## Abstract

A width-modulated silicon waveguide is proposed to realize non-degenerate phase sensitive optical parametric amplification. It is found that the relative phase at the input of the phase sensitive amplifier (PSA) *θ*^{In-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

## 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 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 *θ*^{In-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 de-amplify the signal depending on the relative phase at the input of the PSA *θ*^{In-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 *ω _{p}* passing their energy to a signal wave at frequency

*ω*and an idler wave at frequency

_{s}*ω*[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), free-carrier 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:

_{i}*P*is the power (

_{j}*j*=

*p*,

*s*,

*i*), and

*φ*is the phase of the pump, signal and idle.

_{j}*z*is the propagation distance,

*β*

_{2}

*is the group-velocity dispersion (GVD) coefficient. Time*

_{j}*T = t-z/v*is measured in a reference frame moving with pump pulse traveling at speed

_{gp}*v*. The two walk-off parameters of the signal and idler are defined as

_{gp}*d*

_{s}= β_{1}

_{s}-β_{1}

*and*

_{p}*d*

_{i}= β_{1}

_{i}-β_{1}

*, respectively, where*

_{p}*β*

_{1}

*is the inverse of the group velocity. The nonlinear coefficient*

_{j}*γ*

_{j}= ω_{j}n_{2}

*/cA*, where

_{eff}*n*

_{2}is the nonlinear index coefficient,

*c*is the speed of light in vacuum,

*n*

_{0}is the linear refractive index,

*A*is the effective mode area.

_{eff}*β*is the coefficient of the two photon absorption (TPA). Although n

_{TPA}_{2}and

*β*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 n

_{TPA}_{2}and

*β*as constant [14,25]. Here, we assume n

_{TPA}_{2}= 6 × 10

^{−18}m

^{2}/W and

*β*= 5 × 10

_{TPA}^{−12}m/W over the telecommunication band [24].

*α*accounts for the linear loss and

_{j}*α*represents FCA, where

_{fj}= σ_{j}N_{c}*σ*is the FCA coefficient and

_{j}*N*is the free-carrier density generated by pump, signal and idler pulses.

_{c}*δ*is the free-carrier induced index change. These free-carrier parameters are obtained by solving [14,26]

_{nfj}= ζ_{j}N_{c}*λ*is the wavelength,

_{j}*λ*1550 nm,

_{ref}=*h*is Planck’s constant, and

*τ*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 Δ

_{c}*k =*Δ

*β +*2

*γ*0 [15], where Δ

_{p}P_{pump}=*β = k*2

_{s}+ k_{i}-*k*is the linear phase mismatch, and

_{p}*k*,

_{p}*k*represent the propagation constants of pump, signal and idler waves, respectively. The phase-matching condition Δ

_{s}, k_{i}*k*= 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 *W*_{1} 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 parametric process. The relative phase at the output of the PIA is *θ*^{Out-PIA} = Δ*β*_{1}*L*_{1} + *φ _{s}*(

*L*

_{1}) +

*φ*(

_{i}*L*

_{1})-2

*φ*(

_{p}*L*

_{1}), where Δ

*β*

_{1}is the linear phase mismatch of the silicon waveguide with width of

*W*

_{1}, and

*L*

_{1}is the length of the first segment. The second segment is a strip waveguide with width of

*W*

_{2}, which can be tailored to tune the dispersion of the second segment. The relative phase at the input of the PSA is

*θ*

^{In-PSA}= Δ

*β*

_{1}

*L*

_{1}+ Δ

*β*

_{2}

*L*

_{2}+

*φ*(

_{s}*L*

_{1}+

*L*

_{2}) +

*φ*(

_{i}*L*

_{1}+

*L*

_{2})-2

*φ*(

_{p}*L*

_{1}

*+ L*

_{2}), where Δ

*β*

_{2}is the linear phase mismatch of the silicon waveguide with width of

*W*

_{2}, and

*L*

_{2}is the length of the second segment. By changing the dispersion and length of the second segment, the relative phase

*θ*

^{In-PSA}can be set to an arbitrary value. Then, the pump, signal and idler with relative phase of

*θ*

^{In-PSA}input the third segment strip waveguide with width of

*W*

_{3}. 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

*θ*

^{In-PSA}.

## 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 *W*_{1}, *W*_{2}, and *W*_{3} are different. To determine the widths of the width-modulated waveguide, the linear phase mismatch Δ*β* and GVD coefficient *β*_{2} 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 *β*_{2}<0 when the waveguide width is less than 775 nm. Since the phase matching can be satisfied when the pump wavelength is located in the anomalous GVD regime [14], the width *W*_{1} of the first segment used to PIA and the width *W*_{3} used to PSA should be less than 775 nm. Here, we assume *W*_{1} = 750 nm and *W*_{3} = 767 nm, while the width *W*_{2} of the second segment, which is used to tune the relative phase, is varied from 770 nm to 850 nm.

