We systematically studied the spectral and temporal characteristics of wavelength-sized ultrahigh-Q photonic crystal nanocavities based on width-modulated line defects. By employing accurate measurements, we confirmed that the cavity exhibits an ultra-sharp resonance width (1.23 pm), an ultrahigh-Q (1.28×106), and an ultra-long photon lifetime (1.12 ns). We discussed the correlation between the spectral and temporal measurements for various cavities, and obtained extremely good agreement. In addition, we demonstrated photon trapping for the side-coupling configuration by employing ring-down measurement, which sheds light on another interesting aspect of this phenomenon. Finally, we performed pulse propagation experiments for samples with different waveguide-cavity coupling configurations, and achieved a smallest group velocity of about 4.6 km/s for a novel configuration. These results show that we can effectively trap and delay light by using ultra-small cavities, which can potentially increase the packing density of optical buffers and bit-shifters if applied to coupled-cavity waveguides.
© 2007 Optical Society of America
Various types of high quality factor (Q) microcavities [1–7] are attracting considerable attention due to their small size and strong confinement of light. In particular, with the recent progress on planar semiconductor processing technology, the Q of photonic crystal (PhC) nanocavities fabricated on two-dimensional semiconductor slabs has increased greatly [8–11] and has now reached 2.5 × 105 for GaAs  and 1.2 × 106 for silicon membrane [13,14]. This makes the device an attractive platform for such studies as cavity quantum-electro-dynamics [15–17] and all-optical signal processing [18–22]. To date, the high Q of PhC nanocavities has been demonstrated in both the spectral and time domains. In particular, time domain measurement is a powerful tool for revealing the linear and nonlinear dynamics of materials and photons in an extremely small cavity . Indeed light caging for 1 ns and dynamic Q tuning have been demonstrated in the time domain by employing ring-down measurements .
In addition, we have demonstrated that cavities with a Q of 7.8 × 105, which could be used as basic elements of a coupled resonator based optical waveguide (CROW)  by connecting them in tandem, can exhibit a pulse delay of 1.45 ns if we input an optical pulse with a 1.9-ns width . Owing to the small cavity size, the corresponding group velocity was an extremely small 5.8 km/s. The group velocity demonstrated in a single cavity should correspond to the minimum value that can be obtained in an ideal CROW system. This value is about two orders of magnitude smaller than that obtained for a slow light demonstration with PhC waveguides [25–27]. Although the demonstrated pulse shift (Θ) (the ratio between the delay of the pulse (τ d) and the full width at half-maximum (FWHM) of the output pulse (∆τ out)) at a single PhC nanocavity is not very large, a large delay is expected to be achieved by constructing a CROW system. Indeed, a half-bit shift has been demonstrated by using a microring cavity based CROW at a speed of 1 Gbps . Therefore, it is particularity important to achieve a group velocity that is as small as possible at a single PhC nanocavity to obtain a large pulse shift at an ultrahigh packing density.
The main purpose of this paper is to report extensive and systematic studies of the spectral and temporal characteristics of wavelength-sized ultrahigh Q photonic-crystal nanocavities with various configurations. We recently reported spectral and temporal studies of similar photonic crystal nanocavities , and this work is an extension of those results.
The paper is organized as follows. In the next section, we describe the performance of the latest fabricated PhC nanocavity and discuss the reproducibility and accuracy of spectral and temporal measurements. In addition, we investigate various types of design in terms of the cavity-waveguide coupling, for example direct- and side-coupling. In section 3, we investigate the pulse propagation characteristics of this PhC nanocavity by comparing a pulse incident experiment with a coupled-mode theory simulation. Section 4 describes a newly proposed design for cavity-waveguide coupling, where we observed the lowest group velocity of about 4.6 km/s.
