HILBERT Transformer (HT) plays an important role

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3704 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 20, OCTOBER 15, 2014 Photonic Hilbert Transformer Employing On-Chip Photonic Crystal Nanocavity Jianji Dong, Aoling Zheng, Yong Zhang, Jinsong Xia, Sisi Tan, Ting Yang, and Xinliang Zhang Abstract We propose and experimentally demonstrate a photonic Hilbert transformer using an on-chip silicon photonic crystal nanocavity (PCN). We prove that a PCN can implement photonic Hilbert transformation after certain structure design. The PCN consists of a cavity of three-defect-long (L3) type with a lattice constant of a = 420 nm and a hole radius of r/a =0.3. We demonstrate Hilbert transformations of input Gaussian pulses with different pulsewidth, and all the average deviations are less than 5.8%. The PCN has a small size of 10 μm 10 μm, which may be useful in high-density on-chip spectral shaper. Index Terms Microwave photonics, optical signal processing, photonic crystals. I. INTRODUCTION HILBERT Transformer (HT) plays an important role in communications, computing, information processing, and signal analysis in electronic domain [1]. A photonic Hilbert transformer (PHT) is expected to have a similar range of applications, which would enable the direct processing of microwave signals in optical domain, and overcome the electronic bottleneck. Several recent investigations have been conducted on implementations of a PHT. However, most of these proposed PHTs used discrete and bulky components such as free space optical components [2], Mach Zehnder interferometers with optical delay lines [3], [4], a fiber grating and circulator [5] [8], a programmable wave-shaper [9], and an incoherent delay-line filter operating with multiple lasers [10], [11], etc. These approaches would cause high system complexity, high cost, and lack of compactness. In fact, silicon-based waveguide can offer distinct advantages of increased stability, reliability, compactness, and capability of integration with electronics. Previously, we demonstrated other functionalities for optical computing with integrated silicon waveguides, such as photonic differentiators [12], [13], and differential equation solvers [14], [15], etc. To realize compact optical modules for similar applications, integrated photonic devices are highly desirable. In [16], a PHT Manuscript received December 19, 2013; revised March 6, 2014 and May 3, 2014; accepted January 28, 2014. Date of publication June 1, 2014; date of current version September 1, 2014. This work was supported in parts by the National Natural Science Foundation of China under Grants 60901006, 11174096, and 61177049, the National Basic Research Program of China under Grants 2011CB301704 and 2013CB632104, the Program for New Century Excellent Talents in Ministry of Education of China under Grant NCET-11-0168, and the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant 201139. The authors are with the Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: jinsongxia@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2014.2327644 using a Si 3 N 4 -based ring resonator with a size of 70 μm 6 mm (width length) was reported. The Si 3 N 4 waveguide has a distinct advantage of very low loss with 0.1 db/cm. Thus, a very high Q value of resonators can be obtained. However, due to large bending radius design, the free spectral range of chips is only 15 GHz, corresponding to a small PHT bandwidth (3 GHz 12 GHz). In [17], a PHT based on planar Bragg gratings, fabricated on a silica-on-silicon substrate with a size of 6 μm 10 mm, was experimentally demonstrated. An impressive single sideband modulation (SSB) filter was also demonstrated with integrated Hilbert transformer and flat top reflector. It was claimed that the operation bandwidth could be up to 100 GHz. However, the rejection ratio at resonant wavelength was only about 10 db, which may degrade the PHT performance. Besides, silica waveguide features a large footprint due to its small refractive index difference. In this paper, we experimentally demonstrate a PHT using a silicon-on-insulator (SOI) photonic crystal nanocavity (PCN) with a much smaller size of 10 μm 10 μm. Due to the large refractive index difference in SOI waveguide, the chip can be fabricated with very small size. The PCN we fabricated consists of a cavity of three-defect-long (L3) type with a lattice constant of a = 420 nm and a hole radius of r/a =0.3. APHT is successfully demonstrated using the PCN, and the average deviations are less than 5.8% for Gaussian pulses injection with different pulsewidths. The effective operation bandwidth is from 25 