## Abstract

An efficient and broadband parametric wavelength converter is proposed in the silicon-on-insulator (SOI) waveguide without dispersion engineering. The vertical grating is utilized to achieve the quasi-phase-matching (QPM) of four-wave mixing (FWM). By alternating the phase-mismatch between two values with opposite signs, the parametric attenuation is suppressed. The conversion efficiency at the designated signal wavelength is significantly improved, and the 3-dB conversion bandwidth is also extended effectively. It is demonstrated that the conversion bandwidth is insensitive to both the propagation length and the grating width, which alleviates the tradeoff between the conversion bandwidth and the peak conversion efficiency. For a continuous-wave (CW) pump at 1550 nm, a conversion bandwidth of 331 nm and a peak efficiency of −12.8 dB can be realized in a 1.5-cm-long grating with serious phase-mismatch.

© 2014 Optical Society of America

## 1. Introduction

Silicon-based parametric wavelength conversion has attracted considerable interest for all-optical signal processing and communications [1–4]. As the four-wave mixing (FWM) process is sensitive to the relative phase among the involved waves, the phase-mismatch induced by dispersion and nonlinearity limits the conversion efficiency. Traditional techniques have focused on increasing the optical intensity and minimizing the phase-mismatch to improve the conversion efficiency, such as using the microring resonators [5–7], pumping near the zero-dispersion wavelength (ZDWL) [8,9], and designing novel waveguide geometries with the flat and low dispersion profile [10,11]. However, it is difficult to maintain the perfect phase-matching along the waveguide direction due to the large loss factor in silicon. As a result, the idler power fluctuates periodically with the extension of waveguide length, which indicates the powers of signal and idler back-convert to the pump. By adjusting the relative phase relationship periodically along the propagation path, the quasi-phase-matching (QPM) technique attempts to control the direction of power flow, and offers a new approach to improve the conversion efficiency [12–15]. Since the QPM technique can realize efficient wavelength conversion in the case of serious phase-mismatch, it provides more freedom to design the waveguide geometry and choose the pump wavelength [16,17].

It is a challenge to exploit QPM on the silicon-on-insulator (SOI) platform. The large Kerr parameter and strong nonlinear absorption will result in parametric attenuation and additional power loss while adjusting the phase relationship of the light-waves [16–19]. So the improvement of conversion efficiency is limited, and the device footprint is increased. In addition, the conversion bandwidth cannot be effectively extended in the reported schemes, because the conversion efficiency is only improved around the particular signal wavelengths [16–21]. The narrow 3-dB conversion bandwidth and the feature of discrete wavelength conversion have restricted the applications of these schemes [22–26].

In this paper, a vertically etched silicon grating is utilized to achieve the QPM of FWM. With the assistance of taper connections, the mode is varied adiabatically between the two waveguide widths of a grating. By alternating the phase-mismatch between two values with opposite signs, the parametric attenuation is suppressed. The proposed scheme not only improves the conversion efficiency at a particular signal wavelength, but also extends the 3-dB conversion bandwidth. For a pump wavelength of 1550 nm, compared to the constant-width waveguide with the same phase-mismatch, a 28 dB improvement of the conversion efficiency at 1750 nm can be obtained after 1.5-cm propagation. In addition, the conversion bandwidth is extended by 260 nm. The proposed wavelength converter also offers the advantage that the conversion bandwidth is insensitive to the propagation length and the grating widths, which alleviates the tradeoff between the maximum and bandwidth of the conversion efficiency spectrum. Thus, the efficient and broadband wavelength converter is realized in the SOI waveguide without dispersion engineering.

