Dynamic Characterization and ImpulseResponse Modeling of Amplitude andPhase Response of Silicon NanowiresVolume 5, Number 2, April 2013

Ciaran S. ClearyHua JiJames M. DaileyRoderick P. WebbRobert J. ManningMichael GaliliPalle JeppesenMinhao PuKresten YvindLeif K. Oxenlowe

DOI: 10.1109/JPHOT.2013.22465601943-0655/$31.00 �2013 IEEE

Dynamic Characterization and ImpulseResponse Modeling of Amplitude andPhase Response of Silicon NanowiresCiaran S. Cleary,1 Hua Ji,2 James M. Dailey,1;3 Roderick P. Webb,1Robert J. Manning,1 Michael Galili,2 Palle Jeppesen,2 Minhao Pu,2

Kresten Yvind,2 and Leif K. Oxenlowe2

1Tyndall National Institute, Department of Physics, University College Cork, Lee Maltings, Cork, Ireland2DTU Fotonik, Technical University of Denmark, Kgs. Lyngby 2800, Denmark

3Now with Applied Communication Sciences, Red Bank, NJ 07701 USA

DOI: 10.1109/JPHOT.2013.22465601943-0655/$31.00 �2013 IEEE

Manuscript received January 16, 2013; revised February 6, 2013; accepted February 6, 2013. Date ofpublication February 11, 2013; date of current version February 26, 2013. This work was supported inpart by Science Foundation Ireland Grant 06/IN1969, by EURO-FOS, and by FTP NESTOR.Corresponding author: C. S. Cleary (e-mail: ciaran.cleary@tyndall.ie).

Abstract: Amplitude and phase dynamics of silicon nanowires were measured using time-resolved spectroscopy. Time shifts of the maximum phase change and minimum amplitudeas a function of pump power due to saturation of the free-carrier density were observed. Aphenomenological impulse response model used to fit the experimental data indicated thatthe free-carrier lifetime was between 7:5 � �fc � 16:2 ns, and the two-photon absorptioncoefficient and the Kerr coefficient were 3� 10�12 m.W�1 and 4� 10�18 m2:W�1, respec-tively, for silicon nanowires with lengths varying from 3.6 to 14.9 mm.

Index Terms: Silicon nanowire, pump–probe spectroscopy, two-photon absorption (TPA),free-carrier absorption (FCA).

1. IntroductionSilicon (Si) waveguides with submicron cross-section dimensions show tremendous potential foruse as optical switches in high-speed optical time division multiplexed (OTDM) communicationsystems due to their low linear material loss, high third-order nonlinear coefficient, wide bandwidth,and compact size [1]. The high linear refractive index of silicon at telecommunication wavelengths(3.5 at 1550 nm) [2] leads to high modal confinement in Si waveguides with submicron dimensionsmanufactured using silicon-on-insulator (SOI) technology [3]. Furthermore, SOI waveguides can befabricated using mature complementary metal–oxide–semiconductor (CMOS) technology, openingup the possibility of monolithic integration of electronic and photonic components on a single chip [4].

Ultrafast effects such as four-wave mixing (FWM) [5], cross-phase modulation (XPM) [6], self-phase modulation (SPM) [7], and stimulated Raman scattering (SRS) [8] in Si waveguides havebeen studied in depth [3]. These nonlinear effects have enabled a variety of optical signalprocessing (OSP) functions to be carried out in Si waveguides, including demultiplexing andwaveform sampling at 1.28 Tb.s�1 [9], wavelength conversion at 640 Gb.s�1 [10], and 10-Gb.s�1

signal regeneration [11].However, the efficiency of FWM, XPM, SPM, and SRS is limited by the nonlinear loss

mechanisms of two-photon absorption (TPA) and TPA-induced free-carrier absorption (FCA) [1],[12]. TPA can occur in silicon as photons at telecommunication wavelengths have energy

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(Ephoton ¼ 0:8 eV at 1550 nm) greater than half the indirect band gap for silicon ðEg;indirect ¼ 1:1 eVÞ.As a direct consequence of TPA, free carriers are generated, which can absorb more photons untilthey recombine, primarily via surface recombination [13]. The relatively long FCA lifetime limits thespeed at which OSP functions can be performed. In bulk silicon, the free-carrier lifetime is of theorder of microseconds, but in Si waveguides with submicron cross-section dimensions, the free-carrier lifetime can be much smaller (several nanoseconds [13]) due to an enhanced surfacerecombination rate. The free-carrier lifetime can be further reduced by modifications such as heliumion implantation [14], using porous silicon [15] or polysilicon [16], or incorporation of a p-i-n diodeinto the waveguide [1]. Alternatively, TPA in silicon may be avoided using slot waveguides in silicon-organic-hybrid (SOH) technology, where the field is confined to the slot filled with highly nonlinearorganic material with a lower TPA coefficient [17].

