Influence of spectral broadening on femtosecondwavelength conversion based on four-wave
mixing in silicon waveguides
Zhaolu Wang,1,2 Hongjun Liu,1,* Nan Huang,1 Qibing Sun,1 and Jin Wen1,2
1State Key Laboratory of Transient Optics and Photonics Technology, Xi’an Institute of Opticsand Precision Mechanics, Chinese Academy of Science (CAS), Xi’an, 710119, China
2Graduate University of CAS, Beijing, 100049, China
In recent years, silicon photonics has become a ra-pidly growing area. The nonlinear effects in silicon-on-insulator waveguides are attracting attentionbecause of the large values of the Kerr parameterand the tight confinement of the optical mode [1]. Inorder to better understand the potential of silicon asa nonlinear material, various nonlinear effects suchas stimulated Raman scattering [2,3], two-photon ab-sorption (TPA), free-carrier absorption (FCA) [4–6],self-phase modulation (SPM) [7–10], cross-phasemodulation (XPM) [11–13], and four-wave mixing(FWM) [14–27] have been intensively studied theo-retically and experimentally.
Wavelength conversion based on FWM has beenexplored in silicon waveguides—typically on timescales ranging from the continuous-wave (CW) to
the picosecond regime. In detail, the conversion of40Gb=s data rate using a CW pump has been demon-strated by Ying-Hao Kuo et al. [20]. They enhancethe conversion efficiency to −8:6dB using reversebiased p-i-n rib waveguides. Gao Shi-ming et al. [21]reported C-band wavelength conversion in a siliconwaveguide pumped by picosecond pulses. The pulse-pumped efficiency is demonstrated to be higher thanthe CW-pumped efficiency. However, for many all-optical signal processing applications with high bitrates, such as silicon photonic integrated circuitsfor optical chip-to-chip communications and siliconhighest-speed signals processors for optical commu-nications and computers, silicon-based wavelengthconverters pumped by femtosecond pulses will havea higher efficiency due to low FCA [26].
In this paper, we demonstrate efficient wave-length conversion via degenerate four-wave mixingin a 1:5mm long silicon rib waveguide with fem-tosecond pump and signal pulses. The impact ofspectral broadening on wavelength conversion for
femtosecond pulses is investigated. Moreover, theinfluence of the repetition rate on the conversionefficiency is researched, and high conversion effi-ciency can be obtained by using femtosecond pulses.The conversion bandwidth and the conversion effi-ciency as a function of the pump peak power are alsoinvestigated.
2. Theory
Here, we focus on the degenerate FWM, which typi-cally involves two pump photons at frequency ωp pas-sing their energy to a signal wave at frequency ωsand an idler wave at frequency ωi as the relation2ωp ¼ ωs þ ωi holds. The signal wave is amplifiedand the idler wave is generated during the FWMprocess. Moreover, the phase matching among theinteracting waves is required in the FWM process,which is achieved when the mismatch in the propa-gation constants of the pump, signal, and idlerwaves is compensated by the phase shift due to SPMand XPM, such that Δk ¼ Δβ þ 2γpPpump ¼ 0. Here,Δβ ¼ ks þ ki − 2kp is the phase mismatch due to thelinear dispersion, Ppump is the pump power, γp ¼ωpn2=cAeff is the nonlinear waveguide parameter,n2 ¼ 12π2χð3Þ=n0c is the nonlinear index coefficient,c is the speed of light, n0 is the linear refractive index,and Aeff is the effective area of the propagating mode,respectively. As the nonlinear part (2γpPpump) is po-sitive, phase matching can be realized by locating thepump pulse in the anomalous dispersion regime [25].
