DBS C
9.3 Periodic media with discrete and continuous variation of refractive index
Periodic structures occupy a very important place in optics because of their various possible applications. For example, interference coatings (anti- reflective) are used in antiglare screens and in optical instrumentation to in- crease the light throughput. We may distinguish two broad classes of systems, one with discrete variation of the refractive index, like in the layered me- dia discussed above. The other refers to a continuous variation of the refrac- tive index—for example, a sinusoidal variation along the axis of the optical
176 Wave Optics: Basic Concepts and Contemporary Trends
system. In the first case, in each layer we have forward and backward propagat- ing waves resulting from multiple reflections from each interface, while in the medium with harmonic variation, local inhomogeneity is responsible for the generation of the backward waves. Thus such systems are often referred to as having a distributed feedback. Note that in an infinite homogeneous medium, the forward and backward waves are completely decoupled. The inhomogene- ity leads to the coupling in both the discrete and the continuous cases. In the case of weak coupling, we can develop a perturbative approach, which is known in the literature as thecoupled mode theory, where we retain only the lowest order of scattered waves. More about such systems will be discussed in dealing with the distributed feedback (DFB) systems. However, the case of the discrete variation (layered medium) can be dealt with exactly. In both cases we show the emergence of the band gaps. We calculate the reflection and transmission coefficients for a finite-length periodic medium assuming all the materials to be non-magnetic and non-lossy. The changes for magnetic materials or finite losses can be implemented in a straightforward way.
9.3.1 Discrete variation of refractive index
Consider a periodic layered system like inFig. 9.1. Let each period consist of two layers with refractive indicesna,nband widthsdaanddb, respectively.
For simplicity we restrict ourselves to the case of wave propagation along thez-axis of the periodic system. Denoting the period of the structure by Λ (Λ =da+db) and imposing the periodic boundary conditions, we can relate the output after one period in terms of the input:
(eiµΛI) Hy
Ex
z=0
=Mab
Hy
Ex
z=Λ
, (9.17)
with
Mab=MaMb. (9.18)
The matrix Ma (Mb) denotes the characteristic matrix for an ‘a’ (‘b’) type layer. In writing Eq. (9.17) we made use of the Floquet-Bloch theorem. The fields at the output and input faces are the same except for an overall phase accumulation given by the Bloch wave vector µ (see Ref. [61]). Eq. (9.17) represents a homogeneous system and allows for nontrivial solutions if and only if
A−eiµΛ B
C D−eiµΛ
= 0, (9.19)
where A, B, C and D are the elements of the characteristic matrix for one periodMab:
A= cosζacosζb− pb
pa
sinζasinζb, (9.20)
B=− i
pa
sinζbcosζa+ i pb
sinζacosζb
, (9.21)
C=−(ipasinζacosζb+ipbsinζbcosζa), (9.22) D= cosζacosζb−pa
pb
sinζasinζb, (9.23)
withζa/k0=nada(ζb/k0=kbdb) as the optical width andpa=na (pb=nb), respectively. The approach and notations we follow here are analogous to the ABCD matrix approach for a lens waveguide system [61]. Eq. (9.19) can be rewritten as
ei2µΛ−(A+D)eiµΛ+AD−BC= 0. (9.24) We assume all the layers to be lossless, implying that the characteristic matrix of each layer and also their product matrices are unimodular (i.e.,det(Mi) = 1, withi=a, b, ab). Based on this assumption, Eq. (9.24) can be reduced to
ei2µΛ−(A+D)eiµΛ+ 1 = 0, (9.25) where we have setAD−BC= 1. The roots of Eq. (9.25) are given by
e±iµΛ= A+D
2 ±
s
A+D 2
2
−1. (9.26)
The sum of these two roots gives us eiµΛ+e−iµΛ
2 = cosµΛ = A+D
2 . (9.27)
Making use of Eqs. (9.20), (9.23) and (9.27), we arrive at the dispersion rela- tion for the periodic structure as
cosµΛ = cosζacosζb−1 2
nb
na +na
nb
sinζasinζb. (9.28) We choose the optical pathlengths of the two layers to be same so thatζa = ζb =ζ and Eq. (9.28) simplifies to
cosµΛ = 1−(na+nb)2 2nanb
sin2ζ. (9.29)
The character of wave propagation in the periodic medium is governed by the character of the Bloch wave vectorµ. The inequality|cosµΛ| ≤1 corresponds to real µ, and we have a propagating solution in the structure. In contrast
178 Wave Optics: Basic Concepts and Contemporary Trends
0 0.5 1 1.5 2
−1.5
−1
−0.5 0 0.5 1 1.5
ζ/π
C = 1.2 C = 1.8
1−(C+1)2 2Csin2ζ
FIGURE 9.2: Right-hand side of Eq. (9.29) as a function ofζfor two different values ofC, namely,C = 1.2 (dashed) andC = 1.8 (solid). The dash-dotted lines represent the maximum and minimum of cosµΛ for real arguments.
