Abstract

In a theoretical design study, we propose buried waveguides made of germanium or alloys of germanium and other group-IV elements as a CMOS-compatible platform for robust, high-gain stimulated Brillouin scattering (SBS) applications in the mid-infrared regime. To this end, we present numerical calculations for backward-SBS at 4 μm in germanium waveguides that are buried in silicon nitride. Due to the strong photoelastic anisotropy of germanium, we investigate two different orientations of the germanium crystal with respect to the waveguide’s propagation direction and find considerable differences. The acoustic wave equation is solved including crystal anisotropy; acoustic losses are computed from the acoustic mode patterns and previously published material parameters.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  23. B. G. Helme and P. J. King, “The phonon viscosity tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Sol. (A) 45, K33–K37 (1978).
    [Crossref]
  24. B. A. Auld, Acoustic Fields and Waves in Solids, 2nd. ed. (Krieger Publishing Company, 1990).
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  26. W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
    [Crossref]
  27. W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).
  28. C. Wolff, M. J. Steel, and C. G. Poulton, “Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers,” arXiv:1410.8639 [physics.optics], (2014).
  29. E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
    [Crossref]
  30. K. S. Abedin, “Observation of strong stimulated Brillouin scattering in single-mode As2Se3 chalcogenide fiber,” Opt. Express 13, 10266–10271 (2005).
    [Crossref] [PubMed]
  31. R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thevenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
    [Crossref] [PubMed]

2014 (3)

2013 (4)

J. Li, H. Lee, and K. H. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

R. A. Soref, “Group IV photonics for the mid infrared,” Proc. SPIE 8629, 862902 (2013).
[Crossref]

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

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

2012 (5)

2011 (3)

R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thevenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
[Crossref] [PubMed]

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[Crossref] [PubMed]

N. K. Hon, R. A. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 011301 (2011).
[Crossref]

2010 (2)

2005 (1)

1997 (1)

F. Schäffler, “High-mobility Si and Ge structures,” Semicond. Sci. Technol. 12, 1515–1549 (1997).
[Crossref]

1978 (2)

A. Feldman, R. M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589–2590 (1978).
[Crossref]

B. G. Helme and P. J. King, “The phonon viscosity tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Sol. (A) 45, K33–K37 (1978).
[Crossref]

1974 (1)

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32, 1196–1199 (1974).
[Crossref]

1972 (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

1965 (1)

J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
[Crossref]

Abedin, K. S.

Ahmad, H.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Auld, B. A.

B. A. Auld, Acoustic Fields and Waves in Solids, 2nd. ed. (Krieger Publishing Company, 1990).

Bahl, G.

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

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[Crossref] [PubMed]

Biegelsen, D. K.

D. K. Biegelsen, “Photoelastic tensor of silicon and the volume dependence of the average gap,” Phys. Rev. Lett. 32, 1196–1199 (1974).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd. ed. (Academic, 2003).

Byrnes, A.

Carmon, T.

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

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[Crossref] [PubMed]

Chen, T.

Cheng, F.

X. Yang, F. Cheng, and R. Soref, “Single-mode GeSn mid-infrared waveguides on group-IV substrates,” in Proceedings of the Conference on Lasers and Electro-Optics (Applications and Technology), San Jose, CA (2014), paper JTh2A.57.

Choi, D.-Y.

Chong, W. Y.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Chuwongin, S.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Davids, P.

De Leonardis, F.

Eggleton, B. J.

Evans, R. A.

J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
[Crossref]

Fan, S.

Feldman, A.

A. Feldman, R. M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589–2590 (1978).
[Crossref]

Harun, S. W.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Helme, B. G.

B. G. Helme and P. J. King, “The phonon viscosity tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Sol. (A) 45, K33–K37 (1978).
[Crossref]

Hile, S.

Hon, N. K.

N. K. Hon, R. A. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 011301 (2011).
[Crossref]

Horowitz, D.

A. Feldman, R. M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589–2590 (1978).
[Crossref]

Ippen, E. P.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Jalali, B.

N. K. Hon, R. A. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 011301 (2011).
[Crossref]

King, P. J.

B. G. Helme and P. J. King, “The phonon viscosity tensor of Si, Ge, GaAs, and InSb,” Phys. Stat. Sol. (A) 45, K33–K37 (1978).
[Crossref]

Lee, H.

Li, E.

Li, J.

Luther-Davies, B.

Ma, Z.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Madden, S.

Madden, S. J.

Marquardt, F.

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

Mcfarlane, H.

Melloni, A.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Morichetti, F.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Pant, R.

Passaro, Vand M.N.

Poulton, C. G.

Rakich, P. T.

Schäffler, F.

