Research Article  Open Access
The Coupled Nonlinear Schrödinger Equations Describing Power and Phase for Modeling PhaseSensitive Parametric Amplification in Silicon Waveguides
Abstract
The coupled nonlinear Schrödinger (NLS) equations describing power and phase of the optical waves are used to model phasesensitive (PS) parametric amplification in a widthmodulated silicononinsulator (SOI) channel waveguide. Through solving the coupled NLS equations by the splitstep Fourier and RungeKutta integration methods, the numerical results show that the coupled NLS equations can perfectly describe and character the PS amplification process in silicon waveguides.
1. Introduction
Nowadays, silicon has emerged as a highly attractive material for nonlinear photonic integration [1]. Compared with highly nonlinear fiber, the 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 lowcost fabrication process [2]. Optical parametric amplifications based on fourwavemixing (FWM) in SOI waveguide have been theoretically investigated with the model of the coupled NLS equations describing the slowly varying amplitude of the optical waves [3–5]. The most commonly used numerical scheme for solving the NLSE is the splitstep Fourier (SSF) method due to its simplicity for implementation and high computational efficiency [6–9]. Since phasesensitive amplifiers (PSA) have the potential applications in optical communication, optical processing, photon detection, and optical spectroscopy and sensing [10], it is significant and crucial to model and investigate phasesensitive amplification in SOI waveguide by using the coupled NLS equations describing power and phase of the optical waves for nonlinear photonic integration. The coupled NLS equations describing power and phase have been used to analyze the parametric process in fibers [11, 12]. Particularly, Hansryd et al. used the coupled NLS equations to analyze fiberbased optical parametric amplifiers [11]. Compared with optical fibers, the silicon waveguide has some additional complications, such as twophoton absorption (TPA), freecarrier absorption (FCA), and freecarrier dispersion (FCD). Therefore, we should consider TPA, FCA, and FCD in silicon waveguide to develop the coupled NLS equations.
In this paper, through analyzing the coupled NLS equations describing slowly varying amplitude of the optical waves in FWM process, we develop the model of PS amplification based on coupled NLS equations describing power and phase of the optical waves in a widthmodulated SOI channel waveguide. The model describes the power and phase variation in the silicon waveguide, which can be solved by the splitstep Fourier and RungeKutta integration methods [5, 7–9, 13–15]. The calculation process using the solving methods has been discussed. The numerical results show that the coupled NLS equations can perfectly describe and character the PS amplification process in silicon waveguides.
2. Theory
The PS parametric amplification can be realized using FWM effect. Here, we focus on the degenerate FWM, which typically involves two pump photons at frequency passing their energy to a signal wave at frequency and an idler wave at frequency as the relation holds. The signal wave is amplified and the idler wave is generated during the FWM process. Moreover, the phasematching 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 . Here is the phase mismatch due to the linear dispersion, is the pump power, is the nonlinear waveguide parameter, is the nonlinear index coefficient, is the speed of light, is the linear refractive index, and is the effective area of the propagating mode, respectively.
The pump and signal waves are identically polarized in the fundamental quasiTM mode. To depict the nonlinear optical interaction of the pump, signal, and idler in the waveguide, we use the formulism described in [16–18] and take into account the effects of TPA, FCA, and FCD. The coupled NLS equations describing slowly varying amplitude of the different optical waves read as where is the slowly varying amplitude (), is the propagation distance, and is the GVD coefficient. Time is measured in a reference frame moving with pump pulse traveling at speed . The two walkoff parameters of the signal and idler are defined as and , respectively, where is the inverse of the group velocity. The nonlinear coefficient with m^{2}W^{−1} and mW^{−1} is the coefficient of TPA at the wavelength of 1550 nm [2].
In (1), accounts for the linear loss and represents FCA, where is the FCA coefficient and is the freecarrier density generated by pump, signal, and idler pulses. is the freecarrier induced index change. These freecarrier parameters are obtained by solving [3, 19] where is the wavelength, nm, is Planck’s constant, and ns is the carrier lifetime. In order to describe the power and phase of the different waves, we should find the coupled NLS equations describing power and phase. Let , ; (1) can be rewritten in terms of optical powers and phase: where is the relative phase and , , and are the phases of the pump, signal, and idler. Equations (3) can be solved by splitstep Fourier and RungeKutta integration methods, which are discussed in the following part.