The pump, signal, and idler waves are identicallypolarized in the fundamental quasi-TE mode. To de-scribe the nonlinear optical interaction of the pump,signal, and idler in the waveguide, we use the formu-lism described in [19,24,28] and take into accountthe effects of TPA, FCA, and free-carrier dispersion.Remarkably, the Raman effects are negligible forfemtosecond pulses propagating in silicon wave-guides [29]. The equations that govern the evolutionof the different waves read as
∂Ap
∂zþ iβ2p
2
∂2Ap
∂T2 −β3p6
∂3Ap
∂T3
¼ −12ðαp þ αf cpÞAp þ iγpe
�1þ i
ωp
∂
∂t
�jApj2Ap
þ i2πλp
δnf cpAP þ 2iγpeðjAsj2 þ jAij2ÞAp
þ 2iγpAsAiA�p expðiΔβzÞ; ð1Þ
∂As
∂zþ ds
∂As
∂Tþ iβ2s
2∂2As
∂T2 −β3s6
∂3As
∂T3
¼ −12ðαs þ αf csÞAs þ iγse
�1þ i
ωs
∂
∂t
�jAsj2As
þ i2πλs
δnf csAs þ 2iγseðjApj2 þ jAij2ÞAs
þ iγsA2pA�
i expð−iΔβzÞ; ð2Þ
∂Ai
∂zþ di
∂Ai
∂Tþ iβ2i
2∂2Ai
∂T2 −β3i6
∂3Ai
∂T3
¼ −12ðαi þ αf ciÞAi þ iγie
�1þ i
ωi
∂
∂t
�jAij2Ai
þ i2πλi
δnf ciAi þ 2iγieðjApj2 þ jAsj2ÞAi
þ iγiA2pA�
s expð−iΔβzÞ; ð3Þ
where Aj is the slowly varying amplitude (j ¼ p; s; i),z is the propagation distance, β2j is the group-velocitydispersion (GVD) coefficient, and β3j is the third-order dispersion coefficient. Time T ¼ t − z=vgp ismeasured in a reference frame moving with pumppulse traveling at speed vgp. The two walk-off param-eters of the signal and idler are defined as ds ¼ β1s −β1p and di ¼ β1i − β1p, respectively, where β1j is theinverse of the group velocity. The nonlinear coeffi-cient γje is given by [25]
γje ¼ γj þ iβTPA2Aeff
; ð4Þ
where γj ¼ ωjn2=cAeff with n2 ¼ 6 × 10−18 m2W−1 andβTPA ¼ 5 × 10−12 mW−1 [10] is the coefficient ofTPA. The above coupled equations are solved usingthe split-step Fourier method and a fourth-orderRunge–Kutta solver.
In Eqs. (1)–(3), αj accounts for the linear lossand αf cj ¼ σjNc represents FCA, where σj is the FCAcoefficient and Nc ¼ Ncp þNcs þNci is the free-carrier density generated by pump, signal, and idlerpulses. δnf cj ¼ ζjNc is the free-carrier-induced indexchange. These free-carrier parameters are obtainedby solving [9,25]
σj ¼ 1:45 × 10−21ðλj=λref Þ2 m2;
ζj ¼ −1:35 × 10−27ðλj=λref Þ2 m3; ð5Þ
∂Ncjðz; tÞ∂t
¼ πβTPA2hωjA2
effjAjðz; tÞj4 −
Ncjðz; tÞτc
; ð6Þ
where γj is the wavelength, λref ¼ 1550nm, h isPlanck’s constant, and τc ≈ 1ns is the carrier lifetime.Under the condition of pulsed pump, the repetitionrate R is an important factor impacting on Nc, whichcan be approximately given by [27]
Ncðz; tÞ ≈βTPAT0
2ð1 − e−1=RτcÞhcA2eff
ðλpjApðz; tÞj4 þ λsjAsðz; tÞj4
þ λijAiðz; tÞj4Þ; ð7Þ
where T0 is the half-width (at 1/e-intensity point) ofthe pulse. When Rτc ≪ 1, the free-carrier density ismainly determined by the pulse width, so that thefree-carrier density remains low for femtosecondpulses. However, if Rτc is close to or exceeds 1, thedensity of the carriers increases until a steady state
is reached because carriers produced by a pulse donot have enough time to recombine before the nextpulse arrives.
