Physics · Optics

Polarization

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What you learned

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Key takeaway

Polarization is about wave orientation across the beam, not beam brightness by itself.
For linearly polarized input, the transmitted field is the projection onto the polarizer axis, so the detector follows Malus's law: I_out / I_0 = cos^2(Δθ).

Common misconception

A polarizer is just a dimmer, so any wave should pass if the source is strong enough.

An ideal polarizer does not ask first how strong the wave is. It asks how much of the transverse oscillation points along its axis.

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Equation snapshotField projection onto the axis · Malus's lawShow 2 equations

Field projection onto the axis
$E_{o u t} = E_{0} cos (Δ θ)$
The transmitted field is the component of the incoming transverse oscillation along the polarizer axis.
Malus's law
$I_{o u t} = I_{0} cos^{2} (Δ θ)$
Detector brightness depends on the squared projection because intensity follows the square of field amplitude.

Why it behaves this way

Explanation

Polarization tells you which sideways direction the electric field is oscillating in. The beam still travels to the right, but the electric field can point in different directions across the beam. That is why polarization is a property of transverse waves.

An ideal polarizer acts like a direction filter. It keeps the field component along its axis, blocks the perpendicular part, and leaves the output polarized along that axis. The detector brightness is therefore a projection result: good alignment transmits strongly, a 45° mismatch gives half intensity, and crossed axes ideally give almost none.

Key ideas

01Polarization is about wave orientation across the beam, not beam brightness by itself.

02For linearly polarized input, the transmitted field is the projection onto the polarizer axis, so the detector follows Malus's law: I_out / I_0 = cos^2(Δθ).

03For unpolarized input, one ideal polarizer transmits half the intensity on average and creates linearly polarized output along its axis.

04Polarization is a clean signature of transverse-wave behavior because there must be a sideways oscillation direction to select.

Worked examples

Live polarization checks

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Frozen walkthrough

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current input state and polarizer axis, what field leaves the polarizer and what fraction of the incoming intensity reaches the detector?

state_{in}

Input state

Linear input at 20°

θ_{in}

Input orientation

20° °

θ_{p}

Polarizer axis

50° °

Δ θ

Relative angle

30° °

1. Read the incoming state and axis

The input is Linear input at 20° and the polarizer axis is set to 50°, so the bench treats the transmission as an orientation match problem.

2. Keep only the axis-parallel component

The axis keeps only the cosine projection of the input, so the transmitted field follows the current partial projection case. That gives a transmitted field amplitude of 0.95 arb..

3. Read the transmitted output and detector fraction

The detector receives 75% of the incoming intensity, and the output leaves linearly polarized along 50°.

Current transmitted state

E_{o u t} = 0.95 a r b ., I / I_{0} = 0.75, θ_{o u t} = 50°

The input is only partly aligned with the axis, so the detector reads a partial transmission and the output is reset to 50°.

Quick test

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Accessibility

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The simulation shows a compact transverse cross-section bench with three circular stations: the input, the polarizer, and the detector. A beam-direction guide points horizontally to the right while orientation lines inside the circles show the incoming polarization, the polarizer axis, and the transmitted output.

Optional overlays can call out the transverse cross-section, the axis projection, and the transmitted-versus-blocked detector split. The readout card summarizes the input state, input angle, polarizer angle, relative angle, transmitted field amplitude, and relative detector intensity.

Graph summary

The power-split graph sweeps the polarizer angle and compares transmitted and blocked relative intensity. The field-projection graph sweeps the same angle and compares the axis-aligned and perpendicular field components.

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