Malus's Law
Malus's Law describes how polarized light intensity changes through a polarizer: I = I₀ × cos²(θ), where θ is the angle between polarization and transmission axis.
Did our AI summary help? Let us know.
Unpolarized light loses 50% through the first polarizer. The three-polarizer trick: inserting 45° between crossed polarizers allows 12.5% transmission. Polarized sunglasses block horizontally polarized glare. LCD screens use crossed polarizers with liquid crystals between them.
Ready to run the numbers?
Why: Essential for LCD displays, polarization filters, photography, and optical systems.
How: Polarizers transmit only the component of the electric field aligned with their transmission axis; intensity is proportional to cos²(θ).
Run the calculator when you are ready.
Polarizer Configuration
Step-by-Step Calculation
Visualizations
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
Polarized sunglasses use Malus's Law to block horizontally polarized glare.
— HyperPhysics
LCD displays use two crossed polarizers with liquid crystals between them.
— Physics Classroom
The three-polarizer trick: 0° + 45° + 90° gives 12.5% transmission.
— MIT OCW
Photographers rotate polarizing filters to minimize glare from water or glass.
— NIST
Key Takeaways
- •Malus's Law: I = I₀ × cos²(θ) describes how polarized light intensity changes through a polarizer
- •Maximum transmission: Occurs at 0° (parallel alignment) - 100% of polarized light passes through
- •Half intensity: At 45°, exactly 50% of polarized light is transmitted
- •Complete extinction: At 90° (crossed polarizers), no light passes through
- •Unpolarized light: First polarizer always transmits 50%, regardless of angle
- •Three-polarizer trick: Inserting a 45° polarizer between crossed polarizers allows 12.5% transmission
Did You Know?
🕶️ Polarized Sunglasses
Polarized sunglasses use Malus's Law to block horizontally polarized glare from water and roads. The vertical polarizer blocks reflected light, reducing eye strain and improving visibility.
📺 LCD Screens
Every LCD display uses two crossed polarizers with liquid crystals between them. By rotating the polarization, pixels control light transmission, creating the images you see on your phone, TV, and computer screen.
How It Works
Physical Mechanism
Light is a transverse electromagnetic wave with electric field oscillations perpendicular to the direction of propagation. A polarizer acts like a filter with parallel slits - it only transmits the component of the electric field aligned with its transmission axis.
Since intensity is proportional to the square of the electric field amplitude, we get the cos² dependence.
Polarized Light
Already has a defined polarization direction. The angle θ is measured from this direction to the polarizer's transmission axis. Maximum transmission occurs when they're parallel (0°).
Unpolarized Light
Has random polarization directions. The first polarizer always transmits exactly 50% while creating polarized output aligned with its axis. Subsequent polarizers follow Malus's Law.
Expert Tips
For maximum transmission: Align polarizer with light's polarization direction (0°). For photography, rotate until glare is minimized.
For variable attenuation: Use a rotating polarizer to continuously adjust laser power. Small angle changes create significant intensity variations.
Real polarizers: Have efficiency less than 100% due to absorption. Premium calcite prisms achieve 98-99% efficiency, while sheet polarizers are typically 85-92%.
Three-polarizer trick: Demonstrates that polarizers don't just filter but can also be used to rotate polarization direction, enabling transmission through crossed polarizers.
Comparison Table: Polarizer Types
| Type | Efficiency | Applications | Cost |
|---|---|---|---|
| Sheet Polarizer (HN38) | 85% | Sunglasses, LCD displays | Low |
| Premium Sheet | 92% | Photography filters | Medium |
| Wire Grid | 95% | IR, microwave systems | Medium-High |
| Calcite Prism (Glan-Thompson) | 98-99% | Lasers, precision optics | High |
| Dichroic Glass | 99% | High-power lasers | Very High |
Frequently Asked Questions
Why does unpolarized light lose 50% through the first polarizer?
Unpolarized light contains equal amounts of all polarization directions. A polarizer selects only one direction, transmitting half the total intensity while blocking the perpendicular component.
How does the three-polarizer trick work?
Two crossed polarizers (0° and 90°) block all light. Inserting a 45° polarizer between them creates intermediate polarization states. The first polarizer creates 0° polarization, the 45° rotates it to 45°, and the 90° transmits the 45° component, resulting in 12.5% total transmission.
What's the difference between ideal and real polarizers?
Ideal polarizers have 100% efficiency and perfect extinction. Real polarizers absorb some light (typically 5-15%) and have finite extinction ratios (10³ to 10⁶), meaning some light leaks through even when crossed.
Can I use Malus's Law for circularly polarized light?
Malus's Law applies to linearly polarized light. Circularly polarized light requires different treatment using Jones calculus or Mueller matrices, though a linear polarizer will transmit 50% of circularly polarized light regardless of angle.
Why do photographers rotate polarizing filters?
Rotating the filter changes the angle θ relative to reflected light's polarization. At the optimal angle, glare from water or glass is minimized, improving image clarity and color saturation.
What is extinction ratio?
Extinction ratio is the ratio of maximum transmission (parallel) to minimum transmission (crossed). Higher values indicate better polarizer quality. Ideal polarizers have infinite extinction ratio, while real ones achieve 10³ to 10⁶.
Infographic: Polarization Facts
Official Data Sources
This calculator uses verified data from authoritative sources in optics and polarization physics:
HyperPhysics - Polarization
Comprehensive physics reference for polarization theory
Last Updated: 2024-01-01
NIST - Optical Radiation Measurements
Official optical measurement standards and data
Last Updated: 2024-01-01
Physics Classroom - Polarization
Educational resource on light polarization principles
Last Updated: 2023-01-01
⚠️ Disclaimer
This calculator is for educational purposes. Results assume ideal polarizers and monochromatic light. Real-world applications involve wavelength-dependent effects, imperfect extinction ratios, and absorption losses. For precision optical systems, consult manufacturer specifications and consider all physical factors.
Related Calculators
Bragg's Law Calculator
Calculate X-ray diffraction angles, d-spacings, and wavelengths using Bragg's Law for crystallography.
PhysicsLaser Brightness Calculator
Calculate laser radiance, intensity, and beam quality metrics for CW and pulsed lasers.
PhysicsSnell's Law Calculator
Calculate refraction angles, critical angles, and total internal reflection using Snell's Law.
PhysicsSpeed of Light Calculator
Calculate light travel times, speeds in different media, and relativistic effects like time dilation.
PhysicsAngular Resolution Calculator
Calculate the resolving power of optical systems using Rayleigh criterion and diffraction limits.
PhysicsAperture Area Calculator
Calculate aperture area for circular and non-circular apertures in optical and photography systems.
Physics