On the reflection of light from a regularly stratified medium

The remarkable coloured reflection from certain crystals of chlorate of potash described by Stokes, the colours of old decomposed glass, and probably those of some beetles and butterflies, lend interest to the calculation of reflection from a regular stratification, in which the alternate strata, each uniform and of constant thickness, differ in refractivity. The higher the number of strata, supposed perfectly regular, the nearer is the approach to homogeneity in the light of the favoured wave-lengths. In a crystal of chlorate described by R. W. Wood, the purity observed would require some 700 alternations combined with a very high degree of regularity. A general idea of what is to be expected may be arrived at by considering the case where a single reflection is very feeble, but when the component reflections are more vigorous, or when the number of alternations is very great, a more detailed examination is required. Such is the aim of the present communication. The calculation of the aggregate reflection and transmission by a single parallel plate of transparent material has long been known, but it may be convenient to recapitulate it. At each reflection or refraction the amplitude of the incident wave is supposed to be altered by a certain factor. When the light proceeds at A from the surrounding medium to the plate, the factor for reflection will be supposed to beb' , and for refractionc' ; the corresponding quantities when the progress at B is from the plate to the surrounding medium may be denoted bye' ,f' . Denoting the incident vibration by unity, we have then for the first component of the reflected waveb' for the secondc' e' f' e -ikδ , for the thirdc' e' 3 f' e -2ikδ , and so on, all reckoned as at the first surface A. Hereδ denotes the linear retardation of the second reflection as compared with the first, due to the thickness of the plate, and it is given byδ = 2μ T cosα' , (1) whereμ , is the refractive index, T the thickness, andα' the angle of refraction within the plate. Alsok = 2π /λ ,λ being the wave-length. Adding together the various reflections and summing the infinite geometric series, we findb' +c' e' f' e -ikδ /1 -e' 2 e -ikδ , the incident and transmitted waves being reckoned as at A.