The silicon waveguide used here is a rib wave-guide, which allows for compatibility with electricalcontrol of carrier removal for future devices [3]. Therib height (H) and the etch depth (h) are 300nm and270nm, respectively [17]. We can change the ribwaveguide width (W) to tailor the zero-dispersionwavelength (ZDWL). The core of the waveguide issilicon and the cladding is silica, as shown in Fig. 1.
For the rib waveguide, the TE mode effectiveindices neff are numerically determined using theeffective index method. The dispersion relation isthen calculated from βðωÞ ¼ neff ðωÞω=c. Higher-orderdispersion is finally calculated via numerical differ-entiation from βn ¼ dnβ=dωn, and the results areshown in Fig. 2. The ZDWLs are 1360nm for thesilicon waveguide with a rib width W ¼ 550nm, soit exhibits anomalous group dispersion over thewavelength range studied here. In general, thequasi-phase-matching can be satisfied over a broadband if the pump wavelength is located in the anom-alous GVD regime [23]. Therefore, the broadband
phase-matching condition can be achieved by usingλp ¼ 1550nm pump pulse.
The wavelength conversion is numerically studiedby simultaneously injecting pump pulses centered atλp ¼ 1550nm and signal pulses tunable from 1400 to1540nm with same pulse width TFWHM ðTFWHM ≈
1:763T0Þ and same repetition rate R. Both of thepulses are hyperbolic-secant pulses, which areemitted from mode-locked lasers. The dispersioncoefficients can be obtained from Fig. 2. The corre-sponding value for the effective mode area Aeff isabout 0:09 μm2 and the linear propagation loss ofthe waveguide is about 0:3dB=cm for the threewaves [16].
3. Results and Discussion
In order to realize effectively wavelength conversion,the optimal length of the silicon waveguide shouldbe calculated. Figure 3 depicts the single pulseenergy variation of the pump, signal, and idler alongthe silicon waveguide for signal wavelength at λs ¼1450nm. The pump peak power coupled inside thewaveguide is 5W,while the signal peak power is keptconstant at 1W. The input pulse widths of the pumpand signal are 300 fs. As shown in Fig. 3, the peak ofthe energy for signal and idler pulses appears whenthe propagation length is 1:5mm, then the signal andidler pulses flow back to the pump pulse. Therefore,the maximum conversion efficiency can be achievedusing a 1:5mm long silicon waveguide in this case.For convenience, we assume the rib waveguidelength is 1:5mm long in this paper, although the op-timal length may change with the input peak power
Fig. 1. Rib silicon waveguide dimension.
Fig. 2. (a) Effective indices curve; (b) first-order dispersion curve; (c) group velocity dispersion curve; (d) third-order dispersion curve.
of the pump due to nonlinear phase mismatch.Furthermore, a great deal of the pump energy is con-sumed owing to the combination of the linear deple-tion and nonlinear absorption including TPA and
FCA, and the pump is not depleted due to the inten-sities of both the signal and idler being much smallerthan that of the pump [25].
Figure 4 shows the spectra at the input and outputof the rib waveguide for a 1550nm pump with 5Wpeak power and 1450nm signal when the pumpand signal pulse width are 100 fs, 300 fs, and 500 fs,respectively. It is clear that the signal was efficientlyconverted and amplified into an idler at a wave-length of λi ¼ 1665nm. All of the output spectra arebroadened and spectral overlap appears for the100 fs pulses as shown in Fig. 4. Aside from the SPMof the pump, the XPM effects induced by the signaland idler on the pump also have contribution to thespectral broadening of the pump, while the spectralbroadening of the signal and idler is due to the com-bination between the SPM of themselves and theXPM induced by the pump on the signal and idler.It is clear that the spectral broadening is much largerfor shorter pulses in the femtosecond regime andspectral overlap cannot avoid in this case for pulses,
Fig. 3. (Color online) The relationship between the single pulseenergy of the three waves and silicon waveguide length, whenλs ¼ 1450nm, TFWHM ¼ 300 fs.
