Wavelength conversion in highly nonlinear silicon– organic hybrid slot waveguides Linliang An, Hongjun Liu,* Qibing Sun, Nan Huang, and Zhaolu Wang State Key Laboratory of Transient Optics and Photonics Technology, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science (CAS), Xi’an, 710119, China *Corresponding author: [email protected] Received 29 April 2014; accepted 12 June 2014; posted 18 June 2014 (Doc. ID 211020); published 23 July 2014
Wavelength conversion based on four-wave mixing (FWM) in a silicon–organic hybrid slot waveguide is theoretically investigated in the telecommunication bands. Compared with vertical slot waveguides, the horizontal slot waveguide structure exhibits much flatter dispersion. The maximum nonlinearity coefficient γ of 1.5 × 107 W−1 km−1 and the minimum effective mode area Aeff of 0.065 μm2 are obtained in a horizontal slot waveguide with a 20-nm-thick optically nonlinear layer by controlling the geometric parameters. The wavelength conversion efficiency of 7.45 dB with a pump power of 100 mW in a 4-mm-long horizontal slot waveguide is obtained. This low power on-chip wavelength convertor will have potential applications in highly integrated optical circuits. © 2014 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (190.7110) Ultrafast nonlinear optics; (230.7370) Waveguides; (230.7405) Wavelength conversion devices. http://dx.doi.org/10.1364/AO.53.004886
1. Introduction
Silicon waveguides have attracted much attention because they have large values of Kerr parameter, are transparent at infrared wavelengths, have tight confinement of optical mode [1], and have instantaneous response time for high-speed operation [2]. Silicon is a centrosymmetric material, and its lowest-order optical nonlinearity is third-order. Many optical mechanisms based on third-order nonlinearities, such as self-phase modulation (SPM) [3], cross-phase modulation (XPM) [4], FWM [5], and stimulated Raman scattering (SRS) [6], have all been studied widely for all-optical signal processing. However, the nonlinearity effects in telecommunication bands are severely impaired due to the existence of two-photon absorption (TPA) and free-carrier absorption (FCA) in pure silicon waveguides [7]. Silicon-on-insulator (SOI) waveguides are widely used silicon waveguides, which can take the form 1559-128X/14/224886-08$15.00/0 © 2014 Optical Society of America 4886
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of channel waveguides, ridge waveguides, photoniccrystal waveguides, or slot waveguides. Moreover, the mature technology of complementary metal– oxide–semiconductor (CMOS) used for SOI makes it an attractive platform for photonic integration [2]. To exploit the Kerr coefficient and overcome the limitations from carrier effects, a slot-type geometry was proposed [8–10], where silicon is combined with another low-index high nonlinear coefficient material [11,12]. Compared with other structures of SOI waveguides (such as silicon strip waveguides) and fiber, this kind of slot waveguide shows better performance in nonlinearity integration [8,13], which has the advantages of smaller effective mode area, higher nonlinearity coefficient, and stronger confinement of light. This kind of waveguide is usually divided into horizontal and vertical structures. The candidate materials combined with silicon can be organic or inorganic materials [14], and the silicon–organic hybrid is called a SOH waveguide, where a silicon-slotted waveguide is filled and surrounded with low refractive index, highly nonlinear organic material [2].
In this paper, the SOH waveguide with low-index high nonlinear coefficient organic material of p-toluene sulphonate (PTS) is investigated to realize efficient on-chip wavelength conversion in the telecommunication bands. We compare the dispersion of the horizontal and vertical structures, and calculate the phase-matching condition. Under the optimal geometry, a wavelength conversion efficiency of 7.45 dB via degenerate FWM can be achieved in a 4-mm-long horizontal slot waveguide by using a pump power of 100 mW. Therefore, the horizontal SOH slot waveguide, which possesses a high nonlinear coefficient and small effective mode area, can be used as a highly efficient wavelength convertor with a low pump power and short waveguide length. 2. Structure and Dispersion of the Slot Waveguide
The slot waveguide allows for extremely strong confinement of light in its low-index region, and this feature has attracted considerable attention for the exploitation of third-order nonlinear effects in lowindex media [14]. Silicon–organic hybrid integration is a popular and functional combination, which can make full use of the advantages of these two different media [2]. The highly nonlinear organic material PTS, which has a nonlinear refractive index of n2 2.2 × 10−16 m2 W−1 , is chosen as the candidate organic material [8,14]. For PTS, the large TPA peak is located at 930 nm; thereafter, it decreases monotonically with increasing wavelength until 1500 nm, which nearly turns to zero [15]. Therefore, the combination of silicon and PTS can be highly χ χ nonlinear without suffering from TPA and associated FCA around 1550 nm. For such a SOH waveguide structure, most of the light is confined in the PTS region. This is because, in a high-index contrast interface, the electric field must undergo a large discontinuity with much higher amplitude in the lowindex side, resulting in a high power confinement in the low-index regions [16]. In general, the SOH waveguide can be divided into horizontal and vertical structures, as shown in Fig. 1. Since most of the light is confined in the slot area, reducing the slot width will increase the optical intensity [7]. However, taking into account of the practical production processes, the slot layer cannot be made infinitely small, and the minimum achievable vertical slot width is practically limited to 50 nm [14].