## 2. Operational principle

The single-pump FWM occurs through $2{\omega}_{p}={\omega}_{s}+{\omega}_{i}$, where ${\omega}_{j}$ is the angular frequency and $j=p,s,i$ for the pump, signal, and idler, respectively. For continuous waves in the straight waveguide, the slowly varying amplitude ${A}_{j}$ along the propagation direction $z$ can be described by [27–31]:

*k*,

*l*,

*m*, and

*n*take values

*p*,

*s*, and

*i*, ${n}_{2}=6\times {10}^{-18}$ m

^{2}/W is the nonlinear refractive index, ${\beta}_{TPA}=4.5\times {10}^{-12}$ m/W is the two photon absorption (TPA) coefficient, and ${\overline{a}}_{klmn}$ is the average effective modal area related to multiple light-waves [28]. In addition, ${\alpha}_{j}^{FCA}=1.45\times {10}^{-21}{({\lambda}_{j}/{\lambda}_{ref})}^{2}N$ is the free carrier absorption (FCA) coefficient, and $\Delta {n}_{j}=-5.3\times {10}^{-27}{({\lambda}_{j}/{\lambda}_{ref})}^{2}N$ is the refractive index change due to the free carrier dispersion (FCD) effect, where ${\lambda}_{ref}=1550$ nm, and $N$ is the free carrier density. As the free carrier is generated by TPA, the density can be calculated by $N(z)={\tau}_{0}{\beta}_{TPA}|{A}_{p}{|}^{4}/(2\hslash {\omega}_{p}{a}_{p}^{2})$, where is the effective carrier lifetime [17,27,29]. In Eqs. (1)–(3), all the waves are assumed to be polarized in the fundamental TE mode. The Raman scattering is neglected in the equations, since its bandwidth (105 GHz) is rather small compared with that of the parametric processes [17–19,27,32].

Considering the powers of signal and idler are much smaller than that of the pump, the nonlinear processes induced by the signal and idler including the self-phase modulation (SPM), cross-phase modulation (XPM), and TPA can be neglected. Let ${A}_{j}(z)=\sqrt{{P}_{j}(z)}\mathrm{exp}[i{\varphi}_{j}(z)]$, Eqs. (1)–(3) can be rewritten in terms of the light-wave powers:

As seen from Eqs. (5)–(6), $\theta $ controls the directions of power flow in the FWM process. If $\mathrm{sin}\theta >0$, the power flows from the pump to the signal and idler, corresponding to the parametric amplification. If $\mathrm{sin}\theta <0$, the power is transferred from the signal and idler to the pump, corresponding to the parametric attenuation. Due to the large loss coefficient of silicon, the phase-matching condition $\kappa =0$ cannot be maintained along the propagation path. Thus, the fluctuation of the idler power is inevitable in the constant-width waveguide. In addition, restricted by the fabrication precision, it is difficult to achieve the perfect-phase-matching by the dispersion engineering. We define the wavelength-conversion efficiency ${G}_{i}={P}_{i}^{(out)}/{P}_{s}^{(in)}$.

The linear phase-mismatch $\Delta \beta $ can be calculated by the second-order and fourth-order dispersions at the pump wavelength [9,17]:

In our work, the modulation of $\kappa $ along the waveguide is achieved by a vertical grating, in which the waveguide height $h$ is kept constant and the width $w$ is alternated between two different values ${w}_{1}$ and ${w}_{2}$. The variation of $w$ may induce the mode mismatch, which generates additional losses and reflections. In order to solve this problem, tapers with gradually varying width are used to connect the adjacent waveguide sections. A taper with a length of 25 μm can realize the adiabatic mode conversion within a 120-nm variation of waveguide width [17]. The waveguide structure is conceptually illustrated in Fig. 1(a). The length of the waveguide section with $w={w}_{\xi}$ in a grating period can be approximated as [12]:

where $\xi =1,2$, and ${g}_{\xi}=\sqrt{{({\gamma}_{\xi}{P}_{p})}^{2}-{({\kappa}_{\xi}/2)}^{2}}$. Equation (10) is under the assumption that the pump depletion can be neglected in the waveguide section ${L}_{\xi}$. Otherwise, the length of each waveguide section can be determined by the numeral calculation. To accurately model the light propagation in the tapers, the geometry dependent parameters such as ${\gamma}^{e}$ and $\Delta \beta $ are fitted to polynomial functions of the waveguide widths, as described in [17,19,34,35]. Due to the pump depletion induced by the waveguide loss, the grating period $\Lambda ={L}_{1}+{L}_{2}$ varies gradually with the increase of the propagation length. By adopting the reverse-biased p-i-n structures, the nonlinear power loss can be reduced, and the greater improvement of conversion efficiency can be achieved [36].The conversion bandwidth in each waveguide section ${L}_{\xi}$ can be estimated as the bandwidth for which $\left|{\kappa}_{\xi}{L}_{\xi}\right|<\pi $ [9]. If $\kappa $is predominated by the linear phase-mismatch $\Delta \beta $, compared with Eq. (10), it can be deduced that ${\lambda}_{s,obj}$ is at the edge of the conversion bandwidth in each section of the waveguide. So the conversion bandwidth of the wavelength converter can be approximated as $\Delta \lambda \approx |{\lambda}_{s,obj}-{\lambda}_{i,obj}|$, where is the idler wavelength corresponding to ${\lambda}_{p}$ and ${\lambda}_{s,obj}$ via FWM. It should be noted that the conversion bandwidth tends to be slightly overestimated by the calculation [9]. Therefore, our scheme has the advantage that the conversion bandwidth is insensitive to the length and width of the gratings. In the previously reported schemes, the idler and signal suffer from the parametric attenuation to adjust the relative phase difference $\theta $. The conversion bandwidth is compressed in the parametric attenuation sections, so the overall bandwidth of the converter is decreased. However in our scheme, the parametric attenuation at ${\lambda}_{s,obj}$ is suppressed, and the conversion bandwidth is also extended dramatically.

For practical applications, it is important to clarify the limitations on key parameters such as the grating width and the target signal wavelength. As known, QPM provides more freedom to design the waveguide geometry. However, to meet the requirement of ${\kappa}_{1}{\kappa}_{2}<0$, *w*_{1} and *w*_{2} should be smaller and larger than ${w}_{\kappa =0}$, respectively. ${w}_{\kappa =0}$ is defined as the waveguide width with $\kappa =0$ at ${\lambda}_{s,obj}$, and at low pump power, it is close to ${w}_{\lambda p=ZDWL}$ where the pump wavelength is located at the ZDWL of the waveguide. ${w}_{\kappa =0}$ is varied with the propagation length due to the pump depletion, which limits the minimum of $\Delta w=|{w}_{1}-{w}_{2}|$. In addition, the maximum of $\Delta w$ is restricted by the taper length ${L}_{Taper}$. Larger $\Delta w$ can be achieved by increasing ${L}_{Taper}$ while keeping neglectable reflections in the gratings [37]. As previously discussed, the conversion bandwidth is greatly affected by ${\lambda}_{s,obj}$. As ${L}_{1}$ and ${L}_{2}$ must be larger than ${L}_{Taper}$, ${\lambda}_{s,obj}$ should satisfy the relations ${\kappa}_{\xi}<\pi /{L}_{Taper}$, where $\xi =1,2$ [17]. Reducing $\Delta w$ is an effective approach to extending the variation range of ${\lambda}_{s,obj}$, because ${L}_{Taper}$ can be decreased, and at the same time the effective phase-mismatch is reduced by decreasing the second-order dispersion at the pump wavelength. It can be demonstrated that the conversion bandwidth is sufficient for applications over the entire near-infrared wave-band. Some other measures are also beneficial, such as properly choosing the pump wavelength and using the well-designed waveguides with flat and low dispersion profile [38]. It is suggested that the QPM technique can be employed in combination with the dispersion engineering.

