Optics
Three-polarizer paradox
Insert a third polarizer at 45° between crossed ones and light gets through again — I₀/8.
Three-polarizer paradox — interactive Optics simulation. Insert a third polarizer at 45° between crossed ones and light gets through again — I₀/8. Free browser-based virtual physics lab with live SI measurements and a guided tutorial.
Three polarizers
Insert a third polarizer at 45° between crossed ones and light gets through again — I₀/8.
Inserting a third polarizer at 45° between crossed ones projects light onto an axis neither fully blocks, allowing partial transmission—counter-intuitive but required by sequential Malus factors. Peak transmission can reach I₀/8 for ideal sheets.
- I₀/8 transmitted
- 45° middle
Investigation brief
Plan the question before you open the lab
The brief mirrors the prerendered page: driving question, competing predictions, variable roles, governing laws, setup, analysis and extension prompts remain visible and in this order.
Driving question
Two crossed polarizers (90° apart) block all light. If you insert a THIRD polarizer between them at some angle, does adding another blocking element let MORE light through?
Predictions to weigh
- No — adding any additional polarizer can only block more light, never less.
- Surprisingly, yes — a middle polarizer at an intermediate angle actually lets some light back through, more polarizers paradoxically restoring transmission.
- Adding a third polarizer doesn't change anything if the outer two are still crossed.
Variable roles
What you set:
- Middle polarizer's angle (°)
What you measure:
- Transmitted fraction
How the investigation runs
- Open the three-polarizers preset and press Reset. Outer polarizers are fixed at 0° and 90° (crossed); the middle one starts at 45°.
- Enable the final-transmitted-intensity readout.
- Set the middle polarizer's angle for each trial (outer polarizers fixed at 0° and 90°) and record the fraction of light that makes it through all three.
Governing equation
Three-Polarizer Transmission — I = I₀·cos²θ
Passing through three polarizers in sequence multiplies three factors: the first polarizer's 50% pass-through of unpolarized light, then Malus's law cos² at each subsequent angle step. A middle polarizer at an intermediate angle can transmit far more than two crossed polarizers alone.
What the printable worksheet asks students to work out
- For one trial, compute the transmitted fraction as 0.5·cos²(θ_mid)·cos²(90° − θ_mid), where the initial 0.5 factor accounts for unpolarized light passing the first polarizer, then two Malus's-law steps (0° to θ_mid, then θ_mid to 90°). Compare to the table. Confirm θ_mid = 45° gives exactly 1/8.
- Explain why a middle polarizer at 45° — exactly halfway between the crossed outer ones — maximizes the surprising 'let light back through' effect: each 45° step alone transmits cos²(45°) = 50%, much better than jumping straight from 0° to 90° (which transmits 0%).
Where this shows up beyond the lab
- This 'more filters, more transmission' effect has a famous quantum-mechanics analog (the three-polarizer experiment demonstrating that measurement changes a quantum state). Explain, in classical wave terms, why each polarizer doesn't just 'select' pre-existing light of its own orientation, but actively projects the light onto its own axis, changing its state.
- What happens to the transmitted fraction as the middle polarizer's angle approaches 0° (aligned with the first) or 90° (aligned with the last)? Use your formula to check both limits and explain physically why they make sense.
- AP Physics 2 — Unit 14: Waves, Sound, and Physical Optics
- IB Physics — C.3 Wave phenomena
- General High School Physics — Light & optics
- NGSS High School Physics — Wave properties
- Welcome to Three Polarizers
- Select the middle polarizer
- Press Play
- Middle polarizer lets light through
- Open the Properties panel
- You did it!
Open the interactive simulation to build the scene, press Play, and explore with live measurements and a guided tutorial.