Abstract

We theoretically investigate lasing due to stimulated Brillouin scattering in integrated ring resonators. We give analytic expressions and numerical calculations for the lasing threshold for rings in the presence of for both linear and nonlinear loss. We demonstrate the operation of the ring in the different regimes of amplification and lasing, and show how these regimes depend on the coupling to the ring and on the nonlinear parameters. In the case of nonlinear losses, we find that there can exist an upper threshold to the lasing regime where the losses are dominated by free-carrier absorption. We also find that nonlinear losses can inhibit Brillouin lasing entirely for certain ranges of coupling parameters, and we show how the correct ranges of coupling parameters can be calculated and optimized for the design of integrated Brillouin lasers.

© 2017 Optical Society of America

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References

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    [Crossref]
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2017 (2)

2016 (3)

2015 (3)

2014 (3)

J. Li, H. Lee, and K. J. Vahala, “Low-noise Brillouin laser on a chip at 1064 nm,” Opt. Lett. 39, 287–290 (2014).
[Crossref] [PubMed]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
[Crossref]

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

2013 (2)

2012 (2)

2009 (1)

2008 (1)

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

2007 (1)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

2006 (1)

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[Crossref]

2004 (2)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
[Crossref] [PubMed]

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded raman scattering in ultrahigh-q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004).
[Crossref]

2003 (1)

2002 (1)

S. Spillane, T. Kippenberg, and K. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

1996 (1)

1982 (1)

1972 (1)

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
[Crossref] [PubMed]

Andreani, L. C.

Azzini, S.

Baets, R.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photon 9, 199–203 (2015).
[Crossref]

Bajoni, D.

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
[Crossref] [PubMed]

Baynes, F. N.

Beugnot, J.-C.

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

Blake, M.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[Crossref]

Boyd, R. W.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Büttner, T. F.

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

Camacho, R.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 011008 (2012).

Casas-Bedoya, A.

B. Morrison, A. Casas-Bedoya, G. Ren, K. Vu, Y. Liu, A. Zarifi, T. G. Nguyen, D.-Y. Choi, D. Marpaung, S. J. Madden, A. Mitchell, and B. J. Eggleton, “Compact Brillouin devices through hybrid integration on silicon,” Optica 4, 847–854 (2017).
[Crossref]

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

Chembo, Y. K.

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

Chodorow, M.

Choi, D.-Y.

Chu, S.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Cole, D. C.

Davids, P.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 011008 (2012).

De Leonardis, F.

Debbarma, S.

Delâge, A.

Densmore, A.

Diallo, S.

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

Diddams, S. A.

Duchesne, D.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Eggleton, B. J.

B. Morrison, A. Casas-Bedoya, G. Ren, K. Vu, Y. Liu, A. Zarifi, T. G. Nguyen, D.-Y. Choi, D. Marpaung, S. J. Madden, A. Mitchell, and B. J. Eggleton, “Compact Brillouin devices through hybrid integration on silicon,” Optica 4, 847–854 (2017).
[Crossref]

S. R. Mirnaziry, C. Wolff, M. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated ring resonators,” J. Opt. Soc. Am. B 34, 937–949 (2017).
[Crossref]

S. R. Mirnaziry, C. Wolff, M. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in silicon/chalcogenide slot waveguides,” Opt. Express 24, 4786–4800 (2016).
[Crossref]

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[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]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
[Crossref]

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photon. 5, 536–587 (2013).
[Crossref]

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]

Ferrera, M.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Galli, M.

Gauthier, D. J.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

Geng, J.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[Crossref]

Grassani, D.

Green, A. A.

Gutsche, P.

Helt, L.

Janz, S.

Jiang, S.

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
[Crossref]

Kabakova, I. V.

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
[Crossref]

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]

Kim, B. Y.

Kippenberg, T.

S. Spillane, T. Kippenberg, and K. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Kippenberg, T. J.

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded raman scattering in ultrahigh-q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004).
[Crossref]

Kuyken, B.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photon 9, 199–203 (2015).
[Crossref]

Lee, H.

Li, J.

Lin, G.

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
[Crossref] [PubMed]

Liscidini, M.

S. Azzini, D. Grassani, M. Galli, L. C. Andreani, M. Sorel, M. J. Strain, L. Helt, J. Sipe, M. Liscidini, and D. Bajoni, “From classical four-wave mixing to parametric fluorescence in silicon microring resonators,” Opt. Lett. 37, 3807–3809 (2012).
[Crossref] [PubMed]

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Little, B.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Liu, Y.

Loh, W.

Luther-Davies, B.

Madden, S. J.

Marpaung, D.

