## Abstract

We model the nonlinear response of a silica toroid microcavity using coupled-mode theory and a finite-element method, and successfully obtain Kerr bistable operation that does not suffer from the thermo-optic effect by optimizing the fiber-cavity coupling. Our rigorous analysis reveals the possibility of demonstrating a Kerr bistable memory with a memory holding time of 500 ns at an extremely low energy consumption.

© 2012 Optical Society of America

## 1. INTRODUCTION

Optical bistability is a fundamental physical phenomenon where it is possible for certain devices to have two stable transmission states. It occurs when the refractive index or the absorption of the nonlinear medium in an optical cavity is dependent on the light intensity. Optical bistable devices are considered to be important building blocks in all-optical signal processing for such components as optical memories and optical flip-flops [1], and this phenomenon was extensively studied in the 1970s and 1980s [2]. However, at that time, the size of the cavity and the operating energy were too large to be considered for practical applications.

Recent progress in achieving higher-quality factors ($Q$) in ultrasmall microcavities on-chip [3–6] has refocused attention on optical bistability [7], due to the possibility of achieving denser integration and lower energy consumption. Since the photon density in a cavity scales with $Q/V$, where $V$ is the mode volume, a high-$Q$ cavity with a small $V$ enables us to use various nonlinearities at extremely low input powers; hence it allows optical bistability at an ultralow power.

Soljačić *et al.* [7] demonstrated numerically that optical bistability based on Kerr nonlinearity is possible by using a microcavity at an ultralow driving power of 133 mW. A recent numerical study has reported Kerr bistability at 165 μW [8]. Various experiments have already been reported in silica microspheres [9], silicon photonic crystals [10,11], and silicon microring resonators [12], but by using the thermo-optic (TO) effect. Since the TO effect is accompanied by thermal accumulation, the response is relatively slow. To achieve a faster speed, the carrier-plasma effect has been utilized. This effect is the result of carrier generation [13,14]. The latest research has achieved on 4 bit optical random-access memory operation based on carrier nonlinearity, at a power consumption of only 30 nW, by using InGaAsP photonic-crystal nanocavities [15]. The key to achieving the low power consumption is the smallness of the cavity $V$.

As introduced above, the power required to achieve optical bistability has been significantly reduced in recent years due to the high $Q/V$. However, all of the demonstrations use either TO or carrier-plasma effects to drive the bistability whose nonlinearities are accompanied by photon absorption. Recent advances on linear and nonlinear studies of microcavities and chip-based waveguide devices have made it possible to use them both for classical all-optical processing and for loss-sensitive applications such as quantum information processing [16–19]. With those applications in mind, we need to reduce significantly the power loss (consumption) of the bistable system. The ultimate goal is to use the optical Kerr effect because it does not absorb photons. In addition, the use of an ultra-high $Q$ allows us to reduce the power scattering loss of the signal light from the system, and this will support all-optical information processing for loss-sensitive applications.

Various optical switches and memories using the Kerr effect have been demonstrated experimentally with various microcavities [20–22]. It has been reported that the carrier-plasma effect and the losses related to two- and three-photon absorption are nearly negligible in silica due to its large bandgap [23]. Thus, there have been some preliminary demonstrations on Kerr switching and memory in hybrid silica microspheres [21] and silica bottle microresonators [20]. However, the TO effect can make the Kerr effect difficult to distinguish from other effects in silica. Therefore, we need to carefully analyze whether a Kerr bistable memory is feasible or not in the presence of the TO effect.

In this paper, we demonstrate numerically that an optical bistable memory based on the optical Kerr effect is possible without suffering from the TO effect by controlling the coupling between the cavity and the tapered fiber. We employ a silica toroid microcavity [5] as a platform for the Kerr bistable memory, because it has an ultrahigh $Q$ and is capable of integration on a chip. As discussed above, a high-$Q$ cavity is attractive for achieving optical Kerr bistability and low-loss applications at the same time.

