Abstract

Acousto-optical modulators usually rely on coherent diffraction of light by a moving acoustic wave, leading to bulky devices with a long interaction length. We propose a subwavelength acousto-optical structure that instead relies on a double resonance to achieve strong modulation at near-infrared wavelengths. A periodic array of metal ridges on a piezoelectric substrate defines cavities that create a resonant dip in the optical transmission spectrum. The ridges simultaneously support large flexural vibrations when resonantly excited by a radio-frequency signal, effectively deforming the cavities and leading to strongly nonlinear acousto-optical modulation. The nano-optical structure could find applications in highly compact photonic devices.

© 2017 Optical Society of America

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References

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  1. L. Brillouin, “Diffusion de la lumière et des rayons X par un corps transparent homogène. Influence de l’agitation thermique,” Ann. Phys. 9, 88–122 (1922).
    [Crossref]
  2. P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA. 18, 409–414 (1932).
    [Crossref]
  3. R. Lucas and P. Biquard, “Propriétés optiques des milieux solides et liquides soumis aux vibrations élastiques ultra sonores,” J. Phys. 71, 464–477 (1932).
    [Crossref]
  4. J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).
  5. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [Crossref]
  6. A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref]
  7. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  8. I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
    [Crossref]
  9. 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, 011008 (2012).
    [Crossref]
  10. V. Laude and J.-C. Beugnot, “Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides,” New J. Phys. 17, 125003 (2015).
    [Crossref]
  11. J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
    [Crossref]
  12. Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
    [Crossref]
  13. 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, 066611 (2002).
    [Crossref]
  14. I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
    [Crossref]
  15. D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
    [Crossref]
  16. A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A 9, 745–748 (2007).
    [Crossref]
  17. I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
    [Crossref]
  18. V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
    [Crossref]
  19. M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
    [Crossref]
  20. D. Royer and E. Dieulesaint, Elastic Waves in Solids (Wiley, 1999).
  21. Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
    [Crossref]
  22. L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
    [Crossref]
  23. Q. Rolland, M. Oudich, S. El-Jallal, S. Dupont, Y. Pennec, J. Gazalet, J.-C. Kastelik, G. Leveque, and B. Djafari-Rouhani, “Acousto-optic couplings in two-dimensional phoxonic crystal cavities,” Appl. Phys. Lett. 101, 061109 (2012).
    [Crossref]
  24. A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
    [Crossref]
  25. R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
    [Crossref]

2017 (1)

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

2016 (2)

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
[Crossref]

2015 (1)

V. Laude and J.-C. Beugnot, “Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides,” New J. Phys. 17, 125003 (2015).
[Crossref]

2014 (2)

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

2013 (2)

I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
[Crossref]

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

2012 (2)

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

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, 011008 (2012).
[Crossref]

2011 (2)

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

2010 (1)

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

2009 (2)

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
[Crossref]

2008 (1)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref]

2007 (1)

A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A 9, 745–748 (2007).
[Crossref]

2006 (1)

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

2005 (1)

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[Crossref]

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. E 65, 066611 (2002).
[Crossref]

1932 (2)

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA. 18, 409–414 (1932).
[Crossref]

R. Lucas and P. Biquard, “Propriétés optiques des milieux solides et liquides soumis aux vibrations élastiques ultra sonores,” J. Phys. 71, 464–477 (1932).
[Crossref]

1922 (1)

L. Brillouin, “Diffusion de la lumière et des rayons X par un corps transparent homogène. Influence de l’agitation thermique,” Ann. Phys. 9, 88–122 (1922).
[Crossref]

Achaoui, Y.

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

Adam, P.-M.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Akjouj, A.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Alegre, T. M.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Ballandras, S.

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

Benchabane, S.

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

Beugnot, J. C.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

Beugnot, J.-C.

V. Laude and J.-C. Beugnot, “Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides,” New J. Phys. 17, 125003 (2015).
[Crossref]

Biquard, P.

R. Lucas and P. Biquard, “Propriétés optiques des milieux solides et liquides soumis aux vibrations élastiques ultra sonores,” J. Phys. 71, 464–477 (1932).
[Crossref]

Bonello, B.

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Brillouin, L.

L. Brillouin, “Diffusion de la lumière et des rayons X par un corps transparent homogène. Influence de l’agitation thermique,” Ann. Phys. 9, 88–122 (1922).
[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, 011008 (2012).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Crouse, D. T.

I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
[Crossref]

Daniau, W.

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

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, 011008 (2012).
[Crossref]

Debye, P.

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA. 18, 409–414 (1932).
[Crossref]

Depine, R. A.

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[Crossref]

Dieulesaint, E.

D. Royer and E. Dieulesaint, Elastic Waves in Solids (Wiley, 1999).

Djafari-Rouhani, B.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

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

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Dühring, M. B.

M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
[Crossref]

Dupont, S.

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

Eichenfield, M.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

El Jallal, S.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

El-Jallal, S.

