Silicon nanowire based radio-frequency spectrum analyzer

Transcription

1 Silicon nanowire based radio-frequency spectrum analyzer Bill Corcoran, 1,* Trung D. Vo, 1 Mark D. Pelusi, 1 Christelle Monat, 1 Dan-Xia Xu, 2 Adam Densmore, 2 Rubin Ma, 2 Siegfried Janz, 2 David J. Moss, 1 and Benjamin J. Eggleton 1 1 Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Institute for Photonics and Optical Sciences (IPOS), School of Physics, The University of Sydney, NSW 26, Australia 2 Institute for Microstructural Sciences, National Research Council (NRC-CNRC), Ottawa, ON, K1A-R6, Canada *billc@physics.usyd.edu.au Abstract: We demonstrate a terahertz bandwidth silicon nanowire based radio-frequency spectrum analyzer using cross-phase modulation. We show that the device provides accurate characterization of 64Gbaud on-offkeyed data stream and demonstrate its potential for optical time-division multiplexing optimization and optical performance monitoring of ultrahigh speed signals on a silicon chip. We analyze the impact of free carrier effects on our device, and find that the efficiency of the device is not reduced by two-photon or free-carrier absorption, nor its accuracy compromised by free-carrier cross-chirp. 21 Optical Society of America OCIS codes: (19.436) Nonlinear optics, devices, (6.4256) Networks, network optimization. References and links 1. M. Nakazawa, T. Yamamoto, and K. R. Tamura, 1.28Tbit/s-7 km OTDM transmission using third- and fourthorder simultaneous dispersion compensation with a phase modulator, Electron. Lett. 36(24), (2). 2. H. C. Hansen Mulvad, L. K. Oxenløwe, M. Galili, A. T. Clausen, L. Grüner-Nielsen, and P. Jeppesen, 1.28 Tbit/s single-polarisation serial OOK optical data generation and demultiplexing, Electron. Lett. 45(5), 28 (29). 3. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Silicon-chip-based ultrafast optical oscilloscope, Nature 456(7218), (28). 4. N. K. Fontaine, R. P. Scott, L. J. Zhou, F. M. Soares, J. P. Heritage, and S. J. B. Yoo, Real-time full-field arbitrary optical waveform measurement, Nat. Photonics 4(4), (21). 5. C. Dorrer, and D. N. Maywar, RF spectrum analysis of optical signals using nonlinear optics, J. Lightwave Technol. 22(1), (24). 6. M. Pelusi, F. Luan, T. D. Vo, M. R. E. Lamont, S. J. Madden, D. A. Bulla, D. Y. Choi, B. Luther-Davies, and B. J. Eggleton, Photonic-chip-based radio-frequency spectrum analyser with terahertz bandwidth, Nat. Photonics 3(3), (29). 7. H. P. Weber, Generation and Measurement of Ultrashort Light Pulses, J. Appl. Phys. 39(13), 641 (1968). 8. T. K. Liang, H. K. Tsang, I. E. Day, J. Drake, A. P. Knights, and M. Asghari, Silicon waveguide two-photon absorption detector at 1.5µm wavelength for autocorrelation measurements, Appl. Phys. Lett. 81(7), (22). 9. D. J. Kane, and R. Trebino, Characterization of Arbitrary Femtosecond Pulses Using Frequency-Resolved Optical Gating, J. Quantum Electron. 29(2), (1993). 1. T. D. Vo, H. Hu, M. Galili, E. Palushani, J. Xu, L. K. Oxenløwe, S. J. Madden, D. Y. Choi, D. A. P. Bulla, M. D. Pelusi, J. Schröder, B. Luther-Davies, and B. J. Eggleton, Photonic chip based transmitter optimization and receiver demultiplexing of a 1.28 Tbit/s OTDM signal, Opt. Express 18(16), (21). 11. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, Nonlinear optics in photonic nanowires, Opt. Express 16(2), (28). 12. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, Four-wave mixing in silicon wire waveguides, Opt. Express 13(12), (25). 13. R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, Raman amplification in ultrasmall silicon-on-insulator wire waveguides, Opt. Express 12(16), (24). 14. R. Soref, and J. Larenzo, All-Silicon Active and Passive Guided-Wave Components for λ=1.3 and 1.6µm, IEEE J. Quantum Electron. 22(6), (1986). 15. Q. Lin, O. J. Painter, and G. P. Agrawal, Nonlinear optical phenomena in silicon waveguides: modeling and applications, Opt. Express 15(25), (27). (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 219