Our model combined coupled-mode theory (CMT) and the finite-element method (FEM). The finite-difference time-domain (FDTD) method can model Kerr bistability in a cavity very accurately. However, it has been shown that CMT can describe the behavior of the light in a cavity as accurately as FDTD [24]; thus, we decided to use CMT and FEM. It should be noted that CMT is a powerful tool for analyzing the couping between two cavity modes [25,26].

The paper is organized as follows. First in Section 2, we provide a simple picture of the physics that we are going to employ. In Section 3, we describe a numerical simulation model that combines CMT and FEM. In Section 4, we show the calculation result, and in Section 5 we discuss the energy consumption. We finish with a conclusion.

## 2. SIMPLE MODEL

As described in the previous section, it is still difficult to use the optical Kerr effect in ultrahigh-$Q$ silica microcavities, even though the material has a large bandgap. This is because of the small light absorption at the surface, which is caused by a thin water layer [9,27]. Therefore, we need to analyze this carefully by using rigorous modeling and applying realistic physical parameters to reveal the conditions required for obtaining Kerr bistability. However, before undertaking a rigorous analysis, we start with a simple model to gain an intuitive understanding of the strategy for obtaining Kerr bistability. In this section, we pursue an analytical discussion about how we can obtain the Kerr effect without exhibiting any significant TO effect.

First we derive the relationship between the wavelength shift and the input energy required for Kerr nonlinearity. The energy $U$ of an electromagnetic wave in a dielectric medium is given by [28]

where $\epsilon $, ${E}_{0}$, and $V$ are the dielectric constant, the electric field amplitude, and the mode volume, respectively. Using Eq. (1) and $I=v\epsilon {|{E}_{0}|}^{2}/2$, where $v$ is the light speed in the medium, we obtain the relation between the power density $I$ and the optical energy $U$ as where $c$ is the velocity of light and ${n}_{0}$ is the refractive index of the cavity medium. Then, the refractive index change caused by the Kerr effect is given by where ${n}_{2}$ is the nonlinear refractive index. Equation (3) allows us to calculate the refractive index change as a function of the energy in the cavity.Next we discuss the TO nonlinearity. The refractive index change caused by the TO effect $\mathrm{\Delta}{n}_{\mathrm{TO}}$ is given by

Here, $\xi =(1/n)(\partial n/\partial T)$ is the TO coefficient. The energy required to increase the temperature by 1 K for volume $V$ is given by $C\rho V$, where $C$ is the heat capacity and $\rho $ is the density of the material. Then, using Eq. (4), we obtain where ${U}_{\text{abs}}$ is the energy absorbed by the material. Equations (3) and (5) describe the refractive index change at a given energy for the Kerr and TO effects, respectively. For the Kerr effect to be larger than the TO effect, the following condition is required:To achieve Kerr nonlinearity, we must complete our operation before this time, namely the time that the light is absorbed by the material, has passed. This simple picture is straightforward to understand. If we are to obtain a faster operation speed, we require a smaller ${\tau}_{\text{tot}}$ because it determines the rise and fall time of the light energy in the cavity. ${\tau}_{\text{tot}}$ is given by ${\tau}_{\text{tot}}^{-1}={\tau}_{\text{loss}}^{-1}+{\tau}_{\text{coup}}^{-1}+{\tau}_{\text{abs}}^{-1}$, where ${\tau}_{\text{loss}}$ and ${\tau}_{\text{coup}}$ are the photon lifetimes defined by the loss rate that do not contribute to the generation of heat and coupling to the waveguides. Since ${\tau}_{\text{abs}}$ and ${\tau}_{\text{loss}}$ are mainly determined by the material and the structure that we use, the only parameter that we can control is ${\tau}_{\text{coup}}$. This discussion suggests that optical Kerr operation is possible by controlling the coupling between the cavity and the waveguides, because it enables us to release light into the waveguides before it is absorbed by the material.

In the following section, we perform a rigorous analysis showing that Kerr operation is indeed possible by changing ${\tau}_{\text{coup}}$.

