1. Introduction

Waveguide-based optical ring resonators are key components in integrated optics and fiber optics, which can be used for filtering, sensing, switching, multiplexing/demultiplexing, signal processing, etc. [1]. Although Fabry-Pérot resonators are frequently used as bulk optical elements, they are not structurally adequate to integrated optics applications. Therefore the ring resonators can often serve as a substitute for Fabry-Pérot resonators. Microring resonators, i.e. compact ring resonators with a bending radius of the micrometer order have attracted a lot of interest, which are made of high index contrast waveguides such as Si-wire waveguides [2]. For example, all-optical switching in microrings with a bending radius up to 5 µm has recently been proposed and has been demonstrated using the third-order nonlinearity associated with two-photon absorption or thermal nonlinearity of semiconductors [3,4]. The operation principle of these bistable optical devices is essentially the same as those of the nonlinear fiber ring resonators [5,6]. Although these effects have large apparent nonlinearity based on a change in the induced carrier density, they limit the switching speed of the all-optical devices. To increase the switching speed, we must use materials with as fast nonlinearity as possible and further decrease the ring radius. Of source, an increase in the switching power is required, since there is generally a trade-off between the material response time and the magnitude of nonlinearity. However there is a limit to decrease the ring radius because the radiation loss is increased with decreasing bending radius.

One way to address this problem is to develop ring resonators formed by using photonic crystals (PC) [7], since high transmission through sharp bends in PC waveguides is attainable [8]. Especially PC slabs, i.e. two-dimensional (2D) PCs of finite height are expected to be a platform for a new generation of integrated optical devices and components. The linear transmission properties of the waveguide-coupled PC ring resonators have recently been investigated theoretically and experimentally [9–12]. However Refs. [9–11] are about a dielectric-rod-type PC structure, which is not realistic because line-defect waveguides have no vertical confinement in its PC slab structure and are hence inherently leaky. PC ring resonators without bus waveguides have also been demonstrated for PC ring lasers [13,14]. It is expected that a variety of properties of the air-hole-type PC devices significantly differ from those of the dielectric-rod-type PC devices. Moreover there is not yet an investigation of the nonlinear transmission properties of the waveguide-coupled microrings formed by using air-hole-type PCs.

In this paper we investigate the linear resonance properties of the PC microrings formed in the 2D PC structure of air holes in dielectric and optical bistability in the nonlinear PC microrings. Although the dielectric-rod-type PC waveguides can be easily single mode, air-hole-type PC waveguides tend to be multimode. Therefore it is necessary to engineer the structure of single-mode waveguides, the configuration of ring resonators containing corners, and the coupling between the bus waveguides and the ring resonator. In this paper we propose several types of waveguide-coupled PC microrings and investigate the linear and nonlinear transmission properties by using a 2D plane-wave expansion (PWE) method [15,16] and the 2D finite-difference time-domain (FDTD) method. We have found interesting unique properties such as counter rotation of intracavity fields, non traveling-wave resonance in the ring, and periodic oscillations in the nonlinear domain. We can improve the resonance properties by partly introducing single-mode waveguides into PC microrings and can obtain stable optical bistability without instabilities. In fact, optical bistability in PC microcavities has already been proposed and demonstrated [17–19]. However theoretical investigations have been carried out for dielectric-rod-type PC structures. Moreover a high accuracy is required to fabricate PC microcavities. It seems that the fabrication of bistable optical devices using PC microrings is easier than one using PC microcavities.

2. PC ring resonators made of (10) waveguides

2.1 Structure and linear properties

For simplicity, we consider a 2D photonic crystal, which is a square lattice of air holes drilled in a dielectric medium. We here assume that the dielectric is Ag₂₀As₃₂Se₄₈ chalcogenide glass with a high Kerr nonlinearity, which has the linear refractive index n=3.1 and the nonlinear refractive index n ₂=9.0×10^-17m²/W at 1.05 µm [20], and the radius of the holes is r=0.46a, where a is the lattice constant. The magnitude of n ₂ of this material is 3000 times as large as that of fused silica (n ₂=3.0×10^-20m²/W). We restrict our analysis to TM modes whose electric field is polarized along the axis of the air holes, since the photonic band gap (PBG) for TM modes is wider than that for TE modes. A PC waveguide can usually be created by removing one or several rows of air holes in the (10) direction of the PC. We will call hereafter this waveguide a (10) waveguide. The (10) waveguide consisting of s lines of missing holes is conventionally denoted as Ws.

