Abstract

Stimulated Brillouin scattering has attracted renewed interest with the promise of highly tailorable integration into the silicon photonics platform. However, significant Brillouin amplification in silicon waveguides has yet to be shown. In an effort to engineer a structure with large photon–phonon coupling, we analyzed both forward and backward Brillouin scattering in high-index-contrast silicon slot waveguides. The calculations predict that gradient forces enhance the Brillouin gain in narrow slots. We estimate a currently feasible gain of about 105W1m1 in horizontal slot waveguides, which is an order of magnitude larger than in a stand-alone silicon wire. Such efficient coupling could enable a host of Brillouin technologies on a mass-producible silicon chip.

© 2014 Optical Society of America

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2013

J. Li, H. Lee, and K. Vahala, Nat. Commun. 4, 1 (2013).

B. Eggleton, C. Poulton, and R. Pant, Adv. Opt. Photonics 5, 536 (2013).

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

C. Poulton, R. Pant, and B. Eggleton, J. Opt. Soc. Am. B 30, 1 (2013).

W. Qiu, P. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, Opt. Express 21, 31402 (2013).
[CrossRef]

2012

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

J. Li, H. Lee, T. Chen, and K. Vahala, Opt. Express 20, 369 (2012).

2011

2010

M. Li, W. Pernice, and H. Tang, Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010).
[CrossRef]

D. Van Thourhout and J. Roels, Nat. Photonics 4, 211 (2010).
[CrossRef]

A. Kobyakov, M. Sauer, and D. Chowdhury, Adv. Opt. Photon. 2, 1 (2010).
[CrossRef]

2009

I. Grudinin, A. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef]

D. Braje, L. Hollberg, and S. Diddams, Phys. Rev. Lett. 102, 1 (2009).
[CrossRef]

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

2007

Q. Lin, O. Painter, and G. Agrawal, Opt. Express 15, 416 (2007).

Z. Zhu, D. Gauthier, and R. Boyd, Science 318, 1748 (2007).
[CrossRef]

A. Zadok, A. Eyal, and M. Tur, J. Lightw. Technol. 25, 2168 (2007).
[CrossRef]

J. Beugnot, T. Sylvestre, and H. Maillotte, Opt. Lett. 32, 17 (2007).
[CrossRef]

T. Kippenberg and K. Vahala, Opt. Express 15, 17172 (2007).

R. Sun, P. Dong, N. Feng, C. Hong, J. Michel, M. Lipson, and L. Kimerling, Opt. Express 15, 17967 (2007).
[CrossRef]

2005

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

2004

1985

R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. Lett. 54, 939 (1985).
[CrossRef]

1982

1974

D. Biegelsen, Phys. Rev. Lett. D 32, 1196 (1974).

1972

E. Ippen and R. Stolen, Appl. Phys. Lett. 21, 539 (1972).
[CrossRef]

1965

Y. Shen and N. Bloembergen, Phys. Rev. 290, 1787 (1965).

1964

R. Chiao, C. Townes, and B. Stoicheff, Phys. Rev. Lett. 12, 592 (1964).

Agrawal, G.

Q. Lin, O. Painter, and G. Agrawal, Opt. Express 15, 416 (2007).

Almeida, V.

Bahl, G.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[CrossRef]

Barrios, C.

Bayer, P.

R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. Lett. 54, 939 (1985).
[CrossRef]

Beugnot, J.

Biegelsen, D.

D. Biegelsen, Phys. Rev. Lett. D 32, 1196 (1974).

Bigelow, M.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Bloembergen, N.

Y. Shen and N. Bloembergen, Phys. Rev. 290, 1787 (1965).

Boyd, R.

Z. Zhu, D. Gauthier, and R. Boyd, Science 318, 1748 (2007).
[CrossRef]

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Braje, D.

D. Braje, L. Hollberg, and S. Diddams, Phys. Rev. Lett. 102, 1 (2009).
[CrossRef]

Brenn, A.

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

Butsch, A.

M. Kang, A. Butsch, and P. Russell, Nat. Photonics 5, 549 (2011).
[CrossRef]

Camacho, R.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

Carmon, T.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

Chen, T.

J. Li, H. Lee, T. Chen, and K. Vahala, Opt. Express 20, 369 (2012).

Chiao, R.

R. Chiao, C. Townes, and B. Stoicheff, Phys. Rev. Lett. 12, 592 (1964).

Chodorow, M.

Choi, D.

Chowdhury, D.

Cox, J.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

Dasgupta, S.

S. Dasgupta and F. Poletti, J. Lightw. Technol. 29, 22 (2011).
[CrossRef]

Davids, P.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

P. Rakich, Z. Wang, and P. Davids, Opt. Lett. 36, 217 (2011).
[CrossRef]

Diddams, S.

D. Braje, L. Hollberg, and S. Diddams, Phys. Rev. Lett. 102, 1 (2009).
[CrossRef]

Dong, H.

Dong, P.

Eggleton, B.

Eyal, A.