## 3. RIGOROUS MODELING OF THE OPTICAL KERR EFFECT AND THERMO-OPTIC EFFECT IN A TOROID MICROCAVITY

#### A. CMT in Whispering-Gallery-Mode Resonator

First we describe our master equation based on CMT in a whispering-gallery-mode (WGM) resonator to obtain the linear and nonlinear transmittance. The structure is shown in Fig. 1(a). Because a two-port system (a side-coupled cavity with one waveguide) makes the bistable operation difficult to observe, we focus on a side-coupled four-port system [29] throughout this paper. It consists of a toroid (ring) cavity and two waveguides for input and output light. A detailed discussion comparing side-coupled two- and four-port systems will be provided elsewhere [30]. Briefly, as discussed in Section 2, we need to make the coupling large in order to prevent heat accumulating in the system. However, the transmittance spectrum of a two-port system is shallower in an overcoupled configuration, which makes the switching contrast of the output light very low and difficult to distinguish. Hence, optical-bistable switching with high contrast is difficult to achieve in a two-port system in the presence of the TO effect.

The mode amplitude $a$ in the cavity is given as [31]

When we use a slowly varying envelope approximation, i.e.,

we can rewrite Eq. (10) as#### B. Modeling the Nonlinearities

To describe the nonlinear effects in our model, we take account of the nonlinear refractive index modulation caused by the Kerr effect ($\mathrm{\Delta}{n}_{\text{Kerr}}$) and the TO effect ($\mathrm{\Delta}{n}_{\mathrm{TO}}$) in the master equation. (Note that the carrier-plasma effect is negligible in silica due to its large bandgap.) The nonlinearities in an optical cavity result in a shift in the resonant wavelength because the optical path length changes. Thus, the shift of the resonant wavelength of a cavity is given as

where $\mathrm{\Delta}n(t)$ is the effective nonlinear refractive index change of the cavity. To add the nonlinear effects into Eqs. (11) and (16), we replace ${n}_{0}$ and ${\lambda}_{0}$ by ${n}_{o}+\mathrm{\Delta}n(t)$ and ${\lambda}_{0}+\delta \lambda (t)$, asNext, we describe how we calculated the nonlinear refractive index change $\mathrm{\Delta}{n}_{\text{Kerr}}$ and $\mathrm{\Delta}{n}_{\mathrm{TO}}$. We can directly calculate $\mathrm{\Delta}{n}_{\text{Kerr}}$ from Eq. (3). Taking the spatial dependency into account, we obtain,

The refractive index change $\mathrm{\Delta}{n}_{\mathrm{TO}}$, which is induced by the TO effect, is described as

The cross-sectional temperature distribution is calculated by using two-dimensional FEM (COMSOL Multiphysics). By setting the heat source in the dielectric cavity asFinally, we obtain the effective nonlinear refractive index change $\mathrm{\Delta}n(t)$ as

Now by solving Eq. (18) sequentially using Eqs. (17)–(24), we can obtain the light energy in the cavity ${U}_{p}(t)$ and the output powers ${P}_{\text{out}1}={|{S}_{\text{out}1}|}^{2}$ and ${P}_{\text{out}2}={|{S}_{\text{out}2}|}^{2}$.

#### C. Determining the Absorption and the Photon Lifetimes

The total photon lifetime ${\tau}_{\text{tot}}$ is defined as

Here we describe the photon lifetimes that we used in our analysis. The material absorption of silica at telecom wavelengths is usually very small ($\alpha =0.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}/\mathrm{km}$ [33]), but it is known that silica toroid microcavities have much larger absorption due to the water layer and contamination on their surfaces. And in practice the $Q$ factor is limited by these absorptions [9]. Thus, we decided to consider two cases, which we call an ideal case and a realistic (worst) case. In an ideal case, we assume that the absorption loss is determined by solely the material (this means that the absorption loss is extremely small) and the experimental $Q$ is limited by the losses to the outside of the cavity ${\tau}_{\text{loss}}^{-1}$. On the other hand, we assume that the experimental $Q$ is limited by the absorption loss ${\tau}_{\text{abs}}^{-1}$ in a realistic case. This implies the worst case in terms of thermal accumulation, because a large part of the incident light eventually turns into heat. Any result should be better than that obtained with this latter condition.

