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We fabricate and demonstrate an electro-optic modulator that utilizes the randomness in a photonic crystal waveguide. We exploit a way of using random photonic crystals for device application that involves restricting the area influenced by the randomness. Our random photonic crystal waveguide is in a diffusive regime and the confinement of light is observed only for a W0.98 waveguide (98% of the original width) placed between W1.05 photonic crystal waveguides, where we obtained a transmittance spectrum with an ultra-high Q of 2.4 × 105. A numerical investigation revealed that the experimental yield rate of the appearance of the high-Q confined mode is larger than 80%, by properly designing the length of W0.98. Since the confinement location is predictable, we integrate a p-i-n structure and demonstrate a GHz electro-optic modulation.
© 2016 Optical Society of America
1. Introduction
Silicon photonic crystals (PhCs) have been actively studied due to their high ability to confine light in a small space, which enables us to fabricate various functional devices with small footprints. Ultra-high-Q nanocavities have been demonstrated on a PhC platform [1,2] that can enhance light-matter interaction when the device has a high Q/V, where V is the mode volume. As a result of the strong confinement of light, devices such as ultra-low-power all-silicon Raman lasers [3], all-optical switches [4] and cavity quantum electrodynamic devices [5] have been fabricated. On the other hand, PhC waveguides (WGs) are also attractive for various applications such as slow-light propagation [6,7]. It is shown experimentally that the group velocity is very small when the wavelength is at the cut-off (we call it a mode gap) of the PhC-WG. Propagation with slow light will enhance the light-matter interaction, but in practice, the disorder of the PhC-WG due to fabrication error is limiting the minimum available group velocity. Thus, research has been conducted to understand the effect of the disorder in a PhC-WG [8–13].
On the other hand, the disorder in a photonic crystal is used to realize random lasers [14]. Even without fabricating an artificial cavity structure, light localization occurs and reaches the lasing threshold when the structure contains disorder. However, this device is often difficult to use as a practical application, since it is not possible to control the position or the direction of the light emission. One potential solution in random photonic crystal research is to find a way to control the randomness even though it sounds contradictory.
In this paper, we discuss a way to restrict the area influenced by the randomness in a PhC-WG and consider how to utilize it for practical device applications such as an electro-optic (EO) modulator. In Sec. 2, we explain the idea and perform calculations using the finite-domain time method (FDTD) to discuss the transmittance property of the device. We show that the transmittance is efficiently high when the PhC-WG is in a diffusive regime, which will be achieved when the device is fabricated with the photolithographic process often used in silicon photonics foundries. In Sec. 3, we describe an experimental demonstration of the light confinement of an area-restricted random PhC-WG and show that the result is in good agreement with our numerical calculation. We then discuss the yield rate of the appearance of high-Q resonance both experimentally and numerically since this information is important if we aim to utilize this PhC-WG as a practical device. In Sec. 4, we demonstrate EO modulation at a speed of 1 GHz. Finally, we conclude the paper in Sec. 5.
2. Basic idea and numerical calculations
2.1 Basic idea of controlling randomness
The expression controlling randomness is contradictory, but it is a problem that we have to solve if we want to utilize PhCs that have large fabrication errors. Understanding the effect of randomness is becoming more important, since we have started to use photolithography to fabricate PhC devices. Devices fabricated with this technology usually contain more disorder than those fabricated with electron-beam lithography. Although studies of random PhC lasers have often been driven by a theoretical motivation, it is not easy to use it as a practical device, because the position and the direction of the light emission are completely uncontrollable. When we use randomness, it is impossible to completely control the position, resonance, and yield rate of the appearance of the localization. However, if we can find a way to partly control these properties, we can make it possible to use devices that employ randomness in practical applications.
The structure of our PhC-WG is almost the same as those studied by other researchers, except that a W0.98 PhC-WG (98% of the original width of a W1 PhC-WG) is installed between two W1.05 PhC-WGs, as shown in Fig. 1(a). W1 PhC-WG is defined as a line-defect waveguide in a hexagonal lattice photonic crystal [15,16]. Since W0.98 is less than W1.05, the mode-gap frequency of the W0.98 WG is higher than that of the W1.05 WG, as shown in Fig. 1(b). The mode-gap has the same frequency throughout the W0.98 and W1.05 WGs; hence the input light reflects back at the edge of the W0.98 WG and does not transmit to the other side when the frequency is below the mode gap of W0.98.
