Abstract

We develop a general framework of evaluating the Stimulated Brillouin Scattering (SBS) gain coefficient in optical waveguides via the overlap integral between optical and elastic eigen-modes. This full-vectorial formulation of SBS coupling rigorously accounts for the effects of both radiation pressure and electrostriction within micro- and nano-scale waveguides. We show that both contributions play a critical role in SBS coupling as modal confinement approaches the sub-wavelength scale. Through analysis of each contribution to the optical force, we show that spatial symmetry of the optical force dictates the selection rules of the excitable elastic modes. By applying this method to a rectangular silicon waveguide, we demonstrate how the optical force distribution and elastic modal profiles jointly determine the magnitude and scaling of SBS gains in both forward and backward SBS processes. We further apply this method to the study of intra- and inter-modal SBS processes, and demonstrate that the coupling between distinct optical modes are necessary to excite elastic modes with all possible symmetries. For example, we show that strong inter-polarization coupling can be achieved between the fundamental TE- and TM-like modes of a suspended silicon waveguide.

© 2013 Optical Society of America

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2013 (2)

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]

I. V. Kabakova, R. Pant, D. 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]

2012 (9)

R. Pant, A. Byrnes, C. G. Poulton, E. Li, D. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic-chip-based tunable slow and fast light via stimulated Brillouin scattering,” Opt. Lett.37, 969–971 (2012).
[CrossRef] [PubMed]

R. Pant, A. Byrnes, C. G. Poulton, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic-chip-based tunable slow and fast light via stimulated Brillouin scattering,” Opt. Lett.37, 969–971 (2012).
[CrossRef] [PubMed]

W. C. Jiang, X. Lu, J. Zhang, and Q. Lin, “High-frequency silicon optomechanical oscillator with an ultralow threshold,” Opt. Express20, 15991–15996 (2012).
[CrossRef] [PubMed]

A. Byrnes, R. Pant, E. Li, D.-Y. Choi, C. G. Poulton, S. Fan, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and reconfigurable narrowband microwave photonic filter using stimulated Brillouin scattering,” Opt. Express20, 18836–18845 (2012).
[CrossRef] [PubMed]

J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on silicon,” Opt. Express20, 20170–20180 (2012).
[CrossRef] [PubMed]

C. G. Poulton, R. Pant, A. Byrnes, S. Fan, M. J. Steel, and B. J. Eggleton, “Design for broadband on-chip isolator using stimulated Brillouin scattering in dispersion-engineered chalcogenide waveguides,” Opt. Express20, 21235–21246 (2012).
[CrossRef] [PubMed]

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, and Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X2, 011008 (2012).
[CrossRef]

G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spontaneous Brillouin cooling,” Nat. Phys.8, 203–207 (2012).
[CrossRef]

A. Byrnes, R. Pant, E. Li, D. Choi, C. G. Poulton, S. Fan, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic chip based tunable and reconfigurable narrowband microwave photonic filter using stimulated Brillouin scattering,” Opt. Express20, 18845–18854 (2012).
[CrossRef]

2011 (5)

2010 (6)

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express18, 14439–14453 (2010).
[CrossRef] [PubMed]

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics2, 1–59 (2010).
[CrossRef]

M. S. Kang, A. Brenn, and P. S. J. Russell, “All-optical control of gigahertz acoustic resonances by forward stimulated interpolarization scattering in a photonic crystal fiber,” Phys. Rev. Lett.105, 153901 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics4, 236–242 (2010).
[CrossRef]

P. W. Milonni and R. W. Boyd, “Momentum of light in a dielectric medium,” Adv. Opt. Photonics, 2, 519–553 (2010).
[CrossRef]

M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Youngs Modulus of silicon?,” J. Microelectromech. Syst.19, 229–238 (2010).
[CrossRef]

2009 (8)

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82 (2009).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

J. Roels, I. D. Vlaminck, L. Lagae, B. Maes, D. V. Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (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, 276–280 (2009).
[CrossRef]

