Achromatic Combinations of Half-Wave Plates

The plane of polarization of two or more wavelengths can be rotated by a predetermined amount through the use of two or more identical half-wave plates in series, the axes of which are oriented at predetermined angles with respect to the incident plane of polarization. For two wavelengths a rotation of 90° may be accomplished by the use of two plates with their slow axes at angles of 22.5°+δ and 67.5°−δ, respectively. For three wavelengths a 90° rotation is obtained by using three plates at angles of 11.25°+δ, 45°, and 78.75°−δ respectively. The quantity δ is a small angle usually less than 1° which determines the spectral range of achromatization. Identical half-wave plates are easily obtained by cutting a single splitting of mica or a plane parallel sheet of other birefringent material. The wavelength for which the plates have half-wave retardation is not critical. Rather than having a smaller angular aperture than the single half-wave plate, the three-element achromatic rotator has a larger angular aperture. The use of the stereographic projection of the Poincaré sphere for graphical solution of polarized light problems is discussed briefly.