interaction of the nuclei and electrons will split the energy levels and form the electronic band structures. By the tight-binding approximation, the propagation modes of electron waves of the overall system are closely related to the eigenmodes of individual lattice atoms, and the influences between different lattice points are considered as small perturbations.
Additionally, this method also has been successfully used in various photonic structures similar to the crystalline solids.
In this chapter, we use the extended TBT to analyze the coupling effects of a single CROW in the PCS (PCS-CROW) and the DCs made of CROWs in the PCS (PCS-DCs).
This method assume that the strongly localized electric fields around an isolated defect are not perturbed much by the presence of the other defects, and the finite coupling should exist between successive defect cavities. In addition, the eigenmodes of each point defects are assumed as single modes, and the total fields of propagation modes can be expressed as the linear combination of these defect modes. Hence, the analytical equations of the dispersion relations of these structures can be directly derived, and the design rules of which can be controlled by only few parameters about the coupling strength and the eigenfrequency of these defect cavities.
2-2 TBT in a single PCS-CROW
We begin our analysis by considering two cases : a single CROW consist of reduced rods in the PCS with a square lattice of dielectric rods in air ; and a single CROW consist of enlarged holes in the PCS with a triangular lattice of air holes in a dielectric slab, as shown in Fig. 2.1(a) and Fig. 2.1(b). The lattice constant of the PCS is a, and the propagation direction of waveguides are along x-direction. In addition, the dielectric-rod PCS with square lattice and the air-hole PCS with triangular lattice can lead to relatively large PBG to obtain the behaviors of the defect bands, and the air defects made by reduced rods or enlarged holes can support the single-mode propagation.
(a)
(b)
Fig. 2.1 Structures of a single CROW made in the PCS (a) with a square lattice of dielectric rods in air and (b) with a triangular lattice of air holes in dielectric slab. a is the lattice constant, and C1 , C−1 , C2 , C−2 are the coupling coefficients between the point defects.
By applying the TBT, we assume that each EM mode of the point defects has an eigenfrequency ω0 and is unable to be affected much by the presence of the other defects [13].
The mode of an isolated defect is given by E(r,t) = E0(r) u(t), where E0(r) and u(t) = U0
exp(−iω0 t) are the spatial and the temporal amplitudes of the cavity eigenmode, and U0 is a constant amplitude. For the perturbed mode at defect site n, these amplitudes are En(r) = E0(r − 2nax) and un(t), where x is the unit direction vector along the waveguide. Therefore, the total field of the CROW is the summation of all defect modes as E(r,t) = ∑ En(r) un(t).
Due to the weak coupling of the cavities, we consider the influences only by the nearest and the next-nearest neighboring defects, and the coupled equation to relate un and un+p can be derived by using the slowly varying amplitude approximation [23] as
i (∂un / ∂t) = (ω0 − C0) un − ∑ Cp (un+p + un−p) ,
where C0 represents a small shift to the eigenfrequency of a single point defect due to the dielectric perturbation from the neighboring defects, and Cp = C−p is the coupling coefficient between defects at sites n and n + p. This parameter is defined as
which is proportional to the electric field amplitudes at defect sites, and ∆ε(r) = ε’(r) − ε(r) is the difference of dielectric constants between the perturbed system (coupled defects) ε’(r) and the unperturbed system (an isolated defect) ε(r), and p = 0, 1, 2. Let un(t) = U exp(2ikna − iω1t), where k is the propagation constant, and U is the constant amplitude. By substituting
this into Eq. (2.1), the dispersion relation of a single PCS-CROW can be written as ω1(k) = (ω0 − C0) − ∑ 2Cp cos(2pka),
where C0 stands for the relative shift for all wavevectors to the eigenfrequency of a single point defect, and Cp leads to the sinusoidal modulation of dispersion curves.
n
2-3 TBT in the PCS-DCs
When a second identical PCS-CROW is created to form a symmetric DC, that can be classified into two types from the relative positions of the two waveguides. One is the opposite-type PCS-DC, where the defect sites of both the PCS-CROWs are the same (n, n + 1, n + 2, ... etc) as shown in Fig. 2.2(a) and Fig. 2.2(b). The other is the alternating-type PCS-DC, where the defect sites of one PCS-CROW (n, n + 1, n + 2, ... etc) and of another one (n + 1/2, n + 3/2, ... etc) are shown in Fig. 2.3(a) and Fig. 2.3(b). In addition, the distance between the two waveguides is D, and the slab plane is xy-plane.
