1. Introduction

Optical systems with waveguides and resonators in photonic crystals (PCs) have received special attention recently because of the potential use in design of optical devices such as channel drop filters [1–3], power splitters [4,5] and intersections [6]. Among them, many structures employing several resonators lead to variety of novel effects [7–14] due to the coupling between optical resonators. One of the most interesting and promising effects is the coupled-resonator-induced transparency (CRIT), which results in an extremely narrow resonant transmission on the background of the broader reflection line [7–13]. However, most of the structures designed in PCs are resonant-coupling or side-coupling with the cavities cascaded along the waveguides. In this paper, we propose an optical coupling system to analyze coupled-resonator-induced transparency effect in PC waveguide resonator system. The transmission characteristics of the structure are investigated theoretically by coupled-mode theory in time domain and numerically demonstrated the effect by simulating the propagation of electromagnetic waves in PCs by finite-difference time-domain method.

2. Theoretical considerations

In order to obtain a qualitative understanding of CRIT effect in PC waveguide resonator systems, we provide a theoretical model, as shown in Fig. 1 . The structure is symmetric with respect to the reference plane. The length of the top resonant waveguide is$2l$. The optical energy is coupled between bus waveguide and the resonant waveguide through the side-coupling cavity. The amplitude of the cavity is denoted by a and is normalized to the energy in the modes. The amplitudes of the incoming and outgoing waves into the cavity denoted by $S_{+ij}$ and $S_{-ij}$($i,j=1,2$) (as shown in Fig. 1) and are also normalized to the power carried by the waveguide mode.

 

Fig. 1 Optical system consisting of a cavity side coupled to bus waveguide and resonant waveguide.

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The transmission and reflection properties of such a system can be calculated by the coupled-mode theory in time [15]. When the electromagnetic wave at a frequency ω is launched into the system only from the input port of the bus waveguide, i.e., $S_{+22}=0$, the time evolution of the amplitudes of the cavity in steady state can be described as

In the resonant waveguide, the incoming waves to the cavity should satisfy the relationships in a steady state: $S_{+11}=S_{-11}{e}^{-j2\phi }$and$S_{+12}=S_{-12}{e}^{-j2\phi }$, where φ is the phase shift that the waveguide mode acquired as it propagates from the reference plane to the waveguide ends.

It should be mentioned that the electromagnetic wave coupled into the resonant waveguide exhibits Fabry-Perot oscillations and forms standing-wave modes when the phase-matching condition

When ${\theta }_1-{\theta }_2=0$, the standing-wave modes are even symmetric with respect to the reference plane and then we find from Eq. (1)-(4) that the transmission coefficient T into the output port at steady state can be determined as

From Eq. (6), when $\cot \phi =\pm \infty$, i.e., m is an even integer in Eq. (5), complete transmission can be realized at the resonant frequency ${\omega }_m$ of the standing-wave modes in the resonant waveguide. It should be emphasized that on both sides of ${\omega }_m$ there are two frequencies at which the transmission drops to zero when${\omega }_0=\omega -2\cot \phi /{\tau }_1$.

Examining Eq. (6), we note that the structure behaves as a light reflector because of that the structure can be treated as a single composite cavity with the detuning function $\sigma (\omega )=\left(\omega -{\omega }_0\right){\tau }_2/2-\cot \phi {\tau }_2/{\tau }_1$ [7,8], formed by coupling of the point defect cavity and the waveguide resonator, side-coupled to the bus waveguide. The last term of the detuning function leads to the CRIT effect. Due to the discontinuity of the detuning function at${\omega }_m$, the resonant frequency of the composite cavity splits. To analyze the dependence of the transmission on the decay rates into the bus waveguide and the resonant waveguide, we plot the intensity transmission spectra in Fig. 2 that are calculated from Eq. (6). From Fig. 2, we find that the shapes of the spectrum features depend critically on $1/{\tau }_1$. Small $1/{\tau }_1$ leads to a narrow resonant transmission peak between the two transmission dips.

 

Fig. 2 Theoretical transmission spectra though the optical system as shown in Fig. 1. The spectra are calculated from Eq. (6) with the phase shift $\phi \equiv \omega l/c$. We plot the frequency in the unit of $2\pi c/l$. The resonant frequency of the cavity is ${\omega }_0=0.50(2\pi c/l)$. The decay rates are taken to be $1/{\tau }_1=1/{\tau }_2$ (dash line) and$500/{\tau }_1=1/{\tau }_2$(solid line) with $1/{\tau }_2=0.002(2\pi c/l)$.

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Similarly, when ${\theta }_1-{\theta }_2=\pm \pi$, the standing-wave modes are odd symmetric and the transmission coefficient T can be expressed as

From Eq. (7), when$\tan \phi =\pm \infty$, i.e., m is an odd integer, complete transmission can be realized at the resonant frequency ${\omega }_m$ of the standing-wave modes in the resonant waveguide. On both sides of ${\omega }_m$, there are two frequencies at which the transmission drops to zero when ${\omega }_0=\omega +2\tan \phi /{\tau }_1$.

3. Numerical experiments

Based on the theoretical considerations above, we present two structures in PCs to analyze the dependence of the transmission properties on the cavity resonant frequency and the decay rates. The two-dimensional PC of dielectric rods in air on a square array with lattice constanta, possesses a bandgap for transverse magnetic (electric field parallel to the rods) modes [16]. The rods have a radius of 0.20a and a dielectric constant of 11.56. Waveguides are formed by removing rows or columns of rods in the crystal and cavity is created by reducing the radius of a single cylinder.

