## Abstract

The impact of dispersion profiles of silicon waveguides on femtosecond optical parametric amplification (OPA) is theoretically investigated. It is found that flat quasi-phase-matching, smooth temporal profiles and separable spectra for 200 fs pulses can be obtained by tailoring the cross-section of silicon rib waveguide. We achieve on-chip parametric gain as high as 26.8 dB and idler conversion gain of 25.6 dB for a low pump peak power over a flat bandwidth of 400 nm in a 10-mm-long dispersion engineered silicon waveguide. Our on-chip OPA can find important potential applications in highly integrated optical circuits for all-optical ultrafast signal processing.

©2011 Optical Society of America

## 1. Introduction

In recent years, silicon has emerged as a highly suitable material for the development of integrated photonics circuits [1]. Comparing with highly nonlinear fiber, the silicon-on-insulator (SOI) platform has inherent advantages due to the large values of Kerr parameter, Raman gain coefficient and the tight confinement of optical mode for making nonlinear optical devices, such as Raman amplifier [2], Raman laser [3], wavelength converter [4], and optical switches [5]. However, the SOI also has some additional complications, such as two-photon absorption (TPA), free-carrier absorption (FCA), and free-carrier dispersion (FCD) [6–9]. These effects usually limit device operation because of the strong nonlinear absorption, which results in the attenuation of optical signal. In order to control these detrimental effects and better understand the potential of silicon as a nonlinear material, various nonlinear effects such as self-phase modulation (SPM) [10–12], cross-phase modulation (XPM) [13, 14], and four-wave mixing (FWM) [15–26] have been studied theoretically and experimentally.

Optical parametric amplification based on FWM has been explored in silicon waveguides-typically on time scales ranging from the continuous-wave (CW) to the picosecond regime. In detail, Mark A. Foster *et al* have first reported on-chip optical parametric gain in the telecom-band on picosecond timescale in SOI channel waveguides. They experimentally achieved a net on-chip parametric gain of + 1.8 dB over 60 nm [22]. Xiaoping Liu *et al* have proved mid-infrared optical parametric amplifier in silicon nanophotonic waveguides with picosecond pump pulses and a continuous-wave tunable mid-infrared laser signal, where an on-chip gain of + 25.4 dB over 220 nm was reported [23]. Despite this progress, there is still a strong motivation to investigate femtosecond OPA in silicon waveguides. In the femtosecond regime, the dispersion effects of the silicon waveguides significantly influence the propagation of femtosecond pulses. Therefore, tailoring the dispersion profiles of the silicon waveguide is crucial for the femtosecond OPA.

In this paper, we investigate efficient parametric amplification via degenerate FWM in a 10-mm-long silicon rib waveguide using 200 fs pump and signal pulses, because of the spectral overlap in the OPA process for the input pulse widths close to or less than 100 fs. The impact of dispersion profiles of silicon waveguide on the phase matching, temporal and spectral effects is investigated by tailoring the cross-section of the silicon waveguide. The on-chip parametric gain as high as 26.8 dB and broad bandwidth more than 400 nm are obtained for a pump peak power of 10 W in dispersion engineered silicon rib waveguide. This low power on-chip femtosecond OPA will have potential applications in highly integrated optical circuits.

## 2. FWM theory for silicon waveguides

The degenerate FWM typically involves two pump photons at frequency *ω _{p}* transporting their energy to a signal wave at frequency

*ω*and an idler wave at frequency

_{s}*ω*as the relation 2

_{i}*ω*holds. The pump, signal and idler waves are identically polarized in the fundamental quasi-TE mode. To describe the nonlinear optical interaction of the pump, signal and idler in the waveguide, we use the formulism described in [27] and take into account the effects of TPA, FCA, and FCD. Remarkably, the stimulated Raman scattering (SRS) is negligible for femtosecond pulses propagating in silicon waveguides because the Raman response time is about 3 ps and SRS is only effective for pulses longer than this [28, 29]. The OPA process can be described by the following coupling equations:

_{p}= ω_{s}+ ω_{i}*A*is the slowly varying amplitude (

_{j}*j = p, s, i*),

*z*is the propagation distance,

*β*

_{2}

*is the group-velocity dispersion (GVD) coefficient and*

_{j}*β*

_{3}

*is the third-order dispersion (TOD) coefficient.*

_{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*is given by [25]where

_{je}*γ*

_{j}= ω_{j}n_{2}

*/cA*is the effective nonlinearity of the waveguide,

_{eff}*n*

_{2}= 12

*π*

^{2}

*χ*

^{(3)}/

*n*

_{0}

*c*is the nonlinear index coefficient,

*c*is the speed of light in vacuum,

*n*

_{0}is the linear refractive index,

*A*is the effective area of the propagating mode and

_{eff}*β*

_{TPA}is the coefficient of TPA. Here

*n*

_{2}= 6 × 10

^{−18}m

^{2}W

^{−1}and

*β*

_{TPA}= 5 × 10

^{−12}mW

^{−1}in the 1550-nm regime [12].

