Abstract

We propose a new approach to realize all-optical circulator based on stimulated Brillouin scattering in an integrated microresonator. Stimulated Brillouin scattering is a basic interaction between photon and traveling acoustic wave resulted from electrostriction and photoelastic effects. Due to the phase-matching requirement, the circulating acoustic wave can only couple to probe light which propagating along or opposite to the pump laser direction, thus exhibits a non-reciprocal phase shift. Combined with Mach-Zehnder interferometer, the optical circulator can be realized. Though the bandwidth is relatively small because of the narrow-band nature of microresonator, this magnetic-free all-optical integrated circulator may be applied for future on-chip photonic information processing.

© 2015 Optical Society of America

Full Article  |  PDF Article
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  5. A. F. Koenderin, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
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  6. Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
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  7. C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
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    [Crossref]
  11. W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photon. Technol. Lett. 11, 1012–1014 (1999).
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  12. C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE Trans. Microw. Theory Techn. 13, 15–27 (1965).
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  14. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
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  15. J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
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  16. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
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  21. M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nature Photon. 5, 549–553(2011).
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    [Crossref]
  24. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
    [Crossref] [PubMed]
  25. J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012)
    [Crossref]
  26. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
    [PubMed]
  27. C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
    [Crossref] [PubMed]
  28. J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
    [Crossref]
  29. J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
    [Crossref] [PubMed]
  30. B. Eggleton, C. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photon. 5, 536–587 (2013).
    [Crossref]
  31. P. 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).
  32. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson, A Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale Silicon waveguides,” Nat. Commun. 4, 1944 (2013).
    [Crossref]
  33. I. Kabakova, D. Marpaung, C. Poulton, and B. Eggleton, “Harnessing on-chip SBS,” Opt. Photon. News 26, 34 (2015).
    [Crossref]
  34. C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995).
    [Crossref]
  35. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  36. A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
    [Crossref]

2015 (3)

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

I. Kabakova, D. Marpaung, C. Poulton, and B. Eggleton, “Harnessing on-chip SBS,” Opt. Photon. News 26, 34 (2015).
[Crossref]

2014 (3)

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal devices in silicon photonics,” Sci. Technol. Adv. Mater. 15, 014602 (2014).
[Crossref]

2013 (3)

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

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

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

2012 (4)

P. 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).

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
[Crossref] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012)
[Crossref]

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

2011 (3)

2010 (3)

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
[Crossref]

2009 (6)

A. F. Koenderin, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
[Crossref]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
[Crossref] [PubMed]

2008 (1)

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

2007 (2)

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007).
[Crossref]

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

2000 (1)

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

1999 (1)

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photon. Technol. Lett. 11, 1012–1014 (1999).
[Crossref]

1995 (1)

C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995).
[Crossref]

1991 (1)

Y. Fujii, “High-isolation polarization-independent optical circulator,” J. Lightwave Technol. 9, 1238–1243 (1991).
[Crossref]

1965 (1)

C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE Trans. Microw. Theory Techn. 13, 15–27 (1965).
[Crossref]

Ando, K.

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photon. Technol. Lett. 11, 1012–1014 (1999).
[Crossref]

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Benson, O.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Beugnot, J. C.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Boyd, R. W.

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

Butsch, A.

M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nature Photon. 5, 549–553(2011).
[Crossref]

Camacho, R.

P. 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).

Chen, T.

Chen, X. D.

Chen, Y.

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
[Crossref]

Coldren, L. A.

L. A. Coldren and S. W. Corzine., Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Comstock, R. L.

C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE Trans. Microw. Theory Techn. 13, 15–27 (1965).
[Crossref]

Corzine., S. W.

L. A. Coldren and S. W. Corzine., Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Cox, J. A.

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

Crespi, A.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Cui, J. M.

Davids, P.

P. 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).

Dong, C.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Dong, C. H.

Dtsch, H.

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Eggleton, B.

Fan, S.

X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

Fay, C. E.

C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE Trans. Microw. Theory Techn. 13, 15–27 (1965).
[Crossref]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Fu, W.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Fujii, Y.