A. Zadok, A. Eyal, and M. Tur, J. Lightw. Technol. 25, 2168 (2007).
[CrossRef]

Feng, N.

Gaeta, A.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Gauthier, D.

J. Wang, Y. Zhu, R. Zhang, and D. Gauthier, Opt. Express 19, 5339 (2011).
[CrossRef]

Z. Zhu, D. Gauthier, and R. Boyd, Science 318, 1748 (2007).
[CrossRef]

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Grudinin, I.

I. Grudinin, A. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef]

Hile, S.

Hollberg, L.

D. Braje, L. Hollberg, and S. Diddams, Phys. Rev. Lett. 102, 1 (2009).
[CrossRef]

Hong, C.

Hopcroft, M.

M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010).
[CrossRef]

Ilchenko, V.

Ippen, E.

E. Ippen and R. Stolen, Appl. Phys. Lett. 21, 539 (1972).
[CrossRef]

Jarecki, R.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

Kang, M.

M. Kang, A. Butsch, and P. Russell, Nat. Photonics 5, 549 (2011).
[CrossRef]

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

Kenny, T.

M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010).
[CrossRef]

Kimerling, L.

Kippenberg, T.

Kobyakov, A.

Lee, H.

J. Li, H. Lee, and K. Vahala, Nat. Commun. 4, 1 (2013).

J. Li, H. Lee, T. Chen, and K. Vahala, Opt. Express 20, 369 (2012).

Levenson, M.

R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. Lett. 54, 939 (1985).
[CrossRef]

Li, E.

Li, J.

J. Li, H. Lee, and K. Vahala, Nat. Commun. 4, 1 (2013).

J. Li, H. Lee, T. Chen, and K. Vahala, Opt. Express 20, 369 (2012).

Li, M.

M. Li, W. Pernice, and H. Tang, Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

Lin, Q.

Q. Lin, O. Painter, and G. Agrawal, Opt. Express 15, 416 (2007).

Lipson, M.

Luther-Davies, B.

Madden, S.

Maillotte, H.

Maleki, L.

A. Savchenkov, A. Matsko, V. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[CrossRef]

I. Grudinin, A. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef]

Matsko, A.

A. Savchenkov, A. Matsko, V. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 3338 (2011).
[CrossRef]

I. Grudinin, A. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef]

Mcfarlane, H.

Michel, J.

Nazarkin, A.

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

Nix, W.

M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010).
[CrossRef]

Okawachi, Y.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Olsson, R.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

Painter, O.

Q. Lin, O. Painter, and G. Agrawal, Opt. Express 15, 416 (2007).

Pant, R.

Pernice, W.

M. Li, W. Pernice, and H. Tang, Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

Poletti, F.

S. Dasgupta and F. Poletti, J. Lightw. Technol. 29, 22 (2011).
[CrossRef]

Poulton, C.

Qiu, W.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

W. Qiu, P. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, Opt. Express 21, 31402 (2013).
[CrossRef]

Rakich, P.

W. Qiu, P. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, Opt. Express 21, 31402 (2013).
[CrossRef]

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

P. Rakich, Z. Wang, and P. Davids, Opt. Lett. 36, 217 (2011).
[CrossRef]

Reinke, C.

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

Roels, J.

D. Van Thourhout and J. Roels, Nat. Photonics 4, 211 (2010).
[CrossRef]

Russell, P.

M. Kang, A. Butsch, and P. Russell, Nat. Photonics 5, 549 (2011).
[CrossRef]

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

Sauer, M.

Savchenkov, A.

Schweinsberg, A.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Seidel, D.

Sharping, J.

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Shaw, H.

Shelby, R.

R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. Lett. 54, 939 (1985).
[CrossRef]

Shen, Y.

Y. Shen and N. Bloembergen, Phys. Rev. 290, 1787 (1965).

Shin, H.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

W. Qiu, P. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, Opt. Express 21, 31402 (2013).
[CrossRef]

Soljacic, M.

Starbuck, A.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

Stoicheff, B.

R. Chiao, C. Townes, and B. Stoicheff, Phys. Rev. Lett. 12, 592 (1964).

Stokes, L.

Stolen, R.

E. Ippen and R. Stolen, Appl. Phys. Lett. 21, 539 (1972).
[CrossRef]

Sun, R.

Sylvestre, T.

Tang, H.

M. Li, W. Pernice, and H. Tang, Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

Thevenaz, L.

Tomes, M.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

Townes, C.

R. Chiao, C. Townes, and B. Stoicheff, Phys. Rev. Lett. 12, 592 (1964).

Tur, M.

A. Zadok, A. Eyal, and M. Tur, J. Lightw. Technol. 25, 2168 (2007).
[CrossRef]

Vahala, K.

J. Li, H. Lee, and K. Vahala, Nat. Commun. 4, 1 (2013).

J. Li, H. Lee, T. Chen, and K. Vahala, Opt. Express 20, 369 (2012).

T. Kippenberg and K. Vahala, Opt. Express 15, 17172 (2007).

Van Thourhout, D.