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

Escalante, J. M.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

Favero, I.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

Fink, Y.

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, 066611 (2002).
[Crossref]

Gazalet, J.

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

Golovin, A. B.

I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Ibanescu, M.

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, 066611 (2002).
[Crossref]

Imade, Y.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Joannopoulos, J. D.

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, 066611 (2002).
[Crossref]

Johnson, S. G.

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, 066611 (2002).
[Crossref]

Karrai, K.

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

Kastelik, J.-C.

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

Khelif, A.

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
[Crossref]

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

Kim, H.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref]

Laude, V.

V. Laude and J.-C. Beugnot, “Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides,” New J. Phys. 17, 125003 (2015).
[Crossref]

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
[Crossref]

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

Leveque, G.

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

Lévêque, G.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Lucas, R.

R. Lucas and P. Biquard, “Propriétés optiques des milieux solides et liquides soumis aux vibrations élastiques ultra sonores,” J. Phys. 71, 464–477 (1932).
[Crossref]

Mandel, I. M.

I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
[Crossref]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Martínez, A.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

Matsuda, O.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Maurer, T.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Mrabti, A.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Nicolas, R.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Otsuka, P. H.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Oudich, M.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

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

Painter, O.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Papanikolaou, N.

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Park, G.-W.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Pennec, Y.

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

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

Psarobas, I. E.

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Rakich, P. T.

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, 011008 (2012).
[Crossref]

Reinke, C.

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, 011008 (2012).
[Crossref]

Robert, L.

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

Rolland, Q.

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

Royer, D.

D. Royer and E. Dieulesaint, Elastic Waves in Solids (Wiley, 1999).

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Sakuma, H.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Sears, F. W.

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA. 18, 409–414 (1932).
[Crossref]

Sipe, J. E.

J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
[Crossref]

Skigin, D. C.

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[Crossref]

Skorobogatiy, M. A.

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, 066611 (2002).
[Crossref]

Socié, L.

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

Steel, M. J.

J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
[Crossref]

Stefanou, N.

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Stroud, R.

J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).

Tomoda, M.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Ulbricht, R.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref]

Vial, A.

A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A 9, 745–748 (2007).
[Crossref]

Wang, Z.

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, 011008 (2012).
[Crossref]

Weisberg, O.

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, 066611 (2002).
[Crossref]

Winger, M.

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

Wright, O. B.

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

Xu, J.

J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).

Ann. Phys. (1)

L. Brillouin, “Diffusion de la lumière et des rayons X par un corps transparent homogène. Influence de l’agitation thermique,” Ann. Phys. 9, 88–122 (1922).
[Crossref]

Appl. Phys. Lett. (4)

V. Laude, L. Robert, W. Daniau, A. Khelif, and S. Ballandras, “Surface acoustic wave trapping in a periodic array of mechanical resonators,” Appl. Phys. Lett. 89, 083515 (2006).
[Crossref]

L. Socié, S. Benchabane, L. Robert, A. Khelif, and V. Laude, “Surface acoustic wave guiding in a diffractionless high aspect ratio transducer,” Appl. Phys. Lett. 102, 113508 (2013).
[Crossref]

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

R. Ulbricht, H. Sakuma, Y. Imade, P. H. Otsuka, M. Tomoda, O. Matsuda, H. Kim, G.-W. Park, and O. B. Wright, “Elucidating gigahertz acoustic modulation of extraordinary optical transmission through a two-dimensional array of nano-holes,” Appl. Phys. Lett. 110, 091910 (2017).
[Crossref]

J. Appl. Phys. (1)

M. B. Dühring, V. Laude, and A. Khelif, “Energy storage and dispersion of surface acoustic waves trapped in a periodic array of mechanical resonators,” J. Appl. Phys. 105, 093504 (2009).
[Crossref]

J. Opt. A (1)

A. Vial, “Implementation of the critical points model in the recursive convolution method for modelling dispersive media with the finite-difference time domain method,” J. Opt. A 9, 745–748 (2007).
[Crossref]

J. Phys. (1)

R. Lucas and P. Biquard, “Propriétés optiques des milieux solides et liquides soumis aux vibrations élastiques ultra sonores,” J. Phys. 71, 464–477 (1932).
[Crossref]

Nanophotonics (1)

Y. Pennec, V. Laude, N. Papanikolaou, B. Djafari-Rouhani, M. Oudich, S. El Jallal, J. C. Beugnot, J. M. Escalante, and A. Martínez, “Modeling light-sound interaction in nanoscale cavities and waveguides,” Nanophotonics 3, 413–440 (2014).
[Crossref]

Nat. Photonics (1)

I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3, 201–205 (2009).
[Crossref]

Nature (1)

A. H. Safavi-Naeini, T. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref]

New J. Phys. (2)

V. Laude and J.-C. Beugnot, “Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides,” New J. Phys. 17, 125003 (2015).
[Crossref]

