Abstract

On-chip waveguides on insulator with high stimulated Brillouin gain have wide potential application prospects in the field of nanophotonic structures. We propose a new on-chip hybrid silicon-chalcogenide slot waveguide structure consisting of a chalcogenide As2S3 rectangle core with an air slot and a wrapping layer of silicon. In the new hybrid waveguide, the high radiation pressure and electrostriction force, determined by pump and Stokes optical waves, and the acoustic displacement, determined by acoustic wave, can be achieved by adjusting the dimensions of rectangle core, the thickness of wrapping layers and the width of air slot. Therefore, a strong optomechanical coupling between high radiation pressure and transverse acoustic displacement will be generated. In such a way, a nonlinear gain for backward stimulated Brillouin scattering can be theoretically achieved with a high gain coefficient of 2.88×104 W−1m−1. The enhanced gain coefficient in the proposed waveguide is around 2.4 times as that in an on-chip silicon-chalcogenide hybrid slot waveguide on insulator without the wrapping layer. The Stokes amplification reaches 85.7 dB with the waveguide length of 2.5 cm. Therefore, this method provides a new idea to design nanophotonic waveguides for giant backward stimulated Brillouin scattering.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

2018 (3)

N. T Otterstrom, R. O. Behunin, E. A. Kittlaus, Z. Wang, and P. T. Rakich, “A silicon Brillouin laser,” Science 360(6393), 1113–1116 (2018).
[Crossref]

Z. Yu and X. Sun, “Giant enhancement of stimulated Brillouin scattering with engineered phoxonic crystal waveguides,” Opt. Express 26(2), 1255–1267 (2018).
[Crossref]

S. N. Jouybari, “Brillouin gain enhancement in nano-scale photonic waveguide,” Photonics Nanostructures: Fundam. Appl. 29, 8–14 (2018).
[Crossref]

2017 (4)

2016 (5)

S. Ramadan, K. Kwa, P. King, and A. O’Neill, “Reliable fabrication of sub-10 nm silicon nanowires by optical lithography,” Nanotechnology 27(42), 425302 (2016).
[Crossref]

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

S. R. Mirnaziry, C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in silicon/chalcogenide slot waveguides,” Opt. Express 24(5), 4786–4800 (2016).
[Crossref]

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]

O. Florez, P. F. Jarschel, Y. A. V. Espinel, C. M. B. Cordeiro, T. P. M. Alegre, G. S. Wiederhecker, and P. Dainese, “Brillouin scattering self-cancellation,” Nat. Commun. 7(1), 11759 (2016).
[Crossref]

2015 (2)

C. Wolff, P. Gutsche, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Impact of nonlinear loss on stimulated Brillouin scattering,” J. Opt. Soc. Am. B 32(9), 1968–1978 (2015).
[Crossref]

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]

2014 (4)

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photonics Rev. 8(5), 653–666 (2014).
[Crossref]

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylverstre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5(1), 5242 (2014).
[Crossref]

I. Aryanfar, C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Mode conversion using stimulated Brillouin scattering in nanophotonic silicon waveguides,” Opt. Express 22(23), 29270–29282 (2014).
[Crossref]

R. V. Laer, B. Kuyken, D. V. Thourhout, and R. Baets, “Analysis of enhanced stimulated Brillouin scattering in silicon slot waveguides,” Opt. Lett. 39(5), 1242–1245 (2014).
[Crossref]

2013 (3)

2012 (3)

2010 (3)

M. S. Kang, A. Brenn, and P. St. J. Russell, “All-optical control of gigahertz acoustic resonances by forward stimulated interpolarization scattering in a photonic crystal fiber,” Phys. Rev. Lett. 105(15), 153901 (2010).
[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]

B. Wu, A. Kumar K, and S. Pamarthy, “High aspect ratio silicon etch: A review,” J. Appl. Phys. 108(5), 051101 (2010).
[Crossref]

2009 (2)

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin Lasing with a CaF2 Whispering Gallery Mode Resonator,” Phys. Rev. Lett. 102(4), 043902 (2009).
[Crossref]

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode Thermoelastic Dissipation,” J. Appl. Phys. 105(4), 043505 (2009).
[Crossref]

2008 (2)

2007 (2)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored Light in an Optical Fiber via Stimulated Brillouin Scattering,” Science 318(5857), 1748–1750 (2007).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

2005 (1)

2001 (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]

1973 (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. A 8(1), 14–21 (1973).
[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(21), 592–595 (1964).
[Crossref]

Agarwal, M.

