We demonstrate highly enhanced optical nonlinearity in a coupled-resonator optical waveguide (CROW) in a four-wave mixing experiment. Using a CROW consisting of 200 coupled resonators based on width-modulated photonic crystal nanocavities in a line defect, we obtained an effective nonlinear constant exceeding 10,000 /W/m, thanks to slow light propagation combined with a strong spatial confinement of light achieved by the wavelength-sized cavities.
©2011 Optical Society of America
Slow light on an optical chip [1,2] offers the potential for chip-scale nonlinear optical devices with ultralow power consumption based on the enhanced interaction between matter and the slowly propagating light within. A decade ago, we showed that a line defect in a photonic crystal (PhC) slab exhibits a small group velocity vg near the photonic band edge . It has also been shown that shifted lattices in the vicinity of a line defect of a PhC modulate the waveguide dispersion and this results in a small vg with reduced group-velocity dispersion (GVD) [4,5]. On-chip slow light enhanced optical nonlinearity has recently been intensively studied by using such line-defect-based slow light PhC waveguides; optical pulse propagation with self-phase modulation (SPM) and nonlinear absorption are being investigated in silicon- [4,5] and AlGaAs-based  PhC waveguides. Strong free carrier absorption in silicon achieved optical signal regeneration . Third-harmonic generation realized on-chip green light emission . SPM and strong anomalous waveguide dispersion achieved soliton pulse compression , and four-wave mixing (FWM) is being investigated for all-optical wavelength conversion and switching [10–15].
In addition to a small vg, the strong spatial confinement of light is important for nonlinearity enhancement. It has recently been shown [14,15] that the effective mode area Aeff of line-defect PhC waveguides increases as a function of the group index ng, thus reducing the effective nonlinearity. However, even with such an enlarged spatial field, slow-light PhC waveguides still exhibit strong optical nonlinearity [2,4,5,7,8,10,11,14,15]. Therefore, if the waveguide can support both a small vg and a strong spatial confinement of light, further nonlinearity enhancement can be expected.
A coupled-resonator optical waveguide (CROW), which consists of a one-dimensional periodic array of optical cavities coupled weakly to one another, is a promising candidate for a slow-light waveguide . The waveguiding mode in the CROW underlies the extended mode formed by the nearest-neighbor coupling of the periodically arranged cavities. This is analogous to a situation where the atoms in a crystal lattice are coupled with each other and form a continuous electron band. The group velocity vg in the CROW can be characterized as vg = Lcc/τh, where Lcc is the cavity pitch and τh is the characteristic hopping time of light propagating between neighboring cavities . Here τh is determined by the inter-cavity coupling, which is a variable parameter obtained by setting Lcc and the inter-cavity barrier structure, however, its upper limit is approximately determined by the photon lifetime (~inverse of the cavity Q) for each cavity. If τh is much longer than the individual cavity lifetime, light cannot propagate through the CROW. Therefore, small and high-Q cavities are necessary if we are to achieve a small vg in the CROW.
We have already fabricated a CROW based on width-modulated cavities in a silicon PhC line defect and demonstrated slow-light transmission with a vg as small as c/170, where c is the speed of light in a vacuum . The vg value is the smallest of any CROW yet demonstrated including arrayed point defect silicon photonic crystals , polymer microspheres on a silicon V groove , and a chain of silicon micro-ring resonators [20–22]. This small vg originates from both a tiny (almost wavelength-sized) mode volume and a high Q of as large as one million in our width-modulated cavities [23,24], each of which contributes to the small Lcc and τh, respectively .
Owing to a small vg, our CROW is expected to exhibit the slow light enhancement of optical nonlinearity as in other types of slow light waveguides based on PhC line defects. In this work, first, we reveal that the Aeff of our CROW is very small according to the result of a 3-D FDTD calculation of the resonant field. Next, we perform an FWM experiment to show the high optical nonlinearity of our CROW. FWM has been demonstrated in various slow-light waveguides as described above, thus we can directly compare our and other results. In addition, FWM is important not only in classical applications, but also in quantum information science such as quantum-correlated photon pair generation . Hence, it is worth performing an FWM experiment to explore the future prospects for our CROW. Finally, we compare our experimental results with those for other nonlinear waveguides in terms of the smallness of vg and the spatial confinement of light.