2 16. J. B. Driscoll, W. B. Astar, X. B. Liu, J. I. Dadap, W. M. J. Green, Y. A. Vlasov, G. M. Carter, and J. R. M. Osgood, All-Optical Wavelength Conversion of 1 Gb/s RZ-OOK Data in a Silicon Nanowire via Cross-Phase Modulation: Experiment and Theoretical Investigation, IEEE J. Sel. Top. Quantum Electron. (accepted). 17. B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O Faolain, M. Pelusi, and T. P. White, Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide, Opt. Lett. 35(7), (21). 18. M. D. Pelusi, T. D. Vo, F. Luan, S. J. Madden, D. Y. Choi, D. A. P. Bulla, B. Luther-Davies, and B. J. Eggleton, Terahertz bandwidth RF spectrum analysis of femtosecond pulses using a chalcogenide chip, Opt. Express 17(11), (29). 19. M. Dinu, D. C. Kilper, and H. R. Stuart, Optical performance monitoring using data stream intensity autocorrelation, J. Lightwave Technol. 24(3), (26). 2. T. D. Vo, M. D. Pelusi, J. Schroder, B. Corcoran, and B. J. Eggleton, Multi-Impairment Monitoring at 32 Gb/s Based on Cross-Phase Modulation Radio-Frequency Spectrum Analyzer, IEEE Photon. Technol. Lett. 22(6), (21). 21. R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, Amsterdam; Boston, 28), pp. xix, 613 p. 22. G. P. Agrawal, Nonlinear fiber optics, 3rd ed., Optics and photonics (Academic Press, San Diego, 21), pp. xvi, 466 p. 23. T. D. Vo, M. D. Pelusi, J. Schröder, F. Luan, S. J. Madden, D. Y. Choi, D. A. P. Bulla, B. Luther-Davies, and B. J. Eggleton, Simultaneous multi-impairment monitoring of 64 Gb/s signals using photonic chip based RF spectrum analyzer, Opt. Express 18(4), (21). 24. M. D. Pelusi, T. D. Vo, and B. J. Eggleton, Accuracy of waveform spectrum analysis for ultra-short optical pulses, J. Lightwave Technol. (accepted). 25. M. A. F. Roelens, S. Frisken, J. A. Bolger, D. Abakoumov, G. Baxter, S. Poole, and B. J. Eggleton, Dispersion trimming in a reconfigurable wavelength selective switch, J. 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Introduction Optimal operation of ultrahigh baud rate serial data channels [1,2] requires the temporal characterization of ultra-short optical pulses, a task unable to be achieved by traditional electronic measurement methods. Photonic solutions to this problem have been the focus of recent research, with new techniques such as time-lens assisted optical sampling [3], spectrally sliced coherent detection [4] and cross-phase modulation radio-frequency spectrum analysis (XPM-RFSA) [5,6] being developed. These new techniques are all waveguide based and do not require the large, tunable time delays necessary for well established ultra-short pulse characterization methods such as autocorrelation [7,8] and frequency resolved optical gating (FROG) [9]. Although each of the waveguide-based characterization tools possess their own unique abilities and challenges, XPM-RFSA arguably provides the simplest method for the temporal analysis of pulses, requiring only a highly nonlinear waveguide and optical spectrum analyzer (OSA). Moreover, XPM-RFSA provides a tool for signal optimization and multiple transmission impairment monitoring [1]. Highly nonlinear waveguides, such as tightly confining nanowires in silicon [11 13], are necessary to implement XPM-RFSA efficiently. Silicon possesses a unique advantage over alternative materials, through the billions of research dollars spent to develop this material as a reliable nanofabrication platform for the electronics industry. However, silicon is not an intrinsically ideal material for nonlinear optics in the telecommunications band. A high twophoton absorption (TPA) coefficient and associated generation of free-carriers tend to both hinder the efficiency of Kerr nonlinear effects and complicate optical interactions through free-carrier dispersion (FCD) and free-carrier absorption (FCA) [14 17], which has raised concerns of the usefulness of silicon waveguides as a platform for XPM-RFSA. In this paper, we measure accurate radio-frequency (RF) spectra of 64Gbit/s on-off-keyed (OOK) data gained through cross-phase modulation (XPM) in a silicon nanowire. We also provide examples of the use of the XPM-RFSA technique in optimizing optical time-division (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2191