To solve Eqs. (1)-(7), the parameters are assumed as follows: *τ _{c}* = 1 ns [29,30], and

*α*=

_{p}*α*=

_{s}*α*= 1 dB/cm [15]. The effective mode area and nonlinearity coefficient are calculated for different waveguide widths at the wavelength of 1550 nm as shown in Fig. 3(a). The effective mode area is given by [31],

_{i}*e*is the longitudinal component of the TE mode. It is clear that the effective mode area increases with the increase of the waveguide width, while the nonlinearity coefficient decreases as the waveguide width increases. Figure 3(b) shows the walk-off parameters as a function of the waveguide width. The walk-off parameter

_{t}*d*increases with the increase of the waveguide width, while

_{s}*d*has an opposite trend. The GVD is negligible in the calculation because the GVD length is much longer than the interaction length for 20 ps pump pulse [21].

_{i}The PI parametric amplification process is investigated in a silicon strip waveguide with *W* = 750 nm as shown in Fig. 4(a). In simulations, the initial pump peak power is set to be 6W, while the initial signal power is set to be 1 mW. From Fig. 4(a), it is found that the relative phase *θ* quickly increases to *π*/2, and then decreases along the propagation length of the strip waveguide, because the negative linear phase shift Δ*β*z can counteract the positive nonlinear phase shift *φ _{s}*(

*z*) +

*φ*(

_{i}*z*)-2

*φ*(

_{p}*z*) according to Eq. (7). The signal peak power increases with the increase of the propagation length until the maximum value of 7.8 mW is obtained at the propagation length of 8.2 mm when the relative phase

*θ*decreases to the value of 0.038π. That is because the energy of the pump is transferred to signal and idler for

*θ>0*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<

*θ<*0.038π, and then decreases quickly because of the inverse transformation of FWM for

*θ<0*until the next period of FWM appears as illustrated in Fig. 4(a). Therefore, in the PI parametric amplification process, we cannot change the relative phase

*θ*to tune the gain. The PS parametric amplification process is depicted in a width-modulated silicon waveguide with

*W*

_{1}= 750 nm,

*W*

_{2}= 790 nm and

*W*

_{3}= 767 nm as shown in Fig. 4(b). The relative phase decreases along the propagation length in the first segment of the width-modulated waveguide, and

*θ*

^{Out-PIA}= 0.15

*π*when the length of the first segment is 7 mm. The signal peak power increases as the length of the first segment increases, and signal peak power is 7.3 mW at the output of the first segment waveguide. This parametric amplification process is phase insensitive, which is also described in Fig. 4(a). The second segment of the width-modulated waveguide prevents the decrease of the relative phase and makes it increasing, because the positive linear phase mismatch of the second segment Δ

*β*

_{2}can compensate the negative linear phase mismatch of the first segment Δ

*β*

_{1}. The relative phase is tuned to 0.57

*π*and the signal peak power is increased to 9.5 mW in the second segment of the width-modulated waveguide with length of only 1 mm as shown in Fig. 4(b). The PS parametric amplification occurs in the third segment of the width-modulated waveguide as shown in Fig. 4(b). It exhibits exponential gain, and the signal peak power is amplified to 79.5 mW when the propagation length of the third segment is 10 mm. The relative phase in the third segment decreases along the propagation length due to a negative linear phase mismatch Δ

*β*

_{3}. Therefore, the second segment of the width-modulated waveguide can tune the relative phase at the input of the PSA

*θ*

^{In-PSA}to an appropriate value to realize an effective parametric amplification.

Figure 5 shows the temporal profiles of the pump, signal and idler in a width-modulated waveguide with the same dimension as Fig. 4(b) used. It is clear that the output powers of signal and idler at the output of the third segment waveguide (L = 18 mm) are much larger than that at the output of the first segment waveguide (L = 7mm) due to the phase sensitive amplification process in the third segment waveguide. For pump, signal and idler pulses, the loss of the pulses in the trailing edge is larger than the leading edge as shown in Fig. 5. Because free carriers are accumulated along the time, which means that the trailing edge of the pulses will suffer relatively high free carrier absorption.

The relative phase and signal gain are investigated by tailoring the width and length of the second segment of the width-modulated silicon waveguide when the pump peak power is 6 W and signal power is 1 mW. The relative phase *θ*^{In-PSA} can be tuned from 0.33π to 1.2π by tailoring the waveguide width of the second segment ranging from 770 nm to 850 nm as shown in Fig. 6(a). It is found that the relative higher gain can be obtained when 0.33π<*θ*^{In-PSA}<0.69π. 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. With the increase of the propagation length of the second segment, the relative phase *θ*^{In-PSA} increases from 0.16*π* to 2*π* as shown in Fig. 6(b). The signal gain has a peak and a valley along the propagation length, which depends on the relative phase *θ*^{In-PSA}. The maximum signal gain of 19 dB is obtained when *θ*^{In-PSA} is tuned to be 0.56π for *L*_{2} = 1 mm, while the minimal gain is 0.087 dB when *θ*^{In-PSA} is tuned to be 1.57π for *L*_{2} = 4.8 mm. From Fig. 6, it is clear that the *θ*^{In-PSA} can be tuned by tailoring the waveguide width and length of the second segment. For convenience, the parameters of the width-modulated silicon waveguide used for PSA in the following part are assumed as follows: *W*_{1} = 750 nm, *W*_{2} = 790 nm, *W*_{3} = 767 nm, *L*_{1} = 7 mm, *L*_{2} = 1 mm and *L*_{3} = 10 mm, which can be used to realize a higher gain for the signal wavelength of 1400 nm when the pump peak power is 6 W as illustrated in Fig. 6(b). The silicon strip waveguide with width of 750 nm and length of 8.2 mm can be used as a PIA in the following part as illustrated in Fig. 4(a). Then the signal gain and gain bandwidth of PSA and PIA are compared in the following part.