Fig. 1 (a) Designed structure of PhC-WG, which consists of a W0.98 waveguide in between W1.05 waveguides. This two-dimensional PhC is based on a silicon slab with a hexagonal lattice clad with SiO2. The inset shows the facet structure. (b) Band structure of designed (ideal) PhC-WG. Blue and orange indicate the stop-band of the WG. The red arrow indicates the input light injected from the left side of the structure. (c) As (b) for the fabricated device containing disorder.
However, with our fabricated device, there is disorder in the PhC-WG and it causes a frequency fluctuation at the mode gap. Specifically, we will use photolithography for the fabrication because it is suitable for mass production but has low spatial resolution, and hence we expect our device to have few fabrication errors. When disorder is present, light can tunnel through the W0.98 WG and reach the output if the light frequency is close to the mode gap of the W0.98 WG [Fig. 1(c)]. In addition, light confinement occurs at the W0.98 WG. This W1.05/W0.98/W1.05 design is suitable for limiting the area affected by disorder, since the effect only occurs at the W0.98 waveguide. It also offers some controllability of the frequency, since the light confinement occurs close to the mode-gap frequency of the W0.98 WG. The fluctuation at W1.05 does not influence the transmittance of the device, since the mode-gap frequency is much less than W0.98. Since we can restrict the confinement area at W0.98 by adopting a suitable design, we will be able to fabricate devices such as EO modulators by integrating a p-i-n junction in the W0.98 region.
2.2 Categorizing the randomness in a PhC-WG
Before conducting a numerical study, we discuss the categorization of the random PhC-WG depending on the magnitude of the randomness. There are usually three different regimes, dispersive, diffusive and localized. When the PhC-WG is in a dispersive regime, light propagation is not greatly affected and the WG usually has a low loss [16]. In a diffusive regime, the backscattering in a PhC-WG becomes apparent. Since the amount of randomness is still in a moderate regime, a large part of the backscattered light couples back to the PhC WG mode. Therefore, the propagation loss is larger than the dispersive regime, but it is not severe. When the randomness increases further, the light starts to localize and the WG is now in a localized regime. The transmittance in this regime is usually very low due to disorder induced scattering where the light does not couple back into any PhC WG modes.
The diffusive and localized regimes are distinguished by the relationship between the mean free length lc and the length L of a PhC-WG. When L ≈lc it is regarded as a diffusive regime, since this relationship shows that the light propagates with little backscattering [17]. When L >> lc, the WG is considered to be in a localized state because the light exhibits multiple scattering and is trapped and localized along the PhC-WG.
2.3 Numerical calculation of light confinement
We performed a two-dimensional finite difference time domain (2D FDTD) calculation for 4 different W0.98 waveguide structures (lengths of 22, 28, 34 and 40 periods) as shown in Fig. 1(a). The lattice constant, hole radius, effective refractive index of the slab and the refractive index of SiO2 that fills the PhC holes are, a = 420 nm, r = 0.345a, neff = 2.81 and nsilica = 1.44, respectively. The randomness of the hole radius and the position is taken into account. A schematic is shown in Fig. 2(a). The hole position has a deviation σpos of 2 nm and the radius σrad is 1 nm, both of which are obtained from our scanning electron microscope measurement [18,19]. The hole position shifts at a distance of σpos towards a random angle θ. A calculated transmission spectrum where L = 40a is shown in Fig. 2(b), in which we observe a sharp peak at the mode gap. Figure 2(c) is the mode distribution excited with a frequency at a peak in Fig. 2(b). From these results we confirmed that light confinement occurs in the W0.98 waveguide, and the randomness in W1.05 do not influence the propagation property of this device.
Fig. 2 (a) Schematic illustration of the hole position and radius with deviation σpos and σrad. σpos of 2 nm and σrad of 1 nm is used in our 2D FDTD calculation. (b) Transmission spectrum for W0.98 length of L = 40a. A sharp peak appears close at the mode gap at 1587.5 nm. (c) Calculated spatial distribution of Hz component when light at the wavelength of the peak shown in (b) is excited.