We first consider the propagation of a light wave in PC structure as shown in Fig. 3(a) . The resonant waveguide supports a symmetric standing-wave mode at frequency ${\omega }_m\approx 0.372\left(2\pi c/a\right)$in the frequency range of interest. The point defect cavity supports a single degenerate monopole state. Due to the symmetry of the structure, the decay rates of cavity mode amplitude into the resonant waveguide and into the bus waveguide are equal, i.e., $1/{\tau }_1=1/{\tau }_2$.

 

Fig. 3 (a) Photonic crystal structure. The bus waveguide is formed by removing a single row of rods. The resonant waveguide, created by removing a row of five rods, is $4a$ away from the center of the bus waveguide. A point defect is placed between the center of the two waveguides. (b) Transmission spectra calculated by the finite-difference time-domain method at the resonant frequency of the cavity${\omega }_0=0.3642\left(2\pi /a\right)$.

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2D finite-difference time-domain method with perfectly matched layer boundaries [17] is carried out to study the response of the structure shown in Fig. 3(a). A dipole source located at the input port is used to excite a pulse with a Gaussian envelope in time. The normalized transmission coefficients are then calculated by Fourier transforming the amplitude of the fields at the output port. Figure 3(b) shows the intensity transmission spectra at resonant frequency ${\omega }_0=0.3642(2\pi c/a)$. There are two transmission dips at both sides of ${\omega }_m$ and the CRIT effect can occur at the frequency between the two dips, which is in good agreement with the prediction by the analytic theory.

In order to analyze the dependence of the transmission properties on the decay rates and the positions of resonant frequency, we present another structure in PCs as shown in Fig. 4(a) . The resonant waveguide, created by removing a row of five rods, is $6a$ away from the center of the bus waveguide and the point defect is placed $2a$ away from the center of the bus waveguide on the same side of the resonant waveguide, where $1/{\tau }_1<<1/{\tau }_2$. Cavity is created by reducing the radius of a single cylinder and the resonant frequency ${\omega }_0$ of the cavity can be changed by adjusting the radius of the point defect.

 

Fig. 4 (a) Photonic crystal structure. (b) and (c) Transmission spectra of the structure at different resonant frequencies of the cavity. The solid lines are the transmission spectra through the structure as shown in (a). The dashed line is the transmission spectra for the same structure without the resonant waveguide. The resonant frequencies of the cavity are taken to be: (b)${\omega }_0=0.3722\left(2\pi c/a\right)$, (c)${\omega }_0=0.3702\left(2\pi c/a\right)$.

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Figure 4(b) and 4(c) show the transmission of the structure at different resonant frequencies of the cavity. In comparison, we also show in Fig. 4(b) and 4(c) the transmission spectra for the same structure, except without the resonant waveguide. Comparing Fig. 4(b) and 4(c), we find that the frequency at which CIRT effect occurs is completely determined by the resonant frequency of the standing-wave modes in the resonant waveguide defined by Eq. (5). We also note that the spectra consist of CRIT features superimposed upon a Lorentzian line shape background defined by the side-coupled cavity resonance. The shapes of the CRIT features depend critically on the decay rates of the cavity mode amplitude into the resonant waveguide$1/{\tau }_1$and the relative positions of cavity resonant frequency${\omega }_0$. From Eq. (6), small decay rate $1/{\tau }_1$ reduces the effect of CRIT at the frequencies far from ${\omega }_m=0.3718\left(2\pi c/a\right)$ and the structure works as a side-coupled cavity, but in the immediate vicinity of ${\omega }_m$, i.e., $\cot \phi \to \pm \infty$, the CRIT effect occurs and the transmission reaches 100% which leads to sharp peak in the spectra, as shown in Fig. 4(b). When we vary the resonant frequency ${\omega }_0$ from $0.3722\left(2\pi c/a\right)$ to$0.3702\left(2\pi c/a\right)$, the CRIT transmission spectra display sharp asymmetric line shapes [the solid line in Fig. 4(c)]. The frequency difference between the complete transmission to complete reflection on the other side of ${\omega }_0$ is $0.0001\left(2\pi c/a\right)$, which is far smaller than the full width at half minimum of the cavity resonance. The interaction of the cavity and waveguide resonator is responsible for the line shape of the CRIT which can be employed for various efficient switching devices.

The steady-state field distributions at different frequencies for ${\omega }_0=0.3722\left(2\pi c/a\right)$[spectra see Fig. 4(b)] are shown in Fig. 5 . At frequency$\omega =\text{0 .3725}\left(2\pi c/a\right)$, little energy is coupled into the resonant waveguide and the structure behaves as a light reflector, as shown in Fig. 5(a). This is because of that the frequency of the incident light cannot meet the phase-matching condition described by Eq. (5) and the transmission properties of the structure are determined by the side-coupled cavity. When the frequency of the incident light satisfies the phase-matching condition, i.e. $\omega ={\omega }_m$, complete transmission occurs, as shown in Fig. 5 (b), which is referred as the CRIT effect. In this case, two things should be mentioned: (1) field in the waveguide resonator is strong and almost no energy localizes in the point defect cavity; (2) the reflection phase shift at the ends of the resonant waveguide is $\pi /2$ because the standing-wave mode is even symmetry while m is an even number.

 

Fig. 5 Steady-state electric field distribution in the structure shown as shown in Fig. 4(a) at frequencies: (a) $\omega =\text{0 .3725}\left(2\pi c/a\right)$, (b) $\omega =\text{0 .3718}\left(2\pi c/a\right)$. The resonant frequency of the cavity is ${\omega }_0=0.3722\left(2\pi c/a\right)$.

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4. Summary

In conclusion, we have presented a theoretical CRIT model by the coupled-mode theory to investigate the transmission properties of a waveguide resonator coupling system. Using the theoretical analysis, the CRIT effect in 2D square lattice PCs are numerically studied by the finite-difference time-domain method and an excellent agreement has been found with the present analysis.