Coupled equations of asymmetric photonic crystal waveguides

In Section 2.4, we derived the eigenfrequencies (dispersion relations) and eigenvectors (field amplitudes) of asymmetric PCWs are

where Δ =(ω₂−ω₁) / 2 and χ^± are the amplitude ratios corresponding to frequencies ( )k

ω^± ; ω₁ and ω₂ are the dispersion relations of the sing PCW1 and PCW2 as shown in Fig. 3; g ka( )= +α 2βCos ka( ); α and β are the coupling coefficient between two waveguides.

From the electric field distributions of defects in the square and triangular lattices, shown in Fig. 20, we find that the electric field at the site (x = 0, y = 0) of the square lattice has the same polarity (sign) as its nearest-neighboring site (x = 0, y =2a) and the next nearest-neighboring site (x = a, y = 2a). Because Δε < 0 for the air-defect PCWs in both square and triangular lattices, the coupling coefficients α and β both are negative values.

Here, we assume ω₂>ω₁ in the following discussion; therefore, we shall call the waveguide 2 (waveguide 1) as the high-frequency PCW2 (low-frequency PCW1).

Fig. 20 The electric field distribution (Ez) of a point defect mode in the square lattice for (a) f

=0.364 c/a with a reduced-rod (rd = 0.05a) defect; that in the triangular lattice for (b) eigen frequency f = 0.365 c/a with a defect rod εr = 2.56; and (c) The electric field distribution in the dash lines.

Because |g(ka)| has a maximum value at k = 0, one should expect that the dispersion curves have the largest splitting there. As α and β are negative values discussed before, g(ka) always is a negative value for all k if |2β /α | < 1, and its value can change from the negative to the positive sign as k is increasing from 0 to π when |2β /α |>1. Under this

condition of |2β /α | > 1, the coupler can be decoupled when g(kDa) = 0 at a certain k = kD

and have eigenfrequencies ω⁺ = ω₂and ω⁻ = ω₁ separately; that is, the field launched in PCW1 always will be confined in PCW1 without being coupled to PCW2, and vice versa.

We can simply use the ratio of the maximal field values instead of integrals as Eq. (2.9) to estimate coefficients α, β and |2β /α | by assuming the field distribution is strongly localized near the dielectric rods. Thus, |2β /α |~2E(0,2a)/E(±a,2a) in the square lattice and ~2E(0, 3a)/E(±a, 3a) in the triangular lattice.

Because g(ka) < 0 for 0≤ <k k_Dunder |2β /α | > 1 (or for all k under |2β /α | < 1), the lower frequency mode (ω⁻) has -1 < χ^- < 0; namely, the eigenmode of the coupler displays the PCW1 and PCW2 electric fields not only being out-of-phase but also concentrated on the low-frequency PCW1. This odd-like fundamental (low-frequency) mode is called the

“anti-bonding” mode, borrowed from the molecular physics of two atoms. On the other hand, the high-frequency and even-like mode called the “bonding” mode has χ⁺ > 1; thus, it is superimposed by the in-phase electric fields from both PCWs, where the field strength is concentrated on the high-frequency PCW2.

However, as k > kD under |2β /α | > 1, g(ka) becomes positive and 0 < χ^- < 1. The fundamental mode is a bonding mode, which is superimposed by the in-phase electric fields from both PCWs, where the field strength is concentrated on the low-frequency PCW1. And the high frequency antibonding mode with χ⁺ < -1 has the field strength concentrated on the high-frequency PCW2. We find that the fundamental modes of the asymmetric coupler contain no degenerate state (anti-crossing dispersion relations) and can switch from the antibonding to bonding mode as k varies crossing the decoupling point k_D. As the previous study on the symmetric coupler, we simply can set Δ = 0 to

obtain χ^±=±1 at all k, i.e., the fundamental mode is either odd or even depending upon the sign of g(ka). The dispersion curves of the symmetric coupler can cross at the decoupling point if |2β /α | > 1. Furthermore, upon increasing the separation of PCWs to two rows apart [27], from Eq. (2.9), coupling coefficients α and β become positive values and are smaller than coupling coefficients of the one-row-separation PCWs. The fundamental mode becomes a bonding mode, and whether or not mode switching would happen still is determined by the criterion: |2β /α | > 1.

In order to prove that the derived formula by TBT can explain phenomena gotten by PWEM well, we consider for example a 2D triangular (square) lattice PC made by dielectric rods with dielectric constant εr = 12 and radius = 0.2a in the air. Due to the field symmetry, the coupling coefficient ratio β /α is a larger value in the triangular lattice than in the square lattice. It should be easier to reach the criterion |2β /α | ≈ 2E(0,2a)/E(±a,2a) > 1 of the mode switching behavior in the triangular lattice than in the square lattice, shown in Fig. 20(c). Therefore, we consider a triangular lattice PC, and the line defects forming the PCW1 and PCW2 are created by setting the dielectric constants of defect rods at 2.56 and 2.25, respectively. The eigenfrequencies of a point defect with transverse magnetic field (TM), whose electric field is parallel to the dielectric rods, are ω1 = 0.365 (2πc/a) and ω2 = 0.371 (2πc/a), respectively, where c is the speed of light in vacuum. The decoupling point is located at kD = 0.73π/a where the eigenfrequencies of the PC couplers decouple in the eigenfrequency in single line-defect PCWs, shown in Fig. 21(a). Note that the dispersion curves do not cross in the asymmetric coupler. As shown in Fig. 21(b), the eigenmode of the high (low) frequency band at the wave vectors k < kD are the bonding (anti-bonding) modes, but these modes switch when k > kD, namely, the eigenmode of high (low) frequency band being

anti-bonding (bonding). And the electric field is concentrated on the PCW2 for the high-frequency (ω⁺(k_D)) mode and on the PCW1 for the low frequency (ω^-(k_D)) mode at the decoupling point kD. The mode switching phenomenon at kD is shown easily by plotting the ratios of the eigenmodes (χ=V0/U0) obtained either by the PWEM. We observe that χ's change sign at the decoupling point k_D (see Fig. 21(c)).

Fig. 21 Simulation results of PWEM. (a) Dispersion relations of two isolated PCWs (ε =2.56 and ε =2.25) and the directional coupler in the triangular lattice (shown as the inset). (b) The dispersion curves of the directional coupler and its eigenmode profiles below, above and at the decoupling point. (c) The mode amplitude ratios of the coupler.