The horizontal slot layer thickness can be better controlled for a thinner width about 20 nm. The material dispersion is incorporated via the Sellmeier equations for Si [17], n2Si 11.6858 0.939816∕λ2 8.10461 × 1.10712 ∕λ2 − 1.10712 ;
(1)
and for PTS [18], n2PTS 2.9971 0.6193λ2 ∕λ2 − 0.4044 − 0.2732λ2 : (2) In a SOH waveguide, since FWM-based wavelength conversion is mainly influenced by the phase mismatch, which depends on the dispersion and nonlinearity. An optimized dispersion will contribute to reduce phase mismatch, enhancing the wavelength conversion band and releasing the requirement for high pump power in nonlinear processes [19]. As a result, the chromatic dispersion tailored by altering the waveguide dimensions to realize phase-matching, plays a critical role in the nonlinear processes. By changing the geometry parameter w in SOH waveguide (see Fig. 1), we obtained a series of dispersion distributions as a function of the wavelengths. As shown in Fig. 2(a), when the silicon layer thickness h is 200 nm and the slot thickness s is 20 nm, the zero-dispersion wavelength (ZDW) of the horizontal SOH waveguide shifts from 1400 to 1600 nm as the silicon wires width w increases. The inset shows the specific structure, which has an anomalous dispersion near 1550 nm. One can find that the optimal silicon wires width w of the horizontal SOH waveguide is 600 nm, which has the relatively flat dispersion and slightly anomalous dispersion near 1550 nm. For vertical SOH waveguide, when silicon layer thickness h is 300 nm and slot thickness s is 50 nm, the optimal geometry parameter w is 600 nm, as illustrated in Fig. 2(b). The electric field distribution is calculated with the optimal geometric parameters utilizing the finiteelement method. The fundamental TM-mode profile of the horizontal SOH waveguide at the wavelength of 1550 nm is simulated, as shown in Fig. 3(a), and the Ex and Ey electric field components are shown in Figs. 3(b) and 3(c), respectively. Analogously, for
Fig. 1. (a) Schemes of horizontal and (b) vertical SOH slot waveguides. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 2. Dispersion distribution of (a) horizontal SOH waveguide for different values of silicon wires length w with constant h 200 nm and s 20 nm and (b) vertical SOH waveguide for different values of silicon wires length w with constant h 300 nm and s 50 nm.
vertical SOH waveguide, Fig. 3(d) depicts the fundamental TE-mode profile at 1550 nm, and Figs. 3(e) and 3(f) show the Ex and Ey electric field components, respectively. It is clear that this kind of waveguide can strongly confine light in the middle of the low refractive index region. Moreover, one can intuitively find that the horizontal structure shows better ability in confining light than the vertical structure at 1550 nm, because of the former has a narrower slot thickness than the latter [7]. In resemblance to dispersion, the effective mode area Aeff and the nonlinear coefficient γ e also play important roles in the nonlinear process. The effective mode area and nonlinear coefficient are given by [20,21]
RR Aeff
∞ −∞
RR ∞
2 jFx; yj2 dxdy
−∞
2π γe λ
jFx; yj4 dxdy
;
RR ∞ n x; yjFx; yj4 dxdy −∞ RR ∞2 ; −∞ jFx; yj2 dxdy2
(3)
(4)
where Fx; y is the profile of the field and λ is the wavelength. Figure 4 illustrates the effective mode area Aeff and nonlinear coefficient γ e as a function of the wavelength. The geometry parameters we used here are the optimal values for the horizontal and vertical SOH waveguides mentioned above. As expected, such a SOH waveguide has the advantage
Fig. 3. (a) TM-mode of electric mode and (b) Ex and (c) Ey electric field components of the horizontal SOH waveguide at 1550 nm. (d) The TE-mode of electric mode and (e) Ex and (f) Ey electric field components of the vertical SOH waveguide at 1550 nm. 4888
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Fig. 4. Effective mode area and nonlinear coefficient of the (a) horizontal and (b) vertical SOH waveguide.