In Eqs. (1)-(3), *α _{j}* accounts for the linear loss and

*α*represents FCA, where

_{fcj}= σ_{j}N_{c}*σ*is the free carrier absorption cross section and

_{j}*N*is the free-carrier density. The free-carrier induced index change is

_{c}*δ*. These free-carrier parameters can be obtained by the following equations [25]:

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

_{j}*λ*1550 nm,

_{ref}=*h*is Planck’s constant, and the carrier lifetime is

*τ*1 ns. Here, free carriers induced by the signal and idler are negligible compared with that induced by the pump in the parametric amplification process.

_{c}≈The phase-matching among the interacting waves is required in the FWM process, and the phase mismatch is given by

where Δ*β = k*2

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

_{p}*k*,

_{p}*k*represent the propagation constants of pump, signal and idler waves, respectively. The second term is the nonlinear part, which results from the SPM and XPM introduced by the pump wave [22].

_{s}, k_{i}*P*represents the pump power. As the nonlinear part is positive, a negative linear part is required to achieve phase matching, which can be realized by locating the pump wavelength in the anomalous dispersion regime.

_{p}## 3. Dispersion tailoring in SOI waveguides

The silicon waveguides used here are straight rib waveguides, which allow for compatibility with electrical control of carrier removal for future devices [3]. The core of the rib waveguide is silicon and the cladding is silica as shown in Fig. 1
. The rib height (H) and the etch depth (h) are 300 nm and 260 nm, respectively. Careful choice of the waveguide width (W) is required to obtain anomalous-GVD at the pump center wavelength (1550 nm) and we can change the rib waveguide width to tailor the zero-dispersion wavelength (ZDWL). The widths satisfied the anomalous-GVD condition at the pump wavelength must be a range, and we choose 550 nm, 600 nm and 650 nm from the series of widths as the waveguide widths. The effective mode areas *A _{eff}* are 0.09 μm

^{2}, 0.12 μm

^{2}and 0.15 μm

^{2}for the above-mentioned waveguides, respectively. The linear propagation losses of the three waveguides are assumed to be 0.3 dB/cm, 0.25 dB/cm and 0.22 dB/cm, respectively [24]. In addition, the fundamental quasi-TE mode is only considered in the three multimode waveguides, because the high-order mode does not play an important role in the parametric process and the inverse taper used in each end of the waveguide can ensure only the fundamental mode excited [30, 31].

Figure 2
shows the dispersion parameters for the waveguides with width of 550 nm, 600 nm and 650 nm, respectively. For the rib waveguide, the TE mode effective indices *n _{eff}* are calculated by using the effective index method as described in [24]. The dispersion relation is then calculated from

*β*(

*ω*)

*= n*(

_{eff}*ω*)

*ω/c*. Higher order dispersion is finally calculated via numerical differentiation from

*β*. From Fig. 2 (c), it can be found that the zero dispersion wavelengths (ZDWL) are 1380 nm, 1445 nm and 1520 nm for the above mentioned waveguides and the corresponding GVD parameters are −0.176 ps

_{n}= d^{n}β/dω^{n}^{2}/m, −0.11 ps

^{2}/m, −0.03 ps

^{2}/m for the 1550 nm wavelength, respectively. In general, the quasi-phase-matching can be satisfied if the pump wavelength is located in the anomalous GVD regime [25]. Therefore, the femtosecond OPA can be expected to be achieved by using

*λ*1550 nm pump pulses for the three waveguides.

_{p}=## 4. Results and discussion

The femtosecond OPA is numerically studied by simultaneously injecting pump pulses centered at *λ _{p}* = 1550 nm and tunable signal pulses in the telecommunication band. The pump and signal pulses are taken to be Gaussian pulses with the same pulse width

*T*

_{FWHM}

*=*200 fs and same repetition rate

*R*. In simulations, the pump peak power coupled inside the waveguides ranges from 1 W to 10 W, while the signal peak power is kept constant at 1 mW.