M. Tomes and T. Carmon, “Photonic micro-electromechanical systems vibrating at x-band (11-ghz) rates,” Phys. Rev. Lett.102, 113601 (2009).
[CrossRef] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics3, 91–94 (2009).
[CrossRef]

M. Eichenfield, J. Chan, A. H. Safavi-Naeini, K. J. Vahala, and O. Painter, “Modeling dispersive coupling and-losses of localized optical andmechanical modes in optomechanicalcrystals,” Opt. Express17, 20078–20098 (2009).
[CrossRef] [PubMed]

2008 (4)

R. Pant, M. D. Stenner, M. A. Neifeld, and D. J. Gauthier, “Optimal pump profile designs for broadband sbs slow-light systems,” Opt. Express16, 2764–2777 (2008).
[CrossRef] [PubMed]

A. Schliesser, R. Riviere, G. Anetsberger, O Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys.4, 415–419 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science321, 1172–1176 (2008).
[CrossRef] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics3, 91–94 (2008).
[CrossRef]

2007 (3)

P. T. Rakich, M. A. Popovi, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science318, 1748–1750 (2007).
[CrossRef] [PubMed]

2006 (3)

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

L. S. Hounsome, R. Jones, M. J. Shaw, and P. R. Briddon, “Photoelastic constants in diamond and silicon,” Phys. Status Solidi A203, 3088–3093 (2006).
[CrossRef]

K. Y. Song, K. S. Abedin, K. Hotate, M. G. Herráez, and L. Thévenaz, “Highly efficient Brillouin slow and fast light using As2Se3 chalcogenide fiber,” Opt. Express14, 5860–5865 (2006).
[CrossRef] [PubMed]

2005 (4)

2002 (1)

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. E65, 066611 (2002).
[CrossRef]

1995 (1)

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibres,” Electron. Lett.31, 2032–2034 (1995).
[CrossRef]

1990 (1)

1985 (2)

R. Shelby, M. Levenson, and P. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B31, 5244–5252 (1985).
[CrossRef]

R. Shelby, M. Levenson, and P. Bayer, “Resolved forward Brillouin scattering in optical fibers,” Phys. Rev. Lett.54, 939–942 (1985).
[CrossRef] [PubMed]

1976 (1)

H. O. Hill, B. S. Kawasaki, and D. C. Johnson, “CW Brillouin laser,” Appl. Phys. Lett.28, 608–609 (1976).
[CrossRef]

1973 (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A8, 14–21 (1973).
[CrossRef]

1965 (1)

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev.137, A1787–A1805 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett.12, 592–595 (1964).
[CrossRef]

Abedin, K. S.

Agarwal, M.

S. Chandorkar, M. Agarwal, R. Melamud, R. Candler, K. Goodson, and T. Kenny, “Limits of quality factor in bulk-mode micromechanical resonators,” in IEEE 21st International Conference on Micro Electro Mechanical Systems, 2008. MEMS 2008 (2008), pp. 74–77.

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2006).

Anetsberger, G.

A. Schliesser, R. Riviere, G. Anetsberger, O Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys.4, 415–419 (2008).
[CrossRef]

Arcizet, O

A. Schliesser, R. Riviere, G. Anetsberger, O Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys.4, 415–419 (2008).
[CrossRef]

Baets, R.

J. Roels, I. D. Vlaminck, L. Lagae, B. Maes, D. V. Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

R. Van Laer, D. Van Thourhout, and R. Baets, “Strong stimulated Brillouin scattering in an on-chip silicon slot waveguide,” in CLEO: Science and Innovations, San Jose, CA, 9–14 June 2013.

Bahl, G.

G. Bahl, M. Tomes, F. Marquardt, and T. Carmon, “Observation of spontaneous Brillouin cooling,” Nat. Phys.8, 203–207 (2012).
[CrossRef]

Bahloul, D.

J. Botineau, E. Picholle, and D. Bahloul, “Effective stimulated Brillouin gain in singlemode optical fibres,” Electron. Lett.31, 2032–2034 (1995).
[CrossRef]

Bayer, P.

