Frequency non-degenerate phase-sensitive optical parametric amplification based on fourwave-mixing in width-modulated silicon waveguides Zhaolu Wang,1 Hongjun Liu,1,* Qibing Sun,1 Nan Huang,1 and Xuefeng Li2 1
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science (CAS), Xi'an, 710119, China 2 School of Science, Xi’an University of Post & Telecommunications Xi’an 710121, China * [email protected]
Abstract: A width-modulated silicon waveguide is proposed to realize nondegenerate phase sensitive optical parametric amplification. It is found that the relative phase at the input of the phase sensitive amplifier (PSA) θIn-PSA can be tuned by tailoring the width and length of the second segment of the width-modulated silicon waveguide, which will influence the gain in the parametric amplification process. The maximum gain of PSA is larger by 9 dB compared with the phase insensitive amplifier (PIA) gain, and the gain bandwidth of PSA is larger by 35 nm compared with the gain bandwidth of PIA. Our on-chip PSA can find important potential applications in highly integrated optical circuits for optical chip-to-chip communication and computers. ©2014 Optical Society of America OCIS codes: (190.4390) Nonlinear optics, integrated optics; (190.4380) Nonlinear optics, fourwave mixing; (190.4970) Parametric oscillators and amplifiers; (230.7370) Waveguides.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phasesensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18(2), 1016–1032 (2012). D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24(14), 984–986 (1999). W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999). W. Imajuku, A. Takada, and Y. Yamabayashi, “Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit,” Electron. Lett. 36(1), 63–64 (2000). D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana J. Phys. 56(2-3), 281–285 (2001). Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phasesensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005). R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express 16(12), 9046–9053 (2008). J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phaseinsensitive operation,” Opt. Express 18(5), 4130–4137 (2010). Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett. 36(5), 722–724 (2011). Z. Tong, A. O. J. Wiberg, E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Broadband parametric multicasting via four-mode phase-sensitive interaction,” Opt. Express 20(17), 19363–19373 (2012). J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010). L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007).
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31486
14. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14(11), 4786–4799 (2006). 15. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). 16. X. Sang and O. Boyraz, “Gain and noise characteristics of high-bit-rate silicon parametric amplifiers,” Opt. Express 16(17), 13122–13132 (2008). 17. X. Liu, R. M. Osgood, Jr., Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). 18. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19(24), 24730–24737 (2011). 19. N. Kang, A. Fadil, M. Pu, H. Ji, H. Hu, E. Palushani, D. Vukovic, J. Seoane, H. Ou, K. Rottwitt, and C. Peucheret, “Experimental demonstration of phase sensitive parametric processes in a nanoengineered silicon waveguide,” CLEO:2013 Technical digest CM4D.7. 20. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on four-wave mixing in silicon waveguides,” Appl. Opt. 50(28), 5430–5436 (2011). 21. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (2007). 22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). 23. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007). 24. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). 25. B. Jin, J. Yuan, C. Yu, X. Sang, S. Wei, X. Zhang, Q. Wu, and G. Farrell, “Efficient and broadband parametric wavelength conversion in a vertically etched silicon grating without dispersion engineering,” Opt. Express 22(6), 6257–6268 (2014). 26. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). 27. Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. 31(13), 2113–2121 (2013). 28. T. E. Murphy, software available at http://www.photonics.umd.edu. 29. R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13(11), 4341–4349 (2005). 30. K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanable, T. Shoji, and S. Itabashi, “All-optical efficient wavelength conversion using silicon photonic wire waveguide,” IEEE Photon. Technol. Lett. 18(9), 1046–1048 (2006). 31. S. Afshar V and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009).
