$Light refracting through a prism creating a spectrum$

Malus's law

PHYS 310 · Polarization

Malus's law gives the transmitted intensity of linearly polarized light through a polarizer. This lesson derives the cosine-squared dependence and applies it to multiple polarizers.

Key equations

E=E_0\cos\thetaI\propto E^2I=I_0\cos^2\thetaI=I_0I=0I=I_0\cos^2 45^\circ=\frac{1}{2}I_0I_1=\frac{1}{2}I_0I_2=I_1\cos^2\theta=\frac{1}{2}I_0\cos^2\thetaI_2=I_1\cos^2 45^\circ=\frac{1}{4}I_0I_3=I_2\cos^2 45^\circ=\frac{1}{8}I_0I(\theta)=I_0\cos^2\theta

Learning objectives

State and derive Malus's law.
Apply Malus's law to linearly polarized light.
Handle unpolarized light passing through a first polarizer.
Analyze multiple-polarizer arrangements.
Distinguish electric field projection from intensity transmission.

Polarization projection

Malus's law describes what happens when linearly polarized light passes through an ideal linear polarizer. If the incoming electric field makes angle $heta$ with the polarizer's transmission axis, only the component parallel to that axis is transmitted.

The transmitted electric field amplitude is

$E=E_0cos heta$

Intensity is proportional to the square of electric field amplitude:

$Ipropto E^2$

Therefore the transmitted intensity is

$I=I_0cos^2 heta$

This is Malus's law.

Important angles

If $heta=0^circ$ , the polarization is aligned with the polarizer, and

$I=I_0$

If $heta=90^circ$ , the polarizer is crossed with the incoming polarization, and

$I=0$

If $heta=45^circ$ ,

$I=I_0cos^2 45^circ= rac{1}{2}I_0$

Unpolarized light first

Malus's law applies to linearly polarized incident light. If unpolarized light first passes through an ideal polarizer, the transmitted intensity is

$I_1= rac{1}{2}I_0$

The transmitted light is then linearly polarized along the first polarizer's axis. A second polarizer at angle $heta$ transmits

$I_2=I_1cos^2 heta= rac{1}{2}I_0cos^2 heta$

Three polarizers

A surprising result occurs with three polarizers. Two crossed polarizers transmit no light. But if a third polarizer is placed between them at an intermediate angle, some light can pass.

For example, let the first and third polarizers be crossed, with a middle polarizer at $45^circ$ . Unpolarized incident light first becomes

$I_1= rac{1}{2}I_0$

After the middle polarizer,

$I_2=I_1cos^2 45^circ= rac{1}{4}I_0$

After the final polarizer, another $45^circ$ rotation gives

$I_3=I_2cos^2 45^circ= rac{1}{8}I_0$

The middle polarizer changes the polarization state, allowing partial transmission.

Field versus intensity

A common mistake is to apply cosine instead of cosine squared to intensity. The electric field amplitude projects with $cos heta$ , but intensity depends on amplitude squared. That is why Malus's law uses $cos^2 heta$ .

Real polarizers

Real polarizers are not perfect. They transmit most light along the transmission axis and block most perpendicular light, but not exactly all or none. Performance is described by extinction ratio and transmission efficiency.

High-quality polarizers are important in precision optics, lasers, and imaging.

Rotating analyzer

If a linearly polarized beam passes through a rotating analyzer, the transmitted intensity varies sinusoidally as

$I( heta)=I_0cos^2 heta$

Measuring this variation can determine the polarization direction of the incoming beam.

The big idea

Malus's law comes from projecting the electric field onto a polarizer's transmission axis and then squaring to get intensity. It explains crossed polarizers, intermediate polarizers, polarization measurements, and many optical devices that control light intensity through polarization.

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