This article overviews our recent studies of ultrahigh-Q and ultrasmall photonic-crystal cavities, and their applications to nonlinear optical processing and novel adiabatic control of light. First, we show our latest achievements of ultrahigh-Q photonic-crystal nanocavities, and present extreme slow-light demonstration. Next, we show all-optical bistable switching and memory operations based on enhanced optical nonlinearity in these nanocavities with extremely low power, and discuss their applicability for realizing chip-scale all-optical logic, such as flip-flop. Finally, we introduce adiabatic tuning of high-Q nanocavities, which leads to novel wavelength conversion and another type of optical memories.
© 2007 Optical Society of America
Recently, there has been rapid progress in terms of the cavity quality factor (Q) of miniature-sized optical micro-resonators, such as whispering-gallery-mode cavities  and photonic-crystal (PhC) cavities [3,4, 5, 6, 7]. Of these, PhC cavities have been considered the most advantageous in terms of Q per unit mode volume (V), that is, Q/V, because confinement by the photonic bandgap (PBG) is the most efficient way to confine light in a wavelength-scale volume. Q/V appears in various situations in optics relating to light-matter interactions, and is directly related to the photonic density of states and also to the field intensity (photon density) in a cavity per unit input power. Therefore, if large Q/V cavities were to be realized, various light-matter interactions (which are generally very weak compared with interactions governing electrons) would be greatly enhanced. For example, spontaneous emission rate is enhanced by Q/V (Purcell effect) as a result of the modification of the density of states. Most of optical nonlinear interactions are also enhanced by the field intensity enhancement and long photon lifetime. Extensive studies are being conducted for this purpose, including spontaneous emission control [4,8, 9, 10, 11], and solid-state cavity quantum electrodynamics . This feature is especially important for nonlinear-optic applications, since most optical nonlinear interaction is too weak for practical applications. In addition, large Q/V cavities enable us to employ a novel light-matter interaction of light based on the adiabatic tuning of optical systems, as described later. In addition to their large Q/V, PhC cavities have another distinctive feature compared with other types of cavities. That is, they are highly suited to integration. It is not difficult to integrate many cavities in a tiny chip, and they can be coupled with each other or connected via single-mode PBG waveguides. The flexibility of various coupling forms and the precise controllability of the coupling strength distinguish them from other cavities. We believe that all of these features of PhC cavities make them particularly important for all-optical processing applications in an integrated form. All-optical integration for optical processing has a long research history, but certain fundamental difficulties still remain. We can summarize these difficulties as follows: 1) the circuits require too much power, 2) they are difficult to integrate, and 3) have poor functionality. We believe that PhC-nanocavity-based systems have the potential to overcome these problems.
In this review article, we aim in particular to describe recent progress on PhC cavities and their applications to optical nonlinear control and novel adiabatic control, with a view to convincing readers of their potential as a breakthrough for optical integration. First, we show the latest status of the performance of our ultrahigh-Q PhC nanocavities by using the spectral-and time-domain analysis. We also report the achievement of slow-light propagation in these ultrahigh-Q nanocavities, and discuss the issue of the nanocavity size disorder, which is of great practical importance as regards this system. Second, we employ these cavities for all-optical switching and memory operations based on optical nonlinearity, in which the driving power (energy) is substantially reduced thanks to large Q/V. In the third part, we discuss the possibility of constructing on-chip optical logic based on these bistable nonlinear PhC-nanocavity elements. In the final part, we introduce a novel adiabatic tuning of micro-optical systems with a long photon dwell time, and discuss another form of optical memory based on a pair of PhC nanocavities, where we dynamically change Q by employing adiabatic control of nanocavities.
2. Ultrahigh-Q PhC nanocavities
2.1 Realizing high-Q in 2D PBG systems
As described in the introduction, a large Q/V is one of the most important and promising features of PhC cavities. Conventional optical cavities are always limited by the fundamental trade-off between Q and V -1, but, in principle, PBG cavities do not involve trade-off between Q and V -1. Contrary to this naive expectation, the realization of high-Q and simultaneously small-V cavities in PhCs did not prove easy for two reasons. First, it remains extremely difficult to realize sufficiently-good 3D PBG cavities. Second, if we employ a 2D PBG to realize a high-Q cavity, light easily leaks in the vertical direction where there is no PBG. Owing to this leakage, 2D PBG cavities actually suffer from a Q-V -1 trade-off. At first, it was believed that 3D PBG cavities were essential to overcome the Q-V -1 trade-off. However, this tuned out to be untrue. The vertical leakage can be substantially suppressed by appropriately designing the momentum (k-) space distribution of cavity modes in the 2D plane [13, 14]. The strategy is very simple. If the cavity mode is concentrated outside the light cone of air in the 2D k space, the cavity mode cannot be coupled to the radiation modes. In fact, there are many ways to achieve this situation so, as evidenced by studies of many researches in this area. Here we introduce two of our design examples of ours. The first is a cavity based on a single-missing-hole line defect with local width modulation. The second is a cavity based on a single-missing-hole point defect having a hexapole mode.
