## Abstract

We demonstrate optical bistability in silicon using a high-*Q* (*Q*>10^{5}) one-dimensional photonic crystal nanocavity at an extremely low 1.6 *µ*W input power that is one tenth the previously reported value. Owing to the device’s unique geometrical structure, light and heat efficiently confine in a very small region, enabling strong *thermo-optic confinement*. We also showed with numerical analyses that this device can operate at a speed of ~0.5 *µ*s.

©2009 Optical Society of America

## 1. Introduction

Several approaches have recently been tried in order to create all-optical devices [1–3] that behave similarly to electronics components such as transistors. The interest in integrated all-optical logic devices has been triggered by the idea that the elimination of optical-to-electrical signal conversion will lead to the fabrication of very low-power signal processor systems. Therefore, it is reasonable to require an all-optical logic gate to be very small and to operate at very low power.

Optical bistability is a fundamental physical phenomenon that makes it possible to realize all-optical logic gates [4, 5]. It implies that the optical response of the component is nonlinear, thus the resonant wavelength and absorption depend on the optical power. There are many possible candidate techniques for achieving the nonlinearity, including carrier-plasma dispersion, the optical Kerr effect, saturable absorption, and the thermo-optic effect, which usually appear only when a high optical power density is attained. For this reason, optical bistable operation has been difficult to demonstrate at a reasonably low input power. However, the recent fabrication of micro- and nano-cavities, with favorable designs for integration, has enabled the operation of optical bistability at a reasonably low power [6–8]. This is due to the high quality factor *Q* and small mode volume *V* of the cavities, because the optical power density in the cavity scales with *Q*/*V*.

The two-dimensional (2D) photonic crystal (PhC) nanocavity is a good candidate for an optical bistable device, because some of the best values for *V* [9] and *Q* [10,11] have been obtained with this structure. Indeed, we have demonstrated a low-power optical bistable threshold power *P*
_{tr} of just 25 *µ*W using a 2D silicon (Si) PhC nanocavity [6]. The use of Si is challenging because its material parameters, including its two-photon absorption coefficient, thermo-optic coefficient and thermal capacity, make it more difficult to obtain low power bistability than with other materials. Similar experiments have been undertaken by some research groups using GaAs, and the smallest reported value of *P*
_{tr}=1 *µ*W was obtained by De Rossi *et al*. in a GaAs 2D PhC nanocavity [12]. However, because Si technology fabrication processes are now well known and widely used, and because the development of Si photonics will allow the on-chip integration of electronic and photonic devices, there is still a great interest in Si photonics, despite the material’s intrinsic disadvantages.

In this paper, we focus on Si material and demonstrate thermo-optic bistability at a significantly reduced Ptr by utilizing 1D PhC nanocavities to fulfill the criteria for low power optical processing. 1D PhC nanocavities have interesting geometrical and mechanical properties, [13–16] while their *Q* and *V* are comparable to those of the best 2D PhC cavities. Having fabricated 1D PhC nanocavities [17, 18], we thought it would be interesting to investigate their nonlinear bistable properties, because the structures are well isolated optically, electrically and thermally, and this may allow us to realize a significant reduction in *P*
_{tr}.

The next section begins by describing and discussing an optical bistability experiment in two types of 1D PhC nanocavities with different designs. A better understanding of the influence of the design of a cavity on its efficiency is then provided in §3 by undertaking a numerical analysis of the thermal properties of each design. Finally, the advantage of an 1D cavity compared with a 2D cavity is discussed in §4.