We performed the calculation 18 times for each length. Since lc is given by <ln(T)> = −lc/L, where T is the transmittance of the PhC-WG, we know in which regime this system is located. By taking the average of the 18 calculations, we obtained lc = 18 μm when L was 16.8 μm (L = 40a). This satisfies the requirement that L ≈lc, hence the system is in a diffusive regime. Indeed, the transmission spectrum of the peak that appears in Fig. 2(b) is high, which is one of the characteristics of a diffusive random PhC-WG. Through our numerical analysis, we confirmed that all four lengths are in the diffusive regime and light confinement occurs only in the W0.98 waveguide as shown in Fig. 2(c). Diffusive regime is attractive, because the transmittance of such device is kept high. In addition, we are able to integrate the device with a p-i-n diode, when the light confinement only occurs in a specific area. PhC-WGs with large L have low transmittance and prevent us from having sufficient control of the position of the light confinement. For such a purpose, it is important to know the yield rate of the light confinement, because this gives the probability of the successful fabrication of nanocavity devices. Next, we will report an experimental result, and then discuss the yield rate.
3. Experiment and discussion of demonstration of restricted area light confinement
3.1 Experimental demonstration of light confinement
The photolithographic fabrication of PhC devices is the key to moving the study towards a practical stage. Our PhC device was fabricated in a silicon photonics foundry (IME Singapore) using 248-nm KrF photolithography. The fabricated device is clad with SiO2, which ensures robustness and high stability. We previously reported that a high-Q PhC nanocavity can be realized with this platform [18]. In addition, the process uses well developed silicon photonics technologies, so p-n doping and integration with high transmittance spot-size converters are both available.
Figure 3(a) shows a schematic illustration of our PhC-WG integrated with a p-i-n junction. The doping concentration of the p region is 2.4 × 1017 cm−3 and that of the n region is 1.4 × 1017 cm−3. These two regions are implanted at the center of a W0.98 waveguide with a width of 4 hole periods. The width of the i region is 2.9 μm. From our SEM observation, the r/a ratio of our fabricated devices is 0.310.
Fig. 3 (a) Schematic illustration of the fabricated device. The device is clad with SiO2. (b) A transmission spectrum for a device with L = 40a. The confined mode appears at the mode gap and the Q value is 1.5 × 105. (c)-(e) Infrared camera images taken from the top of the slab. The corresponding input wavelength is shown with arrows in (b). Light is injected from the right of the images. Red rectangles denote the area of the PhC-WG and the blue dotted line shows the W0.98 region. The horizontal red lines are the input/output silicon nanowire waveguides. (f) Magnified image of the peak in (b) with different input powers.
We then conducted transmission measurements for various samples, and one of the transmission spectra is shown in Fig. 3(b), where we obtained a similar shape to the one we obtained with the calculation in Fig. 2(b). The Q of this confined mode is very high at 1.5 × 105, and we obtained an even higher Q of 2.4 × 105 with a different sample. The transmittance of this confined mode is higher than 10%, which indicates that the WG is in a diffusive mode as predicted by the calculation. This is important if we want to put this device to practical use since we prefer high transmittance devices.
Figures 3(c)-3(e) are infrared images taken from the top of the slab at different input wavelengths. By comparing these three images, we find that the light is confined in the W0.98 WG region when the input wavelength is at peak resonance. The optical bistable experiment shown in Fig. 3(f) also indicates that the light is well confined [20–23]. It shows the transmittance spectra of the peak at different input powers, where we observe a clear sign of optical bistability based on the thermo-optic effect induced by two-photon absorption, which only occurs when the light is strongly confined in a small area. These results indicate that our design indeed allows us to obtain light confinement, but only in the W0.98 region.
3.2 Yield rate of light confinement at given randomness
We have successfully restricted the light confinement to a specific area that may allow us to use this device in a controlled way. So the next question is: What is the probability that we can obtain such devices? To answer this question, we conducted transmission measurements on 18 samples of the same design. We repeated the measurement for 4 different W0.98 lengths. In addition to the experiment, we also performed numerical analysis 18 times for each W0.98 length. The results are summarized in Fig. 4. Figures 4(a) and 4(b) show Q values and Figs. 4(c) and 4(d) show the transmittance T at a given W0.98 length L. The highest experimental Q value reached 2.4 × 105 and the other Qs were routinely higher than 104. The percentages shown in the panel represent the probability of the appearance of a sharp peak around the mode gap. Here, a peak is defined when the power difference between the top and bottom of the fitted Lorentzian exceeds 10 dB. When L increases, the yield rate of the appearance of a high-Q peak also increases. The simulation and experiment agree well and it is shown experimentally that more than 80% of the random PhC-WGs exhibit a high Q when L is 40 periods. This yield rate is sufficient to realize practical applications. The transmittance T is high and the device is in the diffusive regime when L = 40a [see Figs. 4(c) and 4(d)]. In other words, although there is a tradeoff between the high yield rate, high transmittance and small L at a given fabrication precision, at the precision used in this work (the precision of a photolithographically fabricated device) the optimum L would be around 40 periods to have a sufficiently high yield rate of > 80%. L = 40a is short enough to obtain a good spatial overlap between the p-i-n junction (at the center of the device) and the area of light confinement.
