Abstract

We derive a set of design guidelines and a figure of merit to aid the engineering process of on-chip waveguides for strong Stimulated Brillouin Scattering (SBS). To this end, we examine the impact of several types of loss on the total amplification of the Stokes wave that can be achieved via SBS. We account for linear loss and nonlinear loss of third order (two-photon absorption, 2PA) and fifth order, most notably 2PA-induced free carrier absorption (FCA). From this, we derive an upper bound for the output power of continuous-wave Brillouin-lasers and show that the optimal operating conditions and maximal realisable Stokes amplification of any given waveguide structure are determined by a dimensionless parameter ℱ involving the SBS-gain and all loss parameters. We provide simple expressions for optimal pump power, waveguide length and realisable amplification and demonstrate their utility in two example systems. Notably, we find that 2PA-induced FCA is a serious limitation to SBS in silicon and germanium for wavelengths shorter than 2200nm and 3600nm, respectively. In contrast, three-photon absorption is of no practical significance.

© 2015 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear optics, 3rd ed. (Academic, 2003).
  2. L. Brillouin, “Diffusion de la lumière par un corps transparent homogène,” Ann. Phys. 17, 88–122 (1922).
  3. R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592 (1964).
    [Crossref]
  4. G. P. Agrawal, Nonlinear fiber optics, 5th ed. (Academic, 2012).
  5. M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
    [Crossref]
  6. R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thévenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
    [Crossref] [PubMed]
  7. W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
    [Crossref]
  8. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
    [Crossref] [PubMed]
  9. R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hyper-sound in a silicon photonic nanowire,” Nature Photon. 9, 199 (2015).
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  10. I. V. Kabakova, R. Pant, D.-Y. Choi, S. Debbarma, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Narrow linewidth Brillouin laser based on chalcogenide photonic chip,” Opt. Lett. 38, 3208–3211 (2013).
    [Crossref] [PubMed]
  11. T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).
  12. X. Huang and S. Fan, “Complete all-optical silica fiber isolator via Stimulated Brillouin Scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
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    [Crossref] [PubMed]
  14. L. Thévenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
    [Crossref]
  15. B. Vidal, M. A. Piqueras, and J. Marti, “Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering,” Opt. Lett. 32, 23–25 (2007).
  16. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
    [PubMed]
  17. B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
    [Crossref]
  18. C. Wolff, P. Gutsche, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Impact of nonlinear loss on Stimulated Brillouin Scattering,” J. Opt. Soc. Am. B 32, 1968–1978 (2015).
    [Crossref]
  19. I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
    [Crossref]
  20. C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin Scattering in integrated photonic waveguides: forces, scattering mechanisms and coupled mode analysis,” Phys. Rev. A 92, 013836 (2015).
    [Crossref]
  21. B.G. Helme and P.J. King, “The Phonon Viscosity Tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Solidi (A) 45, K33 (1978)
    [Crossref]
  22. R. Van Laer, B. Kuyken, R. Baets, and D. Van Thourhout, “Unifying Brillouin scattering and cavity optomechanics,” http://arxiv.org/abs/1503.03044 , (2015).
  23. J.J. Wortman and R.A. Evans, “Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium,” J. Appl. Phys. 36, 153 (1965).
    [Crossref]
  24. D.K. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196 (1974).
    [Crossref]
  25. A. Feldman, R.M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978).
    [Crossref]
  26. A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
    [Crossref]
  27. D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83, 195203 (2011).
    [Crossref]
  28. S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300–3300 nm,” Appl. Phys. Lett. 93, 131102 (2008).
    [Crossref]

2015 (3)

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hyper-sound in a silicon photonic nanowire,” Nature Photon. 9, 199 (2015).
[Crossref]

C. Wolff, P. Gutsche, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Impact of nonlinear loss on Stimulated Brillouin Scattering,” J. Opt. Soc. Am. B 32, 1968–1978 (2015).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin Scattering in integrated photonic waveguides: forces, scattering mechanisms and coupled mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

2014 (3)

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

I. Aryanfar, C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Mode conversion using stimulated Brillouin scattering in nanophotonic silicon waveguides,”; Opt. Express 22, 29270–29282 (2014).
[Crossref] [PubMed]

2013 (4)

2011 (3)

2010 (1)

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

2009 (1)

M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
[Crossref]

2008 (2)

L. Thévenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
[Crossref]

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300–3300 nm,” Appl. Phys. Lett. 93, 131102 (2008).
[Crossref]

2007 (2)

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

B. Vidal, M. A. Piqueras, and J. Marti, “Tunable and reconfigurable photonic microwave filter based on stimulated Brillouin scattering,” Opt. Lett. 32, 23–25 (2007).