of possessing a large nonlinear coefficient, as well as small effective mode area, which makes it suitable for nonlinear applications. Moreover, one can find that the horizontal SOH waveguide has a smaller effective mode area Aeff and a larger nonlinear coefficient γ e than the vertical SOH waveguide. The smaller the effective area provided by the waveguide, the higher the nonlinear interaction [14]. Therefore, through our simulations, the horizontal SOH waveguide shows better performance than the vertical SOH waveguide. The optimal horizontal SOH waveguide is obtained for slot thicknesses of 20 nm with silicon wires of 200 nm height and 600 nm width, which can be used to complete the following wavelength conversion. 3. Wavelength Conversion Based On FWM
FWM is an important nonlinear effect. Its promising applications in optical parametric amplification (OPA), wavelength conversion, and optical parametric oscillator (OPO) have attracted tremendous research interests. Here, we focus on the degenerate FWM, which involves two identical pump photons at frequency ωp passing their energy to signal and idler photons at respective frequency ωs and ωi with the relation of 2ωp ωs ωi. During the FWM process, the signal wave is amplified and the idler wave is generated. The phase mismatch ΔK is given by [20] ΔK Δβ 2γ P ppump ;
(5)
where Ppump is the pump power, γ p ωp n2 ∕cAeff is the nonlinear parameter, Δβ ks ki − 2kp is linear phase-mismatch, and ks , ki , and kp are signal, idler, and pump wave vectors, respectively. Phasematching is achieved under the condition of ΔK 0. Since the nonlinear part (2γ p Ppump ) is positive, the pump pulse should be located in the anomalous dispersion region to achieve phase-matching [22]. We now focus on phase-matching using the horizontal SOH slot waveguide with optimal structure parameters. Figure 5(a) shows the linear phase mismatch for three different pump wavelengths close to the 1.55 μm, and the Δβ nearly turns to zero from 1.46 to 1.64 μm. Figure 5(b) depicts the phase mismatch for different pump powers in the case of injecting pump wavelength at 1.55 μm. Even with a low pump peak power, phase-matching can be easily realized, which is one of the main advantages of this waveguide. Wavelength conversion based on FWM is a promising application in silicon waveguides. The pump, signal, and idler waves are identically polarized in the fundamental quasi-TM-mode in our simulations. The Raman frequency shift of the organic material PTS is about 27 THz [23–25], which is far wider than the frequency (wavelength) difference between pump and signal, so the SRS can be ignored in our simulation. The FWM process can be described by the following coupled wave equations [20,26]:
Fig. 5. (a) Linear phase mismatch with different pump wavelengths. (b) Phase mismatch with different pump power. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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∂Ap iβ2p ∂2 Ap β3p ∂3 Ap − ∂z 2 ∂T 2 6 ∂T 3 1 i ∂ jAp j2 Ap − αp Ap iγ p 1 2 ωp ∂T 2iγ p jAs j2 jAi j2 Ap 2iγ p As Ai Ap expiΔβz; (6) ∂As ∂A iβ ∂2 As β3s ∂3 As ds s 2s − ∂z ∂T 2 ∂T 2 6 ∂T 3 1 i ∂ jAs j2 As − αs As iγ s 1 2 ωs ∂T 2iγ s jAp j2 jAi j2 As iγ s A2p Ai exp−iΔβz;
(7)
∂Ai ∂A iβ ∂2 Ai β3i ∂3 Ai di i 2i − ∂z ∂T 2 ∂T 2 6 ∂T 3 1 i ∂ jAi j2 Ai − αi Ai iγ i 1 2 ωi ∂T 2iγ i jAp j2 jAs j2 Ai iγ i A2p As exp−iΔβz;
(8)
where Aj is the amplitude (j p, s, i), z is the propagation distance, and αj is the linear loss (j p, s, i). The walk-off parameters for the signal and idler are defined as ds β1s − β1p and di β1i − β1p , respectively. βn is the nth-order dispersion coefficient, which is calculated via numerical differentiation from βn dn β∕dωn . The average corresponding value for the effective mode area Aeff is about 0.08 μm2 , the nonlinear coefficient γ e is about 1 × 107 W−1 km−1 , as calculated in Fig. 5(a), and the linear loss is about 4.34 dB∕cm [15]. The above coupled equations are solved using the split-step Fourier method and a fourth-order Runge–Kutta solver. Both the pump and signal pulses injected into the horizontal slot waveguide are hyperbolic–secant pulses. The center wavelength and full width at half-maximum (FWHM) with pump pulses of 1.55 μm and 1 ps. The idler conversion efficiency is defined as the ratio of the output idler power Piout with respect to the incident signal power Psin, and the expression is η 10 log10 Piout ∕Psin . The parametric signal gain is defined as G 10 log10 Psout ∕Psin .
For the purpose of realizing effectively wavelength conversion, the optimal length of this waveguide should be calculated. The pump peak power coupled inside the waveguide is 100 mW while the signal peak power varies for two different values, as depicted in Fig. 6(a). When the peak signal powers are set to 10 and 5 mW, the optimal waveguide lengths are 3.5 and 4 mm, and the corresponding maximum wavelength conversion efficiencies are about 4.85 and 7.45 dB, respectively. Moreover, Fig. 6(b) depicts the energy of the pump, signal, and idler varied along the length of the waveguide with a signal power of 5 mW, which can further describe the generation of optimum length from the point of view of energy. When the propagation length is about 4 mm, the signal and idler reach the maximum energy for the first time. Since then, the inverse transformation of the energy occurs between the pump, signal, and idler inside the waveguide. In this case, the maximum conversion efficiency of 7.45 dB can be achieved using a 4-mmlong waveguide with a signal power of 5 mW. For convenience in this paper, we assume the SOH slot waveguide length is 4 mm. Different signal wavelengths with different pulse widths correspond to different conversion efficiency. Figure 7 shows the conversion efficiency as a function of the signal wavelengths for four different pulse widths. The peak power of the input pump and signal lights are set to be 100 and 5 mW, respectively. The pump is located at 1.55 μm and the signal is tuned from 1.41 to 1.54 μm, which can be converted into idler wave that varies from 1.56 to 1.72 μm. The curves are symmetric around pump wavelength λp 1.55 μm, and only the half is plotted. Therefore, the total conversion bandwidth is about 300 nm in this slot waveguide. For the 0.5, 1, and 10 ps pulses, the conversion efficiency increases first and then tends to saturation as the signal light wavelength increases. This phenomenon is mainly caused by the phase mismatch and the optimal waveguide length. From Fig. 5, one can find that the phase mismatch decreases with the increase of signal wavelength, which induces an increase in conversion efficiency. Moreover, a signal located at 1.46 μm with a power of 5 mW has an optimal waveguide length of
Fig. 6. (a) Conversion efficiency and (b) pulse energy as a function of waveguide length. 4890
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Fig. 7. Conversion efficiency as a function of signal wavelengths for four different pulse widths in a 4-mm waveguide.
