The Coupled Nonlinear Schrödinger Equations Describing Power and Phase for Modeling Phase-Sensitive Parametric Amplification in Silicon Waveguides

The coupled nonlinear Schrödinger (NLS) equations describing power and phase of the optical waves are used to model phasesensitive (PS) parametric amplification in a width-modulated silicon-on-insulator (SOI) channel waveguide. Through solving the coupled NLS equations by the split-step Fourier and Runge-Kutta integrationmethods, the numerical results show that the coupled NLS equations can perfectly describe and character the PS amplification process in silicon waveguides.


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 low-cost fabrication process [2].Optical parametric amplifications based on four-wave-mixing (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][4][5].The most commonly used numerical scheme for solving the NLSE is the split-step Fourier (SSF) method due to its simplicity for implementation and high computational efficiency [6][7][8][9].Since phase-sensitive 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 fiber-based optical parametric amplifiers [11].Compared with optical fibers, the silicon waveguide has some additional complications, such as two-photon absorption (TPA), freecarrier absorption (FCA), and free-carrier 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 width-modulated SOI channel waveguide.The model describes the power and phase variation in the silicon waveguide, which can be solved by the splitstep Fourier and Runge-Kutta integration methods [5,[7][8][9][13][14][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.

Theory
The PS parametric amplification can be realized using FWM effect.Here, we focus on the degenerate FWM, which 2 Journal of Applied Mathematics 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 2  =   +   holds.The signal wave is amplified and the idler wave is generated during the FWM process.Moreover, 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 Δ = Δ+2   pump = 0.
Here Δ =   +   − 2  is the phase mismatch due to the linear dispersion,  pump is the pump power,   =    2 / eff is the nonlinear waveguide parameter,  2 = 12 2  (3) / 0  is the nonlinear index coefficient,  is the speed of light,  0 is the linear refractive index, and  eff is the effective area of the propagating mode, respectively.
The pump and signal waves are identically polarized in the fundamental quasi-TM mode.To depict the nonlinear optical interaction of the pump, signal, and idler in the waveguide, we use the formulism described in [16][17][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  2 is the GVD coefficient.Time  =  − /V  is measured in a reference frame moving with pump pulse traveling at speed V  .The two walk-off parameters of the signal and idler are defined as   =  1 − 1 and   =  1 −  1 , respectively, where  1 is the inverse of the group velocity.The nonlinear coefficient   =    2 / eff with  2 = 6 × 10 −18 m 2 W −1 and  TPA = 5 × 10 −12 m W −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 free-carrier density generated by pump, signal, and idler pulses.  =     is the free-carrier induced index change.These free-carrier parameters are obtained by solving [3,19] where   is the wavelength,  ref = 1550 nm, ℎ is Planck's constant, and   ≈ 1ns 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  = Δ +   +   − 2  is the relative phase and   ,   , and   are the phases of the pump, signal, and idler.Equations ( 3) can be solved by split-step Fourier and Runge-Kutta 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 ℎ, 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  + ℎ.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 Runge-Kutta method, and the results can be expressed as   ( + ℎ)  ,   ( + ℎ)  ,   ( + ℎ)  ,   ( + ℎ)  ,   ( + ℎ)  , and   ( + ℎ)  , respectively.In the second step, dispersion acts alone mathematically, where   represents the Fourier-transform 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 split-step Fourier and Runge-Kutta integration methods.The detailed numerical results in the third part of the paper will show that the splitstep Fourier and Runge-Kutta 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 width-modulated 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  1 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 Δ 1 is the linear phase mismatch of the silicon waveguide with width of  1 and  1 is the length of the first segment.The second segment is channel waveguide with width of  2 , which has a normal dispersion at the pump wavelength.The relative phase at the input of the PSA is , where Δ 2 is the linear phase mismatch of the silicon waveguide with width of  2 and  2 is the length of the second segment.By changing the dispersion and length of the second segment, the relative phase  In-PSA can be set to an arbitrary value.The third silicon waveguide with width of  3 acts as a PSA, which can amplify or deamplify the signal depending on the relative phase at the input of PSA  In-PSA .The widths  1 ,  2 , and  3 of the width-modulated 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 continuous-wave signal at the wavelength of 1360 nm.According to the relation 2  =   +  , the wavelength of the idler is 1801.7 nm.The dispersion parameters of our model are listed in Table 1.

Results and Discussion
The coupled NLS equations of (3) are solved by using splitstep Fourier and Runge-Kutta 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 /2 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 Δ 1 = −1016 m −1 .It is clear that the signal peak power increases with the increase of the propagation length.That is because  > 0 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  < 0, 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  Out-PIA is 0.127 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 Δ 2 can compensate the negative linear phase mismatch of the first segment Δ 1 .The relative phase is tuned to 0.63 and the signal peak power is increased to 7.57 mW through the second segment of the width-modulated waveguide as shown in Figure 2(b).The phase-sensitive 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 Δ 3 = −389.5m −1 .Therefore, the second segment of the width-modulated waveguide can tune the relative phase to an appropriate value to realize an effective PSA.
With the increase of the propagation length of the second segment, the relative phase increases from 0.127 to 2 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  In-PSA .Here, we define the signal gain as the ratio of the output signal power to the input signal power for the width-modulated SOI waveguide.The maximum signal gain of 17.5 dB is obtained when  In-PSA is tuned to be 0.63, while the minimal gain is 1.3 dB for  In-PSA = 1.64.From Figure 3, it is clear that the  In-PSA should be tuned between 0.127 and 0.98 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 split-step Fourier and Runge-Kutta integration methods provide an accurate description of the PS amplification process.
The linear phase mismatch Δ is simulated for different waveguide widths as shown in Figure 4(a), which can be used to simulate the phase-sensitive amplification process for different width of the second segment.The relative phase and signal gain are investigated by tailoring the width  2 of the second segment when the length of the second segment is 2 mm.The relative phase  In-PSA can be tuned from −0.11 to 0.95 by tailoring  2 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 0.41 <  In-PSA < 0.7, which means that  2 should be tailored to satisfy the condition: 610 nm <  2 < 660 nm.

Conclusion
The phase-sensitive parametric amplification process in a width-modulated 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 split-step Fourier and Runge-Kutta integration methods.Numerical results show that the split-step Fourier and Runge-Kutta integration methods provide accurate and stable solution for coupled NLS equations, which can perfectly describe the PS amplification process in silicon waveguides, and the

2 Figure 1 :
Figure 1: Illustration of the phase-sensitive parametric amplification in a width-modulated SOI channel waveguide with identical height.

Figure 2 :
Figure 2: The signal peak power and relative phase as a function of the propagation length for the first segment (a), the second segment (b), and the third segment (c), respectively.

Figure 3 :Figure 4 :
Figure 3: The relative phase at the input of PSA  In-PSA and signal gain as a function of the propagation length for the second segment.