Abstract

Stimulated Brillouin scattering (SBS) is a third-order nonlinear process that involves the interaction of two light fields and an acoustic wave in a medium. It has been exploited for applications of optical communication, sensing, and signal processing. This effect, originally demonstrated in long optical fibers, has recently been realized in silicon waveguides on a chip-scale integrated platform. However, due to the weak per-unit-length SBS gain, the length of the silicon waveguides is usually several centimeters, which prevents device miniaturization for high-density integration. Here, we engineer a phoxonic crystal waveguide structure to achieve significantly enhanced SBS gain in the entire C band, by taking advantage of its simultaneous confinement of slow propagating optical and acoustic waves. The resulting SBS gain coefficient is greater than 3 × 104 W−1 m−1 in the wavelength range of 1520–1565 nm with the highest value beyond 106 W−1 m−1, which is at least an order of magnitude higher than the existing demonstrations. This giant enhancement of SBS gain enables ultracompact and high-performance SBS-based integrated optoelectronic devices such as Brillouin lasers, amplifiers, and signal processors.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2016 (3)

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
[Crossref]

R. Zhang, G. Chen, and J. Sun, “Analysis of acousto-optic interaction based on forward stimulated Brillouin scattering in hybrid phononic-photonic waveguides,” Opt. Express 24(12), 13051–13059 (2016).
[Crossref] [PubMed]

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
[Crossref]

2015 (4)

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(1), 013836 (2015).
[Crossref]

C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. 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,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
[Crossref]

2014 (3)

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. M. Escalante, A. Martínez, and V. Laude, “Design of single-mode waveguides for enhanced light-sound interaction in honeycomb-lattice silicon slabs,” J. Appl. Phys. 115(6), 064302 (2014).
[Crossref]

G. Chen, R. Zhang, J. Sun, H. Xie, Y. Gao, D. Feng, and H. Xiong, “Mode conversion based on forward stimulated Brillouin scattering in a hybrid phononic-photonic waveguide,” Opt. Express 22(26), 32060–32070 (2014).
[Crossref] [PubMed]

2013 (2)

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

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] [PubMed]

2012 (4)

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(1), 011008 (2012).
[Crossref]

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
[Crossref]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

2011 (2)

2010 (4)

2009 (3)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[Crossref]

2008 (1)

2006 (1)

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

2004 (1)

T. Schneider, M. Junker, and D. Hannover, “Generation of millimetre-wave signals by stimulated Brillouin scattering for radio over fibre systems,” Electron. Lett. 40(23), 1500–1502 (2004).
[Crossref]

2002 (2)

T. Tanemura, Y. Takushima, and K. Kikuchi, “Narrowband optical filter, with a variable transmission spectrum, using stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(17), 1552–1554 (2002).
[Crossref] [PubMed]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref] [PubMed]

2001 (1)

2000 (1)

T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
[Crossref]

1999 (1)

1987 (1)

N. Olsson and J. V. D. Ziel, “Characteristics of a semiconductor laser pumped brillouin amplifier with electronically controlled bandwidth,” J. Lightwave Technol. 5(1), 147–153 (1987).
[Crossref]

1983 (1)

1974 (1)

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
[Crossref]

Adibi, A.

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. Photonics 9(3), 199–203 (2015).
[Crossref]

Bahl, G.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
[Crossref]

Benchabane, S.

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Beugnot, J. C.

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Beugnot, J.-C.

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

Biegelsen, D. K.

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32(21), 1196–1199 (1974).
[Crossref]

Braje, D.

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

Brenn, A.

M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
[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(1), 011008 (2012).
[Crossref]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[Crossref] [PubMed]

Chen, G.

Chiu, C.-C.

F.-L. Hsiao, C.-Y. Hsieh, H.-Y. Hsieh, and C.-C. Chiu, “High-efficiency acousto-optical interaction in phoxonic nanobeam waveguide,” Appl. Phys. Lett. 100(17), 171103 (2012).
[Crossref]

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] [PubMed]

Dainese, P.

P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
[Crossref]

Damzen, M. J.

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(1), 011008 (2012).
[Crossref]

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
[Crossref] [PubMed]

Deymier, P. A.

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

Diddams, S.

D. Braje, L. Hollberg, and S. Diddams, “Brillouin-enhanced hyperparametric generation of an optical frequency comb in a monolithic highly nonlinear fiber cavity pumped by a cw laser,” Phys. Rev. Lett. 102(19), 193902 (2009).
[Crossref] [PubMed]

Djafari-Rouhani, B.

S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
[Crossref]

Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
[Crossref]

V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
[Crossref] [PubMed]

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
[Crossref]

Dobrzynski, L.

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
[Crossref]

Dong, C.-H.

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

Dridi, K. H.

T. Søndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B 61(23), 15688–15696 (2000).
[Crossref]

Dupont, S.