Figure 1(a) and 1(b) show the dispersion relations of the W1 and W0.7 waveguides, respectively. We used the 2D plane-wave expansion (PWE) method to calculate the band structures of perfect PCs and used a supercell technique based on the PWE method to calculate the band structures of PC waveguides [15]. In the present case, the PBG for TM modes is found between the angular frequencies ω=0.2588(2πc/a) and ω=0.2971(2πc/a). The gray areas in the figures are the projections in the direction perpendicular to the waveguide of every mode in the band structure of the perfect PC. In order to gain a better understanding of the dispersion relations, solutions of microwave metallic waveguide modes are shown by broken lines in the figures [21]. Moreover the order of the guided modes is given for each branch and field profiles at the position marked by black circles are shown as insets in Fig. 1(a). As can be seen from Fig. 1, the dispersion curves for air-hole-type waveguides are generally complicated since they are folded at the edge of the Brillouin zone [21]. Therefore the air-hole-type W1 waveguide is inherently multimode unlike the W1 waveguide formed in dielectric-rod-type PCs [9,10], i.e. cannot support only a fundamental mode. Moreover the discrete bands must lie below the light line of air (ω=ck ₀) so that the guided modes in actual PC waveguides of finite height are confined in the vertical direction. To operate PC ring resonators appropriately, input and output PC waveguides to be connected to them should be single mode at least. We can obtain a single-mode waveguide by decreasing the width of the line-defect waveguide. In the W0.7 waveguide, there is a single-mode region from 0.2615(2πc/a) to 0.2911(2πc/a). However the frequency region below the light line is narrow and is given by 0.2615(2πc/a) <ω<0.2639(2πc/a).

In many cases, single-mode propagation is required to input and output waveguides of bistable optical devices. It is also difficult to construct PC ring resonators using such a W0.7 waveguide without destroying the symmetry of surrounding PCs. Therefore we construct square PC ring resonators of the multimode W1 waveguide. Figure 2(a) shows a square PC ring resonator laterally coupled to a pair of waveguides, where the two bus waveguides are the W0.7 waveguide, and the ring resonator consists of the W1 waveguide and 3×3 air holes. Figure 2(b) shows a corresponding directional coupler for calculating the coupling strength into the ring resonator, where the two waveguides are separated by a single row of air holes and the length of the coupler L is an integer multiple of the lattice constant a. The patterns of cavity modes to be excited depend strongly on the coupling configuration between the bus waveguide and the ring resonator.

 

Fig. 1. Dispersion curves of the guided modes in (a) a W1 waveguide and (b) a W0.7 waveguide. The PC is a square lattice of air holes (r=0.46a) in dielectric (n=3.1). The gray areas are the projected band structure of the perfect crystal.

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Fig. 2. (a) Square PC ring resonator consisting of the W1 waveguide. The two bus waveguides are the W0.7 waveguide. (b) Corresponding directional coupler for calculating the coupling strength into the ring resonator.