D. Van Thourhout and J. Roels, Nat. Photonics 4, 211 (2010).
[CrossRef]

Wang, J.

Wang, Z.

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

W. Qiu, P. Rakich, H. Shin, H. Dong, M. Soljačić, and Z. Wang, Opt. Express 21, 31402 (2013).
[CrossRef]

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

P. Rakich, Z. Wang, and P. Davids, Opt. Lett. 36, 217 (2011).
[CrossRef]

Xu, Q.

Zadok, A.

A. Zadok, A. Eyal, and M. Tur, J. Lightw. Technol. 25, 2168 (2007).
[CrossRef]

Zehnpfennig, J.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[CrossRef]

Zhang, R.

Zhu, Y.

Zhu, Z.

Z. Zhu, D. Gauthier, and R. Boyd, Science 318, 1748 (2007).
[CrossRef]

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

Adv. Opt. Photon.

Adv. Opt. Photonics

B. Eggleton, C. Poulton, and R. Pant, Adv. Opt. Photonics 5, 536 (2013).

Appl. Phys. Lett.

E. Ippen and R. Stolen, Appl. Phys. Lett. 21, 539 (1972).
[CrossRef]

M. Li, W. Pernice, and H. Tang, Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

J. Lightw. Technol.

S. Dasgupta and F. Poletti, J. Lightw. Technol. 29, 22 (2011).
[CrossRef]

A. Zadok, A. Eyal, and M. Tur, J. Lightw. Technol. 25, 2168 (2007).
[CrossRef]

J. Microelectromech. Syst.

M. Hopcroft, W. Nix, and T. Kenny, J. Microelectromech. Syst. 19, 229 (2010).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Commun.

J. Li, H. Lee, and K. Vahala, Nat. Commun. 4, 1 (2013).

H. Shin, W. Qiu, R. Jarecki, J. Cox, R. Olsson, A. Starbuck, Z. Wang, and P. Rakich, Nat. Commun. 4, 1944 (2013).

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, Nat. Commun. 2, 403 (2011).
[CrossRef]

Nat. Photonics

D. Van Thourhout and J. Roels, Nat. Photonics 4, 211 (2010).
[CrossRef]

M. Kang, A. Butsch, and P. Russell, Nat. Photonics 5, 549 (2011).
[CrossRef]

Nat. Phys.

M. Kang, A. Nazarkin, A. Brenn, and P. Russell, Nat. Phys. 5, 276 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

Y. Shen and N. Bloembergen, Phys. Rev. 290, 1787 (1965).

Phys. Rev. Lett.

D. Braje, L. Hollberg, and S. Diddams, Phys. Rev. Lett. 102, 1 (2009).
[CrossRef]

R. Shelby, M. Levenson, and P. Bayer, Phys. Rev. Lett. 54, 939 (1985).
[CrossRef]

R. Chiao, C. Townes, and B. Stoicheff, Phys. Rev. Lett. 12, 592 (1964).

Y. Okawachi, M. Bigelow, J. Sharping, Z. Zhu, A. Schweinsberg, D. Gauthier, R. Boyd, and A. Gaeta, Phys. Rev. Lett. 153902, 1 (2005).

I. Grudinin, A. Matsko, and L. Maleki, Phys. Rev. Lett. 102, 043902 (2009).
[CrossRef]

M. Tomes and T. Carmon, Phys. Rev. Lett. 102, 113601 (2009).
[CrossRef]

Phys. Rev. Lett. D

D. Biegelsen, Phys. Rev. Lett. D 32, 1196 (1974).

Phys. Rev. X

P. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, Phys. Rev. X 2, 1 (2012).

Science

Z. Zhu, D. Gauthier, and R. Boyd, Science 318, 1748 (2007).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Vertical and (c) horizontal silicon slot waveguides suspended in air, with the corresponding optical mode (b) and (d).

Fig. 2.
Fig. 2.

Typical optical force profile on left beam of vertical slot waveguide: (a) radiation pressure and (b) electrostrictive body force. The radiation pressure is large close to the slot.

Fig. 3.
Fig. 3.

(a) Slot with two mechanically excited beams, (b) slot with just one excited beam, and (c) a stand-alone wire. We work in scenario (b).

Fig. 4.
Fig. 4.

(a)–(c) Brillouin spectrum of a vertical slot waveguide and (b)–(d) the gain of the most promising mode increases rapidly in narrow slots. The color of the modes indicates the sign of u x (red: +, blue: −).

Fig. 5.
Fig. 5.

(a) Gradient forces can be large despite low dispersion; (b)–(d) narrow slots perform better than a stand-alone wire for a range of a ¯ -values; and (c)  G has a clear optimum in the ( a , b ) -plane for the same mode as in (b) with g = 5 nm .

Fig. 6.
Fig. 6.

(a)–(c) Both forward and backward SBS is very efficient in narrow horizontal slots and (d) the flexural mode is sensitive to b ¯ . The color of the modes indicates the sign of u y (red: +, blue: −).

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