The amplitude and phase dynamics of Si nanowires with SiO2 cladding have been studied bySuda et al. [18], [19] who estimated a free-carrier lifetime of 200 ps. They ascribed this short free-carrier lifetime to the high degree of sidewall roughness in their nanowires (propagation loss was20 dB.cm�1) that led to an augmented surface recombination rate. Vallaitis et al.measured the carrierdynamics of SOHwaveguideswith both strip and slot geometries, and estimated a free-carrier lifetimeof 1.2 ns for the strip waveguides [20]. Generally, the free-carrier lifetime in silicon waveguides(without the use of material modifications such as helium implantation or the incorporation of a p-i-ndiode) has been estimated to lie within the range 0.5 to 25 ns [6], [13], [21]–[23].

In this paper, we present time-resolved measurements of the amplitude and phase dynamics ofSi nanowires embedded within a polymer waveguide using a pump–probe spectroscopy test bedcontaining a terahertz optical asymmetric demultiplexer (TOAD) interferometric loop and 3.5-pspump pulses and 2.5-ps probe pulses. We use a phenomenological impulse response model toextract values for the TPA coefficient �TPA and the Kerr coefficient n2 (3� 10�12 m.W�1 and4� 10�18 m2:W�1, respectively), the TPA figure of merit (0.88), and the free-carrier lifetime (7.5 to16.2 ns).

2. Experimental DetailsThe Si nanowires had cross-section dimensions of 450 nm � 240 nm and total lengths of 3.6 mm,4.9 mm, or 14.9 mm, which included tapered sections at both ends of the nanowires that were 0.3 mmlong each. The Si nanowires were embedded within a 3:4 �m� 3:4 �m polymer (SU8-2005)waveguide to decrease the fiber-to-chip coupling loss through better mode-matching with the taperedaccess fibers [10]. SU8 polymer was chosen as the cladding material as it is more convenient tohandle and easier to fabricate in comparison to SiO2 while still allowing a low coupling loss to beachieved. One drawback associated with SU8 is that it has a lower damage threshold power thanSiO2. The nanowires were operated as passive devices so there was no applied drive current orvoltage bias. Total insertion losses for the nanowires varied from 5 to 8 dB, and the propagation losswas 4.5 dB.cm�1.

A schematic of the TOAD loop test bed used to measure the carrier dynamics of the Si nanowiresis displayed in Fig. 1. This type of time-resolved spectroscopy system is a stable, relativelyinexpensive configuration, which allows real-time evaluation of the amplitude and phase evolutionof any nonlinear optical device. For this set of experiments, the 3.5-ps pump and 2.5-ps probepulses were clock streams emitted by actively mode-locked lasers, whose repetition rates werereduced from 10.65 GHz to 665 MHz using LiNbO3 Mach–Zehnder modulators (MZMs). The pumpand probe wavelengths were maintained at 1555 nm and 1540 nm, respectively, for allmeasurements. The pump and probe powers at the Si nanowire input were maintained withinthe ranges 9 dBm � Ppump � 17 dBm and �31 dBm � Pprobe � �27 dBm, respectively. Thesepump and probe powers translate into energy per pulse of 12 pJ � Epump � 75 pJ and1 fJ � Eprobe � 3 fJ, respectively. The polarizations of both the pump and probe pulses wereadjusted so that they were aligned with the TE mode of the Si nanowire under test.

The operational principle of the TOAD loop test bed is described in [24]. The reflected andtransmitted signals measured by the BR[ and BT[ photodiodes can be used to reconstruct both the

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amplitude and phase responses of the Si nanowire devices with �1-ps temporal resolution. It isimportant to note that this measurement system produces the cosine of the phase response. As aresult, the absolute value of the phase response can be measured by this system, but the sign ofthe phase excursion cannot be determined directly.

