Abstract

We derive formal selection rules for Stimulated Brillouin Scattering (SBS) in structured waveguides. Using a group-theoretical approach, we show how the waveguide symmetry determines which optical and acoustic modes interact for both forward and backward SBS. We present a general framework for determining this interaction and give important examples for SBS in waveguides with rectangular, triangular and hexagonal symmetry. The important role played by degeneracy of the optical modes is illustrated. These selection rules are important for SBS-based device design and for a full understanding the physics of SBS in structured waveguides.

© 2014 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics (Academic, 3rd edition, 2003).
  2. L. Brillouin., “Diffusion de la lumière par un corps transparent homogène,” Annals of Physics 17, 88–122 (1922).
  3. R. Y. Chiao, C. H. Townes, and B. P Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592 (1964).
    [Crossref]
  4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 5th edition, 2012).
  5. I. V. Kabakova, R. Pant, D.-Y. Choi, S. Debbarma, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Narrow linewidth Brillouin laser based on chalcogenide photonic chip,” Opt. Lett. 38, 3208–3211 (2013).
    [Crossref] [PubMed]
  6. K. Hu, I. V. Kabakova, T. F. S. Büttner, S. Lefrancois, D. D. Hudson, S. He, and B. J. Eggleton, “Low-threshold Brillouin laser at 2μm based on suspended-core chalcogenide fiber,” Opt. Lett. 39, 4651–4654 (2014).
    [Crossref] [PubMed]
  7. X. Huang and S. Fan, “Complete all-optical silica fiber isolator via Stimulated Brillouin Scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
    [Crossref]
  8. B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
    [Crossref]
  9. P. Dainese, P. St. J. Russell, N. Joly, J. C. Knight, G. S. Wiederhecker, H. L. Fragnito, V. Laude, and A. Khelif, “Stimulated Brillouin scattering from multi-GHz-guided acoustic phonons in nanostructured photonic crystal fibres,” Nat. Phys. 2, 388–392 (2006).
    [Crossref]
  10. M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
    [Crossref]
  11. C. Florea, M. Bashkansky, Z. Dutton, J. Sanghera, P. Pureza, and I. Aggarwal, “Stimulated Brillouin scattering in single-mode As2S3 and As2Se3 chalcogenide fibers,” Opt. Express 14, 12063–12070 (2006).
    [Crossref] [PubMed]
  12. H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
    [Crossref]
  13. P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18, 14439–14453 (2010).
    [Crossref] [PubMed]
  14. 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).
  15. R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thévenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “On-chip stimulated Brillouin scattering,” Opt. Express 19, 8285–8290 (2011).
    [Crossref] [PubMed]
  16. 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).
  17. W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Solja, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
    [Crossref]
  18. P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - I: Summary of Results,” IEEE Trans. Microw. Theory Tech. 23, 421–429 (1975).
    [Crossref]
  19. P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - IT: Theory,” IEEE Trans. Microw. Theory Tech. 23, 429–433 (1975).
    [Crossref]
  20. J. M. Hollas, Modern spectroscopy (Wiley, 4th edition, 2004).
  21. D. F. Nelson and M. Lax, “Theory of the Photoelastic Interaction,” Phys. Rev. B 3, 2778 (1971).
    [Crossref]
  22. M. Lax, Symmetry Principles in Solid State and Molecular Physics (Dover Publications Inc., 1974).
  23. M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Phyics of Condensed Matter (Springer, 2008).
  24. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Optics Express,  10, 1342–1346 (2002).
    [Crossref] [PubMed]
  25. C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
    [Crossref]

2014 (2)

K. Hu, I. V. Kabakova, T. F. S. Büttner, S. Lefrancois, D. D. Hudson, S. He, and B. J. Eggleton, “Low-threshold Brillouin laser at 2μm based on suspended-core chalcogenide fiber,” Opt. Lett. 39, 4651–4654 (2014).
[Crossref] [PubMed]

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

2013 (3)

2012 (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).