## 3. Results and discussion

In this work, vertical gratings with the silica substrate and air cladding are used, and the parameters are as follows: $h=220$ nm, ${\tau}_{0}=1$ ns, and $\alpha =1.5$ dB/cm. The pump and signal with ${P}_{p}(0)=300$ mW and ${P}_{s}(0)=100$μW are launched at ${\lambda}_{p}=1550$ nm and ${\lambda}_{s,obj}=1750$ nm. All the light-waves are polarized in the fundamental TE mode. Considering a strip waveguide with $h=220$ nm and various waveguide widths, the linear phase-mismatch $\Delta \beta $ at ${\lambda}_{s,obj}$ is shown in Fig. 1(b). The second-order and fourth-order dispersions at ${\lambda}_{p}=1550$ nm are obtained by the Sellmeier relations and the finite element method (FEM), and then fitted to the 7th-order polynomials. There is a slight difference between ${w}_{\lambda p=ZDWL}=755$ nm and ${w}_{\Delta \beta =0}\approx 757$ nm as a contribution of the fourth-order dispersion ${\beta}_{4}({\omega}_{p})$, where ${w}_{\Delta \beta =0}$ is defined as the waveguide width with $\Delta \beta =0$ at the signal wavelength ${\lambda}_{s,obj}$.

To analyze the effects of the vertical gratings, Fig. 2 shows the waveguide width, the conversion efficiency, and $\mathrm{sin}\theta $ as functions of the propagation length. As the evolution of $\mathrm{sin}\theta $ is periodic, only the first several periods are shown in Fig. 2(c). Three types of waveguide are evaluated: Grating 1 with ${w}_{1}=695$ nm and ${w}_{2}=815$ nm, Grating 2 with ${w}_{1}=695$ nm and ${w}_{2}=575$ nm, and a constant-width waveguide with $w=695$ nm. As a comparison, Grating 2 adopts the QPM scheme in [17], where ${\kappa}_{1}$ and ${\kappa}_{2}$ are both negative. In the FWM where the dispersion engineering is adopted, the ZDWL is placed near the pump wavelength. Therefore, the ZDWL pumped FWM where the waveguide width is kept at 755 nm is also analyzed as an instance with minor phase-mismatch.

Before the conversion efficiency reaches its maximum at 0.44 mm in the constant-width waveguide, the identical evolution processes can be observed in the three waveguides. As shown in Fig. 2(b), the conversion efficiency decreases and oscillates in the constant-width waveguide during the subsequent propagation. In Grating 2, the idler is amplified and attenuated in the waveguide sections with ${w}_{1}=695$ nm and ${w}_{2}=575$ nm, respectively. The parametric attenuation also leads to a fluctuation of the conversion efficiency. As the idler amplification is larger than the attenuation in every period of Grating 2, an improvement of the conversion efficiency can be obtained with the increase of the grating length. In Grating 1, the parametric attenuation is suppressed, and the idler is amplified in each section of the waveguide. The conversion efficiency at the output of Grating 1 is –16.9 dB, which offers a 17 dB improvement compared with the maximum conversion efficiency in the constant-width waveguide, and a 5 dB improvement compared with that of Grating 2 with the same length of 1.5 cm. Furthermore, to achieve the equivalent conversion efficiency of Grating 2, the grating length can be reduced from 1.5 to 0.3 cm by using Grating 1. The reduction of grating length is especially beneficial for the photonic integration. In the gratings, the nonlinear phase-mismatch is much smaller than the linear one. Thus, both the grating period and the duty cycle are almost constant with the propagation length, as shown in Fig. 2(a). Compared with Grating 2, the number of grating periods is decreased from 27 to 15 in Grating 1. The reduction of period number is beneficial for decreasing the reflection loss. As shown in Fig. 2(c), compared with the constant-width waveguide, the evolution of $\theta $ is accelerated in Grating 2 when . As a result, the length of the parametric attenuation regions is decreased. In Grating 1, is kept positive. decreases from $\pi /2$ to 0 in the first waveguide section, and experiences a phase-shift of $\pi $ in each subsequent section. As the values of $\left|\kappa \right|$ are similar in adjacent sections, the duty cycle is close to 50%.