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We here investigate the linear (n ₂=0) transmission spectra of the PC ring resonator shown in Fig. 2(a) using the FDTD method. The entire structure is surrounded by a perfect matching layer to avoid spurious reflections from its edge. A pulse source is located at the input port and three detectors are placed at output ports B, C, and D. The source is a Gaussian pulse in both space and time, which has a field distribution similar to one inside the W0.7 waveguide and a wide bandwidth. We can easily calculate the transmission spectra by calculating the ratio between the Fourier transforms of the transmitted pulse and the incident pulse. Figure 3(a) shows the linear transmission spectra at three output ports of the ring resonator shown in Fig. 2(a). It is clearly shown that resonance takes place even if the ring waveguide is operated under multimode conditions. It is also found that a part of the incident power comes out of port C unlike ring resonators constructed of conventional dielectric waveguides based on total internal reflection. Although the transmission is greater than unity in Fig. 3(a) (also in Figs. 10 and 15), this is caused by the back reflection from the coupler (or ring resonator) and spurious reflections due to the impedance mismatching at the waveguide terminations [8]. In this example, the sharpest resonance takes place around an angular frequency of 0.2830(2πc/a), which falls within the single-mode region of the bus (W0.7) waveguides. At the other resonant frequencies, resonance is relatively weak. This means that the ring resonator cannot be excited with a high efficiency at these frequencies. It should also be noted that the transmission to ports D and C is nearly zero in the frequency region below ωa/2πc=0.270 because the ring resonator cannot be excited well. Figure 3 (b) shows the field pattern in the ring resonator at ω=0.2830(2πc/a), which was calculated using the FDTD method. For comparison, we calculated the cavity modes in the isolated 3×3 square PC ring resonator using a supercell method [16], which consists of placing a large crystal with the ring resonator into a supercell and repeating it periodically in space. In this case, the supercell has the square symmetry (point group C _4v). We could find many cavity modes with resonant frequencies that fall into the PBG. However the field patterns of these modes are generally very complicated because of the complicated structure and multimode nature of the ring resonator, although they have certain symmetry. Figure 3(c) shows the field in the isolated cavity at ω=0.2796(2πc/a) calculated by the PWE method, which is in agreement with the field pattern in the waveguide-coupled ring resonator at ω=0.2830(2πc/a) shown in Fig. 3(b). Although there is a difference in the expected frequency of the resonant mode, this is caused by the difference in the ring configurations, i.e. one is with the bus waveguides and the other is without them. The field pattern differs remarkably from that in traveling-wave resonators [2] or dielectric-rod-type PC ring resonators [9] and can be considered as a localized state in a large square defect. The movie in Fig. 3(b) shows that the intracavity field propagates counter-clockwise in contrast to the ring resonators made of conventional dielectric waveguides.

 

Fig. 3. (a)Linear transmission spectra at three output ports of the PC ring resonator shown in Fig. 2(a) and (b) field pattern in the ring resonator at ω=0.2830(2πc/a). (c) Mode pattern with a resonance frequency of 0.2796(2πc/a) in the isolated square PC ring resonator, which was calculated by using the PWE method. [Media 1]

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In order to find the origin of the counter-clockwise rotation, we calculate the transmission properties of the directional coupler shown in Fig. 2(b). Figure 4 shows the transmission to three output ports as a function of coupler length L. In this numerical computation, a continuous wave with an angular frequency of 0.2830(2πc/a) was incident into input port A. It is found that a contra-directional coupling takes place, i.e. the input power emerges from port D unlike the directional coupler consisting of the identical two PC waveguides [22]. It should be noted that the obtained results contain the effect of two corners at the ends of the directional coupler. Note also that the coupling section of the 3×3 ring resonator shown in Fig. 3(a) corresponds to the directional coupler with L/a=5. This seems to be the origin of the counter-clockwise rotation.

This phenomenon may also be interpreted by the resonant state symmetry and its degeneracy as discussed in [24]. The field patter shown in Fig. 3(c) has a dipole-like symmetry, which is doubly degenerate [23]. That is, the 90° (or -90°) rotated state of the field pattern shown in Fig. 3(c) is a replica of it. The localized state shown in Fig. 3(c) is the odd mode, as defined with respect to the e (1,1̅) mirror plane and the 90° rotated state is the even mode with respect to this plane. The traveling wave incident from port A will excite the degenerate modes and these modes induce the rotation of the cavity field. The direction of rotation is determined by the phase difference between two degenerate modes. However it is difficult to concretely verify the counter-clockwise rotation, since the phase difference is unknown.

 

Fig. 4. Transmission to three output ports of the directional coupler consisting of the W0.7 and W1 waveguides shown in Fig. 2 (b) as a function of coupler length L.

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2.2 Optical bistability and instability

In order to investigate optical bistability in the nonlinear 3×3 PC ring resonator shown in Fig. 2(a), we launch an intense optical pulse into the system. Transient phenomena always take always place by a change in the input intensity [25]. It should also noted that the steady-state treatment such as an approach used in [17,18] is no longer adequate when the cavity is scanned in a time shorter than or comparable to the cavity build-up time [25,6]. We here assume that the input pulse is a triangular pulse with the following temporal power profile; P(t)=(P ₀/τ)t for 0<t<τ, P(t)=(P ₀/τ)(2τ-t) for τ<t<2τ, and P(t)=0 for t>2τ, where P ₀ is the peak power and τ is the pulse duration (full width at half maximum). In the nonlinear FDTD calculation, we use the intensity-dependent index n(I)=n+n ₂ I=n+(ε ₀ cnn ₂/2)|E|² in place of the linear refractive index n of the dielectric (Ag₂₀As₃₂Se₄₈ glass). In our case, the instantaneous response of the nonlinear material is assumed. Throughout the present nonlinear simulation, the lattice constant a is set to be 0.4387 µm, which leads to the resonance wavelength λ ₀=1.55 µm for the resonance frequency ω ₀=0.2830(2πc/a). Moreover the input pulse width τ is set to be 30 ps.