2.2 Ultrahigh-Q width-modulated line-defect nanocavities and ultrahigh-Q measurements
If we terminate a PhC line-defect waveguide (shown in the left panel of Fig. 1(a)), it forms a cavity. This is something similar to the formation of conventional Fabry-Perot cavities because we fold back propagating waves to form standing waves. If we start from a theoretically lossless waveguide, the design requirement is simply to reduce the effect of this termination to keep the original lossless mode profile in the k space. Our latest design for this strategy is shown in Fig. 1(a), in which a lossless line-defect waveguide is not abruptly terminated but the positions of several holes along the line defect are locally shifted toward the outside to create in-line light confinement. The required hole shift is generally very small, typically several nanometers. In other words, here we locally modify the position of the mode gap of the line-defect waveguides to create the confinement. We proposed this idea before , but the design at that time was not optimized for high Q. The latest design enables us to realize spatially gradual confinement which is effective in preserving the original localized mode distribution in the k space of the starting waveguide. The basic mechanism is similar to that used in hetero-structure cavities where the lattice constant of the background PhC is altered . In our case, we introduced only local modification of the background PhC, which is suited for integration. The use of the mode gap for creating cavities was also reported in different designs. We numerically examined this type of cavities using the finite-difference time-domain (FDTD) method, and found that after optimizing the hole shift values, the theoretical Q is higher than 108 and the mode volume is 1.1~1.7(λ/n)3 where n is the refractive index.
We fabricated this type of cavities coupled to input/output waveguides in silicon PhC slabs by electron-beam lithography and dry etching. Figure 1(b) shows the transmission spectrum of the sample, which exhibits extremely sharp resonance as a result of resonant transmission via the cavity. The measured transmission width is as narrow as 1.2 pm, which corresponds to a Q value of 1.3 million.[6,17] As is clear from its definition, Q can be also deduced from independent time-domain measurements, which become more accurate as Q becomes higher. We performed time-domain ring-down measurements to deduce the cavity Q for the same cavity.[6,17] This method, in which we abruptly switch off the CW input and monitor the temporal output from the output waveguide, is the most accurate way to determine the photon lifetime of a cavity . If we have a linear Lorentzian cavity, we expect single exponential decay whose time constant is τ=Q/ω. Figure 1(c) shows ring-down measurement results. The deduced photon lifetime is 1.1 ns.
With such small and high-Q cavities, both of spectral and time-domain measurements are easily perturbed by small fluctuations in the environment or samples and this may limit the accuracy and reproducibility. Thus, we made a substantial effort to confirm the accuracy and reproducibility of our Q estimation. We performed a series of measurements for the same cavity to clarify the reproducibility and statistical error of our measurements. As a result, we found that the photon lifetime τ ph=1.07±0.05 ns for 12 independent spectral-domain measurements and τ ph=1.12 ±0.07 ns for 16 independent time-domain measurements, which directly proves that both measurements provide good accuracy and reproducibility. In addition, we systematically checked the correlation between the spectral and time-domain measurements, and confirmed that both methods give us approximately identical results (Q and τ ph) as long as Q>105. When Q<105, the response time of our detector limits the resolution of the time-domain measurement (which is 70 ps). Details of the accuracy and resolution of these measurements can be found in . We have recently measured the transmission spectra of ultrahigh-Q cavities using a single-side-band frequency shifter, by which offers further better spectral resolution than a conventional tunable laser.  We obtained essentially the same spectrum with both methods. All of these experimental results clearly prove the accuracy of our Q measurements.
2.3. Ultrahigh-Q hexapole-mode point-defect nanocavities
If a cavity possesses symmetrical multi-nodes in the 2D plane, the vertical radiation perpendicular to the plane will be reduced by destructive interference. This mechanism was originally proposed for 1D PhCs , and then applied to quadrupole modes in 2D square PhCs . Some years ago, we further extended it to hexapole modes in 2D hexagonal PhCs. We assumed that hexagonal PhCs would be most suited for realizing a high-Q cavity because it has the largest 2D PBG (in TE polarization), and the hexapole mode is the most symmetric multi-node mode in hexagonal PhCs (to be rigorous, this restriction will be relaxed if we employ aperiodic lattices, such as photonic quasicrystals that can have any rotational symmetry ). After some numerical calculations, we found that theoretical Q is over a million with V~(λ/n)3 . Figure 2(a) shows the structural design of this cavity. Six nearest-neighbor holes are shifted toward the outside from a point defect. This modification makes hexapole modes to be located in the middle of the PBG. Interestingly, this particular cavity has a strange characteristic as regards the waveguide coupling. It shows null waveguide coupling if it is side-coupled or in-line end-coupled to the waveguide. Thus, it took us some time to find appropriate structures (the answer is off-aligned end-coupling, as shown in the figure) for the experimental verification of their high-Q . Very recently, we succeeded in measuring the Q value of the sample shown in Fig. 2(a) . Figure 2(b) shows the transmission spectrum of the hexapole-mode cavity through the input waveguide to the output waveguide. We observe a sharp resonance peak at 1547.52 nm with a width of 4.8 pm, which corresponds to a Q of 3.2×105. Figure 2(c) shows a ring-down measurement result. The deduced lifetime is 300 ps, which leads to a Q of 3.65×105. This is sufficiently close to the value we deduced from the spectral domain measurement. This Q is the largest value reported for point-defect type cavities, as far as we know.
2.4 Slow-light application of nanocavities
Recently, slow-light media, in which the group velocity of light is greatly reduced, have attracted much attention . They are considered to be possible candidates for optical buffer memories/quantum memories, and they are also expected to be efficient tools for the huge enhancement of light-matter interaction. We have reported a reduction in the group velocity to approximately c/100 in W1 PhC waveguides (W1: a single-missing-hole line defect without adjusting the width in a hexagonal PhC) owing to their huge dispersion in the vicinity of the mode edge.[26,15] However, it is difficult to slow down the pulse propagation in W1 waveguides since these waveguides have too much group-velocity dispersion (GVD). Recently, we performed pulse propagation experiments using dispersion-managed slow-light PhC waveguides, and observed a group delay of 180 ps. There are several ways to reduce the GVD for slow-light PhC waveguides. One of the simplest ways is to employ a cavity to delay the pulse. Generally a cavity has a Lorentzian spectral response, which leads to a cosine-like phase response. It is easily shown that the group delay of a single cavity is 2τ ph (=2Q/ω) and simultaneously GVD=0 at the resonance frequency. Thus, a cavity produces a substantially large group delay with zero GVD if Q is high. Coupled-resonator optical waveguides (CROWs) have the same feature only except that the delay is multiplied by the number of the cavities. Another important issues is that the resultant group velocity should be scaled to the cavity size. Thus, an ultrahigh-Q and simultaneously ultrasmall cavity is a good candidate for slow-light media.
