Abstract

We theoretically investigate a new class of silicon waveguides for achieving Stimulated Brillouin Scattering (SBS) in the mid-infrared (MIR). The waveguide consists of a rectangular core supporting a low-loss optical mode, suspended in air by a series of transverse ribs. The ribs are patterned to form a finite quasi-one-dimensional phononic crystal, with the complete stopband suppressing the transverse leakage of acoustic waves, confining them to the core of the waveguide. We derive a theoretical formalism that can be used to compute the opto-acoustic interaction in such periodic structures, and find forward intramodal-SBS gains up to 1750 m−1W−1, which compares favorably with the proposed MIR SBS designs based on buried germanium waveguides. This large gain is achieved thanks to the nearly complete suppression of acoustic radiative losses.

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

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References

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2018 (3)

M. Merklein, B. Stiller, and B. J. Eggleton, “Brillouin-based light storage and delay techniques,” J. Opt. 20, 083003 (2018).
[Crossref]

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

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

2017 (4)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774 (2017).
[Crossref]

R. Zhang and J. Sun, “Design of silicon phoxonic crystal waveguides for slow light enhanced forward stimulated Brillouin scattering,” J. Light. Technol. 35, 2917–2925 (2017).
[Crossref]

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

2016 (6)

2015 (4)

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

D. Marpaung, B. Morrison, M. Pagani, R. Pant, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Low-power, chip-based stimulated Brillouin scattering microwave photonic filter with ultrahigh selectivity,” Optica 2, 76–83 (2015).
[Crossref]

2014 (4)

2013 (2)

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

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

2012 (1)

P.-L. Yu, T. Purdy, and C. Regal, “Control of material damping in high-Q membrane microresonators,” Phys. Rev. Lett. 108, 083603 (2012).
[Crossref]

2011 (3)

N. K. Hon, R. 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]

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]

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. L. 23, 1775 (2011).
[Crossref]

2010 (2)

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

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

2007 (1)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

2006 (1)

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006).
[Crossref]

1980 (1)

H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[Crossref]

Alonso-Ramos, C.

Alù, A.

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774 (2017).
[Crossref]

Asatryan, A. A.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Baets, R.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

Barg, A.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

Bazin, A.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

Behunin, R. O.

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

Bereyhi, M. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Bontemps, J.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Boodhoo, L. A.

Botten, L. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Boyd, R. W.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

R. W. Boyd, Nonlinear optics (Elsevier, 2003).

Büttner, T. F.

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

Cheben, P.

Chen, P. Y.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Choi, D.-Y.

Cox, J. A.

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

de Sterke, C. M.

M. J. A. Smith, B. T. Kuhlmey, C. M. de Sterke, C. Wolff, M. Lapine, and C. G. Poulton, “Metamaterial control of stimulated Brillouin scattering,” Opt. Lett. 41, 2338–2341 (2016).
[Crossref]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Dieterle, P. B.

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Eggleton, B.

Eggleton, B. J.

M. Merklein, B. Stiller, and B. J. Eggleton, “Brillouin-based light storage and delay techniques,” J. Opt. 20, 083003 (2018).
[Crossref]

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

D. Marpaung, B. Morrison, M. Pagani, R. Pant, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Low-power, chip-based stimulated Brillouin scattering microwave photonic filter with ultrahigh selectivity,” Optica 2, 76–83 (2015).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

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

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]

Engelsen, N. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Fedorov, S. A.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Fink, J. M.

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Gauthier, D. J.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

Ghadimi, A. H.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Gröblacher, S.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

Halir, R.

Hile, S.

Hill, J. T.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

Hon, N. K.

N. K. Hon, R. 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]

Jalali, B.

N. K. Hon, R. 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]

Jarecki, R.

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

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University Press, 2011).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University Press, 2011).

Kabakova, I. V.

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

Kalaee, M.

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Kang, S.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Khokhar, A.

Khokhar, A. Z.

Kippenberg, T. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Kittlaus, E. A.

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

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10, 463 (2016).
[Crossref]

Koning, J.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Kuhlmey, B. T.

Kuyken, B.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

Lapine, M.

Leonardis, F. D.

Li, E.

Li, H.

H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[Crossref]

Luther-Davies, B.

Madden, S. J.

Maldovan, M.

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006).
[Crossref]

Marpaung, D.

Mashanovich, G. Z.

Mcfarlane, H.

McPhedran, R. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University Press, 2011).

Meenehan, S.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

Merklein, M.

