Abstract

For the first time detailed interactions between optical and acoustic modes in a silicon slot waveguide are presented. A new computer code has been developed by using a full-vectorial formulation to study the acoustic modes in optical waveguides. The results have shown that the acoustic modes in an optical slot waveguide are not purely longitudinal or transverse but fully hybrid in nature. The model allows the effects of Stimulated Brillouin Scattering and the associated frequency shift due to the interaction of these hybrid acoustic modes with the fully hybrid optical mode also to be presented.

© 2014 Optical Society of America

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References

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  1. K. Hotate, M. Tanaka, “Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation based continuous-wave technique,” IEEE Photon. Technol. Lett. 14(2), 179–181 (2002).
    [CrossRef]
  2. K. Y. Song, K. S. Abedin, K. Hotate, M. González Herráez, L. Thévenaz, “Highly efficient Brillouin slow and fast light using As2Se3 chalcogenide fiber,” Opt. Express 14(13), 5860–5865 (2006).
    [CrossRef] [PubMed]
  3. V. R. Almeida, Q. Xu, C. A. Barrios, M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
    [CrossRef] [PubMed]
  4. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
    [CrossRef]
  5. S. Wang, J. Hu, H. Guo, X. Zeng, “Optical Cherenkov radiation in an As2S3 slot waveguide with four zero-dispersion wavelengths,” Opt. Express 21(3), 3067–3072 (2013).
    [CrossRef] [PubMed]
  6. F. Dell’Olio, V. M. N. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15(8), 4977–4993 (2007).
    [CrossRef] [PubMed]
  7. T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
    [CrossRef]
  8. J. T. Robinson, K. Preston, O. Painter, M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008).
    [CrossRef] [PubMed]
  9. R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. K. J. Lee, H. C. Park, B. Y. Kim, “Highly efficient all-fiber tunable polarization filter using torsional acoustic wave,” Opt. Express 15(19), 12362–12367 (2007).
    [CrossRef] [PubMed]
  12. W. Zou, Z. He, K. Hotate, “Acoustic modal analysis and control in W-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008).
    [CrossRef] [PubMed]
  13. Y. S. Mamdem, E. Burov, L. A. de Montmorillon, Y. Jaouën, G. Moreau, R. Gabet, F. Taillade, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” Opt. Express 20(2), 1790–1797 (2012).
    [CrossRef] [PubMed]
  14. A. Safaai-Jazi, R. O. Claus, “Acoustic modes in optical fiberlike waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 619–627 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  16. G. O. Stone, “Higher-order finite elements for inhomogeneous acoustic guiding structures,” IEEE Trans. Microw. Theory Tech. 21(8), 538–542 (1973).
    [CrossRef]
  17. B. M. A. Rahman, J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. 2(5), 682–688 (1984).
    [CrossRef]
  18. D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express 18(8), 8528–8539 (2010).
    [CrossRef] [PubMed]
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2013 (1)

2012 (2)

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

Y. S. Mamdem, E. Burov, L. A. de Montmorillon, Y. Jaouën, G. Moreau, R. Gabet, F. Taillade, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” Opt. Express 20(2), 1790–1797 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (1)

2008 (3)

2007 (2)

2006 (1)

2005 (1)

2004 (1)

2002 (1)

K. Hotate, M. Tanaka, “Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation based continuous-wave technique,” IEEE Photon. Technol. Lett. 14(2), 179–181 (2002).
[CrossRef]

1988 (1)

A. Safaai-Jazi, R. O. Claus, “Acoustic modes in optical fiberlike waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 619–627 (1988).
[CrossRef] [PubMed]

1985 (1)

R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
[CrossRef] [PubMed]

1984 (1)

B. M. A. Rahman, J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. 2(5), 682–688 (1984).
[CrossRef]

1973 (2)

P. E. Lagasse, “Higher-order finite element analysis of topographic guides supporting elastic surface waves,” J. Acoust. Soc. Am. 53(4), 1116–1122 (1973).
[CrossRef]

G. O. Stone, “Higher-order finite elements for inhomogeneous acoustic guiding structures,” IEEE Trans. Microw. Theory Tech. 21(8), 538–542 (1973).
[CrossRef]

Abedin, K. S.