Figure 5 shows the temporal shapes of the transmitted pulses from three output ports for a few values of the peak power P ₀ and initial detuning Δω=ω ₀-ω, where the resonant frequency is ω ₀=0.2830(2πc/a). As is well known, we must somewhat shift the operating point (i.e., operating frequency ω) from a resonant point (i.e., resonant frequency ω ₀) to obtain optical bistability. It is found that nonlinear switching takes place at input power levels from 10 to 50 W/µm. The optical power carried in the direction of propagation by the guided wave is given by integrating the optical intensity (i.e., the power per unit area) and it becomes infinite in a 2D model. Therefore the optical power is here expressed as the power per a unit length (1 µm) along the axis of the air holes. However the transmitted powers from three output ports C-D begin an almost sinusoidal oscillation after switching. It is also found that the ring resonator reveals transient properties even though the duration of the input pulse is long. The incident power emerges from ports C and D after the incident pulse passes through port B, i.e. for t>60 ps. Since these general features are similar to those of transient nonlinear Fabry-Perot resonators, please refer to [25].

 

Fig. 5. Temporal shapes of the transmitted pulses from three output ports of the nonlinear PC ring resonator (n=3.1 and n ₂=9.0×10^-17m²/W) shown in Fig. 2(a) when a triangular pulse is incident into it for a few values of the peak power P ₀ and initial detuning Δω=ω ₀-ω.

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Figure 6(a) shows the enlarged temporal variation of the transmitted pulse from port D for Δω=0.002 ω ₀ and P ₀=24 W/µm in Fig. 5. The output oscillates with a period of 0.17 ps. Such a periodic oscillation is not ascribed to numerical instability of the FDTD method used in our numerical simulation. This phenomenon is similar to the Ikeda instability that occurs in the higher output power branch in the nonlinear fiber ring resonators [25,26]. However it seems to be not due to the Ikeda instability, since the Ikeda instability takes place under single-mode conditions and the present PC ring waveguide is operated multimode. Figure 6(b) shows the instantaneous electric field distribution in the PC ring resonator, which is different from one in a linear case (Fig. 3(b) or 3(c)). The movie in Fig. 6(b) shows that the optical pulse as shown in Fig. 6(b) rotates counter-clockwise in the ring resonator as a solitary wave and the output from port D reaches the greatest value when its peak approaches the output coupler connected by the bus waveguide CD. Therefore the round-trip time of the intracavity pulse corresponds to the period of the transmitted power from each output port. A round-trip time of 0.17 ps means that the pulse propagating in the ring is threefold slower than the guided wave propagating in the W1 waveguide. Such an instability is possibly caused by the strong dispersion and nonlinearity, but the details are unknown at the present stage. It is very important to find methods for suppressing such an instability. One way without improvement of the ring configurations is to decrease the pulse width τ̣ and to increase the peak power P ₀. Figure 7 shows gurations is to decrease the pulse width τ̣ and to increase the peak power P ₀. Figure 7 shows a typical example when a triangular pulse with τ=10 ps and P ₀=226 W/µm is incident into the ring resonator with Δω=0.002 ω ₀. The figure shows clearly the disappearance of the instability. We observe a clockwise hysteresis loop in input-output characteristics between port A and port D although the response is more transient than that for τ=30 ps. It is also expected that the instability disappears when the medium response time is more than the cavity round-trip time [25]. Moreover we will propose other ring configurations to suppress the instability in the following two sections.

 

Fig. 6. (a) Enlarged temporal variation of the transmitted pulse from port D for Δω=0.002 ω ₀ and P ₀=24 W/µm in Fig. 5. (b) Instantaneous electric field distribution in the PC ring resonator corresponding to (a). [Media 2]

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Fig. 7. (a) Temporal shape of the transmitted pulse from port D when a triangular pulse with τ=10 ps and P ₀=226 W/µm is incident into the PC ring resonator with Δω=0.002 ω ₀. (b) Corresponding input-output characteristics.