Fig. 4 Yield of a sharp peak for a simulation (a) and an experiment (b). Transmittance, T, corresponding to those peaks are shown in (c) and (d). Red circles exhibit the average and error bars are corresponding to standard deviation.
Figure 5(a) shows one of the calculated transmission spectra of a PhC-WG with an L of 100 periods. Two sharp peaks appear close to the mode gap although just one peak appears when L is in the 22a to 40a range. The peak is split into two, the transmittance is low, and the light confinement occurs far from the center of the structure as shown in Fig. 5(b). Since the misalignment of the position of the light confinement is so large, it is difficult to ensure that there is an overlap between the optical mode and the p-i-n junction. We assessed that our narrow-section length L of 40a lies in the optimal zone of this parameter for our case of a device obtained by photolithography. This optimal length is confirmed by simulation as stemming from the underlying (Q, T, L) statistics associated with the process accuracy.
Fig. 5 (a) Transmittance spectrum of a PhC-WG with L of 100 periods, which is obtained with a 2D FDTD calculation. (b) Field distribution when the structure is excited with 1586.6 nm ± 0.1 nm wavelength light.
4. Electro-optic modulator based on random photonic crystal waveguide structure
Finally, we discuss our demonstration of EO modulation using these devices. We employ the same device as that measured in Fig. 3(f). Figure 6(b) shows the transmission spectra of the device when we apply different forward bias voltages between the p and n junctions. The resonance shifts towards a shorter wavelength due to the carrier-plasma effect, and it decreases due to free carrier absorption. These results indicate that the overlap between the confined mode and the p-i-n junction is sufficiently large.
Fig. 6 (a) Schematic of the EO modulation demonstration. Input continuous wave laser light is modulated at a random PhC-WG device integrated with a pin junction. TLD: Tunable laser diode, EDFA: Erbium doped fiber amplifier, BPF: Bandpass filter (1-nm width), OSO: Optical sampling oscilloscope (Agilent 86103A), PPG: Pulse pattern generator (Keysight 81134A). (b) Transmittance spectrum of the confined mode at different forward bias voltages. (c) Detected output signals when a 500 MHz radio-frequency signal is applied. The red line is at the peak resonance, and the black line is when the input laser is slightly detuned at a wavelength shorter than the resonance. (d) As (c) but with 1-GHz modulation.
Next we performed an EO modulation demonstration using the setup shown in Fig. 6(b). An input continuous-wave laser light is modulated at the random PhC-WG with an electrical radio-frequency (RF) signal with a square pulse. The optical output is detected and shown in Figs. 5(c) and 5(d), where the input RF signal has repetition rate frequencies of 500 MHz and 1 GHz, respectively. The amplitude is 2 Vpp (−0.5 to 1.5 V). We observe a clear ON to OFF contrast in both cases. Since similar speed is obtained with the device for which we fabricated a controlled PhC cavity structure, the origin of the limiting speed is not due to the random property of the device. Also, the Q value is not the limiting factor since the photon lifetime supports a modulation speed of up to ~10 GHz. The speed of this device is somewhat low, which is due to the unoptimized p-i-n structure. The device structure is similar to that of an EO modulator based on a Mach-Zehnder interferometric slow-light PhC-WG modulator [24], except that our device is much shorter than devices where L is about ~500 periods. So we should be able to obtain a similar or higher speed (~10 GHz) by optimizing the p-i-n structure of our device, such as by optimizing the length of the insulator regime and changing the doping concentration. The key advantage of our device is that it may allow us to realize a small footprint.
5. Summary
We studied the conditions for the appearance of a high-Q mode at a mode gap using experiments and numerical analysis. The results are in good agreement and they showed that a high Q exceeding 104 can be obtained at a yield rate of over 80%, simply by fabricating the PhC-WGs with a photolithographic process compatible with silicon photonics. The key to limiting the position of the light confinement is to connect a W1.05 WG on either side of a W0.98 WG, whose length is sufficiently long to yield a high-Q mode.
By using these devices, we integrated a p-i-n structure in the W0.98 WG region and demonstrated GHz EO modulation. This demonstration opens the possibility of utilizing randomness in a PhC structure for practical devices, which is particularly important when we employ photolithographic fabrication methods.