## 2. Optical properties of 1D PhC nanocavities

#### 2.1. Describing the structure

Figure 1 shows schematic illustrations and scanning electron microscope images of fabricated 1D Si PhC nanocavities. We call Fig. 1(a) a “stack” cavity on SiO_{2} and 1(d) an air-bridged “ladder” cavity. Figures 1(b) and 1(e) are scanning electron microscope images of the fabricated samples. Figures 1(c) and 1(f) are the mode profiles calculated by 3D finite-difference time-domain method. Stack cavities can be simply described as Si boxes laid on SiO_{2}, whereas in air-bridged ladder cavities, Si boxes are connected by two Si bridges, which allows the underlying SiO2 to be removed and the cavity to be suspended in air. The position and size of the Si boxes are modulated, which enables the creation of a mode gap and allows light to be confined [13]. The theoretical *Q* and *V* for these two types of cavities are; *Q*=1.9×10^{7} with *V*⋍2.0(*λ*/*n*)^{3} for a stack cavity [17, 18], and *Q*=2.0×10^{8} with *V*⋍1.4(*λ*/*n*)^{3} for an air-bridged ladder cavity [13]. The coupling between the cavity and the input/output waveguides is controlled by changing the number of boxes. A typical sample has about 30 boxes, which gives a total length of ~15 *µ*m. Further details of a numerical study and the fabrication of these cavities have been published elsewhere [17, 18].

We employed these two structures as candidates for ultra low power thermo-optic bistable devices because they should have superior heat confinement characteristics owing to the very small thermal conductivities of SiO_{2} and the air that surrounds the cavity.

## 2.2. Demonstration of low-power thermo-optic bistability

Our experiment for demonstrating thermo-optic bistability consists of sweeping a continuous-wave laser light from short to long wavelengths at a speed of 0.5 nm/s and measuring the spectral response of the cavity with a power meter. The cavity *Q* and power transmittance *T*
_{r} are measured at a very low input power (about 100 nW in the waveguide) to ensure that there is no nonlinear effect that can alter these values. The fabricated stack cavity exhibits a very high *Q* of 8.2×10^{4} (*T*
_{r}⋍3%) and an even higher *Q* of 2.5×10^{5} (*T*
_{r}⋍0.3%) is obtained for an air-bridged ladder cavity. As noted before, higher *T*
_{r} is also possible by reducing the number of boxes (at the cost of slightly lowering the high-*Q*).

The high concentration of optical energy in high-*Q* cavities results in non-negligible two photon absorption (TPA), which leads to the generation of heat through the relaxation of the TPA carriers. Thus the temperature *T* of Si increases. Since the refractive index *n* of Si is described as follows,

the resonant wavelength of a Si nanocavity should become longer when the input power increases (red shift). At an efficient high input power, the cavity resonance can lock to the wavelength of the input light, which leads to the modification of the measured transmittance spectrum. As regards thermo-optic bistability, it is well known that the resonance of the cavity follows the wavelength of the input and drops sharply at a certain wavelength when the input is swept from shorter to longer wavelengths. Such a sharp drop in the spectrum is direct evidence of the existence of optical bistability [4, 6]. We observed this spectrum shape in our cavities, and the results with the lowest *P*
_{tr} are plotted in Fig. 2

*P*
_{tr} is experimentally obtained by considering the lowest input power *P*
_{in} for which a sharp drop can be observed. *P*
_{in} is the power at the PhC waveguide. It is obtained by measuring the power at the input fiber and subtracting the coupling efficiency. The coupling efficiency is estimated by measuring the transmittance of a sample without a cavity (only a PhC waveguide).

In order to achieve high accuracy, the fiber aligner is automated, which enables an alignment reproducibility of less than 0.2 dB. Since we can obtain a hysteresis curve of the input and output power by increasing and decreasing the power at a fixed wavelength, we can determine *P*
_{tr} from the power threshold appearing in a hysteresis curve. However the value obtained by using this method is usually not as accurate as that obtained from the nonlinear spectra as we employed in this study. It is because the determination of Ptr by nonlinear spectra measurement is more robust to the temperature fluctuation, particularly when the linewidth of the cavity is very small.

In the stack cavity [Fig. 2(a)], *P*
_{tr} appears to be 35 *µ*W, giving a result comparable to that of 25 *µ*W obtained in our previous experiment in 2D PhC [6]. But the air-bridged ladder cavity gives a result that is 20 times lower; a drop is visible at an input power of exactly 1.6 *µ*W [Fig. 2(b)]. To the best of our knowledge, this is the lowest value ever reported for an optical bistable threshold in Si and is even comparable to the lowest value of 1 *µ*W in PhC obtained using GaAs [12]. From our experiment, the 1D air-bridged cavity appears to be a good candidate for the fabrication of optical bistable devices operating at an extremely low power, because of its unique geometrical structure that enables *thermo-optic confinement*.

