## Modal theory of slow light enhanced third-order nonlinear effects in photonic crystal waveguides |

Optics Express, Vol. 20, Issue 18, pp. 20043-20058 (2012)

http://dx.doi.org/10.1364/OE.20.020043

Acrobat PDF (2899 KB)

### Abstract

In this paper, we derive the couple-mode equations for third-order nonlinear effects in photonic crystal waveguides by employing the modal theory. These nonlinear interactions include self-phase modulation, cross-phase modulation and degenerate four-wave mixing. The equations similar to that in nonlinear fiber optics could be expanded and applied for third-order nonlinear processes in other periodic waveguides. Based on the equations, we systematically analyze the group-velocity dispersion, optical propagation loss, effective interaction area, slow light enhanced factor and phase mismatch for a slow light engineered silicon photonic crystal waveguide. Considering the two-photon and free-carrier absorptions, the wavelength conversion efficiencies in two low-dispersion regions are numerically simulated by utilizing finite difference method. Finally, we investigate the influence of slow light enhanced multiple four-wave-mixing process on the conversion efficiency.

© 2012 OSA

## 1. Introduction

1. T. Baba, “Slow light in photonic crystals,” Nat. Photonics **2**, 465–473 (2008). [CrossRef]

3. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 344–356 (2010). [CrossRef]

4. A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express **17**, 552–557 (2009). [CrossRef] [PubMed]

5. S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. **95**, 221108 (2009). [CrossRef]

6. K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express **17**, 7206–7216 (2009). [CrossRef] [PubMed]

2. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express **17**, 2944–2953 (2009). [CrossRef] [PubMed]

7. J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett. **93**, 251105 (2008). [CrossRef]

8. X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B **82**, 041308 (2010). [CrossRef]

9. B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics **3**, 206–210 (2009). [CrossRef]

10. C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express **18**, 6831–6840 (2010). [CrossRef] [PubMed]

11. J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express **18**, 15484–15497 (2010). [CrossRef] [PubMed]

12. C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett. **36**, 3413–3415 (2011). [CrossRef] [PubMed]

13. P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics **4**, 862–868 (2010). [CrossRef]

14. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express **17**, 18340–18353 (2009). [CrossRef] [PubMed]

15. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

16. S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. **12**, 104004 (2010). [CrossRef]

17. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. **35**, 1440–1442 (2010). [CrossRef] [PubMed]

18. I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express **19**, 6093–6099 (2011). [CrossRef] [PubMed]

20. P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett. **36**, 2629–2631 (2011). [CrossRef] [PubMed]

*μ*m or 96

*μ*m) dispersion-engineered silicon PCWs [21

21. C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express **18**, 22915–22927 (2010). [CrossRef] [PubMed]

22. B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett. **36**, 1728–1730 (2011). [CrossRef] [PubMed]

23. J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express **19**, 4458–4463 (2011). [CrossRef] [PubMed]

24. N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 257–266 (2010). [CrossRef]

25. M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express **18**, 21024–21029 (2010). [CrossRef] [PubMed]

26. S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J. **4**, 224–233 (2012). [CrossRef]

28. N. Matsuda, T. Kato, K. Harada, H. Takesue, E. Kuramochi, H. Taniyama, and M. Notomi, “Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide,” Opt. Express **19**, 19861–19874 (2011). [CrossRef] [PubMed]

29. C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett. **24**, 7–9 (2012). [CrossRef]

## 2. Modal theory for third-order nonlinear effects

*x*direction in the PCW. The

*p*,

*s*,

*i*refer to the pump, signal and idler waves for degenerate FWM (2

*ω*=

_{p}*ω*+

_{s}*ω*).

_{i}**e**

*(*

_{p,s,i}*x*,

*y*,

*z*) and

**h**

*(*

_{p,s,i}*x*,

*y*,

*z*) denote the lattice-periodic electric and magnetic field component. The

*z*dependence of

**e**

*(*

_{p,s,i}*x*,

*y*,

*z*) and

**h**

*(*

_{p,s,i}*x*,

*y*,

*z*) are similar to the fundamental mode of slab waveguide mode, the

*x*and

*y*dependence are consistent with the modulation behavior of PCW in the

*x*and

*y*directions [30

30. A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B **81**, 155115 (2010). [CrossRef]

*k*(

_{p,s,i}*ω*) is the respective Bloch vectors.

**P**

*we define the perturbed electromagnetic fields in the time domain*

_{NL}*ω*,

_{p}*ω*,

_{s}*ω*are the central frequencies of the three waves, and

_{i}*k*,

_{p}*k*,

_{s}*k*are the corresponding central Bloch vectors.