With such features in mind, we performed pulse transmission experiments using our ultrahigh-Q nanocavities based on width-modulated line-defects [6, 17]. Figure 3 shows the experimental setup and results. We observed a group delay of 1.45 ns by comparison with the output from the reference straight PhC waveguide. From this value we estimated the group velocity of this pulse to be 5.8 km/s, which is approximately c/50,000. To the best of our knowledge, this is the smallest group velocity ever reported for all-dielectric slow-light media. Note that this group velocity was obtained via direct pulse transmission experiments. In the past, the group velocity has been obtained in indirect ways (such as an interference method) for many of the all-dielectric slow-light waveguides. Both of the small footprint and high-Q contribute to this small group velocity, and thus this result clearly demonstrates one of the advantages of ultrahigh-Q nanocavities.
Although the above result shows promising potential of ultrahigh-Q nanocavities in slowing light, there are still many things to be overcome considering the real applications. This single cavity is not very practical, because it can delay the pulse by approximately the same length as the input pulse length. However, if we cascade a number of cavities to form a CROW, we can increase the group delay or extend the bandwidth. In terms of the delay (not the group velocity) bandwidth product, apparently cascaded long devices are more advantageous than a single cavity. In terms of the group velocity itself, the above result gives us a very rough estimate of the lower limit for the achievable group velocity in CROWs based on the same cavity. Concerning the transmission intensity, there is a trade-off with the group delay because higher loaded Q means low transmittance and longer group delay. In practice, the transmission loss may limit the degree of cascadability. Currently, we are investigating coupled resonator structures based on the similar cavities for slow-light investigation.
2.5 Disorder issues with waveguides and cavities
As described above, experimental Q is always smaller than the theoretical Q with our high-Q PhC cavities. We believe this difference to be due to the disorder-induced scattering in fabricated samples. Before discussing disorder issues with cavities, we briefly summarize the disorder issue for PhC waveguides. Recently, the propagation loss of PhC waveguides has been greatly reduced. We carefully studied this problem both experimentally and theoretically, and found that a disorder-induced scattering process dominates the propagation loss of fabricated PhC waveguides . Figure 4 shows our latest record as regards propagation loss measured for W1 PhC waveguides . It shows a pronounced wavelength dependence that has been well explained by theory , and the lowest loss is 2dB/cm which is the lowest value for a single-mode PhC waveguides. A rough estimate of the disorder in terms of the RMS of the width fluctuation is less than 2 nm, which is consistent with the scanning electron microscope observation.
Considering the fact that ultrahigh-Q line-defect cavities are based on the same W1 waveguides and are fabricated by the same lithography-and-etching process, it is naturally expected that Q is primarily limited by the lowest propagation loss of W1 waveguides. However, this is not true. If we assume a loss of 2dB/cm and a group refractive index of ng=6, the estimated photon lifetime is shorter than 300 ps. Thus, Q should be limited to below 4×105. This will be something of an underestimation because the loss should be much larger if the group index is larger than ng=6 , which must be the case for the line-defect cavity mode. Note that this cavity mode is located significantly close to the mode gap of the W1 waveguide where the group index should be substantially large. (In other words, the line-defect cavity mode is based on slow-light modes in the W1 waveguide.) Contrary to this estimation of the photon lifetime from the waveguide loss, we observed much longer photon lifetime (1.1 ns) for fabricated cavities, as described in the previous section. This means that cavities are much less sensitive to disorder than waveguides, at least in the present situation. We have not yet investigated this issue in detail, but we believe that the propagation loss of PhC waveguides is dominated by disorder with a sufficiently long correlation length, and therefore the same disorder does not affect the cavity Q. Another issue worth pointing out is the effect of backscattering. The backscattering can be significant in a PhC waveguide  but may not be so in a cavity, which could contribute to the difference in loss.
We have numerically investigated the effect of disorder on the cavity Q using the 3D FDTD method. We assumed a set of random distributions (Gaussian) in terms of the hole radius for all the air holes in the PhC cavities, and calculated Q with the standard statistical method. We performed this calculation for three different cavities, namely a width-modulated line-defect cavity (cavity A) with Q=4.2×106, a hexapole-mode point-defect cavity (cavity B) with Q=1.8×106, and a five-point end-hole-shifted cavity (cavity C) with Q=2×105 . Figure 5 summarizes the results. If the size variation is large, all the cavities have practically the same Q. However, if the size variation is less than 5 nm, there is a large difference between different cavities. For PhC waveguides, we roughly estimated that the width variation is less than 2 nm. If we use the same value for the radius variation, the Q values for the disordered cavies are Q=1.5×106, Q=5×105 and Q=1×105 for cavities A, B, and C, respectively. In fact, these values are not so different from the experimentally observed Q values for these cavities (1.3×106, 3×105 and 0.9×105). Although the estimation of the radius variation is very crude, we can guess that the experimentally observed Q for our PhC cavities is limited by the hole radius variation. It is worth noting that as long as the variation is sufficiently small, a higher theoretical Q leads to a higher experimental Q.