3 multiplexing (OTDM) as well as measuring residual chromatic dispersion. Utilizing a simplified pulse propagation model, we analyze theoretically the relative magnitude of FCDbased cross-chirp to Kerr-based XPM and investigate the effect of nonlinear losses on high baud rate data. We show that although TPA and FCA are a priori expected to reduce the overall efficiency of silicon based XPM-RFSA, these effects are negligible at powers needed to implement this scheme for characterizing ultrafast optical data. Likewise, FCD is found to have more than two orders of magnitude less effect than Kerr-based XPM, and so does not compromise our measurements of high speed data. 2. Background Radio-frequency (RF) spectral monitoring is a tool commonly used to analyze the temporal characteristics of signals. The RF spectrum is a power spectrum of signal intensity, i.e. the squared magnitude of the Fourier transform of time varying intensity. By analyzing the RF spectrum, one can monitor signal distortions that manifest as temporal variations in intensity. Additionally, through the Wiener-Khintchine theorem, the inverse Fourier transform of a signals RF spectrum provides the autocorrelation of that signal [18]. It has been shown previously that analysis of certain characteristics of a signals autocorrelation can give information about different signal transmission impairments simultaneously, such as optical signal-to-noise ratio (OSNR) and residual group velocity dispersion (GVD) [19,2]. Fig. 1. Silicon-based XPM-RFSA concept. A signal with optical spectrum (S SIG) and a monochromatic CW probe co-propagate in a silicon waveguide. Through cross-phase modulation, the RF spectrum of the signal is superimposed onto the probe wave. The broadened optical spectrum S PRO of the modulated probe is measured with an optical spectrum analyzer (OSA), and the signal RF spectrum (S RF) extracted from S PRO. I denotes a Fourier transform. Traditionally, RF spectra of optical signals have been obtained by measuring varying intensity with a photodetector, followed by analysis in the electronic domain. The limited bandwidth (<1GHz) of photodetectors hampers their usefulness in analyzing the ultra-short optical pulses used in ultrahigh speed serial communications. It is possible to overcome this bandwidth restriction by using all-optical techniques that rely on the ultrafast Kerr nonlinearity, which has a response time in silicon below 1fs [15,21,22], intrinsically able to respond to pulses of bandwidths >1THz. More specifically, all-optical RF spectrum information can be gathered by analyzing the modulation of a CW probe through XPM induced by the optical signal I SIG (t) (I = E 2 denoting signal intensity) to be measured. Given some assumptions, Eq. (1) shows that the amplitude of the modulation sidebands generated on the probe are proportional to the RF spectrum of the signal (S RF (f) - see Fig. 1). 2 ( ) = ( 1 + ϕ( )) exp 2 π ( ) ( ϕ( ) = 2 2 (, ). ) (1) SPRO f t i f f t dt t i ksn ISIG t z dz In Eq. (1), φ(t) is the accumulated cross phase shift on the probe (probe centre frequency f ), the free-space wavenumber and nonlinear index at the signal frequency are given as k S and n 2 respectively. Equation (1) represents the spectral broadening of the probe due to XPM, under the condition that dispersive effects are negligible, the probe is weak compared to the (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2192

4 signal and cross-phase modulation of the probe is small. The modulation of the probe can be considered to be small when φ(t) <1. Additionally, spectral broadening of the probe must be predominantly from Kerr-based XPM as opposed to free carrier cross-chirp. For a derivation of Eq. (1), refer to [5] and references therein. As φ(t) is related to the intensity of the signal, the sidebands on the probe are proportional to the RF spectrum of the signal (Fig. 1). For return-to-zero (RZ) signals, the sidebands themselves present strong tones detuned from the CW probe by a frequency equal to the baud rate of the signal (represented by the spikes on the probe sidebands depicted in Fig. 1). The amplitude of these tones reflects the integrity of the data encoded on the optical signal, a property we make use of to monitor residual dispersion and for OTDM optimization. 