4 mm, as shown in Fig. 6(b). The output pump power has minimum values around 1.46 μm, where the idler obtains the maximum energy. Therefore, the maximum conversion efficiency appears around the wavelength of 1.46 μm. However, the conversion efficiency of the 0.1 ps pulses is lower than others due to the large spectral broadening of the pump pulse. Because part of the pump energy is used to generate new spectral components, the effective pump power used to FWM is reduced [27]. To find the suitable width of pulses for propagating in the SOH slot waveguide, we simulate the temporal characteristics of the signal and idler for four different pulse widths with signal pulses centered at 1.46 μm. The input pump and signal peak power are set to be 100 and 5 mW, respectively. It is clear
that the output pulses from the waveguide with 0.1 ps are seriously distorted, whereas the output pulses with input pulse widths of 0.5, 1, and 10 ps are relatively smooth and without obvious distortion, as shown in Fig. 8. Compared with the pulse with a relatively wide pulse width, the shorter pulse has a smaller dispersion length, which means a larger influence of the dispersion effects [20,28]. Moreover, the role of the XPM effect cannot be ignored for the output signal and idler pulses [28]. Therefore, the distortion of the output signal and idler pulses of the 0.1 ps pulses are mainly caused by the combination of large dispersion and XPM, and it is difficult to achieve a wavelength converter when the input pulse widths are close to or less than 0.1 ps. The conversion efficiency is also influenced by the input pump power. In the following simulation, the input pump peak power varies from 10 to 220 mW with a signal peak power of 5 mW. Figure 9(a) shows the relationship between conversion efficiency and pump peak power for three different pulses widths with the same signal wavelength of 1.46 μm. For picosecond pulses, with increasing the pump peak power, the conversion efficiency saturates gradually, and drops after maximum efficiency eventually due to the increase of the phase mismatch [27,29]. The maximum idler conversion efficiency of the 1 ps pulses is 9.67 dB when the pump peak power reaches 160 mW. For the 10 ps pulses, the maximum idler conversion efficiency is 8.55 dB with the pump peak power of 130 mW. This kind of SOH slot waveguide can be applied to not only wavelength conversion but also parametric amplification, as shown in Fig. 9(b).
Fig. 8. Output temporal profiles from the input pulse widths of (a) 0.1 ps, (b) 0.5 ps, (c) 1 ps, and (d) 10 ps. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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has two structures: vertical and horizontal. This waveguide can confine the light in its low-index region without severe impairment by TPA. The chromatic dispersion, effective mode area, and nonlinear coefficient of vertical and horizontal SOH waveguide structures have been analyzed. By comparison, our results indicate that the horizontal SOH waveguide shows better performance and is more suitable for nonlinear interaction than the vertical SOH waveguide. The wavelength conversion has been numerically investigated in the optimized horizontal SOH waveguide. Under the optimal structural, we obtain efficient wavelength conversion of 7.45 dB via degenerate FWM in a 4-mm-long horizontal SOH slot waveguide with a pump peak power of 100 mW. This work was supported by the National Natural Science Foundation of China under Grant 61178023 and 61275134. References
Fig. 9. (a) Idler conversion efficiency, (b) signal gain, and (c) output pump power as a function of pump power.
It is found that the gain becomes larger as the pump power increases, and gain saturation appears when the pump peak power exceeds about 120 mW for the picosecond pulses. Both Figs. 9(a) and 9(b) show the 0.5 ps pulses require more pumping power to saturation. Because XPM- and SPM-induced spectral broadening have a large impact on shorter pulse, the pump power of shorter pulse used for FWM is reduced [27,28]. Therefore, the wavelength conversion efficiency is declined for short pulses. As shown in Fig. 9(c), the output pump power of the picosecond pulses sharply increase after 120 mW due to the saturation, but the femtosecond pulses still consume the pump and have a lower output power. Therefore, the femtosecond pulses need higher pump power to reach saturation. 4. Conclusion
In this paper, we have presented a theoretical study of wavelength conversion in a SOH waveguide, which 4892
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1. J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010). 2. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009). 3. H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, twophoton absorption and self-phase modulation in silicon waveguides at 1.5 μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002). 4. R. Dekker, A. Driessen, T. Wahlbrink, C. Moormann, J. Niehusmann, and M. Först, “Ultrafast Kerr-induced alloptical wavelength conversion in silicon waveguides using 1.55 μm femtosecond pulses,” Opt. Express 14, 8336–8346 (2006). 5. V. Raghunathan, R. Claps, D. Dimitropoulos, and B. Jalali, “Wavelength conversion in silicon using Raman-induced four-wave mixing,” Appl. Phys. Lett. 85, 34–36 (2004). 6. H. Rong, Y. H. Kuo, S. Xu, A. Liu, R. Jones, M. Paniccia, O. Cohen, and O. Raday, “Monolithic integrated Raman silicon laser,” Opt. Express 14, 6705–6712 (2006). 7. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear high index-contrast waveguides with optimum geometry,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2007), paper JWA2. 8. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). 9. P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon-based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol. 25, 1298–1305 (2007). 10. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon–nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012). 11. T. Vallaitis, S. Bogatscher, L. Alloatti, P. Dumon, R. Baets, M. L. Scimeca, I. Biaggio, F. Diederich, C. Koos, W. Freude, and J. Leuthold, “Optical properties of highly nonlinear silicon organic hybrid (SOH) waveguide geometries,” Opt. Express 17, 17357–17368 (2009). 12. J. Matres, C. Lacava, G. C. Ballesteros, P. Minzioni, I. Cristiani, J. M. Fédéli, J. Martí, and C. J. Oton, “Low TPA and free-carrier effects in silicon nanocrystal-based horizontal slot waveguides,” Opt. Express 20, 23838–23845 (2012). 13. X. Liu, Nonlinear Applications Using Silicon Nanophotonic Wires (Columbia University, 2011).
14. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). 15. S. Polyakov, F. Yoshino, M. Liu, and G. Stegeman, “Nonlinear refraction and multiphoton absorption in polydiacetylenes from 1200 to 2200 nm,” Phys. Rev. B 69, 115421 (2004). 16. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1290–1291 (2004). 17. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998). 18. A. K. Bhowmik and M. Thakur, “Self-phase modulation in polydiacetylene single crystal measured at 720–1064 nm,” Opt. Lett. 26, 902–904 (2001). 19. L. Zhang, Y. Yue, Y. X. Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18, 13187–13193 (2010). 20. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2006). 21. M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express 20, 15899–15907 (2012). 22. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultra-broadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14, 4786–4799 (2006).
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Wavelength conversion based on four-wave mixing (FWM) in a silicon–organic hybrid slot waveguide is theoretically investigated in the telecommunication bands. Compared with vertical slot waveguides, the horizontal slot waveguide structure exhibits much flatter dispersion. The maximum nonlinearity coefficient γ of 1.5 × 107 W−1 km−1 and the minimum effective mode area Aeff of 0.065 μm2 are obtained in a horizontal slot waveguide with a 20-nm-thick optically nonlinear layer by controlling the geometric parameters. The wavelength conversion efficiency of 7.45 dB with a pump power of 100 mW in a 4-mm-long horizontal slot waveguide is obtained. This low power on-chip wavelength convertor will have potential applications in highly integrated optical circuits. © 2014 Optical Society of America OCIS codes: (190.4380) Nonlinear optics, four-wave mixing; (190.7110) Ultrafast nonlinear optics; (230.7370) Waveguides; (230.7405) Wavelength conversion devices. http://dx.doi.org/10.1364/AO.53.004886
1. Introduction
Silicon waveguides have attracted much attention because they have large values of Kerr parameter, are transparent at infrared wavelengths, have tight confinement of optical mode [1], and have instantaneous response time for high-speed operation [2]. Silicon is a centrosymmetric material, and its lowest-order optical nonlinearity is third-order. Many optical mechanisms based on third-order nonlinearities, such as self-phase modulation (SPM) [3], cross-phase modulation (XPM) [4], FWM [5], and stimulated Raman scattering (SRS) [6], have all been studied widely for all-optical signal processing. However, the nonlinearity effects in telecommunication bands are severely impaired due to the existence of two-photon absorption (TPA) and free-carrier absorption (FCA) in pure silicon waveguides [7]. Silicon-on-insulator (SOI) waveguides are widely used silicon waveguides, which can take the form 1559-128X/14/224886-08$15.00/0 © 2014 Optical Society of America 4886
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of channel waveguides, ridge waveguides, photoniccrystal waveguides, or slot waveguides. Moreover, the mature technology of complementary metal– oxide–semiconductor (CMOS) used for SOI makes it an attractive platform for photonic integration [2]. To exploit the Kerr coefficient and overcome the limitations from carrier effects, a slot-type geometry was proposed [8–10], where silicon is combined with another low-index high nonlinear coefficient material [11,12]. Compared with other structures of SOI waveguides (such as silicon strip waveguides) and fiber, this kind of slot waveguide shows better performance in nonlinearity integration [8,13], which has the advantages of smaller effective mode area, higher nonlinearity coefficient, and stronger confinement of light. This kind of waveguide is usually divided into horizontal and vertical structures. The candidate materials combined with silicon can be organic or inorganic materials [14], and the silicon–organic hybrid is called a SOH waveguide, where a silicon-slotted waveguide is filled and surrounded with low refractive index, highly nonlinear organic material [2].
In this paper, the SOH waveguide with low-index high nonlinear coefficient organic material of p-toluene sulphonate (PTS) is investigated to realize efficient on-chip wavelength conversion in the telecommunication bands. We compare the dispersion of the horizontal and vertical structures, and calculate the phase-matching condition. Under the optimal geometry, a wavelength conversion efficiency of 7.45 dB via degenerate FWM can be achieved in a 4-mm-long horizontal slot waveguide by using a pump power of 100 mW. Therefore, the horizontal SOH slot waveguide, which possesses a high nonlinear coefficient and small effective mode area, can be used as a highly efficient wavelength convertor with a low pump power and short waveguide length. 2. Structure and Dispersion of the Slot Waveguide
The slot waveguide allows for extremely strong confinement of light in its low-index region, and this feature has attracted considerable attention for the exploitation of third-order nonlinear effects in lowindex media [14]. Silicon–organic hybrid integration is a popular and functional combination, which can make full use of the advantages of these two different media [2]. The highly nonlinear organic material PTS, which has a nonlinear refractive index of n2 2.2 × 10−16 m2 W−1 , is chosen as the candidate organic material [8,14]. For PTS, the large TPA peak is located at 930 nm; thereafter, it decreases monotonically with increasing wavelength until 1500 nm, which nearly turns to zero [15]. Therefore, the combination of silicon and PTS can be highly χ χ nonlinear without suffering from TPA and associated FCA around 1550 nm. For such a SOH waveguide structure, most of the light is confined in the PTS region. This is because, in a high-index contrast interface, the electric field must undergo a large discontinuity with much higher amplitude in the lowindex side, resulting in a high power confinement in the low-index regions [16]. In general, the SOH waveguide can be divided into horizontal and vertical structures, as shown in Fig. 1. Since most of the light is confined in the slot area, reducing the slot width will increase the optical intensity [7]. However, taking into account of the practical production processes, the slot layer cannot be made infinitely small, and the minimum achievable vertical slot width is practically limited to 50 nm [14].
