 Solution

A simple polariscope contains a fixed polarizer and a rotatable optical analyzer, with a space for a specimen between them. A historically suitable implementation uses two Nicol prisms. Each selects one linearly polarized component from the incident light through the crystal's optical action. A lamp and a collimating arrangement illuminate the first prism, and the observer examines the transmitted field through the second.

With no specimen and ideal elements, Malus's law predictswhere $I_0$ is the intensity after the polarizer and $\theta$ is the relative transmission-axis angle. Parallel settings are brightest; crossed settings give extinction. Rotating the analyser supplies an immediate check on the first element's polarization.

Insert a transparent birefringent plate between crossed elements with its principal directions oblique to their axes. The two components can acquire different phases, so they need not recombine into the original linear polarization; light can then pass the crossed analyser. With white light, wavelength-dependent retardation gives colours. Rotation of the plate, and a check without it, distinguish this response from simple sample opacity or a poor extinction adjustment. An optically active sugar solution can instead rotate the polarization plane, as in Biot's saccharimeter. The instrument therefore reveals changes of polarization through an analysed intensity, rather than making polarization directly visible to the eye.