To consider those two cases, first we fix the intrinsic photon lifetime ${\tau}_{\mathrm{int}}={({\tau}_{\text{abs}}^{-1}+{\tau}_{\text{loss}}^{-1})}^{-1}$ at 329 ns (corresponding to ${Q}_{\mathrm{int}}=4\times {10}^{8}$ [34], where ${Q}_{\mathrm{int}}$ is the intrinsic $Q$), since this is the record largest experimental ${\tau}_{\mathrm{int}}$ yet obtained. In the ideal case we disregard losses other than the intrinsic material absorption; so we use ${\tau}_{\text{abs}}=164\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$ (corresponding to ${Q}_{\text{abs}}\simeq {Q}_{\text{mat}}=2\times {10}^{11}$ [35]) and ${\tau}_{\text{loss}}=330\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$; i.e., ${\tau}_{\mathrm{int}}^{-1}\simeq {\tau}_{\text{loss}}^{-1}\gg {\tau}_{\text{abs}}^{-1}$. On the other hand, we use ${\tau}_{\text{abs}}=329\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$ (corresponding to ${Q}_{\text{abs}}=4\times {10}^{8}$ [27,34]) and ${\tau}_{\text{loss}}=\infty $ for the realistic case. This ${Q}_{\text{abs}}$ value is the same as the highest experimentally obtained ${Q}_{\mathrm{int}}$, for the reason discussed above; i.e., ${\tau}_{\mathrm{int}}^{-1}\simeq {\tau}_{\text{abs}}^{-1}\gg {\tau}_{\text{loss}}^{-1}$.

Although ${\tau}_{\mathrm{int}}$ is determined by the material and structure, we can control ${\tau}_{\text{coup}1}$ and ${\tau}_{\text{coup}2}$ by adjusting the distance between the fiber and the cavity [36]. With this in mind, we conducted numerical simulations for various ${\tau}_{\text{coup}2}$ values. Note that ${\tau}_{\text{coup}1}$ is controlled to satisfy ${\tau}_{\text{coup}1}={({\tau}_{\mathrm{int}}^{-1}+{\tau}_{\text{coup}2}^{-1})}^{-1}$ and thus achieve critical coupling [24,31,37] between the cavity and the lower waveguide. In this condition, the power transmittance through the lower waveguide ${T}_{\text{out}1}={P}_{\text{out}1}/{P}_{\text{in}}$ decreases to zero on resonance, and high contrast can be obtained between two output states [30].

## 4. NUMERICAL CALCULATIONS

#### A. An Ideal Case: Small Material Absorption

In this section, we show that a Kerr bistable memory is easily feasible, without careful adjustment of ${\tau}_{\text{coup}}$, if only the inherent material absorption of silica is present.

First, we input a rectangular pulse to investigate the refractive index changes $\mathrm{\Delta}{n}_{\text{Kerr}}$ and $\mathrm{\Delta}{n}_{\mathrm{TO}}$. The result is shown in Fig. 2(a), where ${\tau}_{\text{coup}2}$ is equal to ${\tau}_{\mathrm{int}}$. It shows that $\mathrm{\Delta}{n}_{\text{Kerr}}$ is always larger than $\mathrm{\Delta}{n}_{\mathrm{TO}}$, which tells us that the influence of the absorption-induced thermal generation is nearly negligible. Hence the Kerr effect is easily obtained without it suffering from the TO effect.

Next, we input a triangular pulse to investigate the relationship between the input and output of the system. To allow us to charge and discharge the cavity gradually, we set the pulse rising/falling rate of the triangular inputs at $d{P}_{\text{in}}/dt=62.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nW}/\mathrm{\mu s}$. Figures 2(b) and 2(c) are plotted from the input–output response of a triangular input with different detuning values $\delta $. The lower coupling photon lifetime ${\tau}_{\text{coup}1}$ is set equal to ${({\tau}_{\mathrm{int}}^{-1}+{\tau}_{\text{coup}2}^{-1})}^{-1}$ to achieve critical coupling. When $\delta $ is greater than 20 fm, clear hysteresis is observed, which is direct evidence of optical bistability. Optical bistability is observed in the longer side of the wavelength detuning, because the Kerr effect increases the refractive index (shifts the resonance toward the longer wavelength). Note that we obtained a large contrast between the two bistable states in Fig. 2(b) because the cavity transmittance ${P}_{\text{out}1}$ falls to zero on resonance in the critical coupling [24].

