## Phase-matching and nonlinear optical processes in silicon waveguides

Optics Express, Vol. 12, Issue 1, pp. 149-160 (2004)

http://dx.doi.org/10.1364/OPEX.12.000149

Acrobat PDF (170 KB)

### Abstract

The efficiency of four-wave-mixing arising from Raman and non-resonant nonlinear susceptibilities in silicon waveguides is studied in the 1.3–1.8µm regime. The wavelength conversion efficiency is dominated by the Raman contribution to the nonlinear susceptibility, and high conversion efficiencies can be achieved under the phase-matching condition. In this context, dispersion in silicon waveguides is analyzed and it is shown that phase-matching is achieved in properly engineered waveguides where birefringence compensates for material dispersion. Finally the sensitivity of the phase mismatch to fabrication-induced errors in waveguide dimensions is quantified.

© 2004 Optical Society of America

## 1. Introduction

1. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, “Advances in Silicon-on-Insulator Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. **4**, 938–947 (1998). [CrossRef]

_{2}. This renders SOI ideal for photonic band structures as it minimizes radiation into the substrate even when extreme index modulation is performed [2

2. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. **18**, 1402–1411 (2000). [CrossRef]

3. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express **11**, 1731–1739 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731. [CrossRef] [PubMed]

^{4}times higher in silicon than in silica. Additionally, SOI waveguides can confine the optical field to an area that is approximately 100 times smaller than the modal area in a silica fiber. The combination of these properties will make SRS observable over millimeter-scale interaction lengths encountered in an integrated optic device.

4. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express **11**, 2862–2872 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2862. [CrossRef] [PubMed]

^{rd}order optical nonlinearity in the medium. Coherent Anti-Stokes Raman Scattering (CARS) occurs through coherent interaction of the two waves with a high intensity pump. Wavelength conversion is of paramount importance in optical networks because it enables optical packet switching, a key milestone in the realization of all-optical networks. While the recent demonstration [4

4. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express **11**, 2862–2872 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2862. [CrossRef] [PubMed]

^{-5}will have to be drastically increased before a practical device can be realized. Since the limited efficiency is due to the phase mismatch inside the SOI rib waveguide, the key problem is how to engineer the Group Velocity Dispersion (GVD) inside the waveguide in order to attain phase matching.

^{2}in a 2cm long waveguide with a loss of 1dB/cm. In Section 2, the relevant nonlinear optical properties of silicon are summarized. Section 3 outlines the phase matching condition and describes various contributions to the overall GVD. Section 4 provides the design of phase matched SOI waveguides. Section 5 considers the sensitivity of phase matching to variations in waveguide dimension caused by fabrication errors.

## 2. Non-linear processes in SOI waveguides

^{rd}order nonlinear optical response. The nonlinear response of the electrons to the optical field (electronic — non-resonant susceptibility) [5

5. S.S. Jha and N. Bloembergen, “Nonlinear Optical Susceptibilities in Group-IV and III–V semiconductors,” Phys. Rev. **171**, 891–898 (1968). [CrossRef]

6. R. Loudon, “Theory of first-order Raman effect in crystals,” Proc. Roy. Soc. (London) A **275**, 218–232 (1963). [CrossRef]

_{R}=15.6THz and the resonance half-width is Γ

_{R}=2π×53GHz. In the crystallographic axes system x,y,z the resonant susceptibility has nonzero components

*ω*is the frequency shift with respect to the pump frequency and the peak value (Δ

*ω*=Ω

_{R}) is

*jξ*

_{R}=-

*j*11.2×10

^{-18}(m

^{2}V

^{-2}). For comparison, the other contribution to the 3

^{rd}order nonlinear susceptibility, i.e., the electronic (non-resonant) susceptibility in silicon is

^{-18}with

7. J.J. Wynne, “Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. **178**, 1295–1303 (1969). [CrossRef]

3. R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express **11**, 1731–1739 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731. [CrossRef] [PubMed]

^{2}.

