Malus’s law

Ideal linear polarizers. P1 is at 0°. The source is either already aligned with P1 (no first-filter loss) or unpolarized (P1 transmits one half). Two filters give I=I₁cos²θ. Inserting P2 between P1 and the final P3 gives I=I₁cos²φcos²(θ−φ). Crossed outer filters with a 45° middle filter transmit 25% of aligned incident light, or 12.5% of unpolarized light.

The disks show each filter axis separately, in optical order. The graph is final-filter angle versus intensity; values share arbitrary linear intensity units. Zero incident intensity is allowed, but its transmission ratio is undefined. Sweep runs at 55° per second, independent of display frame rate; Stop and hiding the tool stop it.

Sample curve records a simulated point, not a real measurement. You can instead paste real rows as angle; intensity (semicolon permits decimal commas), or comma-separated decimal-point values. Fit uses I=A·factor(θ)+B with a background offset and unweighted least squares. Points at extinction are included. RMSE is the residual root mean square, not an uncertainty estimate. A negative amplitude or insufficient factor variation is rejected.

Changing intensity, source or middle-filter geometry clears active points, leaving pasted text available to refit. Angle changes keep points. Typed inputs and pasted data text stay in this browser, including unfinished entries. Sampled points, fits and sweep angles are temporary and are not restored. Reset restores the optical defaults and clears points, keeping pasted text. Ideal filters: no imperfect extinction, wavelength dependence, detector saturation or noise simulation. Simulated samples are exact evaluations, not measured evidence.