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3. PC ring resonators made of decreased-index waveguides

3.1 Structure and linear properties

We try to construct a single-mode PC waveguide by decreasing the difference in refractive index between the core (i.e., guiding region) and the cladding. For example, we can create a line-defect waveguide by decreasing the radii r_s of a line of air holes instead of removing them perfectly, as shown in Fig. 8(a). We hereafter call this waveguide a decreased-index waveguide. Figure 8(b) shows the dispersion relations for the decreased-index waveguide with r_s=0.35a. We have a single-mode region from 0.2588(2πc/a) to 0.2818(2πc/a), which lies below the light line of air. Figures 9(a) and 9(b) show the two kinds of waveguide-coupled ring resonators constructed of the decreased-index waveguide with r_s=0.35a. In the 5×5 ring resonator shown in Fig. 9(b), four air-holes are added at corners as a reflector to improve the transmission properties of the ring resonator shown in Fig. 9(a).

 

Fig. 8. (a) Single-mode PC waveguide (decreased-index waveguide) where the radii r_s of a line of air holes is decreased. (b) Its dispersion curves of the guided modes for r_s=0.35a.

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Fig. 9. Two kinds of waveguide-coupled PC ring resonators constructed of the decreased-index waveguide with r_s=0.35a.

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Figures 10(a) and 10(b) show the linear (n ₂=0) transmission spectra at three output ports for the PC ring resonators shown in Figs. 9(a) and 9(b), respectively. The spectrum in the transmission region and resonant curve at port B are not smooth compared to those shown in Fig. 3(a). Although we can obtain a deep drop in the transmission to port B, the transmission to port D or port C is generally much lower than that shown in Fig. 3(a). In order to have a better understanding of these properties, we calculate the electric field distributions in the ring resonators. Figures 11(a), 11(b) and 11(c) show the instantaneous electric field distributions in the ring resonators at resonant points A in Fig. 10(a), B and C in Fig. 10(b), respectively. These resonant points fall into the single-mode region from 0.2588(2πc/a) to 0.2818(2πc/a). The movie in Fig. 11(a) shows that the ring resonator shown in Fig. 10(a) seems to operate as a parallel-coupled finite waveguide of a length of 5a at resonant point A since the coupled-in fields cannot propagate inside the ring because of the strong reflection at its two corners. The movies in Figs. 11(b) and 11(c) also show that the fields such as those shown in Fig. 11(a) are alternatively changed between the facing sides of the square ring with time and the coupled-in fields do never go around the ring. Such no circulation of the intracavity fields is caused by high reflection at each corner. Although an air hole was added at each corner as a reflector in the ring resonator shown in Fig. 9(b), we were able to obtain high transmission at corners. The present result agrees with one given by [27]. For comparison, we calculated the cavity modes in the isolated square PC ring resonators using the supercell method [16]. However the field patterns of these resonant modes did not agree with those shown in Fig. 11. This is because the resonant modes cannot be excited by the coupling method shown in Fig. 9.

 

Fig. 10. Linear transmission spectra (a) and (b) at three output ports of the PC ring resonators shown in Figs. 9(a) and 9(b), respectively.

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Fig. 11. Instantaneous electric field distributions in the PC ring resonators at resonant points A-C marked in Fig. 10. Figures (a), (b), and (c) correspond to resonant points A, B, and C, respectively.[Media 3][Media 4][Media 5]