1. Introduction As is well known, phase-sensitive (PS) fiber optical parametric amplifiers (FOPAs), which can be used to realize noiseless optical amplification, have the potential applications in optical communication, optical processing, photon detection, optical spectroscopy and sensing [1]. Two types of PS-FOPAs have been investigated so far: frequency-degenerate (signal and idler frequencies are identical) [2–5] and non-degenerate (frequencies are different) [6–11]. Since degenerate PS-FOPA is usually difficult to implement with high gain, and can only amplify one optical wavelength strip for a fixed pump configuration, non-degenerate PS-FOPA is a more promising solution for future ultra-low noise amplifiers, which can achieve exponential gain and simultaneous multistrip amplification [6]. However, it is challenging to phase-lock pump, signal and idler at different wavelength for non-degenerate PS-FOPA. To cope with this, Tang et al. proposed a cascaded PS-FOPA by simply inserting a standard single mode fiber in between two dispersion-shifted fiber spools to realize frequency non-degenerate PS parametric amplification [7]. Then, the method of cascaded PS-FOPA has been widely used to realize high gain, low noise and multi-channel PS parametric amplification in fiber [8–11]. Despite of those progresses, as the development of the silicon photonic integrated circuits for optical chip-to-chip communication and computers, there is still a strong motivation to investigate PS-OPA in silicon waveguides. Nowadays, silicon has emerged as a highly attractive material for nonlinear photonic integration [12]. Compared with highly nonlinear fiber, the silicon-on-insulator (SOI) platform has inherent advantages due to the large values of Kerr parameter and Raman gain coefficient, the tight confinement of the optical mode, and the mature and low-cost fabrication process [13]. OPAs based on four-wave-mixing in silicon waveguides have been studied theoretically and experimentally [14–18]. Until now, there are few reports to investigate the
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31487
PSA with silicon waveguides [19], thus it will be very interesting to investigate the gain characteristics of PS parametric amplification in silicon waveguides. In this paper, we propose a width-modulated silicon waveguide to investigate the gain characteristics of a frequency non-degenerate PSA, which is composed of three segment silicon strip waveguides. The first segment of the width-modulated waveguide acts as a PIA, which can amplify the signal and generate the idler. The generated idler automatically acquires a certain phase relationship with the pump and signal through this parametric process. The second segment of the width-modulated waveguide can be used to tune the relative phase at the input of the PSA θIn-PSA between the pump, signal and idler due to the linear and nonlinear phase shifts. The third segment acts as a PSA, which can amplify or deamplify the signal depending on the relative phase at the input of the PSA θIn-PSA. The PS parametric amplification can be realized in a width-modulated silicon waveguide with length of only 1.8 cm. This on-chip silicon based PSA will have potential applications in highly integrated optical circuits. 2. Theory The OPA can be described by the degenerate FWM process, in which typically involves two pump photons at frequency ωp passing their energy to a signal wave at frequency ωs and an idler wave at frequency ωi [20,21]. The pump and signal waves are assumed to be polarized in the fundamental quasi-TE mode. To describe the nonlinear optical interaction of the pump, signal and idler in silicon waveguides, we use the formalism described in [7] and take into account the effects of two-photon absorption (TPA), free-carrier absorption (FCA), freecarrier dispersion (FCD), and the dispersion terms for picosecond pulse pump. The Raman scattering can be negligible when the frequency detuning between pump and signal does not satisfy the Raman shift of 15.6 THz [22]. The coupled equations describing power and phase of the different optical waves read as: ∂Pp ∂z
= β2 p
∂Pp ∂ϕ p ∂T ∂T
+ β 2 p Pp
∂ 2ϕ p ∂T
2
− (α p + α fp ) Pp −
βTPA Aeff
Pp2
1/ 2 2β − TPA ( Ps + Pi ) Pp − 4γ p ( Ps Pi Pp2 ) sin θ , Aeff
∂Ps ∂P ∂P ∂ϕ s ∂ 2ϕ s β = −d s s + β 2 s s + β 2 s Ps − (α s + α fs ) Ps − TPA Ps2 ∂z ∂T ∂T ∂T Aeff ∂T 2 1/ 2 2β − TPA ( Pp + Pi ) Ps + 2γ s ( Ps Pi Pp2 ) sin θ , Aeff
∂Pi ∂P ∂P ∂ϕi ∂ 2ϕi β = −di i + β 2i i + β 2i Pi − (α i + α fi ) Pi − TPA Pi 2 2 ∂z ∂T ∂T ∂T Aeff ∂T −
1/ 2 2βTPA ( Pp + Ps ) Pi + 2γ i ( Ps Pi Pp2 ) sin θ , Aeff