Acknowledgment
Part of this work was supported by the Strategic Information and Communications R&D Promotion Programme (SCOPE), from the Ministry of Internal Affairs and Communications, Japan.
References and links
1. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]
2. H. Sekoguchi, Y. Takahashi, T. Asano, and S. Noda, “Photonic crystal nanocavity with a Q-factor of ~9 million,” Opt. Express 22(1), 916–924 (2014). [CrossRef] [PubMed]
3. Y. Takahashi, Y. Inui, M. Chihara, T. Asano, R. Terawaki, and S. Noda, “A micrometre-scale Raman silicon laser with a microwatt threshold,” Nature 498(7455), 470–474 (2013). [CrossRef] [PubMed]
4. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87(15), 151112 (2005). [CrossRef]
5. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007). [CrossRef] [PubMed]
6. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
7. T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2(8), 448–450 (2008). [CrossRef]
8. J. Topolancik, F. Vollmer, and B. Ilic, “Random high- Q cavities in disordered photonic crystal waveguides,” Appl. Phys. Lett. 91(20), 201102 (2007). [CrossRef]
9. A. Parini, P. Hamel, A. De Rossi, S. Combrié, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouën, and G. Vadalà, “Time-wavelength reflectance maps of photonic crystal waveguides: A new view on disorder-induced scattering,” J. Lightwave Technol. 26(23), 3794–3802 (2008). [CrossRef]
10. M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer-Lambert law,” Phys. Rev. B 80(19), 195305 (2009). [CrossRef]
11. M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-Induced Coherent Scattering in Slow-Light Photonic Crystal Waveguides,” Phys. Rev. Lett. 102(25), 253903 (2009). [CrossRef] [PubMed]
12. N. Le Thomas, H. Zhang, J. Jágerská, V. Zabelin, R. Houdré, I. Sagnes, and A. Talneau, “Light transport regimes in slow light photonic crystal waveguides,” Phys. Rev. B 80(12), 125332 (2009). [CrossRef]
13. N. Mann, A. Javadi, P. D. García, P. Lodahl, and S. Hughes, “Theory and experiments of disorder-induced resonance shifts and mode-edge broadening in deliberately disordered photonic crystal waveguides,” Phys. Rev. A 92(2), 023849 (2015). [CrossRef]
14. J. Liu, P. D. Garcia, S. Ek, N. Gregersen, T. Suhr, M. Schubert, J. Mørk, S. Stobbe, and P. Lodahl, “Random nanolasing in the Anderson localized regime,” Nat. Nanotechnol. 9(4), 285–289 (2014). [CrossRef] [PubMed]
15. 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(25), 253902 (2001). [CrossRef] [PubMed]
16. 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(16), 161318 (2005). [CrossRef]
17. Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, “Different regimes of light localization in a disordered photonic crystal,” Phys. Rev. B 60(3), 1555–1562 (1999). [CrossRef]
18. Y. Ooka, T. Tetsumoto, A. Fushimi, W. Yoshiki, and T. Tanabe, “CMOS compatible high-Q photonic crystal nanocavity fabricated with photolithography on silicon photonic platform,” Sci. Rep. 5, 11312 (2015). [CrossRef] [PubMed]
19. T. Tetsumoto, Y. Ooka, and T. Takasumi, “High-Q coupled resonances on a PhC waveguide using a tapered nanofiber with high coupling efficiency,” Opt. Express 23(12), 16256–16263 (2015). [CrossRef] [PubMed]
20. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13(7), 2678–2687 (2005). [CrossRef] [PubMed]
21. T. Uesugi, B.-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14(1), 377–386 (2006). [CrossRef] [PubMed]
22. X. Yang, C. Husko, C. W. Wong, M. Yu, and D.-L. Kwong, “Observation of femtojoule optical bistability involving Fano resonances in high-Q/Vm silicon photonic crystal nanocavities,” Appl. Phys. Lett. 91(5), 051113 (2007). [CrossRef]
23. L.-D. Haret, T. Tanabe, E. Kuramochi, and M. Notomi, “Extremely low power optical bistability in silicon demonstrated using 1D photonic crystal nanocavity,” Opt. Express 17(23), 21108–21117 (2009). [CrossRef] [PubMed]
24. H. C. Nguyen, Y. Sakai, M. Shinkawa, N. Ishikura, and T. Baba, “10 Gb/s operation of photonic crystal silicon optical modulators,” Opt. Express 19(14), 13000–13007 (2011). [CrossRef] [PubMed]