3. Experimental ResultsThe amplitude and phase response dynamics of the 3.6-mm-long Si nanowire measured using theTOAD loop pump–probe spectroscopy test bed are shown in Fig. 2. The corresponding evolutionsfor the 4.9-mm-long and 14.9-mm-long nanowires were similar to those for the 3.6-mm-longnanowire, except that the amplitude modulation depth and the maximum phase shift were greaterfor the longer nanowires. The amplitude curves shown in Fig. 2(a) were determined using the powermeasured by the BPP[ photodiode, normalized relative to the power level immediately before the

Fig. 2. (a) Amplitude and (b) phase dynamics of 3.6-mm-long Si nanowire, with probe input power of�27 dBm and pump input power varied from 9 dBm to 17 dBm at the input to the chip.

Fig. 1. TOAD loop system for measuring amplitude and phase dynamics of a silicon nanowire within theinterferometric loop ðSi wg¼silicon waveguide/nanowire;BPF¼bandpass filter;EDFA¼erbium dopedfiber amplifier; PC ¼ polarization controllerÞ.

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arrival of the pump pulse. The phase evolutions shown in Fig. 2(b) were calculated using thereflection and transmission coefficients measured by the BR[ and BT[ photodiodes shown in Fig. 1.

It can be seen from Fig. 2(a) that the amplitude response consisted of an initial ultrafastcomponent followed by a long-lived recovery tail, which we ascribe to TPA and FCA, respectively.Fig. 2(b) demonstrates that the associated phase response consisted of an initial ultrafastcomponent succeeded by a long-lived component, which we believe to result from the Kerr effectand the free-carrier plasma effect (FCPE). The TPA and Kerr components had subpicosecondrecovery times, whereas the FCA and FCPE components had a recovery time of severalnanoseconds. The phase shift due to TPA was positive, whereas the phase shift due to FCPE wasnegative [25], [26], although this is not immediately apparent from Fig. 2(b) as the TOAD loop testbed only measures the absolute value of the phase. However, at high pump powers (9 11 dBm),there was a sharp dip in phase following the TPA component, and this marks the point where thephase flips from positive to negative values. It is important to note that the individual curves in Fig. 2have been aligned so that the rising edge of each curve started at the same point in time to facilitatedirect comparison of the magnitude of each response. However, in reality, the point of maximumphase change and minimum amplitude in each curve shifted in time as a function of pump power, asdemonstrated in Fig. 3.

It is evident from Fig. 3 that the points of minimum amplitude and maximum phase change shiftedin time as a function of pump power. The exact origin of this effect is unclear, but it may beattributable to carrier screening within the device at high pump powers (9 11 dBm). Moreover, Fig. 3indicates that there was a relative delay between the point of minimum amplitude (i.e., maximumamplitude change) and the point of maximum phase change at a given pump power. In Fig. 4(a), the

Fig. 3. Amplitude and phase dynamics of 3.6-mm-long Si nanowire, without alignment of the responsesat the rising edge of each response.

Fig. 4. (a) Maximum phase change and loge (minimum amplitude) as a function of energy per pumppulse for 3.6-mm-long Si nanowire. (b) Point of minimum amplitude, point of maximum phase changeand the relative delay time between these points as a function of energy per pump pulse for 3.6-mm-long Si nanowire.

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maximum phase change and natural logarithm of the minimum amplitude are plotted as a functionof energy per pump pulse. In Fig. 4(b), the point of minimum amplitude, the point of maximum phasechange, and the difference between these values are plotted as a function of energy per pumppulse.

It can be perceived in Fig. 4(a) that the maximum phase shift and the minimum amplitude beganto saturate at an energy per pump pulse 9 40 pJ. In Fig. 4(b), there was a near-linear shift of thepoint of maximum phase shift and the point of minimum amplitude (as well as an increasingrelative delay between these points) as energy per pump pulse increased. We attribute the relativetime delay to an instantaneous phase response with negligible associated TPA in the polymerwaveguide.

The FCA recovery time was estimated using the unnormalized probe amplitude power measuredby the BPP[ photodiode displayed in Fig. 5.