(a)
(b)
Fig. 2.2 Structures of the opposite-type PCS-DCs made of (a) a square lattice of dielectric rods in air, and (b) a triangular lattice of air holes in dielectric slab. α and β are the coupling coefficients of one PCS-CROW induced by the nearest neighboring and the next-nearest neighboring defects of another PCS-CROW.
(a)
(b)
Fig. 2.3 Structures of the alternating-type PCS-DCs made of (a) a square lattice of dielectric rods in air, and (b) a triangular lattice of air holes in dielectric slab.
2-3.1 Opposite-type PCS-DC
In the opposite-type PCS-DC, un(t) and vn(t) are the temporal amplitudes of the perturbed mode at defect site n of the two PCS-CROWs, and the coupled equation can be derived in the same way as
i (∂un / ∂t) = (ω0 − C0) un − ∑ Cp (un+p + un−p) − α vn − β (vn+1 + vn−1)
i (∂vn / ∂t) = (ω0 − C0) vn − ∑ Cp (vn+p + vn−p) − α un − β (un+1 + un−1) ,
where α and β are the coupling coefficients of a defect at site n in one PCS-CROW induced by the nearest neighboring (at site n) and the next-nearest neighboring (at sites n + 1) defects of another PCS-CROW, and have the same form as Cp in Eq. 2.2. These coupling coefficients are defined as
p = 1
p = 1 2
2
(2.4)
(2.5)
where y is the unit direction vector along y-direction. Let un(t) = U exp(2ikna − iω2 t) and vn(t) dispersion relation of the opposite-type PCS-DC can be expressed as
ω2(k) = ω1(k) + [α + 2β cos(2ka)],
where α causes the frequency shift and β leads to the sinusoidal modulation of the dispersion curves. Under the condition |β/α| > 1/2, these curves will cross at a wave vector k = [cos-1(−α / 2β)] / 2a , where α + 2β cos(2ka) = 0.
2-3.2 Alternating-type PCS-DC
On the other hand, in the alternating-type PCS-DC, un(t) is the temporal amplitude of the perturbed mode at defect site n of one PCS-CROW, and vn+1/2(t) is the same thing at defect site n + 1/2 of another PCS-CROW. Similarly, the coupled equation can be written as
i (∂un / ∂t) = (ω0 − C0) un − ∑ Cp (un+p + un−p)
i (∂vn+1/2 / ∂t) = (ω0 − C0) vn+1/2 − ∑ Cp (vn+1/2+p + vn+1/2−p)
− α (un + un+1) − β (un+2 + un−1) ,
where α and β are the coupling coefficients of a defect at site n in one PCS-CROW induced by the nearest neighboring (at sites n + 1/2) and the next-nearest neighboring (at sites n + 3/2) defects of another PCS-CROW, and are defined as
Let un(t) = U exp(2ikna − iω2 t) and vn+1/2(t) = V exp[2ik(n + 1/2)a − iω2 t], where U and V are the constant amplitudes, and substituting these into Eq. (2.11) and Eq. (2.12). We can gain
U (ω2 − ω1) + 2V [α cos(ka) + β cos(3ka)] = 0 V (ω2 − ω1) + 2U [α cos(ka) + β cos(3ka)] = 0 , and the dispersion relation of the alternating-type PCS-DC can be derived as
ω2(k) = ω1(k) + 2[α cos(ka) + β cos(3ka)] ,