Fundamentals of Modern Optics
series 12
20.01.2017
to be returned: 27.01.2017, at the beginning of the lecture
Problem 1 - Optical layer (1+3+3+2 points)
Let us consider a single optical layer as shown in the gure a. For simplicity, we consider light in TE
polarization only.
a) What is the angle φt of the transmitted beam √
as a function of the angle of the incident
beam φi ?
√
√
Assume the refractive indices of substrate ns = s , lm nf = f , and cladding nc = c , and the
thickness of the layer d.
b) Compute the coecients for reection and transmission as functions of the angle of incidence φi .
c) Compute the reectivity and transmissivity of the single layer, and show that they add up to 1. For
simplicity of calculation, assume that f > s sin2 (φi ) and c > s sin2 (φi ) are true.
q
d) Consider the special case of kf x d = d kf2 − kz2 = π/2 (λ/4-layer) and compute the reectivity. Now
assume the incident light is perpendicular to the layer (φi = 0). Find a condition between the refractive
indices to obtain minimum reectivity.
Problem 2 - Stratied Media (4+2+2 points)
A beam of light (with free-space wave-vector of k0 = ω/c) is passing through a periodic stack of alternating
layers, odd layers with refractive index na and thickness da , even layers with refractive index nb with
thickness db (gure b). For the case of the innitely long periodic stack, the eld in the structure follows
the Bloch theorem, which states that this structure supports Bloch modes of wave-vector K . For such a
mode the eld at the beginning of layer N is connected to the eld at the beginning of layer N + 1 through
EN +1 = exp(iKΛ)EN with Λ = da + db . For simplicity consider the case of Normal incidence φi = 0.
a) Derive the dispersion relation for K as a function of na , nb , da , db , and k0 .
b) A purely real K indicates a propagating mode and a complex-valued one indicates a decaying mode.
Determine under what conditions we get a propagating or a decaying Bloch mode.
c) Determine the frequencies at which the decay is the strongest (in other words when the imaginary part
of K is the largest). For simplicity assume na da = nb db .