∂ϕ p ∂z
=−
β 2 p ∂ 2 Pp 2 Pp
∂T
2
+
β 2 p ∂ϕ p 1/ 2
(2)
(3)
2
2π δ n fp + γ p Pp + 2 ∂T λp
+2γ p ( Ps + Pi ) + 2γ p ( Ps Pi )
(1)
(4)
cos θ ,
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31488
2
∂ϕ s ∂ϕ β ∂ 2 Ps β 2 s ∂ϕ s 2π = −d s s − 2 s + + γ s Ps + δ n fs ∂z ∂T 2 Ps ∂T 2 2 ∂T λs 1/ 2
Pp2 Pi +2γ s ( Pp + Pi ) + γ s Ps
(5)
cos θ , 2
∂ϕi ∂ϕ β ∂ 2 Pi β 2i ∂ϕi 2π δ n fi = −di i − 2i + + γ i Pi + 2 λi ∂z ∂T 2 Pi ∂T 2 ∂T Pp2 Ps +2γ i ( Pp + Ps ) + γ i Pi
1/ 2
(6)
cos θ ,
θ ( z ) = Δβ z + ϕ s ( z ) + ϕi ( z ) − 2ϕ p ( z )
(7)
where Pj is the power (j = p,s,i), and φj is the phase of the pump, signal and idle. z is the propagation distance, β2j is the group-velocity dispersion (GVD) coefficient. Time T = t-z/vgp is measured in a reference frame moving with pump pulse traveling at speed vgp. The two walk-off parameters of the signal and idler are defined as ds = β1s-β1p and di = β1i-β1p, respectively, where β1j is the inverse of the group velocity. The nonlinear coefficient γj = ωjn2/cAeff, where n2 is the nonlinear index coefficient, c is the speed of light in vacuum, n0 is the linear refractive index, Aeff is the effective mode area. βTPA is the coefficient of the two photon absorption (TPA). Although n2 and βTPA may change with wavelength [14,23,24], the values of them have the same order of magnitude over the telecommunication band and some theoretical papers treat n2 and βTPA as constant [14,25]. Here, we assume n2 = 6 × 10−18 m2/W and βTPA = 5 × 10−12 m/W over the telecommunication band [24]. αj accounts for the linear loss and αfj = σjNc represents FCA, where σj is the FCA coefficient and Nc is the free-carrier density generated by pump, signal and idler pulses. δnfj = ζjNc is the free-carrier induced index change. These free-carrier parameters are obtained by solving [14,26] σ j = 1.45 × 10
−21
(λ
j
∂N c ( z , t ) ∂t
λref =
)
2
m , ζ j = −1.35 × 10 2
πβ TPA hω p Aeff 2
Pp ( z , t ) 2 −
−27
(λ
Nc ( z, t )
τc
j
,
λref
)
2
3
m ,
(8) (9)
where λj is the wavelength, λref = 1550 nm, h is Planck’s constant, and τc is the carrier lifetime. The phase-matching among the interacting waves is required in the FWM process, which is achieved when the mismatch in the propagation constants of the pump, signal and idler waves is compensated by the phase shift due to SPM and XPM, such that Δk = Δβ + 2γpPpump = 0 [15], where Δβ = ks + ki-2kp is the linear phase mismatch, and kp, ks, ki represent the propagation constants of pump, signal and idler waves, respectively. The phase-matching condition Δk = 0 cannot be maintained along the propagation length due to the pump depletion [25]. The above coupled equations are solved using the split-step Fourier method and a fourth-order Runge-Kutta solver. The PS parametric amplification is investigated in a width-modulated SOI waveguide, which is comprised of three segments of strip waveguides with different widths and identical height as shown in Fig. 1. Two tapers are used to connect the three segments to avoid the mode mismatch induced by the variation of width [25]. The length of a taper is set to 25μm which is at least 200 times larger than any shift in width in this paper [25,27]. Since the power and relative phase of the optical waves have negligible change in a taper with length of 25 μm in a FWM process, the light propagation in the two tapers can be neglected. The first segment with width of W1 acts as a PIA to amplify the signal and generate an idler. The idler automatically acquires a certain phase relationship with the pump and signal through this #223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31489
parametric process. The relative phase at the output of the PIA is θOut-PIA = Δβ1L1 + φs(L1) + φi(L1)-2φp(L1), where Δβ1 is the linear phase mismatch of the silicon waveguide with width of W1, and L1 is the length of the first segment. The second segment is a strip waveguide with width of W2, which can be tailored to tune the dispersion of the second segment. The relative phase at the input of the PSA is θIn-PSA = Δβ1L1 + Δβ2L2 + φs(L1 + L2) + φi(L1 + L2)-2φp(L1 + L2), where Δβ2 is the linear phase mismatch of the silicon waveguide with width of W2, and L2 is the length of the second segment. By changing the dispersion and length of the second segment, the relative phase θIn-PSA can be set to an arbitrary value. Then, the pump, signal and idler with relative phase of θIn-PSA input the third segment strip waveguide with width of W3. The PS parametric amplification occurs in this segment, which can amplify or de-amplify the signal depending on the relative phase at the input of PSA θIn-PSA.