3. All-optical switching and memory
3.1 Nonlinear switch based on high-Q nanocavities
As has been studied in various forms, all-optical switches can be realized using optical resonators, where a control optical pulse induces a resonance shift via optical nonlinear effects. For such a resonator-based switch, there is a two-fold enhancement in terms of the switching power if a small cavity with a high Q is employed. First, the light intensity inside the cavity should be proportional to Q/V. Second, the required wavelength shift is proportional to 1/Q. In total, the switching power should be reduced by (Q2/V), which can be significantly large for PhC nanocavities. Although the switching mechanism itself is basically similar to that of previous resonator-based switches, such as nonlinear etalons,  this large enhancement has had an important impact on optical integration since most optical switching components require too much power for realistic integration. In addition, resonator-based optical switches are well known to exhibit optical bistability, and thus they can be used for optical memory and all-optical logic. Such functionality is one of the most important functions missing from existing photonic devices. Thus, we believe that all-optical bistable switches based on PhC cavities are important candidates for future optical integration. Besides high Q/V cavities, we can expect enhancement of nonlinearity using slow-light media as well. Application of slow light for all-optical switching and logic has been reported by Asakawa’s group. 
3.2 Bistable operation by thermo-optic nonlinearity
First, we investigate an all-optical bistable switching operation employing the thermo-optic nonlinearity induced by two-photon absorption (TPA) in silicon. It is worthwhile to note that silicon is not an efficient nonlinear material in comparison with III/V semiconductors. For this study, we designed an end-hole shifted four-point PhC cavity (shown in Fig. 6)  having two resonant modes, one of which we used for a control (mode A) and the other for a signal (mode B). The injection of the control light (mode A) with appropriate detuning (δA) pulls in the mode A as a result of the nonlinear shift of the index in the cavity, and the mode A is switched to ON state. This type of switching using a resonator is known to exhibit bistability . Simultaneously, we inject the signal light (mode B) with another detuning (δB). The mode B shifts as a result of the index change induced by the bistable switching in the mode A. In total, the output signal light shows bistable switching by varying the input control light. The condition for both detuning is shown in the lower-left panel in Fig. 6. Note that we can select switching parity (OFF to ON or ON to OFF) by selecting δB. The right panel in Fig. 6 shows the output power for mode B as a function of the input power for mode A, which exhibits an apparent bistable switching behavior for two different detuning conditions. This operation is basically what we expect for so-called all-optical transistors, and will be basis for various logic functions. The detail of this operation is described in . For example, we demonstrated that we can amplify an AC signal using this device. The most noteworthy point regarding this switching is its switching power, which is as small as 40 µW. This value is remarkably smaller that of bulk-type thermo-optic nonlinear etalons (a few to several tens mW)  and also smaller than that of recent miniature-sized thermo-optic silicon micro-ring resonator devices (~0.8 mW). In addition, TPA occurs only in the cavity, and therefore we can easily integrate this device with transparent waveguides in the same chip. Although the bistable operation itself is similar to that of nonlinear etalon switches, these PhC switches can be clearly distinguished in terms of the operating power and capability for integration. The mode volume of this cavity is only approximately 0.1 µm3. This small footprint is of course advantageous for integration, but it is also beneficial for reducing the switching speed because our device is limited by the thermal diffusion process. The relaxation time of our switch is approximately 100 ns, which is much shorter than that of conventional thermo-optic switches (~msec).
3.3 Bistable operation by carrier-plasma nonlinearity and memory action
These thermo-optic nonlinear bistable switches clearly demonstrate that large Q/V PhC cavities are very effective in improving the operation power and speed. However, the speed itself is still not very fast, which is limited by the intrinsically slow thermo-optic effect. To realize much faster all-optical switches, here we employ another nonlinear effect, namely the carrier-plasma effect . This process is also based on the same TPA process in silicon. Thus, most of the arguments concerning their advantages are similar to 3.2. For this experiment, we used basically similar PhC cavity devices with a control pulse input. If the duration of the control pulse is sufficiently short, we can avoid thermal heating and may be able to observe only carrier-plasma nonlinearity. In fact, we observed a clear blue shift in the resonance when we injected a 6-ps pulse into this device, which is consistent with the expected shift induced by carrier-plasma nonlinearity. Figure 7 shows the time-resolved output intensity for the signal mode when a 6-ps control pulse is input . We observed clear all-optical switching from OFF to ON (ON to OFF) for the detuning of 0.45 nm (0.01 nm). The required switching energy is only a few hundred fJ, which is much smaller than that of ring-cavity-based silicon all-optical switches. In addition, numerical estimations showed that the carrier relaxation time (which limits the switching speed of this device) is approximately 80 ps. This relaxation time is greatly shorter than the conventional carrier lifetime in silicon (~µs). The model simulation tells us that the diffusion process in our tiny devices is significantly fast, and thus the relaxation time is determined by the fast carrier diffusion time not by the carrier recombination time. Note that this short carrier relaxation time is much shorter than that in other silicon photonic micro-devices.  That is, the small footprint of the device is again effective in improving the operating speed.