It is apparent in Fig. 5 that, when 665-MHz pump pulses were launched into the Si nanowire, theprobe transmission power did not return to the steady-state level corresponding to no pump input tothe device. This implies that the free-carrier density did not fully decay within the pump pulse period(1.5 ns) and, as a result, the amplitude and phase evolutions had sawtooth-like waveforms. Boththe steady-state amplitude level when no pump was incident on the nanowire ðA0Þ and theunnormalized amplitude when pump pulses were input to the nanowire ðAppðtÞÞ were measured.The difference between the arrival of the pump pulse and the point where the extrapolation of theFCA recovery tail in AppðtÞ intersected with A0 was taken to be approximately equal to the FCAlifetime (see Fig. 6).

Using the technique described above, the FCA recovery lifetime was determined to be 16.2 ns forthe 3.6-mm-long nanowire, 8.3 ns for the 4.9-mm-long nanowire, and 7.5 ns for the 14.9-mm-longnanowire.

4. Phenomenological Impulse Response ModelA phenomenological impulse response model was developed to explain the main features of theamplitude and phase evolution of the Si nanowires, using a similar approach to the impulseresponse model for carrier dynamics in semiconductor optical amplifiers (SOAs) described byHall et al. [27]. For the Si nanowire modeling, contributions from TPA, Kerr effect, FCA, and FCPEprocesseswere all taken into consideration because FCAand FCPEbecame significant at high pumppowers (9 11 dBm). The TPA impulse response and the Kerr impulse response were not representedby instantaneous delta functions, but were instead assumed to be exponential functions with a shortlifetime (0.6 ps) in order to capture hot carrier relaxation effects [28]. The FCA and FCPE impulse

Fig. 5. Unnormalized transmission power evolution of 3.6-mm-long Si nanowire, with probe input powerset to �27 dBm and pump input power varied from 9 dBm to 17 dBm at the chip input.

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responses were modeled as step functions with constant recovery rates to simulate sawtoothoperation. The TPA and FCA amplitude impulse responses Rampl;tpa and Rampl;fca were describedby (1) and (2), respectively, where Qampl;tpa determined the magnitude of the TPA response, �hc wasthe hot carrier relaxation time, �Hampl;fca determined the magnitude of the FCA response, Campl;fca

could be used to adjust the slope of the FCA response, and �fc was the free-carrier lifetime (16 nsfor the 3.6-mm-long Si nanowire)

Rampl;tpa ¼Qampl;tpaexp�t�hc

� �(1)

Rampl;fca ¼�Hampl;fca � Campl;fcat�fc

� �: (2)

Similarly, the Kerr effect and FCPE impulse responses Rphase;Kerr and Rphase;fcpe were describedby (3) and (4), respectively, where Qphase;Kerr determined the magnitude of the Kerr effect response,�Hphase;fcpe determined the magnitude of the FCPE response, and Cphase;fcpe could be used toadjust the slope of the FCPE response

Rphase;Kerr ¼Qphase;Kerrexp�t�hc

� �(3)

Rphase;fcpe ¼ ��Hphase;fcpe þ Cphase;fcpet�fc

� �: (4)

To reproduce the modal amplitude and phase change (gconv;pumpðtÞ and �’conv;pumpðtÞ,respectively) of the nanowire in response to a pump pulse, the TPA and Kerr impulse responseswere initially convolved with a modeled 3.5-ps sech2 pump pulse. This convolution was performedbecause the ultrafast effects detected by the TOAD loop system were due to TPA of one pump andone probe photon, and the changes in absorption ��tpa and refractive index �ntpa due to TPA andthe Kerr effect are described by (5) and (6), where �TPA is the TPA coefficient and n2 is the Kerrcoefficient [29]. Although TPA of two pump photons occurred, this process was not directlydetectable by the TOAD loop system

��tpa ¼ �tpaIpump (5)

�ntpa ¼ n2Ipump: (6)

However, the FCA and FCPE impulse responses were convolved with the modeled 3.5 ps sech2

pump pulse squared, because the vast majority of free carriers were generated by TPA of two pump

Fig. 6. Unnormalized probe transmission power vs. time for 3.6-mm-long Si nanowire (blue line) for apump power of 15 dBm at the nanowire input. The green line displays the mean unnormalized amplitudelevel when there is no input pump power, and the red line is the least squares fit to the long FCArecovery tail (the equation of this fitted line is shown in the graph).