Almeida, V. R.

Annunziata, F.

Baehr-Jones, T.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Barrios, C. A.

Bayer, P. W.

R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
[CrossRef] [PubMed]

Burov, E.

Chen, X.

Claus, R. O.

A. Safaai-Jazi, R. O. Claus, “Acoustic modes in optical fiberlike waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 619–627 (1988).
[CrossRef] [PubMed]

Dalton, L.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Davies, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Davies, J. B.

B. M. A. Rahman, J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. 2(5), 682–688 (1984).
[CrossRef]

de Montmorillon, L. A.

Dell’Olio, F.

Gabet, R.

González Herráez, M.

Grattan, K. T.

Grattan, K. T. V.

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

Guo, H.

He, Z.

Hochberg, M.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Hotate, K.

Hu, J.

Huang, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Jaouën, Y.

Jen, A.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Kejalakshmy, N.

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express 18(8), 8528–8539 (2010).
[CrossRef] [PubMed]

Kim, B. Y.

Kim, T.-D.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Kobyakov, A.

Lagasse, P. E.

P. E. Lagasse, “Higher-order finite element analysis of topographic guides supporting elastic surface waves,” J. Acoust. Soc. Am. 53(4), 1116–1122 (1973).
[CrossRef]

Lee, K. J.

Leung, D. M. H.

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express 18(8), 8528–8539 (2010).
[CrossRef] [PubMed]

Levenson, M. D.

R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
[CrossRef] [PubMed]

Li, M. J.

Lipson, M.

Luo, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Mamdem, Y. S.

McElhenny, J. E.

Moreau, G.

Painter, O.

Park, H. C.

Passaro, V. M. N.

Pattnaik, R. K.

Penkov, B.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Preston, K.

Rahman, B. M. A.

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express 18(8), 8528–8539 (2010).
[CrossRef] [PubMed]

B. M. A. Rahman, J. B. Davies, “Finite element solution of integrated optical waveguides,” J. Lightwave Technol. 2(5), 682–688 (1984).
[CrossRef]

Robinson, J. T.

Ruffin, A. B.

Safaai-Jazi, A.

A. Safaai-Jazi, R. O. Claus, “Acoustic modes in optical fiberlike waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 619–627 (1988).
[CrossRef] [PubMed]

Scherer, A.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Shelby, R. M.

R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
[CrossRef] [PubMed]

Song, K. Y.

Stone, G. O.

G. O. Stone, “Higher-order finite elements for inhomogeneous acoustic guiding structures,” IEEE Trans. Microw. Theory Tech. 21(8), 538–542 (1973).
[CrossRef]

Sullivan, P.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Taillade, F.

Takayesu, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Tanaka, M.

K. Hotate, M. Tanaka, “Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation based continuous-wave technique,” IEEE Photon. Technol. Lett. 14(2), 179–181 (2002).
[CrossRef]

Thévenaz, L.

Toulouse, J.

Wang, S.

Xu, Q.

Zeng, X.

Zou, W.

Appl. Phys. Lett. (1)

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. Hotate, M. Tanaka, “Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation based continuous-wave technique,” IEEE Photon. Technol. Lett. 14(2), 179–181 (2002).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

G. O. Stone, “Higher-order finite elements for inhomogeneous acoustic guiding structures,” IEEE Trans. Microw. Theory Tech. 21(8), 538–542 (1973).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

A. Safaai-Jazi, R. O. Claus, “Acoustic modes in optical fiberlike waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 35(5), 619–627 (1988).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

P. E. Lagasse, “Higher-order finite element analysis of topographic guides supporting elastic surface waves,” J. Acoust. Soc. Am. 53(4), 1116–1122 (1973).
[CrossRef]