to 100 GHz, suitable for millimeter-wave processing. II. OPERATION PRINCIPLE The transfer function of an ideal HT is defined as [1] j, if ω>ω 0 H(ω) = j sgn (ω ω 0 )= 0, if ω = ω 0 (1) j, if ω<ω 0 where j = 1, and ω and ω 0 represent the angular frequency and central angular frequency, respectively. It can be seen that the phase response of HT has a π shift at ω = ω 0, whereas the amplitude remains constant, as shown in Fig. 1. In what follows, we aim to prove that the transfer function of an on-chip PCN accords with Eq. (1) at certain approximation. The schematic of a 2-D photonic crystal slab including a cavity and a waveguide is shown in Fig. 2. The amplitudes of the incoming wave coupled to the waveguide and the outgoing wave reflected by the point-defect cavity are denoted by S +1 and S 1, respectively. The amplitudes of the outgoing wave to output facet and the cavity mode are denoted by S 2 and a, respectively. The decay rates from the cavity into the waveguide and into free 0733-8724 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

DONG et al.: PHOTONIC HILBERT TRANSFORMER EMPLOYING ON-CHIP PHOTONIC CRYSTAL NANOCAVITY 3705 Fig. 1. The amplitude response and phase response of an ideal Hilbert transformer. Fig. 3. The simulated amplitude response and phase response of PCN. Fig. 2. Schematic of a 2-D photonic crystal slab including a cavity and a waveguide. space are denoted by 1/τ in and 1/τ v, respectively. The decay rates are related to the in-plane quality factor (Q in ) and vertical quality factor (Q v )byq in = τ in ω 0 /2 and Q v = τ v ω 0 /2. The propagation constant is denoted by β. Positions of the three holes adjacent to the cavity are optimized to obtain a high Q v [18]. Then, the equations for the evolution of the cavity modes and the outgoing waves in time domain are governed as follows [18], [19]: ( da dt = jω 0 1 1 τ v τ in 1 ) 1 a + e jβd S +1 (2) τ in S 1 = e jβd a (3) τ in ) 1 S 2 = e (S j2βd +1 e jβd a. (4) τ in Based on Fourier transformation, frequency response of the cavity can be expressed by H cavity (ω) = S 2 = e j2βd j (ω ω 0 )+1/τ v (5) S +1 j (ω ω 0 )+1/τ v +1/τ in where e j2βd implies a constant phase delay, which can be ignored for simplicity. From Eq. (5), we can get a π phase shift at ω = ω 0 approximately when 1/τ in 1/τ v.fig.3showsthe amplitude response and phase response of the PCN, where the Fig. 4. (a)the deviations with different value of τ v /τ in, (b) temporal waveforms of input Gaussian pulse, ideal Hilbert transformer and the designed PHT. resonant wavelength is set at 1555.25 nm, τ v is set at 0.67 ns, and τ v /τ in is set at 500. As shown in Fig. 3, the amplitude response approximates a constant except for a narrow notch at the central frequency, and the phase shift does exist. A nonzero slope of the phase profile is also observed, which is caused by the constant phase delay mentioned earlier. And a calibrated phase response with a zero-slope profile is also shown, with a frequency range from 50 to 50 GHz. Comparing Figs. 1 and 3, we can see that the PCN can implement a PHT at certain approximation. To analyze the accuracy of the designed PHT, we define an average deviation as the mean absolute deviation of the output power of the designed PHT from the ideal one on certain pulse period [20]. Fig. 4(a) shows the calculated deviations as a function of different value of τ v /τ in, where the full width at half maximum (FWHM) of the input pulse is 9 ps, τ v is set at 0.67 ns, and the integral time is about 300 ps. As shown in Fig. 4(a), the deviation decreases with the increase of the value of τ v /τ in, that is to say, the PCN structure should have a low Q in and a high Q v, where the latter is determined by the

3706 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 20, OCTOBER 15, 2014 Fig. 6. Measured and simulated transmission spectra for the PCN, (a) measured amplitude response for the PCN fabricated, (c) simulated phase response, (b) and (d) contrast of measured and simulated transmission spectra. Fig. 5. (Color online) (a) SEM image of PCN, (b) (e) zoom-in SEM images of the grating coupler, coupling region, and details of the cavity region, respectively. coupling loss to free space. We optimize the positions of the three holes adjacent to the cavity to decrease the coupling loss so as to obtain a high τ v /τ in [18]. Fig. 4(b) shows the temporal waveforms of input Gaussian pulse, ideal Hilbert transformer, and the designed PHT, where the FWHM of the input pulse is 15 ps. The deviation is caused by the approximation of the transfer function of the PCN, and the asymmetry of the output waveform of the designed PHT in Fig. 4(b) is mainly caused by the phase shift