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. Eggleton1 1Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Institute for Photonics and Optical Sciences
(IPOS), School of Physics, The University of Sydney, NSW 2006, Australia
2Institute for Microstructural Sciences, National Research Council (NRC-CNRC), Ottawa, ON, K1A-0R6, 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 640Gbaud on-off-keyed 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.
©2010 Optical Society of America
OCIS codes: (190.4360) Nonlinear optics, devices, (060.4256) Networks, network optimization.
References and links
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1. 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 [10].
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 two-photon 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 640Gbit/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
multiplexing (OTDM) as well as measuring residual chromatic dispersion. Utilizing a simplified pulse propagation model, we analyze theoretically the relative magnitude of FCD-based cross-chirp to Kerr-FCD-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) [19,20].
Fig. 1. Silicon-based XPM-RFSA concept. A signal with optical spectrum (SSIG) 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 SPRO of the modulated probe is measured with an optical spectrum
analyzer (OSA), and the signal RF spectrum (SRF) extracted from SPRO. 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 (<100GHz) 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 10fs [15,21,22], intrinsically able to respond to pulses of bandwidths >100THz. 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 ISIG(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 (SRF(f) - see Fig. 1).
( )
(
)
(
)
2(
)
0 2
1 ( ) exp 2 ( ) 2 ( , ).
PRO S SIG
S f =
∫
+ϕt ⋅ −πi f−f t dt⋅ ϕt = i k n I∫
t z dz (1) In Eq. (1), φ(t) is the accumulated cross phase shift on the probe (probe centre frequencyf0), the free-space wavenumber and nonlinear index at the signal frequency are given as kS and
n2 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
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, SiO2 clad, 450nm wide by 260nm high, 1.5cm in length (Fig. 2b). The TE mode of this waveguide in the optical communications C-band has an effective index neff ~2.5 and effective area Aeff ~0.15µm2, 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 640Gbit/s OOK signals. (b) Cross-section schematic of the silicon nanowire waveguide used in experiments.
The RF spectrum of a 640Gbaud, 27-1 pseudo-random bit sequence (PRBS), 33% duty cycle, OOK signal is measured in our device. The signal is constructed from a 40GHz pulse train from a mode-locked fiber laser. The pulses are nonlinearly compressed (similar to ref [23]), and OOK data encoded with a 231-1 PRBS through at 40Gbit/s. This data stream is then time-division multiplexed (time interleaved with a 27-1 bit delay) up to 640Gbit/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 640Gbit/s 33% duty cycle OOK signal about the probe central frequency. Note the strong tone at 640GHz. (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).
The output spectrum of the waveguide is measured on a high resolution OSA (specified resolution bandwidth ∆λ in C-band of 10pm) 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 λ0 and dω≈2π.λ02/∆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 640GHz 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 320Gbit/s OOK signals of unequal amplitude. Note the 320GHz sub-harmonic tones overlayed on the 640GHz 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 640Gbaud signal synthesized from two time-division multiplexed 320Gbaud signals of unequal amplitude. In addition to the strong 640GHz tone, sub-harmonic tones spaced 320GHz from this main 640GHz tone appear (i.e. appearing at 320 and 960GHz). From inspection of the eye diagram [inset Fig. 4(a)], a clear 320GHz modulation appears on the 640GHz eye pattern, intuitively giving rise to the 320GHz 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 640GHz tone as a function of group velocity dispersion (GVD), applied to the signal using an optical arbitrary waveform generator [25]. The 640GHz 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 >20dB,
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 640GHz 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 (fS) with a third probe wave detuned from that signal (at frequency fP) [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 fP 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 fs, and a detuned CW
probe at fp. (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∆f3dB=κ π/
(
.Leff. .D∆λ)
, where κ~1.3915 (κ is such that sinc2(κ) = 0.5), Leff is the effectivemeasured 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 >100THz 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<<T02/|β2| [22]. For
a Gaussian pulse with 0.52ps FWHM (33% duty cycle width of a 640Gbaud 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 τC~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 τC, are much larger than the pulse width of high speed serial data, our analysis below remains valid and independent of this particular choice of τC.
Fig. 7. Illustration of free carrier effects in silicon (33% duty, repetition period τC/20). (a)
Inter-pulse effects - Free carrier density over many Inter-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 carriers has two notable effects on pulse propagation, namely free-carrier absorption (FCA) and free-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
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 intra-pulse 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 (rX) of cross-chirp produced on the probe wave
through FCD to Kerr-based XPM is given for a Gaussian pulse in ref [15]. 0
2 0 /
( )
/ 4 2
free carrirer C TPA pulse pulse
X
XPM eff eff eff eff
z n E E r z cn k n A n A ϕ σ β θ λ ϕ − ∂ ∂ = ≈ = ∂ ∂ ℏ (2)
The material parameters σC, βTPA, n0 and n2 are the free-carrier chirp coefficient,
two-photon 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. Epulse is the pulse energy and k0 = 2π/λ0 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 ~ 0.021 (using the values of Table 1).