First, we assume the propagation length is divided into a large number of segments. Each segment length is defined as step length , which is small enough that the nonlinear effects and dispersion can be treated independently. Two steps should be carried out for the propagation length from to . In the first step, the nonlinearity acts alone, and (3) are simplified as ordinary differential equations: The above ordinary differential equations can be solved by RungeKutta method, and the results can be expressed as , , , , , and , respectively. In the second step, dispersion acts alone mathematically, where represents the Fouriertransform operation and is the frequency in the Fourier domain [18]. Therefore, the coupled NLS equations describing powers and phase of different optical waves are solved by splitstep Fourier and RungeKutta integration methods. The detailed numerical results in the third part of the paper will show that the splitstep Fourier and RungeKutta integration methods provide accurate and stable solution for the coupled NLS equations describing power and phase.
Here the coupled NLS equations describing power and phase are used to model and simulate the PS parametric amplification in a widthmodulated SOI waveguide, which is comprised of three segments of channel waveguides with different widths and identical height as shown in Figure 1. Tapers are used to connect the three segments to avoid the mode mismatch induced by the variation of width [20]. In the first segment, the SOI waveguide with width of has an anomalous dispersion at the pump wavelength of 1550 nm, which acts as a PIA to amplify the signal and generate an idler. The relative phase at the output of the PIA is , where is the linear phase mismatch of the silicon waveguide with width of and is the length of the first segment. The second segment is channel waveguide with width of , which has a normal dispersion at the pump wavelength. The relative phase at the input of the PSA is , where is the linear phase mismatch of the silicon waveguide with width of and is the length of the second segment. By changing the dispersion and length of the second segment, the relative phase can be set to an arbitrary value. The third silicon waveguide with width of acts as a PSA, which can amplify or deamplify the signal depending on the relative phase at the input of PSA .
The widths , , and of the widthmodulated SOI waveguide are assumed as 500 nm, 650 nm, and 580 nm, respectively, while the height is 800 nm. The linear propagation losses of the three segments are set to be 0.3 dB/cm, 0.2 dB/cm, and 0.25 dB/cm, respectively [21]. The PSA process is theoretically investigated with pump pulse of 20 ps at the wavelength of 1550 nm and continuouswave signal at the wavelength of 1360 nm. According to the relation , the wavelength of the idler is 1801.7 nm. The dispersion parameters of our model are listed in Table 1.

3. Results and Discussion
The coupled NLS equations of (3) are solved by using splitstep Fourier and RungeKutta integration methods to investigate the PS parametric amplification process in the widthmodulated SOI channel waveguide as shown in Figure 2. The initial pump peak power is set to be 5 W, while the initial signal power is set to be 1 mW. From Figure 2(a), it is found that the relative phase quickly increases to when the initial phases of pump and idler are zero due to the generation of the idler and then decreases as the propagation length increases because of the large negative linear phase mismatch m^{−1}. It is clear that the signal peak power increases with the increase of the propagation length. That is because along the propagation length, and the energy of the pump is transferred to signal and idler according to (3). It is expected that the signal peak power will decrease when further increasing the propagation length leads to , because the energy will be transferred back to pump. The output signal peak power is up to 5.07 mW, and relative phase at the output of the PIA is when the length of the first segment is 8 mm. The second segment of the waveguide stops the decrease of the relative phase, because the positive linear phase mismatch of the second segment can compensate the negative linear phase mismatch of the first segment . The relative phase is tuned to and the signal peak power is increased to 7.57 mW through the second segment of the widthmodulated waveguide as shown in Figure 2(b). The phasesensitive amplification occurs in the third segment as shown in Figure 2(c). It exhibits exponential gain, and the signal peak power is amplified to 56.8 mW when the propagation length of the third segment is 10 mm. The relative phase in the third segment decreases as the propagation length increases due to a negative linear phase mismatch m^{−1}. Therefore, the second segment of the widthmodulated waveguide can tune the relative phase to an appropriate value to realize an effective PSA.
(a)
(b)
(c)
With the increase of the propagation length of the second segment, the relative phase increases from to as shown in Figure 3. However, the signal gain has a peak and a valley along the propagation length, which depends on the relative phase . Here, we define the signal gain as the ratio of the output signal power to the input signal power for the widthmodulated SOI waveguide. The maximum signal gain of 17.5 dB is obtained when is tuned to be , while the minimal gain is 1.3 dB for . From Figure 3, it is clear that the should be tuned between and for the PSA to obtain a gain larger than 15 dB, which means the length of the second segment should be less than 4.6 mm. From Figures 2 and 3, it is clear that the coupled NLS equations describing power and phase of the optical waves solved by splitstep Fourier and RungeKutta integration methods provide an accurate description of the PS amplification process.
The linear phase mismatch is simulated for different waveguide widths as shown in Figure 4(a), which can be used to simulate the phasesensitive amplification process for different width of the second segment. The relative phase and signal gain are investigated by tailoring the width of the second segment when the length of the second segment is 2 mm. The relative phase can be tuned from to by tailoring ranging from 500 nm to 700 nm as shown in Figure 4(b). It is found that the relative higher gain over 17 dB can be obtained for , which means that should be tailored to satisfy the condition: .