Y. Fujii, “High-isolation polarization-independent optical circulator,” J. Lightwave Technol. 9, 1238–1243 (1991).
[Crossref]

Fujita, J.

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
[Crossref]

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Gotzinger, S.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
[Crossref] [PubMed]

Guo, G.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Guo, G. C.

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[Crossref] [PubMed]

Guo, X.

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007).
[Crossref]

Hafezi, M.

Han, K.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Han, Z. F.

Hsieh, I.

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

Huang, X.

Jarecki, R.

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

Kabakova, I.

I. Kabakova, D. Marpaung, C. Poulton, and B. Eggleton, “Harnessing on-chip SBS,” Opt. Photon. News 26, 34 (2015).
[Crossref]

Kang, M. S.

M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nature Photon. 5, 549–553(2011).
[Crossref]

Kim, J.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Koenderin, A. F.

A. F. Koenderin, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
[Crossref]

Koenderink, A. F.

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
[Crossref]

Kuzyk, M. C.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Laude, V.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Lebrun, S.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Lee, H.

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

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012)
[Crossref]

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Levy, M.

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Li, J.

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

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012)
[Crossref]

Lipson, M.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Liu, J.-M.

J.-M. Liu, Photonic Devices (Cambridge University, 2005)
[Crossref]

Lodahl, P.

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
[Crossref]

Maillotte, H.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Maleki, L.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
[Crossref] [PubMed]

Marpaung, D.

I. Kabakova, D. Marpaung, C. Poulton, and B. Eggleton, “Harnessing on-chip SBS,” Opt. Photon. News 26, 34 (2015).
[Crossref]

Mataloni, P.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Matsko, A. B.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
[Crossref] [PubMed]

Matthews, J. C. F.

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[Crossref]

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

Mazzei, A.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Menezes, L. de S.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Mizumoto, T.

Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal devices in silicon photonics,” Sci. Technol. Adv. Mater. 15, 014602 (2014).
[Crossref]

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

O’Brien, J. L.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
[Crossref]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[Crossref]

Olsson, R. H.

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

Osellame, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Osgood, R. M.

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Pant, R.

Pauliat, G.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Politi, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[Crossref]

Poulton, C.

Qiu, W.

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

Rabl, P.

Rakich, P.

P. 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).

Rakich, P. T.

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

Ram, R. J.

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007).
[Crossref]

Ramponi, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Reed, G. T.

C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995).
[Crossref]

Reinke, C.

P. 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).

Ren, X. F.

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[Crossref] [PubMed]

Russell, P. St. J.

M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nature Photon. 5, 549–553(2011).
[Crossref]

Sandoghdar, V.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Sansoni, L.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Sciarrino, F.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Shen, Z.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Shin, H.

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

Shoji, Y.

Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal devices in silicon photonics,” Sci. Technol. Adv. Mater. 15, 014602 (2014).
[Crossref]

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

Starbuck, A

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

Stefanov, A.

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

Sun, F. W.

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[Crossref] [PubMed]

Sylvestre, T.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

Tang, C. K.

C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995).
[Crossref]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Vahala, K. J.

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

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express 20, 20170–20180 (2012)
[Crossref]

Vallone, G.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
[Crossref]

Wang, H.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Wang, Z.

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

P. 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).

Wilkens, L.

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Xiong, X.

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

Yokoi, H.

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

Yu, Z.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

Zaets, W.

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photon. Technol. Lett. 11, 1012–1014 (1999).
[Crossref]

Zaman, T. R.

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007).
[Crossref]

Zhang, Y.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Zou, C.

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

Zou, C. L.

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[Crossref] [PubMed]

Zumofen, G.

