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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 26 — Dec. 24, 2007
  • pp: 17458–17481
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Nonlinear and adiabatic control of high-Q photonic crystal nanocavities

M. Notomi, T. Tanabe, A. Shinya, E. Kuramochi, H. Taniyama, S. Mitsugi, and M. Morita  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 17458-17481 (2007)
http://dx.doi.org/10.1364/OE.15.017458


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Abstract

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

1. Introduction

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

13. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002). [CrossRef]

, 14

14. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670 [PubMed]

]. 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.[5

5. E. Kuramochi, M. Notomi, M. Mitsugi, A. Shinya, and T. Tanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]

] 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 [15

15. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, “Waveguides, resonators, and their coupled elements in photonic crystal slabs”, Opt. Express 12, 1551–1561 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1551 [CrossRef] [PubMed]

], 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 [3

3. B-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mat. 4, 207–210 (2005). [CrossRef]

]. 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.[16

16. K. Inoshita and T. Baba, “Lasing at bend, branch and intersection of photonic crystal waveguides,” Electron. Lett. 39, 844 (2003). [CrossRef]

] 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.

Fig. 1. Width-modulated line-defect PhC cavities. (a) Cavity design: (from left to right) a starting straight line defect waveguide without theoretical loss and cavities with gradual light confinement. The rightmost cavity has the highest theoretical Q. The hole shifts are typically 9 nm (red holes), 6 nm (green holes), and 3 nm (blue holes). (b) Spectral measurement of a nanocavity fabricated in a silicon hexagonal air-hole photonic slab with a=420 nm and 2r=216 nm. The transmission spectrum of a cavity with a second-stage hole-shift. The inner and outer hole shifts are 8 and 4 nm, respectively. (c) Time-domain ring-down measurement. The time decay of the output light intensity from the same cavity as (b). Details can be found in [17].

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

6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nature Photon. 1, 49–52 (2007). [CrossRef]

,17

17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, “Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities,” Opt. Express. 15, 7826–7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]

] 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

6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nature Photon. 1, 49–52 (2007). [CrossRef]

,17

17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, “Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities,” Opt. Express. 15, 7826–7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]

] 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 [2

2. D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]

]. 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.[17

17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, “Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities,” Opt. Express. 15, 7826–7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]

] 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 [17

17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, “Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities,” Opt. Express. 15, 7826–7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]

]. 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. [18

18. T. Tanabe, M. Notomi, and E. Kuramochi, “Measurement of an ultra-high-Q photonic crystal nanocavity using a single-side-band frequency modulator,” Electron. Lett. 43, 187–188 (2007). [CrossRef]

] 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 [19

19. S.G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

], and then applied to quadrupole modes in 2D square PhCs [20

20. H-Y. Ryu, S-H. Kim, H-G. Park, J-K. Hwang, and Y-H Lee, “Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,” Appl. Phys. Lett. 80, 3883–3885 (2002). [CrossRef]

]. 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 [21

21. M. Notomi, H. Suzuki, T. Tamamura, and K. Edagawa, “Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice”, Phys. Rev. Lett. 92, 123906 (2004). [CrossRef] [PubMed]

]). After some numerical calculations, we found that theoretical Q is over a million with V~(λ/n)3 [22

22. H-Y. Ryu, M. Notomi, and Y-H. Lee, “High quality-factor and small mode-volume hexapole modes in photonic crystal slab nano-cavities,” Appl. Phys. Lett. 83, 4294–4296 (2003). [CrossRef]

]. 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 [23

23. G-H. Kim, Y-H. Lee, A. Shinya, and M. Notomi, “Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode,” Opt. Express 12, 6624–6631 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-26-6624 [CrossRef] [PubMed]

]. Very recently, we succeeded in measuring the Q value of the sample shown in Fig. 2(a) [24

24. T. Tanabe, A. Shinya, E. Kuramochi, S. Kondo, H. Taniyama, and M. Notomi, “Single point defect photonic crystal nanocavity with ultrahigh quality factor achieved by using hexapole mode,” Appl. Phys. Lett. 91, 021110 (2007). [CrossRef]

]. 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.