2. Samples and FDTD simulation
Figure 1(a) is a schematic of our CROW. The CROW is based on width-modulated line-defect cavities in a two-dimensional silicon photonic crystal with a triangular lattice of air holes fabricated by electron-beam lithography and dry etching [23,24]. The lattice constant a of our silicon PhC chip is 420 nm, the hole radius is 0.25a and the thickness of the PhC slab is 0.5a. Local width modulation of a 0.98a wide line defect achieves high-Q and wavelength-sized cavity. The displacements of the red and green holes toward the outside are 8 and 4 nm, respectively. The shifted holes create optical nanocavities with a mode volume of 1.7(λ/n)3, where λ is the wavelength of light and n is the refractive index of silicon . Here we chose Lcc as 5a, which yields ng ~40 as shown later, so that CROW nonlinearity can be compared with that of the slow-light-engineered PhC line defect waveguides exhibiting a comparable ng [5,11]. Here ng increases monotonically with increase of Lcc, as the intercavity hopping time decreases by increased neighboring decoupling. The relationship between Lcc and ng including the loss is discussed in  in detail. The cavity number Ncav is 200, thus the total CROW length becomes 420 μm. To couple the light into the CROW, line-defect PhC waveguides are introduced at both ends of the CROW as indicated by purple holes. The width of the access line defect is 1.05a, yielding ng of approximately 5. The length of each element is shown in Fig. 1(a). In addition, the PhC waveguides are connected to silicon-wire waveguides (SWWs) (600 nm wide and 200 nm high). As a reference waveguide, we prepared another PhC waveguide whose CROW section we replaced with a W1.05 silicon PhC waveguide as shown in Fig. 1(b). As shown in Fig. 1, light is guided from left to right by lensed fibers for the coupling. For the following discussion, we label the optical powers at several important positions as shown in Fig. 1. Please note that Pi and Po indicate the powers in the lensed fibers.
First, we estimate the Aeff of our CROW using a 3-D FDTD calculation. Figure 2(a) shows examples of simulated resonant electromagnetic fields (lower half) of the CROW for various Ncav values, shown with the corresponding structures used in each calculation (upper half). Assuming the spatial mode has a uniform waveguiding structure along the propagation axis, we calculate Aeff asFig. 2 (a)). For x and y axis the integration range took over the entire space considered. The denominator is the corresponding integration length of |zNcav − z1|. As a definition of Veff, we used a well-used definition for the optical cavities 
We then derived Aeff in this way for various Ncav. The results are plotted in Fig. 2(b) with the corresponding Veff. We performed the calculation in the Ncav < 10 range due to the long calculation time required for a large-scale CROW. We see that Veff is proportional to Ncav – 1 since for a single cavity whose zNcav = z1 (i.e. no integral region in our definition). We also obtained an almost constant Aeff against the variation of Ncav, with an average Aeff value of 0.04 μm2. To compare our result with conventional line-defect waveguides, we evaluated Aeff of a PhC W1 waveguide in our definition. In this case, we calculated Veff by the integration of propagating field based on Eq. (1) using a finite period of the waveguide along z axis as the integral range, and then divided the Veff by the length of the period. The obtained Aeff was 0.04 μm2 at ng = 5 and 0.08 μm2 at ng = 25. Hence, our CROW exhibited Aeff value comparable to a line-defect waveguide in the fast light region and smaller to that of the slow light region. Such a small Aeff of our CROW can be regarded as the result of the strong (wavelength-sized) confinement of light achieved by individual cavities. The fluctuation of Aeff in Fig. 2(b) is because the fraction of the optical field outside the integration range varies depending on the shape of the resonant field determined by Ncav. Thus, the fluctuation is expected to become small for a larger Ncav with the fraction outside the range. We note that our definition of Veff in Eq. (2) is different from the definition7]. Our definition tends to exhibit relatively smaller values.