In the same way as thermo-optic switching, carrier-plasma switching also provides bistable operation. Figure 8 shows bistable operations realized by employing a pair of set and reset pulses. When a set pulse is fed into the input waveguide, the output signal is switched from OFF to ON and remains ON even after the set pulse exits (green curve). When a pair of set and reset pulses is applied, the output is switched from OFF to ON by the set pulse and then ON to OFF by the reset pulse (blue curve). This is simply a memory operation using optical bistability. The energy of the set pulse is less than 100 fJ, and the DC bias input for sustaining the ON/OFF states is only 0.4 mW. These small values are primarily the results of the large Q/V ratio of the PhC cavity. It is worth noting that the largest Q/V should always result in the smallest switching power, but the operation speed can be limited by Q. In the present situation, the switching speed is still limited by the carrier relaxation time, and thus a large Q/V is preferable. In the case when the photon lifetime limits the operation speed, we have to choose appropriate loaded Q for the required speed. Even in such a case, it is better to have high unloaded Q because loaded Q can be controlled by changing the cavity-waveguide coupling, and high unloaded Q means low loss of the device. The best design of out device would be a device with the smallest volume, the lowest transmission loss, and the designated loaded Q (depending on the operation speed). The lowest loss with the designated loaded Q can be obtained only when we employs an ultrahigh unloaded Q cavity.
Compared with other types of all-optical memories, this device has several advantages, such as small footprint, low energy consumption, and the capability for integration. The fact that all the light signals used for the operation are transparent in waveguides is important for the application, which is fundamentally different from bistable-laser-based optical memories.
3.4 High speed operation
As described above, although carrier-induced nonlinearity is generally considered to be a slow process, the present all-optical switches based on carrier-induced nonlinearity can operate at significantly high speed. In fact, we have recently demonstrated the 5GHz operation of all-optical switching as shown in Fig. 9. In this demonstration, a 5GHz clock signal (A) is modulated by a random bit stream (B) using a PhC nanocavity switch (similar to that used in 3.3). In the case for the detuning of 0.06 nm, the device operates as a “NOT” gate, and the resultant output is NOT of A and B. In the case for the detuning of -0.2 nm, it operates as an “AND” gate, and the resultant output is AND of A and B.
If we wish to increase the operation speed further, we have to decrease the carrier relaxation time. To do this, we have recently employed an Ar-ion implantation process in order to introduce extremely fast non-radiative recombination centers into silicon. If the carrier recombination time becomes faster than the diffusion time, we can expect an improvement in the operation speed. When we implanted silicon PhC nanocavity switches with Ar+ dose of 2.0×1014 cm-2 and an acceleration voltage of 100 keV, we observed a significant improvement in switching speed. In the case of detuning for an AND gate, the switching time was reduced from 220 ps to 70 ps. In the case of detuning for an NOT gate, it was reduced from 110 ps to 50 ps. The detail has been reported elsewhere. 
4. Towards all-optical logic
4.1 Flip-flop operation by double nanocavities
In the previous section, we showed that a single PhC cavity coupled to waveguides functions as a bistable switch or a memory. If we couple two or more bistable cavities, we can create much more complex logic functions, in the same way as with transistor-based logic in electronics. As an example, here we show our numerical design for an all-optical SR (set and reset) flip-flop consisting of two bistable cavities integrated in a PhC. Asakawa’s group proposed different type of flip-flop operation using symmetric Mach-Zehnder switches implemented in PhCs.
It has been proposed that all-optical flip-flops be realized by using two nonlinear etalons with appropriate cross-feedback, but this proposal is unsuitable for on-chip integration. Here we propose a different design using two PhC nanocavities. Figure 10(a) shows an actual design implemented in a 2D PhC and Fig. 10 (b) shows a schematic of the design concept. Each of two bistable cavities (CvR and CvS) has two resonant modes (lower and upper modes) and one of them (lower mode) is common for two cavities (Fig. 10 (d)). Each cavity exhibits bistable switching, and we set the bias input for the lower mode at the OFF state in the bistable regime with appropriate detuning as shown in the left panels of Fig. 10(d). At such condition, we can switch each cavity to the ON state by injecting a light pulse closely resonant to the upper mode (CS or CR). The dotted vertical lines in the right panels of Fig. 10(d) schematically shows appropriate detuning required for the three inputs (B, CS, and CR). Each operation is equivalent to bistable switching using two wavelengths described in Fig. 6. The crucial point is that here we introduce cross-feedback between these two bistable cavities. The cross-feedback is introduced by making two cavities coupled to the same input waveguide. Therefore, two cavities share the same single CW bias input (B) at λB for achieving their own bistable operation, which leads to the cross-feedback. That is, if one cavity is switched to ON, then the bias input for the other cavity is reduced. This leads to flip-flop operation, as we will describe below.
To realize required operation, there are some essential points to this design. First, two nanocavities are located very close to each other, but they are decoupled because the parity of the two cavities is different. This is advantageous for reducing the size. Second, the input and output waveguides have specific transmission windows by which we can selectively couple each cavity mode to a different waveguide channel. This simplifies the system very much because we do not need additional wavelength filters.
Next, we explain the operation sequence. As described, both cavities have two resonant modes. The common lower mode is used for the CW bias input (B). The other upper modes are used for the control set pulse inputs (CS and CR) for each of cavities. CS and CR are close to resonant to CvS and CvR, respectively. Fig. 10(g) explains the operation sequence in terms of three inputs (B, CS, and CR) and two outputs (Q and Q̄). Suppose that both cavities are initially in the OFF state. First, we send a set pulse CS, the cavity CvS is switched to ON and it remains ON. Then, we send a set pulse CR, then the cavity CvR is switched ON and simultaneously cavity CvS is switched down to OFF because the DC input (B) is now shared by two cavities and this is insufficient to hold the ON state of cavity CvS. Next, we send a pulseCS, then cavity CvS is ON and cavity CvR is OFF. This is nothing but a typical SR flip-flop operation. Note that this operation is equivalent to conventional SR flip-flop in electronic circuits as shown in Fig. 10(c).