Fig. 2. Hexapole-mode single-point-defect silicon PhC cavities. (a) FDTD simulation of the field intensity profile for a hexapole cavity coupled to input and output waveguides. The inset shows the geometrical design of the hexapole cavity. (b) Spectral measurement of a hexapole cavity fabricated in a silicon hexagonal air-hole photonic slab with a=420 nm and 2r=176 nm The transmission spectrum across the input and output waveguides is shown. The hole shift is 0.23a. (c) Time-domain ring-down measurement. The time decay of the output light intensity from the same cavity as (b). The solid line is an exponential fit for the data. Ref is the reference data without the cavity (showing the time resolution of our set up).

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 [25

25. T. Baba and D. Mori, “Slowlight engineering in photonic crystals,” J. Phys. D: Appl. Phys. , 40, 2659–2665 (2007). [CrossRef]

]. 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

26. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]

,15

15. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, “Waveguides, resonators, and their coupled elements in photonic crystal slabs”, Opt. Express 12, 1551–1561 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1551 [CrossRef] [PubMed]

] 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.[27

27. S. C. Huang, M. Kato, E. Kuramochi, C. P. Lee, and M. Notomi, “Time-domain and spectral-domain investigation of inflection-point slow-light modes in photonic crystal coupled waveguides,” Opt. Express 15, 3543–3549 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3543 [CrossRef] [PubMed]

] 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.[28

28. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

] 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

6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nature Photon. 1, 49–52 (2007). [CrossRef]

, 17

17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, “Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities,” Opt. Express. 15, 7826–7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]

]. 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.

Fig. 3. Slow-light propagation measurement of a width-modulated line-defect cavity coupled to input and output waveguides. Sample and measurement setup (left). Measured output intensity as a function of time (right). The cavity is a width-modulated line-defect cavity with a three-stage hole shift. The shifts are 9, 6, and 3 nm in Fig. 1(a). The vertical scale for two curves is normalized. The transmittance via a cavity is less than 10% of the reference.

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 [29

29. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B72, 161318(R) (2005).

]. Figure 4 shows our latest record as regards propagation loss measured for W1 PhC waveguides [30

30. E. Kuramochi, M. Notomi, S. Hughes, L. Ramunno, G. Kira, S. Mitsugi, A. Shinya, and T. Watanabe, “Scattering loss of photonic crystal waveguides and nanocavities induced by structural disorder,” Pacific Rim Conference on Lasers and Electro-Optics (CLEO-PR), Japan, July 11–15, CTuE1-1, 2005. (pp. 10–11)

]. It shows a pronounced wavelength dependence that has been well explained by theory [29

29. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs,” Phys. Rev. B72, 161318(R) (2005).

], 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.

Fig. 4. Propagation loss measurement of W1 waveguides fabricated in silicon hexagonal airhole PhCs with a=430 nm. The loss was determined from the transmitted light intensity as a function of the waveguide length (left). The loss spectrum (right). The minimum loss is 2dB/cm around the center of the transmission window. The horizontal axis in the right plot is normalized angular frequency ω, which is deduced as a/λ (a is the lattice constant). The loss measurement scheme is the same as that reported in [29].

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 [31

31. S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, “Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs,” in Proceedings of 16th Annual Meeting of IEEE LEOS (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 214–215.

]. 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.

Fig. 5. Effect of size disorder on Q for various PhC cavities. Cavity A is a width-modulated line-defect cavity. (a=432 nm, 2r=230 nm, shift=9, 6, 3 nm). Cavity B is a hexapole cavity. (a=420 nm, 2r=168 nm, shift=0.23a). Cavity C is an end-hole shifted cavity. (a=420 nm, 2r=230 nm, shift=55 nm, 2r for the shifted holes=126 nm).

3. All-optical switching and memory

3.1 Nonlinear switch based on high-Q nanocavities

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.[35

35. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678 [CrossRef] [PubMed]

] 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) [31

31. S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, “Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs,” in Proceedings of 16th Annual Meeting of IEEE LEOS (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 214–215.