Finally, we performed optical memory operations as shown in Fig. 2(d). Again, we set ${\tau}_{\text{coup}2}$ equal to ${\tau}_{\mathrm{int}}$ and the detuning $\delta $ equal to 27 fm. The solid curve is an input with a drive power ${P}_{\text{in}}^{\text{drive}}$ of 3.2 μW. The peak power of the 0.8 μs square set pulse is ${P}_{\text{in}}^{\text{set}}=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu W}$. To reset the system, we reduce the input power to ${P}_{\text{in}}^{\text{reset}}=2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu W}$ for a duration of 0.8 μs, which we call a reset pulse. The duration of the negative reset pulse must be longer than the discharging time of the cavity, which is equal to ${\tau}_{\text{tot}}$; otherwise the cavity does not reset. The set and reset pulses are input at $t=8$ and 20.8 μs, respectively. ${P}_{\text{out}2}$ (shown as the dashed curve) rises to high (ON) state when the set pulse is input. It maintains the ON state until the reset pulse is injected. After the reset pulse has been input, ${P}_{\text{out}2}$ drops to the low (OFF) state and maintains it. ${P}_{\text{out}1}$ [indicated by the dotted curve in Fig. 2(d)] shows the inverse behavior of ${P}_{\text{out}2}$. Figure 2(d) clearly shows optical memory operation, which is based on Kerr nonlinearity. Although the “memory holding time” demonstrated with this calculation [shown in Fig. 2(d)] is about 30 μs, it can be much larger, since ${P}_{\text{out}1}$ and ${P}_{\text{out}2}$ exhibit an almost plateau response due to the small material absorption.

#### B. A Realistic Case: Large Material Absorption

As shown in Section 4.A, the realization of a Kerr bistable memory is feasible without it suffering from the TO effect, even when the coupling is not large, if only the inherent absorption of silica is present in the cavity. In reality, however, other sources of absorption occur in a microcavity, such as surface absorptions caused by water and contamination. Thus, in this section, we use ${\tau}_{\text{abs}}=329\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$ to simulate a case with faster thermal generation. As discussed above, here we assume that the ${Q}_{\mathrm{int}}$ of the cavity is limited by absorption and not by losses to the outside of the cavity, since it appears to be the worst (but realistic case) for toroid microcavities. We employ ${\tau}_{\text{abs}}$, which is derived from the highest experimental ${Q}_{\mathrm{int}}$ (${\tau}_{\text{abs}}=329\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ns}$ corresponds to ${Q}_{\mathrm{int}}=4\times {10}^{8}$). If we can clarify the requirements for demonstrating a Kerr bistable memory under this condition, it is a significant step toward the experimental realization of a Kerr bistable memory in silica toroid microcavities.

First, in a similar way to that shown in Fig. 2(a), we employ a rectangular pulse to obtain the refractive index change $\mathrm{\Delta}{n}_{\text{Kerr}}$ and $\mathrm{\Delta}{n}_{\mathrm{TO}}$. The calculation results are shown in Fig. 3 for three different ${\tau}_{\text{coup}2}$ values (${\tau}_{\text{coup}1}$ is adjusted to satisfy ${\tau}_{\text{coup}1}^{-1}={\tau}_{\text{coup}2}^{-1}+{\tau}_{\mathrm{int}}^{-1}$). Figure 3 shows that $\mathrm{\Delta}{n}_{\mathrm{TO}}$ is larger than $\mathrm{\Delta}{n}_{\text{Kerr}}$ in all three cases when $t$ is larger than $\sim 2.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. This number gives us the upper limit of the Kerr memory holding time without the memory suffering from the TO effect. It also tells us that this number is insensitive to ${\tau}_{\text{coup}2}$. This result is consistent with Eq. (8) obtained from a simple model, where the equation is dependent on ${\tau}_{\text{abs}}$ but independent of ${\tau}_{\text{coup}}$. Figure 3 also shows the effect of the different charging speeds caused by different ${\tau}_{\text{coup}2}$ values. The cavity charging time is much faster for ${\tau}_{\text{coup}2}={\tau}_{\mathrm{int}}/100$, which allows the cavity to reach a plateau $\mathrm{\Delta}{n}_{\text{Kerr}}$ domain much faster. This enables us to have a longer “Kerr memory usable” regime, which allows us to use the cavity for longer as an optical Kerr memory. Figure 3 shows that we can maximize the Kerr dominant “Kerr memory usable” regime by setting ${\tau}_{\text{coup}2}$ as small as possible. Again, this result is consistent with that obtained from the simple analysis described in Section 2.