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode Thermoelastic Dissipation,” J. Appl. Phys. 105(4), 043505 (2009).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics , IV (Academic, 2006).

Alegre, T. P. M.

O. Florez, P. F. Jarschel, Y. A. V. Espinel, C. M. B. Cordeiro, T. P. M. Alegre, G. S. Wiederhecker, and P. Dainese, “Brillouin scattering self-cancellation,” Nat. Commun. 7(1), 11759 (2016).
[Crossref]

Aryanfar, I.

Baets, R.

Bao, X.

Bauters, J. F.

Behunin, R.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

Behunin, R. O.

N. T Otterstrom, R. O. Behunin, E. A. Kittlaus, Z. Wang, and P. T. Rakich, “A silicon Brillouin laser,” Science 360(6393), 1113–1116 (2018).
[Crossref]

Benschop, J.

A. Yen, H. Meiling, and J. Benschop, “Enabling manufacturing of sub-10nm generations of integrated circuits with EUV lithography,” 2019 Electron Devices Technology and Manufacturing Conference (EDTM), Singapore, Singapore, 2019, pp. 475–477.

Beugnot, J. C.

J. C. Beugnot, S. Lebrun, G. Pauliat, H. Maillotte, V. Laude, and T. Sylverstre, “Brillouin light scattering from surface acoustic waves in a subwavelength-diameter optical fibre,” Nat. Commun. 5(1), 5242 (2014).
[Crossref]

Blumenthal, D. J.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

Bohley, C.

P. W. Nolte, C. Bohley, and J. Schilling, “Degenerate four wave mixing in racetrack resonators formed by Chalcogenide infiltrated silicon slot waveguides,” in 2014 IEEE 11th International Conference on Group IV Photonics (GFP) (IEEE, 2014), pp. 118–119.

Bose, D.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

Bowers, J. E.

Boyd, R. W.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored Light in an Optical Fiber via Stimulated Brillouin Scattering,” Science 318(5857), 1748–1750 (2007).
[Crossref]

R. W. Boyd, Nonlinear Optics (Academic, 2009), III ed., Chap. 9.

Bremner, T.

Brenn, A.

M. S. Kang, A. Brenn, and P. St. J. Russell, “All-optical control of gigahertz acoustic resonances by forward stimulated interpolarization scattering in a photonic crystal fiber,” Phys. Rev. Lett. 105(15), 153901 (2010).
[Crossref]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007).
[Crossref]

Brodnik, G. M.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

Brown, A.

Byrnes, A.

Cabrera-Granado, E.

Calderón, O. G.

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]

Candler, R. N.

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode Thermoelastic Dissipation,” J. Appl. Phys. 105(4), 043505 (2009).
[Crossref]

Casas-Bedoya, A.

Chandorkar, S. A.

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode Thermoelastic Dissipation,” J. Appl. Phys. 105(4), 043505 (2009).
[Crossref]

Chauhan, N.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

Chen, A.

Chen, T.

Chiao, R. Y.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin Scattering and Coherent Generation of Intense Hypersonic Waves,” Phys. Rev. Lett. 12(21), 592–595 (1964).
[Crossref]

Choi, D.

Choudhary, A.

Cordeiro, C. M. B.

O. Florez, P. F. Jarschel, Y. A. V. Espinel, C. M. B. Cordeiro, T. P. M. Alegre, G. S. Wiederhecker, and P. Dainese, “Brillouin scattering self-cancellation,” Nat. Commun. 7(1), 11759 (2016).
[Crossref]

Dainese, P.

O. Florez, P. F. Jarschel, Y. A. V. Espinel, C. M. B. Cordeiro, T. P. M. Alegre, G. S. Wiederhecker, and P. Dainese, “Brillouin scattering self-cancellation,” Nat. Commun. 7(1), 11759 (2016).
[Crossref]

Davenport, M. L.

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]

Debnath, K.