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

Adv. Opt. Photon. (1)

App. Phys. Lett. (1)

T. R. Zaman, X. Guo, and R. J. Ram, “Faraday rotation in an InP waveguide,” App. Phys. Lett. 90, 023514 (2007).
[Crossref]

Appl. Phys. Lett. (3)

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dtsch, “Waveguide optical isolator based on Mach–Zehnder interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
[Crossref]

Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

X. Guo, C. L. Zou, X. F. Ren, F. W. Sun, and G. C. Guo, “Broadband opto-mechanical phase shifter for photonic integrated circuits,” Appl. Phys. Lett. 101, 071114 (2012).
[Crossref]

Electron. Lett. (1)

C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451–452 (1995).
[Crossref]

IEEE Photon. Technol. Lett. (2)

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photon. Technol. Lett. 11, 1012–1014 (1999).
[Crossref]

X. Xiong, C. L. Zou, X. F. Ren, and G. C. Guo, “Broadband plasmonic absorber for photonic integrated circuits,” IEEE Photon. Technol. Lett. 26, 1726–1729 (2014).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

C. E. Fay and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE Trans. Microw. Theory Techn. 13, 15–27 (1965).
[Crossref]

J. Lightwave Technol. (2)

Y. Fujii, “High-isolation polarization-independent optical circulator,” J. Lightwave Technol. 9, 1238–1243 (1991).
[Crossref]

X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
[Crossref]

Nano Lett. (1)

A. F. Koenderin, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
[Crossref]

Nat. Commun. (4)

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

C. Dong, Z. Shen, C. Zou, Y. Zhang, W. Fu, and G. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylvestre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5, 5242 (2014).
[Crossref] [PubMed]

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

Nature Photon. (5)

J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photon. 3, 346–350(2009).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vuckovic, “Photonic quantum technologies,” Nature Photon. 3, 687–695 (2009).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nature Photon. 4, 37–40 (2010).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon. 3, 91–94 (2009).
[Crossref]

M. S. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nature Photon. 5, 549–553(2011).
[Crossref]

Nature Phys. (1)

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nature Phys. 11, 275–280 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Opt. Photon. News (1)

I. Kabakova, D. Marpaung, C. Poulton, and B. Eggleton, “Harnessing on-chip SBS,” Opt. Photon. News 26, 34 (2015).
[Crossref]

Phys. Rev. B (1)

Y. Chen, P. Lodahl, and A. F. Koenderink, “Dynamically reconfigurable directionality of plasmon-based single photon sources,” Phys. Rev. B 82, 081402 (2010).
[Crossref]

Phys. Rev. Lett. (3)

A. Mazzei, S. Gotzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled coupling of counter-propagating whispering-gallery modes by a single Rayleigh scatterer: a classical problem in a quantum optical light,” Phys. Rev. Lett. 99, 173603 (2007).
[Crossref]

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009)
[Crossref] [PubMed]

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled states measurement on a chip,” Phys. Rev. Lett. 105, 200503(2010).
[Crossref]

Phys. Rev. X (1)

P. 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).

Sci. Technol. Adv. Mater. (1)

Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal devices in silicon photonics,” Sci. Technol. Adv. Mater. 15, 014602 (2014).
[Crossref]

Science (1)

A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221 (2009).
[Crossref]

Other (3)

L. A. Coldren and S. W. Corzine., Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

J.-M. Liu, Photonic Devices (Cambridge University, 2005)
[Crossref]

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

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Figures (4)

Fig. 1
Fig. 1 Schematic illustration of a integrated all-optical circulator: a whispering-gallery microresonator is coupling to one arm of a four-port Mach-Zehnder interferometer. The microresonator enables non-reciprocal SBS, which is serving as non-reciprocal phase shifter. The phase modulator on the upper arm can modulate paths lenth difference of two arm.
Fig. 2
Fig. 2 (a). Non-reciprocal phase shifter: A strong control laser in the forward direction causes the difference of phase shift between forward and backward laser. Transmission and phase shift of the nonreciprocal phase shifter for forward and backward direction in critically coupled regime (b) and in over-coupled regime (c) with C = 100 and Γ = 0.02κ 0.
Fig. 3
Fig. 3 (a):Transmittance of this interfering system when operates as an optical circulator, and for this plot κex = 100κ 0, C = 1000. (b) The circulator bandwidth of 30 dB is approximately linear to the C factor. (c) The circulator bandwidth of 30 dB and loss after light intensity of two arm are balanced.
Fig. 4
Fig. 4 (a) The transmittance for our optical circulator when backscattering exists. For this plot β = 1 and C = 1000. (b) Contour for the dip in the transmission spectra of light 4-1 and 2–3, which is the most prominent impact of backscattering. (c) Probe light is backscattered and propagates back and for this plot C = 1000.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