3. Linear transmission property
Figure 3(a) shows the linear transmission spectrum of the CROW and the reference line-defect waveguide, measured with a wavelength-tunable cw laser emitting TE polarized light at around 1.55 μm. First, for the reference waveguide, the average transmission loss is approximately 18 dB, which consists of in- and out-coupling losses ηRef ( = PiR/Pi and Po/PoR in Fig. 1) of 8 dB per facet and a linear propagation loss αRef in a W1.05 PhC waveguide of 2 dB/mm measured by the cut-back method in a similar sample. The fringe in the spectrum of the reference waveguide originates from the Fabry-Perot interference caused by reflections at both facets . Meanwhile, the CROW exhibited a clear transmission band of 4 to 5 nm, which is much wider than that of a single nanocavity (~1 pm) [23,24]. Such a wide transmission band can be obtained with the coupled-cavity structure [16,17]. Spectral peaks, whose number corresponds in principle to the number of cavities, are the fingerprint of the extended mode . Comparing the average transmittances of the reference waveguide and that of the CROW at around the band center (as indicated by the dashed lines in Fig. 3(a)), the CROW’s average additional loss is estimated to be approximately 7 dB. The excess loss is presumably due to additional coupling losses ηC at both ends of the CROW (corresponding to PiR/PiC and PoC/PoR in Fig. 1) and the propagation loss in the CROW αCROW that arises from scattering and intrinsic decays at each cavity.
We measured the dispersion property of the CROW using the time-of-flight (TOF) method with optical pulses. Figure 3(b) is a density plot of the output-waveform spectra of optical pulses with a duration of 100 ps. The optical pulses were obtained with a wavelength-tunable cw laser followed by an electro-absorption (EA) modulator. Output optical pulses from the samples were measured directly with a high-speed optical sampling oscilloscope (bandwidth: 80 GHz) synchronized with a gate signal applied to the EA modulator. The horizontal axis in Fig. 3(b) shows the center frequency of each optical pulse, while the vertical axis is the time relative to the gate signal’s arrival. The intensities in each figure are normalized independently. Compared with the reference waveguide, an obvious propagation delay of greater than 50 ps can be obtained for the entire CROW band. It is also observed that the group velocity becomes slower as the wavelength approaches to the band edges of the CROW, as expected from the tight-binding approximation . Although we could evaluate the dispersion property of the CROW also using spectral peak positions as shown in , the method is not effective when Ncav is as large as our present device whose several spectral peaks united so are indistinguishable. This time we solved this problem by adopting TOF method. The phase-shift method is widely used in usual dispersion measurements , however, the use of this approach resulted in a noisy group delay spectrum in the presence of strong reflection at the facets in our CROW.
Next, we evaluate the ng value of our CROW from the difference between the traveling times of two samples. Here, the CROW length of 420 μm is taken into account. An extracted group-index spectrum obtained using the data in Fig. 3(b) is shown in Fig. 3(c). A group index ng of more than 30 was obtained across the CROW band. In accordance with the tight-binding model , in principle, the group velocity is at its largest at the band center and decreases toward zero around the band edge. However, an asymmetric group-index spectrum was obtained in our CROW. This is presumably due to the original anomalous dispersion in the W1 waveguide  that is as wide as the width-modulated section in our CROW. Note that not only resonators, but also periodic dielectric components yield the tight-binding mode. Hence, our CROW could be understood also as a periodic structure of W1 line-defect waveguides. We infer our asymmetric ng spectrum is a result of the symmetric spectrum in the tight binding structure incorporated with a strong anomalous dispersion with a considerable variation of ng up to 100  in the W1 waveguide. On the other hand, the finite vg around the band edges is due the finite length of the CROW and localization of light along the propagation axis as reported in . We fitted our result with a polynomial function of wavelength λ (up to the third order) shown as a solid curve. The fitted function is set at ng(λ) for the FWM phase matching calculation as shown in Sec. 4.1.
4. Four-wave mixing experiments
4.1 Wavelength dependence of FWM
We used two independent wavelength-tunable cw lasers as a pump (p) and a signal (s) source in the FWM experiment. Although the transmission band of the CROW is high, here we chose the cw experiment rather than the pulsed experiment. Using cw lasers we can reduce errors in the nonlinearity estimation, which requires an extra estimation of the peak intensity in the pulsed case. The pump laser was amplified by an erbium-doped fiber amplifier, and subsequently filtered by a tunable band-pass filter consisting of a fiber circulator and a fiber-Bragg grating with a full-width at half maximum (FWHM) of 0.5 nm to suppress background noise. The pump and the signal beams were combined by a 50/50 directional coupler, and then TE polarized by a half and a quarter waveplate before being coupled to the PhC waveguides with a lensed fiber. They were then collected from the waveguide output by another lensed fiber, and the overall output spectrum was measured directly with an optical spectrum analyzer (OSA) with a spectral resolution of 0.05 nm.