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3.2 Bistability

We can expect to the generation of optical bistability in the PC ring resonator constructed of decreased-index waveguides since resonance takes place although it never operates as a traveling-wave resonator. In order to investigate input-output characteristics of the nonlinear ring resonator, we launch an intense triangular pulse into the system. The parameters of the input pulse used in this section are identical with ones used in Sec. 2.2. We here focus on resonance at ω ₀=0.2763(2πc/a), which is marked by C in Fig. 10(b). Figures 12(a) and 12(b) show the temporal variation of the transmitted pulses from ports B and D for two cases; case-1, Δω=0.002ω ₀ and P ₀=8.5 W/µm and case-2, Δω=0.004ω ₀ and P ₀=65 W/µm. Figures 12(c) and 12(d) show the input-output characteristics corresponding to Figs. 12(a) and 12(b), respectively. In these figures, the output power from ports B and D is drown as a function of the input power at port A for two cases. It is clearly found that a counterclockwise hysteresis loop is observed in input-output characteristics between port A and port B and a clockwise loop is observed between port A and port D. In a double-coupler fiber ring resonator, ports B and D operate as reflection and transmission ports of the Fabry-Pérot resonator, respectively [5]. The switching power is 2 W/µm for the detuning Δω=0.002 ω ₀. The switching power and the width of the hysteresis loop are increased with increasing detuning. These general tendencies agree with ones for the double-coupler fiber ring resonator [5]. The drawback of this ring resonator is the low transmission from port D, as expected from the results shown in Fig. 10.

 

Fig. 12. (a) and (b) Temporal shapes of the transmitted pulses from ports B and D of the nonlinear PC ring resonator shown in Fig. 9(b), respectively, when the triangular pulse in incident into it for two values of Δω=ω ₀-ω (where ω ₀=0.2763(2πc/a), which is marked by C in Fig. 10(b)) and P ₀. (c) and (d) Input-output characteristics corresponding to (a) and (b), respectively.

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Fig. 13. (a) (11) waveguide defined by removing one row of air holes in the (11) direction of the PC. (b)Its dispersion relations.

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4. PC ring resonators containing (11) waveguides

4.1 Structure and linear properties

Next we construct a single-mode waveguide by removing one row of air holes in the (11) direction of the PC as shown in Fig. 13(a). We will call hereafter this waveguide a (11) waveguide. Figure 13(b) shows the dispersion relations for the (11) waveguide, where the edge of Brillouin zone is at ka/2π=2^1/2/4 since the period of the PC in the (11) direction is 2^1/2 a. We have a single-mode region from 0.2766(2πc/a) to 0.2850(2πc/a) although the dispersion curve lies above the light line of air. We here introduce the (11) waveguide into ring resonators although it is leaky. It is expected that leakage losses are negligible because the (11) waveguide used is very short.

Figure 14 shows three types of PC ring resonators, where the (11) waveguide is incorporated at each corner or is used as each side of a square ring. In all the PC ring configurations, the W0.7 waveguide is used as two bus waveguides to ensure their single-mode propagation. Ring R1 shown in Fig. 14(a) is just one where an air hole is added as a reflector at each corner in the 3×3 PC ring resonator shown in Fig. 2(a). Ring R2 is a 5×5 ring resonator with four reflectors of a length of two air holes. In ring resonators R1 and R2, four sides are formed of the W1 waveguide. On the other hand, ring 3 is a 4×4 ring resonator consisting of the only (11) waveguide. Figure 15 shows the linear transmission spectra of three PC ring resonators shown in Fig. 14. The instantaneous electric field distributions at typical resonant points are also shown as insets in the figure. These resonance frequencies are 0.2776(2πc/a), 0.2774(2πc/a), and 0.2846(2πc/a), which fall into the single-mode region of the (11) waveguide. It is found that the fundamental mode partly or entirely propagates in the ring resonator. The movies in Fig. 15 show that the intracavity field rotates clockwise in ring resonators R1 and R2, while it rotates counterclockwise in ring resonator R3. It is found that the rotation direction of the fields in the ring resonator shown in Fig. 2(a) can be converted by adding an air hole at each corner as shown in Fig. 14(a). In all cases, the ring resonators act as a traveling-wave resonator in contrast to ring resonators constructed of the decreased-index waveguide. Therefore the transmission spectra except for resonance frequencies are smooth and around unity compared with those shown in Fig. 10. It is very important to properly design the structure of single-mode waveguides, the configuration of ring resonators containing corners, and the coupling between bus waveguides and the ring resonator.

 

Fig. 14. Three types (R1, R2, and R3) of PC ring resonators incorporating the (11) waveguides.

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Fig. 15. Linear transmission spectra of the three PC ring resonators shown in Fig. 13.[Media 6][Media 7][Media 8]

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4.2 Bistability

We first compare with the nonlinear properties of three PC ring resonators at the resonance frequencies marked in Fig. 15. For given values of the initial detuning Δω=0.02ω ₀ and the peak power of the incident triangular pulse P ₀=48.3 W/µm, it is found that ring resonator R3 has the largest output power, the lowest switching power, and the fastest switching time. Therefore we investigate the nonlinear properties for R3 only.