Fig. 1. Illustration of the phase-sensitive parametric amplification in a width-modulated SOI strip waveguide with identical height
3. Results and discussion The PS parametric amplification is numerically studied with pump pulse of 20 ps at the wavelength of 1550 nm and continuous-wave (CW) signal at the wavelength of 1400 nm in a width-modulated silicon waveguide. The height of the width-modulated waveguide is 340 nm, while the widths W1, W2, and W3 are different. To determine the widths of the widthmodulated waveguide, the linear phase mismatch Δβ and GVD coefficient β2 at the pump wavelength are simulated for different strip waveguide widths as shown in Fig. 2. The dispersion coefficients are determined using a finite-difference mode solver [28]. It is clear that β20 in the FWM process according to Eqs. (1)-(6). After that, the signal peak power decreases first due to the linear loss and nonlinear losses for 0
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Science (CAS), Xi'an, 710119, China 2 School of Science, Xi’an University of Post & Telecommunications Xi’an 710121, China * [email protected]
Abstract: A width-modulated silicon waveguide is proposed to realize nondegenerate phase sensitive optical parametric amplification. It is found that the relative phase at the input of the phase sensitive amplifier (PSA) θIn-PSA can be tuned by tailoring the width and length of the second segment of the width-modulated silicon waveguide, which will influence the gain in the parametric amplification process. The maximum gain of PSA is larger by 9 dB compared with the phase insensitive amplifier (PIA) gain, and the gain bandwidth of PSA is larger by 35 nm compared with the gain bandwidth of PIA. Our on-chip PSA can find important potential applications in highly integrated optical circuits for optical chip-to-chip communication and computers. ©2014 Optical Society of America OCIS codes: (190.4390) Nonlinear optics, integrated optics; (190.4380) Nonlinear optics, fourwave mixing; (190.4970) Parametric oscillators and amplifiers; (230.7370) Waveguides.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phasesensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18(2), 1016–1032 (2012). D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24(14), 984–986 (1999). W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3 dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999). W. Imajuku, A. Takada, and Y. Yamabayashi, “Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit,” Electron. Lett. 36(1), 63–64 (2000). D. Levandovsky, M. Vasilyev, and P. Kumar, “Near-noiseless amplification of light by a phase-sensitive fibre amplifier,” Pramana J. Phys. 56(2-3), 281–285 (2001). Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phasesensitive amplifiers,” Nat. Photonics 5(7), 430–436 (2011). R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005). R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express 16(12), 9046–9053 (2008). J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phaseinsensitive operation,” Opt. Express 18(5), 4130–4137 (2010). Z. Tong, C. Lundström, M. Karlsson, M. Vasilyev, and P. A. Andrekson, “Noise performance of a frequency nondegenerate phase-sensitive amplifier with unequalized inputs,” Opt. Lett. 36(5), 722–724 (2011). Z. Tong, A. O. J. Wiberg, E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Broadband parametric multicasting via four-mode phase-sensitive interaction,” Opt. Express 20(17), 19363–19373 (2012). J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010). L. Yin and G. P. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. 32(14), 2031–2033 (2007).