## 2.3. Simple analysis of thermal properties obtained from experiment

Here we try to understand the property of thermo-optic confinement in two different types of 1D PhC nanocavities. Since the experiments were performed with two types of 1D nanocavities whose *Q*, *T*
_{r} and geometrical structures were different, we should discuss the effects of optical confinement (i.e. *Q* and *T*
_{r}) and thermal confinement (i.e. geometrical effect) separately.

To understand the physics involved, and in line with our experimental results, we accept that the wavelength shift *δλ* approximately scales with the energy in the cavity as follows,

A complete model of the physics will take account of such effects as free-carrier absorption and inherent linear absorption, as found in [19] and [20], and should make it possible to obtain a more accurate model for the calculation of *δλ*. However, as suggested by the experiment, the following study is undertaken using the simple relation Eq. (2). Theoretically, the wavelength shift needed to reach a bistable threshold is $\delta {\lambda}_{\mathrm{tr}}=\frac{\sqrt{3}}{2}\Delta \lambda =\frac{\sqrt{3}}{2}{\lambda}_{0}\u2044Q$, where Δ*λ* is the resonance width [4, 19]. Thus our simple model expresses the threshold power as,

where *r* is the ratio of the optical energy converted into thermal energy and *R*
_{th} is the thermal resistance of the cavity.

*Optical characteristic*: For the same *R*
_{th}, we should obtain a small *P*
_{tr} when the √*T*
_{r}
*Q*
^{2} product is large. This product is a figure of merit that reflects the optical confinement property of a cavity. In our experiment, this product is 3 times higher for an air-bridged ladder cavity than for a stack cavity, which means the air-bridged ladder cavity has better overall optical characteristics for exhibiting low power optical bistability. Indeed, an air-bridged ladder cavity has a higher unloaded *Q* of ~2.4×10^{5} compared with the unloaded *Q* of 8.2×10^{4} for a stack cavity.

*Thermal characteristic*: The thermal resistance quantifies the intrinsic ability of the cavity to convert stored energy into an effective wavelength shift, and its value has no clear dependence on *Q* or *T*
_{r}. An experimental estimation of *rR*
_{th} is given by $\sqrt{{T}_{\text{r}}}{Q}^{2}$. The results allow the next comparison: *rR*
^{ladder}
_{th}=10 *rR*
^{stack}
_{th}. The same global tendencies are reached when the results obtained with 30 different samples are averaged. This shows that the design properties of the air-bridged ladder cavity make the operation of thermo-optic bistability much easier. To confirm this, we performed a numerical analysis of the thermal properties of an 1D nanocavity.

## 3. Numerical analysis of thermal properties of 1D photonic crystal nanocavities

#### 3.1. Comparison of stack cavity and ladder cavity

The purpose now is to evaluate the thermal capacity of the cavity to convert the heat source into an effective temperature increase.

We use the following considerations in our elaboration of the numerical model. The calculations are made in the steady-state regime. TPA is considered to be a heat source located in a position where the electromagnetic field is intense. With the stack cavity, the heat source is assumed to be uniformly distributed in the three Si boxes at the center of the cavity, where the optical mode is located. This is sufficiently accurate because the carriers cannot diffuse outside the box. With the air-bridged ladder cavity, we also assume that the heat source is uniformly distributed at the center of the cavity. It should be noted that there is the possibility of the carriers diffusing. However, a numerical simulation of the carriers taking the surface recombination into account [21] shows that, even in this case, the free carriers (and therefore the heat source) remain in the majority located at the center of the cavity, which indicates that our approximations are sufficiently accurate. The differential equation of the model is the classic heat equation

with *k* being the conductivity of Si or SiO_{2}, and *p* the power source density (W/m^{3}). The source term is null except at the center of the device. Before solving this, we must choose appropriate boundary conditions. The temperatures of Si and SiO_{2} far from the cavity are assumed to be room temperature *T*
_{0}. Also, it can be confirmed numerically that the effect of air convection is totally negligible, and henceforth air will be considered a perfect insulator. A numerical simulation gives a temperature increase *δT* taken at the center, from which the thermal resistance *R*
_{th}=*δT*/Φ is deduced. Φ is the total power of the source.