3. XPM-RFSA accuracy and applications In this investigation we use a highly nonlinear silicon waveguide to implement a silicon-chip based XPM-RFSA. The waveguide is a silicon-on-insulator nanowire, SiO 2 clad, 45nm wide by 26nm high, 1.5cm in length (Fig. 2b). The TE mode of this waveguide in the optical communications C-band has an effective index n eff ~2.5 and effective area A eff ~.15µm 2, as calculated by a finite element method mode solver (RSoft FEMsim). Coupling loss to the TE mode via a lensed fiber is estimated ~8dB per facet and propagation losses ~3dB/cm. Fig. 2. (a) Schematic of the experimental setup for XPM-RFSA of 64Gbit/s OOK signals. (b) Cross-section schematic of the silicon nanowire waveguide used in experiments. The RF spectrum of a 64Gbaud, pseudo-random bit sequence (PRBS), 33% duty cycle, OOK signal is measured in our device. The signal is constructed from a 4GHz pulse train from a mode-locked fiber laser. The pulses are nonlinearly compressed (similar to ref [23]), and OOK data encoded with a PRBS through at 4Gbit/s. This data stream is then time-division multiplexed (time interleaved with a bit delay) up to 64Gbit/s. The signal and CW probe are amplified, filtered and combined before being launched into the Si nanowire [schematic Fig. 2(a)]. The average coupled signal and probe powers are ~13dBm and ~7dBm respectively. Fig. 3. (a) XPM-RF spectrum of 64Gbit/s 33% duty cycle OOK signal about the probe central frequency. Note the strong tone at 64GHz. (Inset data eye diagram captured using an optical sampling oscilloscope) (b) Signal autocorrelation measured from a commercial SHG autocorrelator (red solid) and extracted from the Si-based XPM-RFSA device (dotted blue). (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2193

5 The output spectrum of the waveguide is measured on a high resolution OSA (specified resolution bandwidth λ in C-band of 1pm) around the frequency of the probe [shown Fig. 3(a)]. The autocorrelation of the signal is extracted from the inverse Fourier transform of the low frequency sideband of the probe. The temporal resolution of the retrieved waveform is inversely proportional to the spectral width of the captured RF spectrum. The resolution of the retrieval is dt = 2π/N.dω, where N is the number of samples in the captured spectrum around a central wavelength λ and dω 2π.λ 2 / λ.c the resolution bandwidth [18]. In these experiments, the temporal resolution of the reconstruction was ~85fs. By comparing an autocorrelation trace generated using this method to one directly measured with a commercial second-harmonic generation (SHG) autocorrelator, we can qualitatively estimate the accuracy of the XPM-RFSA. Figure 3(b) shows a close match between the autocorrelation traces gathered through both XPM and SHG. The central peak FWHM for the XPM autocorrelation is 1.3ps versus 1.1ps for the SHG autocorrelation. The discrepancy in these values is likely due to a slight roll-off in the magnitude of XPM (see section 4) over the measured RF spectrum bandwidth (for more details, see [24]). The bit period measured by both methods matches up very well, reflecting the match between the 64GHz tone on the RF spectrum displayed in Fig. 3(a) with the baud rate of the signal. Fig. 4. (a) XPM-RF spectrum of a time-division multiplexed signal comprised of two 32Gbit/s OOK signals of unequal amplitude. Note the 32GHz sub-harmonic tones overlayed on the 64GHz tone (Inset data eye diagram captured using an optical sampling oscilloscope) (b) SHG (solid red trace) and XPM (dotted blue) autocorrelation traces of the un-optimized signal. The information directly gained from the signal RF spectrum can be used for several different signal monitoring applications. Un-optimized optical time-division multiplexed signals have a characteristic signature in the RF spectrum. Figure 4(a) shows an RF spectrum of a 64Gbaud signal synthesized from two time-division multiplexed 32Gbaud signals of unequal amplitude. In addition to the strong 64GHz tone, sub-harmonic tones spaced 32GHz from this main 64GHz tone appear (i.e. appearing at 32 and 96GHz). From inspection of the eye diagram [inset Fig. 4(a)], a clear 32GHz modulation appears on the 64GHz eye pattern, intuitively giving rise to the 32GHz sub-harmonic tones seen on the RF spectrum. The measurement of the RF spectrum can therefore be directly used for real-time OTDM optimization by minimizing these sub-harmonic tones to provide a spectrum closer to Fig. 3(a). The accuracy of the Si-based XPM-RFSA measurements is again qualitatively illustrated with the close match between the central peak FWHM and bit-period of the SHG and XPM signal autocorrelations [Fig. 4(b)]. By analyzing the autocorrelation trace reconstructed from the RF spectrum one can measure multiple transmission impairments simultaneously [23]. It is also possible by measuring the amplitude of RF tones [see schematic, Fig. 5(a)], to gain a simple, high dynamic range method of monitoring isolated transmission impairments [6]. Figure 5(b) shows the amplitude of the 64GHz tone as a function of group velocity dispersion (GVD), applied to the signal using an optical arbitrary waveform generator [25]. The 64GHz tone power predictably drops with applied dispersion, down to a minimum before rising again due to the temporal Talbot effect [26]. The amplitude of the monitored tone changes by >2dB, (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2194