By scanning the wavelength of the signal from 1400 nm to 1700 nm, Fig. 3
shows the phase mismatch for the three waveguides with the pump peak power of 2 W and 5 W, respectively. When the pump peak power is 2 W, the broad phase matching can be achieved for the waveguide with width of 650 nm, and the phase mismatch can be controlled to be within ± 6 cm^{−1} for the signal wavelength from 1450 nm to 1650 nm for the other two waveguides. According to Eq. (7), the phase mismatch increases as the pump peak power increases because the nonlinear part of the phase mismatch becomes larger, which is illustrated by Fig. 3(b). The phase mismatch increases for the three waveguides, when the pump peak power is 5 W. From Fig. 3, one can find that the phase mismatch curve of the waveguide with the width of 650 nm is more flat than the other two waveguides. Therefore, a flat and broad phase-matching (>300 nm) can be achieved by tailoring the cross-section of the silicon waveguide.

We simulate the temporal and spectral characteristics of the pump, signal and idler using the three waveguides with signal pulses centered at 1450 nm. The lengths of the three waveguides are 10 mm and the input pump peak power is 5 W. The repetition rate of pump and signal pulses is 0.1 GHz, which means low free carrier effects for femtosecond pulses [29]. The output temporal profiles of the pump, signal and idler from the three waveguides are shown in Fig. 4 . It is clear that the output pulses from the waveguide with 550 nm width are seriously distorted, while the output pulses from the waveguide with 600 nm width show slight distortion and the output pulses from waveguide with 650 nm width are relative smooth.

The pulse-propagation dynamics is strongly influenced by the combined action of optical nonlinearity and dispersion effects for femtosecond four-wave mixing. The main nonlinear effects in the FWM process are SPM and XPM introduced by the pump wave as described in Eqs. (1)-(3). The SPM of the signal and idler can be ignored for the low power of them. The characteristic lengths are GVD length, TOD length, walk-off length, and nonlinear length, which are defined as [27]

*T*

_{0}

*≈T*

_{FWHM}/1.665 is the half-width of the pulse,

*d*is the maximal walk-off parameter between

*d*and

_{s}*d*. The dispersion parameters can be obtained from Fig. 2. The characteristic lengths for the three waveguides are listed in Table 1 . Clearly, the GVD and walk off effects cannot be ignored in the femtosecond OPA process for the waveguide with width of 550 nm, while these effects can be ignored when the width increase to 650 nm. When the width of the waveguide is 550 nm, the interplay between the SPM and GVD compresses the pump pulse in the anomalous GVD regime, and the distortion of the output signal and idler pulses is mainly caused by the combination of XPM and GVD as shown in Fig. 4(a) [27]. With increasing the width of the waveguide, the dispersion length

_{i}*L*becomes longer, and the influence of GVD on the propagation of the femtosecond pulses becomes weaker. Consequently, we can suppress pulse distortion by tailoring the cross-section of the waveguide to change the dispersion profiles of the silicon waveguide.

_{D}Figure 5
shows the spectra of the pump, signal and idler from the three waveguides. It is clear that the spectrum of the pump is greatly broadened when the width of the waveguide is 550 nm, drowning the signal and idler spectra as shown in Fig. 5(a). With increasing the width of the waveguide, the degree of spectral broadening is decreasing. Figure 5(c) shows the signal and idler spectra only have little overlap with the pump spectrum. To explain these phenomena, we should introduce the peak of the frequency chirping *δω*_{max} induced by SPM, which can estimate the magnitude of spectral broadening. For an unchirped Gaussian pulse, we can get [27]

*φ*

_{max}

*=*ln(1

*+*2

*rγ*)/2

_{p}P_{p}L_{eff}*r*represents the maximum nonlinear phase shift at the pulse center of the pump,

*r = β*

_{TPA}

*/*(2

*k*

_{p}n_{2}) represents the TPA parameter, and

*L*is the effective length of the waveguide [12]. When the width of the waveguide is 550 nm, the pump pulses have a higher peak power due to pulse compression as shown in Fig. 4(a), so the maximum nonlinear phase shift

_{eff}*φ*

_{max}of the pump pulses induced by SPM is larger compared with the

*φ*

_{max}of pump pulses from the other two waveguides. The maximum nonlinear phase shift