Fig. 4. (Color online) The input and output spectrum in the 1:5mm long silicon rib waveguide for a 1550nm pump and a 1450nm signalwith (a) 100 fs, (b) 300 fs, and (c) 500 fs pulse widths, respectively.
which are less than 100 fs. Therefore, it is difficult toachieve a wavelength converter when the inputpump and signal pulse widths are close to or lessthan 100 fs because it is difficult to separate thesignal and idler from the pump. In addition, theconversion efficiency must be influenced by the spec-tral broadening, which can be explained by the fol-lowing part.
Conversion efficiency as a function of input pulsewidth is shown in Fig. 5 for a 1450nm signal and1550nm pump. The pump peak power coupled insidethe waveguide is 6W. Here, we define the conversionefficiency as the ratio of the pulse energy of the out-put idler Eiout with respect to the pulse energy of theinput signal Esin:
η ¼ 10 log10ðEiout=EsinÞ: ð8ÞEquation (7) indicates that shorter pulses means
lower loss caused by FCA and higher conversion ef-ficiency under the same pump peak power and thesame repetition rate. However, Fig. 5(a) shows thatthe conversion efficiencies of the 100 fs and 200 fspulses are lower than the 300 fs pulses when therepetition rate is low (0:1GHz). The serious spec-tral broadening for 100 fs pulses, as shown in Fig. 4,can explain this phenomenon. As part of the pumpenergy used to generate new spectral components,the effective pump power used to FWM is reduced.Therefore, the spectral broadening reduces theconversion efficiency and this can be validated byFig. 5(b), which shows that the conversion efficiency
increases with increasing the pulse width when theFCA is neglected. It also can be seen from Fig. 5(b)that the influence of the spectral broadening on con-version efficiency can be neglected when the pulsewidth is longer than 500 fs. The distinction of the con-version efficiency (about 0:15dB) for the femtosecondpulses is small when the repetition rate is low, asshown in Fig. 5(a). Therefore, compared with TPA,the combination of the spectral broadening and FCAonly has slight contribution to the reduction of theconversion efficiency for a low repetition rate [25].However, the FCA effect must play an importantrole for a high repetition rate (>1GHz), which isillustrated by Fig. 6.
Figure 6 shows the relationship of conversionefficiency and repetition rate with different pulsewidths. The pump peak power is 6W and the carrierlifetime is 1ns. It is shown that the conversion effi-ciency is without significant change when the repeti-tion rate is below 1GHz. That is because carriersproduced by a pulse have enough time to recombinebefore the next pulse arrives as long as Rτc < 1 andthe FCA is determined by the pulse width. Theconversion efficiency begins to decrease when therepetition increases due to the enhanced FCA. Highconversion efficiency can be achieved for femtosecondpulses when the repetition rate is 100GHz, while thepicosecond pulses have low conversion efficiency.Therefore, the wavelength converter pumped byfemtosecond pulses is more efficienct than theconverter pumped by picosecond pulses for optical
Fig. 5. Conversion efficiency versus pulse width in the cases of (a) low repetition rate (0:1GHz) and (b) neglecting FCA.
Fig. 6. Conversion efficiency versus the repetition rate fordifferent pulse widths.
Fig. 7. Conversion efficiency versus signal wavelength for twodifferent pulse widths when the repetition rate is 100GHz.
telecommunication and signal processing applica-tions with bit rates as high as 100Gb=s.