E out = M E in ,
M 21 ( 43 ) = 1 2 [ 1 + T f e i ϕ f ]
M 12 ( 34 ) = 1 2 [ 1 + T b e i ϕ b ]
M 41 ( 23 ) = 1 2 [ 1 T f e i ϕ f ]
M 12 ( 34 ) = 1 2 [ 1 T b e i ϕ b ] .
M ideal = ( 0 0 0 a b 0 0 0 b c 0 0 0 0 d 0 ) ,
H = H modes + H int + H in ,
H modes = ω 1 a c , f a c , f + ω 2 a p , f a p , f + ω 2 a p , b a p , b + Ω b b ,
H int = g ( a p , f b a c , f + a p , f b a c , f ) ,
H in = i κ ex E c , f , in ( a c , f e i ω c t a c , f e i ω c t ) + i κ ex E p , f , in ( a p , f e i ω p t a p , f e i ω p t ) + i κ ex E p , b , in ( a p , b e i ω p t a p , b e i ω p t ) .
d d t a c , f = ( i ω 1 κ / 2 ) a c , f i g a p , f b + κ e x E c , f , in e i ω c t .
a c , f = N c e i ω c t = | κ ex E c , f , in i Δ 1 κ / 2 | e i ω c t .
H ˜ = Δ 1 a c , f a c , f Δ 2 a p , f a p , f Δ 2 a p , b a p , b δ ω b b + g N c ( a p , f b + a p , f b ) + i κ ex E p , b , in ( a p , b a p , b ) + i κ ex E p , f , in ( a p , f a p , f ) .
d d t a p , f = [ i Δ 2 κ / 2 ] a p , f i g N c b + κ ex E p , f , in ,
d d t a p , b = [ i Δ 2 κ / 2 ] a p , b + κ ex E p , b , in ,
d d t b = [ i δ ω Γ / 2 ] b i g N c a p , f ,
a p , f = κ ex E p , f , in i Δ 2 κ / 2 + g 2 N c i Δ 2 Γ / 2 .
a p , b = κ ex E p , b , in i Δ 2 κ / 2 .
E p , out = E p , in κ ex a p .
ϕ = arg ( E p , out E p , in ) ,
T = | E p , out E p , in | 2 .
E out ( Δ ) = 1 2 E p , in [ L 1 / 2 T 1 / 2 ( Δ ) e ϕ f ( Δ ) ] .
E out ( Δ ) = E out ( 0 ) + E out ( 0 ) Δ + o ( Δ ) E out ( 0 ) Δ .
| M 41 ( 23 ) | 2 = 10 3 | M 14 ( 32 ) | 2 .
Bandwidth = 10 3 / 2 ( 1 + C ) 2 Γ C
H ˜ op = H op + ω 1 a c , b a c , b
H ˜ mech = Ω b f b f + Ω b b b b
H ˜ int = H int + β a c , f a c , b + β * a c , f a c , b + β a p , f a p , b + β * a p , f a p , b + g ( a p , b b b a c , b + a p , b b c , b a c , b )
d d t a c , f = ( i Δ 1 κ / 2 ) a c , f i g a p , f b f + κ ex E c , f , in i β * a c , b
d d t a p , f = ( i Δ 2 κ / 2 ) a p , f i g a c , f b f + κ ex E p , f , in i β * a p , b
d d t a c , b = ( i Δ 1 κ / 2 ) a c , b i g a p , b b b i β a c , f
d d t a p , b = [ i Δ 2 κ / 2 ] a p , f i g a c , b b b + κ ex E p , b , in i β a p , f
d d t b f = [ i δ ω Γ / 2 ] b f i g a p , f a c , f
d d t b b = [ i δ ω Γ / 2 ] b b i g a p , b a c , b .

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