2. Ultrahigh-Q width-modulated line defect PhC nanocavity with shoulder coupling
2.1. Ultrahigh-Q demonstration in spectral domain
We fabricated a width-modulated line defect PhC nanocavity, whose design is shown schematically in Fig. 1(a) We shifted some holes towards the outside to enable modegap confinement. The cavity is shoulder coupled with input and output waveguides. First, we obtained the loaded Q of the cavity by carefully measuring the transmittance spectrum. Figure 1(b) shows the experimental result, where there is a transmission width of 1.22 pm, which corresponds to a Q of 1.29 × 106. According to a three-dimensional finite difference time domain calculation, this cavity has a theoretical Q of 1.3 × 107 and a mode volume of Va = 1.65(λ 0/n)3, where λ 0 is the wavelength of the resonance light and n the refractive index of silicon. This mode volume is slightly larger than that of the cavity described in Ref. 13, which has a value of Vb = 1.51(λ 0/n)3. This is due to the smaller potential barrier, because a smaller number of holes are shifted. (Note: The Vb value is slightly different from the value we reported in Ref. 13, since we recently found that our previous value was overestimated.)
The Q’s of ultrahigh-Q PhC cavities have been extensively studied by spectral measurement, but it has been pointed out that stable measurements are very difficult to perform for these cavities [9,29] because they require a stable setup with excellent wavelength resolution. Therefore we would like to describe briefly how we confirmed the stability and accuracy of our spectrum measurement system to obtain the graph shown in Fig. 1(b).
The wavelength resolution of our setup is 50 fm. This value is obtained from Fig. 2(a), where we measured the transmission width of a well characterized high-finesse Fabry-Pérot cavity that has a transmission width of 4.5 MHz. This resolution is sufficient for the spectral measurement of an ultrahigh-Q PhC nanocavity. Another important issue as regards obtaining high accuracy is to achieve high temperature stability, because different temperatures yield different resonant wavelengths for the PhC nanocavity due to the thermo-optic effect. The stability was checked by setting the laser wavelength at the peak in Fig. 1(b) and recording the transmitted power. Figure 2(b) shows the monitored result obtained at room temperature, where less than 1 dB fluctuation is achieved when the device is thermally insulated from the outside environment. We also provide some notes about the Lorentz fitting that we performed in Fig. 1(b). We employed the maximum transmittance and the FWHM from the experimental data as the initial values. We placed the maximum transmittance at the center of the data array and used 1024 plots for the fitting. We employed the Levenberg-Marquardt algorithm and carried out a maximum of 500 iterations. Hence, there are no freely adjustable parameters, and the fitting results are always the same for a given data set.
We performed the spectrum measurement 12 times on the same ultrahigh-Q PhC nanocavity to confirm its reproducibility. The measurement interval was 30 sec, and Fig. 3(a-f) show the first 6 spectra. We obtained good reproducibility for 12 graphs, which had an average spectrum width of 1.23 pm. The standard deviation was 0.06 pm, which is reasonable because the relative wavelength resolution is 50 fm. The corresponding Q was 1.28×106 and the photon lifetime was 1.07 ns, with standard deviations of 0.06 × 106 and 0.05 ns, respectively. To the best of our knowledge this Q value is the highest achieved with a PhC nanocavity. In addition, we would like to mention that we have fabricated many PhC nanocavities with a similar design, and obtained similar Q values.
2.2. Ultrahigh-Q demonstration in time domain
Although spectral measurement provides information on the cavity Q, demonstrating a high-Q in the time domain offers a direct view of the dynamic behavior of photons and the material property inside the cavity. As regards time resolved measurement, ring-down measurement, which is the most direct and intuitive way of obtaining the photon lifetime, has previously been applied for the characterization of ultrahigh-Q troid microcavities . Recently we applied this method to an ultrahigh-Q PhC nanocavity to demonstrate the ultra-long photon lifetime and obtain information about the dynamic tuning of the Q . Another group reported a pulsed experiment for both low Q  and ultrahigh-Q  PhC nanocavities that was designed to obtain their impulse responses and information about photon lifetimes. As we demonstrated, the nonlinear behavior that results from the strong confinement of the photons can only be revealed through time domain measurement. In addition, since a high-Q system has a long photon lifetime, a time-domain measurement is potentially more accurate than a spectrum measurement in terms of characterizing the photon lifetime of ultrahigh-Q systems.