The effect of the grating widths on the conversion efficiency is described in Fig. 3. We define the average waveguide width ${w}_{dc}=({w}_{1}+{w}_{2})/2$, and the amplitude of width variation $\Delta w=|{w}_{1}-{w}_{2}|$. In Fig. 3(a), ${w}_{dc}$ is kept at 755 nm and $\Delta w$ is varied from 0 to 120 nm. $\Delta w=0$ corresponds to the ZDWL pumped FWM, where the conversion efficiency is 6 dB lower than that of the proposed wavelength converter. In the gratings, the increasing $\Delta w$ will result in a decrease of the conversion efficiency, as the phase-mismatch is increased in each waveguide section. As the effective phase-mismatch $\kappa $ is also affected by the high-order dispersion and the intensity relevant nonlinear effects, ${w}_{\lambda p=ZDWL}$ is just an approximation of ${w}_{\kappa =0}$. Therefore, to satisfy the requirement of ${\kappa}_{1}{\kappa}_{2}<0$, the minimum of $\Delta w$ is limited. In Fig. 3(b), $\Delta w$ is kept at 120 nm while ${w}_{dc}$ is varied from 705 to 805 nm. When ${w}_{dc}$ approaches its extremes, one of the two waveguide sections approaches the width of 755 nm, which approximates to the perfect phase-match and leads to higher idler power at the output.

By approaching the waveguide width of 755 nm, the ZDWL can be located near the pump wavelength, which corresponds to the FWM scheme using the dispersion engineering. As seen from Fig. 3(a), when ${w}_{dc}$ is kept at 755 nm, the conversion efficiency is decreased with $\Delta w=0$. The evolution of the conversion efficiency for the 755-nm-wide waveguide is presented in Fig. 4. Similar process can be observed compared to the 695-nm-wide waveguide which is shown in Fig. 2. The conversion efficiency first increases up to the maximum, and then decreases during the propagation as a result of the parametric attenuation. As the phase-mismatch is reduced, the evolution of $\theta $ is much slower. Although the conversion efficiency is increased with respect to the 695-nm-wide waveguide, it is still smaller than that of the QPM schemes in Fig. 3(a). To suppress the parametric attenuation, a vertical grating with ${w}_{1}=755$ nm and ${w}_{2}=815$ nm is used. Thus, the conversion efficiency is increased to −15.6 dB. It is demonstrated that the improvement of conversion efficiency can be increased by using the proposed QPM scheme in combination with the dispersion engineering. As shown from the red curve in Fig. 4, the conversion efficiency is decreased slightly between the adjacent peaks. As $\mathrm{sin}\theta $ is kept positive in the 1.5-cm-long waveguide, the dips are caused by the linear loss, TPA and FCA. Due to the large phase-mismatch, the improvement of conversion efficiency is limited in the waveguide section with ${w}_{2}=815$ nm.

The conversion efficiency spectra for different waveguides are shown in Fig. 5. Each waveguide has a length of 1.5 cm. Although the four spectra have similar maxima, their bandwidths are very different from each other. The 3-dB bandwidths of the 695-nm-wide waveguide, 755-nm-wide waveguide and Grating 1 are 70 nm, 301 nm and 331 nm, respectively. Thus, the continuously tunable parametric wavelength conversion can be achieved from 1402 nm to 1733 nm in our scheme. Although the phase-mismatch is much more serious, the conversion bandwidth of Grating 1 has exceeded that of the ZDWL pumped FWM. By increasing the target signal wavelength and the propagation length, the bandwidth advantage of the proposed wavelength converter will become more significant.