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31486
14. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14(11), 4786–4799 (2006). 15. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). 16. X. Sang and O. Boyraz, “Gain and noise characteristics of high-bit-rate silicon parametric amplifiers,” Opt. Express 16(17), 13122–13132 (2008). 17. X. Liu, R. M. Osgood, Jr., Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). 18. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19(24), 24730–24737 (2011). 19. N. Kang, A. Fadil, M. Pu, H. Ji, H. Hu, E. Palushani, D. Vukovic, J. Seoane, H. Ou, K. Rottwitt, and C. Peucheret, “Experimental demonstration of phase sensitive parametric processes in a nanoengineered silicon waveguide,” CLEO:2013 Technical digest CM4D.7. 20. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on four-wave mixing in silicon waveguides,” Appl. Opt. 50(28), 5430–5436 (2011). 21. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (2007). 22. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007). 23. Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91(2), 021111 (2007). 24. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). 25. B. Jin, J. Yuan, C. Yu, X. Sang, S. Wei, X. Zhang, Q. Wu, and G. Farrell, “Efficient and broadband parametric wavelength conversion in a vertically etched silicon grating without dispersion engineering,” Opt. Express 22(6), 6257–6268 (2014). 26. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). 27. Y. Lefevre, N. Vermeulen, and H. Thienpont, “Quasi-phase-matching of four-wave-mixing-based wavelength conversion by phase-mismatch switching,” J. Lightwave Technol. 31(13), 2113–2121 (2013). 28. T. E. Murphy, software available at http://www.photonics.umd.edu. 29. R. L. Espinola, J. I. Dadap, R. M. Osgood, Jr., S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13(11), 4341–4349 (2005). 30. K. Yamada, H. Fukuda, T. Tsuchizawa, T. Watanable, T. Shoji, and S. Itabashi, “All-optical efficient wavelength conversion using silicon photonic wire waveguide,” IEEE Photon. Technol. Lett. 18(9), 1046–1048 (2006). 31. S. Afshar V and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009).
1. Introduction As is well known, phase-sensitive (PS) fiber optical parametric amplifiers (FOPAs), which can be used to realize noiseless optical amplification, have the potential applications in optical communication, optical processing, photon detection, optical spectroscopy and sensing [1]. Two types of PS-FOPAs have been investigated so far: frequency-degenerate (signal and idler frequencies are identical) [2–5] and non-degenerate (frequencies are different) [6–11]. Since degenerate PS-FOPA is usually difficult to implement with high gain, and can only amplify one optical wavelength strip for a fixed pump configuration, non-degenerate PS-FOPA is a more promising solution for future ultra-low noise amplifiers, which can achieve exponential gain and simultaneous multistrip amplification [6]. However, it is challenging to phase-lock pump, signal and idler at different wavelength for non-degenerate PS-FOPA. To cope with this, Tang et al. proposed a cascaded PS-FOPA by simply inserting a standard single mode fiber in between two dispersion-shifted fiber spools to realize frequency non-degenerate PS parametric amplification [7]. Then, the method of cascaded PS-FOPA has been widely used to realize high gain, low noise and multi-channel PS parametric amplification in fiber [8–11]. Despite of those progresses, as the development of the silicon photonic integrated circuits for optical chip-to-chip communication and computers, there is still a strong motivation to investigate PS-OPA in silicon waveguides. Nowadays, silicon has emerged as a highly attractive material for nonlinear photonic integration [12]. Compared with highly nonlinear fiber, the silicon-on-insulator (SOI) platform has inherent advantages due to the large values of Kerr parameter and Raman gain coefficient, the tight confinement of the optical mode, and the mature and low-cost fabrication process [13]. OPAs based on four-wave-mixing in silicon waveguides have been studied theoretically and experimentally [14–18]. Until now, there are few reports to investigate the