] 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 [33

33. H.M. Gibbs, Optical bistability: controlling light with light. (Academic Press, Orlando, 1985).

]. 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 [35

35. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-7-2678 [CrossRef] [PubMed]

]. 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) [36

36. G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. Macleod, and F. Van Milligen, “Microsecond room-temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt powers,” Appl. Phys. Lett. 45, 1031–1033 (1984). [CrossRef]

] and also smaller than that of recent miniature-sized thermo-optic silicon micro-ring resonator devices (~0.8 mW).[37

37. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29, 2387–2389 (2004). [CrossRef] [PubMed]

] 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).

Fig. 6. All-optical bistable switching in a silicon hexagonal air-hole PhC nanocavity realized by the thermo-optic nonlinearity induced by two-photon absorption in silicon. a=420 nm, 2r=0.55a. The radius of end-holes of the cavity is 0.125a. The radius of end-holes of the waveguide is 0.15a. The output is switched from ON to OFF with δB=20 pm, and OFF to ON with δB=260 pm. Both show similar bistable switching.

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 [38

38. P. M. Johnson, A. F. Koenderinc, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B66, 081102 (R) (2002).

]. 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 [39

39. T. Tanabe, M. Notomi, A. Shinya, S. Mitsugi, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87, 151112 (2005). [CrossRef]

]. 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.[40

40. V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, Nature 431, 1081–1083 (2004). [CrossRef] [PubMed]

] 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. [40

40. V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, Nature 431, 1081–1083 (2004). [CrossRef] [PubMed]

] That is, the small footprint of the device is again effective in improving the operating speed.

Fig. 7. All-optical switching in a silicon PhC nanocavity realized by carrier-plasma nonlinearity induced by two-photon absorption in silicon. The right panel shows the output intensity of the signal light when applying a 6-ps control pulse with two different detuning conditions.

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.[41

41. T. Tanabe, M. Notomi, A. Shinya, S. Mitsugi, and E. Kuramochi, “Fast bistable all-optical switch and memory on silicon photonic crystal on-chip,” Opt. Lett. 30, 2575–2577 (2005). [CrossRef] [PubMed]

] 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.

Fig. 8. All-optical bistable memory operation in a silicon PhC nanocavity realized by the carrier-plasma nonlinearity induced by two-photon absorption in silicon. (left) Injected control light consisting of a pair of set and reset pulses. (right) Output signal intensity as a function of time for three different cases: with no set/reset pulses (red curve), with set pulse only (green curve), and with set and reset pulses (blue curve).

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. [42

42. T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90, 031115 (2007). [CrossRef]

]

Fig. 9. All-optical 5Gb/s demultiplexing operation by a random bit stream using a silicon PhC nanocavity switch.

4. Towards all-optical logic

4.1 Flip-flop operation by double nanocavities

It has been proposed that all-optical flip-flops be realized by using two nonlinear etalons with appropriate cross-feedback,[44

44. H. Tsuda and T. Kurokawa, “Construction of an all-optical flip-flop by combination of two optical triodes,” Appl. Phys. Lett. 57, 1724–1726 (1990). [CrossRef]

] but this proposal is unsuitable for on-chip integration. Here we propose a different design using two PhC nanocavities.[45

45. S. Mitsugi, A. Shinya, T. Tanabe, M. Notomi, and I. Yokohama, “Design and FDTD analysis of micro photonic flip-flop based on 2D photonic crystal slab,” in Extended abstracts of the 52nd spring meeting of the Japan Society of Applied Physics, 30p-YV-11, p.1197, Mar. 30, 2005.

] 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.

Fig. 10. All-optical SR flip-flop consisting of two bistable cavities coupled to waveguides. (a) Structural design based on a hexagonal air-hole 2D PhC. The air-hole diameter for the lattice is 0.55a. Two cavities are both seven-point end-hole shifted cavities. The end hole is shifted by -0.30a with 2r=0.24a. (b) Schematic of the design. (c) Equivalent electronic SR flip-flop. (d) Schematic operation of two bistable cavities. (e) Detailed design of CvS and CvR. (f) Detailed design of WG2. The hole diameter in the waveguide is 0.60a. (g) Time sequence of three inputs (bias, and set clock pulse, and reset clock pulse) and two outputs. (h) Simulated operation using 2D FDTD. A blue and red curves correspond to the output intensity of the two ports. The bottom plots are snapshots of intensity profiles in the device.

