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Walk-through

How to Use This Calculator

3 steps

1

Enter the initial intensity and angle

On the Intensity tab, enter I0 (the intensity of light already passed through a first polarizer) and the angle θ between that polarizer's axis and a second polarizer (the analyzer). The calculator applies I = I0·cos²θ and updates the transmitted intensity instantly, or pick one of the common-angle presets (0°, 30°, 45°, 60°, 90°).

2

Solve for the angle instead

Switch to the Solve Angle tab when you know I0 and the transmitted intensity I but need the angle between the polarizers. The calculator rearranges the formula to θ = acos(√(I/I0)).

3

Chain multiple polarizers

Use the Multiple Polarizers tab to see what happens with three or more polarizers in sequence — each stage's output becomes the next stage's I0. This reveals the classic 'crossed polarizers plus a middle one' demonstration, where inserting a third polarizer between two crossed ones unlocks transmission that would otherwise be zero.

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Reference

Formula & Methodology

2 formulas

Malus's law

I = I0 · cos²(θ)

I0 is the intensity of light already polarized (for example, after passing through a first polarizer); θ is the angle between that light's polarization axis and the transmission axis of the next polarizer (the analyzer); I is the resulting transmitted intensity. Rearranged for the angle: θ = acos(√(I / I0)).

Chained polarizers

I_n = I_{n-1} · cos²(θ_n)

For a sequence of polarizers, apply Malus's law repeatedly: each stage's transmitted intensity becomes the incident intensity for the next stage, using the angle between that polarizer and the one immediately before it.

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Glossary

Key Terms Explained

6 terms

Malus's law The relationship I = I0·cos²θ, describing how the intensity of polarized light drops as it passes through a polarizer whose transmission axis is rotated by angle θ from the light's own polarization direction. Named after French physicist Étienne-Louis Malus, who discovered it in 1808.

Polarizer An optical filter that only transmits light whose electric field oscillates along a specific direction (its transmission axis), blocking the rest. Common in sunglasses, camera filters, and LCD screens.

Polarized light Light whose electric field oscillates in a single, well-defined plane, rather than in random directions (as in ordinary unpolarized light from the sun or a bulb).

Transmitted intensity The intensity (I) of light that successfully passes through a polarizer, always less than or equal to the incident intensity (I0) since a polarizer can only remove light, never add it.

Analyzer The second (or later) polarizer in a sequence, used to measure or filter light that has already been polarized by an earlier polarizer. Its angle relative to the incoming polarization determines how much light gets through.

Extinction The condition where a polarizer's transmission axis sits at 90° to the incoming light's polarization, transmitting essentially zero intensity — also called 'crossed polarizers'.

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Scenarios

Real-World Examples

3 worked examples

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Photographer with a polarizing filter

Analyzer rotated 45°

Initial intensity I0 100 Angle θ 45°

I = 100 × cos²(45°) = 100 × 0.5 = 50 — exactly half the light passes through. This is why rotating a polarizing filter by 45° from fully-open cuts brightness in half, well before it looks fully dark.

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Two stacked polarizing sunglasses lenses

Crossed polarizers at 90°

Initial intensity I0 100 Angle θ 90°

I = 100 × cos²(90°) = 100 × 0 = 0 — no light gets through at all. This is the classic 'crossed polarizers' demonstration: two polarizing filters rotated 90° apart block essentially all light, which is how you can tell if a pair of sunglasses (or an LCD screen) uses polarization.

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Physics classroom demonstration

A third polarizer 'unlocks' crossed polarizers

Initial intensity I0 100 Polarizer 1 angle 45° Polarizer 2 angle 45°

Two polarizers 90° apart transmit zero light. But inserting a third polarizer between them at 45° to each changes the outcome: I = 100 × cos²(45°) × cos²(45°) = 100 × 0.5 × 0.5 = 25 — a quarter of the original light now gets through, even though the first and last polarizers are still 90° apart. This counterintuitive result is a textbook demonstration of Malus's law applied in sequence.

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Deep Dive

Understanding Malus's Law

3 chapters · 4 min read

Malus's law describes exactly how much light survives passing through a polarizing filter, based only on the angle between the light's polarization direction and the filter's transmission axis. It explains why polarizing sunglasses cut glare, why rotating a camera filter dims the sky, and why stacking two polarizers at 90° blocks light completely — while a third polarizer placed between them can bring some of that light back.