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31487
PSA with silicon waveguides [19], thus it will be very interesting to investigate the gain characteristics of PS parametric amplification in silicon waveguides. In this paper, we propose a width-modulated silicon waveguide to investigate the gain characteristics of a frequency non-degenerate PSA, which is composed of three segment silicon strip waveguides. The first segment of the width-modulated waveguide acts as a PIA, which can amplify the signal and generate the idler. The generated idler automatically acquires a certain phase relationship with the pump and signal through this parametric process. The second segment of the width-modulated waveguide can be used to tune the relative phase at the input of the PSA θIn-PSA between the pump, signal and idler due to the linear and nonlinear phase shifts. The third segment acts as a PSA, which can amplify or deamplify the signal depending on the relative phase at the input of the PSA θIn-PSA. The PS parametric amplification can be realized in a width-modulated silicon waveguide with length of only 1.8 cm. This on-chip silicon based PSA will have potential applications in highly integrated optical circuits. 2. Theory The OPA can be described by the degenerate FWM process, in which typically involves two pump photons at frequency ωp passing their energy to a signal wave at frequency ωs and an idler wave at frequency ωi [20,21]. The pump and signal waves are assumed to be polarized in the fundamental quasi-TE mode. To describe the nonlinear optical interaction of the pump, signal and idler in silicon waveguides, we use the formalism described in [7] and take into account the effects of two-photon absorption (TPA), free-carrier absorption (FCA), freecarrier dispersion (FCD), and the dispersion terms for picosecond pulse pump. The Raman scattering can be negligible when the frequency detuning between pump and signal does not satisfy the Raman shift of 15.6 THz [22]. The coupled equations describing power and phase of the different optical waves read as: ∂Pp ∂z
= β2 p
∂Pp ∂ϕ p ∂T ∂T
+ β 2 p Pp
∂ 2ϕ p ∂T
2
− (α p + α fp ) Pp −
βTPA Aeff
Pp2
1/ 2 2β − TPA ( Ps + Pi ) Pp − 4γ p ( Ps Pi Pp2 ) sin θ , Aeff
∂Ps ∂P ∂P ∂ϕ s ∂ 2ϕ s β = −d s s + β 2 s s + β 2 s Ps − (α s + α fs ) Ps − TPA Ps2 ∂z ∂T ∂T ∂T Aeff ∂T 2 1/ 2 2β − TPA ( Pp + Pi ) Ps + 2γ s ( Ps Pi Pp2 ) sin θ , Aeff
∂Pi ∂P ∂P ∂ϕi ∂ 2ϕi β = −di i + β 2i i + β 2i Pi − (α i + α fi ) Pi − TPA Pi 2 2 ∂z ∂T ∂T ∂T Aeff ∂T −
1/ 2 2βTPA ( Pp + Ps ) Pi + 2γ i ( Ps Pi Pp2 ) sin θ , Aeff
∂ϕ p ∂z
=−
β 2 p ∂ 2 Pp 2 Pp
∂T
2
+
β 2 p ∂ϕ p 1/ 2
(2)
(3)
2
2π δ n fp + γ p Pp + 2 ∂T λp
+2γ p ( Ps + Pi ) + 2γ p ( Ps Pi )
(1)
(4)
cos θ ,
#223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31488
2
∂ϕ s ∂ϕ β ∂ 2 Ps β 2 s ∂ϕ s 2π = −d s s − 2 s + + γ s Ps + δ n fs ∂z ∂T 2 Ps ∂T 2 2 ∂T λs 1/ 2
Pp2 Pi +2γ s ( Pp + Pi ) + γ s Ps
(5)
cos θ , 2
∂ϕi ∂ϕ β ∂ 2 Pi β 2i ∂ϕi 2π δ n fi = −di i − 2i + + γ i Pi + 2 λi ∂z ∂T 2 Pi ∂T 2 ∂T Pp2 Ps +2γ i ( Pp + Ps ) + γ i Pi
1/ 2
(6)
cos θ ,
θ ( z ) = Δβ z + ϕ s ( z ) + ϕi ( z ) − 2ϕ p ( z )
(7)
where Pj is the power (j = p,s,i), and φj is the phase of the pump, signal and idle. z is the propagation distance, β2j is the group-velocity dispersion (GVD) coefficient. Time T = t-z/vgp is measured in a reference frame moving with pump pulse traveling at speed vgp. The two walk-off parameters of the signal and idler are defined as ds = β1s-β1p and di = β1i-β1p, respectively, where β1j is the inverse of the group velocity. The nonlinear coefficient γj = ωjn2/cAeff, where n2 is the nonlinear index coefficient, c is the speed of light in vacuum, n0 is the linear refractive index, Aeff is the effective mode area. βTPA is the coefficient of the two photon absorption (TPA). Although n2 and βTPA may change with wavelength [14,23,24], the values of them have the same order of magnitude over the telecommunication band and some theoretical papers treat n2 and βTPA as constant [14,25]. Here, we assume n2 = 6 × 10−18 m2/W and βTPA = 5 × 10−12 m/W over the telecommunication band [24]. αj accounts for the linear loss and αfj = σjNc represents FCA, where σj is the FCA coefficient and Nc is the free-carrier density generated by pump, signal and idler pulses. δnfj = ζjNc is the free-carrier induced index change. These free-carrier parameters are obtained by solving [14,26] σ j = 1.45 × 10