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 [31

31. S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, “Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs,” in Proceedings of 16th Annual Meeting of IEEE LEOS (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 214–215.

] 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 [15

15. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, “Waveguides, resonators, and their coupled elements in photonic crystal slabs”, Opt. Express 12, 1551–1561 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1551 [CrossRef] [PubMed]

], 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. [46

46. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002). [CrossRef]

] 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.

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.

Fig. 11. All-optical retiming circuit based on two bistable cavities. (a) Design based on a hexagonal air-hole 2D PhC with a=400 nm and 2r=0.55a. Two waveguides in the upper area (PA and PC) are W1 and the other two in the lower area (PB and PD) is W0.8. (b) Simulated operation.

5. Photon DRAM by adiabatic control of nanocavities

5.1 Adiabatic tuning of high-Q nanocavities

Recently, we have shown that the simple dynamic tuning of a cavity within the photon lifetime leads to adiabatic wavelength conversion, [50

50. B.P.J. Bret, T.L. Sonnemans, and T.W. Hijmans, “Capturing a light pulse in a short high-finesse cavity,” Phys. Rev. A 68, 023807 (2003). [CrossRef]

, 52

52. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97, 023903 (2006). [CrossRef] [PubMed]

] 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 [53

53. S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nature Photon. 1, 293 (2007). [CrossRef]

].

In addition, we have also found that this conversion process can be employed for enhancing opto-mechanical interaction. [52

52. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs,” Phys. Rev. Lett. 97, 023903 (2006). [CrossRef] [PubMed]

] 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.

Fig. 12. Adiabatic wavelength conversion. (a) A five-point end-hole shifted PhC cavity used for the simulation. (b) Tuning of the refractive index for the tuned area in (a) as a function of time. (c) Wavelength spectra with and without tuning obtained by 3D FDTD calculation. (d) U, Δλ, and U/ω obtained by FDTD calculations. (e) Examples of classical oscillators, for which dynamic tuning is easily realized.

5.2 Photon DRAM based on directly-coupled double cavities

Fig. 13. Photonic memory based on a directly-coupled cavity pair. (a) Design based on a 2D hexagonal air-hole PhC with a=400 nm and 2r=0.55a. Cavity M is a four-point-long cavity and Cavity G is a two-point-long cavity. (b) The resonant wavelength versus the detuning of the gate cavity calculated by FDTD. (c) Q versus the detuning calculated by FDTD. (d) A model for coupled-mode theory calculation. (e) The resonant wavelength versus the detuning calculated by the coupled-mode theory. (f) Q versus the detuning calculated by the coupled-mode theory.

To change the Q of the optical system dynamically, we employ a pair of cavities. Here we show two ways to do this [54

54. M. Notomi, T. Tanabe, A. Shinya, S. Mitsugi, E. Kuramochi, and M. Morita, “Dynamic nonlinear control of resonator-waveguide coupled system in photonic crystals,” Pacific Rim Conference on Lasers and Electro-Optics (CLEO-PR), Japan, July 11–15, CWe4-1, 2005. (pp. 1020–1021).

,55

55. M. Morita, M. Notomi, S. Mitsugi, and A. Shinya, “Dynamic Q control in photonic-crystal-slab resonator-waveguide coupled system,” in Extended abstracts of the 52nd spring meeting of the Japan Society of Applied Physics, 31p-YV-15, p.1208, Mar. 30, 2005.

, 56

56. M. Morita, M. Notomi, S. Mitsugi, and A. Shinya, “Dynamic Q control in photonic-crystal-slab resonator-waveguide coupled system (2),” in Extended abstracts of the 66th autumn meeting of the Japan Society of Applied Physics, 9p-H-11, p.924, Sept. 9, 2005.