Figure 4(a) is a schematic diagram of the cavity ring-down measurement. We input square shaped pulses to charge and discharge the cavity. Once we had filled the cavity with photons, we suddenly turned the input light off and monitored the signal discharging from the PhC nanocavity at the output waveguide using a time correlated single photon counter (TCSPC) . Since the photon density scales as Q/V for a given input energy, it is important to employ a weak input signal to prevent the cavity from exhibiting any nonlinear effects such as two-photon absorption [20,32,33]. Figure 4(b) shows the measured waveform, where a smooth exponential decay was observed. We fitted the data by using the following function, y 0 + Aexp[-(t - t 0)/τph], where y 0, A, t 0, and τph are the background, amplitude, temporal offset and photon lifetime, respectively. The obtained photon lifetime was 1.12 ns for Fig. 4(b), where there is good agreement with the value obtained using the spectral domain measurement. We performed the same measurement 16 times to validate its accuracy. The first 6 results are shown in Fig. 5(a-f). We obtained good reproducibility for the 16 graphs, where the mean value was 1.12 ns with a standard deviation of 0.08 ns. Note that our setup has a measurement resolution of about 70 ps. Again, this photon lifetime is the longest yet demonstrated for an ultrahigh-Q PhC nanocavity.
Here, we discuss the correlation between the temporal resolution of the ring-down measurement and the accuracy of the obtained τ ph value. We prepared different PhC nanocavities with different Q values, and the results of the spectral measurement are shown in Fig. 6(a-e). We performed a ring-down measurement for these PhC cavities, and the results are shown in Fig. 7. The cavity discharging time agrees well with the photon lifetime obtained with the spectral domain measurement shown in Fig. 6. Two waveforms with slow decay [Fig. 7(a) and (b)] were obtained by using TCSPC with a time resolution of 70 ps. In contrast, fast-decaying waveforms [Fig. 7(c-e)] were acquired by using a digital sampling oscilloscope (DSO) with a 28 GHz optical head. With TCSPC, we observed a lifetime difference of a few tens of ps between Fig. 6 and 7, which is not a significant error for a cavity with a relatively high Q. However, it becomes critical when the cavity Q is of the order of 105. In fact, with TCSPC we measured a photon lifetime of 360 ps for a cavity that had a Q spec of 3.9×105 (Q spec is the Q derived from spectral measurement), which corresponds to a photon lifetime of 320 ps in the spectral domain. The difference becomes more pronounced in Fig. 6(d). TCSPC results in a discharging decay of 260 ps for this cavity, which does not give an accurate Q value. Fortunately, for a lower cavity Q, we can input higher optical intensity without inducing a nonlinear effect. Therefore, we can use a fast InGaAs detector and obtain reasonably accurate results even for lower Q values. Figure 8 summarizes the Qs values obtained in the spectral and time domains, which shows that good agreement can be achieved between the spectral and time domain measurements by choosing an appropriate detector.
We would like to emphasize that by carefully comparing the spectral and temporal measurements, we were able to confirm the accuracy of our measurement results. For low-Q systems, the Q time/Q spec value obtained in Fig. 8(b) is greater than 1, because the temporal resolution of the time domain measurement is less than the corresponding bandwidth of the spectral measurement. In contrast, the value should be less than 1 for the extremely high-Q systems owing to the limited wavelength resolution of the spectral measurement. Therefore, we expect time-domain measurement to become an indispensable tool for the characterization of future ultrahigh-Q PhC nanocavities, when we recall that the Q values of PhC nanocavities are rapidly increasing.
2.3. High-Q measurement of a side-coupled cavity in time domain
In previous sections, we discussed direct-coupled cavities where the transmittance spectrum exhibits a peak. In contrast, side-coupled cavities exhibit a spectral dip at the resonance. Side-coupled cavities are useful configurations for extending PhC nanocavities to such applications as wavelength division add drop filters. However, for such cavities, it is sometimes difficult to characterize the Q in the spectral domain. Therefore, here we would like to describe the time-domain measurement of a side-coupled cavity designed to achieve an accurate characterization of the Q.