6 showing similar sensitivity as demonstrations of this same method in chalcogenide waveguides [6]. Fig. 5. Residual dispersion monitoring. a) Illustration of the GVD monitoring scheme b) Measured variation of the 64GHz tone amplitude with applied GVD. 4. XPM-RFSA bandwidth Although the intrinsic bandwidth of the XPM-RFSA is set by the response time of the Kerr effect, in waveguides the group velocity mismatch between the signal and probe may result in walk-off, which limits the bandwidth over which XPM is effective. This bandwidth limitation can be measured by mixing two beating CW signal waves equally spaced about a central frequency (f S ) with a third probe wave detuned from that signal (at frequency f P ) [5]. The beating waves provide a signal with a modulated envelope which oscillates at a frequency equal to the frequency separation of the two CW waves (df). When mixed with the weak CW probe, XPM RF spectral tones detuned from f P by ± df are created [Fig. 6(a)]. By measuring the power amplitude of these tones when varying the detuning df between the CW signal waves, one can determine the XPM bandwidth. For the bandwidth measurement, the two CW signal waves are detuned around 1551nm, with the probe wave at 1584nm to avoid spectral overlap between signal and XPM tone on the probe at high beat frequencies df. Fig. 6. XPM bandwidth measurement. (a) Schematic of the output spectrum obtained when launching a signal comprised of two beating CW waves centered about f s, and a detuned CW probe at f p. (b) Measurement of the sidetone amplitude generated on the probe versus beat signal detuning df. The waveguide 3dB bandwidth for 33nm signal/probe detuning is ~1.6THz. Figure 6(b) shows that the measured XPM 3dB bandwidth of the nanowire is ~1.6THz. The limitation on bandwidth due to walk-off can be calculated (developed from Ref. [5]) as f3 db = κ /( π. Leff. D. λ), where κ~ (κ is such that sinc 2 (κ) =.5), L eff is the effective guide length [22], D the group velocity dispersion and λ the signal/probe separation. The (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2195

7 measured bandwidth of 1.6THz lets us infer a chromatic dispersion of around D~898 ps/nm.km. Although this measured bandwidth could be increased simply by decreasing the detuning between signal and probe waves, this may result in the overlap of the measured spectrum of the signal with the spectrum of the broadened probe, preventing extraction of the RF spectrum. To increase device performance, one could engineer the dispersion of the silicon nanowire (as per [11]), which was not done here. For example, four-wave mixing which as a phase matched process is much more sensitive to dispersion, has been recently demonstrated over >1THz of bandwidth in silicon nanowires [27]. Additionally, XPM-RFSA requires that the temporal shape of the signal does not change significantly when co-propagating with the CW probe, so the dispersion length of the waveguide needs to be significantly greater than the physical length i.e. L<<T 2 / β 2 [22]. For a Gaussian pulse with.52ps FWHM (33% duty cycle width of a 64Gbaud signal), the dispersion length is ~8.5cm for D = 898 ps/nm.km, much greater than the physical length of our waveguide. These figures confirm that dispersion is not a barrier in using our Si nanowires for XPM-RFSA of high speed optical signals. 