The PSA gain and PIA gain versus pump peak power are shown in Fig. 7 for signal wavelength of 1400 nm. From Fig. 7, it is clear that the PSA gain is much larger than the PIA gain, because the second segment of the width-modulated waveguide tunes the relative phase to an appropriate value that leads to the increase of the gain as illustrated in Fig. 4(b). The maximum gain of PSA is about 25 dB when the pump peak power is 10 W, which is larger by 9 dB compared with the PIA gain as shown in Fig. 7. The gain saturation appears as the pump peak power increases due to the increase of the nonlinear losses [18].

To investigate the signal gain bandwidth, we calculate the first-order dispersion and linear phase mismatch by scanning the wavelength of the signal from 1.3 μm to 1.9 μm for the three segment of the width-modulated silicon waveguide as shown in Fig. 8. From Fig. 8(b), one can find that Δ*β*_{2} (linear phase mismatch for *W*_{2} = 790 nm) is larger than zero for the wavelength ranging from 1.3 μm to 1.9 μm, while Δ*β*_{1} (linear phase mismatch for *W*_{1} = 750 nm) and Δ*β*_{3} (linear phase mismatch for *W*_{3} = 767 nm) are less than zero. The relative phase induced by Δ*β*_{2} together with the relative phase induced by SPM and XPM can counteract the relative phase induced by Δ*β*_{1}, which can stop the decrease of the relative phase and tune the relative phase to an appropriate value to obtain a higher gain as illustrated in Fig. 4(b). Since the pump peak power in the third segment is smaller than that in the first segment, we use a relative smaller linear phase mismatch Δ*β*_{3} compared with Δ*β*_{1} to realize phase matching in the third segment of the width-modulated waveguide.

The PSA and PIA gain spectra are investigated with the pump peak power of 6 W, which are symmetric around the pump wavelength as shown in Fig. 9(a). From Fig. 9(a), one can find that two peak gains as high as 20 dB for PSA are obtained, which are larger by 8 dB compared with the two peak gains of the PIA. The gain bandwidth of PSA spans the range of 1380 nm-1490 nm and 1615 nm-1765 nm, while the gain bandwidth of PIA spans the range of 1390 nm-1490 nm and 1615 nm-1740 nm. Therefore the gain bandwidth of PSA is larger by 35 nm compared with the gain bandwidth of PIA. Since the PSA gain spectrum depends on the relative phase at the input of the PSA *θ*^{In-PSA}, the variation trend of the PSA gain spectrum is consistent with the sine values of *θ*^{In-PSA} as shown in Fig. 9(b).

## 4. Fabrication tolerances analysis

In practice, deviations can occur in the fabrication process due to fabrication errors. The SOI waveguide we designed has a standard height of 340 nm, and the waveguide width can vary with fabrication error ΔW. Figure 10 shows the influence of error in waveguide width on signal gain. Error in waveguide width ΔW = 0 represents the waveguide mentioned above with parametric: *W*_{1} = 750 nm, *W*_{2} = 790 nm, *W*_{3} = 767 nm, *L*_{1} = 7 mm, *L*_{2} = 1 mm and *L*_{3} = 10 mm. Therefore, ΔW = 5 means that the waveguide with widths: *W*_{1} = 755 nm, *W*_{2} = 795 nm and *W*_{3} = 772 nm. From Fig. 10, the signal gain of 19 dB is obtained for ΔW = 0, which is consistent with the result of Fig. 6. However, the gain drops to 10.7 dB for ΔW = −5 nm, and drops to 11.2 dB for ΔW = 10 nm. The result suggest that in order to maintain a higher gain, the fabrication error in with should be satisfy the condition: −5 nm<ΔW<10 nm, which can be easily realized because the fabrication errors can be controlled around ± 5 nm.

## 5. Conclusion

The gain characteristics of PS parametric amplification are theoretically investigated in a width-modulated silicon waveguide. Numerical results show that the relative phase between pump, signal and idler can be tuned by the second segment of the width-modulated silicon waveguide, which will influence the gain and bandwidth of the parametric process. The PSA in the width-modulated silicon waveguide can obtain higher gain and broader bandwidth than PIA in the silicon strip waveguide. This low power on-chip PSA in a width-modulated silicon waveguide can find important applications in highly integrated optical circuits for all-optical signal processing.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61178023 and 61275134.

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