We implemented this design in a 2D hexagonal air-hole (2r=0.6a) PhC slab (n eff=2.8) with a=400 nm. We employ relatively long (seven-point-defect) end-hole shifted cavities  as shown in Fig. 10(e), and set the first-order mode in CvS and the second-order mode in CvR to have almost the same resonant wavelengths at 1620.80 nm and 1620.88 nm (lower modes). Therefore, these two cavities share the same resonant wavelength, but the mutual coupling is sufficiently reduced. For S and R, we use the third-order modes (1563.61 nm and 1578.52 nm) in CvS and CvR, respectively (upper modes). For adjusting the position of the modes , we varied the width (w) of both cavities by -0.02a and +0.018a for CvS and CvR, respectively. Next, we design the waveguides. B should exit only from Q and Q̄. S and R should exit from B. For this requirement, we employ three different waveguides that have a different transmission window. WG1 is a W1 waveguide that transmits all the resonant modes in cavities. WG2 is a W3 waveguides filled with five holes in the core as shown in Fig. 10(f), which transmits only lower modes (~1621 nm) and rejects other upper modes. WG3 is a modified W1 whose width is narrowed by 0.06a. WG3 transmits two upper modes, but reject lower modes. Thus they meet our requirement. Finally, we adjust the coupling between waveguides and cavities by adjusting the distance and the size of end holes. The resultant Qs are 1000–3000 for all modes.
We numerically simulated this operation using the 2D FDTD method assuming Kerr nonlinearity.  The detuning is set at +2.5 nm, respectively. Figure 10(h) shows the simulated output for Q and Q̄, which shows expected SR flip-flop operation at a repetition rate of approximately 44GHz. The intensity profiles show snapshots obtained at different times. Although this design is not yet optimized (for example, the output intensity is not constant for the Q=1 state) and thus the operation quality is still poorer than that of the electronic counterpart, the present result demonstrates that flip-flop operation is possible by using double bistable cavities appropriately coupled to waveguides in a PhC platform. Note that if we have an SR flip-flop, we can realize various much complex logic processing based on it.
4.2 Retiming circuit based on Flip-flop operation
A typical example of the flip-flop operation in the high-speed information processing is a retiming circuit, which corrects the timing jitter of an information bit stream and synchronizes it with the clock pulses. This function is normally accomplished by high-speed electronic circuits, but if it can be done all-optically, it will be advantageous for future ultrahigh-speed data transmission. Although this operation is basically possible by cascading several SR flip-flops, here we propose another much simpler design for realizing the retiming function.
Figure 11(a) shows a design for the retiming circuit. Its operation principle detailed in our previous report . The coupled cavities (C1 and C2) have one common resonant mode (λ2=1548.48 nm, Q2=4500) extended to both cavities and two modes (λ1=1493.73 nm, Q1=6100, and λ3=1463.46 nm, Q3=4100) localized in each cavity. Here, we use two bistable switching operations for C1 and C2. The cross feedback is realized as follows. C1 is switched ON only when λ1 and λ2 are both applied (PIN1 and PIN2 are ON). C2 is ON only when λ3 are applied (PIN3 is ON) and simultaneously λ2 is supplied from C1 (which means C1 is ON). Thus, the output signal of λ3 (POUT3) becomes ON only if PIN3 is turned ON when C1 is already ON in advance. This results achieves retiming process. We set PIN1 and PIN3 as two different clock signals as shown in Fig. 11(b), and assume PIN2 to be bit stream NRZ (non-return- to-zero) data with finite timing jitter. The resultant POUT3 is precisely synchronized to the clock signals and is actually an RZ (return-to-zero) data stream converted from PIN2 with jitter corrected.
We designed this function in a PhC slab system, and numerically simulated its operation. The structural parameters are shown in the figure caption. We assumed realistic material parameters (with a Kerr coefficient χ (3)/ε 0=4.1×10-19 (m2/V2), a typical value for AlGaAs) and the instantaneous driving power is assumed to be 60 mW for all three inputs. Figure 11(b) shows three input signals (a data stream with jitter, and two clock pulses), and the output from PD (POUT3). As seen in this plot, POUT3 is the RZ signal of the input with the jitter corrected. We confirmed that the operation speed corresponds to 50GHz operation. Note that this work was intended to demonstrate the operation principle and the structure has not yet been optimized.
5. Photon DRAM by adiabatic control of nanocavities
5.1 Adiabatic tuning of high-Q nanocavities
The previous sections concerned the enhancement of the light-matter interaction especially with respect to optical nonlinearity. These are not the only advantages for high-Q and ultrasmall cavities in terms of optical processing. In principle, various kinds of light-matter interaction can be enhanced, such as light amplification or light emitting processes. In addition, high-Q and ultrasmall cavities can produce novel functionalities. We look at this aspect in this final section. If the photon dwell-time in an optical system is long and its size is small, then we can change the optical system within the photon dwell-time. This process is sometimes called “dynamic tuning”. [48,49, 50, 51, 52] Since the light velocity in the material is so fast, such tuning is normally difficult. However, it becomes meaningful for high-Q ultrasmall cavities or slow-light media. Recently, it has been clarified that this dynamic tuning allows light to be controlled in various surprising ways.