5. Impact of Free-Carriers on XPM-RFSA Intensity dependent nonlinear optical effects in silicon are generally complicated by the presence of photogenerated free-carriers, created via TPA [15,16]. In this section, we estimate the potential impact of these effects on high-speed optical pulse trains through a simplified propagation model to determine whether they influence the operation of our XPM-RFSA. We assume a free-carrier recombination time of τ C ~1ns, which is a conservatively large estimate in order to give the worst-case for FCA. This free-carrier lifetime is still within the typical range of values found for undoped silicon nanowire waveguides [13,16,28]. Because all typical values for τ C, are much larger than the pulse width of high speed serial data, our analysis below remains valid and independent of this particular choice of τ C. Fig. 7. Illustration of free carrier effects in silicon (33% duty, repetition period τ C/2). (a) Interpulse effects - Free carrier density over many pulses. The blue trace is a numerical result from Eq. (3), the red trace the inter-pulse average value of the free carrier density (Inset free carrier density variation over one signal period in the steady-state regime). (b) Intra-pulse effects - Nonlinear phase shifts due to both Kerr (dashed red trace) and Free Carrier (dotted blue) effects. Photogeneration of free-carriers has two notable effects on pulse propagation, namely freecarrier absorption (FCA) and free-carrier dispersion (FCD). FCA is dependent on the density of free carriers, and becomes greater as free carriers build up after many pulses. As such, FCA is sensitive to inter-pulse effects, significant when considering pulse trains or high-speed optical data [Fig. 7(a)]. Note that to effectively illustrate the fluctuations of free-carrier density on Fig. 7 (in particular the intra-pulse variation) we used a lower bit rate than in experiment. FCD, unlike FCA, is sensitive to intra-pulse effects as FCD-induced spectral broadening is proportional on the change in free carrier density with time. Figure 7(b) illustrates the difference in Kerr-based and free-carrier based phase shifts, indicating that FCD (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2196

8 can create an asymmetric spectral broadening as opposed to the symmetric Kerr-based XPM. In the context of XPM-RFSA, inter-pulse FCA may reduce device efficiency, while intrapulse FCD can cause a cross-chirp that modifies and distorts the XPM spectra on the probe [16]. As such, the ratio of FCD-based cross-chirp to XPM needs to be small for the signal RF spectrum to be accurately mapped onto the probe. We next investigate the magnitudes of FCD and nonlinear loss separately in order to estimate their impact on XPM-RFSA. 5.1 Effect of Free carrier dispersion (FCD) A simple analytical expression for the ratio (r X ) of cross-chirp produced on the probe wave through FCD to Kerr-based XPM is given for a Gaussian pulse in ref [15]. ϕ free carrirer / z σ C βtpa n E pulse Epulse rx = = θ ( λ) (2) ϕxpm / z 4 2cn n 2kħ eff Aeff neff Aeff The material parameters σ C, β TPA, n and n 2 are the free-carrier chirp coefficient, twophoton absorption coefficient and the linear and nonlinear refractive indices respectively. The parameters c and ħ are the vacuum speed of light and Plank s constant. E pulse is the pulse energy and k = 2π/λ the central wavenumber of the pulse. The material parameters at a given wavelength can be included in a single constant θ(λ). For operation in a crystalline silicon waveguide in the optical communications C-band, this constant is equal to θ ~.21 (using the values of Table 1). Table 1. Material coefficients used for crystalline silicon in optical communications C-band TPA (β TPA) FCA (σ A) FCD (σ C) Nonlinear index (n 2) Linear index (n ) 5 12 x1 m/w x1 m x1 m x1 m 2 /W 3.48 Equation (2) therefore allows us to estimate the relative magnitude of the effects of FCD and XPM on the probe wave in XPM-RFSA experiments. With the coupled peak power of the 64Gbit/s 33% duty cycle signal in our experiment ~21dBm, r X ~.27. From this, we estimate free-carrier cross-chirp to have produced ~4 times less of an effect on the probe than XPM, such that the effect of FCD is negligible under our operating conditions. This confirms that FCD does not prevent us from accurately mapping the signals RF spectrum onto the probe wave. More generally, the effect of FCD increases linearly with decreasing bit rate for OOK signals with constant duty cycle and peak power i.e. a 4Gbit/s signal will experience 16 times more FCD than a 64Gbit/s signal. This is further illustrated in Fig. 8, where r X is plotted against coupled signal input peak power P SIGpeak. The range of P SIGpeak represented in Fig. 8 corresponds to the regime of the XPM-RFSA operation in our device for an ideal case of lossless propagation. Evaluating φ(t)<1 in the lossless case gives φ(t) = 2γP SIGpeak L<1 (where γ = k S n 2 /A eff ), a common estimate of cross phase shift [22]. From this, we can infer that FCD may be of some concern for bit rates approaching 4Gbit/s and below but it should not impede XPM-RFSA in our waveguides for ultrahigh bit rates where this technique is of interest. (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2197