−21
(λ
j
∂N c ( z , t ) ∂t
λref =
)
2
m , ζ j = −1.35 × 10 2
πβ TPA hω p Aeff 2
Pp ( z , t ) 2 −
−27
(λ
Nc ( z, t )
τc
j
,
λref
)
2
3
m ,
(8) (9)
where λj is the wavelength, λref = 1550 nm, h is Planck’s constant, and τc is the carrier lifetime. The phase-matching among the interacting waves is required in the FWM process, which is achieved when the mismatch in the propagation constants of the pump, signal and idler waves is compensated by the phase shift due to SPM and XPM, such that Δk = Δβ + 2γpPpump = 0 [15], where Δβ = ks + ki-2kp is the linear phase mismatch, and kp, ks, ki represent the propagation constants of pump, signal and idler waves, respectively. The phase-matching condition Δk = 0 cannot be maintained along the propagation length due to the pump depletion [25]. The above coupled equations are solved using the split-step Fourier method and a fourth-order Runge-Kutta solver. The PS parametric amplification is investigated in a width-modulated SOI waveguide, which is comprised of three segments of strip waveguides with different widths and identical height as shown in Fig. 1. Two tapers are used to connect the three segments to avoid the mode mismatch induced by the variation of width [25]. The length of a taper is set to 25μm which is at least 200 times larger than any shift in width in this paper [25,27]. Since the power and relative phase of the optical waves have negligible change in a taper with length of 25 μm in a FWM process, the light propagation in the two tapers can be neglected. The first segment with width of W1 acts as a PIA to amplify the signal and generate an idler. The idler automatically acquires a certain phase relationship with the pump and signal through this #223668 - $15.00 USD Received 23 Sep 2014; revised 24 Nov 2014; accepted 25 Nov 2014; published 12 Dec 2014 (C) 2014 OSA 15 Dec 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031486 | OPTICS EXPRESS 31489
parametric process. The relative phase at the output of the PIA is θOut-PIA = Δβ1L1 + φs(L1) + φi(L1)-2φp(L1), where Δβ1 is the linear phase mismatch of the silicon waveguide with width of W1, and L1 is the length of the first segment. The second segment is a strip waveguide with width of W2, which can be tailored to tune the dispersion of the second segment. The relative phase at the input of the PSA is θIn-PSA = Δβ1L1 + Δβ2L2 + φs(L1 + L2) + φi(L1 + L2)-2φp(L1 + L2), where Δβ2 is the linear phase mismatch of the silicon waveguide with width of W2, and L2 is the length of the second segment. By changing the dispersion and length of the second segment, the relative phase θIn-PSA can be set to an arbitrary value. Then, the pump, signal and idler with relative phase of θIn-PSA input the third segment strip waveguide with width of W3. The PS parametric amplification occurs in this segment, which can amplify or de-amplify the signal depending on the relative phase at the input of PSA θIn-PSA.
Fig. 1. Illustration of the phase-sensitive parametric amplification in a width-modulated SOI strip waveguide with identical height
3. Results and discussion The PS parametric amplification is numerically studied with pump pulse of 20 ps at the wavelength of 1550 nm and continuous-wave (CW) signal at the wavelength of 1400 nm in a width-modulated silicon waveguide. The height of the width-modulated waveguide is 340 nm, while the widths W1, W2, and W3 are different. To determine the widths of the widthmodulated waveguide, the linear phase mismatch Δβ and GVD coefficient β2 at the pump wavelength are simulated for different strip waveguide widths as shown in Fig. 2. The dispersion coefficients are determined using a finite-difference mode solver [28]. It is clear that β20 in the FWM process according to Eqs. (1)-(6). After that, the signal peak power decreases first due to the linear loss and nonlinear losses for 0