Figure 9(a) shows the transmittance spectrum from a side-coupled PhC nanocavity, which is resonant at a wavelength of 1577.65 nm. Because there is interference between the PhC nanocavity output and the transmitted light, resonant light is reflected back toward the input waveguide. Although it should be possible to characterize the Q of the cavity by the width of the spectral dip, it is sometimes difficult to measure this accurately. This is due to the Fabry-Pérot oscillation caused by the reflection at the facets ends. We would like to emphasize that time-domain measurement is a powerful tool with which to characterize Q even in such a case, because it can separate information of the Fabry-Pérot reflection from that of charging and discharging signals of the PhC nanocavity. Figure 9(b) shows the output waveform from the PhC τ ph = 220 ps, which corresponds to a Q of 2.6 × 105 and a spectral width of 6.0 pm. According to the result shown in Fig. 8, this Q value should be accurate although it cannot be derived accurately from the spectrum shown in Fig. 9(a).
3. Pulse delay of high-Q cavity system
3.1. Experimental setup and principle
Recently, a maximum half-bit shift was demonstrated for a 1 Gb/s optical pulse using the CROW system . Since the basic element of the CROW is a high-Q optical cavity, the largest group velocity demonstrated by a single cavity should correspond to the maximum performance of the CROW. Previously, we demonstrated a pulse delay of 1.45 ns (5.8 km/s) by using a cavity with Q = 7.8×105. An indication of a pulse delay has also been observed at a cavity with a lower Q . Here, we carefully investigated the pulse delay phenomenon of the PhC nanocavity with the highest Q by employing experiments and numerical calculations to try to reveal the optimum performance achievable with a single cavity.
The experiment was performed by comparing the pulse output of a cavity and that of a reference waveguide as shown in Fig. 10. The peak of the pulse is measured as a delay. Although the pulse delay and the photon trapping of the cavity are closely related, fundamentally they should be explained in different ways. It should be noted that the dispersion property of the cavity should be considered to obtain an understanding of the pulse delay. Figure 11(a) and (b) show the transmittance and the dispersion property of a Fabry-Pérot cavity with respect to the normalized frequency. As shown in Fig. 11(a), because the transmission bandwidth of a high-Q cavity is limited, short pulses exhibit pulse width broadening due to the Fourier bandwidth limitation. However, this does not explain the pulse delay phenomenon. The pulse delay is caused by the dispersion of the cavity as shown in Fig. 11(b), which has a Kramers-Krönigrelation with Fig. 11(a). By calculating the differential of the dispersion curve we can obtain the pulse delay with respect to the frequency as shown in Fig. 11(c). Since the tilt of the dispersion curve at the center wavelength is 2τ ph, it should be possible to achieve a maximum delay of 2τ ph for a relatively long input pulse that has the same center wavelength as the cavity. As regards the dispersion property, the pulse delaying phenomena of PhC nanocavities have a similar physical appearance to those of slow light physics with respect to electromagnetically induced transparency [34, 35] in an atomic system.
3.2. Experimental result
According to the discussion in Fig. 11, a larger pulse delay should be possible for an input pulse with a narrower bandwidth. Thus, we generated Gaussian-like optical pulses with various pulse widths by employing a lithium-niobate electro-optic modulator and an electrical pulse shaper and used them to analyze the pulse response of a PhC nanocavity. The measured results are shown in Fig. 12 for different input pulse widths. We observed a larger pulse delay for wider input pulses. It should be recalled that the τ ph value is 1.12 ns for this cavity. We obtained a maximum delay of ~ 1.5 ns, which is very large if we consider the size of the cavity (a few μm). A pulse shift of Θ ~ 0.44 was obtained in Fig. 12(b). It should be noted that even though this pulse shift does not appear sufficiently large for useful applications, in terms of the basic element of the CROW system, we should be able to achieve a much larger delay at an ultrahigh packing density.
3.3. Numerical analysis and discussion
To discuss the experimental result, we performed a numerical analysis based on coupled mode theory . We modeled the input and output of the cavity as follows:
Pin, Pout, and ω 0 are the input power, output power, and center angular frequency of the cavity resonance, respectively. For simplicity, we assumed a two-dimensional model, where the cavity Q is determined by the coupling between the cavity and the waveguides. The calculated Pin and Pout with respect to the normalized time are shown in Fig. 13. As expected, a larger delay is obtained for a larger input pulse width.
