A Theoretical Treatment of the Reflexion of Light by Multilayer Structures

1. A convenient method is presented for calculating the reflectance of a stack of dielectric layers consisting of a series of identical repeats of any particular sequence of layers. The method is closely related to that published by Lord Rayleigh in 1917. 2. In this method, two quadratic equations are formed from the thicknesses and refractive indices of the layers composing a single repeat unit. The reflectance is obtained by substituting the solutions of these equations into an explicit formula. 3. Particularly simple formulae result for the case of a stack of p plates, optical thickness λo/2π, uniformly spaced in an infinite medium with spaces of the same optical thickness. If r is the amplitude reflexion coefficient at a single interface, the reflectance of the whole stack is as follows: ( a ) when cos 2 φ 2 , reflectance = 1/1+4m 2 (r 2 -cos 2 φ)/r 2 (1-m 2 ) 2 , where m=(1-√r 2 -cos 2 φ/sin 2 φ/1+√r 2 -cos 2 φ/sin 2 φ) p ; ( b ) when cos 2 φ > r 2 reflectance = 1/1+cos 2 φ-r 2 /r 2 sin 2 pθ, Where cosθ = cos2φ-r 2 /1-r 2 ; ( c )when the number of repeats in the stack is large ( p →∞), reflexion is complete so long as cos 2 φ 2 . Outside this range the reflections is 1-√(1-r 2 cos 2 φ). 4. These results are extended to cover: ( a ) unequal optical thicknesses in plates and spaces; ( b ) oblique incidence; ( c ) layers of materials of other refractive indices above and below the stack itself; and ( d ) stacks consisting of repeats of more complex units.