2011 (2)

2010 (1)

2009 (1)

M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
[Crossref]

2006 (2)

C. Florea, M. Bashkansky, Z. Dutton, J. Sanghera, P. Pureza, and I. Aggarwal, “Stimulated Brillouin scattering in single-mode As2S3 and As2Se3 chalcogenide fibers,” Opt. Express 14, 12063–12070 (2006).
[Crossref] [PubMed]

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

2002 (1)

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Optics Express,  10, 1342–1346 (2002).
[Crossref] [PubMed]

1994 (1)

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

1975 (2)

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - I: Summary of Results,” IEEE Trans. Microw. Theory Tech. 23, 421–429 (1975).
[Crossref]

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - IT: Theory,” IEEE Trans. Microw. Theory Tech. 23, 429–433 (1975).
[Crossref]

1971 (1)

D. F. Nelson and M. Lax, “Theory of the Photoelastic Interaction,” Phys. Rev. B 3, 2778 (1971).
[Crossref]

1964 (1)

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

1922 (1)

L. Brillouin., “Diffusion de la lumière par un corps transparent homogène,” Annals of Physics 17, 88–122 (1922).

Aggarwal, I.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 5th edition, 2012).

Argyros, A.

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Optics Express,  10, 1342–1346 (2002).
[Crossref] [PubMed]

Bashkansky, M.

Bassett, I. M.

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Optics Express,  10, 1342–1346 (2002).
[Crossref] [PubMed]

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

Boyd, R. W.

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

Brenn, A.

M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
[Crossref]

Brillouin., L.

L. Brillouin., “Diffusion de la lumière par un corps transparent homogène,” Annals of Physics 17, 88–122 (1922).

Büttner, T. F. S.

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).

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, 592 (1964).
[Crossref]

Choi, D.-Y.

Cox, J. A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

Dainese, P.

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

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).

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18, 14439–14453 (2010).
[Crossref] [PubMed]

de Sterke, C. M.

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

Debbarma, S.

Dong, H.

Dresselhaus, G.

M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Phyics of Condensed Matter (Springer, 2008).

Dresselhaus, M. S.

M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Phyics of Condensed Matter (Springer, 2008).

Dutton, Z.

Eggleton, B. J.

Fan, S.

Florea, C.

Fragnito, H. L.

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

He, S.

Hile, S.

Hollas, J. M.

J. M. Hollas, Modern spectroscopy (Wiley, 4th edition, 2004).

Hu, K.

Huang, X.

Hudson, D. D.

Jarecki, R.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

Joly, N.

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

Jorio, A.

M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Phyics of Condensed Matter (Springer, 2008).

Kabakova, I. V.

Kang, M. S.

M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
[Crossref]

Khelif, A.

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

Knight, J. C.

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

Laude, V.

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

Lax, M.

D. F. Nelson and M. Lax, “Theory of the Photoelastic Interaction,” Phys. Rev. B 3, 2778 (1971).
[Crossref]

M. Lax, Symmetry Principles in Solid State and Molecular Physics (Dover Publications Inc., 1974).

Lefrancois, S.

Li, E.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

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

Luther-Davies, B.

Madden, S.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Madden, S. J.

Marpaung, D.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Mcfarlane, H.

McIsaac, P. R.

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - I: Summary of Results,” IEEE Trans. Microw. Theory Tech. 23, 421–429 (1975).
[Crossref]

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - IT: Theory,” IEEE Trans. Microw. Theory Tech. 23, 429–433 (1975).
[Crossref]

Morrison, B.

B. Morrison, D. Marpaung, R. Pant, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Tunable microwave photonic notch filter using on-chip stimulated Brillouin scattering,” Opt. Commun. 313, 85–89 (2014).
[Crossref]

Nazarkin, A.

M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
[Crossref]

Nelson, D. F.

D. F. Nelson and M. Lax, “Theory of the Photoelastic Interaction,” Phys. Rev. B 3, 2778 (1971).
[Crossref]

Olsson Ill, R. H.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

Pant, R.

Poulton, C. G.

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

Pureza, P.

Qiu, W.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Solja, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[Crossref]

Rakich, P. T.

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Solja, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[Crossref]

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[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).

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18, 14439–14453 (2010).
[Crossref] [PubMed]

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).

Russell, P. St. J.

M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. 5, 276–280 (2009).
[Crossref]

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

Sanghera, J.

Shin, H.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Solja, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[Crossref]

Solja, M.

Starbuck, A.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

Steel, M. J.

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).

Stoicheff, B. P

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

Street, A. G.

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

Thévenaz, L.

Townes, C. H.

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

Wang, Z.

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

W. Qiu, P. T. Rakich, H. Shin, H. Dong, M. Solja, and Z. Wang, “Stimulated Brillouin scattering in nanoscale silicon step-index waveguides: a general framework of selection rules and calculating SBS gain,” Opt. Express 21, 31402–31419 (2013).
[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).

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express 18, 14439–14453 (2010).
[Crossref] [PubMed]

Wiederhecker, G. S.

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

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).