Why intensity follows a cosine-squared curve

A polarizer only transmits the component of the light's electric field that lines up with its transmission axis. If the incoming light's amplitude is E0 and the analyzer is rotated by angle θ, only E0·cosθ of the amplitude survives — the rest is absorbed or reflected by the filter. Since intensity is proportional to amplitude squared, the transmitted intensity is I = I0·cos²θ. This is why the drop-off is gradual near 0° (small angle changes barely affect cos²θ) but steep near 90° (where cos²θ is falling fastest).

The two-polarizer and three-polarizer cases

With just two polarizers, rotating the second one from 0° to 90° smoothly dims the light from 100% to 0% transmission — at 45° exactly half the light passes through, not a linear midpoint but the natural consequence of cos²(45°) = 0.5. The famous 'paradox' appears with three polarizers: two polarizers crossed at 90° transmit nothing, since cos²(90°) = 0. But inserting a third polarizer in between, at say 45° to each, transmits cos²(45°) × cos²(45°) = 25% of the original light — because each individual stage only has to survive a 45° rotation, not a full 90° one. Adding a filter can paradoxically let more light through.

Limits and edge cases

This calculator assumes ideal polarizers with no absorption loss beyond the cos²θ term, and it assumes the incident light entering each stage is already fully polarized (as it is after passing through a first polarizer). Real polarizing filters absorb a small additional fraction of light even when fully aligned (θ = 0°), and unpolarized light hitting a single polarizer for the first time transmits at most 50% of its intensity regardless of orientation — a separate rule from Malus's law, which only applies to already-polarized light meeting a subsequent polarizer.

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Questions

Frequently Asked Questions

6 questions

What is Malus's law?+

Malus's law, I = I0·cos²θ, gives the intensity (I) of polarized light that passes through a polarizer, based on the incident intensity (I0) and the angle (θ) between the light's polarization direction and the polarizer's transmission axis. It was discovered by Étienne-Louis Malus in 1808.

What happens at 45 degrees?+

At θ = 45°, cos²(45°) = 0.5, so exactly half the incident intensity is transmitted — I = 0.5 × I0. This is a common reference point since it's the halfway mark in intensity, even though 45° is the halfway angle between 0° and 90°.

Why does 90 degrees block all the light?+

At θ = 90° ('crossed polarizers'), cos(90°) = 0, so cos²(90°) = 0 and no light gets through at all. This is the classic demonstration of stacking two polarizing filters rotated perpendicular to each other.

How does Malus's law work with multiple polarizers?+

Apply the formula in a chain: the transmitted intensity of one polarizer becomes the incident intensity (I0) for the next, using the angle between each polarizer and the one before it. This is why inserting a third polarizer between two crossed ones can 'unlock' transmission that would otherwise be zero — each individual stage only has to survive a smaller angle change.

What units does this calculator use?+

Intensity (I0 and I) can be entered in any consistent unit — the calculator only works with the ratio between them, so watts per square meter, arbitrary sensor units, or a percentage all work the same way. Angles are entered and displayed in degrees.

Can I solve for the angle instead of the intensity?+

Yes — use the Solve Angle tab. Given I0 and a target transmitted intensity I (which must be between 0 and I0), the calculator rearranges the formula to θ = acos(√(I / I0)) and returns the angle in degrees.

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Malus's Law Calculator

Inputs

Transmitted intensity

Solve for the angle

Angle θ

Chain of polarizers

Final transmitted intensity

How to Use This Calculator

Enter the initial intensity and angle

Solve for the angle instead

Chain multiple polarizers

Formula & Methodology

Key Terms Explained

Real-World Examples

Photographer with a polarizing filter

Two stacked polarizing sunglasses lenses

Physics classroom demonstration

Understanding Malus's Law

Why intensity follows a cosine-squared curve

The two-polarizer and three-polarizer cases

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Malus's Law Calculator

Inputs

Transmitted intensity

Solve for the angle

Angle θ

Chain of polarizers

Final transmitted intensity

How to Use This Calculator

Enter the initial intensity and angle

Solve for the angle instead

Chain multiple polarizers

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding Malus's Law

Why intensity follows a cosine-squared curve

The two-polarizer and three-polarizer cases

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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