Recently, we have shown that the simple dynamic tuning of a cavity within the photon lifetime leads to adiabatic wavelength conversion, [50, 52] which is completely different from conventional wavelength conversion using optical nonlinear (χ (2) or χ (3)) crystals. We investigated the following situation. When a light pulse is stored in a PhC cavity (we assumed five-point end-hole shifted cavities) shown in Fig. 12(a), we change the resonance frequency of the cavity as a function of time by tuning the refractive index as shown in Fig. 12(b). Using FDTD simulations, we found that the optical spectrum of the light in a cavity shifts after the tuning, as shown in Fig. 12(c). The important thing is that this wavelength shift does not depend on the tuning rate, and is completely determined by the shift of the resonance frequency. Thus, this process is fundamentally different from the conventional χ (3) process. In fact, this process is analogous to the adiabatic tuning of classical oscillators, such as a guitar. This is verified by the fact that U/ω is preserved in this process, which is a signature of adiabatic tuning process. Such tuning is very trivial in sonic vibrations, but it has not been seriously considered in optics because such tuning is rather difficult to achieve in conventional optical systems. However, it is possible in high-Q microcavities, such as PhC cavities. This means that small optical systems with high Q enable us to realize novel ways of controlling light. Very recently our prediction was experimentally confirmed in a silicon microcavity .
In addition, we have also found that this conversion process can be employed for enhancing opto-mechanical interaction.  We numerically confirmed that high-Q PhC double-layer cavities can convert optical energies to mechanical energies extremely efficiently, and it may be possible to employ this phenomenon in some types of optical micro-machines. This efficient energy conversion is made possible by adiabatic optomechanical wavelength conversion in a cavity.
5.2 Photon DRAM based on directly-coupled double cavities
In the following two sections, we show another aspect of dynamic tuning, namely the dynamic control of Q, which may be useful for future all-optical processing using nanocavities. We have already shown that high-Q nanocavities are useful for enhancing light-matter interaction. But it is not always advantageous to have high Q because high-Q means a slow response and a narrow bandwidth. If Q is a static value within its photon lifetime, this is a fundamental limitation. Here, we will show that high-Q does not necessarily mean a slow response or a narrow bandwidth. In addition, this dynamic control of Q leads to a novel type of photon memory, in which we can store (or trap) photons in a cavity. In the previous section, we showed that optical bistability in nanocavities leads to optical memory operation, which can be employed in various types of optical logic. The photon dynamic memory that we introduce here is somewhat different from a bistable memory because the latter memorizes the state of the optical system, not the photon itself.
To change the Q of the optical system dynamically, we employ a pair of cavities. Here we show two ways to do this [54,55, 56]. The first example is shown in Fig. 13(a). The system consists of a gate and memory cavities. The memory cavity (CM) is coupled to the waveguide only via coupling to the gate cavity (CG). If CM is resonant to CG, CM can be coupled to the waveguide. Thus, we can switch on and off the coupling of CM to the waveguide by tuning the resonance frequency of CG. In other words, we can change the loaded Q of the cavity by tuning the cavity-waveguide coupling.
This explanation of the operation mechanism is slightly over-simplified, and in reality we have to handle this system accurately as a doubly-coupled cavity system. We calculated the resonance wavelength and cavity-Q of the whole system (including the waveguide) as a function of the refractive index detuning of the gate cavity ΔnG by the 2D FDTD method, as shown in Fig. 13(b, c). Note that since it does not include the vertical radiation loss, all the cavity Qs are determined by the coupling to the waveguide, which is a good approximation for ultrahigh-Q cavities. The result in Fig. 13(b) shows a typical behavior of a coupled-resonator system. Figure 13(c) shows that the Q of the two modes sensitively depends on ΔnG. Under large detuning conditions, the two cavities are well decoupled, and the memory cavity’s Q (QM) is over 1.5×105. When the detuning becomes small, QM drastically decreases. With zero detuning, QM falls to 3×103. This clearly shows that the tuning of the gate cavity switches on and off the inter-cavity coupling. As shown in Fig. 13(b), the low-QM state and high-QM state are on the same branch of the coupled cavity system, and thus we can adiabatically change the system from low-QM to high-QM and vice versa by tuning ΔnG.
Since this is a simple coupled-resonator system connected to a single bus line, it is relatively easy to analyze with the coupled-mode theory established by Haus  as shown in Fig. 13(d). The coupled-mode equation is given by
where a, ω, γ, κ are the field amplitude in a cavity, the resonance frequency, the decay rate, and the coupling rate, respectively. s1+ is the input power. The calculated solution is shown in Fig. 13(e, f), where we set γG/ω0=0.0002 and κ/ω0=0.0017. The behavior in Fig. 13 (b, c) is well explained by Fig. 13 (d, e), although we do not discuss more quantitative comparison of this analysis.
Next, we investigate write/read operations using the time-dependent tuning of this cavity. First, we numerically simulate the read-out operation with the 2D FDTD method, as shown in Fig. 14(a). Initially, there is a light pulse stored in a cavity, and then we change the refractive index as shown by the gray broken line. The green line is the field amplitude in the memory cavity without tuning, which shows a single exponential decay with Q=1.2×105, as expected. When the index is tuned, the amplitude decays faster as shown by the red line. This clearly shows that Q is switched from 1.2×105 to 4.9×103 by this tuning. Figure 14(b) shows the write-in operation where the index is tuned when a light pulse arrives at the gate cavity. This shows that Q is switched from 3.7×103 to 4.7×104. Figure 14(c) shows the write-read operation (that is, the memory operation). A signal light pulse is injected into the input waveguide. When the pulse arrives at the gate cavity, nG is switched from n G1 to n G2. After a certain time period, n G is switched back from n G2 to n G1. Figure 14(c) clearly shows that the optical pulse is trapped in the cavity after the first switching, and then it is released after the second switching. This is exactly the expected operation for a photon dynamic memory. The upper limit of the memory time is determined by the highest Q M and the switching speed is limited by the lowest Q M. Finally, we add a comment on the bandwidth of the pulse. In our process, the bandwidth of the pulse is equivalently scaled to 1/Q. In the reading-out process, the bandwidth is expanded. In the write-in process, it is squeezed. As was discussed in , the pulse bandwidth is varied during the adiabatic tuning process. It also occurs in our situation, and that is why we can keep a wide-bandwidth pulse within a cavity having the narrow bandwidth.