Now, what can we expect? The stack cavity has one advantage in terms of achieving high thermal resistance. Because there is no physical connection between the Si boxes, heat can only be evacuated through the underlying SiO_{2}. By contrast, heat can escape via the bridge in the air-bridged ladder cavity. But the air-bridged ladder cavity also has an advantage; the cavity is completely surrounded by air, whereas the box type cavity is on SiO_{2}, and SiO_{2} is not as good an insulator as air. The question is which of these two factors has the most important effect.

To solve the equation, we modeled the structure using the 3D finite element method (FEM), and obtained a solution at the point of thermal equilibrium (COMSOL multiphysics). The numerical results are presented in Fig. 3, and show that the air-bridged cavity [Fig. 3(c)] confines heat twice as well as the stack cavity [Fig. 3(a)]. The thermal flux lines drawn in red confirm that a significant proportion of heat is evacuated by the Si bridge in the ladder cavity [Fig. 3(d)] and escapes into SiO_{2} [Fig. 3(b)] in the stack cavity. This clearly demonstrates and explains the conclusion of the thermal analysis. The air-bridged ladder cavity has better thermal properties, independent of its superior optical property (i.e. higher *Q*), because the cavity is suspended in air. The conjunction of these two properties made it possible to obtain the low *P*
_{tr} value. The thermal resistance depends on materials and interfaces, and also on geometrical parameters, as described in the next section.

## 3.2. Geometrical study of air-bridged cavity

Heat can only escape from the air-bridged ladder cavity by diffusing into the Si bridge. Intuitively, the thermal energy left in the cavity will be larger if the only path allowing it to escape is thinner. Mathematically speaking, when e is the bridge width, 2*l* the length of the cavity, *h* the slab thickness and *δ T* the temperature difference between the center and the edge of the cavity, Fourier’s law **j**
_{th}=-*k*
_{Si}∇*T* gives

where **j**
_{th} is the heat flux. Moreover, if *𝒱* represents the volume of the entire air-bridged cavity and *𝓢* its boundary, conservation of thermal energy gives,

from which we derive

Equation (7) predicts a linear increase in temperature with *e*
^{-1}. As this is an approximation that does not take all geometric details into account, we also performed 3D FEM calculations of the cavity thermal resistance for several e values, and the results are shown in Fig. 4 (black dots). Figure 4 also shows the thermal resistance versus width *e* curve (in blue) derived from Eq. (7), which reveals the good agreement between our simple model and the numerical results. A thin bridge is the key to achieving thermal confinement. We can see in Fig. 4 that if *e* is larger than 250 nm, the thermal resistance of the air-bridged ladder cavity is lower than that of the stack cavity. Since *e* is 96 nm in our case, we succeeded in achieving a lower *P*
_{tr} for air-bridged ladder cavity. In addition, Eq. (7) shows that a very long bridge would realize better insolation. We performed experiments with cavities whose lengths varied from 14.5 to 17 *µ*m. *P*
_{tr} decreases with *l*, thus reaching its minimum value for *l*=17 *µ*m. (*P*
_{tr}=63,6.3,2.5, and 1.6 *µ*W for *l*=14.6,15.4,16.2, and 17, respectively.) Although the fabrication of a very long thin bridge poses a challenge, we may be able to reduce the operating power even further by taking these factors into account.

Here we would like to make a brief comment about the accuracy of the model given by Eq. (7). As the inset of Fig. 3(d) shows, the heat straightly flows along the Si bridge without passing through the Si ladders. Therefore, the structure can be regarded as equivalent to two thin beams, which makes Eq. (7) a good approximation. On the other hand, when we separately consider the thermal resistance for the a and b region [Fig. 3(d) inset] and connect them in series, the total resistance is given as,

Although Eq. (8) seems to give a better fit of the first point, the resulting fit, shown by the green line in Fig. 4, is not as good as the one given by Eq. (7), which is due to the considerations about thermal flux discussed above.