1978 (2)

B.G. Helme and P.J. King, “The Phonon Viscosity Tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Solidi (A) 45, K33 (1978)
[Crossref]

A. Feldman, R.M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978).
[Crossref]

1974 (1)

D.K. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196 (1974).
[Crossref]

1965 (1)

J.J. Wortman and R.A. Evans, “Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium,” J. Appl. Phys. 36, 153 (1965).
[Crossref]

1964 (1)

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592 (1964).
[Crossref]

1922 (1)

L. Brillouin, “Diffusion de la lumière par un corps transparent homogène,” Ann. Phys. 17, 88–122 (1922).

Agrawal, G. P.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

G. P. Agrawal, Nonlinear fiber optics, 5th ed. (Academic, 2012).

Aryanfar, I.

Baets, R.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hyper-sound in a silicon photonic nanowire,” Nature Photon. 9, 199 (2015).
[Crossref]

Biegelsen, D.K.

D.K. Biegelsen, “Photoelastic Tensor of Silicon and the Volume Dependence of the Average Gap,” Phys. Rev. Lett. 32, 1196 (1974).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic, 2003).

Brenn, A.

M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
[Crossref]

Brillouin, L.

L. Brillouin, “Diffusion de la lumière par un corps transparent homogène,” Ann. Phys. 17, 88–122 (1922).

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Buettner, T. F. S.

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

Chiao, R. Y.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592 (1964).
[Crossref]

Choi, D.-Y.

Cohen, P. I.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83, 195203 (2011).
[Crossref]

Cox, J. A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Debbarma, S.

Dong, H.

Eggleton, B. J.

C. Wolff, P. Gutsche, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Impact of nonlinear loss on Stimulated Brillouin Scattering,” J. Opt. Soc. Am. B 32, 1968–1978 (2015).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin Scattering in integrated photonic waveguides: forces, scattering mechanisms and coupled mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

I. Aryanfar, C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Mode conversion using stimulated Brillouin scattering in nanophotonic silicon waveguides,”; Opt. Express 22, 29270–29282 (2014).
[Crossref] [PubMed]

I. V. Kabakova, R. Pant, D.-Y. Choi, S. Debbarma, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Narrow linewidth Brillouin laser based on chalcogenide photonic chip,” Opt. Lett. 38, 3208–3211 (2013).
[Crossref] [PubMed]

R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thévenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
[Crossref] [PubMed]

Evans, R.A.

J.J. Wortman and R.A. Evans, “Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon and Germanium,” J. Appl. Phys. 36, 153 (1965).
[Crossref]

Fan, S.

Feldman, A.

A. Feldman, R.M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978).
[Crossref]

Feldman, L. C.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83, 195203 (2011).
[Crossref]

Gregory, J. M.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83, 195203 (2011).
[Crossref]

Gutsche, P.

Helme, B.G.

B.G. Helme and P.J. King, “The Phonon Viscosity Tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Solidi (A) 45, K33 (1978)
[Crossref]

Hile, S.

Horowitz, D.

A. Feldman, R.M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589 (1978).
[Crossref]

Huang, X.

Hudson, D. D.

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

Jarecki, R.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Judge, A.

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

Kabakova, I. V.

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

I. V. Kabakova, R. Pant, D.-Y. Choi, S. Debbarma, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Narrow linewidth Brillouin laser based on chalcogenide photonic chip,” Opt. Lett. 38, 3208–3211 (2013).
[Crossref] [PubMed]

Kang, M. S.

M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
[Crossref]

King, P.J.

B.G. Helme and P.J. King, “The Phonon Viscosity Tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Solidi (A) 45, K33 (1978)
[Crossref]

Kuyken, B.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hyper-sound in a silicon photonic nanowire,” Nature Photon. 9, 199 (2015).
[Crossref]

Lee, H.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

Li, E.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thévenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
[Crossref] [PubMed]

Li, J.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

Luther-Davies, B.

Madden, S.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Madden, S. J.

Marpaung, D.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Marti, J.

Mcfarlane, H.

Morrison, B.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Nazarkin, A.