approximation. The phase shift of PCN is always less than π, as shown in Fig. 3. We then designed and fabricated a PCN on an SOI wafer for the purpose of PHT, as shown in Fig. 5. The PCN was fabricated on a SOI wafer (220 nm thick silicon on 3 μm thick silica). We used E-beam lithography (Vistec EBPG 5000 Plus) to define the photonic crystal structures on a ZEP520A resist. Then, the pattern was transferred to the top silicon layer by inductively coupled plasma (ICP) etching using SF 6 and C 4 F 8 gases. The upper silicon layer was etched downward for 220 nm to form a silicon ridge waveguide and etched downward for 70 nm to form input/output grating couplers. The couplers have a period of 630 nm, and the duty cycle is 50%. The coupling efficiency for one coupler is about 5.1 db, thus the maximum transmission includes two couplers is about 10.2 db [21]. In order to strengthen optical confinement in normal direction and increase symmetry of the structure, the buried silicon oxide (BOX) was removed with a dilute hydrofluoric acid solution. Finally, we fabricated a L3 cavity with a lattice constant of a = 400 nm and a hole radius of r/a =0.3. We optimized the positions of the three holes adjacent to the cavity to obtain a Q v value as large as possible. Fig. 5(a) shows the scanning electron microscope (SEM) image of the fabricated PCN, and (b) (e) are SEM images of grating coupler, cavity region, and details of the cavity region, respectively. The measured transmission spectrum of the PCN was illustrated in Fig. 6(a), while the zoom-in spectrum in Fig. 6(b), from which we can see that the resonant wavelength is 1555.26 nm. The notch bandwidth (marked as B Notch ) at resonant region is about 0.2 nm (25 GHz), which determines the minimum operating frequency of the proposed PHT, because the operation state is more like a photonic differentiator if input pulse bandwidth is less than 25 GHz according to the theory of [22] and [23]. Besides, the rejection ratio at resonant wavelength is as large as 20 db. The measured Q v value for the PCN is about 45 000. The total loss is about 17 db if the coupling loss (about 10.2 db) is taken into account. Fig. 6(d) shows the measured phase response from 8 to 8 GHz. One can see a π-shift at the resonant wavelength. The method of phase measurement can be found in [24]. Fig. 6(c) shows the simulated phase response with a frequency range from 50 to 50 GHz. One can see that the phase response has a linear approximation region (marked as B Phase ), which limits the maximum operating frequency of the PHT. The PHT deviation becomes very large if input pulse bandwidth is larger than B Phase. Therefore, the notch bandwidth of the PCN and the linear phase response restrain the PHT operation bandwidth, i.e., the bandwidth of the input pulse is restrained between B Notch and B Phase. Fig. 6(b) and (d) also shows the contrast of measured and simulated transmission spectra, where the simulated spectrum is calculated by Eq. (5) when τ v is set at 0.67 ns and τ v /τ in is set at 200. One can see the theoretical phase response accords well with the measured one. And the amplitude response in simulation has a larger rejection ratio than that in measurement. Due to the limited resolution of

DONG et al.: PHOTONIC HILBERT TRANSFORMER EMPLOYING ON-CHIP PHOTONIC CRYSTAL NANOCAVITY 3707 Fig. 7. Experimental setup for PHT with a PCN. optical spectrum analyzer (AQ6370B), the real rejection ratio should be larger than the measured value (20 db). III. EXPERIMENTAL RESULTS In order to verify the theoretical analysis and the simulation results, we carried out experiments with the configuration depicted in Fig. 7. A continuous wave (CW) beam generated by the tunable laser diode (TLD) with a precisely tuning resolution of 100 MHz, was externally modulated using a Mach-Zehnder modulator (MZM) and phase modulator (PM) successively. The frequency of the CW beam was aligned to the resonance frequency of the PCN according to the device operation principle. Two polarization controllers (PCs) were placed before the MZM to optimize the incident polarization states. The MZM and PM have a bandwidth of 20 and 40 GHz, respectively, and were both driven by a tunable radio frequency (RF) signal (frequency range: 6 16 GHz). A single mode fiber (SMF, 5 km) was used to compensate the incident chirps so as to generate a Gaussian pulse with narrow pulsewidth. The pulsewidth of input Gaussian pulse can be tuned by changing the frequency of the RF signal. There were two erbium doped fiber amplifiers (EDFA) in the system, of which the first one was used to boost the input optical power, while the second one was used to compensate the power attenuation