M. Merklein, B. Stiller, and B. J. Eggleton, “Brillouin-based light storage and delay techniques,” J. Opt. 20, 083003 (2018).
[Crossref]

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

Minasian, R. A.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. L. 23, 1775 (2011).
[Crossref]

Mirhosseni, M.

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Molina-Fernández, I.

Morrison, B.

Nedeljkovic, M.

Olsson, R. H.

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

Ortega-Moñux, A.

Otterstrom, N. T.

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

Pagani, M.

Painter, O.

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Pant, R.

Passaro, V. M. N.

Penadés, J. S.

Peruzzo, M.

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

Polzik, E. S.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

Poulton, C.

Poulton, C. G.

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

M. J. A. Smith, B. T. Kuhlmey, C. M. de Sterke, C. Wolff, M. Lapine, and C. G. Poulton, “Metamaterial control of stimulated Brillouin scattering,” Opt. Lett. 41, 2338–2341 (2016).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

C. Wolff, M. J. Steel, and C. G. Poulton, “Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers,” Opt. Express 22, 32489–32501 (2014).
[Crossref]

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

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]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Purdy, T.

P.-L. Yu, T. Purdy, and C. Regal, “Control of material damping in high-Q membrane microresonators,” Phys. Rev. Lett. 108, 083603 (2012).
[Crossref]

Qiu, W.

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

Qu, Z.

Rakich, P. T.

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

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10, 463 (2016).
[Crossref]

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

Regal, C.

P.-L. Yu, T. Purdy, and C. Regal, “Control of material damping in high-Q membrane microresonators,” Phys. Rev. Lett. 108, 083603 (2012).
[Crossref]

Ruigrok, J.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Safavi-Naeini, A. H.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016).
[Crossref]

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

Sarabalis, C. J.

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016).
[Crossref]

Schilling, R.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Schliesser, A.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

Shin, H.

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10, 463 (2016).
[Crossref]

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

Sipe, J.

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

Smith, M. J. A.

Soref, R.

C. Wolff, R. Soref, C. Poulton, and B. Eggleton, “Germanium as a material for stimulated Brillouin scattering in the mid-infrared,” Opt. Express 22, 30735–30747 (2014).
[Crossref]

N. K. Hon, R. 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]

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

Soref, R. A.

Sounas, D. L.

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774 (2017).
[Crossref]

Starbuck, A.

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

Steel, M.

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

Steel, M. J.

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

C. Wolff, M. J. Steel, and C. G. Poulton, “Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers,” Opt. Express 22, 32489–32501 (2014).
[Crossref]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

Steeneken, P.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Stiller, B.

M. Merklein, B. Stiller, and B. J. Eggleton, “Brillouin-based light storage and delay techniques,” J. Opt. 20, 083003 (2018).
[Crossref]

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

Sun, J.

R. Zhang and J. Sun, “Design of silicon phoxonic crystal waveguides for slow light enhanced forward stimulated Brillouin scattering,” J. Light. Technol. 35, 2917–2925 (2017).
[Crossref]

Thevenaz, L.

Thomas, E. L.

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006).
[Crossref]

Troia, B.

Tsaturyan, Y.

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

Van Beek, J.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

Van Laer, R.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

Van Thourhout, D.

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

Wang, Z.

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

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

Wangüemert-Pérez, J.

Wilson, D. J.

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University Press, 2011).

Wolff, C.

Yu, P.-L.

P.-L. Yu, T. Purdy, and C. Regal, “Control of material damping in high-Q membrane microresonators,” Phys. Rev. Lett. 108, 083603 (2012).
[Crossref]

Zhang, R.

R. Zhang and J. Sun, “Design of silicon phoxonic crystal waveguides for slow light enhanced forward stimulated Brillouin scattering,” J. Light. Technol. 35, 2917–2925 (2017).
[Crossref]

Zhang, W.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. L. 23, 1775 (2011).
[Crossref]

Zhu, Z.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

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

APL Photonics (1)

C. J. Sarabalis, J. T. Hill, and A. H. Safavi-Naeini, “Guided acoustic and optical waves in silicon-on-insulator for Brillouin scattering and optomechanics,” APL Photonics 1, 071301 (2016).
[Crossref]

Appl. Phys. Lett. (1)

M. Maldovan and E. L. Thomas, “Simultaneous localization of photons and phonons in two-dimensional periodic structures,” Appl. Phys. Lett. 88, 251907 (2006).
[Crossref]

IEEE Photonics Technol. L. (1)

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photonics Technol. L. 23, 1775 (2011).
[Crossref]

