Understanding Light Polarization
Light consists of oscillating electric and magnetic fields travelling through space as an electromagnetic wave. In ordinary unpolarized light, these oscillations occur in all directions perpendicular to the wave's path. Polarized light, by contrast, has electric field oscillations confined to a single plane.
Linearly polarized light can be created through several methods: reflection from a surface at Brewster's angle, transmission through polarizing filters, or scattering from certain materials. The direction of polarization is defined by the orientation of the electric field vector. When polarized light encounters a second polarizing filter (called an analyzer), the amount of light transmitted depends critically on the relative orientation of the two polarization directions.
How Polarizers Work
A polarizer is typically made from aligned microscopic molecules or crystals that preferentially transmit or absorb light based on its polarization direction. When incident light's polarization aligns with the polarizer's transmission axis, maximum light passes through. When polarization is perpendicular to the axis, nearly all light is absorbed.
At intermediate angles, only a fraction of the incident light is transmitted. This partial transmission follows a precise mathematical relationship discovered by Étienne-Louis Malus in the early 19th century. The transmitted intensity decreases with the square of the cosine of the angle between the incoming polarization and the polarizer axis.
Common polarizers include:
- Dichroic polarizers — absorb light in one polarization direction, transmit the other
- Polarizing beamsplitters — separate light into two orthogonal polarization components
- Wire-grid polarizers — reflect one polarization, transmit the other using parallel metallic wires
Malus Law Equation
The transmitted intensity through an ideal polarizer is calculated using Malus law, which relates the output intensity to the input intensity and the angle between the incident polarization and the polarizer's transmission axis.
I = I₀ × cos²(θ)
I— Transmitted light intensity (W/m²)I₀— Initial incident light intensity (W/m²)θ— Angle between incident light polarization and polarizer axis (degrees)
Practical Application: Rotating Polarizers
Consider a beam of polarized light with initial intensity of 6 W/m² passing through a rotatable polarizer. As you rotate the polarizer relative to the light's polarization direction, the transmitted intensity changes dramatically:
- At
θ = 0°:I = 6 × cos²(0°) = 6 W/m²(maximum transmission) - At
θ = 30°:I = 6 × cos²(30°) = 4.5 W/m² - At
θ = 45°:I = 6 × cos²(45°) = 3.0 W/m² - At
θ = 60°:I = 6 × cos²(60°) = 1.5 W/m² - At
θ = 90°:I = 6 × cos²(90°) = 0 W/m²(complete extinction)
This non-linear relationship means that small rotations near perpendicular alignment cause rapid intensity changes, while rotations near parallel alignment produce only modest intensity variations.
Common Considerations When Using Malus Law
Apply these practical insights when working with polarizers and interpreting Malus law calculations.
- Real polarizers are never ideal — Practical polarizers have transmission losses and don't achieve perfect extinction at 90°. Dichroic polarizers typically transmit 38–42% of light in their preferred direction and absorb 99.9% of the orthogonal component, not 100%.
- Multiple polarizers multiply transmission losses — Two polarizers aligned at 45° to each other transmit 25% of the initial light (0.5 × 0.5). Three polarizers can actually transmit more than two, demonstrating that the angle between successive filters is critical.
- Depolarization doesn't follow Malus law — Unpolarized or partially polarized light behaves differently. Malus law applies only to fully linearly polarized light. Depolarization effects from scattering or birefringent media must be considered separately.
- Intensity and irradiance are context-dependent — Malus law expresses intensity as power per unit area (W/m²). In photometry, visible light is often weighted by human eye sensitivity and measured in lux, requiring additional conversion factors beyond the basic cos² relationship.