The horizontal slot layer thickness can be better controlled for a thinner width about 20 nm. The material dispersion is incorporated via the Sellmeier equations for Si [17], n2Si 11.6858 0.939816∕λ2 8.10461 × 1.10712 ∕λ2 − 1.10712 ;
(1)
and for PTS [18], n2PTS 2.9971 0.6193λ2 ∕λ2 − 0.4044 − 0.2732λ2 : (2) In a SOH waveguide, since FWM-based wavelength conversion is mainly influenced by the phase mismatch, which depends on the dispersion and nonlinearity. An optimized dispersion will contribute to reduce phase mismatch, enhancing the wavelength conversion band and releasing the requirement for high pump power in nonlinear processes [19]. As a result, the chromatic dispersion tailored by altering the waveguide dimensions to realize phase-matching, plays a critical role in the nonlinear processes. By changing the geometry parameter w in SOH waveguide (see Fig. 1), we obtained a series of dispersion distributions as a function of the wavelengths. As shown in Fig. 2(a), when the silicon layer thickness h is 200 nm and the slot thickness s is 20 nm, the zero-dispersion wavelength (ZDW) of the horizontal SOH waveguide shifts from 1400 to 1600 nm as the silicon wires width w increases. The inset shows the specific structure, which has an anomalous dispersion near 1550 nm. One can find that the optimal silicon wires width w of the horizontal SOH waveguide is 600 nm, which has the relatively flat dispersion and slightly anomalous dispersion near 1550 nm. For vertical SOH waveguide, when silicon layer thickness h is 300 nm and slot thickness s is 50 nm, the optimal geometry parameter w is 600 nm, as illustrated in Fig. 2(b). The electric field distribution is calculated with the optimal geometric parameters utilizing the finiteelement method. The fundamental TM-mode profile of the horizontal SOH waveguide at the wavelength of 1550 nm is simulated, as shown in Fig. 3(a), and the Ex and Ey electric field components are shown in Figs. 3(b) and 3(c), respectively. Analogously, for
Fig. 1. (a) Schemes of horizontal and (b) vertical SOH slot waveguides. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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Fig. 2. Dispersion distribution of (a) horizontal SOH waveguide for different values of silicon wires length w with constant h 200 nm and s 20 nm and (b) vertical SOH waveguide for different values of silicon wires length w with constant h 300 nm and s 50 nm.
vertical SOH waveguide, Fig. 3(d) depicts the fundamental TE-mode profile at 1550 nm, and Figs. 3(e) and 3(f) show the Ex and Ey electric field components, respectively. It is clear that this kind of waveguide can strongly confine light in the middle of the low refractive index region. Moreover, one can intuitively find that the horizontal structure shows better ability in confining light than the vertical structure at 1550 nm, because of the former has a narrower slot thickness than the latter [7]. In resemblance to dispersion, the effective mode area Aeff and the nonlinear coefficient γ e also play important roles in the nonlinear process. The effective mode area and nonlinear coefficient are given by [20,21]
RR Aeff
∞ −∞
RR ∞
2 jFx; yj2 dxdy
−∞
2π γe λ
jFx; yj4 dxdy
;
RR ∞ n x; yjFx; yj4 dxdy −∞ RR ∞2 ; −∞ jFx; yj2 dxdy2
(3)
(4)
where Fx; y is the profile of the field and λ is the wavelength. Figure 4 illustrates the effective mode area Aeff and nonlinear coefficient γ e as a function of the wavelength. The geometry parameters we used here are the optimal values for the horizontal and vertical SOH waveguides mentioned above. As expected, such a SOH waveguide has the advantage
Fig. 3. (a) TM-mode of electric mode and (b) Ex and (c) Ey electric field components of the horizontal SOH waveguide at 1550 nm. (d) The TE-mode of electric mode and (e) Ex and (f) Ey electric field components of the vertical SOH waveguide at 1550 nm. 4888
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Fig. 4. Effective mode area and nonlinear coefficient of the (a) horizontal and (b) vertical SOH waveguide.