J. Lightwave Technol. (2)

J. Nonlinear Opt. Phys. (1)

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. V. Grattan, “Rigorous numerical analysis and characterization of silicon vertical-slot nano-waveguide,” J. Nonlinear Opt. Phys. 21(01), 1250007 (2012).
[CrossRef]

Opt. Express (8)

Y. S. Mamdem, E. Burov, L. A. de Montmorillon, Y. Jaouën, G. Moreau, R. Gabet, F. Taillade, “Importance of residual stresses in the Brillouin gain spectrum of single mode optical fibers,” Opt. Express 20(2), 1790–1797 (2012).
[CrossRef] [PubMed]

S. Wang, J. Hu, H. Guo, X. Zeng, “Optical Cherenkov radiation in an As2S3 slot waveguide with four zero-dispersion wavelengths,” Opt. Express 21(3), 3067–3072 (2013).
[CrossRef] [PubMed]

K. Y. Song, K. S. Abedin, K. Hotate, M. González Herráez, L. Thévenaz, “Highly efficient Brillouin slow and fast light using As2Se3 chalcogenide fiber,” Opt. Express 14(13), 5860–5865 (2006).
[CrossRef] [PubMed]

F. Dell’Olio, V. M. N. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15(8), 4977–4993 (2007).
[CrossRef] [PubMed]

K. J. Lee, H. C. Park, B. Y. Kim, “Highly efficient all-fiber tunable polarization filter using torsional acoustic wave,” Opt. Express 15(19), 12362–12367 (2007).
[CrossRef] [PubMed]

W. Zou, Z. He, K. Hotate, “Acoustic modal analysis and control in W-shaped triple-layer optical fibers with highly-germanium-doped core and F-doped inner cladding,” Opt. Express 16(14), 10006–10017 (2008).
[CrossRef] [PubMed]

J. T. Robinson, K. Preston, O. Painter, M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16(21), 16659–16669 (2008).
[CrossRef] [PubMed]

D. M. H. Leung, N. Kejalakshmy, B. M. A. Rahman, K. T. Grattan, “Rigorous modal analysis of silicon strip nanoscale waveguides,” Opt. Express 18(8), 8528–8539 (2010).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. B Condens. Matter (1)

R. M. Shelby, M. D. Levenson, P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B Condens. Matter 31(8), 5244–5252 (1985).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Variation of the phase velocity of the Uy acoustic mode with the frequency for a vertical slot waveguide.

Fig. 2
Fig. 2

The displacement vector plots of the Uy mode in a slot waveguide with (a) Uy (b) Ux and (c) Uz displacement vector profiles at ka = 20 μm−1.

Fig. 3
Fig. 3

Variations of displacement vector Uy of the Uy mode along the y-axis for ka = 20.0 μm−1 and ka = 40.0 μm−1.

Fig. 4
Fig. 4

Variations of the effective index, neff, and effective area, Aeff, with the slot width, Ws.

Fig. 5
Fig. 5

Variations of the SBS frequency shift and modal overlap with the slot width, Ws.

Equations (10)

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

U i =u( u x , u y ,j u z )exp{j( ω a t k a z)}
S=u
.T=ρ 2 u t 2
T ij = c ijkl S kl ;i,j,k,l=x,y,z
[T]=[c][S]
u={ u x u y u z }=[ N 1 0 0 0 N 1 0 0 0 N 1 | N 2 0 0 0 N 2 0 0 0 N 2 | N 3 0 0 0 N 3 0 0 0 N 3 ][ u x 1 u y 1 u z 1 u x 2 u y 2 u z 2 u x 3 u y 3 u z 3 ]
([A] ω a 2 [B])U=F
ω o 2 = [ ( ×H ) * ε ^ 1 ( ×H )+p ( H ) * ( H ) ] dxdy H * μ ^ H dxdy
f B =2 n eff V a / λ o
k a =2 β 0

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