8. D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in Silicon-On-Insulator waveguides,” Opt. Lett. **28**, 1–3 (2003). [CrossRef]

9. H.K. Tsang, C.S. Wong, T.K. Liang, I.E. Day, S.W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase-modulation in silicon waveguides at 1.54µm wavelength,” Appl. Phys. Lett. **80**, 416–418 (2002). [CrossRef]

^{-18}, which is very close to the value reported for bulk silicon in [7

7. J.J. Wynne, “Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. **178**, 1295–1303 (1969). [CrossRef]

10. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. **82**, 2954–2956 (2003). [CrossRef]

*ω*

_{1},

*ω*

_{2}=

*ω*

_{1}-Ω+Δ

*ω*

_{2}, and

*ω*

_{3}=

*ω*

_{1}+Ω+Δ

*ω*

_{3}, where the optical power of first wave is much higher than the power in the other two. It is understood that frequency deviations from resonance, Δ

*ω*

_{2}, Δ

*ω*

_{3}are smaller than the half-width Γ

_{R}. More specifically the amplitude conversion efficiency between the Anti-Stokes and Stokes waves are [11

11. E. Golovchenko, P.V. Mamyshev, A.N. Pilipetskii, and E.M. Dianov, “Mutual Influence of the Parametric Effects and Stimulated Raman Scattering in Optical Fibers,” IEEE J. Quantum Electron. **26(10)** p.1815–1820 (1990). [CrossRef]

*x*is the propagation length [11

11. E. Golovchenko, P.V. Mamyshev, A.N. Pilipetskii, and E.M. Dianov, “Mutual Influence of the Parametric Effects and Stimulated Raman Scattering in Optical Fibers,” IEEE J. Quantum Electron. **26(10)** p.1815–1820 (1990). [CrossRef]

*β*is the phase mismatch (to be formally defined later). The importance of phase-matching lies in the fact that the conversion efficiency between the two-waves varies as ~ sin

*c*

^{2}(Δ

*βx*/2) when Δ

*β*≫

*κ*

_{22},

*κ*

_{33}as can be seen from Eq. (2). Referring to Equation (4), since|

*χ*

^{R}|/|

*χ*

^{NR}|~45, the conversion efficiency is dominated by the Raman contribution to the nonlinear susceptibility.

*β*term, as will be shown below. The efficiency for Stokes to anti-Stokes power conversion is plotted in Fig. 1 for a 2cm long waveguide with 4µm

^{2}cross-section and for 400mW monochromatic pump at 1434nm. 0.1 and 1 dB/cm losses have been assumed in this example. A peak efficiency of ~18% is obtained at the phase matched condition. It shows that the conversion efficiency is a strong function of phase mismatch. For example even a modest phase mismatch of 10 cm

^{-1}reduces the conversion efficiency by approximately 20dB. It should be acknowledged that this analysis assumes a zero linewidth pump. The finite pump linewidth will lower the overall efficiency.

## 3. The phase-matching condition in SOI waveguides

12. R.A. Soref, J. Schmidtchen, and K. Petermann, “Large single-mode rib waveguides in GeSi-Si and Si-on-SiO_{2},” IEEE J. Quantum Electron. **27**, 1971–1974 (1991). [CrossRef]

*w*and

*h*, even for a wavelength

*λ*≪

*w,h*. When the single-mode condition is satisfied the waveguides support two modes, one approximately TE polarized (E field parallel to the horizontal) and the other approximately TM polarized (H field parallel to the vertical). Since the index difference between Si and SiO

_{2}is large, the mode lies almost entirely in the Si area and we expect the dispersion of these waveguides to be very low.

*β*

_{B}induced by the waveguide birefringence, Δ

*β*

_{WD}induced by the waveguide dispersion, Δ

*β*

_{MAT}induced by the material dispersion, and Δ

*β*

_{SPM,XPM}is induced by the self-phase modulation of the pump wave at

*ω*

_{1}and the cross-phase modulation of the sidebands due to the pump wave. In the normal dispersion regime of the material Δ

*β*

_{MAT}<0 and in the anomalous regime Δ

*β*

_{MAT}>0. Δ

*β*

_{SPM,XPM}<0, when the nonlinear susceptibility has positive sign, as is the case in silicon. The birefringence contribution can be positive or negative depending on the relative polarization of the sidebands with respect to the pump.