The conversion efficiency as a function of the signalwavelength for different pulse widths with a repeti-tion rate of 100GHz is shown in Fig. 7. The pumppeak power is 6W and the signal peak power is 1W.The curves are symmetric around pump wavelengthλp ¼ 1550nm, so only the half is plotted. From Fig. 7,one can find that the conversion efficiency pumpedby the 500 fs pulses is more than 3dB higher thanthat pumped by the 1ps pulses. The conversion band-width spans the range from 1430 to 1540nm asshown in Fig. 7. Therefore, the total conversion band-width is about 220nm and it can cover the S-band,C-band, L-band, and U-band.
Figure 8 shows the relationship of conversionefficiency and pump peak power for two differentpulse widths with 100GHz repetition rate for a sig-nal wavelength at λs ¼ 1480nm. The input pumppeak power varies from 1 to 12W, while the signalpeak power is 1W. With increasing the pump peakpower, the conversion efficiency gradually saturatesand eventually drops after maximum efficiency isachieved. The reason for this phenomenon is theincreasing nonlinear losses and phase mismatchas the pump peak power increases [27]. The con-version efficiency of the 1ps pulses more quicklyreaches saturation because the FCA is larger for 1pspulses. The maximum efficiency is obtained by using6W pump peak power for 500 fs pulses, and using4W pump peak power for 1ps pulses.
4. Conclusion
We have presented a theoretical study of wavelengthconversion in telecommunication band for femto-second pulses in silicon rib waveguides. The simula-tion model allows us to show clearly the importanceof SPM and XPM effects induced spectral broadeningin the FWM process. First, it is difficult to achievea wavelength converter when the pump and signalpulse widths are close to or less than 100 fs due tospectral overlap. Moreover, the spectral broadeningof the femtosecond pulses reduces the conversionefficiency. Furthermore, compared with picosecondpulses, high conversion efficiency can be obtained
by using femtosecond pulses for a high (approxi-mately 1–100GHz) repetition rate. Finally, ultra-broadband wavelength conversion in the siliconrib waveguide is demonstrated with bandwidth of220nm and peak conversion efficiency of −8dB for500 fs pulses.
This work was supported by the National NaturalScience Foundation of China (NSFC) under grants60878060 and 61078029.
References
1. J. Leuthold, C. Koos, and W. Freude, “Nonlinear siliconphotonics,” Nat. Photon. 4, 535–544 (2010).
2. V. Raghunathan, O. Boyraz, and B. Jalali, “20dB on-offRaman amplification in silicon waveguides,” in Proceedingsof the Conference on Lasers and Electro-optics (CLEO), Vol. 1(IEEE, 2005), pp. 349–351.
3. H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M.Paniccia, “A continuous-wave Raman silicon laser,” Nature433, 725–728 (2005).
4. T. K. Liang and H. K. Tsang, “Silicon waveguide two-photonabsorption detector at 1:5 μm wavelength for auto-correlationmeasurements,” Appl. Phys. Lett. 81, 1323–1325 (2002).
5. D. Dimitripoulos, R. Jhavery, R. Claps, J. C. S. Woo, andB. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86,071115 (2005).
6. Y. Liu and H. Tsang, “Time dependent density of free carriersgenerated by two photon absorption in silicon waveguides,”Appl. Phys. Lett. 90, 211105 (2007).
7. H. K. Tsang, C. S.Wong, T. K. Liang, I. E. Day, S.W. Roberts, A.Harpin, J. Drake, and M. Asghari, “Optical dispersion,two-photon absorption, and self-phase modulation in siliconwaveguides at 1:5um wavelength,” Appl. Phys. Lett. 80,416–418 (2002).
8. I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M.Osgood Jr., “Ultrafast-pulse self-phase modulation and third-order dispersion in Si photonic wire-waveguides,” Opt.Express 14, 12380–12387 (2006).
9. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and super-continuum generation in silicon waveguides,” Opt. Lett. 32,391–393 (2007).