|cosµΛ|>1 would mean a complexµand corresponding damped waves. We thus have propagating solutions if
−1≤1−(na+nb)2 2nanb
sin2ζ≤1, (9.30)
or if
0≤ (C+ 1)2
4C sin2ζ≤1, (9.31)
whereC=na/nbis the refractive index contrast between the two constituent media. We assume that na > nb so that C > 1. The band gap occurs in the region where the inequality Eq. (9.31) is violated. The stop gap for two values ofCis shown in Fig. 9.2, where we have plotted the right-hand side of Eq. (9.29) as a function ofζ. As can be seen from Fig. (9.2), the stop gap is centered atζ= (2m+1)π/2 and its width is proportional to the contrast. With increasing contrast the band gap broadens. This can be easily seen from the following estimate. For low contrast (C ≈1) the range of ζ (in the principal domain) corresponding to a stop gap is given by
π 2 −∆ζ
2 < ζ < π 2 +∆ζ
2 , (9.32)
where ∆ζ is the width of the band gap. It can be shown that ∆ζvaries as
∆ζ= δC
C , (9.33)
withδC=|C−1|.
9.3.2 Continuous variation of refractive index: DFB structures
Consider the continuous periodic variation of refractive index along the direction of propagationz as
n(z) =n0(1 +n1cosKz), with K= 2π/Λ, (9.34) wheren0 is the background refractive index andn1 is the modulation ampli- tude while Λ denotes the period. We consider the medium to be nonmagnetic and lossless. Further, the periodic variation is assumed to be a small pertur- bation to the background (i.e.,n1 ≪1). The dielectric function can then be expressed as
ǫ(z) =n20+ 2n0n1cosKz, (9.35)
where we have ignored the term containingn21. The solution to the Helmholtz equation can be written as a superposition of forward and backward propa- gating waves given by
E(z) =A+(z)eikz+A−(z)e−ikz, k= (ω/c)n0, (9.36) withA+(z) (A−(z)) as a forward-propagating (backward-propagating) slowly varying wave amplitude. Note that in absence of modulation both these am- plitudes are constants, and the inhomogeneity due to modulation scatters the forward waves into the backward ones. In principle all the scattering events will lead to the spatial harmonicsks:
ks=k+mK, m= 0,±1,±2· · ·, (9.37) with m being the order of the spatial harmonic. These waves exchange en- ergy among themselves as they propagate through the medium. However, for small modulation, significant contribution comes from only the lower-order harmonics. In the spirit of coupled mode theory, we retain only the lowest- order spatial harmonics. Substituting Eq. (9.36) in the Helmholtz equation, we obtain
d2A+
dz2 eikz+d2A− dz2 e−ikz
+n0
n1
hA−eikze−i(Kz+2kz)+A+e−ikzei(Kz+2kz)i +
2ik∂A+
∂z +n0
n1
k2A−ei(Kz−2kz)
eikz
−
2ik∂A−
∂z −n0
n1
k2A+e−i(Kz−2kz)
e−ikz = 0. (9.38) The slowly varying envelopes A± satisfy the conditions |d2dzA2±|≪k|dAdz±|≪
k2|A±|, and we can neglect the terms in the first square brackets in Eq. (9.38).