]. 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 [57

57. H.A. Haus, Waves and fields in optoelectronics (Prince-Hall, New Jersey, 1984).

] as shown in Fig. 13(d). The coupled-mode equation is given by

daMdt=iωGa+iκaG
(1)
daGdt=(iωGγG)aG+iκaM+2γGs1+
(2)

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 [49

49. M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004). [CrossRef] [PubMed]

], 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.

Fig. 14. Temporal operation of the photonic memory simulated by FDTD. (a) Read out. A stored pulse is read out by the index tuning. The green line is without index tuning otherwise the condition is the same as the red line. (b) Write in. An injected pulse is stored by index tuning. (d) Read and write. The combination of (a) and (b) results in the memory operation.

5.3 Photon DRAM based on indirectly-coupled double cavities

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).

Fig. 15. Photonic memory based on an indirectly-coupled cavity pair. (a) Design based on a 2D hexagonal air-hole PhC with a=400 nm and 2r=0.55a. (b) The resonant wavelength versus the detuning of the gate cavity calculated by FDTD. (c) Q versus the detuning calculated by FDTD. (d) A model for coupled-mode theory calculation. (e) The resonant wavelength versus the detuning calculated by the coupled-mode theory. (f) Q versus the detuning calculated by the coupled-mode theory. There is slight deviation between low-Q modes in (b, c) and (e, f), which might be due to numerical errors in FDTD, since it becomes difficult to resolve a low-Q mode when a high-Q mode coexists.

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

daSdt=(iωSγS(1cos2ϕ))aS+2γSγSeiϕaE+γS(1e2iϕ)s1+
(3)
daEdt=(iωEγE)aE2γEγSeiϕaS+2γEeiϕs1+
(4)

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.

Fig. 16. Temporal operation of the photonic memory simulated by FDTD. (a) Read out. A stored pulse is read out by the index tuning. The red curve is the light intensity in cavity E, and the dark yellow line is the light intensity at the waveguide. The monitoring positions are marked by crosses in Fig. 15(a). It is clearly seen that the light pulse is released from the cavity to the waveguide after the tuning. (b) Write in. An injected pulse is stored by the index tuning, and there is very little leak into the waveguide. (c) Read and write. The combination of (a) and (b) results in the memory operation.

6. Conclusion

Acknowledgments

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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OCIS Codes
(190.1450) Nonlinear optics : Bistability
(200.4660) Optics in computing : Optical logic
(230.5750) Optical devices : Resonators
(250.5300) Optoelectronics : Photonic integrated circuits
(230.5298) Optical devices : Photonic crystals

ToC Category:
Nonlinear Optics for Functional Devices and Applications

History
Original Manuscript: October 11, 2007
Revised Manuscript: November 29, 2007
Manuscript Accepted: December 3, 2007
Published: December 11, 2007

Virtual Issues
Focus Serial: Frontiers of Nonlinear Optics (2007) Optics Express

Citation
M. Notomi, T. Tanabe, A. Shinya, E. Kuramochi, H. Taniyama, S. Mitsugi, and M. Morita, "Nonlinear and adiabatic control of high-Q photonic crystal nanocavities," Opt. Express 15, 17458-17481 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17458


