Quantum Experiments · I · Observations and models · Chapter 05
Why can adding a polarizer increase transmission through crossed filters?
Pass a beam through a horizontal linear polarizer (\(\theta_1=0^\circ\), preparing state \(\lvert H\rangle\)) followed by a crossed vertical linear polarizer (\(\theta_3=90^\circ\), projecting onto \(\lvert V\rangle\)). For ideal polarizers, the orthogonal overlap \(\langle V\vert H\rangle=0\) blocks the beam completely. Now insert a third linear polarizer between them with its transmission axis tilted at angle \(\theta\) from horizontal: up to \(1/4\) of the horizontally polarized flux now emerges through the final vertical polarizer. Rather than merely culling a subset of pre-existing vertically polarized particles, the middle polarizer projects survivors onto the new polarization state \(\lvert\theta\rangle\), which has a nonzero component along \(\lvert V\rangle\).
Linear optical polarization provides a direct mathematical bridge between classical vector wave optics and two-level quantum mechanics. A classical monochromatic electric field \(\mathbf{E}=E_0\hat{\mathbf{x}}\) projected successively onto unit vectors \(\hat{\mathbf{e}}_\theta\) and \(\hat{\mathbf{y}}\) obeys the exact same inner-product projection rules as a single photon prepared in \(\lvert H\rangle\). Consequently, observing light restoration through three polarizers with a macroscopic laser pointer demonstrates noncommuting vector projections, while interpreting the transmitted fraction as a single-photon survival probability requires a verified single-photon source.
Our analytical model assumes lossless linear polarizers with infinite extinction ratio across a collimated beam. Real sheet and crystal polarizers exhibit finite absorption along the transmission axis (\(10\%\) to \(30\%\) insertion loss in inexpensive film) and a nonzero crossed leakage ratio (\(10^{-3}\) to \(10^{-6}\)). When performing tabletop optics demonstrations, never look directly into a laser beam or specular reflection; the interactive simulation below requires no physical laser hardware.
Visual animation · Section 5.1
Crossed polarizers and intermediate state projection
Represent horizontal and vertical linear polarization by the orthonormal basis vectors \(\lvert H\rangle=(1,0)^T\) and \(\lvert V\rangle=(0,1)^T\). A linear polarizer whose transmission axis makes physical angle \(\theta\) with the horizontal projects onto the state
\[\lvert\theta\rangle = \cos\theta\,\lvert H\rangle + \sin\theta\,\lvert V\rangle.\]
Unlike spin-\(1/2\) where orthogonal states lie \(180^\circ\) apart in physical space, orthogonal linear polarizations \(\lvert H\rangle\) and \(\lvert V\rangle\) lie \(90^\circ\) apart in space, so the full angle \(\theta\) appears directly: the transmission probability from \(\lvert H\rangle\) through the middle filter is Malus’s law, \(P(H\to\theta)=|\langle\theta\vert H\rangle|^2=\cos^2\theta\).
Every photon transmitted by the middle filter emerges in the state \(\lvert\theta\rangle\). Its conditional probability of passing the final vertical polarizer \(\lvert V\rangle\) is \(P(\theta\to V)=|\langle V\vert\theta\rangle|^2=\sin^2\theta\). Multiplying the two sequential stage probabilities gives the net transmission relative to the \(\lvert H\rangle\)-prepared beam:
\[P_{H\to\theta\to V} = \cos^2\theta\,\sin^2\theta = \frac{1}{4}\sin^2(2\theta),\]
which vanishes at \(\theta=0^\circ\) and \(\theta=90^\circ\) and reaches its maximum value of \(1/4\) at \(\theta=45^\circ\).
The physical order of the filters is decisive because orthogonal projections do not commute. Writing \(P_\theta=\lvert\theta\rangle\langle\theta\rvert\) and \(P_V=\lvert V\rangle\langle V\rvert\), the middle-filter configuration applies \(P_V P_\theta\lvert H\rangle=\sin\theta\cos\theta\lvert V\rangle\neq 0\) for \(\theta\in(0^\circ,90^\circ)\), whereas placing the \(\theta\) filter after the crossed \(H\) and \(V\) pair applies \(P_\theta P_V\lvert H\rangle = 0\): once the crossed pair extinguishes the amplitude, no downstream passive filter can restore it.