Figure 16 shows the temporal variation of the transmitted pulses from output ports B, C, and D when a triangular pulse with τ=30 ps and P ₀=58 W/µm is incident into port A of ring resonator R3 for ω ₀=0.2846(2πc/a) and three values of the initial detuning Δω. The temporal variation of the incident pulse is also drawn in the figure. It is clearly seen that optical switching between low transmission and high transmission, and overshoot and ringing on the switched-up power. Although the output from port C is less than one from port D, overshoot and ringing at port C are greater than those at port D. Figure 17 shows the corresponding input-output characteristics of ring resonator R3. We can obtain optical bistability without instabilities. A counterclockwise hysteresis loop is observed in input-output characteristics between port A and port B and a clockwise loop is observed between port A and port D or C. The input-output characteristics of the PC ring resonator have a strong resemblance to those of the double-coupler fiber ring resonator since this PC ring resonator acts as a traveling-wave resonator [5]. In [5], a similarity between the nonlinear fiber ring resonator and the nonlinear Fabry- Pérot resonator has also been pointed out. Ports D and B correspond to transmission and reflection ports of the Fabry-Pérot resonator, respectively. A difference between the PC ring resonator and the fiber ring resonator is that a part of the incident power emerges from port C in the PC ring resonators and back reflection into input port A takes place. In this example, the switch-up power is 10–15 W/µm and the switching speed is 2–3 ps for the detuning Δω=0.002ω ₀. The switching speed of the bistable device is generally determined by the cavity decay (build-up) time and medium response time (in this case, instantaneous Kerr nonlinearity is assumed). The cavity decay time of the traveling-wave resonators is given by the cavity length and the coupling coefficient between the bus waveguide and the ring. As can be seen from Fig. 17, the switching power (or the width of the hysteresis loop) depends on the detuning. The switch-up power is decreased with decreasing detuning. On the other hand, the small difference between high transmission and low transmission in reflection bistability at port B can be improved by increasing the detuning.

 

Fig. 16. Temporal variation of the transmitted pulses from ports B, C, and D when the triangular pulse with τ=30 ps and P ₀=58 W/µm is incident into port A of ring resonator R3 with ω ₀=0.2846(2πc/a) for three values of the initial detuning Δω.

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Fig. 17. Input-output characteristics of ring resonator R3 corresponding to Fig. 16.

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5. Conclusion

Optical bistability in nonlinear 2D-PC microrings and related linear resonance properties have been numerically investigated by using the plane-wave expansion method and the FDTD method. In contrast to previous works [9–11], where the dielectric-rod-type PC structure without vertical confinement was treated, the PC structure of air holes in dielectric has been considered as a more realistic model. In this paper, Ag-As-Se chalcogenide glass has been assumed as dielectric, which is one of glasses with the highest third-order nonlinearity. The air-hole-type PC waveguides tend to be multimode in contrast to the dielectric-rod-type PC waveguides. It has been found that the waveguide-coupled PC ring resonators operated under multimode conditions have interesting unique properties such as counter rotation of intracavity fields, transmission to all output ports, non traveling-wave resonance in the ring, and periodic oscillation in the nonlinear domain. These facts reflect the configuration of ring resonators containing corners and the coupling between the bus waveguides and the ring resonator. The resonance properties can be improved by incorporating single-mode waveguides into the resonator and we can obtain stable optical bistability. However the dropping efficiency in port D (or C) of air-hole-type PC ring resonators is generally low because of weak coupling efficiency in the ring compared with that of rod-type PC resonators [10]. The further improvement of coupling structures and ring structures is required.

Although the infinite 2D-PC structures have here been treated for simplicity, actual counterparts will be constructed with PC slabs of finite height as mentioned before. Exact theoretical modeling of such PC structures requires fully three-dimensional (3D) calculations such as a 3D-FDTD. However we can approximately treat them as 2D calculations by introducing the concept of effective index to account for the vertical confinement [29,30,21]. In the effective index method, the refractive index of the dielectric material is replaced by the effective index of guided modes in the unperturbed slab waveguide. Therefore the results obtained in this work are expected to be useful in developing bistable PC devices.