R. Shelby, M. Levenson, and P. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B31, 5244–5252 (1985).
[CrossRef]

R. Shelby, M. Levenson, and P. Bayer, “Resolved forward Brillouin scattering in optical fibers,” Phys. Rev. Lett.54, 939–942 (1985).
[CrossRef] [PubMed]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev.137, A1787–A1805 (1965).
[CrossRef]

Botineau, J.

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Figures (4)

Fig. 1
Fig. 1

The guided optical and elastic modes of a silicon rectangular waveguide. Optical frequency is in unit of 2πc/a, while elastic frequency is in unit of 2πVL/a. V L = E / ρ = 8.54 × 10 3 m / s is the velocity of longitudinal elastic waves in bulk silicon. (a) Dispersion relation of optical modes Ey11 and Ez11. (b) Dispersion relation of elastic modes which have even symmetry with respect to both y = 0 and z = 0 planes. E-modes (black lines) are the eigen-modes of the actual silicon waveguide, with silicon-air interfaces treated as free boundaries. For comparison, the dispersion relations of purely longitudinal modes (designated as P-modes, blue curves) and purely transverse modes (designated as S-modes, red curves) are included. They are constrained respectively with x-only displacement, and y-z-only movements. At q = 0, E-modes manifest as either P-modes or S-modes. (c) The displacement profiles of mode E1 through E5 at q = 0, with the peak deformation shown. The color represents y-displacement (uy) for S-like E modes and x-displacement(ux) for P-like E modes. Blue, white, and red correspond to negative, zero, and positive values respectively. Mode E1 experience a DC longitudinal offset at Ω = 0.

Fig. 2
Fig. 2

Optical force distributions and the resultant gain coefficients of the Forward SBS. In panels (a) and (b), the width of the waveguide is a = 315nm, and the incident optical waves have ω = 0.203(2πc/a), and k = 0.75(π/a). The elastic waves are generated at q = 0. (a) The force distribution of electrostriction body force density, electrostriction surface pressure, and radiation pressure respectively. All three types of optical forces are transverse. (b) Calculated FSBS gains of the elastic modes, assuming mechanical Q = 1000. Blue, red, and green bars represent FSBS gains under three conditions: electrostriction-only, radiation-pressure-only, and the combined effects. Only the S-like E modes have non-zero gains. (c) The scaling relation of FSBS gains as the device dimension a is varied from 0.25μm to 2.5μm. Solid and dotted curves correspond to the gain coefficients for mode E2 and E5 respectively.

Fig. 3
Fig. 3

Optical force distributions and the resultant gain coefficients of the Backward SBS. In panels (a) and (b), the width of the waveguide is a = 315nm, and the incident optical waves have ω = 0.203(2πc/a), and k = 0.75(π/a). The elastic waves are generated at q = 1.5(π/a). (a) The force distribution of electrostriction body force density, electrostriction surface pressure, and radiation pressure respectively. Electrostriction have both longitudinal and transverse components. Radiation pressure are purely transverse. (b) Calculated BSBS gains of the elastic modes, assuming mechanical Q = 1000. Blue, red, and green bars represent FSBS gains under three conditions: electrostriction-only, radiation-pressure-only, and the combined effects.(c) The scaling relation of BSBS gains related to mode E1 as a is varied from 0.25μm to 2.5μm, color-coded similar to panel (b). For comparison, gain coefficients predicted by conventional fiber BSBS theory are shown as the solid black curve. The dotted black curve represents the electrostriction-only BSBS gain of the constrained mode P1. Black circles represent the largest electrostriction-only BSBS gain coefficient among all E-modes for a given a. (d) BSBS spectra near the anti-crossing between mode E4 and E5 around q = 1.66(π/a). The mechanical quality factor Q is assumed to be 100. The red lines represent the total BSBS gain. The blue and green lines represent contributions from mode E4 and E5.