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J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
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V. Laude, J.-C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, J. M. Escalante, and A. Martinez, “Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs,” Opt. Express 19(10), 9690–9698 (2011).
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V. Laude, J. C. Beugnot, S. Benchabane, Y. Pennec, B. Djafari-Rouhani, N. Papanikolaou, and A. Martinez, “Design of waveguides in silicon phoxonic crystal slabs,” in 2010 IEEE International Ultrasonics Symposium(2010), pp. 527–530.
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Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J. C. Kastelik, G. Lévêque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101(6), 061109 (2012).
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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).
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M. S. Kang, A. Nazarkin, A. Brenn, and P. S. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5(4), 276–280 (2009).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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Safavi-Naeini, A. H.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
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A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112(15), 153603 (2014).
[Crossref] [PubMed]

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
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E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
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R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
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Van Thourhout, D.

R. Van Laer, B. Kuyken, D. Van Thourhout, and R. Baets, “Interaction between light and highly confined hypersound in a silicon photonic nanowire,” Nat. Photonics 9(3), 199–203 (2015).
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Vasseur, J. O.

Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
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Wang, H.

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
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Wang, J.

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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).
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P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
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S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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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(1), 013836 (2015).
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Xiong, H.

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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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APL Photonics (1)

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1(7), 071301 (2016).
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Appl. Phys. Lett. (3)

J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter, “Optimized optomechanical crystal cavity with acoustic radiation shield,” Appl. Phys. Lett. 101(8), 081115 (2012).
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C.-H. Dong, Z. Shen, C.-L. Zou, Y.-L. Zhang, W. Fu, and G.-C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
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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).
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E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10(7), 463–467 (2016).
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[Crossref]

Nat. Phys. (3)

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11(3), 275–280 (2015).
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P. Dainese, P. S. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2(6), 388–392 (2006).
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Nature (1)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
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Opt. Express (9)

T. Sakamoto, T. Yamamoto, K. Shiraki, and T. Kurashima, “Low distortion slow light in flat Brillouin gain spectrum by using optical frequency comb,” Opt. Express 16(11), 8026–8032 (2008).
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S. Mohammadi, A. A. Eftekhar, A. Khelif, and A. Adibi, “Simultaneous two-dimensional phononic and photonic band gaps in opto-mechanical crystal slabs,” Opt. Express 18(9), 9164–9172 (2010).
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P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18(14), 14439–14453 (2010).
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A. H. Safavi-Naeini and O. Painter, “Design of optomechanical cavities and waveguides on a simultaneous bandgap phononic-photonic crystal slab,” Opt. Express 18(14), 14926–14943 (2010).
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J. Wang, Y. Zhu, R. Zhang, and D. J. Gauthier, “FSBS resonances observed in a standard highly nonlinear fiber,” Opt. Express 19(6), 5339–5349 (2011).
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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(1), 013836 (2015).
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S. El-Jallal, M. Oudich, Y. Pennec, B. Djafari-Rouhani, V. Laude, J.-C. Beugnot, A. Martínez, J. M. Escalante, and A. Makhoute, “Analysis of optomechanical coupling in two-dimensional square lattice phoxonic crystal slab cavities,” Phys. Rev. B 88(20), 205410 (2013).
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S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, “Two-dimensional phononic crystals: Examples and applications,” Surf. Sci. Rep. 65(8), 229–291 (2010).
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Figures (5)

Fig. 1
Fig. 1 (a) Unit cell of a two-dimensional silicon phoxonic crystal structure. The structural parameters are h = 340 nm and a = 490 nm. (b, c) Calculated quasi-TE photonic (b) and phononic (c) band diagrams of the phoxonic crystal structure with the unit cell shown in (a). The bandgaps are marked in red and the light cone is marked in gray.
Fig. 2
Fig. 2 (a) Left: Schematic of the W1 phoxonic crystal waveguide structure. Right: Profiles of the defect-guided optical mode (|Ey|2) and acoustic mode (Uy). (b) Photonic band diagram of the W1 phoxonic crystal waveguide structure. The cyan-shaded regions denote the slab-guided band continuum, where the modes in the dielectric (air) band reside mostly in the material (air holes). The black and red lines in the bandgap denote the defect-guided optical modes which do not cross each other in the entire kx range. (c) Phononic band diagram of the W1 phoxonic crystal waveguide structure. The light blue regions denote the phononic bandgaps. The red line denotes the defect-guided acoustic mode. (d) Group velocities of the defect-guided optical (vl,g, in red) and acoustic (vs,g, in blue) modes propagating in the phoxonic crystal waveguide.
Fig. 3
Fig. 3 Illustration showing the phase matching among the pump light, the Stokes light, and the acoustic wave in the engineered phoxonic crystal waveguide. The pump light and Stokes light are both in the slow-light region of the defect-guided mode. In the right zoomed-in diagram, the green (red) line denotes the dispersion curve of the defect-guided optical (acoustic) mode traveling along the x direction in the phoxonic crystal waveguide. The phononic band diagram is superimposed onto the photonic band diagram with the origin of the former aligned to the operating point of the pump light in the latter.
Fig. 4
Fig. 4 (a) Calculated group velocities of the pump and Stokes light vl,g (red) and acoustic wave vs,g (blue) at the phase-matching condition with the pump light wavelength in the C band. (b) Calculated SBS gain coefficient (red) and optomechanical coupling coefficient gOM (blue) at the phase-matching condition with the pump light wavelength in the C band.
Fig. 5
Fig. 5 Evolution of the amplitude of the acoustic wave |b(x)| along the propagation direction, where red solid and blue dash-dotted lines correspond to acoustic waves of slow (vg = 1.08 m/s) and normal (vg = 5,330 m/s) group velocities, respectively.