Table 1. Material coefficients used for crystalline silicon in optical communications C-band
TPA (βTPA) FCA (σA) FCD (σC) Nonlinear index (n2) Linear index (n0)
5x10−12 m/W 1.45x10−21 m2 −5.3x10−27 m3 6x10−18 m2/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 640Gbit/s 33% duty cycle signal in our experiment ~21dBm, rX~0.0027. From this, we
estimate free-carrier cross-chirp to have produced ~400 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 40Gbit/s signal will experience 16 times more FCD than a 640Gbit/s signal. This is further illustrated in Fig. 8, where rX is
plotted against coupled signal input peak power PSIGpeak. The range of PSIGpeak 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) = 2γPSIGpeakL<1
(where γ = kSn2/Aeff), a common estimate of cross phase shift [22]. From this, we can infer that
FCD may be of some concern for bit rates approaching 40Gbit/s and below but it should not impede XPM-RFSA in our waveguides for ultrahigh bit rates where this technique is of interest.
Fig. 8. Chirp ratio rX for different coupled peak powers, limited to an ideal XPM-RFSA
operation regime where 2γPpeakL<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. φmax<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]:
2 ( , )
( , ) ( , ) ( , )
SIG
TPA SIG A SIG SIG
I z t I z t NI z t I z t z β σ α ∂ = − − − ∂ (3a) 2 0 ( , ) ( , ) ( , ) 2 TPA SIG C SIG N z t N z t I z t t β τ ω ∂ = − + ∂ ℏ (3b)
In Eqs. (3a) and (3b), ISIG(t,z) is the time varying intensity of the signal pulses in the
propagation (z) direction along the waveguide, ωSIG0 = 2πfSIG0 the angular central frequency of
the pulse, N(z,t) is the free-carrier density, σA the free-carrier absorption coefficient and α the
linear loss coefficient. Equations (3a) and 3(b) are in a moving reference frame, with t = 0 always the start of the pulse train. Note that the effect of self-phase modulation and free-carrier 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 ISIG(z,t) analytically for one isolated pulse by
setting the initial value for N(z,t) as 0 [30]. 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 NSS(z) that is reached asymptotically at a
time t>5τC as illustrated in Fig. 7(a). To find NSS(z) we first apply a time average over one
repetition period, denoted as the <..> operator, to Eq. (3b). This gives: 2 0 ( , ) 1 ( , ) ( , ) 2 TPA SIG C N z t N z t I z t t β τ ω ∂ − = + ∂ ℏ (4)
where ISIG(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:
/ 2 2 2 0 2 / 2 2 0 0 ( , ). exp / . ( , ) ( , ). . 2 2 T P T SIG P I z t t t dt t T I z t I z t Erf T T t π − − = =
∫
(5) where t0 is the pulse 1/e half-width (the pulse FWHM is t0·2√ln [2]), and IP(z,t) the peakintensity 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>→0, <N(z,t)>→NSS, and
the peak intensity of the pulses IP(z)→IPSS(z). So, in evaluating Eq. (4) asymptotically we find:
2 2 0 0 0 ( ) . . ( ) ( ) 2 2 2 TPA SS C PSS SS PSS t T N z Erf I z I z T t β π τ υ ω = = ℏ (6) By further substituting N(z,t) = NSS(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:
2 3 ( ) ( ) ( ) ( ) PSS TPA PSS A SS PSS PSS I z I z I z I z z β σ υ α ∂ = − − − ∂ (7)
Solving Eq. (7) for IPSS(z) provides an implicit solution, similar to that found in ref [30].
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 [30]. 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 IPSS(z).
0 0 2 2 0 ( ) ( 0) 1 (1 ) / z PSS PSS z A SS I e I z where I I z I e α α σ υ α − − = = = + − (8)
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:
(
)
1 1 max 0 0 2 0 2 2 2 0 0 2 2 0 0 2 2 0 ( ) (0) ( ) ( ) tan ( ) (0) tan . (0) ( ) exp[2 ] ( 1) ( ) 1 exp[ 2 ]/ z PSS A SS A SS A SS A SS A SS SS u z u k n I z z k n u z v z u v I I u z z I I v z I z ϕ δ σ υ σ υ α α σ υ σ υ α σ υ α α συ − − = ⋅ = ⋅ ⋅ − ⋅ = ⋅ ⋅ + − = − − +∫
(9)As the maximum value of φ(t) (which we call φmax, 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 0.37 [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 640Gbit/s, 33% duty cycle OOK signal, nonlinear loss has a negligible effect for the whole range of signal power where φmax<1. As the steady-state background free carrier density NSS(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. >40Gbit/s), nonlinear loss has negligible impact upon device operation.
Fig. 9. Free carrier absorption effects on a 33% duty cycle, 640GHz 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 φmax<1.
6. Conclusion
We demonstrate a silicon waveguide based XPM-RFSA, accurately measuring RF spectra for 640Gbit/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 all-optical signal processes with the well established power of silicon integrated electronics.