Fig. 4
Fig. 4

Optical force distributions, relavant elastic modes, and the resultant gain coefficients of inter-modal FSBS between Ey11 (pump) and Ez11 (Stokes). The width of the waveguide is set to be a = 315nm. The incident optical waves have ω = 0.203(2πc/a), with the pump-wave propagation constant at kp = 0.750(π/a), and the Stokes-wave propagation constant at ks = 0.665(π/a). The elastic waves are generated at q = 0.085(π/a). (a) The force distribution of electrostriction body force density, electrostriction surface pressure, and radiation pressure respectively. The longitudinal forces (not shown here) are negligible, in comparison to the transverse forces. All optical forces are anti-symmetric with respect to plane y = 0 and plane z = 0, exciting elastic modes with the matching symmetry (designated as O-modes). (b) Calculated inter-modal SBS gains, assuming mechanical Q = 1000. The insets illustrate the displacement profiles of mode O1 through O5 at q = 0.085(π/a), at peak deformation. ”Jet” color map is used to shown the amplitude of total displacement. Blue and red correspond to zero and maximum respectively.

Equations (24)

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d P s d x = g P p P s α s P s .
g ( Ω ) = ω s 2 Ω P p P s Re f , d u d t ,
A , B A * B d s .
g ( Ω ) = 2 ω s v g p v g s Im f , u E p , ε E p E s , ε E s .
ρ Ω 2 u i = x j c i j k l u l x k + f i .
u m , ρ u n = δ m n u m , ρ u m .
b m = u m , f u m , ρ u m 1 Ω m 2 Ω 2 .
b m = u m , f u m , ρ u m 1 Ω m Γ m Γ m / 2 Ω m Ω i Γ m / 2 .
g ( Ω ) = m G m ( Γ m / 2 ) 2 ( Ω Ω m ) 2 + ( Γ m / 2 ) 2 .
G m = 2 ω Q m Ω m 2 v g p v g s | f , u m | 2 E p , ε E p E s , ε E s u m , ρ u m .
f , u m = n f n , u m .
σ i j = 1 2 ε 0 n 4 p i j k l E k E l .
σ i j = 1 4 ε 0 n 4 p i j k l ( E p k E s l * + E p l E s k * ) .
[ σ x x σ y y σ z z σ y z σ x z σ x y ] = 1 2 ε 0 n 4 [ p 11 p 12 p 13 p 12 p 22 p 23 p 13 p 23 p 33 p 44 p 55 p 66 ] [ E p x E s x * E p y E s y * E p z E s z * E p y E s z * + E p z E s y * E p x E s z * + E p z E s x * E p x E s y * + E p y E s x * ] .
f x E S = i q σ x x y σ x y z σ x z f y E S = i q σ x y y σ y y z σ y z f z E S = i q σ x z y σ z y z σ z z .
F i E S = ( σ 1 i j σ 2 i j ) n j .
T i j = ε 0 ε ( E i E j 1 2 δ i j E 2 ) .
F i R P = ( T 2 i j T 1 i j ) n j .
F R P = 1 2 ε 0 E t 2 ( ε 2 ε 1 ) n + 1 2 ε 0 1 D n 2 ( ε 2 1 ε 1 1 ) n .
F R P = 1 2 ε 0 E p t E s t * ( ε 2 ε 1 ) n + 1 2 ε 0 1 D p n D s n * ( ε 2 1 ε 1 1 ) n .
f x E S = i q σ x x = 1 2 i q ε 0 n 4 p 12 E y 2 .
G 0 = ω 2 n 7 p 12 2 c 3 ρ V L Γ 1 A ,
[ σ x x σ y y σ z z σ y z σ x z σ x y ] = 1 2 ε 0 n 4 [ p 11 p 12 p 13 p 12 p 22 p 23 p 13 p 23 p 33 p 44 p 55 p 66 ] [ | E x | 2 | E y | 2 | E z | 2 2 Re ( E y E z * ) 0 0 ] .
[ σ x x σ y y σ z z σ y z σ x z σ x y ] = 1 2 ε 0 n 4 [ p 11 p 12 p 13 p 12 p 22 p 23 p 13 p 23 p 33 p 44 p 55 p 66 ] [ E x 2 E y 2 E z 2 2 E y E z 2 E x E z 2 E x E y ] .

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