Equations (36)

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E p ( x , y , z , t ) = a p ( x , t ) e p ( x , y , z , t ) + c . c . ,
E s ( x , y , z , t ) = a s ( x , t ) e s ( x , y , z , t ) + c . c . ,
e p ( x , y , z , t ) = e ˜ p ( x , y , z ) exp [ i ( ω p t k p x ) ] ,
e s ( x , y , z , t ) = e ˜ s ( x , y , z ) exp [ i ( ω s t k s x ) ] .
P p = 2 x ^ { [ e p ( x , y , z , t ) ] * × h p ( x , y , z , t ) } d y d z ,
P s = 2 x ^ { [ e s ( x , y , z , t ) ] * × h s ( x , y , z , t ) } d y d z ,
ξ p = 2 0 a d x ε ( x , y , z ) [ e p ( x , y , z , t ) ] * e p ( x , y , z , t ) d y d z ,
ξ s = 2 0 a d x ε ( x , y , z ) [ e s ( x , y , z , t ) ] * e s ( x , y , z , t ) d y d z ,
U ( x , y , z , t ) = b ( x , t ) u ( x , y , z , t ) + c .c . ,
u ( x , y , z , t ) = u ˜ ( x , y , z ) exp [ i ( Ω t q x ) ] .
P b = 2 i Ω j m l c x j m l u j * u l m d y d z ,
ξ b = 2 Ω 2 0 a d x ρ | u | 2 d y d z ,
× × ( E + Δ E ) + μ 0 ( D + Δ D ) t = 0.
Δ E = Δ e s a p b * + Δ e p a s b + c .c .,
Δ D = Δ d s a p b * + Δ d p a s b + c .c ..
0 = × × ( a s e s + Δ e s a p b * ) + μ 0 ( a s d s + a p b * Δ d s ) t ,
0 = a s ( × × e s + μ 0 d s t ) + [ x ^ × ( × e s ) + × ( x ^ × e s ) ] a s x 2 i ω s μ 0 d s a s t + a p b * ( × × Δ e s + μ 0 2 Δ d s t 2 ) + h .o .t . + c .c . ,
i ω s a P s a s x i ω s P s v s a s t = a p b * x 0 x 0 + a d x [ ( e s ) * ( i ω ) 2 Δ d s ( i ω ) 2 d s Δ e s ] d y d z .
v s a s x + a s t = i ω s Q OM s ξ s a p b * ,
v p a p x + a p t = i ω p Q OM p ξ p a s b ,
Q OM p = x 0 x 0 + a d x [ ( e p ) * Δ d p d p Δ e p ] d y d z ,
Q OM s = x 0 x 0 + a d x [ ( e s ) * Δ d s d s Δ e s ] d y d z .
ρ 2 U i t 2 + j m l j [ c i j m l + η i j m l t ] m U l = F i ,
F ( r , t ) = f ( r , t ) a s * ( x , t ) a p ( x , t ) + c .c ..
i Ω j m l [ ( c i x m l m + j c i j x l ) u l b x 2 i Ω ρ u i b t + j η i j m l u l m b + f i a s * a p ] + c .c . = 0.
v b b x + α b b + b t = i Ω Q OM b ξ b a s * a p ,
Q OM b = x 0 x 0 + a d x u * f d y d z .
Q OM s = ( Q OM p ) * = Q OM b ,
v s a s ( x ) x = i ω s Q OM * ξ s a p ( x ) b * ( x ) ,
v p a p ( x ) x = i ω p Q OM ξ p a s ( x ) b ( x ) ,
v b b ( x ) x + α b v b b ( x ) = i Ω Q OM * ξ b a s * ( x ) a p ( x ) .
Q PE = ε 0 V d 3 r i j m l ε r 2 ( E i p ) * E j s p i j m l S m l ,
Q MB = S d 2 r ( u * n ^ ) [ ( ε r ε air ) ε 0 ( n ^ × E p ) * ( n ^ × E s ) ( ε r 1 ε air 1 ) ε 0 1 ( n ^ D p ) * ( n ^ D s ) ] ,
g OM = ω Q OM s Q OM p / ( ξ s ξ p ) ,
G = Ω | g OM | 2 ω v p v s v b α b ( ξ b / a ) ,
b ( x ) = i Ω g OM 0 + d x [ a s * ( x x ) a p ( x x ) exp ( α b x ) ] = i g OM a s * ( x ) a p ( x ) α b v b .

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