M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
[Crossref]

Olsson Ill, R. H.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Pant, R.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

T. F. S. Buettner, I. V. Kabakova, D. D. Hudson, R. Pant, C. G. Poulton, A. Judge, and B. J. Eggleton, “Phase-locking in Multi-Frequency Brillouin Oscillator via Four Wave Mixing,” Sci. Rep. 4, 5032 (2014).

I. V. Kabakova, R. Pant, D.-Y. Choi, S. Debbarma, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Narrow linewidth Brillouin laser based on chalcogenide photonic chip,” Opt. Lett. 38, 3208–3211 (2013).
[Crossref] [PubMed]

R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thévenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
[Crossref] [PubMed]

Pearl, S.

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300–3300 nm,” Appl. Phys. Lett. 93, 131102 (2008).
[Crossref]

Piqueras, M. A.

Poulton, C. G.

Premaratne, M.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

Qiu, W.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[Crossref]

Rakich, P. T.

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[Crossref]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Commun. 4, 1944 (2013).
[Crossref] [PubMed]

Rotenberg, N.

S. Pearl, N. Rotenberg, and H. M. van Driel, “Three photon absorption in silicon for 2300–3300 nm,” Appl. Phys. Lett. 93, 131102 (2008).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. Van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Rukhlenko, I. D.

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear Silicon Photonics: Analytical Tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

Russell, J.

M. S. Kang, A. Nazarkin, A. Brenn, P. St, and J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Ramanoscillators, ”; Nat. Phys. 5, 276–280 (2009).
[Crossref]

Seo, D.

D. Seo, J. M. Gregory, L. C. Feldman, N. H. Tolk, and P. I. Cohen, “Multiphoton absorption in germanium using pulsed infrared free-electron laser radiation,” Phys. Rev. B 83, 195203 (2011).
[Crossref]

Shin, H.

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Figures (3)

Fig. 1
Fig. 1

Optimal pump power (left panel) and optimal waveguide length (right panel) as functions of the figure of merit ℱ [see Eq. (12)]; numerical results (black solid lines) in comparison to the approximate expressions Eq. (14) and Eq. (13) (red dashed lines) The optimal pump power is between P / 2 and P and the optimal waveguide length is of order α−1.

Fig. 2
Fig. 2

The maximally realisable Stokes amplification for FCA-dominated setups as a function of the figure of merit ℱ [see Eq. (12)] computed numerically (black solid line) and approximate fit according to Eq. (15) (red dashed line). Materials and waveguide designs with ℱ < 1 cannot amplify an injected Stokes wave.

Fig. 3
Fig. 3

Waveguide geometries and effective indices of the optical modes that are studied for backward SBS in Table 2. The color-plot depicts the modulus of the modal electric field in arbitrary units, the black arrows indicate the in-plane electric field components. The waveguides consist of silicon in [100]-orientation; their cross-sections are shown as a thin black rectangle and are scaled with the light’s vacuum wavelength λ0.

Tables (2)

Tables Icon

Table 1 Table of nonlinear coefficients (Γ, β, γ), acoustic frequency Ω/2π, Brillouin line width ΔΩ/2π, SBS-figure of merit ℱ, maximally realisable Stokes amplification A ( max ) and the natural unit of power P (four times the max. SBS-laser output power in case of ℱ > 1) for bulk silicon and germanium at wavelengths on the red and the blue side of the 2PA-threshold. We assumed a linear loss of α = 0.1 dB/cm and a carrier life time of 10ns throughout. The quantities Γ and Ω were computed from literature expressions [20] assuming wave propagation along the [100]-direction and purely electrostrictive coupling based on literature photoelastic coefficients [24, 25]. The Brillouin line width was computed as described in the main text using literature values for the dynamic viscosity [21]. The bulk 2PA-, 3PA- and FCA-coefficients are literature values [26–28] (annotated in the square brackets). As mentioned in the main text, this table lists bulk parameters, which are naturally expressed as power intensities (units of W/m2) rather than powers; this is reflected in the units for A, β, γ and P and P(opt), which differ from those in Table 2. Note that the photo-elastic tensor and the dynamic viscosity are available in the literature only for selected wavelengths, so we assumed them to be nondispersive, leading to a wavelength-independent prediction of the SBS-gain.