The observed FWM spectrum is shown as a density plot in Fig. 4(a) (left, CROW; right, reference waveguide). Here the signal wavelength λs (vertical axis) was scanned across the CROW transmission band, while the pump wavelength λp was fixed at 1545.45 nm, which was around the band center of the CROW. Here, the pump and signal powers coupled to the W1.05 PhC waveguide PiR(p) and PiR(s) were 1.6 and 0.47 mW, respectively. These values were obtained from the coupling efficiency to the reference waveguide ηR and the power before the PhC chip Pi. In addition, we assumed these values as also the coupled powers to the first W1.05 line-defect section of the waveguide containing the CROW as shown in Fig. 1(a). For both samples, we see almost straight lines consisting of created idler peaks (as indicated), which appear almost symmetric with the signal line with respect to the pump wavelength. This idler frequency can be explained by the energy conservation of the FWM;
Next, in Fig. 4(b) we show measured idler intensities (background subtracted) as a function of λs obtained from the result in Fig. 4(a). Here the vertical axis corresponds to Po (the power coupled to the output lensed fiber) in Fig. 1. It is noteworthy that the idler peaks of the CROW were more than one order of magnitude larger than that of the reference waveguide. The only difference between these two measurements is the presence of the CROW section in the sample as shown in Fig. 1. In addition, the CROW has a larger coupling loss than the reference waveguide because of the extra connection loss of ηC ( = PiR/PiC and PoC/PoR in Fig. 1). Therefore, the pump and signal intensities coupled to the CROW (PiC(p) and PiC(s)) should be smaller than those coupled to the reference waveguide (PiR(p) and PiR(s)). Taking these factors into account, we can conclude that the CROW exhibited higher nonlinearity than the reference waveguide.
To evaluate the nonlinearity quantitatively, first, we estimate the FWM conversion efficiency ηFWM from the result. ηFWM is usually defined as the ratio of the output idler power to the input signal power inside the waveguide. Therefore, for the ηFWM of the CROW, we should use the powers in the CROW PoC(i) and PiC(s). However, instead we use the estimated powers in the W1.05 reference waveguide for the CROW’s nonlinearity as
If we used the powers coupled to the CROW section, ηFWM = PoC(i)/PiC(s) = Po(i)/Pi(s)/(ηR2ηC2), which cause an overestimation by the uncertain ηC in the present experiment. The ηFWM of Eq. (5) gives us the conversion efficiency that is always lower than the actual value, thus it never gives us overestimated nonlinearity.
The conversion efficiency ηFWM obtained from the experiment is plotted in Fig. 5 , as a function of λs for various λp values in the CROW band. Here the data at λp = 1545.45 nm correspond to the result in Fig. 4. We see that there is an FWM gain band for all λp values. We also see that the top of the ηFWM band increases with increases in λp. This corresponds to the measured dispersion property in Fig. 3(c), in which vg becomes smaller as λp becomes longer. This indicates slow light enhanced nonlinearity. Meanwhile, the FWM gain band becomes narrow with λp around the band edges because the dispersion slope becomes steep near the band edges. The spectral fluctuation in the measured conversion efficiency is due to the fringe of the transmission spectrum in Fig. 3(a).
Next, we try to fit the experimental data in Fig. 5 to estimate waveguide nonlinearity. For the calculation, we assume that our CROW has a uniform waveguide structure along the propagation axis to evaluate the “effective” waveguide nonlinearity as we did with the Aeff calculation. In this case, ηFWM is governed by [10–13]Fig. 2(c) as β2m(ω) = (1/c)d(2m-1)ng(ω)/dω(2m-1). In the range of parameters used in our experiment, the term (sinh(gL)/(gL))2 determines the FWM bandwidth since it becomes unity for a finite bandwidth. Therefore, the ηFWM value is governed exclusively by (γeffPiR(p)Leff)2 exp(−αRefL). Here we used PiR and αRef rather than PiC (<PiR) and αCROW (>αRef), respectively, to avoid overestimating γeff.