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References

  1. K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003). [CrossRef] [PubMed]
  2. D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003). [CrossRef] [PubMed]
  3. B-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Mat. 4, 207-210 (2005). [CrossRef]
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  5. E. Kuramochi, M. Notomi, M. Mitsugi, A. Shinya, and T. Tanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
  6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, "Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity," Nat. Photonics 1, 49-52 (2007). [CrossRef]
  7. R. Herrmann, T. Sunner, T. Hein, A. Loffler, M. Kamp, and A. Forchel, "Ultrahigh-quality photonic crystal cavity in GaAs," Opt. Lett. 31, 1229-1231 (2006). [CrossRef] [PubMed]
  8. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  9. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, "Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal," Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]
  10. H-Y. Ryu, M. Notomi, E. Kuramochi, and T. Segawa, "Large spontaneous emission factor (>0.1) in the photonic crystal monopole-mode laser," Appl. Phys. Lett. 84, 1067 (2004). [CrossRef]
  11. K. Nozaki, S. Kita, and T. Baba, "Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser," Opt. Express 15, 7506-7514 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7506 [CrossRef] [PubMed]
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  14. K. Srinivasan and O. Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-15-670 [PubMed]
  15. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H-Y. Ryu, "Waveguides, resonators, and their coupled elements in photonic crystal slabs," Opt. Express 12, 1551-1561 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1551 [CrossRef] [PubMed]
  16. K. Inoshita and T. Baba, "Lasing at bend, branch and intersection of photonic crystal waveguides," Electron. Lett. 39, 844 (2003). [CrossRef]
  17. T. Tanabe, M. Notomi, E. Kuramochi, and H. Taniyama, "Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities," Opt. Express. 15, 7826-7839 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-12-7826 [CrossRef] [PubMed]
  18. T. Tanabe, M. Notomi, and E. Kuramochi, "Measurement of an ultra-high-Q photonic crystal nanocavity using a single-side-band frequency modulator," Electron. Lett. 43, 187-188 (2007). [CrossRef]
  19. S.G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, "Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap," Appl. Phys. Lett. 78, 3388-3390 (2001). [CrossRef]
  20. H-Y. Ryu, S-H. Kim, H-G. Park, J-K. Hwang, and Y-H Lee, "Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode," Appl. Phys. Lett. 80, 3883-3885 (2002). [CrossRef]
  21. M. Notomi, H. Suzuki, T. Tamamura, K. Edagawa, "Lasing action due to the two-dimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice," Phys. Rev. Lett. 92, 123906 (2004). [CrossRef] [PubMed]
  22. H-Y. Ryu, M. Notomi, and Y-H. Lee, "High quality-factor and small mode-volume hexapole modes in photonic crystal slab nano-cavities," Appl. Phys. Lett. 83, 4294-4296 (2003). [CrossRef]
  23. G-H. Kim, Y-H. Lee, A. Shinya, and M. Notomi, "Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode," Opt. Express 12, 6624-6631 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-26-6624 [CrossRef] [PubMed]
  24. T. Tanabe, A. Shinya, E. Kuramochi, S. Kondo, H. Taniyama, and M. Notomi, "Single point defect photonic crystal nanocavity with ultrahigh quality factor achieved by using hexapole mode," Appl. Phys. Lett. 91, 021110 (2007). [CrossRef]
  25. T. Baba and D. Mori, "Slowlight engineering in photonic crystals," J. Phys. D: Appl. Phys.  40, 2659-2665 (2007). [CrossRef]
  26. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
  27. S. C. Huang, M. Kato, E. Kuramochi, C. P. Lee, and M. Notomi, "Time-domain and spectral-domain investigation of inflection-point slow-light modes in photonic crystal coupled waveguides," Opt. Express 15, 3543-3549 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3543 [CrossRef] [PubMed]
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  29. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, "Disorder-induced scattering loss of line-defect waveguides in photonic crystal slabs," Phys. Rev. B 72, 161318(R) (2005).
  30. E. Kuramochi, M. Notomi, S. Hughes, L. Ramunno, G. Kira, S. Mitsugi, A. Shinya, and T. Watanabe, "Scattering loss of photonic crystal waveguides and nanocavities induced by structural disorder," Pacific Rim Conference on Lasers and Electro-Optics (CLEO-PR), Japan, July 11-15, CTuE1-1, 2005. (pp. 10-11)
  31. S. Mitsugi, A. Shinya, E. Kuramochi, M. Notomi, T. Tsuchizawa, and T. Watanabe, "Resonant tunneling wavelength filters with high Q and high transmittance based on photonic crystal slabs," in Proceedings of 16th Annual Meeting of IEEE LEOS (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 214-215.
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  36. G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. Macleod, and F. Van Milligen, "Microsecond roomtemperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt powers," Appl. Phys. Lett. 45, 1031-1033 (1984). [CrossRef]
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  39. T. Tanabe, M. Notomi, A. Shinya, S. Mitsugi, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005). [CrossRef]
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