Tables Icon

Table 2 Table of SBS-resonance parameters, loss parameters, SBS-figure of merit, maximally realisable Stokes-amplification and optimal operating conditions for two simple suspended nanowire designs (see Fig. 3) operated in backward SBS configuration in the 2PA-regime. The effective waveguide coefficients were computed in analogy to and using material parameters listed in Table 1, yet using the specified optical and best-suited acoustic eigenmodes. The SBS-gain coefficient includes both electrostriction and radiation pressure terms as described in the literature [20]. Furthermore, we show data for an experimentally studied geometry [9], where we adopted the SBS-parameters (gain, shift, linewidth), the linear loss and the carrier life time published for that particular structure and computed the nonlinear loss and from this ℱ, P and and extrapolated A ( max ). Optimal operation conditions cannot be provided, because this structure does not provide net Stokes amplification.

Equations (23)

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s d d z P ( 1 ) = ( Γ 2 β γ P ( 2 ) ) P ( 2 ) P ( 1 ) α P ( 1 ) + C 1 ,
d d z P ( 2 ) = ( β + γ P ( 2 ) ) [ P ( 2 ) ] 2 α P ( 2 ) + C 2 ,
C 1 = ( β + γ P ( 1 ) + 4 γ P ( 2 ) ) [ P ( 1 ) ] 2 ,
C 2 = ( 2 β + Γ + 4 γ P ( 2 ) + γ P ( 1 ) ) P ( 1 ) P ( 2 )
s d d z P ( 1 ) = ( Γ 2 β γ P ( 2 ) ) P ( 2 ) P ( 1 ) α P ( 1 ) ( β + γ P ( 1 ) + 4 γ P ( 2 ) ) [ P ( 1 ) ] 2 ,
s d d z P ( 1 ) < [ Γ 2 β 4 γ P ( 1 ) γ P ( 2 ) ] P ( 1 ) P ( 2 )
P ( 1 ) < Γ 2 β 4 γ P ( 2 ) 4 γ < P 4 ,
P = ( Γ 2 β ) / γ ;
Λ = γ / ( Γ 2 β ) 2 .
A ( L , P 0 ) = 10 log 10 [ P ( 1 ) ( L ) P ( 1 ) ( 0 ) ] dB
= 10 ln 10 { 1 α Λ [ tan 1 ( 1 + ψ ) exp ( 2 α L ) 1 tan 1 ψ ] 1 2 ln [ ( 1 + ψ ) exp ( 2 α L ) L ψ ] } dB ,
P ( 2 ) = 1 2 γ [ ( Γ 2 β ) ± ( 2 β Γ ) 2 4 α γ ] > 0 ; Γ 2 β 2 α γ > 1.
= Γ 2 β 2 α γ
L ( opt ) ( 2 ln ) 0.713 α 1 ,
P ( opt ) ( 1 0.25 2 0.25 6 ) P ,
A ( max ) 13.0 [ ( 1 ) + 3.0 3.0 ] dB .
s d d z P ( 1 ) = ( Γ 2 β 12 γ 122 P ( 2 ) ) P ( 2 ) P ( 1 ) α 1 P ( 1 ) small signal terms ( β 11 γ 112 P ( 1 ) + 4 γ 111 P ( 2 ) ) [ P ( 1 ) ] 2 large signal corrections ,
d d z P ( 2 ) = ( β 22 + γ 222 P ( 2 ) ) [ P ( 2 ) ] 2 α 2 P ( 2 ) small signal terms ( 2 β 21 + Γ + γ 221 P ( 2 ) + γ 211 P ( 1 ) ) P ( 1 ) P ( 2 ) large signal corrections .
α i = 2 ε 0 ω P ( i ) d 2 r | e ˜ ( i ) | 2 { ε r } ,
β i j = 2 P ( i ) P ( j ) d 2 r ( | e ˜ ( i ) e ˜ ( j ) | 2 + | e ˜ ( i ) ( e ˜ ( j ) ) * | 2 + | e ˜ ( i ) | 2 | e ˜ ( j ) | 2 ) 2 PA ,
γ i j k = 2 P ( i ) P ( j ) P ( k ) d 2 r | e ˜ ( i ) | 2 [ | e ˜ ( j ) e ˜ ( k ) | 2 + | e ˜ ( j ) ( e ˜ ( k ) ) * | 2 + | e ˜ ( j ) | 2 | e ˜ ( k ) | 2 ] FCA .
P ( 1 ) < Γ 2 β 12 4 γ 112 .
= Γ 2 β 12 2 α 1 γ 122 .

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