_{i}*a*(

_{p}*x*,

*t*),

*a*(

_{s}*x*,

*t*),

*a*(

_{i}*x*,

*t*) denote the slowly varying envelopes. The

**P**

*,*

_{NL}**E**

_{2},

**H**

_{2}satisfy the Maxwell’s equations, and we can obtain their relationship for non-absorptive materials [32

32. D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E **68**, 065601–065601 (2003). [CrossRef]

*S*is the surface of the volume

*V*. The three waves are TE guided modes. Although the TE mode of PCW have a polarization component along the

*x*direction, it contains only a small fraction of incident power, and the nonlinear effects are dominated by the transverse

*y*component. For this reason, we simplify the problem considerably by adopting a scalar approach and writing the total electric field the third-order nonlinear polarization

**P**

*for degenerate FWM in the form The*

_{NL}**P**

*,*

_{p}**P**

*,*

_{s}**P**

*are the envelopes of the polarization at*

_{i}*ω*,

_{p}*ω*,

_{s}*ω*and given by

_{i}*x*-component electric field is relatively small. For simplicity, we neglect the

*e*=

_{p,s,i}**e**

*(*

_{p,s,i}*x*,

*y*,

*z*).

*χ*depicts the varied third-order nonlinearity as a function of three-dimensional coordinate (

*x*,

*y*,

*z*). The first two terms on the right-hand side describe the nonlinear phase shift due to SPM and XPM. The last term describes the energy transfer between the interacting fields. The THG and other FWM are neglected by assuming that they will not be phase matched.

*ω*≈

*ω*≈

_{p}*ω*≈

_{s}*ω*. Here, we mainly deduce the differential equations for the degenerate FWM in PCW, and these equations are analogous to those in the nonlinear fiber optics. In addition, the deduced process for the nondegenerate FWM is similar to it.

_{i}## 3. Parameters of coupled-mode equations

### 3.1. Characteristics of slow light engineered PCW

39. L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express **18**, 27627–27638 (2010). [CrossRef]

*a*= 410 nm, thickness

*h*= 220 nm and hole radius

*r*= 0.27

*a*. The dispersion is engineered by shifting the first two rows of holes either side of the waveguide as shown in Fig. 2(a), and the displacement is

*s*= −0.127

*a*[15

15. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

*x*,

*y*, and

*z*axis, respectively. The computational step size along the

*x*,

*y*, and

*z*directions are

*a*/16,

*a*/16, and

*a*/30, respectively. Figure 2(a) shows that the PCW has two TE-like guiding modes located in the frequency bandgap by using 3D PWE method. In order to utilize the even single mode propagation, only the normalized frequency in the thick solid red line of Fig. 2(a) is investigated.

40. S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. **94**, 033903 (2005). [CrossRef] [PubMed]

41. L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express **15**, 13129–13138 (2007). [CrossRef]

39. L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express **18**, 27627–27638 (2010). [CrossRef]

16. S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. **12**, 104004 (2010). [CrossRef]

*x*-axis, the Bloch theorem implies that the propagation constant, which is oriented along the

*x*-axis, may be presented by

*m*is an integer and represents the high-order Bloch harmonics and

*k*

_{0}is restricted to the first Brillouin zone −0.5 ≤

*k*

_{0}≤ 0.5 [30

30. A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B **81**, 155115 (2010). [CrossRef]

42. E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B **78**, 235101 (2008). [CrossRef]

*β*

_{1}=

*∂k*/

*∂ω*= 1/

*v*defines the GV of a pulse with frequency

^{g}*ω*, we consider the GV and group index to be positive. Thus, the Bloch wavevector may be taken in the range

*β*

_{2}=

*∂*

^{2}

*k*/

*∂ω*

^{2}) is negative. These wavelength dependence of dispersion coefficients (

*β*

_{1}and

*β*

_{2}), which are calculated for the TE even mode, are shown in Fig. 2(c). There are two low-dispersion regions, one is from 1560 nm to 1580 and the other is from 1595 nm to 1605 nm. The available wavelength range in small group index region is larger than the one in high group index region.

### 3.2. Effective interaction areas

*λ*=

_{s}*λ*− Δ

_{P}*λ*and

*λ*=

_{i}*λ*+ Δ

_{P}*λ*are assumed. Figure 3(d) depicts the calculated effective interaction areas for different nonlinear effects and Δ

*λ*. The effective interaction areas for SPM, FWM and XPM with Δ

*λ*= 1 nm are about the same. We also calculated the FWM effective interaction areas with Δ

*λ*= 3 nm and Δ

*λ*= 7 nm. It is found that the

*λ*increases. If the Δ

*λ*is large, we can not replace

*y*direction.

3. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 344–356 (2010). [CrossRef]

### 3.3. Slow light enhanced factors and phase mismatches

*τ*,

_{pppp}*τ*and

_{ppss}*τ*, respectively. Figure 4(a) shows the calculated slow light enhanced factors for different nonlinear interactions and Δ

_{ppsi}*λ*. Similar to the effective interaction areas, the slow light enhanced factors for SPM, XPM and FWM with Δ

*λ*= 1 nm are close to each other. The XPM enhanced factor

*τ*changes and is far away from the SPM enhanced factor

_{ppss}*τ*as the Δ

_{pppp}*λ*increases, while the FWM enhanced factor is consistent with the SPM enhanced factor.

*k*=

*k*+

_{s}*k*− 2

_{i}*k*) for different Δ

_{p}*λ*. The two low-dispersion regions correspond to the small phase mismatch. Meanwhile, the phase mismatch increases following the increasing Δ

*λ*. In FWM process, the phase match is very important for the conversion efficiency and conversion bandwidth, so the wavelengths in such two regions is widely used.

## 4. Theoretical analysis and numerical simulation

### 4.1. Degenerate FWM in Si PCW

24. N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 257–266 (2010). [CrossRef]

*γ*

_{1234}of SPM and XPM with

*β*is the coefficient of TPA. The Eqs. for degenerate FWM in Si PCW are obtained as

_{TPA}*α*(

_{lq}*q*=

*p*,

*s*,

*i*) represents the linear propagation loss for the pump, signal and idler waves.

*α*=

_{fcq}*σ*represents the nonlinear loss from FCA, where

_{q}N_{c}*σ*is the FCA cross section and

_{q}*N*is the free-carrier density. The

_{c}*k*is the FC-induced index change.

_{qc}N_{c}24. N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. **16**, 257–266 (2010). [CrossRef]

2. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express **17**, 2944–2953 (2009). [CrossRef] [PubMed]

43. L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. **32**, 2031–2033 (2007). [CrossRef] [PubMed]

*λ*(nm) is the wavelength,

_{q}*v*is the central frequency of the pump.

_{p}*τ*is the carrier lifetime (typically 100 ps∼1 ns) [14

_{c}14. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express **17**, 18340–18353 (2009). [CrossRef] [PubMed]

*λ*= 1 nm. In this case, the differences of effective interaction areas and slow light enhanced factors among the SPM, XPM and FWM can be neglected as shown in Fig. 3(d) and Fig. 4(a). The Kerr and TPA coefficients of Si are

*n*

_{2}= 6 × 10

^{−6}

*μ*m

^{2}/W and

*β*= 5 × 10

_{TPA}^{−6}

*μ*m/W [43

43. L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. **32**, 2031–2033 (2007). [CrossRef] [PubMed]

*τ*and input CW signal power are supposed to be 200 ps and 1 mW. We numerically solve the above FWM Eqs. (14) for a CW pump by using the finite difference method, and the FWM conversion efficiency is calculated with

_{c}*L*= 300

*μ*m and Δ

*λ*= 1 nm for signal power

*P*= 1 mW. It is found that the TPA and FCA may reduce the conversion efficiency seriously when the pump power is high. There is an oscillatory behavior with

_{s}*P*= 0.5 W when wavelength is larger than 1606 nm. Although the enhanced factor increases with the increasing group index, the loss raises more rapidly. So the loss limits the increase of conversion efficiency and the conversion efficiency will decrease when group index is big enough. Figure 5(b) illustrates the conversion efficiency for different device lengths. With the propagation loss and nonlinear losses, the wavelengths in the inflection points of curve diminish for increasing

_{p}*L*. That is, the available wavelength range for FWM is reduced correspondingly.

*P*is assumed to be 100 mW, and it may be reached experimentally [23

_{p}23. J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express **19**, 4458–4463 (2011). [CrossRef] [PubMed]

*λ*may be achieved. Since the SPM and FWM effective interaction areas and enhanced factors almost are equal, we mainly investigate the XPM effective interaction areas and enhanced factors as shown in Figs. 6(a) and 6(b).

*λ*= 3 nm because of the losses and large phase mismatch, and the optimized waveguide length is restricted within 400

*μ*m. This length may be increased through engineering the dispersions of the two regions individually. The waveguide lengths for the two regions are chosen as

*L*= 400

*μ*m and

*L*= 1 mm, respectively. Figure 7(b) and 7(d) show the relationship between conversion efficiency and wavelength detuning (Δ

*λ*). In the two regions, the 3-dB bandwidth of Δ

*λ*for wavelength with smaller group index is wider than the one with bigger group index. Although the maximum conversion efficiency in high group index region is larger than the small one, the coupled pump power of latter may be higher since its mode field and group index match the ridge access waveguides better [44

44. Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. **95**, 061105 (2009). [CrossRef]