It is known that γeff is proportional to ng2 [1,2]. This is due to the slow light effect and can be explained as a prolonged interaction length by ng combined with the enhanced peak intensity of the shrunken optical pulses also by ng. Therefore, γeff(λ) = κ0 ng(λ)2 where κ0 can be a constant in the small detuning of our experiment. We chose the κ0 value so that ηFWM in Eq. (6) well describes the experimental results. The calculated ηFWM is represented by the solid curves in Fig. 5 for κ0 = 5. Although the model of the introduced waveguide structure is simple, the calculation captures the experimental results well, especially the dependence of the peak gain values on λp. As a result, we obtained γeff = 7,200 /W/m at λp = 1545.2 nm (ng = 36) and 13,000 /W/m at λp = 1547.2 nm (ng = 49). These values are compared with those of other nonlinear waveguides in Sec. 5. To the best of our knowledge, this is the first experimental report of a γeff value exceeding 10,000 /W/m in silicon-based nonlinear waveguides (note that γeff = 13,439 at ng = 66 is estimated by using a theoretical simulation of a silicon slow-light waveguide ). On the other hand, the calculated and experimental gain bandwidths are different especially at λp = 1545.15 nm. The possible reason for this is a discrepancy between the measured ng data and the fitted function (in Fig. 3(c)) used for the ηFWM calculation. The difference becomes large at λ < 1544 nm, which corresponds to the region where the data and the ηFWM simulation diverge. Therefore, an imperfect fitted function may have caused the discrepancy. Another possible reason for the discrepancy is that Eq. (6) is insufficiently accurate to describe our device. Although we assumed that the CROW has a uniform waveguiding field along the propagation axis, the extended modes are not exactly uniform as seen in Fig. 2(a). Further study will require a comparison of these results with detailed theory such as that provided in [29–31].
4.2 Pump power dependence of the FWM
The measured pump-power dependence of the conversion efficiency ηFWM in our CROW is shown in Fig. 6 . Here λp = 1545.08 nm (ng = 36) and λs = 1544.75 nm for the blue symbols and λp = 1547.13 nm (ng = 49) and λs = 1546.75 nm for the red symbols. We chose the conditions so that the pump, signal and idler wavelengths would exhibit high transmittance simultaneously. Again, we used a pump power PiR(p) (>PiC(p)) to avoid possible overestimation of nonlinearity. The experimental results show clear P2 dependence, corresponding to the two-photon excitation process of the FWM. The solid curves are calculated values based on Eq. (6) with γeff as a free fitting parameter. From the fitting, we have obtained significantly high nonlinearities of γeff = 18,600 /W/m at λp = 1545.08 nm and γeff = 25,000 /W/m at λp = 1547.13 nm. We confirmed that the reason for the depletion at high excitation for λp = 1547.13 nm was the depletion of the output pump power caused by the red shift of the high transmission peak resulting from the thermo-optic effect, as reported for other silicon nonlinear devices in the cw regime [32–34].
5. Discussions and conclusion
We observed the highly enhanced optical nonlinearity of our CROW. The parameters we obtained are summarized in Table 1 along with those of other nonlinear waveguides whose cores are crystalline silicon. By using silicon as the core material, we can directly compare structural nonlinearities.