The bandwidth differences among Grating 1, Grating 2, and the 695-nm-wide waveguide can be explained as follows. for some signal wavelengths between ${\lambda}_{p}$ and ${\lambda}_{s,obj}$, since the grating period and duty cycle are both optimized for ${\lambda}_{s,obj}$, Grating 2 will cause a decrease of the amplification regions and an increase of the parametric attenuation regions. As a result, the 3-dB conversion bandwidth is compressed compared with the constant-width waveguide, and only the side lobes centered at 1750 nm and 1391 nm are amplified. As the peak values of the side lobes are not large enough, the two side lobes are both beyond the conversion bandwidth of the wavelength converter. On the contrary, for any signal wavelength between ${\lambda}_{p}$ and ${\lambda}_{s,obj}$, the idler will not experience any parametric attenuation in Grating 1, which results in a much wider bandwidth.

As mentioned above, the conversion bandwidth of the proposed wavelength converter is insensitive to the grating dimensions. In Table 1, four gratings with 1.5-cm length and different widths are considered. It can be seen that the conversion bandwidth varies in the range of 4 nm with various ${w}_{dc}$ and $\Delta w$. The maximum and 3-dB bandwidth of the conversion efficiency spectrum are compared in various gratings, as shown in Table 2. The grating lengths are varied from 1 period to 15 periods, while the grating widths are kept as ${w}_{1}=695$ nm and ${w}_{2}=815$ nm. It can be seen that the conversion bandwidth is only decreased by 9 nm, while the peak conversion efficiency is increased by 12 dB. Figure 6 shows the evolution of the conversion efficiency spectra from $z=0.5$ mm to $z=4.5$ mm. For clarity, only a half of the spectra where ${\lambda}_{s}>{\lambda}_{p}$ is considered. FWM in the constant-width waveguide has a trade-off between the peak conversion efficiency and the conversion bandwidth. The extension of waveguide length is required to achieve the higher conversion efficiency, which will result in a decrease of the conversion bandwidth. As discussed before, its bandwidth is inversely proportional to the square root of the propagation length [9]. In Grating 2, the peaks centered at ${\lambda}_{p}$ and ${\lambda}_{s,obj}$ become sharper with the increase of the propagation length. In Grating 1, however, the conversion bandwidth is kept almost constant, while the peak conversion efficiency is kept increasing along the waveguide.

The output of the FWM process is very sensitive to the waveguide geometry in the constant-width waveguide. If the waveguide height is kept constant, the efficient and broadband FWM process can be realized only in a small range of the waveguide widths, as shown in Fig. 7. Traditionally, the waveguide geometry should be tailored carefully for the dispersion engineering, so that the ZDWL can be located near the pump wavelength. On the contrary, as shown in Fig. 3, the conversion efficiency is varied in the range of 1.5 dB in our scheme with various ${w}_{dc}$ and $\Delta w$. In addition, the grating width has little impact on the conversion bandwidth as shown in Table 1. Therefore, our QPM scheme based on the vertical grating is more robust compared to the scheme based on the dispersion engineered constant-width waveguide.

## 4. Conclusion

In summary, an efficient and broadband wavelength converter is proposed in the vertically etched silicon gratings without dispersion engineering. For the target signal wavelength, the QPM of FWM is achieved by properly tailoring the period and duty cycle of the gratings, by which the parametric attenuation is suppressed and the conversion efficiency is improved. It is also demonstrated that the target signal wavelength is close to the edge of the 3-dB conversion bandwidth. Thus, the conversion bandwidth of the proposed wavelength converter is insensitive to both the propagation length and the grating width. With the increase of the propagation length, the larger conversion bandwidth can be obtained compared to the ZDWL pumped FWM. The variation range of the target signal wavelength is adequate for the near-infrared applications, although it is restricted by the pump wavelength, the dispersion profile of the waveguide, and the variation amplitude of the grating width. The proposed scheme provides a new way to realize efficient wavelength conversion in the case of serious phase-mismatch, and is beneficial for the broadband signal processing on the low-cost SOI platform without dispersion engineering.