### 4.2. Numerical Simulation for multiple FWM

17. V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. **35**, 1440–1442 (2010). [CrossRef] [PubMed]

23. J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express **19**, 4458–4463 (2011). [CrossRef] [PubMed]

45. X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A **77**, 043818 (2008). [CrossRef]

*A*|

_{p}^{2}, |

*A*|

_{s}^{2}≫ |

*A*|

_{i}^{2}, |

*A*|

_{j}^{2}, we neglect the terms including |

*A*|

_{i}^{2}and |

*A*|

_{j}^{2}. According to Eqs. (14), one can establish the multiple FWM Eqs. as

*k*=

_{ppsi}*k*+

_{s}*k*− 2

_{i}*k*, Δ

_{p}*k*=

_{sspj}*k*+

_{p}*k*− 2

_{j}*k*and Δ

_{s}*k*=

_{spij}*k*+

_{i}*k*−

_{j}*k*−

_{p}*k*. The TPA and FCA parameters are from Eqs. (15). The FC lifetime

_{s}*τ*is long enough and the time between two pulses is long compared to

_{c}*τ*, we can neglect FC recombination within the picosecond duration of the probe pulses. The

_{c}*N*can be calculated by integrating the TPA-induced FC generation term over the pulse duration.

_{c}*ω*= 2

_{i}*ω*−

_{p}*ω*and

_{s}*ω*= 2

_{j}*ω*−

_{s}*ω*processes. The nondegenerate FWM process is the

_{p}*ω*+

_{j}*ω*=

_{i}*ω*+

_{p}*ω*. Since the generated two idler wave power are very low, the nondegenerate FWM process is not important. We numerically solve the above multiple FWM Eqs. for a 100 ps wide Gaussian pulse pump centered at 1604 nm and a CW probe signal at 1602 nm also by using the finite difference method. The corresponding

_{s}*λ*= 1606 nm and

_{i}*λ*= 1600 nm. To clearly display the pulse evolution within the waveguide, the pump, signal and two idler waves propagating along the waveguide are simulated in Fig. 9. From Fig. 9(b), the CW signal wave gradually evolves into a pulsed wave and the pulse is decreased owing to large propagation loss. Figures 9(c) and 9(d) display the two idler waves evolutions in the multiple FWM process. It is found that there is an optimized waveguide length with loss for the wavelength conversion in the FWM process. The

_{j}*L*= 400

*μ*m is considered in the rest of the discussion.

*λ*idler wave has the same pulse width to pump pulse when the

_{j}*λ*idler wave exhibits pulse narrowing of about 30% to 70 ps. The two idler wave widths depend on the two degenerate FWM processes. The

_{i}*λ*idler wave generation is determined by the time dependent pump pulse power, and gain is not a constant and will compress the

_{i}*λ*idler pulse. On the other hand, the

_{i}*λ*idler wave generation is approximate constant in time domain owing to the CW signal, so the temporal waveform is basically unchanged.

_{j}*λ*increase more rapidly than

_{j}*λ*power. As shown in Fig. 11(b), the peak conversion efficiency for

_{i}*λ*increases with the increasing input signal power when the one for

_{j}*λ*keep changelessly. It seems that the generation of

_{i}*λ*has nothing to do with the wavelength conversion of

_{j}*λ*for small signal multiple FWM process. This phenomenons can also be understood through the two degenerate FWM processes.

_{i}## 5. Conclusion

## Acknowledgment

## References and links

1. | T. Baba, “Slow light in photonic crystals,” Nat. Photonics |

2. | C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express |

3. | C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. |

4. | A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express |

5. | S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. |

6. | K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express |

7. | J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett. |

8. | X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B |

9. | B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics |

10. | C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express |

11. | J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express |

12. | C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett. |

13. | P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics |

14. | M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express |

15. | J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express |

16. | S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. |

17. | V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett. |

18. | I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express |

19. | I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett. |

20. | P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett. |

21. | C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express |

22. | B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett. |

23. | J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express |

24. | N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron. |

25. | M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express |

26. | S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J. |

27. | G. Agrawal, |

28. | N. Matsuda, T. Kato, K. Harada, H. Takesue, E. Kuramochi, H. Taniyama, and M. Notomi, “Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide,” Opt. Express |

29. | C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett. |

30. | A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B |

31. | A. Snyder and J. Love, |

32. | D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E |

33. | S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

34. | R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B |

35. | R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A |

36. | A. Yariv, |

37. | T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. |

38. | M. Soljacic and J. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. |

39. | L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express |

40. | S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. |

41. | L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express |

42. | E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B |

43. | L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett. |

44. | Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. |

45. | X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A |

**OCIS Codes**

(160.6000) Materials : Semiconductor materials

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(190.4223) Nonlinear optics : Nonlinear wave mixing

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 23, 2012

Revised Manuscript: May 13, 2012

Manuscript Accepted: August 6, 2012

Published: August 17, 2012

**Citation**

Tao Chen, Junqiang Sun, and Linsen Li, "Modal theory of slow light enhanced third-order nonlinear effects in photonic crystal waveguides," Opt. Express **20**, 20043-20058 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20043

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### References

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