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Journal of Experimental Biology 48,227-245 (1968)
Published by Company of Biologists 1968

A Theoretical Treatment of the Reflexion of Light by Multilayer Structures

A. F. HUXLEY 1

1 Department of Physiology, University College London, Gower Street, London, W.C.1

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 {lambda}ø/2{pi}, 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 cos2{varphi}<r2,

reflectance = 1/1+4m2(r2-cos2{varphi})/r2(1-m2)2,

where m=(1-{surd}r2-cos2{varphi}/sin2{varphi}/1+{surd}r2-cos2{varphi}/sin2{varphi})p;

(b) when cos2{varphi} > r2

reflectance = 1/1+cos2{varphi}-r2/r2sin2p{theta},

Where cos{theta} = cos2{varphi}-r2/1-r2;

(c)when the number of repeats in the stack is large (p->{infty}), reflexion is complete so long as cos2{varphi}<r2. Outside this range the reflections is 1-{surd}(1-r2cos2{varphi}).

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.

Submitted on September 30, 1966




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© The Company of Biologists Ltd 1968