5.3 Photon DRAM based on indirectly-coupled double cavities
In this section, we describe another design of the photon memory based on a pair of cavities, as shown in Fig. 15(a). Here, a cavity is side-coupled to the waveguide, and another cavity is end-coupled to the waveguide. Both two cavities are interacting with each other via the waveguide, and they form effectively a doubly-coupled cavity system, which is similar to that described in 5.2. In contrast to the case in 5.2 where the inter-cavity interaction is evanescent-wave coupling, the inter-cavity interaction in this case involves propagating-wave coupling via the waveguide. Thus, this coupling can be switched on and off by managing the interference condition of the propagating waves. When propagating waves from two cavities destructively interfere perfectly, both cavities are decoupled from the waveguide, and thus the loaded Q of the coupled cavity system becomes infinitely high (when we ignore the intrinsic loss of cavities). If we dynamically change the resonance wavelength of one of the two cavities, we can change this interference condition dynamically, which should lead to dynamic tuning of the total Q. It is worthwhile noting that although the configuration is different, the physical mechanism of this interference effect itself is similar to the previous work , which discusses the dynamic tuning of coupled-cavity waveguides for stopping light. They use interference to change the coupling between cavities and a waveguide. Very recently, dynamic Q tuning was experimentally demonstrated using a pair of ring cavities  and a single cavity with a reflection mirror in a PhC slab . They also use a similar interference effect for tuning Q.
We calculated the resonance wavelength and cavity-Q of the whole system as a function of the refractive index detuning of the end-coupled cavity (CE) as shown in Fig. 15(b, c). The resonance wavelength plot shows typical behavior for coupled resonators similar to Fig. 13(b), and the Q of the entire system sensitively depends on the detuning whose behavior is different from that in Fig. 13(c). Under large detuning condition, two cavities are independently coupled to the waveguide, and Q is substantially low (3,500 at minimum). When the detuning becomes small, Q for the upper mode increases greatly. At zero detuning, this mode is completely decoupled from the waveguide, and Q reaches up to 9.2x107. This clearly shows that the tuning of the end cavity can change Q significantly. As shown in Fig. 15(c), the low-Q state and high-Q state are on the same branch of the coupled cavity system, and thus we can adiabatically change the system from low-Q to high-Q and vice versa by tuning ΔnE. (Of course, we can do the same thing by tuning the side-coupled cavity).
We also analyze this system with the simplified model shown in Fig. 15(d) using the coupled-mode theory. In this case, the coupled-mode equations are given by
where S and E denote side-coupled and end-coupled cavities. ϕ is the phase difference determined by the distance between two cavities. As with the case for directly-coupled memories (5.2), we also confirmed that the FDTD simulation is well explained by this simple mode, as shown in Fig. 15(e and f).
Next, we investigate write/read operations using time-dependent tuning of this cavity in a similar way to that undertaken for a directly-coupled cavity memory in Fig. 14. Figures 16(a) and (b) show that we can switch Q from high to low and from low to high by index tuning. Unlike Fig. 14, the required index shift is much smaller and the Q contrast is much larger than those in Fig. 15. Figure 16(c) shows the write-and-read operation (memory operation). A signal light pulse is injected into the input waveguide. When the pulse arrives at the end cavity, n E is switched from n E1 to n E2. After a certain time period, n E is switched back from n E2 to n E1. The simulated intensity inside the end-coupled cavity shown in Fig. 16(c) clearly reveals that the optical pulse is trapped in the cavity after the first switching, and then it is released after the second switching. This memory operation is similar to Fig. 14(c) but the operation in Fig.16(c) requires a smaller index change and a longer memory time. This is because the achievable Q is much higher and Q is more sensitive to the resonance wavelength detuning than directly-coupled memories.
We have described our latest results for ultrahigh-Q PhC nanocavities and their applications for optical nonlinear processing and the adiabatic control of light. It is now becoming possible to confine light in a wavelength-scale volume for over a nanosecond. In addition, we can introduce various types of coupling between nanocavities and with waveguides in a single chip. This has scarcely been possible in any previous optical systems. When we consider some form of all-optical processing, the weak light-matter interaction and difficulty in integration generally limit its applicability. PhC nanocavities can potentially overcome this problem, or at least offer a significant advantage over other approaches. As described in the last part, strong light confinement is also offering novel functionalities that are realized by the dynamic tuning of optical systems. Considering these three features, namely the enhancement of the light-matter interaction, the potential for integration and the novel functionality, we believe that these nanocavities in PhCs are now providing new opportunities for photonics technology. In terms of the integration, our work is still very limited in a small scale. For pursuing large-scale optical integrated circuits, it will become important to realize cascading many elements with low coupling loss which will be a hard task. It is worth noting that ultrahigh-Q cavities are also effective in reducing the coupling loss.
We are grateful for invaluable support and collaborations by T. Tamamura, I. Yokohama, Y. Hirayama, S. Kawanishi, M. Kato, S.C. Huang, G-K. Kim, H-Y. Ryu, Y-H. Lee, D. Takagi, S. Kondo, G. Kira. K. Nishiguchi, H. Inokawa, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi. Part of this work was supported by CREST-JST.
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