by the chip loss. We employed vertical grating coupler to couple the optical signal between fiber and silicon waveguide. Finally, the temporal waveform was analyzed by a high speed oscilloscope (OSC) with a bandwidth of 500 GHz (EYE-1000C). First, the frequency of the RF signal was fixed at 13 GHz. Hence, a Gaussian pulse train with a repetition rate of 13 GHz and a pulsewidth of 10.70 ps was generated, as illustrated in Fig. 8(a). The launched peak power before the chip was about 13 dbm. When we fine-tuned the TLD wavelength to align with the resonant notch of the PCN, we measured a temporal waveform of first order Hilbert transform, as illustrated in Fig. 8(b). Then, the frequency of the RF signal was fixed at 11 and 9 GHz, and the pulse generator generated Gaussian pulse trains with repetition rates of 11 and 9 GHz, and pulse widths of 12.69 and 15.40 ps, respectively, as illustrated in Fig. 8(c) and (e). Then, the output waveforms of Hilbert transform were measured, respectively, as illustrated in Fig. 8(d) and (f), respectively. The simulated waveforms of ideal PHT are also illustrated as the dash lines. It can be seen that the measured temporal waveforms of Hilbert transform fit well with the simulated waveforms, ex- Fig. 8. Measured waveforms for Hilbert transform, (a), (c), and (e) are input Gaussian pulses with different pulsewidths, (b), (d), and (f) are output waveforms of Hilbert transform, respectively. cept for a small discrepancy. The deviation may be caused by the imperfect phase response of the chip. To analyze the Hilbert transform accuracy, an average deviation is defined as the mean absolute deviation of the measured PHT power from the calculated one on certain pulse period [20]. The calculated average deviations for Hilbert transform of input pulse widths of 10.70, 12.69, and 15.40 ps are 5.75%, 4.50%, and 5.32%, respectively. Fig. 9 shows the evolution of transmission spectra when implementing the PHT. The frequency of the driving RF signal is 13 GHz. One can see the central frequency of the input Gaussian pulse is aligned with the resonant notch of the PCN. The 3 db bandwidth of the transmission spectra of the PCN is about 25 GHz, and the spectral bandwidth of input pulse train is about 125 GHz, so the PCN mainly functions as a broadband 180 phase shifter between bilateral bands detached by the notch, which is exactly the transfer function of a Hilbert transformer. The single side-band (SSB) modulation is one of the most important applications of PHT [17], [25]. Fig. 10 shows a schematic of SSB modulation process based on PHT. We then simulate optical SSB signal generation with a microwave frequency from 10 to 60 GHz, as illustrated in Fig. 11. One can see that a sideband suppression ratio (SSR) as large as 21.6 db is

3708 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 20, OCTOBER 15, 2014 the average deviations are less than 5.8%. The PCN has a small size of 10 μm 10 μm, which may be useful in high-density on-chip spectral shaper. ACKNOWLEDGMENT The authors would like to thank Dr. Q. Huang in the Center of Micro-Fabrication and Characterization (CMFC) of Wuhan National Laboratory for Optoelectronics (WNLO) for the assistance in device fabrication. Fig. 9. The transmission spectra of the PCN, the spectrum of input pulse train of repetition rate of 13 GHz, and the spectrum of output waveform. Fig. 10. Principle to generate an optical single sideband signal using a PHT. Fig. 11. The sideband suppression ratio for signals with different frequency, and optical SSB spectrum for signal of 30 GHz with sideband suppression ratio of 18.2 db. achieved when the microwave frequency is about 30 GHz, and the SSR decreases to less than 10 db when microwave frequency is larger than 45 GHz. Thus, the linear approximation region of the phase response (B Phase ) mentioned earlier is about 90 GHz. One should notice that when microwave frequency is less than 10 GHz, there is large power attenuation due to the transmission null of the PCN. Thus, the PHT is expected to realize SSB signal generation with a minimum operating frequency of 10 GHz and a maximum operating frequency of 45 GHz. IV. CONCLUSION We have proposed and experimentally demonstrated a PHT using an on-chip PCN. The PCN consists of a cavity of three defect-long (L3) type with a lattice constant of a = 420 nm, and a hole radius of r/a =0.3. We demonstrate Hilbert transformation of input Gaussian pulses of different pulsewidths, and all REFERENCES [1] L. H. Stefan, Hilbert transforms, in The Transforms and Applications Handbook, 2nd ed. Boca Raton, FL, USA: CRC Press, 2000. [2] A. D. McAulay, Hilbert transform and mirror-image optical correlators, Appl. Opt., vol. 39, pp. 2300 2309, 2000. [3] K. 