Visual animation · Section 5.2
Malus’s law and noncommuting projections
The three-polarizer experiment cleanly illustrates projective state update \(P_\theta\lvert H\rangle=\cos\theta\lvert\theta\rangle\) and noncommutativity \(P_V P_\theta\neq P_\theta P_V\). Because classical Maxwell fields obey the same projection algebra, however, this intensity ratio alone neither proves field quantization nor tests Bell nonlocality; distinguishing single photons from classical waves requires coincidence counting as in Chapter 6.
When comparing theoretical formulas with photodetector readings, always record whether the denominator is the flux immediately after the first horizontal polarizer \(\lvert H\rangle\) or the unpolarized lamp flux before it. Because an ideal first polarizer transmits \(1/2\) of an unpolarized beam, the three-filter peak at \(\theta=45^\circ\) equals \(1/4\) of the \(\lvert H\rangle\)-prepared flux and \((1/2)\times(1/4)=1/8\) of the initial unpolarized flux.
In the interactive apparatus below, rotating the middle filter axis \(\theta\) updates both the intermediate transmission \(\cos^2\theta\) and the final output \(\frac{1}{4}\sin^2(2\theta)\) across \(\theta\in[0^\circ,90^\circ]\).
Visual animation · Section 5.3
Separating wave projection from photon statistics
Sweep the middle polarizer angle \(\theta\) from \(0^\circ\) to \(90^\circ\) to locate the maximum transmission \(P_{H\to\theta\to V}=1/4\) at \(\theta=45^\circ\). Verify why the final output vanishes at both endpoints \(\theta=0^\circ\) and \(\theta=90^\circ\) for different stage-by-stage reasons.
Interactive model · Idealized predictions
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Suppose \(N=1{,}000\) single photons prepared in \(\lvert H\rangle\) enter an ideal three-polarizer train whose middle filter is set to \(\theta=30^\circ\). What are the expected photon counts after the middle filter and after the final vertical filter?
At \(\theta=30^\circ\), the middle filter transmits with probability \(\cos^2(30^\circ)=(\sqrt{3}/2)^2=3/4=0.75\), so an average of \(1{,}000\times(3/4)=750\) photons pass the middle stage in state \(\lvert 30^\circ\rangle\). The final vertical polarizer transmits a conditional fraction \(\sin^2(30^\circ)=(1/2)^2=1/4\) of those survivors, yielding a mean final count of \(1{,}000\times(3/16)=187.5\) (with binomial standard deviation \(\sqrt{1{,}000(3/16)(13/16)}\approx 12.3\)).
What is the maximum possible transmission probability \(P_{H\to\theta\to V}\) relative to an \(\lvert H\rangle\)-prepared input using a single intermediate linear polarizer, and at what angle \(\theta\in[0^\circ,90^\circ]\) does it occur?
Maximize \(\frac{1}{4}\sin^2(2\theta)\).
Because \(\sin^2(2\theta)\le 1\) with equality when \(2\theta=90^\circ\), the maximum transmission is \(1/4=0.25\) at \(\theta=45^\circ\).
Does observing a \(1/4\) intensity restoration with a classroom laser pointer and three polarizing films prove that the light beam consists of indivisible photons?
Compare the classical electric-field projection \((\hat{\mathbf{y}}\cdot\hat{\mathbf{e}}_\theta)(\hat{\mathbf{e}}_\theta\cdot\hat{\mathbf{x}})E_0\) with the quantum amplitude.
No. Classical transverse electromagnetic waves obey the identical Malus projection law \(I_{\mathrm{out}}/I_H=\cos^2\theta\sin^2\theta\). Demonstrating photon indivisibility requires correlation measurements such as \(g^{(2)}(0)<1\).
Why does placing the \(45^\circ\) polarizer after the crossed \(H\) and \(V\) polarizers yield zero transmission instead of \(1/4\)?
Evaluate \(P_{45^\circ}P_V\lvert H\rangle\) in temporal order from right to left.
When the \(\lvert H\rangle\) state hits the vertical polarizer first, \(P_V\lvert H\rangle=\lvert V\rangle\langle V\vert H\rangle=0\). With zero transmitted amplitude leaving the \(V\) polarizer, the downstream \(45^\circ\) polarizer receives nothing to project.
Projective filters both select outcomes and rotate the surviving state vector, so operator order matters (\(P_V P_\theta\lvert H\rangle\neq P_\theta P_V\lvert H\rangle\)). Always distinguish two-component wave projection from genuinely nonclassical photon-counting statistics.
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.