Use of Chebychev polynomials in thin film computations

Abstract

Fron. H erpin 's ex press ion for t he m th power of a mul til ayer matrix, very s impl e closed formu las a re derived for t he matrices and optical constants of any mul t ilayer with a p eriodi c s t ru cture.Acco rdin g to Epstein 's t heore m, a ny sy mm etri cal mu ltilayer is eq uivalen t to a fictitious monolayer. A simple ex press ion for th e equiva len t index and t hi ckn ess of this monolayer is d edu ced for t he case of a p eriodic and sy mmetrical sequen ce of equally t hick film s.As compared to a ny oth er method of numerical computation , the s ugges ted formul ation provides a cons iderable sa vin g of time a nd work . In a num eric al example, th is saving a mounts to a bou t 80 percen t.

Electromagnetic Waves in Stratified Media

“…The second advantage is that the matrices M have unity determinant. Hot only does this provide a useful numerical check at any stage hut it also enables simple closed for mulae for 'periodic' multilayers to be obtained (Mielenz, 1959).…”

Tolerances for layer thicknesses in dielectric multilayer coatings and interference filters

“…Consider an odd number, N =2 m +1, of layers of nominally equal optical thickness, alternately of high index n H and low index n L , with a high index on the outside. If all layers are a quarter wave thick at a wavelength λ 0 , eq (2) may be written as with S m and S m −1 are Chebychev polynomials of the argument defined by etc., see [ 4 ]. From eqs (12) to ( 14 ), and ( 7 ), with …”

“…Consider an error in thickness in one of the high-index layers, v = 2k +1. Then, eq (28) yields According to [ 4 ], and therefore, …”

Comment on Multilayer Dielectric Filters

“…It has already been shown (Armstrong 1956a(Armstrong , 1956b how such a representation can be applied to a great variety of problems. In case the layers happen to be of two kinds placed alternately (and the Kronig-Penny model could be considered as such a situation) one will have results involving powers of matrices, and there are some special relations available to be applied to these (Pease 1952 ;Armstrong 1953Armstrong , 1956bMielenz 1959).…”

“…It is interesting that, about the same time that Gascoigne's article appeared, there was another, about half-way around the world (Mielenz 1959) dealing with much the same problem. In this latter article matrix analysis is used, as well as some special ways of dealing with powers of matrices.…”