## Acknowledgments

This work is partly supported by the National Basic Research Program (2010CB327601, 2010CB327605 and 2010CB328300), the National High-Technology Research and Development Program of China (2013AA031501), the National Natural Science Foundation of China (61307109), the Specialized Research Fund for the Doctoral Program of Higher Education (20120005120021), the Fundamental Research Funds for the Central Universities (2013RC1202), the Program for New Century Excellent Talents in University (NECT-11-0596), Beijing Nova Program (2011066), the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) P. R. China, the Research Grant Council of the Hong Kong Special Administrative Region China (project PolyU5272/12E), and the Science Foundation Ireland (SFI) under the International Strategic Cooperation Award Grant Number SFI/13/ISCA/2845.

## References and links

**1. **L. K. Oxenlowe, M. Hua Ji, M. Galili, Minhao Pu, H. C. H. Hao Hu, K. Mulvad, J. M. Yvind, A. T. Hvam, Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. **18**(2), 996–1005 (2012). [CrossRef]

**2. **Y. Xie, S. Gao, and S. He, “All-optical wavelength conversion and multicasting for polarization-multiplexed signal using angled pumps in a silicon waveguide,” Opt. Lett. **37**(11), 1898–1900 (2012). [CrossRef] [PubMed]

**3. **N. Ophir, R. K. W. Lau, M. Menard, R. Salem, K. Padmaraju, Y. Okawachi, M. Lipson, A. L. Gaeta, and K. Bergman, “First demonstration of a 10-Gb/s RZ end-to-end four-wave mixing-based link at 1884 nm using silicon nanowaveguides,” IEEE Photon. Technol. Lett. **24**(4), 276–278 (2012). [CrossRef]

**4. **J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics **4**(8), 535–544 (2010). [CrossRef]

**5. **F. Morichetti, A. Canciamilla, C. Ferrari, A. Samarelli, M. Sorel, and A. Melloni, “Travelling-wave resonant four-wave mixing breaks the limits of cavity-enhanced all-optical wavelength conversion,” Nat Commun **2**, 296 (2011). [CrossRef] [PubMed]

**6. **F. Li, M. Pelusi, D.-X. Xu, R. Ma, S. Janz, B. J. Eggleton, and D. J. Moss, “All-optical wavelength conversion for 10 Gb/s DPSK signals in a silicon ring resonator,” Opt. Express **19**(23), 22410–22416 (2011). [CrossRef] [PubMed]

**7. **A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express **16**(7), 4881–4887 (2008). [CrossRef] [PubMed]

**8. **A. C. Turner-Foster, M. A. Foster, R. Salem, A. L. Gaeta, and M. Lipson, “Frequency conversion over two-thirds of an octave in silicon nanowaveguides,” Opt. Express **18**(3), 1904–1908 (2010). [CrossRef] [PubMed]

**9. **M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express **15**(20), 12949–12958 (2007). [CrossRef] [PubMed]

**10. **M. Zhu, H. Liu, X. Li, N. Huang, Q. Sun, J. Wen, and Z. Wang, “Ultrabroadband flat dispersion tailoring of dual-slot silicon waveguides,” Opt. Express **20**(14), 15899–15907 (2012). [CrossRef] [PubMed]

**11. **L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express **20**(2), 1685–1690 (2012). [CrossRef] [PubMed]

**12. **J. Kim, Ö. Boyraz, J. H. Lim, and M. N. Islam, “Gain enhancement in cascaded fiber parametric amplifier with quasi-phase matching: theory and experiment,” J. Lightwave Technol. **19**(2), 247–251 (2001). [CrossRef]

**13. **M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B **25**(4), 463–480 (2008). [CrossRef]

**14. **N. Satyan, G. Rakuljic, and A. Yariv, “Chirp multiplication by four wave mixing for wideband swept-frequency sources for high resolution imaging,” J. Lightwave Technol. **28**(14), 2077–2083 (2010). [CrossRef]

**15. **H. Zhu, B. Luo, W. Pan, L. Yan, S. Xiang, and K. Wen, “Gain enhancement of fiber optical parametric amplifier via introducing phase-shifted fiber Bragg grating for phase matching,” J. Opt. Soc. Am. B **29**(6), 1497–1502 (2012). [CrossRef]