180 Wave Optics: Basic Concepts and Contemporary Trends
The scattering event that is of interest to us corresponds to 2k=K, and we can rule out the other higher-order events in the second square brackets. Col- lecting the coefficients ofeikz ande−ikzin Eq. (9.38), we arrive at the coupled mode equations
dA+
dz =iβA−e−iδz, (9.39)
dA−
dz =−iβA+eiδz, (9.40)
where
β= ωn1
2c , δ= 2ω
cn0−K. (9.41)
Under the transformation A+(z)
A−(z)
=V(z)
A¯+(z) A¯−(z)
, where V(z) =
e−iδz/2 0 0 eiδz/2
, (9.42) Eqs. (9.39) and (9.40) can be written in terms of transformed variables as
d dz
A¯+(z) A¯−(z)
=iµM
A¯+(z) A¯−(z)
, (9.43)
where
M= 1 µ
δ/2 β
−β −δ/2
, µ2=−β2+δ2/4. (9.44) The solution to Eq. (9.43) is given by
A¯+(z) A¯−(z)
= exp (iµMz)
A¯+(0) A¯−(0)
. (9.45)
It is worth noting that if−β2+δ2/4<0 in Eq. (9.44), thenµwill be purely imaginary (taking only the positive root in order to ensure causality); the waves inside the medium will be evanescent. This means that the forward propagating wave becomes evanescent by giving its energy to the backward propagating wave, resulting in reflection. The interval in which µ becomes imaginary corresponds to the the optical stopgap. For clarity we have plotted real and imaginary parts ofµin Fig. 9.3. It is clear that near the band edge there is considerable dispersion, while inside the gap the Bloch vector becomes purely imaginary. Note also that the gap width is proportional to the coupling strengthβ (∆δ= 4β). Noting thatM2=I(I= identity matrix), we have
exp(iµMz) =I
1−(µz)2
2! +(µz)4 4! − · · ·
+iM
µz−(µz)3
3! +(µz)5 5! − · · ·
,
=Icos(µz) +iMsin(µz). (9.46)
−1 −0.5 0 0.5 1
−1
−0.5 0 0.5 1
δ/2
Re (µ), Im(µ)
Im (µ) Re (µ)
FIGURE 9.3: Re(µ) and Im(µ) as a function ofδ/2 forβ= 0.25.
Rewriting the solution (Eq. (9.45)) in terms ofA±(z), we have A+(z)
A−(z)
=V(z) exp(iµMz)V−1(0)
A+(0) A−(0)
=U(z)
A+(0) A−(0)
. (9.47)
MatrixU(z) gives the spatial evolution of the amplitudes inside the periodic medium. The elements of matrixU(L) are given by
A=
cos(µL) + iδ
2µsin(µL)
e−iδL/2, (9.48)
B=iβ
µ sin(µL)e−iδL/2, (9.49)
C=−iβ
µ sin(µL)eiδL/2, (9.50)
D=
cos(µL) + iδ
2µsin(µL)
eiδL/2. (9.51)
TheABCDmatrix represents the same spirit as in the case of a lens guide or a discrete periodic structure. Like in the earlier cases, we haveAD−BC = 1 for a lossless system. For a finite segment of such a medium, we can define the amplitude reflection coefficient as
r= A−(0) A+(0) =−C
D (9.52)
182 Wave Optics: Basic Concepts and Contemporary Trends and the amplitude transmission coefficient as
t= A+(L)
A+(0) = AD−BC
D = 1
D. (9.53)
Thus, the expressions for reflection and transmission turn out to be r= iβsinh(sL)
scosh(sL)−iδ2sinh(sL), (9.54) t= se−iδL/2
scosh(sL)−iδ2sinh(sL). (9.55) In Eqs. (9.54) and (9.55) we replacedµ byis in order to make contact with known expressions in the literature [66]. The magnitude of the amplitude reflection coefficient r attains its maximum value at δ = 0, implying K = 2k; this is exactly Bragg’s condition for reflection from the periodic medium.
Considering the ambient media to be same and homogeneous, the intensity reflection and transmission coefficients are given by
R=|r|2, T =|t|2. (9.56)