Next, we show how we can obtain Kerr optical bistability when the absorption is large. Figures 4(a) and 4(b) show the input and output power relationships (${P}_{\text{out}1}$ and ${P}_{\text{out}2}$) for different ${\tau}_{\text{coup}2}$ values when a triangular pulse is input. As shown in Figs. 4(a) and 4(b), a hysteresis loop is observed when ${\tau}_{\text{coup}2}={\tau}_{\mathrm{int}}/100$, but the loops are deformed when ${\tau}_{\text{coup}2}\ge {\tau}_{\mathrm{int}}/10$. This is because the light cannot charge and discharge the cavity quickly enough before the heat accumulates in the system when ${\tau}_{\text{coup}2}$ is large. Again, we obtained a clear hysteresis loop only when we made the coupling strong (i.e., ${\tau}_{\text{coup}2}$ is small).

Finally, the memory operation for various ${\tau}_{\text{coup}2}$ values is shown in Fig. 5. Since the response speeds of ${P}_{\text{out}1}$ and ${P}_{\text{out}2}$ depend on the total photon lifetime ${\tau}_{\text{tot}}$, we normalized the temporal axis $t$ by ${\tau}_{\text{tot}}$. Figure 5 shows clearly that the reset pulse does not work when ${\tau}_{\text{coup}2}={\tau}_{\mathrm{int}}$ and ${\tau}_{\mathrm{int}}/10$, and the memory operation cannot be obtained under this condition. If the system is operating in the Kerr dominant regime, we should be able to reset the state by injecting a negative reset pulse. The pulsewidth needed for the reset pulse is $>{\tau}_{\text{tot}}$, since we can discharge the cavity within this time. However, Fig. 5 shows that significant heat is accumulating in the system, which prevents the system from resetting because $\mathrm{\Delta}{n}_{\mathrm{TO}}$ cannot be reset by such a short negative pulse due to its much longer relaxation time.

On the other hand, when ${\tau}_{\text{coup}2}\le {\tau}_{\mathrm{int}}/100$, we can successfully set and reset the system, and use the device as a Kerr bistable memory. However, the TO effect cannot be eliminated completely even in this case, and thus there is a holding time. ${P}_{\text{out}1}$ is automatically switched from ON to OFF, or vice versa for ${P}_{\text{out}2}$, due to the thermal accumulation. Figure 5 shows that the memory holding time for the realistic case is about 500 ns. This value is inconsistent with the length of the “Kerr memory usable” regime shown in Fig. 3 since the thermal nonlinearity depends on the accumulation of ${U}_{\text{abs}}$ while the Kerr nonlinearity depends on the instantaneous ${U}_{p}$.

In this section we achieved Kerr bistable memory operation and obtained a sufficiently long memory holding time of 500 ns by allowing the system to charge and discharge quickly by adjusting ${\tau}_{\text{coup}2}$.

## 5. DISCUSSION: POWER CONSUMPTION

We showed in previous sections that Kerr bistable operation is possible by adopting a large coupling constant. The basic idea is to allow the light to charge and discharge before it turns into heat. However, stronger coupling with waveguides (i.e., a smaller ${\tau}_{\text{coup}2}$) results in a lower total $Q$, which decreases the photon density in a cavity and makes the nonlinearity small. Hence, there is a tradeoff between operating speed and operating power.