of possessing a large nonlinear coefficient, as well as small effective mode area, which makes it suitable for nonlinear applications. Moreover, one can find that the horizontal SOH waveguide has a smaller effective mode area Aeff and a larger nonlinear coefficient γ e than the vertical SOH waveguide. The smaller the effective area provided by the waveguide, the higher the nonlinear interaction [14]. Therefore, through our simulations, the horizontal SOH waveguide shows better performance than the vertical SOH waveguide. The optimal horizontal SOH waveguide is obtained for slot thicknesses of 20 nm with silicon wires of 200 nm height and 600 nm width, which can be used to complete the following wavelength conversion. 3. Wavelength Conversion Based On FWM
FWM is an important nonlinear effect. Its promising applications in optical parametric amplification (OPA), wavelength conversion, and optical parametric oscillator (OPO) have attracted tremendous research interests. Here, we focus on the degenerate FWM, which involves two identical pump photons at frequency ωp passing their energy to signal and idler photons at respective frequency ωs and ωi with the relation of 2ωp ωs ωi. During the FWM process, the signal wave is amplified and the idler wave is generated. The phase mismatch ΔK is given by [20] ΔK Δβ 2γ P ppump ;
(5)
where Ppump is the pump power, γ p ωp n2 ∕cAeff is the nonlinear parameter, Δβ ks ki − 2kp is linear phase-mismatch, and ks , ki , and kp are signal, idler, and pump wave vectors, respectively. Phasematching is achieved under the condition of ΔK 0. Since the nonlinear part (2γ p Ppump ) is positive, the pump pulse should be located in the anomalous dispersion region to achieve phase-matching [22]. We now focus on phase-matching using the horizontal SOH slot waveguide with optimal structure parameters. Figure 5(a) shows the linear phase mismatch for three different pump wavelengths close to the 1.55 μm, and the Δβ nearly turns to zero from 1.46 to 1.64 μm. Figure 5(b) depicts the phase mismatch for different pump powers in the case of injecting pump wavelength at 1.55 μm. Even with a low pump peak power, phase-matching can be easily realized, which is one of the main advantages of this waveguide. Wavelength conversion based on FWM is a promising application in silicon waveguides. The pump, signal, and idler waves are identically polarized in the fundamental quasi-TM-mode in our simulations. The Raman frequency shift of the organic material PTS is about 27 THz [23–25], which is far wider than the frequency (wavelength) difference between pump and signal, so the SRS can be ignored in our simulation. The FWM process can be described by the following coupled wave equations [20,26]:
Fig. 5. (a) Linear phase mismatch with different pump wavelengths. (b) Phase mismatch with different pump power. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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∂Ap iβ2p ∂2 Ap β3p ∂3 Ap − ∂z 2 ∂T 2 6 ∂T 3 1 i ∂ jAp j2 Ap − αp Ap iγ p 1 2 ωp ∂T 2iγ p jAs j2 jAi j2 Ap 2iγ p As Ai Ap expiΔβz; (6) ∂As ∂A iβ ∂2 As β3s ∂3 As ds s 2s − ∂z ∂T 2 ∂T 2 6 ∂T 3 1 i ∂ jAs j2 As − αs As iγ s 1 2 ωs ∂T 2iγ s jAp j2 jAi j2 As iγ s A2p Ai exp−iΔβz;
(7)
∂Ai ∂A iβ ∂2 Ai β3i ∂3 Ai di i 2i − ∂z ∂T 2 ∂T 2 6 ∂T 3 1 i ∂ jAi j2 Ai − αi Ai iγ i 1 2 ωi ∂T 2iγ i jAp j2 jAs j2 Ai iγ i A2p As exp−iΔβz;
(8)
where Aj is the amplitude (j p, s, i), z is the propagation distance, and αj is the linear loss (j p, s, i). The walk-off parameters for the signal and idler are defined as ds β1s − β1p and di β1i − β1p , respectively. βn is the nth-order dispersion coefficient, which is calculated via numerical differentiation from βn dn β∕dωn . The average corresponding value for the effective mode area Aeff is about 0.08 μm2 , the nonlinear coefficient γ e is about 1 × 107 W−1 km−1 , as calculated in Fig. 5(a), and the linear loss is about 4.34 dB∕cm [15]. The above coupled equations are solved using the split-step Fourier method and a fourth-order Runge–Kutta solver. Both the pump and signal pulses injected into the horizontal slot waveguide are hyperbolic–secant pulses. The center wavelength and full width at half-maximum (FWHM) with pump pulses of 1.55 μm and 1 ps. The idler conversion efficiency is defined as the ratio of the output idler power Piout with respect to the incident signal power Psin, and the expression is η 10 log10 Piout ∕Psin . The parametric signal gain is defined as G 10 log10 Psout ∕Psin .