*φ*

_{ma}

*for the three waveguides are 4.5π, 2.5π and 1.5π, respectively, which can be obtained by counting the number of peaks in the output spectrum of the pump using the relation*

_{x}*φ*

_{max}

*≈*(

*M*-1/2)π, where M is the number of the peaks in the spectrum. A large

*φ*

_{max}means a large

*δω*

_{max}according to Eq. (9), hence the magnitude of spectral broadening of the pump is great and even covers the signal and idler spectra as illustrated in Fig. 5(a). By tailoring the cross-section of the silicon waveguide, the extent of spectral broadening is limited as shown in Fig. 5(c). Therefore, separable spectra can be achieved by tailoring the dispersion profiles of the silicon waveguide. In addition, the spectral broadening of the signal and idler pulses is mainly caused by the interplay between FWM and XPM.

The waveguide with width of 650 nm can be used to amplify signal pulses centered at 1450 nm as described above, and we use this waveguide in the following part. However, spectral overlap will take place due to the spectral broadening when the center wavelength of signal pulse is longer than 1450 nm, while the signal pulse with wavelength less than 1450 nm can be amplified without spectral overlap. Figure 6 shows the output spectra for a pump wavelength of 1550 nm and a signal wavelength of 1400 nm. It is clear that the FWM bandwidth becomes wider and the intensities of signal and idler become higher with the increase of pump peak power. The signal spectrum width is about 70 nm when the pump power is 10 W, while the idler spectrum is larger than 100 nm. The pump, signal and idler spectra are separated completely. Therefore, higher parametric gain can be obtained when the pump peak power is 10 W.

The relationship between the on-chip parametric gain and signal center wavelength is shown in Fig. 7(a)
for a pump peak power of 10 W. The gain curves are symmetric around pump wavelength, so only the halves of them are plotted. The signal center wavelengths that vary from 1350 nm to 1450 nm are considered in the following simulations for longer wavelengths that ranging from 1460 nm to 1540 nm will result in spectral overlap. Here, we define the on-chip parametric signal gain (idler conversion gain) as the ratio of the output pulse energy of the signal (idler) to the input signal pulse energy, such that: *G _{s}* = 10

*log*

_{10}(

*E*),

_{sout}/E_{sin}*G*= 10

_{s}*log*

_{10}(

*E*). From Fig. 7(a), one can find that a broad bandwidth of the femtosecond OPA can be obtained in the silicon rib waveguide with well gain flatness. As can be seen from Fig. 7(a), when the signal wavelength is 1450 nm, the maximum signal and idler on-chip gains reach 26.8 dB and 25.6 dB, respectively. When the signal wavelength is 1350 nm, the minimal gains are 21.8 dB and 20.5 dB, respectively. It is clear that the on-chip gain is large enough to overcome the fiber-chip coupling losses that are about 13 dB [23]. Considering the whole gain curve, the FWM bandwidth is larger than 400 nm for a pump peak power of 10 W.

_{iout}/E_{sin}The on-chip parametric signal gain and idler conversion gain versus pump peak power are shown in Fig. 7(b) for signal wavelength of 1400 nm. It is found that the gains become larger as the pump power increases, and gain saturation appears when the pump peak power exceeds 6 W due to the increasing nonlinear losses and phase mismatch as the pump peak power increases [26]. Remarkably, FCA plays a significant role for a high repetition rate (>1 GHz). In this case, the on-chip gain will decrease, and when repetition rate reaches 100 GHz the gain may even to be zero. One way to overcome the obstacle for high repetition rate OPA is to use a reverse-biased p-i-n diode structure to reduce the carrier lifetime.

## 5. Conclusion

The complete simulation model allows us to show clearly the importance of dispersion profiles of silicon waveguides for the femtosecond OPA. The flat quasi-phase-matching, smooth temporal profiles and separable spectra can be obtained by tailoring the cross-section of silicon waveguide. Our femtosecond OPA has an on-chip parametric signal gain of 26.8 dB and idler conversion gain of 25.6 dB for a pump peak power of 10 W over a flat bandwidth of 400 nm in a 10-mm-long dispersion engineered silicon waveguide. This low power on-chip parametric gain in a dispersion engineered silicon waveguide can find important applications in highly integrated optical circuits for all-optical ultrafast signal processing.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 60878060 and 61078029.

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