Here, we consider two measures for evaluating the loss of our system: ${Q}_{\mathrm{int}}$ and ${U}_{\text{cons}}$. ${Q}_{\mathrm{int}}$ is the figure of merit of a cavity, whose value gives the cavity loss, and ${U}_{\text{cons}}$ is the energy consumed for the operation. We often want to increase ${Q}_{\mathrm{int}}$ and reduce ${U}_{\text{cons}}$ to build a lossless system. Table 1 compares these values for different systems. A bistable memory based on an ultrahigh-$Q$ silica toroid microcavity with Kerr nonlinearity has a clear advantage over other schemes in terms of both losses. The lossless nature of this memory is the advantage of this system.

First, a system with a high ${Q}_{\mathrm{int}}$ yields a low loss, because ${Q}_{\mathrm{int}}$ corresponds to the fundamental loss characteristics of a cavity. As shown in Table 1, the ${Q}_{\mathrm{int}}$ of a toroid cavity is much higher than that of other types of cavities. Furthermore, we would like to note that both linear and nonlinear losses are significant in other systems, especially those that use TO and carrier-based nonlinearities to achieve bistability. This is because carrier generation is unavoidable. Free carrier absorption (FCA) significantly decreases both the ${Q}_{\mathrm{int}}$ and the transmittance of the system [38]. On the other hand, a Kerr bistable memory does not generate carriers and hence the system does not suffer from FCA loss.

Secondly, ${U}_{\text{cons}}$ is extremely low in our system because Kerr nonlinearity does not absorb photons. ${U}_{\text{cons}}$ is the energy consumed for a bistable operation. We can estimate ${U}_{\text{cons}}$ from the required refractive index change, which is needed to obtain the cavity resonance shift for the operation. The values are large in memories based on heat and carrier even though they have large nonlinear efficiencies, which is because they incorporate photon absorption. On the other hand, the total photon number for the Kerr effect remains unchanged after the operation. Only the wavelength (energy) of the photons changes, which results in an extremely low energy consumption. Hence ${U}_{\text{cons}}$ is almost negligible as shown in Table 1.

As shown in Fig. 5, we need ${P}_{\text{in}}^{\text{drive}}=7.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mW}$ to drive the system, which is significantly smaller than the value shown in [7], which uses the same Kerr nonlinearity. This is due to the high ${Q}_{\mathrm{int}}$ of our system. However, this value is still larger than the experimentally demonstrated value in a photonic-crystal nanocavity using carrier-based nonlinearity [15]. Since the coefficients of carrier nonlinearities in semiconductor materials are normally larger than the Kerr nonlinearity in ${\mathrm{SiO}}_{2}$, it is not an easy task to reduce the driving power of our system to the same level. However, glass has a low propagation loss [39,40], which is usually difficult to obtain with devices made of semiconductors.

Although the driving power ${P}_{\text{drive}}$ of our system is not necessarily the smallest, the linear, nonlinear, and consumption losses are significantly smaller than those of other devices. With this in mind, this device is attractive for applications that require high efficiency $\eta ={P}_{\text{out}}/{P}_{\text{in}}$, such as quantum information processing [18].

## 6. CONCLUSION

We rigorously modeled the Kerr and the TO effects in a silica toroid microcavity by combining CMT and FEM. We gained a clear understanding of the impact of adjusting the coupling, and showed that Kerr optical bistable memory operation is possible by adjusting the coupling between the cavity and the waveguides. The memory holding time was about 500 ns. Although the driving power was 7.3 mW, the energy consumed by the system was extremely low. This is because unlike other nonlinearities such as carriers of TO effects, Kerr nonlinearity does not absorb photons. In addition due to the ultrahigh-$Q$ of the system, the energy loss outside the system is also low. Our Kerr bistable memory in a silica toroid microcavity exhibits extremely low loss and is thus suitable for applications such as quantum signal processing.

## ACKNOWLEDGMENTS

This work is supported in part by the Strategic Information and Communications R&D Promotion Programme (SCOPE) from the Ministry of Internal Affairs and Communications, the Canon Foundation, the Support Center for Advanced Telecommunications Technology Research, Foundation, and the Leading Graduate School program for “Science for Development of Super Mature Society” from the Ministry of Education, Culture, Sports, Science, and Technology in Japan.

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