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photons. Thus, the effects of FCA and FCPE were proportional to the square of the pump intensity.It was necessary to include a relative time delay between the onset of the TPA response and theonset of the Kerr effect response, varying from 0.6 to 1.3 ps. The modal amplitude was convertedinto total amplitude Gconv;pumpðtÞ using (7)

Gconv;pumpðtÞ ¼ exp �gconv;pumpðtÞ� �

: (7)

As will be outlined further on in this section, the convolutions of the TPA, Kerr effect, FCA, andFCPE impulse responses with the pump were translated into reflection and transmissioncoefficients for the TOAD loop. In turn, these coefficients were convolved with the probe beforefinally calculating the modeled amplitude and phase evolutions for the Si nanowire. This methodfacilitated direct comparison between the experimentally measured data and the data derived fromthe impulse response model.

Both Gconv;pumpðtÞ and �’conv;pumpðtÞ when the pump power was 15 dBm, as well as theparameters used to model the amplitude and phase responses, are displayed in Fig. 7. It was onlynecessary to change the peak pump power parameter and the relative time delay between theonset of TPA and the onset of the Kerr effect for modeling the amplitude and phase evolution of theSi nanowire as the pump power was varied from 9 to 17 dBm, while all other parameters in the fittingroutine remained approximately constant (i.e., they had similar values to those shown in Fig. 7).

As depicted in Fig. 7(b), the modeled Kerr phase response was positive whereas the modeledFCPE phase response was negative following the injection of a pump pulse to the device.

Following the calculation of the amplitude and phase dynamics of the Si nanowire in response toa pump pulse, the reflection and transmission coefficients (Rconv;1 and Tconv;1) for the TOAD loopwere calculated from Gconv;pumpðtÞ and �’conv;pumpðtÞ using equations similar to those described in[24]. The residual free-carrier density generated by previous pump pulses was represented in theimpulse response model by a phase offset term. The reflection and transmission coefficients(Rconv;1 and Tconv;1) were convolved with a modeled 2.5-ps sech2 probe pulse to produce newvalues for these coefficients, i.e., Rconv;2 and Tconv;2, so as to include the effect of the probe. Finally,Rconv;2 and Tconv;2 were employed to calculate the amplitude and phase evolutions, which permitteda direct comparison to be made between the measured data and the modeled data.

In Fig. 8, the measured amplitude and phase evolutions of the 3.6-mm-long Si nanowire with aninput pump power of 15 dBm are compared with the corresponding modeled evolutions. Note thatthe experimental amplitude evolution shown in Fig. 8(a) was derived from the TOAD loop reflectionand transmission coefficients measured by the BR[ and BT[ photodiodes, which slightlyunderestimated the FCA component compared to the amplitude evolution measured by the BPP[photodiode. This underestimation of the FCA recovery tail may have been due to inaccuracies inthe calibration of the measurement system.

Fig. 7. (a) Modeled amplitude and (b) phase change dynamics of 3.6-mm-long Si nanowire in responseto input pump pulse, with pump power set to 15 dBm.

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It can be seen from Fig. 8 that a close fit between the experimental phase data and the modeledphase curve could be obtained, thereby suggesting that the simple impulse response model, whichincluded the effects of TPA, the Kerr effect, FCA, and FCPE accounted for the main features in theamplitude and phase evolutions of the Si nanowires as measured by the TOAD loop test bed. Thus,without the use of complicated rate equations, it was possible to fit the experimental data for a rangeof input pump powers by only varying the peak pump power parameter, and the relative time delaybetween the onset of TPA and the onset of the Kerr effect, while all other parameters remained fixed.

Values for the TPA coefficient �TPA and the Kerr coefficient n2 were estimated using theamplitude and phase impulse responses at low pump powers (� 11 dBm). At pump powers below11 dBm, individual pump pulses contributed only a small addition to the mean free-carrierpopulation. At low pump powers, the experimental data shows that the FCA can be approximatedas a simple addition to the linear propagation loss. This approximation can be used to calculate thenormalized probe power transmission TP and the total phase change �’tot for the nanowire for lowinput powers [20]. The expressions for TP and �’tot in this reference can be rearranged to giveexpressions for �tpa and n2 [see (8) and (9)] [20], [30]