*β*

_{SPM,XPM}, or if the fiber is polarization maintaining, one can launch the waves such that

*ω*

_{2},

*ω*

_{3}are on the slow axis and Δ

*β*

_{B}<0.

*ω*

_{2}=Δ

*ω*

_{3}=0. Since the optical fields have frequencies below the bandgap of silicon, the dispersion is normal (

*d*

^{2}

*β*/

*dω*

^{2})

_{MAT}>0.

*ε*

_{1}=11.6858,

*A*=0.939816,

*B*=8.10461×10

^{-3}and

*λ*

_{1}=1.1071

*µm*. The third term is the contribution of the indirect bandgap and the first two represent contributions from all other resonances. The wavevector mismatch that this dispersion contributes with respect to the pump wavelength is shown in Fig. 3. For comparison, the material GVD induced mismatch for silica fiber is also shown. The model used for fiber is the three resonance Sellmeier model in [14].

^{nd}order material dispersion is sufficient and

*cm*

^{-1}. As a side note, while in fiber the material dispersion is much lower than in silicon, this does not mean that phase matching is easier for the CARS. As can be seen from Eqs. (2) and (3) the regime for which the conversion efficiencies are particularly high are for Δ

*β*/Im(

*κ*

_{22}+

*κ*

_{33})~1. Thus, because in fiber the Raman susceptibility is much lower than silicon, the condition for phase-matching is more stringent in fiber.

*cm*

^{-1}. Hence it is insufficient for compensating for the material dispersion. Because the non-resonant susceptibility of silicon is very small, SPM and XPM contributions to Δ

*β*are negligible. Indeed,

*I*

_{1}is the pump intensity and

*χ*

_{NR}the non-resonant susceptibility. Even for intensities of the order of 100

*MW/cm*

^{2}, the nonlinear contribution to the phase mismatch is Δ

*β*

_{SPM,XPM}~-0.2

*cm*

^{-1}, a negligible value.

*β*

_{TE}(

*ω*

_{1})>

*β*

_{TM}(

*ω*

_{1}) and for the polarizations configuration in the experiments of Reference [4

4. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express **11**, 2862–2872 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2862. [CrossRef] [PubMed]

*β*

_{B}>0. As simulations in the next section reveal, for this configuration, this term is positive and is in the range of 50–600

*cm*

^{-1}, depending on the specific rib dimensions. Thus the birefringence can be used to cancel the material dispersion and phase-match the interacting waves. Should one use TM polarization for the pump and TM for the Stokes and Anti-Stokes waves, the birefringence term will have the opposite sign and it adds up to the material dispersion.

*ω*

_{1}, the phase mismatch in Eq. (5) can be written as:

*ω*

_{2}, Δ

*ω*

_{3}the added contribution to the above expression is (to 1

^{st}order in Δ

*ω*

_{2}, Δ

*ω*

_{3}):

## 4. Simulation results

^{TM}Version 4.1.6) which uses the mode-matching technique to calculate the propagation constant. All the waveguides chosen for the simulations satisfy the single-mode condition [12

12. R.A. Soref, J. Schmidtchen, and K. Petermann, “Large single-mode rib waveguides in GeSi-Si and Si-on-SiO_{2},” IEEE J. Quantum Electron. **27**, 1971–1974 (1991). [CrossRef]

*H*=1.4um was chosen and the ratio

*h/H*was varied from 0.5 to 0.7, with the rib width

*w*taking values 1.0, 1.2 and 1.4µm. The results for the total wavevector mismatch in CARS are shown in Figs. 4 and 5, for anti-Stokes (1334.8nm) to Stokes (1550nm) conversion and Stokes (1847.9 nm) to Anti-Stokes (1550nm) respectively. The mismatch is different in the two cases because the wavelengths involved are different. Larger pump wavelength leads to higher birefringence and waveguide dispersion.