B. Morrison, A. Casas-Bedoya, G. Ren, K. Vu, Y. Liu, A. Zarifi, T. G. Nguyen, D.-Y. Choi, D. Marpaung, S. J. Madden, A. Mitchell, and B. J. Eggleton, “Compact Brillouin devices through hybrid integration on silicon,” Optica 4, 847–854 (2017).
[Crossref]

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
[Crossref]

Martinenghi, R.

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

McKinnon, W. R.

Merklein, M.

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

Min, B.

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded raman scattering in ultrahigh-q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004).
[Crossref]

Mirnaziry, S. R.

Mitchell, A.

Morandotti, R.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Morrison, B.

B. Morrison, A. Casas-Bedoya, G. Ren, K. Vu, Y. Liu, A. Zarifi, T. G. Nguyen, D.-Y. Choi, D. Marpaung, S. J. Madden, A. Mitchell, and B. J. Eggleton, “Compact Brillouin devices through hybrid integration on silicon,” Optica 4, 847–854 (2017).
[Crossref]

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
[Crossref]

Moss, D.

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
[Crossref]

Nguyen, T. G.

Nikles, M.

Pagani, M.

M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
[Crossref] [PubMed]

Pant, R.

Papp, S. B.

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M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
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P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 011008 (2012).

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Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. Lin, S. Diallo, K. Saleh, R. Martinenghi, J.-C. Beugnot, T. Sylvestre, and Y. K. Chembo, “Cascaded Brillouin lasing in monolithic barium fluoride whispering gallery mode resonators,” Appl. Phys. Lett. 105, 231103 (2014).
[Crossref]

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

T. J. Kippenberg, S. M. Spillane, B. Min, and K. J. Vahala, “Theoretical and experimental study of stimulated and cascaded raman scattering in ultrahigh-q optical microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1219–1228 (2004).
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M. Merklein, A. Casas-Bedoya, D. Marpaung, T. F. Büttner, M. Pagani, B. Morrison, I. V. Kabakova, and B. J. Eggleton, “Stimulated Brillouin scattering in photonic integrated circuits: novel applications and devices,” IEEE J. Sel. Top. Quantum Electron. 22, 336–346 (2016).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Geng, S. Staines, Z. Wang, J. Zong, M. Blake, and S. Jiang, “Highly stable low-noise Brillouin fiber laser with ultranarrow spectral linewidth,” IEEE Photon. Technol. Lett. 18, 1813–1815 (2006).
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J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photon. Rev. 8, 653–666 (2014).
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Nat. Photon (1)

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photon 9, 199–203 (2015).
[Crossref]

Nat. Photon. (1)

M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. Sipe, S. Chu, B. Little, and D. Moss, “Low-power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nat. Photon. 2, 737–740 (2008).
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Nature (2)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on silicon chip,” Nature 431, 1081 (2004).
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S. Spillane, T. Kippenberg, and K. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
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Opt. Express (2)

Opt. Lett. (6)

Optica (2)

Phys. Rev. X (1)

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 011008 (2012).

Science (1)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

Other (2)

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

D. G. Rabus, Integrated Ring Resonators (Springer, 2007).

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Figures (9)

Fig. 1
Fig. 1

Schematic of a ring resonator in vicinity of a straight coupler.

Fig. 2
Fig. 2

Output Stokes power as a function of input pump power at the lasing region and resonant condition in the presence of (a) linear losses and (b) both linear and nonlinear losses. In (a) Γ = 500 W−1m−1, R = 10−11 and κ = 0.31. In (b) αL = 0.2; γ = 1.8 × 105 W−2m−1, β = 10 W−1m−1, κ = 0.16 and Γ = 4000 W−1m−1. The length L = 10.879 mm corresponds a ring resonator with free spectral range equal to a Brillouin frequency shift of 10 GHz.

Fig. 3
Fig. 3

(a) Schematic variation of the round-trip gain in a ring resonator with linear loss within the SSA and full model. (b) The corresponding Stokes total amplification of a ring resonator with the parameters described in Fig. 2(a). The solid circle lines show �� for the small signal model.

Fig. 4
Fig. 4

(a) Contours of the lasing threshold as a function of αL and the coupling coefficient. The dashed line shows the critical coupling. (b) The threshold difference Δ P p in , th between the power obtained by Eq. (13) with the threshold estimated from 15, plotted for a range of the coupling coefficient |κ| and for different values of αL. Δ P p in , th is normalized to the exact theoretical value of the threshold (i.e. Eq. (13)) and is plotted in percentage.