�tpa ¼1ffiffiffiffiffiTPp � 1� �Ip0Leff

(8)

n2 ¼�pump�’tot

4�RLwg0

IpumpðzÞdz: (9)

In (8) and (9), Ip0 refers to the pump intensity at the input, �pump is the pump wavelength, Lwg is thelength of the active region, Leff is the typical parameter, which refers to the effective interactionlength in the waveguide, and z is the distance traveled by the pump pulse within the waveguide. Forthe 3.6-mm-long Si nanowire, �tpa and n2 were calculated to be 3� 10�12 m.W�1 and 4 �10�18 m2:W�1, respectively, using �pump ¼ 1555 nm, Ip;0 ¼ 1:2� 1013 W.m�2, � ¼ 4:5 dB.cm�1,TP ¼ 0:857, and �’tot ¼ 0:137� rad (when pump input power was 9 dBm). These calculatedvalues for �tpa and n2 have the same order of magnitude as the range of values cited in the literature[31]–[34], although they were marginally lower than expected, possibly due to the difficulties indetermining the waveguide effective area and the taper loss. The associated TPA figure of meritðFOMtpaÞ was 0.9, calculated using the definition used by Dinu et al. [32]. In all-optical switchingdevices, which exploit optical nonlinearities, a TPA figure of merit 9 2 is generally desirable [32].Although the TPA figure of merit for the Si nanowires in this paper is below this value, the sametypes of nanowires have already been employed to demonstrate high-speed ð� 640 Gb.s�1Þdemultiplexing, waveform sampling, and wavelength conversion [9], [10], [35]. In Table 1, thefree-carrier lifetimes, TPA coefficients and FCA coefficients for silicon waveguides (with no

Fig. 8. Measured and modeled (a) amplitude and (b) phase response of the 3.6-mm-long Si nanowire(15 dBm input pump power).

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material modifications and were not incorporated into a p-i-n diode structure) in the literature arecompared with the values for these parameters measured by the authors to place the researchpresented in this paper into context.

5. ConclusionWe have reported the amplitude and phase responses for Si nanowires with various lengths (3.6 mmto 14.9 mm) embedded in a polymer waveguide using a pump–probe test bed incorporating a TOADloop interferometer.We have shown that the points ofmaximumultrafast phase change andminimumultrafast amplitude shifted in time as a function of pump power, which suggested that the carrierdensity in the Si nanowires was saturating at high pump powers (9 11 dBm). Moreover, we haveobserved a relative time delay between the points of maximum ultrafast phase change and minimumultrafast amplitude that increased as a function of pumppower, whichwe attribute to an instantaneousphase response in the SU8-2005 polymer.

The amplitude and phase evolutions of the Si nanowires have been fitted using a simple impulseresponse formalism, which accounted for TPA, the Kerr effect, FCA, and FCPE, without requiringthe use of detailed rate equations for the carrier dynamics. Based on this impulse response model,we have presented values for �TPA and n2 (3� 10�12 m.W�1 and 4� 10�18 m2:W�1, respectively),which have the same order of magnitude to values given in the literature, and the associated TPAfigure of merit was 0.9. Although this figure of merit is less than the optimum value of 2, these typesof Si nanowires can perform a variety of high-speed OSP functionalities. The long values for thefree-carrier lifetime (7.5 to 16.2 ns) may have been due to a combination of the relatively low degreeof sidewall roughness (which led to a moderate propagation loss of 4.5 dB.cm�1) and passivation ofthe Si nanowire surface by the SU8-2005 polymer [38].

The reversal from a positive phase shift to a negative phase shift at the point when TPA givesway to FCA leads to wavelength chirping, where the red and blue shifts have the same order ofmagnitude. This phenomenon has been exploited to perform two-copy wavelength conversion at80 Gb.s�1 [39], [40] using similar Si nanowires to the devices experimentally measured in thispaper, demonstrating that these devices are suitable for performing high-speed OSP functionsprovided that the contribution of FCA to the total absorption is minimized by maintaining the energyper pump pulse below �20 pJ.

TABLE 1

Comparison between the results presented in this paper and results reported in the literature(Nw ¼ nanowire, Rib ¼ rib waveguide, Slot ¼ slot waveguide, w ¼ waveguide width, h ¼ waveguideheight, L ¼ waveguide length, C.M. ¼ cladding material, and the dash means that the quantity was notreported)

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