For the purpose of realizing effectively wavelength conversion, the optimal length of this waveguide should be calculated. The pump peak power coupled inside the waveguide is 100 mW while the signal peak power varies for two different values, as depicted in Fig. 6(a). When the peak signal powers are set to 10 and 5 mW, the optimal waveguide lengths are 3.5 and 4 mm, and the corresponding maximum wavelength conversion efficiencies are about 4.85 and 7.45 dB, respectively. Moreover, Fig. 6(b) depicts the energy of the pump, signal, and idler varied along the length of the waveguide with a signal power of 5 mW, which can further describe the generation of optimum length from the point of view of energy. When the propagation length is about 4 mm, the signal and idler reach the maximum energy for the first time. Since then, the inverse transformation of the energy occurs between the pump, signal, and idler inside the waveguide. In this case, the maximum conversion efficiency of 7.45 dB can be achieved using a 4-mmlong waveguide with a signal power of 5 mW. For convenience in this paper, we assume the SOH slot waveguide length is 4 mm. Different signal wavelengths with different pulse widths correspond to different conversion efficiency. Figure 7 shows the conversion efficiency as a function of the signal wavelengths for four different pulse widths. The peak power of the input pump and signal lights are set to be 100 and 5 mW, respectively. The pump is located at 1.55 μm and the signal is tuned from 1.41 to 1.54 μm, which can be converted into idler wave that varies from 1.56 to 1.72 μm. The curves are symmetric around pump wavelength λp 1.55 μm, and only the half is plotted. Therefore, the total conversion bandwidth is about 300 nm in this slot waveguide. For the 0.5, 1, and 10 ps pulses, the conversion efficiency increases first and then tends to saturation as the signal light wavelength increases. This phenomenon is mainly caused by the phase mismatch and the optimal waveguide length. From Fig. 5, one can find that the phase mismatch decreases with the increase of signal wavelength, which induces an increase in conversion efficiency. Moreover, a signal located at 1.46 μm with a power of 5 mW has an optimal waveguide length of
Fig. 6. (a) Conversion efficiency and (b) pulse energy as a function of waveguide length. 4890
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Fig. 7. Conversion efficiency as a function of signal wavelengths for four different pulse widths in a 4-mm waveguide.
4 mm, as shown in Fig. 6(b). The output pump power has minimum values around 1.46 μm, where the idler obtains the maximum energy. Therefore, the maximum conversion efficiency appears around the wavelength of 1.46 μm. However, the conversion efficiency of the 0.1 ps pulses is lower than others due to the large spectral broadening of the pump pulse. Because part of the pump energy is used to generate new spectral components, the effective pump power used to FWM is reduced [27]. To find the suitable width of pulses for propagating in the SOH slot waveguide, we simulate the temporal characteristics of the signal and idler for four different pulse widths with signal pulses centered at 1.46 μm. The input pump and signal peak power are set to be 100 and 5 mW, respectively. It is clear
that the output pulses from the waveguide with 0.1 ps are seriously distorted, whereas the output pulses with input pulse widths of 0.5, 1, and 10 ps are relatively smooth and without obvious distortion, as shown in Fig. 8. Compared with the pulse with a relatively wide pulse width, the shorter pulse has a smaller dispersion length, which means a larger influence of the dispersion effects [20,28]. Moreover, the role of the XPM effect cannot be ignored for the output signal and idler pulses [28]. Therefore, the distortion of the output signal and idler pulses of the 0.1 ps pulses are mainly caused by the combination of large dispersion and XPM, and it is difficult to achieve a wavelength converter when the input pulse widths are close to or less than 0.1 ps. The conversion efficiency is also influenced by the input pump power. In the following simulation, the input pump peak power varies from 10 to 220 mW with a signal peak power of 5 mW. Figure 9(a) shows the relationship between conversion efficiency and pump peak power for three different pulses widths with the same signal wavelength of 1.46 μm. For picosecond pulses, with increasing the pump peak power, the conversion efficiency saturates gradually, and drops after maximum efficiency eventually due to the increase of the phase mismatch [27,29]. The maximum idler conversion efficiency of the 1 ps pulses is 9.67 dB when the pump peak power reaches 160 mW. For the 10 ps pulses, the maximum idler conversion efficiency is 8.55 dB with the pump peak power of 130 mW. This kind of SOH slot waveguide can be applied to not only wavelength conversion but also parametric amplification, as shown in Fig. 9(b).
Fig. 8. Output temporal profiles from the input pulse widths of (a) 0.1 ps, (b) 0.5 ps, (c) 1 ps, and (d) 10 ps. 1 August 2014 / Vol. 53, No. 22 / APPLIED OPTICS
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has two structures: vertical and horizontal. This waveguide can confine the light in its low-index region without severe impairment by TPA. The chromatic dispersion, effective mode area, and nonlinear coefficient of vertical and horizontal SOH waveguide structures have been analyzed. By comparison, our results indicate that the horizontal SOH waveguide shows better performance and is more suitable for nonlinear interaction than the vertical SOH waveguide. The wavelength conversion has been numerically investigated in the optimized horizontal SOH waveguide. Under the optimal structural, we obtain efficient wavelength conversion of 7.45 dB via degenerate FWM in a 4-mm-long horizontal SOH slot waveguide with a pump peak power of 100 mW. This work was supported by the National Natural Science Foundation of China under Grant 61178023 and 61275134. References
Fig. 9. (a) Idler conversion efficiency, (b) signal gain, and (c) output pump power as a function of pump power.
It is found that the gain becomes larger as the pump power increases, and gain saturation appears when the pump peak power exceeds about 120 mW for the picosecond pulses. Both Figs. 9(a) and 9(b) show the 0.5 ps pulses require more pumping power to saturation. Because XPM- and SPM-induced spectral broadening have a large impact on shorter pulse, the pump power of shorter pulse used for FWM is reduced [27,28]. Therefore, the wavelength conversion efficiency is declined for short pulses. As shown in Fig. 9(c), the output pump power of the picosecond pulses sharply increase after 120 mW due to the saturation, but the femtosecond pulses still consume the pump and have a lower output power. Therefore, the femtosecond pulses need higher pump power to reach saturation. 4. Conclusion
In this paper, we have presented a theoretical study of wavelength conversion in a SOH waveguide, which 4892
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