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DONG et al.: PHOTONIC HILBERT TRANSFORMER EMPLOYING ON-CHIP PHOTONIC CRYSTAL NANOCAVITY 3709 [21] D. Taillaert et al., Grating couplers for coupling between optical fibers and nanophotonic waveguides, Japanese J. Appl. Phys., vol. 45, p. 6071, 2006. [22] T. Yang et al., Experimental observation of optical differentiation and optical Hilbert transformation using a single SOI microdisk chip, Sci. Rep., vol. 4, 2014. [23] N. Q. Ngo and Y. Song, On the interrelations between an optical differentiator and an optical Hilbert transformer, Opt. Lett., vol. 36, pp. 915 917Mar. 15, 2011. [24] Z.Tang et al., A high resolution optical vector network analyzer based on a wideband and wavelength-tunable optical single-sideband modulator, Opt. Exp., vol. 20, pp. 6555 6560, 2012. [25] Z. Li, H. Chi, X. Zhang, and J. Yao, Optical single-sideband modulation using a fiber-bragg-grating-based optical Hilbert transformer, IEEE Photon. Technol. Lett., vol. 23, no. 9, pp. 558 560, May 2011. Yong Zhang received the B.S. degree in optical information science and technology from Northwestern Polytechnical University, Xi an, China, in 2010. He is currently working toward the Ph.D. degree in Wuhan National Laboratory for Optoelectronics, School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, China. His current research interests include design, fabrication and application of photonic crystal cavity. Jinsong Xia received the B.S. degree in physics from the University of Science and Technology of China, Hefei, China, in 1999, and the Ph. D. degree in microelectronics and solid-state electronics from the Institute of Semiconductors, Chinese Academy of Sciences, Beijing, China, in 2004. In 2004, he joined the Advanced Research Laboratories, Tokyo City University as a Postdoctoral Research Fellow. Since 2010, he has been with the Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China, as a Professor. His research interests include siliconbased photonic devices. Jianji Dong received the Ph.D. degree in optoelectronics engineering from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 2008. From Feb. 2006 to Aug. 2006, he worked in Network Technology Research Centre, Nanyang Technological University, Singapore, as an Exchange Student. From Nov. 2008 to Feb. 2010, he worked in the Centre of Advanced Photonics and Electronics, Cambridge University, U.K., as a Research Associate. He is currently an Associate Professor in the Wuhan National Laboratory for Optoelectronics, HUST. He is working on silicon photonics devices, novel spatial division multiplexing, and integrated microwave photonics. He has made some contributions in broad areas, such as ultrafast photonic differentiator using silicon waveguides, microwave signal processing, ultra-wide band signal generation and modulation, and so forth. He has published more than 80 journal papers, including OSA, IEEE group. He is the Principal Investigator of two funds from the National Natural Science Foundation of China. He received the National Excellent Doctor Dissertations Award in 2010, which is the highest award for Ph. D. students in China. Sisi Tan received the B.E. degree in optoelectronics engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2011, where she is currently working toward the M.E. degree in Wuhan National Laboratory for Optoelectronics, School of Optical and Electronic Information. She is currently working on optical temporal differentiator. Ting Yang received the B.S. degree in optical information science and technology from the Wuhan University of Technology, Wuhan, China, in 2011. She is currently working toward the M.E. degree in Wuhan National Laboratory for Optoelectronics, School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, China. She is currently working on arbitrary waveform generation. Aoling Zheng received the B.E. degree in optoelectronics engineering from the Huazhong University of Science and Technology, Wuhan, China, in 2012, where he is currently working toward the M.E. degree in Wuhan National Laboratory for Optoelectronics, School of Optical and Electronic Information. He is currently working on application of silicon devices. Xinliang Zhang received the Ph.D. degree from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 2001. He is currently a Professor with the Wuhan National Laboratory for Optoelectronics and the Dean of School of Optical and Electronic Information, HUST. He has published 60 related papers in international journals or conferences proceedings. His current research interests include all-optical signal processing and related components.