This simulation models how light interacts with polarizing filters, dielectric interfaces and birefringent crystals. In Malus's Law mode, unpolarized light is halved in intensity by the first polarizer, and each subsequent polarizer attenuates the beam according to I = I₀·cos²θ, where θ is the angle between transmission axes. The Brewster mode solves the full Fresnel equations for s- and p-polarized reflectance at a glass interface, locating the angle at which reflected light is purely s-polarized. Wave View animates the transverse electric field to distinguish linear, circular and elliptical states, while Birefringence mode decomposes an incident wave into fast- and slow-axis components and recombines them after a wave-plate retardation Γ.
Four linked optics phenomena: Malus's Law transmission through up to three rotating polarizers, Brewster's-angle reflection/transmission computed from the Fresnel equations, the oscillating transverse E-field of linear/circular/elliptical/unpolarized light, and how a birefringent wave plate changes polarization state by retarding one axis relative to the other.
Switch between the Malus's Law, Brewster, Wave View and Birefringence tabs. In Malus's Law drag the Polarizer 1/2/3 angle sliders (0–180°) or enable the third polarizer and try the Parallel, Crossed, 45° and Magic Middle presets. In Brewster mode drag the Incident angle slider (0–89°) and pick a medium (glass, water, diamond) to find θ_B = arctan(n₂/n₁). In Wave View choose a polarization type and adjust amplitude, phase offset δ and polarization angle. In Birefringence pick a plate type (λ/2, λ/4, λ, custom), then adjust the Retardation Γ and Fast axis angle sliders.
The "Magic Middle" preset places a third polarizer between two crossed ones — instead of blocking all light (as two crossed polarizers alone would), the middle filter rotates the polarization axis in two smaller cos²θ steps, letting light through. This counter-intuitive result is a favourite demonstration in undergraduate optics courses.
Malus's Law states that when polarized light of intensity I₀ passes through an ideal polarizer whose transmission axis is at angle θ to the light's polarization direction, the transmitted intensity is I = I₀·cos²θ. In the simulation, unpolarized light first loses half its intensity passing through Polarizer 1 (since an unpolarized beam has no preferred axis), and each subsequent polarizer then follows cos²θ based on the angle difference from the previous one. Stacking three polarizers lets you see this multiplicative attenuation happen twice in sequence.
Two polarizers at 90° to each other (crossed) transmit zero light because cos²(90°) = 0. But if you insert a third polarizer at, say, 45° between them, the light first passes through a 45° rotation (cos²45° = 0.5 transmission) and then another 45° rotation to reach the final 90° axis (another cos²45° = 0.5), giving 25% transmission overall instead of 0%. This is the "Magic Middle" preset in the simulation — each individual step only rotates the polarization axis a little, so no single stage fully blocks the beam.
Brewster's angle θ_B = arctan(n₂/n₁) is the angle of incidence at which the Fresnel reflection coefficient for p-polarized light (electric field in the plane of incidence) becomes exactly zero. At this angle, only s-polarized light (electric field perpendicular to the plane of incidence) is reflected, so the reflected beam is purely s-polarized even though the incident light was unpolarized. The simulation computes both Rs and Rp from the exact Fresnel equations and highlights when the incident angle matches θ_B for the chosen medium.
These describe the path traced by the tip of the electric field vector as the wave propagates. Linear polarization oscillates along a single fixed axis. Circular polarization has two orthogonal components of equal amplitude with a 90° phase difference, so the field vector traces a circle (right- or left-handed depending on the sign of that phase). Elliptical polarization is the general case, where the components have unequal amplitudes or a phase difference other than 90°, tracing an ellipse. Unpolarized light has no fixed relationship between components — the phase and relative amplitude fluctuate randomly.
A birefringent crystal has different refractive indices along its "fast" and "slow" optical axes, so light polarized along one axis travels at a different speed than light polarized along the other. This introduces a phase retardation Γ between the two components as the light exits the crystal. A quarter-wave plate (Γ = 90°) converts linear polarization into circular (or elliptical) polarization; a half-wave plate (Γ = 180°) flips the polarization axis, mirroring it about the fast axis. The simulation lets you set the retardation directly or choose a preset plate type and rotate the fast-axis angle to see the resulting output state.