9 Fig. 8. Chirp ratio r X for different coupled peak powers, limited to an ideal XPM-RFSA operation regime where 2γP peakl<1 for our waveguide parameters. 5.2 Effect of Free-carrier absorption (FCA) Nonlinear absorption, through both TPA and FCA, reduces the intensity of high power signals travelling through silicon waveguides. However, in XPM-RFSA the allowed signal intensity is limited to an upper bound such that the maximum accumulated cross-phase shift is small (i.e. φ max <1 from Eq. (1). In this section, we investigate whether nonlinear losses constitute a significant effect in our device, given the power restriction imposed for XPM-RFSA. In order to do this, the intensity of the signal while propagating along the waveguide needs to be investigated. From [29]: I SIG ( z, t) 2 = βtpaisig ( z, t) σ ANISIG ( z, t) α I SIG ( z, t) z N ( z, t) N( z, t) βtpa 2 = + ISIG ( z, t) (3b) t τ C 2ħωSIG In Eqs. (3a) and (3b), I SIG (t,z) is the time varying intensity of the signal pulses in the propagation (z) direction along the waveguide, ω SIG = 2πf SIG the angular central frequency of the pulse, N(z,t) is the free-carrier density, σ A the free-carrier absorption coefficient and α the linear loss coefficient. Equations (3a) and 3(b) are in a moving reference frame, with t = always the start of the pulse train. Note that the effect of self-phase modulation and freecarrier dispersion are neglected in 3a as we are interested in the low modulation regime (i.e. φ(t)<1) and FCD was found to be negligible in the previous section. Chromatic dispersion is also neglected since the physical length of the waveguide we investigate is much less than the dispersion length. The probe can be considered weak compared to the signal, so we also neglect cross-free-carrier absorption. Equations (3a) and (3b) can be solved for I SIG (z,t) analytically for one isolated pulse by setting the initial value for N(z,t) as [3]. However, this is not a valid assumption for our waveguide when analyzing propagation effects on pulses in a pulse train at GHz-THz repetition rates. In this case, a background average density of free-carriers is built up over subsequent pulses due to the relatively slow recombination rate of free carriers in the waveguide [see Fig. 7(a)]. In addition, for high bit-rate signals, this background density of free-carriers generated through inter-pulse dynamics rapidly becomes much larger than the faster intra-pulse variation of free carrier density over a single bit period [inset of Fig. 7(a)]. We therefore analyze the effect of nonlinear loss on the signal intensity by evaluating the steady-state background average free-carrier density N SS (z) that is reached asymptotically at a (3a) (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2198

10 time t>5τ C as illustrated in Fig. 7(a). To find N SS (z) we first apply a time average over one repetition period, denoted as the <..> operator, to Eq. (3b). This gives: N ( z, t) 1 β = N z t + I z t t τ C TPA 2 (, ) SIG (, ) 2ħω where I SIG (z,t) is the peak pulse intensity and T is the repetition period of the pulse train. If we again assume Gaussian pulses, then the integral of the squared intensity of a pulse in the train can be evaluated as: T / (, ). exp 2 P / /2. T 2 SIG P I z t t t dt t π T I ( z, t) = = I ( z, t).. Erf (5) T T 2 2t where t is the pulse 1/e half-width (the pulse FWHM is t 2 ln [2]), and I P (z,t) the peak intensity of a pulse at some time t in the pulse train. As the average free-carrier density asymptotes to a steady-state value, for increasing time t, < N(z,t)/ t>, <N(z,t)> N SS, and the peak intensity of the pulses I P (z) I (z). So, in evaluating Eq. (4) asymptotically we find: βtpa t π T 2 2 NSS ( z) =.. Erf τ C I ( z) = υss I ( z) (6) 2 ω T 2 2t ħ By further substituting N(z,t) = N SS (z) from Eq. (6) into