J. E. Sipe and M. J. Steel, “A Hamiltonian treatment of stimulated Brillouin scattering in nanoscale integrated waveguides,” New J. Phys. 18, 045004 (2016).
[Crossref]

Phys. Rev. A (1)

I. M. Mandel, A. B. Golovin, and D. T. Crouse, “Fano phase resonances in multilayer metal-dielectric compound gratings,” Phys. Rev. A 87, 053847 (2013).
[Crossref]

Phys. Rev. B (3)

I. E. Psarobas, N. Papanikolaou, N. Stefanou, B. Djafari-Rouhani, B. Bonello, and V. Laude, “Enhanced acousto-optic interactions in a one-dimensional phoxonic cavity,” Phys. Rev. B 82, 174303 (2010).
[Crossref]

Y. Achaoui, A. Khelif, S. Benchabane, L. Robert, and V. Laude, “Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars,” Phys. Rev. B 83, 104201 (2011).
[Crossref]

A. Mrabti, G. Lévêque, A. Akjouj, Y. Pennec, B. Djafari-Rouhani, R. Nicolas, T. Maurer, and P.-M. Adam, “Elastoplasmonic interaction in metal-insulator-metal localized surface plasmon systems,” Phys. Rev. B 94, 075405 (2016).
[Crossref]

Phys. Rev. E (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. E 65, 066611 (2002).
[Crossref]

Phys. Rev. Lett. (1)

D. C. Skigin and R. A. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. Lett. 95, 217402 (2005).
[Crossref]

Phys. Rev. X (1)

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, 011008 (2012).
[Crossref]

Proc. Natl. Acad. Sci. USA. (1)

P. Debye and F. W. Sears, “On the scattering of light by supersonic waves,” Proc. Natl. Acad. Sci. USA. 18, 409–414 (1932).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Science (1)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref]

Other (2)

D. Royer and E. Dieulesaint, Elastic Waves in Solids (Wiley, 1999).

J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).

Supplementary Material (3)

NameDescription
» Supplement 1       Supplemental document
» Visualization 1       Acousto-optical modulation of the transmission spectrum
» Visualization 2       Acousto-optical modulation at a fixed optical wavelength

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Figures (5)

Fig. 1.
Fig. 1.

Doubly resonant AO structure. w1 and w2 are cavity widths, w is the metal width, p=2w+w1+w2 is the period of the grating, and h is the metal thickness. The imbalance parameter e=w1w2 measures the departure from equal cavity widths. The piezoelectric substrate supporting the structure is semi-infinite. Metal ridges are alternatively connected to a current source, thus forming an elastic wave transducer. A time-harmonic electric potential difference V(Ω) is applied to drive mechanical motion at an acoustic frequency Ω.

Fig. 2.
Fig. 2.

Optical transmission around the Fano resonance. The structure of silver ridges on a lithium niobate substrate has parameters h=270  nm, w=100  nm, p=640  nm, and e=80  nm, with reference to Fig. 1. A normally incident TM plane wave travels upward. Transmission (T) and reflection (R) are shown in (a) as a function of optical wavelength in the region of the Fano resonance. The electric field intensity (color map) and the Poynting vector (arrows) are shown in (b) at the resonance wavelength (λ=1460.5  nm), i.e., for the minimum transmission.

Fig. 3.
Fig. 3.

Piezoelastic response of the structure. The maximum absolute displacements are shown as a function of frequency Ω/(2π) for an applied electric potential difference of 1 V, in (a) for longitudinal and in (b) for shear-vertical components. The yellow region marks the sound cone for the substrate. Acoustic modal shapes are illustrated in (c) for the four main resonances. The color scale is for the normalized total displacement. Note that vibration amplitudes have been adjusted for display purposes (see text).

Fig. 4.
Fig. 4.

AO modulation under small deformations. The modulation of the optical transmission caused by the various acoustic resonances is shown as a function of time in (a) and (b) for a maximum displacement limited to 100 pm and 1 nm, respectively. Normalized time is defined as Ωmt/(2π), with Ωm the resonant acoustic frequency. The optical wavelength is selected at the left inflection point in Fig. 2(a), or λ=1459.5  nm, in order to obtain the maximum modulation.

Fig. 5.
Fig. 5.

AO modulation under full flexural motion. The modulation of the optical transmission spectrum is evaluated for the fundamental acoustic mode. The instantaneous optical spectrum is shown in (a) as a function of time (see Visualization 1). Three cross sections at constant optical wavelength are shown in (b) following the dotted lines in (a) (see Visualization 2). The time-averaged transmission spectrum is compared in (c) to transmission without deformation.

Tables (1)

Tables Icon

Table 1. Resonant Frequencies and Maximum Absolute Displacements Are Given for the First Four Surface Acoustic Modes of Fig. 3, for a Potential Difference of 1  V

Equations (1)

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

αmR[ui(r;Ωm)exp(iΩmt)],

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