**16. **N. Vermeulen, J. E. Sipe, Y. Lefevre, C. Debaes, and H. Thienpont, “Wavelength conversion based on Raman- and non-resonant four-wave mixing in silicon nanowire rings without dispersion engineering,” IEEE J. Sel. Top. Quantum Electron. **17**(4), 1078–1091 (2011). [CrossRef]

**17. **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). [CrossRef]

**18. **J. B. Driscoll, R. R. Grote, X. P. Liu, J. I. Dadap, N. C. Panoiu, and R. M. Osgood Jr., “Directionally anisotropic Si nanowires: on-chip nonlinear grating devices in uniform waveguides,” Opt. Lett. **36**(8), 1416–1418 (2011). [CrossRef] [PubMed]

**19. **J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express **20**(8), 9227–9242 (2012). [CrossRef] [PubMed]

**20. **Y. Huang, E.-K. Tien, S. Gao, S. K. Kalyoncu, Q. Song, F. Qian, and O. Boyraz, “Quasi phase matching in SOI and SOS based parametric wavelength converters,” Proc. SPIE **8120**(81200F), 81200F (2011). [CrossRef]

**21. **B. Jin, C. Yu, J. Yuan, X. Sang, X. Xiang, Z. Liu, and S. Wei, “Efficient wavelength conversion based on quasi-phase-matched four-wave mixing in a silicon waveguide with a microring phase shifter,” J. Opt. Soc. Am. B **30**(9), 2491–2497 (2013). [CrossRef]

**22. **E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express **18**(21), 21981–21989 (2010). [CrossRef] [PubMed]

**23. **T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature **416**(6877), 233–237 (2002). [CrossRef] [PubMed]

**24. **M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature **456**(7218), 81–84 (2008). [CrossRef] [PubMed]

**25. **M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Ultrafast waveform compression using a time-domain telescope,” Nat. Photonics **3**(10), 581–585 (2009). [CrossRef]

**26. **A. E. Willner, O. F. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. **17**(2), 320–332 (2011). [CrossRef]

**27. **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). [CrossRef] [PubMed]

**28. **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). [CrossRef] [PubMed]

**29. **X. Sang and O. Boyraz, “Gain and noise characteristics of high-bit-rate silicon parametric amplifiers,” Opt. Express **16**(17), 13122–13132 (2008). [CrossRef] [PubMed]

**30. **F. D. Leonardis and V. M. N. Passaro, “Efficient wavelength conversion in optimized SOI waveguides via pulsed four-wave mixing,” J. Lightwave Technol. **29**(23), 3523–3535 (2011). [CrossRef]

**31. **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). [CrossRef] [PubMed]

**32. **I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. **16**(1), 200–215 (2010). [CrossRef]

**33. **A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express **14**(10), 4357–4362 (2006). [CrossRef] [PubMed]

**34. **W. Mathlouthi, H. Rong, and M. Paniccia, “Characterization of efficient wavelength conversion by four-wave mixing in sub-micron silicon waveguides,” Opt. Express **16**(21), 16735–16745 (2008). [CrossRef] [PubMed]

**35. **S. Lavdas, J. B. Driscoll, H. Jiang, R. R. Grote, R. M. Osgood Jr, and N. C. Panoiu, “Generation of parabolic similaritons in tapered silicon photonic wires: comparison of pulse dynamics at telecom and mid-infrared wavelengths,” Opt. Lett. **38**(19), 3953–3956 (2013). [CrossRef] [PubMed]

**36. **A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express **18**(4), 3582–3591 (2010). [CrossRef] [PubMed]

**37. **V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper for compact mode conversion,” Opt. Lett. **28**(15), 1302–1304 (2003). [CrossRef] [PubMed]

**38. **L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express **18**(19), 20529–20534 (2010). [CrossRef] [PubMed]