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

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References

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. of Phys. & Chem. Ref. Data

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).

Appl. Phys. Lett.

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]

IEEE J. Quantum Electron

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]

IEEE J. Sel. Top. Quantum Electron

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]

J. Lightwave Technol.

Opt. Express

Opt. Lett.

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]

Phys. Rev.

J.J. Wynne, �??Optical Third-Order Mixing in GaAs, Ge, Si, and InAs,�?? Phys. Rev. 178, 1295 �?? 1303 (1969).
[CrossRef]

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

Proc. Roy. Soc.

R. Loudon, �??Theory of first-order Raman effect in crystals,�?? Proc. Roy. Soc. (London) A 275, 218 �?? 232 (1963).
[CrossRef]

Other

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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Figures (10)

Fig. 1.
Fig. 1.

Conversion efficiency in a 2cm waveguide at 10MW/cm2 pump intensity with linear losses 0.1 dB/cm (solid line) and 1dB/cm (dashed line). Stokes to Anti-Stokes efficiency is shown for pump centered at 1434nm.

Fig. 2.
Fig. 2.

SOI rib waveguide geometry.

Fig. 3.
Fig. 3.

Material induced wavevector mismatch with respect to the pump wavelength for the CARS process. Both the case of silicon (solid line) and the case of fiber (dashed line) are shown for comparison.

Fig. 4.
Fig. 4.

Waveguide induced wavector mismatch for anti-Stokes (1.3348µm) to Stokes (1.55µm) down-conversion.

Fig. 5.
Fig. 5.

Waveguide induced wavevector mismatch for Stokes (1847.9nm) to anti-Stokes (1550nm) up-conversion.

Fig. 6.
Fig. 6.

Total momentum mismatch vs. dimension increase for both Stokes to Anti-Stokes (solid line) and Anti-Stokes to Stokes conversion (dashed line).

Fig. 7.
Fig. 7.

Momentum mismatch for waveguide w=H=2.3µm, h=0.6H versus Stokes wavelength in CARS : total mismatch (solid line), material dispersion contribution (dotted line), waveguide dispersion contribution (dashed line) and birefringence contribution.

Fig. 8.
Fig. 8.

Waveguide profiles fabricated by LOCOS and DRIE (solid lines) and the targeted profile (dashed line).

Fig. 9.
Fig. 9.

Phase-mismatch variation with respect to the error in slab height (error in the etching depth) for the phase-matched waveguide w=H=2.3µm, h=1.38µm at center wavelength 1520nm.

Fig. 10.
Fig. 10.

Phase-mismatch variation with respect to the error in rib width (over-cutting the rib) for the phase-matched waveguide w=H=2.3µm, h=1.38µm at center wavelength 1520nm.

Equations (11)

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χ ijkl R ( ω 1 Δ ω , ω 1 ± Δ ω , ω 1 , ω 1 ) = 2 Ω R Γ R ξ R 2 j Γ R Δ ω + Ω R 2 ( Δ ω ) 2
C AS S = κ 23 e κ 23 x e s x e s + x s s + , C S AS = κ 32 e κ 33 x e λ x e λ + x λ λ +
λ = s ± j Δ β + κ 22 + κ 33
s ± = j Δ β κ 22 κ 33 ± ( j Δ β κ 22 κ 33 ) 2 4 κ 23 κ 32 2
κ 22 = j k 2 2 2 β 2 χ R A 1 2 , κ 23 = j k 3 2 2 β 2 ( 2 χ NR + χ R ) A 1 2
κ 33 = j k 3 2 2 β 3 χ R A 1 2 , κ 32 = j k 2 2 2 β 3 ( 2 χ NR + χ R ) ( A 1 * ) 2
Δ β = 2 β ( ω 1 ) β ( ω 2 ) β ( ω 3 )
Δ β = Δ β B + Δ β WD + Δ β MAT + Δ β SPM , XPM
n 2 = ε 1 + A λ 2 + B λ 1 2 λ 2 λ 1 2
Δ β 2 ( β TE ( ω 1 ) β TM ( ω 1 ) ) ( d 2 β TM d ω 2 ) WG + MAT Ω 2
( d β TM d ω ) WG + MAT ( Δ ω 2 + Δ ω 3 ) + ( d 2 β TM d ω 2 ) WG + MAT Ω ( Δ ω 2 Δ ω 3 )

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