F. Schäffler, “High-mobility Si and Ge structures,” Semicond. Sci. Technol. 12, 1515–1549 (1997).
[Crossref]

F. Schäffler, in Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe, M. E. Levinshtein, S. L. Rumyantsev, and M. S. Shur, eds. (John Wiley & Sons, Inc., 2001), pp. 149–188.

Seo, J. H.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Shuai, Y. C.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Soref, R.

R. Soref, “Silicon-based silicon-germanium-tin heterostructure photonics,” Phil. Trans. R. Soc. A 372, 20130113 (2014).
[Crossref] [PubMed]

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4, 495–497 (2010).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

X. Yang, F. Cheng, and R. Soref, “Single-mode GeSn mid-infrared waveguides on group-IV substrates,” in Proceedings of the Conference on Lasers and Electro-Optics (Applications and Technology), San Jose, CA (2014), paper JTh2A.57.

Soref, R. A.

R. A. Soref, “Group IV photonics for the mid infrared,” Proc. SPIE 8629, 862902 (2013).
[Crossref]

N. K. Hon, R. A. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1−xGex in the midwave and longwave infrared,” J. Appl. Phys. 110, 011301 (2011).
[Crossref]

Steel, M. J.

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. Express 20, 21235–21246 (2012).
[Crossref] [PubMed]

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,” arXiv:1407.3521 [physics.optics], (2014).

C. Wolff, M. J. Steel, and C. G. Poulton, “Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers,” arXiv:1410.8639 [physics.optics], (2014).

Stolen, R. H.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Thevenaz, L.

Tomes, M.

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

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[Crossref] [PubMed]

Troia, B.

Vahala, K. H.

J. Li, H. Lee, and K. H. Vahala, “Low-noise Brillouin laser on a chip at 1064 nm,” Opt. Lett. 39, 287–290 (2014).
[Crossref] [PubMed]

J. Li, H. Lee, and K. H. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).
[PubMed]

Vahala, K. J.

Wang, Z.

Waxler, R. M.

A. Feldman, R. M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589–2590 (1978).
[Crossref]

Wolff, C.

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,” arXiv:1407.3521 [physics.optics], (2014).

C. Wolff, M. J. Steel, and C. G. Poulton, “Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers,” arXiv:1410.8639 [physics.optics], (2014).

Wortman, J. J.

J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
[Crossref]

Yang, H.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Yang, W.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Yang, X.

X. Yang, F. Cheng, and R. Soref, “Single-mode GeSn mid-infrared waveguides on group-IV substrates,” in Proceedings of the Conference on Lasers and Electro-Optics (Applications and Technology), San Jose, CA (2014), paper JTh2A.57.

Zehnpfennig, J.

G. Bahl, J. Zehnpfennig, M. Tomes, and T. Carmon, “Stimulated optomechanical excitation of surface acoustic waves in a microdevice,” Nat. Commun. 2, 403 (2011).
[Crossref] [PubMed]

Zhao, D.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Zhou, W.

W. Zhou, Z. Ma, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and W. Yang, “Semiconductor nanomembranes for integrated silicon photonics and flexible Photonics,” Opt. Quant. Electron. 44, 12–13 (2012).
[Crossref]

W. Zhou, Z. Ma, W. Yang, S. Chuwongin, Y. C. Shuai, J. H. Seo, D. Zhao, H. Yang, and R. Soref, “Semiconductor nanomembranes for integrated and flexible photonics,” in IEEE 2011 ICO International Conference on Information Photonics (IP), Ottawa (18 May 2011).

Zulkifli, M. Z.

M. Z. Zulkifli, W. Y. Chong, A. Melloni, F. Morichetti, S. W. Harun, and H. Ahmad, “Extraction of a single Stokes line from a Brillouin fibre laser using a silicon oxynitride microring filter,” Laser Phys. 23, 095102 (2013).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

J. Appl. Phys. (3)

J. J. Wortman and R. A. Evans, “Young’s modulus, shear modulus, and Poisson’s ratio in silicon and germanium,” J. Appl. Phys. 36, 153–156 (1965).
[Crossref]