*h/H*=0.5 towards the slab waveguide limit of

*h/H*=1, the mismatch increases. This is due to the fact that this mismatch is dominated by the birefringence which assumes its highest value when the slab geometry is approached (in the other extreme, a square waveguide has zero birefringence).

*H*=1.4um waveguides the minimum wavevector mismatch achieved with the above variations is ~360cm

^{-1}. We next consider fixed ratios of

*h/H*=0.6 and

*w/H*=1 and vary

*w*from 2 to 3.5um.

*w*=2.35µm), where the birefringence and material dispersion will cancel.

*w=H*=2.3µm,

*h*=0.6

*H*the different contributions to the wavevector mismatch and the total mismatch are shown in Fig. 7 for Stokes wavelength between 1500–1600nm (pump wavelengths 1391–1477nm). Phase-matching is achieved for Stokes wavelength around 1520nm. The calculated waveguide dispersion is ~11cm

^{-1}, very low in comparison with the birefringence. We have considered it to be constant over the range of frequencies used. In general, for different waveguide profiles the dispersion is on the order of ~5–10 cm

^{-1}.

## 5. Sensitivity to fabrication errors

1. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, “Advances in Silicon-on-Insulator Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. **4**, 938–947 (1998). [CrossRef]

*Δβ*, as a function of the variations in rib height Δ

*h*and width Δ

*w*, respectively. The waveguide considered has

*w=H*=2.3µm and

*h*=1.38µm. The results suggest that in order to maintain the phase mismatch below 1.3 cm

^{-1}, the variations in width or etch depth must be less than 70nm and 10nm, respectively. While these are stringent requirements, they are within the capability of silicon manufacturing which is currently producing microprocessors with tens of millions of transistors with 90nm feature size [15

15. Intel Technology Journal, Vol. 06 (02), ISSN 1535-766X. http://www.intel.com/technology/itj/2002/volume06issue02/art01_130nmlogic/p01_abstract.htm

## 5. Conclusions

^{2}in a 2cm long waveguide with a loss of 1dB/cm. Group velocity dispersion and birefringence in silicon waveguides was quantified. It was shown the waveguide dispersion and the SPM induced phase mismatch are much smaller than the contributions of the birefringence and the material dispersion, and that phase-matched waveguides can be realized by using the birefringence to compensate for the material dispersion. The sensitivity of the phase matching condition to fabrication error was studied and it was shown that accurate control of rib width and etch depth must be achieved in order to realize a high conversion efficiency.

## Acknowledgments

## References and links

1. | B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, “Advances in Silicon-on-Insulator Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. |

2. | M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. |

3. | R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express |

4. | R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express |

5. | S.S. Jha and N. Bloembergen, “Nonlinear Optical Susceptibilities in Group-IV and III–V semiconductors,” Phys. Rev. |

6. | R. Loudon, “Theory of first-order Raman effect in crystals,” Proc. Roy. Soc. (London) A |

7. | J.J. Wynne, “Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,” Phys. Rev. |

8. | D. Dimitropoulos, B. Houshmand, R. Claps, and B. Jalali, “Coupled-mode theory of the Raman effect in Silicon-On-Insulator waveguides,” Opt. Lett. |

9. | H.K. Tsang, C.S. Wong, T.K. Liang, I.E. Day, S.W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase-modulation in silicon waveguides at 1.54µm wavelength,” Appl. Phys. Lett. |

10. | M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. |

11. | E. Golovchenko, P.V. Mamyshev, A.N. Pilipetskii, and E.M. Dianov, “Mutual Influence of the Parametric Effects and Stimulated Raman Scattering in Optical Fibers,” IEEE J. Quantum Electron. |

12. | R.A. Soref, J. Schmidtchen, and K. Petermann, “Large single-mode rib waveguides in GeSi-Si and Si-on-SiO |

13. | H.H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. of Phys. & Chem. Ref. Data , |

14. | G.P. Agrawal, |

15. | Intel Technology Journal, Vol. 06 (02), ISSN 1535-766X. http://www.intel.com/technology/itj/2002/volume06issue02/art01_130nmlogic/p01_abstract.htm |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 11, 2003

Revised Manuscript: December 23, 2003

Published: January 12, 2004

**Citation**

Dimitrios Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, "Phase-matching and Nonlinear Optical Processes in Silicon Waveguides," Opt. Express **12**, 149-160 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-149

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### References

- B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, �??Advances in Silicon-on- Insulator Optoelectronics,�?? IEEE J. Sel. Top. Quantum Electron. 4, 938 �?? 947 (1998). [CrossRef]
- M. Loncar, T. Doll, J. Vuckovic, A. Scherer, �??Design and fabrication of silicon photonic crystal optical waveguides,�?? J. Lightwave Technol. 18, 1402-1411 (2000). [CrossRef]
- R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, B. Jalali, �??Observation of stimulated Raman amplification in silicon waveguides,�?? Opt. Express 11, 1731-1739 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731</a> [CrossRef] [PubMed]
- R. Claps, V. Raghunathan, D. Dimitropoulos, B. Jalali, �??Anti-Stokes Raman conversion in Silicon waveguides,�?? Opt. Express 11, 2862-2872 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2862">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2862</a>. [CrossRef] [PubMed]
- S.S. Jha, N. Bloembergen, �??Nonlinear Optical Susceptibilities in Group-IV and III-V semiconductors,�?? Phys. Rev. 171, 891 �?? 898 (1968). [CrossRef]
- R. Loudon, �??Theory of first-order Raman effect in crystals,�?? Proc. Roy. Soc. (London) A 275, 218 �?? 232 (1963). [CrossRef]
- J.J. Wynne, �??Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,�?? Phys. Rev. 178, 1295 �?? 1303 (1969). [CrossRef]
- D. Dimitropoulos, B. Houshmand, R. Claps, B. Jalali, �??Coupled-mode theory of the Raman effect in Silicon-On-Insulator waveguides,�?? Opt. Lett. 28, 1-3 (2003). [CrossRef]
- H.K. Tsang, C.S. Wong, T.K. Liang, I.E. Day, S.W. Roberts, A. Harpin, J. Drake, M. Asghari, �??Optical dispersion, two-photon absorption and self-phase-modulation in silicon waveguides at 1.54µm wavelength,�?? Appl. Phys. Lett. 80, 416 �?? 418 (2002). [CrossRef]
- M. Dinu, F. Quochi, H. Garcia, �??Third-order nonlinearities in silicon at telecom wavelengths,�?? Appl. Phys. Lett. 82 , 2954 �?? 2956 (2003). [CrossRef]
- E. Golovchenko, P.V. Mamyshev, A.N. Pilipetskii, E.M. Dianov, �??Mutual Influence of the Parametric Effects and Stimulated Raman Scattering in Optical Fibers,�?? IEEE J. Quantum Electron. 26(10) p.1815- 1820 (1990). [CrossRef]
- R.A. Soref, J. Schmidtchen, K. Petermann, �??Large single-mode rib waveguides in GeSi-Si and Si-on-SiO2,�?? IEEE J. Quantum Electron. 27, 1971 �?? 1974 (1991). [CrossRef]
- H.H. Li, �??Refractive index of silicon and germanium and its wavelength and temperature derivatives,�?? J. of Phys. & Chem. Ref. Data, 9, p.591-658 (1980).
- G.P. Agrawal, Nonlinear Fiber Optics, (Academic Press, San Diego, 2001) ISBN 0-12-045143-3.
- Intel Technology Journal, Vol. 06 (02), ISSN 1535-766X. <a href="http://www.intel.com/technology/itj/2002/volume06issue02/art01_130nmlogic/p01_abstract.htm">http://www.intel.com/technology/itj/2002/volume06issue02/art01_130nmlogic/p01_abstract.htm</a>

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