## 4. Comparison with 2D PhC cavities

#### 4.1. Thermal resistance

It is interesting to compare the results below with those of our previous experiment using an air suspended 2D PhC [6]. The latter device has a thickness *h*=200 nm, the lattice constant a is 420 nm, and the air hole radius *r* is 0.275 *a*. The structure is depicted in Fig. 5(a). The optical parameters of this experiment can be summarized by *Q*=3.3×10^{4}, *T*
_{r}=40% and *P*
_{tr}=25 *µ*W. By using the same reasoning as in § 2.3, we consider the quantity ${\left(r{R}_{\mathrm{th}}\right)}_{2D}={\left({P}_{\mathrm{tr}}\sqrt{{T}_{\text{r}}}{Q}^{2}{n}_{T}\right)}^{-1}=0.25{\left(r{R}_{\mathrm{th}}\right)}_{\mathrm{ladder}}$. Then by performing a numerical simulation similar to that carried out in § 3.1, we find the thermal resistance of the 2D PhC cavity to be approximately 10 times less than that of the 1D ladder cavity. Figure 5(b), which shows the heat distribution for a 2D case, illustrates an obvious but a crucial fact; unlike the 1D case, heat diffuses in all the in-plane directions.

The chart in Table 1 summarizes important parameters and results for the three cavity types. The possibility of fabricating high-*Q* 1D photonic cavities, combined with better heat confinement than with 2D, explains our low *P*
_{tr} value.

## 4.2. Operation time

In the introduction we listed two criteria that must be met for all-optical components to be of practical use; small size and low operating power, and we have dealt with them in previous sections. However, operating speed is also an important feature of an all-optical device. An experimental measurement has shown that the effective thermal relaxation time *τ* is ~100 ns for a thermo-optic bistable switch fabricated in Si 2D PhC [6]. We used our numerical model to calculate *τ* and obtained almost the same value. First the heat source continuously excites the device until it reaches thermal equilibrium, and then the source is turned off to obtain the thermal relaxation dynamics of the cavity. We evaluate the relaxation time by considering the time needed for the temperature to be divided by *e*=2.718. Then we performed the simulation in an 1D air-bridged ladder cavity and this time obtained a value of 0.5 *µ*s. Figure 6 shows a logarithmic plot of the temperature decay for both cavities. Since the 1D cavity confines heat efficiently, due to the higher *R*
_{th}, it was expected to have a longer *τ*. At the cost of a slower operating speed (~×1/5), the 1D PhC nanocavity can operate at very low power (~×1/10) compared with 2D PhC nanocavities. However, the operating speed is still much faster than other types of thermo-optic switches, typically a fraction of a millisecond [22]. In our case, the small mode volume makes the relaxation time very short.

## 5. Conclusion

We demonstrated low input power bistable behavior using 1D PhC nanocavities. These cavities have a noteworthy property; they can confine both light and heat very efficiently. In the air-bridged ladder cavity, a high *Q* and a favorable design allowed us to reach an optical bistable threshold value lower than 1.6 *µ*W, which we believe to be the lowest reported value for Si. This result depends crucially on the sample design, particularly the width of the Si bridge (96 nm). We also note that we may be able to reduce the operating power further by taking advantage of the material parameters of, for example, InP [3] or GaAs [12, 23].

In addition, we estimated the speed of this switch to be about 0.5 µs, which is very fast for a thermo-optic device. Although this value still remains large compared with current electronics standards, we may be able to increase the speed by employing the carrier-plasma dispersion effect [7]. We can expect very low power carrier-plasma bistability, because the similar structures of the heat diffusion equation and free carrier diffusion equation indicate that the good thermal properties of the air-bridged 1D structure should be converted into good carrier confinement.

We believe this type of design will become more generally applicable for achieving viable all-optical signal processors.

## Acknowledgment

The first author thanks Prof. H. Benisty for his support, and NTT Basic Research Laboratories for financing his internship program.

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