Warning: this question may come across as too specific. Maybe it would help if I set values for the probability parameters below - they'd be based on the Fresnel equations. But anyway, I think this is an interesting problem - or at least, I'd like to know how to find the answer.
The Fibonacci word is formed by concatenation in the same manner that the Fibonacci sequence is formed by addition, thus:
W1 = 0, W2 = 01, W3 = W2W1 = 010, Wn = Wn-1Wn-2
W∞ = 0100101001001010010100100101001001...
A quasicrystal is a quasiperiodic structure. One of the simplest is the Fibonacci quasicrystal, a one-dimensional structure which can be defined by replacing each 0 in W∞ with one type of optically transparent layer and each 1 with another type. Partial reflection occurs at the interface between each pair of adjacent layers, with a probability dependent on the types of the layers. Light which is not reflected passes instead into the next layer.
So we have five parameters determining the probabilities of reflection when passing from one medium into another: p (air into 0), q (0 into 1), r (1 into 0), s (0 into 0), t (0 into air).
Light of wavelength λ undergoes a phase change of Λ/λ after passing through a 1 layer, and φΛ/λ after passing through a 0 layer, where φ is the golden ratio (1+√5)/2.
Finally, light reflected from the quasicrystal interferes with itself so that the resulting waveform for wavelength λ is
f(τ) = p cos(2πcτ/λ) + (1-p)q(1-t) cos(2(πcτ+φΛ)/λ) + (1-p)(1-q)r(1-r)(1-t) cos(2(πcτ+(φ+1)Λ)/λ) + ...
and the associated intensity is the root mean square of f over the entire time domain.
Pretty horrific, I know. But (in theory at least) it can all be written in terms of trig functions of πcτ, φΛ/λ and Λ/λ along with sums of products of the probabilities. The question is, how can I actually evaluate the intensity associated to λ? Is it even vaguely possible without lots of numerical computation?