A. Feldman, R. M. Waxler, and D. Horowitz, “Photoelastic constants of germanium,” J. Appl. Phys. 49, 2589–2590 (1978).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Schematic of the backward-SBS systems we investigate in this study: A germanium-waveguide on a substrate and covered in a cladding material. The waveguide is illuminated by a pump field (blue wave) around 4 μm. A resonant acoustic wave (green wave) acts as a travelling grating and scatters the pump into a Stokes wave (red wave), which is also red-shifted due to the Doppler effect. Due to conservation of energy and momentum, the sound wave is amplified by the back-scattering process.
Fig. 2
Fig. 2 Sketches of a waveguide (gray) composed of a semiconductor with cubic symmetry on an isotropic substrate (cyan) for the two crystal orientations that we consider in this paper. We generally chose the Cartesian coordinate system according to the system’s geometry, i.e. along the substrate normal (we chose this to be the y-axis) and the waveguide’s propagation direction (we chose this to be the z-axis). We focus on two cases for two orientations of the material tensors’ principal axes: a) shows a waveguide with the all cubic axes aligned with the Cartesian frame of reference. We refer to this (slightly inaccurately) as the waveguide being composed of a [100]-semiconductor; b) shows a waveguide where the cubic crystal is rotated by 45° with respect to the Cartesian frame of reference around the surface normal (y-axis). We refer to this as a waveguide composed of a [110]-semiconductor.
Fig. 3
Fig. 3 Results for backward-SBS of light with a vacuum wavelength of 4 microns in a germanium waveguide in [100]-orientation buried in Si3N4. In the top right corner, we show a sketch of the structure. The top plot to the left depicts the total power gain according to Eq. (7) as a function of the waveguide width while the waveguide height is fixed at 550nm. Among the acoustic modes to which coupling is permitted due to symmetry considerations [28] we show the three modes with lowest frequencies. The other two plots show the corresponding acoustic frequencies and the acoustic decay lengths α−1, i.e. the inverse decay parameters according to Eq. (8), neglecting mechanical loss inside the cladding. Below the system sketch, we show the optical and acoustic modes for points highlighted in the gain plot, i.e. for a waveguide width of 1200nm. The colorscale (in arbitrary units) is the z-component of the time-averaged Poynting vector 〈S〉 and the longitudinal component uz of the mechanical displacement field for the optical and acoustic mode plots, respectively. The arrows indicate the in-plane electric field distribution and the in-plane components of the displacement field, the acoustic mode plots also contain iso-contours of the in-plane acoustic displacement field norm | u x | 2 + | u y | 2 (thin blue lines). The colors in the line-art plots and the frames around the mode plots match for identical acoustic modes.
Fig. 4
Fig. 4 Results for backward-SBS of light with a vacuum wavelength of 4 microns in a germanium waveguide in [110]-orientation buried in Si3N4. See the caption of Fig. 3 and the main text for details.

Tables (1)

Tables Icon

Table 1 This table contains the material parameters for the structures that we investigate in this paper as well as values for the velocities vp, vs and vq that we calculated [24] from them for quasi-longitudinal, shear and quasi-shear waves travelling through the bulk material along the z-direction, respectively. For the [100]-orientation of the cubic materials and for silicon nitride, the values for the stiffness [20] tensor c, the photoelastic [21,22] tensor p, and the viscosity [23] tensor η were taken from their respective references. See main text for further information on how the values for the [110]-orientation were obtained. The parameters in this table are valid for light of vacuum wavelength around 3500nm and for acoustic frequencies in the GHz-range. We were unable to retrieve reliable data for some material parameters of amorphous Si3N4 in the target frequency range.

Equations (9)

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E = e ( 1 ) ( r , t ) a ( 1 ) ( z , t ) + e ( 2 ) ( r , t ) a ( 2 ) ( z , t ) + c . c . ,
= e ˜ ( 1 ) a ( 1 ) exp ( i β ( 1 ) z i ω ( 1 ) t ) + e ˜ ( 2 ) a ( 2 ) exp ( i β ( 1 ) z i ω ( 1 ) t ) + c . c . ,
e ˜ ( 1 ) [ e ˜ ( 2 ) ] * ; β ( 1 ) β ( 2 ) ; 𝒫 ( 1 ) 𝒫 ( 2 ) .
U ( r , t ) = u ( r , t ) b ( z , t ) + c . c . = u ˜ ( x , y ) b ( z , t ) exp ( i q z i Ω t ) + c . c . ;
with ρ Ω 2 u ˜ i + i j k l ( + i q z ^ ) j c i j k l ( + i q z ^ ) k u ˜ l = 0 .
q = β ( 2 ) β ( 1 ) 2 β ( 2 ) , Ω = ω ( 2 ) ω ( 1 ) .
z P ( 1 ) = Γ P ( 1 ) P ( 2 ) , z P ( 2 ) = Γ P ( 1 ) P ( 2 ) , Γ = 2 ω Ω | Q | 2 𝒫 ( 1 ) 𝒫 ( 2 ) 𝒫 b α
α = Ω 2 𝒫 b { d 2 r j k l u i * j η i j k l k u l } ,
Q = 𝒞 d r ( u * n ^ ) [ ( ε a ε b ) ε 0 ( n ^ × e ( 2 ) ) ( n ^ × e ( 2 ) ) ( ε b 1 ε a 1 ) ε 0 1 ( n ^ d ( 2 ) ) ( n ^ d ( 2 ) ) ] + ε 0 d 2 r i j k l e i ( 2 ) e j ( 2 ) ε r 2 p i j k l k u l * + i Ω μ 0 ε 0 d 2 r ( ε r 1 ) u * ( e ( 2 ) × h ( 2 ) ) .

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