10. L. Yin and G. P. Agrawal, “Impact of two-photon absorption onself-phase modulation in silicon waveguides,” Opt. Lett. 32,2031–2033 (2007).
11. M. Khorasaninejad and S. S. Saini, “All-optical logic gatesusing nonlinear effects in silicon-on-insulator waveguides,”Appl. Opt. 48, F31–F36 (2009).
12. I.-W. Hsieh, X. Chen, J. I. Dadap, N. C. Panoiu, and R. M.Osgood Jr., “Cross-phase modulation-induced spectral andtemporal modulation effects on co-propagating femtosecondpulses in silicon photonic wires,” Opt. Express 15, 1135–1146(2007).
13. L. Yin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Opticalswitching using nonlinear polarization rotation inside siliconwaveguides,” Opt. Lett. 34, 476–478 (2009).
14. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa,T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mix-ing in silicon wire waveguides,” Opt. Express 13, 4629–4637(2005).
15. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J.Painter, “A proposal for highly tunable optical parametricoscillation in silicon micro-resonators,” Opt. Express 16,10596–10610 (2008).
Fig. 8. Conversion efficiency versus pump peak power for twodifferent pulse widths when the repetition rate is 100GHz.
16. H. Rong, Y. Kuo, A. Liu, and M. Paniccia, “High efficiencywavelength conversion of 10Gb=s data in silicon waveguides,”Opt. Express 14, 1182–1188 (2006).
17. A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta,and M. Lipson, “Frequency conversion over two-thirds ofan octave in silicon nanowaveguides,” Opt. Express 18,1904–1908 (2010).
18. S. Zlatanovic, J. S. Park, S. Moro, J. M. C. Boggio, I. B.Divliansky, N. Alic, S. Mookherjea, and S. Radic, “Mid-infraredwavelength conversion in silicon waveguides using ultracom-pact telecom-band-derived pump source,” Nat. Photon. 4,561–564 (2010).
19. R. L. Espinola, J. I. Dadap, R. M. Osgood Jr., S. J. McNab, andY. A. Vlasov, “C-band wavelength conversion in silicon photo-nic wire waveguides,” Opt. Express 13, 4341–4349 (2005).
20. Y.-H. Kuo, H. Rong, V. Sih, S. Xu, M. Paniccia, and O. Cohen,“Demonstration of wavelength conversion at 40Gb=s datarate in silicon waveguides,” Opt. Express 14, 11721–11726(2006).
21. S.-M. Gao, E.-K. Tien, Q. Song, Y.-W. Huang, and S. K.Kalyoncu, “Experiment of C-band wavelength conversion ina silicon waveguide pumped by dispersed femtosecond laserpulse,” Chin. Phys. Lett. 27, 124206 (2010).
22. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M.Lipson, and A. L. Gaeta, “Broad-band optical parametric gainon a silicon photonic chip,” Nature 441, 960–963 (2006).
23. X. Liu, R. M. Osgood Jr., Y. A. Vlasov, and W. M. J. Green,“Mid-infrared optical parametric amplifier using silicon nano-photonic waveguides,” Nat. Photon. 4, 557–560 (2010).
24. W. Mathlouthi, H. Rong, and M. Paniccia, “Characterizationof efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express 16, 16735–16745(2008).
25. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultra-broadband parametric generation and wavelength conversionin silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
26. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear opticalphenomena in silicon waveguides: Modeling and applica-tions,” Opt. Express 15, 16604–16644 (2007).
27. X. Sang and O. Boyraz, “Gain and noise characteristics ofhigh-bit-rate silicon parametric amplifiers,” Opt. Express16, 13122–13132 (2008).
28. G. P. Agrawal,Nonlinear Fiber Optics, 4th ed. (Elsevier, 2007).29. N. C. Panoiu, X. Liu, and R. M. Osgood Jr., “Self-steepening of
ultrashort pulses in silicon photonic nanowires,”Opt. Lett. 34,947–949 (2009).
