Three polarizers are stacked: the first produces linearly polarized light, the second is at 45° to the first, and the third is at 90° to the first (crossed). Without the middle polarizer, no light passes. With it, light passes through. Why?
AThe middle polarizer amplifies the light intensity, allowing enough to overcome the crossed configuration
BThe middle polarizer creates a new 45° polarization axis; by Malus' law each stage passes cos²(45°) = 50% of the light it receives, so light reaches the third polarizer with non-zero intensity at an angle that is no longer 90°
CThe middle polarizer rotates the polarization direction of individual photons, gradually steering them parallel to the third polarizer's axis
DThe middle polarizer scatters some light into the transmission direction of the third polarizer
This is the classic three-polarizer paradox. Without the middle polarizer, the first and third are crossed at 90° and block all light. With the middle polarizer at 45°: stage 1→2 passes I₀ cos²(45°) = I₀/2; stage 2→3 passes (I₀/2) cos²(45°) = I₀/4. The key is that after the second polarizer, the light is polarized at 45° — which is only 45° from the third polarizer's axis, not 90°. Each polarizer resets the polarization direction, and the 90° relationship between first and third no longer matters. The middle polarizer does not amplify; it reorients.
Question 2 Multiple Choice
Unpolarized light of intensity I₀ passes through an ideal linear polarizer. What intensity emerges, and does the answer depend on the polarizer's orientation?
AI₀ — an ideal polarizer transmits all light without absorption
BI₀/2, regardless of the polarizer's orientation, because unpolarized light carries equal energy in all oscillation planes
CI₀ cos²θ, where θ is the angle between the polarizer axis and the dominant oscillation direction of the incident light
DIt depends on the polarizer's orientation — rotating it changes how much unpolarized light is absorbed
Unpolarized light is, by definition, light with equal intensity distributed uniformly across all oscillation planes. No matter which direction you orient the polarizer's transmission axis, it passes the component of electric field along that axis and blocks the rest. For a uniform distribution, exactly half the total intensity is carried by oscillations in any one direction. So an ideal polarizer always transmits I₀/2 from unpolarized light, regardless of orientation. This is why Malus' law I = I₀ cos²θ only applies when the incident light is already linearly polarized — for unpolarized light, the input is always I₀/2.
Question 3 True / False
Two ideal polarizers with their transmission axes crossed at exactly 90° will transmit essentially no light, because the polarization direction of the light from the first polarizer is perpendicular to the transmission axis of the second.
TTrue
FFalse
Answer: True
Malus' law gives I = I₀ cos²(90°) = I₀ · 0 = 0. The electric field component along the second polarizer's axis is exactly zero: the first polarizer produces light oscillating in one plane, and the second polarizer's transmission axis is perpendicular to that plane. There is no component to transmit. In practice, with real polarizers, a tiny amount of light leaks through due to imperfections in alignment and absorption, but an ideal crossed-polarizer configuration produces zero transmission.
Question 4 True / False
When linearly polarized light passes through an analyzer at angle θ, the transmitted electric field amplitude is reduced by the factor cos²θ.
TTrue
FFalse
Answer: False
The electric field amplitude is reduced by cosθ, not cos²θ. Malus' law states that the transmitted intensity I = I₀ cos²θ. Since intensity is proportional to amplitude squared (I ∝ E²), we have E_transmitted = E₀ cosθ. The cos²θ factor applies to intensity, not amplitude. This distinction matters for understanding interference phenomena, where electric field amplitudes add (not intensities), and for calculating what happens when polarized light passes through multiple optical elements.
Question 5 Short Answer
Why does inserting a polarizer at 45° between two crossed polarizers allow light to pass through the system, when the two crossed polarizers alone block all light?
Think about your answer, then reveal below.
Model answer: The middle polarizer resets the polarization direction to 45°. After the first polarizer, light is polarized in one direction (call it 0°). The middle polarizer at 45° transmits the component of this light along its 45° axis — I₀/2 passes, now polarized at 45°. The third polarizer at 90° then receives light polarized at 45°, which is only 45° away from its transmission axis (not 90°). It transmits I₀/2 · cos²(45°) = I₀/4. Without the middle polarizer, the light from the first polarizer (0°) hits the third polarizer (90°) at 90° — zero transmission. The middle polarizer breaks the perpendicularity relationship by introducing an intermediate polarization state.
This result is counterintuitive because adding an element that blocks half the light at each stage somehow produces more light out the other end than the system without it. The resolution is that it is not the light that passes through the first two polarizers that matters — it is the polarization direction of the light arriving at the third polarizer. The middle polarizer changes that direction from 0° to 45°, and 45° has a non-zero projection onto the 90° axis. More generally, any polarizer inserted between crossed polarizers at an angle other than 0° or 90° will transmit some light.