K. Debnath, A. Z. Khokhar, G. T. Reed, and S. Saito, “Fabrication of Arbitrarily Narrow Vertical Dielectric Slots in Silicon Waveguides,” IEEE Photonics Technol. Lett. 29(15), 1269–1272 (2017).
[Crossref]

Demerchant, M.

Dong, H.

Doylend, J. K.

Duan, J.

Y. Zheng, P. Gao, B. Xia, X. Wu, Y. Yan, and J. Duan, “Experimental Research on Silicon Optical Waveguide and Focus Coupling Grating,” 2018 IEEE 3rd Optoelectronics Global Conference (OGC), Shenzhen, 2018, pp. 72–75.

Duwel, A.

S. A. Chandorkar, R. N. Candler, A. Duwel, R. Melamud, M. Agarwal, K. E. Goodson, and T. W. Kenny, “Multimode Thermoelastic Dissipation,” J. Appl. Phys. 105(4), 043505 (2009).
[Crossref]

Eggleton, B. J.

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Pamarthy, S.

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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(5), 969–971 (2012).
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[Crossref]

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]

R. Pant, D. Marpaung, I. V. Kabakova, B. Morrison, C. G. Poulton, and B. J. Eggleton, “On-chip stimulated Brillouin scattering for microwave signal processing and generation,” Laser Photonics Rev. 8(5), 653–666 (2014).
[Crossref]

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B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
[Crossref]

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(5), 969–971 (2012).
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S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
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Rakich, P. T.

S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-Hz Linewidth Photonic-Integrated Brillouin Laser,” Nat. Photonics 13(1), 60–67 (2019).
[Crossref]

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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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K. Debnath, A. Z. Khokhar, G. T. Reed, and S. Saito, “Fabrication of Arbitrarily Narrow Vertical Dielectric Slots in Silicon Waveguides,” IEEE Photonics Technol. Lett. 29(15), 1269–1272 (2017).
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K. Debnath, A. Z. Khokhar, G. T. Reed, and S. Saito, “Fabrication of Arbitrarily Narrow Vertical Dielectric Slots in Silicon Waveguides,” IEEE Photonics Technol. Lett. 29(15), 1269–1272 (2017).
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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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Adv. Opt. Photonics (1)

B. J. Eggleton, C. G. Poulton, and R. Pant, “Inducing and harnessing stimulated Brillouin scattering in photonic integrated circuits,” Adv. Opt. Photonics 5(4), 536–587 (2013).
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Figures (12)

