A Special Case of the Superposition of Crystal Plates between Crossed Polars and its Bearing on the Microscopy of Cellulosic Fibres

Abstract

WHEN monochromatic light is passed at perpendicular incidence through two superposed and differently orientated transparent crystal plates between crossed polars, the intensity of the light transmitted by the analyser relative to that of the light entering the plates is given by the following expression, as was shown by Fresnel: where I is the relative intensity as just defined, ψ₁, ψ₂ are, respectively, the angles made by the corresponding vibration directions of the plates (that is, either the ‘slow’ or the ‘fast’ directions) with the vibration direction of the polarizer, and δ₁, δ₂ are the phase differences produced by the plates. (The expression ignores any reduction in the intensity due to absorption by the analyser, such as occurs in polaroid, but for a given analyser this reduction is by a constant factor, and does not affect the arguments which follow.) If the plates are of equal thickness and birefringence, δ₁ = δ₂ = δ, and the fourth term in the expression vanishes. If also we put 2θ for the angle between the corresponding vibration directions, say the ‘slow’ directions, of the plates, and α for the angle between the bisectrix of 2θ and the vibration direction of the polariser (Fig. 1), the expression can be reduced to the form: (This involves making use of the following equalities: