Abstract

We analytically characterize the temperature dependence of Raman amplification, Raman attenuation, and parametric Raman wavelength conversion (PRWC) in submicrometer silicon waveguides (WGs) over the temperature range of 100 to 500K, near the O-band and C-band wavelengths of 1.33 and 1.55μm. The efficiencies of Raman amplification/attenuation and PRWC are studied in the context of how the interplay among the Raman gain, two-photon absorption, free-carrier absorption, sidewall roughness, pump-signal-input intensity ratio, and phase matching condition influences the wave propagation in the submicrometer WG at different temperatures. Our results show that the effects of temperature variation can be harnessed to enhance and tune the Raman amplification/attenuation and PRWC. This offers a more dynamic control of the Raman performances of submicrometer silicon WG devices as compared to conventional silicon Raman WG devices operating at a fixed (room) temperature.

© 2011 Optical Society of America

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References

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  1. G. T. Reed, “The optical age of silicon,” Nature 427, 595–596(2004).
    [CrossRef] [PubMed]
  2. B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
    [CrossRef]
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    [PubMed]
  4. 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).
    [CrossRef] [PubMed]
  5. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
    [CrossRef] [PubMed]
  6. O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. K. R. Lewelling and P. J. McCann, “Finite element modeling predicts possibility of thermoelectrically-cooled lead-salt diode lasers,” IEEE Photon. Technol. Lett. 9, 297–299 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

2009

2008

K. K. Tsia, S. Fathpour, and B. Jalali, “Electrical control of parametric processes in silicon waveguides,” Opt. Express 16, 9838–9843 (2008).
[CrossRef] [PubMed]

L. Yang, D. Dai, and S. He, “Thermal analysis for a photonic Si ridge wire with a submicron metal heater,” Opt. Commun. 281, 2467–2471 (2008).
[CrossRef]

2007

S. T. Lim, Y. L. Ang, C. E. Png, and Eng A. Ong, “Single mode, polarization-independent submicron silicon waveguides based on geometrical adjustments,” Opt. Express 15, 11061–11072(2007).
[CrossRef] [PubMed]

P. Wang and A. Bar-Cohen, “On-chip hot spot cooling using silicon thermoelectric microcoolers,” J. Appl. Phys. 102, 034503(2007).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS—Part 1: theoretical analysis,” IEEE J. Sel. Top. Quantum Electron. 13, 770–787 (2007).
[CrossRef]

2006

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
[CrossRef]

2005

2004

2003

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

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).
[CrossRef] [PubMed]

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in silicon waveguides,” Opt. Express 11, 2862–2872 (2003).
[CrossRef] [PubMed]

2002

1997

K. R. Lewelling and P. J. McCann, “Finite element modeling predicts possibility of thermoelectrically-cooled lead-salt diode lasers,” IEEE Photon. Technol. Lett. 9, 297–299 (1997).
[CrossRef]

1983

M. Balkanski, R. F. Wallis, and E. Haro, “Anharmornic effects in light scattering due to optical phonon in silicon,” Phys. Rev. B 28, 1928–1934 (1983).
[CrossRef]

1980

1966

T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. 148, 845–848 (1966).
[CrossRef]

Abdollahi, S.

Aggarwal, R. L.

T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. 148, 845–848 (1966).
[CrossRef]

Ahmad, I.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Ang, Y. L.

Balkanski, M.

M. Balkanski, R. F. Wallis, and E. Haro, “Anharmornic effects in light scattering due to optical phonon in silicon,” Phys. Rev. B 28, 1928–1934 (1983).
[CrossRef]

Bar-Cohen, A.

P. Wang and A. Bar-Cohen, “On-chip hot spot cooling using silicon thermoelectric microcoolers,” J. Appl. Phys. 102, 034503(2007).
[CrossRef]

Boyraz, O.

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004).
[CrossRef] [PubMed]

Brinkmeyer, E.

Claps, R.

Colli, A.

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Dadap, J.

Dai, D.

L. Yang, D. Dai, and S. He, “Thermal analysis for a photonic Si ridge wire with a submicron metal heater,” Opt. Commun. 281, 2467–2471 (2008).
[CrossRef]

Dallas, T.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Dasgupta, P.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Datta, A.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Debaes, C.

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS—Part 1: theoretical analysis,” IEEE J. Sel. Top. Quantum Electron. 13, 770–787 (2007).
[CrossRef]

Dhar, A.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Dimitropoulos, D.

Edwards, D. F.

Eom, I.-Y.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Espinola, R.

Fathpour, S.

Gangopadhyay, S.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Han, Y.

Haro, E.

M. Balkanski, R. F. Wallis, and E. Haro, “Anharmornic effects in light scattering due to optical phonon in silicon,” Phys. Rev. B 28, 1928–1934 (1983).
[CrossRef]

Hart, T. R.

T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. 148, 845–848 (1966).
[CrossRef]

He, S.

L. Yang, D. Dai, and S. He, “Thermal analysis for a photonic Si ridge wire with a submicron metal heater,” Opt. Commun. 281, 2467–2471 (2008).
[CrossRef]

Holtmannspoetter, M.

M. Holtmannspoetter, E. Pitschujew, and B. Schmauss, “Broadband, spectrally controlled Raman-active attenuator,” in 35th European Conference on Optical Communication, 2009 (IEEE, 2009), pp. 1–2.

Holtz, M.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Jalali, B.

Khachadorian, S.

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Krause, M.

Kuban, P.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Lax, B.

T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. 148, 845–848 (1966).
[CrossRef]

Lewelling, K. R.

K. R. Lewelling and P. J. McCann, “Finite element modeling predicts possibility of thermoelectrically-cooled lead-salt diode lasers,” IEEE Photon. Technol. Lett. 9, 297–299 (1997).
[CrossRef]

Liang, T. K.

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Quantum Electron. 10, 1149–1153 (2004).
[CrossRef]

Lim, S. T.

Manor, R.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

McCann, P. J.

K. R. Lewelling and P. J. McCann, “Finite element modeling predicts possibility of thermoelectrically-cooled lead-salt diode lasers,” IEEE Photon. Technol. Lett. 9, 297–299 (1997).
[CrossRef]

McNab, S.

Moravvej-Farshi, M. K.

Niu, J.

J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
[CrossRef]

Ochoa, E.

Ong, Eng A.

Osgood, R.

Pitschujew, E.

M. Holtmannspoetter, E. Pitschujew, and B. Schmauss, “Broadband, spectrally controlled Raman-active attenuator,” in 35th European Conference on Optical Communication, 2009 (IEEE, 2009), pp. 1–2.

Png, C. E.

Raghunathan, V.

Reed, G. T.

G. T. Reed, “The optical age of silicon,” Nature 427, 595–596(2004).
[CrossRef] [PubMed]

Renner, H.

Scheel, H.

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Schmauss, B.

M. Holtmannspoetter, E. Pitschujew, and B. Schmauss, “Broadband, spectrally controlled Raman-active attenuator,” in 35th European Conference on Optical Communication, 2009 (IEEE, 2009), pp. 1–2.

Sha, J.

J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
[CrossRef]

Temkin, H.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

Thienpont, H.

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS—Part 1: theoretical analysis,” IEEE J. Sel. Top. Quantum Electron. 13, 770–787 (2007).
[CrossRef]

Thomsen, C.

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Tsang, H. K.

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Quantum Electron. 10, 1149–1153 (2004).
[CrossRef]

Tsia, K. K.

Vermeulen, N.

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS—Part 1: theoretical analysis,” IEEE J. Sel. Top. Quantum Electron. 13, 770–787 (2007).
[CrossRef]

Vierck, A.

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Vlasov, Y.

Wallis, R. F.

M. Balkanski, R. F. Wallis, and E. Haro, “Anharmornic effects in light scattering due to optical phonon in silicon,” Phys. Rev. B 28, 1928–1934 (1983).
[CrossRef]

Wang, P.

P. Wang and A. Bar-Cohen, “On-chip hot spot cooling using silicon thermoelectric microcoolers,” J. Appl. Phys. 102, 034503(2007).
[CrossRef]

Way, W. I.

Wu, M.

Yang, D.

J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
[CrossRef]

Yang, L.

L. Yang, D. Dai, and S. He, “Thermal analysis for a photonic Si ridge wire with a submicron metal heater,” Opt. Commun. 281, 2467–2471 (2008).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

T. K. Liang and H. K. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Quantum Electron. 10, 1149–1153 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

N. Vermeulen, C. Debaes, and H. Thienpont, “Mitigating heat dissipation in near- and mid-infrared silicon-based Raman lasers using CARS—Part 1: theoretical analysis,” IEEE J. Sel. Top. Quantum Electron. 13, 770–787 (2007).
[CrossRef]

B. Jalali, V. Raghunathan, D. Dimitropoulos, and O. Boyraz, “Raman-based silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 412–421 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

K. R. Lewelling and P. J. McCann, “Finite element modeling predicts possibility of thermoelectrically-cooled lead-salt diode lasers,” IEEE Photon. Technol. Lett. 9, 297–299 (1997).
[CrossRef]

IEEE Sens. J.

A. Datta, I.-Y. Eom, A. Dhar, P. Kuban, R. Manor, I. Ahmad, S. Gangopadhyay, T. Dallas, M. Holtz, H. Temkin, and P. Dasgupta, “Microfabrication and characterization of Teflon AF-coated liquid core waveguide channels in silicon,” IEEE Sens. J. 3, 788–795 (2003).
[CrossRef]

J. Appl. Phys.

P. Wang and A. Bar-Cohen, “On-chip hot spot cooling using silicon thermoelectric microcoolers,” J. Appl. Phys. 102, 034503(2007).
[CrossRef]

J. Lightwave Technol.

Laser Photon. Rev.

N. Vermeulen, C. Debaes, and H. Thienpont, “Coherent anti-Stokes Raman scattering in Raman lasers and Raman wavelength converters,” Laser Photon. Rev. 4, 656–670 (2010).
[CrossRef]

Nature

G. T. Reed, “The optical age of silicon,” Nature 427, 595–596(2004).
[CrossRef] [PubMed]

Opt. Commun.

L. Yang, D. Dai, and S. He, “Thermal analysis for a photonic Si ridge wire with a submicron metal heater,” Opt. Commun. 281, 2467–2471 (2008).
[CrossRef]

Opt. Express

Phys. Rev.

T. R. Hart, R. L. Aggarwal, and B. Lax, “Temperature dependence of Raman scattering in silicon,” Phys. Rev. 148, 845–848 (1966).
[CrossRef]

Phys. Rev. B

M. Balkanski, R. F. Wallis, and E. Haro, “Anharmornic effects in light scattering due to optical phonon in silicon,” Phys. Rev. B 28, 1928–1934 (1983).
[CrossRef]

Phys. Status Solidi B

S. Khachadorian, H. Scheel, A. Colli, A. Vierck, and C. Thomsen, “Temperature dependence of first- and second-order Raman scattering in silicon nanowires,” Phys. Status Solidi B 247, 3084–3088 (2010).
[CrossRef]

Scr. Mater.

J. Niu, J. Sha, and D. Yang, “Temperature dependence of first-order Raman scattering in silicon nanowire,” Scr. Mater. 55, 183–186 (2006).
[CrossRef]

Other

M. Holtmannspoetter, E. Pitschujew, and B. Schmauss, “Broadband, spectrally controlled Raman-active attenuator,” in 35th European Conference on Optical Communication, 2009 (IEEE, 2009), pp. 1–2.

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Figures (8)

Fig. 1
Fig. 1

(a) Three-dimensional view of a submicrometer silicon-on-insulator (SOI) channel waveguide. The upper cladding of silicon dioxide of thickness l 1 that covers the silicon core is not shown here. (b) Computed TE 00 and TM 00 modal profiles at λ = 1.55 μm and T = 298 K for the SOI channel waveguide with w = 0.445 μm , h c = 0.22 μm , l 1 = l 2 = 3 μm , and L = 2 μm .

Fig. 2
Fig. 2

Temperature dependence of the various parameters that affect Raman scattering: (a)  f v ( T ) , (b)  2 Γ ( T ) , (c)  n s i ( λ , T ) for λ = 1.3285 μm and λ = 1.5413 μm , and (d)  g eff ( λ , T ) for λ = 1.3285 μm and λ = 1.5413 μm .

Fig. 3
Fig. 3

Temperature dependence of the real part of the Raman susceptibility Re [ χ R ( Ω d , T ) ] = χ R ( Ω d , T ) and the imaginary part of the Raman susceptibility Im [ χ R ( Ω d , T ) ] = χ R ( Ω d , T ) as a function of the (a) anti-Stokes wavelength and (b) Stokes wavelength for the pump wavelength of λ p = 1.427 μm .

Fig. 4
Fig. 4

Longitudinal view of the submicrometer silicon WG of length L, functioning as (a) Raman amplifier/attenuator if | Δ β | 0 and (b) Raman wavelength converter if | Δ β | 0 .

Fig. 5
Fig. 5

Effects of varying the pump intensity I p on the relationship of the phase matching condition Δ β with the (a) conversion efficiency (CE) and (b) amplification efficiency (AE) of the submicrometer silicon WG at different temperatures T for Case A (see Subsection 3A for definition of Case A). Note that I p increases in the direction of the arrows.

Fig. 6
Fig. 6

Effects of varying the pump intensity I p on the relationship of the phase matching condition Δ β with the (ai)–(aiii) conversion efficiency (CE) and (bi)–(biii) amplification efficiency (AE) of the submicrometer silicon WG at different temperatures T for Case B (see Subsection 3A for definition of Case B). Note that I p increases in the direction of the arrows.

Fig. 7
Fig. 7

Demonstration of harnessing the effects of temperature T variation to improve the Raman performances of the submicrometer silicon waveguide for Case A (see Subsection 3A for definition of Case A). The CE (at | Δ β | = 0 ) is shown in (ai) and (aii), while the AE (at | Δ β | = 2000 m 1 ) is shown in (b).

Fig. 8
Fig. 8

Demonstration of harnessing the effects of temperature T variation to improve the Raman performances of the submicrometer silicon waveguide for Case B (see Subsection 3A for definition of Case B). The conversion efficiency (at | Δ β | = 0 ) is shown in (ai) and (aii), while the amplification efficiency (at | Δ β | = 2000 m 1 ) is shown in (bi) and (bii).

Tables (1)

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Table 1 Raman Performances of Submicrometer Silicon WG via Temperature (T) Variation

Equations (22)

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χ i j l q ( 3 ) = χ i j l q N R + χ i j l q R ,
f s = f p f v ( T ) λ s = c ( f p f v ( T ) ) 1 ,
f a = f p + f v ( T ) λ a = c ( f p + f v ( T ) ) 1 ,
χ i j l q R ( Ω d , T ) = 2 Γ ( T ) f v ( T ) 2 c n s i ( f d , T ) g eff ( T ) f d ( μ o / ε o ) 1 / 2 m = 1 , 2 , 3 ( R i j ) m ( R l q ) m ( f v 2 ( T ) Ω d 2 j 2 Γ ( T ) Ω d ) ,
Ω d = f p f d , g eff ( f d , T ) = 2 c 2 h π f d 3 S n s i 2 ( f d , T ) ( N ( f v ( T ) , T ) + 1 ) 2 Γ ( T ) 1 1 + Δ f p / 2 Γ ( T ) ,
N ( f v ( T ) , T ) = { exp ( h f v ( T ) / [ K B T ] ) 1 } 1 ,
R m = 1 = [ 0 0 0 0 0 1 0 1 0 ] , R m = 2 = [ 0 0 1 0 0 0 1 0 0 ] , R m = 3 = [ 0 1 0 1 0 0 0 0 0 ] ,
f v ( T ) = f 0 + Δ f v ( T ) ,
Δ f v ( T ) = A ( 1 + 2 exp ( h f 0 / 2 K B T ) 1 ) + B ( 1 + 3 exp ( h f 0 / 3 K B T ) 1 + 3 ( exp ( h f 0 / 3 K B T ) 1 ) 2 ) ,
2 Γ ( T ) = C ( 1 + 2 exp ( h f 0 / 2 K B T ) 1 ) + D ( 1 + 3 exp ( h f 0 / 3 K B T ) 1 + 3 ( exp ( h f 0 / 3 K B T ) 1 ) 2 ) ,
n s i ( λ , T ) = 3.41696 + 0.138497 ( ( λ × 10 6 ) 2 0.028 ) + 0.013924 ( ( λ × 10 6 ) 2 0.028 ) 2 2.09 × 10 5 ( λ × 10 6 ) 2 + 1.48 × 10 7 ( λ × 10 6 ) 4 + 1.5 × 10 4 × ( T 298 ) ,
± d E p ± d z = { α p 2 φ 2 λ p 2 N c b 2 A wg ( E p ± 2 + 2 E s + 2 + 2 E s 2 + 2 E a + 2 + 2 E a 2 + 2 E p 2 ) κ s s 2 A wg λ s λ p ( E s + 2 + E s 2 ) + κ a a 2 A wg λ a λ p ( E a + 2 + E a 2 ) } E p ± ,
± d E s ± d z = { α s 2 φ 2 λ s 2 N c b 2 A wg ( E s ± 2 + 2 E p + 2 + 2 E p 2 + 2 E a + 2 + 2 E a 2 + 2 E s 2 ) + κ s s 2 A wg ( E p + 2 + E p 2 ) } E s ± + κ s a 2 A wg ( E p ± 2 ) ( E a ± 2 ) * exp ( ± j Δ β z ) ,
± d E a ± d z = { α a 2 φ 2 λ a 2 N c b 2 A wg ( E a ± 2 + 2 E p + 2 + 2 E p 2 + 2 E s + 2 + 2 E s 2 + 2 E a 2 ) κ a a 2 A wg ( E p + 2 + E p 2 ) } E a ± + κ a s 2 A wg ( E p ± 2 ) ( E s ± ) * exp ( ± j Δ β z ) ,
κ s s ( Ω d , T ) = j 2 π f s 2 c 2 k s ( 2 χ i j l q N R ( Ω d , T ) + χ i j l q R ( Ω d , T ) ) ,
κ s a ( Ω d , T ) = j 2 π f s 2 c 2 k s ( χ i j l q N R ( Ω d , T ) + χ i j l q R ( Ω d , T ) ) ,
κ a a ( Ω d , T ) = j 2 π f a 2 c 2 k a ( 2 χ i j l q N R ( Ω d , T ) + χ i j l q R ( Ω d , T ) ) ,
κ a s ( Ω d , T ) = j 2 π f a 2 c 2 k a ( χ i j l q N R ( Ω d , T ) + χ i j l q R ( Ω d , T ) ) ,
A wg = [ I ( x , y ) d x d y ] 2 / | I 2 ( x , y ) d x d y | ,
N c = η { E p + 4 + E p 4 + E s + 4 + E s 4 + E a + 4 + E a 4 + 4 ( E p + 2 E s 2 + E p + 2 E a 2 + E p + 2 E s + 2 + E p + 2 E a + 2 + E p + 2 E p 2 ) + 4 ( E s + 2 E p 2 + E s + 2 E a 2 + E s 2 E p 2 + E s + 2 E a + 2 + E s + 2 E s 2 ) + 4 ( E a + 2 E p 2 + E a + 2 E s 2 + E a 2 E p 2 + E a 2 E s 2 + E a + 2 E a 2 ) } ,
CE = I a ( L ) / I s ( 0 ) ,
AE = I s ( L ) / I s ( 0 ) ,

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