Annals of Physics (1)

L. Brillouin., “Diffusion de la lumière par un corps transparent homogène,” Annals of Physics 17, 88–122 (1922).

IEEE Trans. Microw. Theory Tech. (2)

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - I: Summary of Results,” IEEE Trans. Microw. Theory Tech. 23, 421–429 (1975).
[Crossref]

P. R. McIsaac, “Symmetry-Induced Modal Characteristics of Uniform Waveguides - IT: Theory,” IEEE Trans. Microw. Theory Tech. 23, 429–433 (1975).
[Crossref]

J. Appl. Phys. (1)

C. M. de Sterke, I. M. Bassett, and A. G. Street, “Differential losses in Bragg fibers,” J. Appl. Phys. 76, 680–688 (1994).
[Crossref]

J. Lightwave Technol. (1)

Nat. Comm. (1)

H. Shin, W. Qiu, R. Jarecki, J. A. Cox, R. H. Olsson Ill, A. Starbuck, Z. Wang, and P. T. Rakich, “Tailorable stimulated Brillouin scattering in nanoscale silicon waveguides,” Nat. Comm. 4, 1944 (2013).
[Crossref]

Nat. Phys. (2)

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Figures (5)

Fig. 1
Fig. 1

Illustration of three different waveguide groups, divided according to the waveguide symmetry; a) rectangular waveguides (��2v group), b) triangular waveguides (��3v group) and c) hexagonal waveguides (��6v group).

Fig. 2
Fig. 2

Electric field patterns for the fundamental optical modes in a structured silica fiber with hexagonal symmetry. The arrows represent the transversal field components. The color represents the time-averaged power density directed along the waveguide axis (i.e. the z-component of the time-averaged Poynting vector). The color scale is in arbitrary units.

Fig. 3
Fig. 3

Examples for the symmetries of acoustic modes in a structured silica fiber with hexagonal symmetry. The arrows and the deformation plot indicate the in-plane components u = (ux, uy, 0) of the mechanical displacement field. The color represents the absolute value of the total displacement field | u | = | u x | 2 + | u y | 2 + | u z | 2. The color scale is in arbitrary units.

Fig. 4
Fig. 4

Typical examples for combinations of symmetric eigenmodes that can have non-zero opto-acoustic overlap for the point groups ��2v (left hand side) and ��3v (right hand side). Each line contains sketches of two optical modes (left and middle column) and one acoustic mode (right column). The symmetry of the undistorted waveguide is represented by a black polygon, the effect of distortion is sketched in light gray. Red arrows indicate the general behavior of the major component of the optical modes’ electric fields. Blue arrows indicate the general behavior of the acoustic displacement field.

Fig. 5
Fig. 5

Typical examples for combinations of symmetric eigenmodes that can have non-zero opto-acoustic overlap for the point group ��6v. See the caption of Fig. 4 for a more detailed description of what is shown here.

Tables (2)

Tables Icon

Table 1 Relevant parts of the character tables for the groups ��2v, ��3v and ��6v. These tables are standard in many textbooks on group theory and we reproduce them here for convenience following the notation of [22].

Tables Icon

Table 2 Relevant parts of the product decomposition tables for ��3v and ��6v. The abbreviations “i.d.m.” and “o.d.m.” in the decomposition tables stand for “identical degenerate modes” and “orthogonal degenerate modes”, respectively. See Section 3. for details. Again, these tables are standard in many textbooks on group theory such as [22].

Equations (8)

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

Q = d 2 r u * ( r ) f ( r ) = d 2 r w ( r ) ,
Γ = 2 ω Ω | Q | 2 𝒫 3 { 1 α i κ } ,
w ( r ) = ε 0 [ ε r ( r ) ] 2 i j k l p i j k l ( r ) [ e i ( 1 ) ] * e j ( 2 ) k u l * electrostriction + ε 0 ( ε r ( a ) ε r ( b ) ) i j k l M i j k l [ e i ( 1 ) ] * e j ( 2 ) n k u l * radiation pressure .
M i j k l ( r ) = [ ( 1 δ i k ) ( 1 δ j k ) ε r ( a ) ε r ( b ) δ i k δ j k ] δ k l δ ( ) = M j i k l = M k l i j .
R ^ ϕ ( r ) = φ ( 1 r ) .
[ R ^ v ( r ) ] i = j v j ( 1 r ) j i ; [ R ^ T _ ( r ) ] i j = k l T k l ( 1 r ) k i l j .
R ^ f ( i ) = j f ( j ) 𝒟 ( R ^ ) j i .
w = i j k N i j k [ e i ( 1 ) ] * e j ( 2 ) u k * ,

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