(a)
(b)
4. Conclusion
The phasesensitive parametric amplification process in a widthmodulated silicon waveguide is described by the model of coupled NLS equations describing power and phase of the optical waves, which can be solved by the splitstep Fourier and RungeKutta integration methods. Numerical results show that the splitstep Fourier and RungeKutta integration methods provide accurate and stable solution for coupled NLS equations, which can perfectly describe the PS amplification process in silicon waveguides, and the PS amplification can be achieved in our designed widthmodulated silicon waveguide.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grant no. 61275134.
References
 J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nature Photonics, vol. 4, no. 8, pp. 535–544, 2010. View at: Publisher Site  Google Scholar
 L. Yin and G. P. Agrawal, “Impact of twophoton absorption on selfphase modulation in silicon waveguides,” Optics Letters, vol. 32, no. 14, pp. 2031–2033, 2007. View at: Publisher Site  Google Scholar
 Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Optics Express, vol. 14, no. 11, pp. 4786–4799, 2006. View at: Publisher Site  Google Scholar
 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,” Optics Express, vol. 19, no. 24, pp. 24730–24737, 2011. View at: Publisher Site  Google Scholar
 X. Li, Z. Wang, and H. Liu, “Optimizing initial chirp for efficient femtosecond wavelength conversion in silicon waveguide by splitstep Fourier method,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11970–11975, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q. Zhang and M. I. Hayee, “Symmetrized splitstep fourier scheme to control global simulation accuracy in fiberoptic communication systems,” Journal of Lightwave Technology, vol. 26, no. 2, pp. 302–316, 2008. View at: Publisher Site  Google Scholar
 H. Wang, “Numerical studies on the splitstep finite difference method for nonlinear Schrödinger equations,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 17–35, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 X. Xu and T. Taha, “Parallel splitstep Fourier methods for nonlinear Schrödingertype equations,” Journal of Mathematical Modelling and Algorithms, vol. 2, no. 3, pp. 185–201, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 G. M. Muslu and H. A. Erbay, “Higherorder splitstep Fourier schemes for the generalized nonlinear Schrödinger equation,” Mathematics and Computers in Simulation, vol. 67, no. 6, pp. 581–595, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 Z. Tong, C. Lundstrm, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phasesensitive optical amplifiers, and their applications,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 18, no. 2, pp. 1016–1032, 2012. View at: Publisher Site  Google Scholar
 J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiberbased optical parametric amplifiers and their applications,” IEEE Journal on Selected Topics in Quantum Electronics, vol. 8, no. 3, pp. 506–520, 2002. View at: Publisher Site  Google Scholar
 R. Hang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phasesensitive fiberoptic parametric amplifier with phase selfstabilized input,” Optics Express, vol. 13, no. 26, pp. 10483–10493, 2005. View at: Publisher Site  Google Scholar
 S. Zhang, Z. Deng, and W. Li, “A precise RungeKutta integration and its application for solving nonlinear dynamical systems,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 496–502, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Z. Liu, S. F. Ma, and Z. W. Yang, “Stability analysis of RungeKutta methods for unbounded retarded differential equations with piecewise continuous arguments,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 57–66, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 B. S. Attili, K. Furati, and M. I. Syam, “An efficient implicit RungeKutta method for second order systems,” Applied Mathematics and Computation, vol. 178, no. 2, pp. 229–238, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Influence of spectral broadening on femtosecond wavelength conversion based on fourwave mixing in silicon waveguides,” Applied Optics, vol. 50, no. 28, pp. 5430–5436, 2011. View at: Publisher Site  Google Scholar
 R. L. Espinola, J. I. Dadap, R. M. Osgood Jr., S. J. McNab, and Y. A. Vlasov, “Cband wavelength conversion in silicon photonic wire waveguides,” Optics Express, vol. 13, no. 11, pp. 4341–4349, 2005. View at: Publisher Site  Google Scholar
 G. P. Agrawal, Nonlinear Fiber Optics, 4th edition, 2007.
 L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Optics Letters, vol. 32, no. 4, pp. 391–393, 2007. View at: Publisher Site  Google Scholar
 B. Jin, J. Yuan, C. Yu et al., “Efficient and broadband parametric wavelength conversion in a vertically etched silicon grating without dispersion engineering,” Optics Express, vol. 22, pp. 6257–6268, 2014. View at: Google Scholar
 H. Rong, Y. Kuo, A. Liu, M. Paniccia, and O. Cohen, “High efficiency wavelength conversion of 10 Gb/s data in silicon waveguides,” Optics Express, vol. 14, no. 3, pp. 1182–1188, 2006. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Xuefeng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.