Fig. 1.
Fig. 1. Sketch of the cross section of the hybrid on-chip structure. a represents the silicon layer width, b represents the silicon layer height, c is the As2S3 rectangle core width, d is As2S3 rectangle core height, and e is the air slot width. f and g represent the width and height of SiO2 substrate, respectively. h describes the thickness of the top and bottom silicon wrapping layers.
Fig. 2.
Fig. 2. Optical modes and radiation pressure distribution of the proposed waveguide. Figure 2(a) is the sketch of radiation pressure distribution on (i) to (iv) boundaries of left As2S3 rectangle core. Radiation pressure distribution on boundaries of right As2S3 rectangle core is symmetry to the y-axis (in the center of air slot) and is not presented. Figs. 2(b)–2(d) are the guided transverse profiles of the fundamental optical mode for the Ex, Ey, and Ez field components.
Fig. 3.
Fig. 3. Five lowest order hybrid acoustic modes of the proposed hybrid waveguide. Figs. 3(a)–3(e) show the transverse profile of the normalized lowest first order to fifth order hybrid acoustic waves (A1-A5) for the ux, uy, uz field components.
Fig. 4.
Fig. 4. Calculated BSBS gain of A1-A5 acoustic modes by assuming the acoustic quality factor of 1000. (a) The BSBS total gain spectrum versus the acoustic angular frequency. The BSBS total gain values of A2 and A4 are presented near the marks A2 and A4, respectively. (b) The BSBS gains coefficient. Column with different three colors represent BSBS gain coefficients under three conditions: Electrostriction force-only, radiation pressure-only, and the total effects. The individual acoustic frequency is in the top of every grouped column.
Fig. 5.
Fig. 5. Normalized transverse and total displacements of modes A1, A3 and A5 with parameters Smx and Sm.
Fig. 6.
Fig. 6. (a) Computed profiles and intensities of BSBS gain coefficient (in W−1m−1) by varying both chalcogenide As2S3 core width c from 140 nm to 230 nm and height d from 300 nm to 365 nm with e = 10 nm. The region of high BSBS gain coefficient within the black dashed lines (where c covers from 160 nm to 200 nm and d varies from 320 nm to 364 nm) is presented. BSBS gain coefficient of structure without top and bottom wrapping layer (b= d = 365 nm and h= 0) is also presented in the top edge of the figure. (b) The effect of top and bottom thickness h of silicon layers on BSBS total gain coefficients and S parameter when h varies from 0 nm to 23.5 nm (corresponding to d varies from 300 nm to 365 nm) in c = 180 nm. (c) The ux-component profiles of an acoustic mode in h = 2.5 nm (d = 360 nm) and h = 22.5 nm (d = 320 nm) respectively, with size of c = 180 nm and e = 10 nm.
Fig. 7.
Fig. 7. (a) Computed BSBS gain coefficients as the air slot width e is varied. Different colors and shapes of scatters denote gain coefficients in three individual condition. (b) Normalized COT, |QCRP|2 and radiation pressure gain coefficient vary with different e, respectively. (c) Normalized CFV, CEF and COT vary with different e. (d) |QCRP| on boundaries (i)-(iv) vary with different e, respectively. The e is varied from 5 nm to 35 nm.
Fig. 8.
Fig. 8. Computed magnitude of |FRP| and corresponding |ux| on boundary (ii) when y position changed from −174 nm to 174 nm with e = 5 nm, e = 15 nm, e = 25 nm, e = 35 nm, respectively. (a) The magnitude of |FRP|. (b) The magnitude of |ux|.
Fig. 9.
Fig. 9. The variation of nonlinear loss coefficients with the increase of air slot width (range from 10 nm to 35 nm) with a = 450 nm, b = 365 nm, c = 180 nm, d = 352 nm and h = 6.5 nm. (a) The variation of β. (b) The variation of γ.
Fig. 10.
Fig. 10. The figure of merit $\mathscr{F}$ for BSBS with the change of air slot width (range from 10 nm to 35 nm) with a = 450 nm, b = 365 nm, c = 180 nm, d = 352 nm and h = 6.5 nm. The linear loss α=11.5m−1. The unit of $\mathscr{F}$ is dimensionless and is omitted.
Fig. 11.
Fig. 11. Left: the Stokes amplification versus waveguide length L ranges from 0 to 35 cm. Right: a clearer plot of Stokes amplification varying as L ranges from 0 to 2.5 cm marked by the yellow dotted box in the left plot. The air slot width e = 10 nm.
Fig. 12.
Fig. 12. The fabrication processes of proposed waveguide with an air slot of 10 nm.

Tables (2)

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Table 1. Optical and Acoustic Properties of the Materials [19]

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Table 2. The effect of size variations on BSBS gain coefficient and Stokes Amplification

Equations (19)

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

Ω   =   ω p ω s ,
q = k p k s ,
d P p d z = ( α + β P p + γ P P 2 ) P p ,
d P s d z = α P s + ( g + 2 β ) P p P s + γ P p 2 P s ,
g ( Ω ) = m G m ( Γ m / 2 ) 2 ( Ω Ω m ) + ( 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 ,
ρ t 2 u i + j k l j c i j k l k u l = f i ,
G m = C O T m | Q C m | 2 ,
Q C m = n f n , u m .
f i e s = i j j σ i j ,
σ i j = 1 4 ε 0 ε r 2 p i j k l ( E p k E s l + E p l E s k ) ,
F i R P = ( T 2 i j T 1 i j ) n j ,
T i j = ε 0 ε r ( E i E j 1 2 δ i j E 2 ) .
Q C m R P = F R P ( e ) , u m ( e , h ) ,
S m ( a , b , c , h , e ) = U m T r a n s . ( a , b , c , h , e ) U m T o t a l ( a , b , c , h , e ) ,
F = g 2 β 2 α γ > 1.
A ( L , P s ( 0 ) ) = 10 log 10 ( P s ( 0 ) / P s ( L ) ) ,
P o p t = α γ ( F + F 2 1 ) .
L o p t = 1 2 α ln ( F + F 2 1 F F 2 1 ) .

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