To investigate the pulse delay in detail, we plotted the pulse delay as a function of the input pulse width. The result is shown by a black curve in Fig. 14(a). As the width of the input pulse increases, the delay τ d gradually approaches 2τ c, as predicted by Fig. 11(c). However, it appears possible to achieve a pulse delay of about 1.6τ c in a practical situation. In fact, the calculation result agrees well with the experimental result shown in Fig. 12. Figure 14(a) also shows the pulse shift as a function of the input pulse width. Although the pulse width becomes larger for wider input pulses, the pulse shift exhibits a peak at an input pulse width of ~ τ ph.
The maximum pulse shift is 0.43, which agrees well with our experimental results. In addition, to investigate the influence of detuning between the input light wavelength and the cavity resonance, we calculated the delay with respect to detuning in Fig. 14(b) for an input pulse width of τ ph. The pulse shift is not very sensitive to wavelength detuning, however a small effect is apparent. If the detuning is about 2∆λ, where ∆λ is the FWHM of the cavity transmittance spectrum, the pulse delay exhibits a clear reduction. Therefore, the center wavelength of the input pulse should be adjusted carefully with the resonance of the PhC nanocavity to obtain a large τ d. From a technical viewpoint, it is particularly important with an ultrahigh-Q cavity system to minimize any temperature fluctuation during the pulse delay measurement to obtain an accurate τ d.
In the future, we expect to obtain a pulse shift of greater than 1 by connecting ultrahigh-Q PhC nanocavities in tandem to construct a CROW system. As discussed in this section, we need a high-Q cavity system to achieve a large delay at a high packing density.
4. Ultrahigh-Q width-modulated line defect PhC nanocavity with closely positioned waveguides
Although we obtained small group velocities by employing the PhC nanocavity with the highest Q, an alternative way to achieve a small velocity is to reduce the device size. Since the distance between the input and output waveguides is almost to the same as the closest position of another cavity, there is the potential to achieve a high packing density for CROW if we can reduce this value.
With this in mind, we redesigned the coupling of the cavity with the waveguides. Fig. 15(a) shows the designed PhC nanocavity where the input and output waveguides are coupled on both sides of a PhC nanocavity. The PhC nanocavity design is the same as that in Fig. 1(a); therefore, the unloaded Q value is also the same. The measured transmittance from this cavity has a spectral width of 2.3 pm and a Q of 6.7 × 105. The corresponding photon lifetime is 560 ps. Such a high loaded Q is achieved, because the coupling between the waveguides and the cavity remains small even though the waveguide ends are designed to be closer to the cavity than those in Fig. 1(a). The distance between the two waveguide ends is 5.0 μm. The coupling is small because the mode profile of the cavity is stretched in the longitudinal direction and a smaller portion of the optical mode penetrates in a direction perpendicular to the line defect direction.
Finally we demonstrated the pulse response of this cavity. The measurement results are shown in Fig. 15(b) and (c) for different input pulse widths. The obtained pulse delays are 0.7 and ~1.1 ns, which correspond to a group velocities of 7.2 and ~ 4.6 km/s, respectively. Since the group velocity obtained in Fig. 15(c) is the smallest value achieved on a silicon chip, we believe it to be a promising candidate for the basic element of an ultra-compact CROW system.
We systematically investigated the time domain and spectral domain response of the highest Q of any previously reported PhC cavity. We provided information about the highly accurate measurement of a high-Q PhC nanocavity system, and showed that good agreement is obtained between the spectral and temporal Q. In addition, by employing the cavity with the highest Q, we achieved an extremely large pulse delay of ~1.5 ns. Furthermore, we tried to obtain a higher packing density by redesigning the position of the cavity and the input and output waveguide ends, where we obtained a group velocity of ~4.6 km/s. As regards the use of this cavity as a basic element of the CROW system, the small group velocity enables us to realize a high packing density for an optical buffer or an optical bit-shifter on-chip.
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