Eq. (3a), since the highest impact of FCA is expected from the inter-pulse build up of free-carriers, we find that the peak intensity of later pulses changes as: I ( z) 2 3 = βtpai ( z) σ Aυ SS I ( z) α I ( z) (7) z Solving Eq. (7) for I (z) provides an implicit solution, similar to that found in ref [3]. Rukhlenko et al use a variational technique to solve coupled equations for Raman signal amplification analytically, which may also be of use for analyzing XPM [3]. However in order to simplify our analysis in this paper, we evaluate the ratio of the TPA and FCA terms in Eq. (7) to determine whether either nonlinear loss term is dominant [15]. For the waveguide and signal parameters from our experiment, and for signal powers where nonlinear loss is an appreciable effect, the loss due to FCA is greater than the loss due TPA. Dropping the TPA term from Eq. (7) allows a closed form solution to Eq. (7) for I (z). I e I z where I I z α z ( ) = = ( = ) 2 2α z 1 + σ Aυ SS I (1 e ) / α Equation (8) provides a good approximate solution to Eq. (7), with numerically calculated transmission from Eq. (7) deviating (due to the neglected effect of TPA) by less than 2% from the transmission derived from Eq. (8). Figure 9(a) shows that for the coupled signal peak power used in our experiment (~21dBm), nonlinear losses (i.e. both TPA and FCA) have a negligible effect on transmission. Thus we expect FCA to have a negligible impact on the accumulated cross-phase modulation of the probe for the signal powers we used. However, FCA may still impact on XPM for higher signal powers. The maximum phase shift on the pulse, evaluated with Eq. (8) is given by Eq. (9) below: (4) (8) (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 2199

11 ϕ z 1 1 max = k n2 I ( z) δz kn2 u( z) tan v( z) u() tan =. v() 2 2 σaυ SS I σaυ SS I u( z) = α exp[2 αz] ( σ υ I + 1) σ υ I 2 2 A SS A SS ( ) v( z) = α σ υ I 1 exp[ 2 αz]/ α + συ 2 2 A SS SS u( z) u() As the maximum value of φ(t) (which we call φ max, occurring at the peak of the pulse) must be less than 1 for XPM-RFSA [see Eq. (1)], this sets a bound on the maximum allowed signal power for this scheme to be effective. In the particular case of the coupled signal power used in our experiment, we expect the maximum phase shift on the probe to be less than.37 [see Fig. 9(b)], indicating that the spectrum superimposed on the probe provides an accurate representation of the signals RF spectrum, in agreement with our experimental observations. In a more general way, Fig. 9(b) shows that for a 64Gbit/s, 33% duty cycle OOK signal, nonlinear loss has a negligible effect for the whole range of signal power where φ max <1. As the steady-state background free carrier density N SS (z) remains the same for signals of the same duty cycle and average power, we can further infer that for 33% duty cycle OOK signals, at bit rates where XPM-RFSA is interesting (i.e. >4Gbit/s), nonlinear loss has negligible impact upon device operation. (9) 6. Conclusion Fig. 9. Free carrier absorption effects on a 33% duty cycle, 64GHz Gaussian pulse train in our waveguide. (a) Propagation loss through the waveguide. TPA (dotted green) and full Eq. (7) (solid blue) traces are numerical solutions of Eqs. (3a) and (3b), the FCA (dashed red) trace is derived from Eq. (8). (b) Accumulated peak cross-phase change on the CW probe for signal input powers where φ max<1. We demonstrate a silicon waveguide based XPM-RFSA, accurately measuring RF spectra for 64Gbit/s OOK signals without degradation from free-carrier effects. Although silicon has known problems as a platform for applied nonlinear optics, we show through analytical modeling that the problems of nonlinear absorption and free-carrier dispersion do not impede the use of highly nonlinear silicon waveguides for XPM-RFSA of high speed optical data. XPM in Si nanowires therefore provides accurate RF spectra of terabaud optical signals, allowing signal monitoring functionalities such as OTDM optimization and signal impairment monitoring. Our work further establishes the usefulness of silicon as a platform for ultrafast nonlinear optics, opening possibilities for combining the unique abilities of high speed alloptical signal processes with the well established power of silicon integrated electronics. (C) 21 OSA 13 September 21 / Vol. 18, No. 19 / OPTICS EXPRESS 22