This 2D companion does not animate a static rotated arrow like a 3D scene would — it actually decomposes the plane-polarized beam into its two physical circular components and sums them back up every frame. A linearly polarized wave is exactly the sum of a right-circular and a left-circular component of equal amplitude:
E_R(z,t) = ( cos φ_R, sin φ_R), φ_R = ωt − k_R z
E_L(z,t) = ( cos φ_L, −sin φ_L), φ_L = ωt − k_L z
E_R + E_L = 2cos(φ̄) · ( cos θ(z), sin θ(z) )
where φ̄ = (φ_R+φ_L)/2, θ(z) = (φ_R−φ_L)/2 = Δk·z / 2
Because a chiral solution has a different refractive index for left- and right-circular light (circular birefringence, Δn = nL − nR), the two components pick up different phase per metre travelled, Δk = 2πΔn/λ, so their sum's direction θ(z) rotates steadily with depth — the resultant's magnitude oscillates in time (the real field flips sign like any wave) but its axis stays fixed at θ(z), which is exactly the polarization plane a polarimeter measures. At the far end (z = l) this reduces to Biot's law:
α = θ(l) = Δk·l/2 = π·Δn·l/λ ⇔ α = [α]λ·l·c (Biot's law, same result)
Δn itself is solved for from the compound's tabulated specific rotation and the Drude single-term dispersion relation [α](λ) = A/(λ²−λc²) — the phasor panel (top-right of the beam) then draws the actual ER and EL vectors spinning at the exit face so you can see the mechanism, not just the outcome. A racemic 50/50 mixture has Δn = 0: the two circular components travel at identical speed and the plane never rotates, even though every individual molecule is still chiral.
- Compound — sets the sign and size of Δn (dextro-/levorotatory/racemic).
- Concentration & tube length — both scale Δk·l linearly, per Biot's law.
- Wavelength buttons — recompute [α] (and hence Δn) from Drude dispersion, and recolor the beam.
- Analyzer angle — a second polarizer; transmitted intensity follows Malus's law, I = I₀cos²(θa − α), exactly how a real saccharimeter finds α by hunting for the extinction angle.
Real-world relevance: this is the working principle behind saccharimeters used in the sugar industry, and behind pharmaceutical QC checks that confirm a drug is the correct enantiomer.