Fig. 5
Fig. 5

(a) Schematic variation of the round-trip gain in a ring resonator with nonlinear loss in three different operating regimes shown in pink (with two lasing thresholds), green (with single threshold) and blue (no lasing). Dashed lines show the result of the small signal model and the solid lines are expected in the full model. (b) An example of the Stokes output power for a ring with two lasing thresholds. The black dotted lines shows the SSA. The results of the full model are also shown for different values of the power ratio R. for a ring with the SBS gain and loss parameters described in Fig. 2(b). (c) The Stokes output power for a ring with parameters which leads to a single lasing threshold. Γ = 5970 W−1m−1; |κ| = 0.24; α = 40.9 m−1 and γ = 1.8 × 105 W−2m−1. (d) The Stokes output power in a ring with parameters that leads to only Stokes amplification. Γ = 5700 W−1m−1 and the loss and coupling parameters are as in (c).

Fig. 6
Fig. 6

Vmin and Vmax at the lasing threshold as a function of κ and αL for (a,b) = 1.1 and (c,d) = 1.5.

Fig. 7
Fig. 7

Minimum/Maximum values of the lasing threshold in mW as a function of κ and αL for = 1.1 (a,b) and = 1.5 (c,d) for α γ = 2 × 10 4 W 2.

Fig. 8
Fig. 8

Lasing thresholds as a function of the coupling coefficient for different values of . αL is assumed to be 0.3 and α/γ = 2 × 10−4 W2.

Fig. 9
Fig. 9

The normalized input pump power corresponding to the maximum Stokes output in the lasing regime for a range of αL and SBS figure of merit. |τ| is assumed to be 0.9. (b) The maximum Stokes output in mW. The initial Stokes is assumed to be 1 pW.

Equations (24)

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d P p d z = ( α + β P p + γ P p 2 ) P p ( 2 β + 4 γ P p + γ P s + Γ ) P p P s ,
d P s d z = ( α + β P p + γ P s 2 ) P s + ( 2 β + 4 γ P s + γ P p Γ ) P p P s ,
P p in = 1 | κ | 2 ( P p ( 0 ) | τ | P p ( L ) ) 2 ,
P s in = 1 | κ | 2 ( P s ( L ) | τ | P s ( 0 ) ) 2 .
R = P s in P p in = v g h f s P p in ,
d P p ( z ) d z = ( α + β P p ( z ) + γ P p 2 ( z ) ) P p ( z ) ,
d P s ( z ) d z = ( α + γ P p 2 ( z ) ( Γ 2 β ) P p ( z ) ) P s ( z ) .
G = P s ( 0 ) P s ( L ) .
𝒜 : = P s out P s in = | | | τ | G | 1 | τ | G | 2 ,
G = exp [ 1 2 0 L g ( z ) d z ] .
| τ | exp [ 1 2 0 L g ( z ) d z ] = 1 .
G = exp [ α L 2 + Γ 2 α P p ( 0 ) ( 1 e α L ) ] .
P p in , th = ( α L 2 ln | τ | ) ( 1 e α L ) α Γ ( 1 | α | e ( α L / 2 ) ) 2 | κ | 2 ,
Q L = 2 π L n eff 2 λ 1 + | τ | 2 e α L 1 | τ | e α L 2 ,
P p in , th = π 2 n eff 2 λ 2 L Γ Q c 1 Q L 3 ,
Q c = 2 π L n eff 2 λ 1 + τ 2 1 τ .
P p ( z ) = α γ V ( 1 + V 2 ) e 2 α z V 2 .
G = ( 1 V 2 + 1 V 2 e 2 α L ) 1 4 exp [ α L 2 arctan ( V V ( V 2 + 1 ) e 2 α L V 2 V 2 + ( V 2 + 1 ) e 2 α L V 2 ) ] ,
P p in , th = α γ V th | κ | ( 1 | τ | 1 ( ( V th ) 2 + 1 ) e 2 α L ( V th ) 2 4 ) ,
T p = ( | τ | e α L 2 ) 2 + | τ | e α L 2 ( Δ θ 2 ) 2 ( 1 | τ | e α L 2 ) 2 + | τ | e α L 2 ( Δ θ 2 ) 2 ,
T p = 1 2 ( T max + T min ) ,
T max = ( | τ | + e α L 2 ( 1 + | τ | e α L 2 ) ) 2 , T min = ( | τ | e α L 2 1 | τ | e α L 2 ) 2 .
Δ θ FWHM = 2 2 ( 1 1 | τ | e α L 2 ) 1 + | τ | 2 e α L .
Q L = ω 0 Δ ω FWHM = 2 π L n eff 2 λ 1 + | τ | 2 e α L 1 | τ | e α L 2 ,

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