It is clear from the γeff value that our CROW exhibited the highest nonlinearity. In addition, this is the first report of γeff exceeding 10,000 /W/m in a silicon waveguide. This significant nonlinearity enhancement is achieved in our waveguide where the slow-light mode co-exists with a small mode area, thanks to the strong light confinement achieved by wavelength-sized cavities. To estimate the degree of enhancement, we compare our γeff with that of a PhC line-defect slow-light waveguide based on a lattice shift. In Sec. 2, we roughly estimated that Aeff of our CROW is almost half of that of the W1 waveguide operating at slow light regime. Here we assume that the line-defect dispersion engineered slow light waveguide  exhibits similar Aeff to the W1 waveguide in large ng regime. Taking account of ng2/Aeff dependence of γeff as described above, γeff = 2,930 /W/m at ng = 30 in  can be extrapolated as γeff ~8,400 /W/m at ng = 36 with and γeff ~16,000 /W/m at ng = 49. These estimated values show a good agreement with our experimental resultse. However, we infer that our CROW’s nonlinearity is potentially higher than these values, since we neglected the CROW’s additional losses of ηC and αCROW yielding the total loss of 7 dB. In this work, we treated the CROW as a uniform waveguide along the propagation axis and characterized nonlinearity by ng and Aeff for simplicity. However, the local field enhancement of the individual cavity must be treated based on more accurate characterization [29–31]. We believe that cavity enhancement effect become significant in our CROW with larger cavity pitch, which results in a significant discretization (separation) of the spatial supermode along the propagation axis significant. Hence, evaluation of optical nonlinearity in such CROW would be of interest.
Although the CROW’s nonlinearity was significantly high, the FWM gain bandwidth was lower than that of conventional nonlinear waveguides as seen in Table 1. This is because the dispersion was stronger than that of the other waveguides. However, this problem could be overcome. In Section 2 we mentioned that the asymmetric ng spectrum in our CROW is because of the additional dispersion imposed by the original W1 line defect constructing the CROW. Conversely, the dispersion property can be controlled by modifying the structure of our CROW. This dispersion-engineered CROW will be the subject of further study. Even without a flattened dispersion, the CROW can still be used for such applications as chip-scale nonlinear pulse compression, where a strong dispersion property is key as well as high nonlinearity .
In silicon, high nonlinearity is accompanied by strong free-carrier effects. To avoid these effects we used cw lasers for an accurate evaluation of waveguide nonlinearity. Since free carriers are excited by two pump photons at the telecom wavelength, the FWM efficiency will be degraded during high excitation by the free-carrier absorption effect. However, our CROW structure is not restricted by its material. Chalcogenide glasses [35,36] and compound semiconductors with wide bandgaps [6,9,12] are used for PhCs with nonlinear functions without any free-carrier effects, and these are applicable to our CROW structure. Of course, efficient free-carrier effects in silicon without substituting the material have been attracting a lot of attention for a variety of applications including optical regeneration, pulse compression and variable delay [7,37,38].
On the application of CROWs to such all-optical signal processing, the oscillating spectrum is a considerable issue. In principle, if the number of cavities is infinite and the propagation loss is negligible, the CROW exhibits flat transmission property. In practice, a finite number of cavities, a propagation loss due to realistic structure, non-uniformity (disorder) of fabricated cavities and a finite coupling strength to the access waveguides cause the oscillating spectrum. Nevertheless, some factors could be improved. We have demonstrated that flatter transmission spectrum can be obtained by the “apodization” of the cavity pitch at the end of the CROW to increase the CROW-waveguide coupling strength . Although our present CROW was not apodized since Lcc of 5a is in a minimum design of the pitch, a slower CROW with larger Lcc can be apodized. In addition, we have realized the largest Ncav (up to 400) in realized CROWs, thanks to the individual cavity’s high Q of 106 . Large Ncav also improves the flatness. Thus, along with these techniques incorporated with the improvement of the fabrication technology, the spectral response of very large scale CROW could be improved to some extent.
In summary, we have demonstrated significantly high waveguide nonlinearity in a CROW consisting of width-modulated nanocavities in a line defect of a silicon PhC slab. A FWM experiment revealed inherent strong nonlinearity with an effective nonlinear constant γeff exceeding 10,000 /W/m. This value is the highest yet reported for silicon-core nonlinear waveguides with an integrated structure. The remarkably high nonlinearity was obtained because the slow-light mode of the CROW exhibited the small effective modal area due to the wavelength-sized confinement of individual cavities. This CROW has the potential to provide a new stage for integrated waveguide photonics because of its noteworthy nonlinearity with a view to realizing on-chip devices with an ultralow power consumption.
We are grateful to Dr. Yasuhiro Tokura, Dr. Kaoru Shimizu, Dr. Takasumi Tanabe and Dr. Akihiko Shinya for fruitful discussions. This work was supported in part the Japan Society for the Promotion of Science with a Grant-in-Aid for Scientific Research (No. 22360034).
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