since the middle sheet is at 45° and the third is at 90° to the first, the angle between the middle and third sheet is: θ = 90° - 45° = 45°.
  4. Calculate transmitted intensity: I3 = I2 cos²(45°) = (0.5 I1) × 0.5 = 0.25 I1.
  5. Express the fraction of initial intensity transmitted through the entire three-sheet system: I3 / I1 = 0.25 (or 25%).

Final Derived Answer: Fraction of Light Transmitted = 0.25 (or 25% of incident polarized light).

Example 3 Problem Statement

At what angle must an analyzer be set relative to a polarizer so that the transmitted intensity is 75% of the incident polarized intensity?

View Mathematical Solution Steps
  1. Recall Malus's law: I = I0 cos²θ.
  2. Identify target transmission ratio: I / I0 = 0.75.
  3. Solve for cosθ: cosθ = √(0.75) = √3 / 2 ≈ 0.866.
  4. Solve for angle θ: θ = cos⁻¹(0.866) = 30°.

Final Derived Answer: Orientation Angle θ = 30°.

Self-Check Questions

Question 1

Define polarization of light and explain why sound waves in air cannot be polarized.

Show Answer & Explanation

Polarization is the restriction of wave vibrations to a single plane perpendicular to the direction of wave travel. Sound waves in air are longitudinal waves, meaning their vibrations are parallel to wave travel. Since they have no transverse vibrational components, they cannot undergo polarization.

Question 2

State Malus's law and write its mathematical equation, explaining all variables.

Show Answer & Explanation

Malus's law states that when completely plane polarized light of intensity I₀ is incident on an analyzer, the transmitted intensity (I) varies as the square of the cosine of the angle (θ) between the axes of the polarizer and analyzer: I = I₀ cos²θ.

Question 3

How do polarized sunglasses eliminate horizontal glare from lake and road surfaces?

Show Answer & Explanation

Glare is caused by sunlight reflecting horizontally from flat surfaces, which makes the reflected light horizontally polarized. Polarized sunglasses are made with vertical transmission axes. This vertical orientation completely blocks the horizontally polarized glare while letting useful vertical light waves pass.

Question 4

What happens to unpolarized light intensity when it passes through a single ideal polarizing filter?

Show Answer & Explanation

Unpolarized light has waves vibrating in all possible directions. A single ideal polarizer transmits only the component of each wave that is parallel to its transmission axis, which reduces the total light intensity by exactly half (I = I₀/2).

Question 5

Explain how the rope and vertical slit analogy works as a physical model for polarization.

Show Answer & Explanation

If a rope is passed through a vertical slit and shaken, waves vibrating vertically pass through easily. However, waves vibrating horizontally are blocked by the vertical slit. The vertical slit acts as a polarizer, demonstrating that only transverse wave vibrations can be filtered based on orientation.

Question 6

What is Brewster's angle, and what happens to the reflected ray at this angle?

Show Answer & Explanation

Brewster's angle (θ_p) is the specific angle of incidence at which light reflecting from a boundary is completely plane polarized. At this angle, the reflected light ray and the refracted light ray are perpendicular (at 90 degrees) to each other, and tanθ_p = n.