J. Appl. Phys. (1)

N. K. Hon, R. 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]

J. Light. Technol. (1)

R. Zhang and J. Sun, “Design of silicon phoxonic crystal waveguides for slow light enhanced forward stimulated Brillouin scattering,” J. Light. Technol. 35, 2917–2925 (2017).
[Crossref]

J. Opt. (1)

M. Merklein, B. Stiller, and B. J. Eggleton, “Brillouin-based light storage and delay techniques,” J. Opt. 20, 083003 (2018).
[Crossref]

J. Phys. Chem. Ref. Data (1)

H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[Crossref]

Nat. Commun (2)

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

M. Merklein, I. V. Kabakova, T. F. Büttner, D.-Y. Choi, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, “Enhancing and inhibiting stimulated Brillouin scattering in photonic integrated circuits,” Nat. Commun 6, 6396 (2015).
[Crossref]

Nat. Nanotechnol. (1)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nanotechnol. 12, 776 (2017).
[Crossref]

Nat. Photonics (3)

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

E. A. Kittlaus, H. Shin, and P. T. Rakich, “Large Brillouin amplification in silicon,” Nat. Photonics 10, 463 (2016).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774 (2017).
[Crossref]

New J. Phys. (3)

R. Van Laer, A. Bazin, B. Kuyken, R. Baets, and D. Van Thourhout, “Net on-chip Brillouin gain based on suspended silicon nanowires,” New J. Phys. 17, 115005 (2015).
[Crossref]

C. Wolff, B. Stiller, B. J. Eggleton, M. J. Steel, and C. G. Poulton, “Cascaded forward Brillouin scattering to all Stokes orders,” New J. Phys. 19, 023021 (2017).
[Crossref]

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

Opt. Express (4)

Opt. Lett. (3)

Optica (1)

Phys. Rev. A (2)

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, “Group velocity in lossy periodic structured media,” Phys. Rev. A 82, 053825 (2010).
[Crossref]

C. Wolff, M. J. Steel, B. J. Eggleton, and C. G. Poulton, “Stimulated Brillouin scattering in integrated photonic waveguides: forces, scattering mechanisms, and coupled-mode analysis,” Phys. Rev. A 92, 013836 (2015).
[Crossref]

Phys. Rev. Lett. (2)

A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, “Two-dimensional phononic-photonic band gap optomechanical crystal cavity,” Phys. Rev. Lett. 112, 153603 (2014).
[Crossref]

P.-L. Yu, T. Purdy, and C. Regal, “Control of material damping in high-Q membrane microresonators,” Phys. Rev. Lett. 108, 083603 (2012).
[Crossref]

Science (3)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[Crossref] [PubMed]

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

A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, “Elastic strain engineering for ultralow mechanical dissipation,” Science 360, 764–768 (2018).
[Crossref]

Other (5)

COMSOL, AB, COMSOL Multiphysics v4.4.

P. Steeneken, J. Ruigrok, S. Kang, J. Van Beek, J. Bontemps, and J. Koning, “Parameter extraction and support-loss in MEMS resonators,” arXiv preprint arXiv:1304.7953 (2013).

M. Kalaee, M. Mirhosseni, P. B. Dieterle, M. Peruzzo, J. M. Fink, and O. Painter, “Quantum electromechanics of a hypersonic crystal,” arXiv preprint arXiv:1808.04874 (2018).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton University Press, 2011).

R. W. Boyd, Nonlinear optics (Elsevier, 2003).

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Figures (5)

Fig. 1
Fig. 1 (a) A section of the suspended waveguide setup, with 4 unit cells along z axis, and a zoomed in section of the waveguide. Patterned ribs, spaced by Λ, connect the central waveguide with rectangular cross section (ax × ay), with the slab region. While the entire structure is made of silicon (Si), Matching Layers (ML) are introduced in the slab region, using built-in COMSOL procedures for optics [24], and following the method described in [25] for acoustics. Unless otherwise specified, throughout this work we will be discussing structures with (ax, ay, Λ) = (1.2, 0.3, 1.25) μm (see the discussion in the main text), and ribs with square cross section (0.18 μm)2. The patterning of ribs is discussed in Section 2. (b) Phase-matching in intra-mode Forward Stimulated Brillouin Scattering (intra-mode FBS [14, 19]). Coupling between the optical pump and Stokes waves is mediated by the co-propagating acoustic wave, fulfilling the phase-matching conditions kS + q = kp and ωS + Ω = ωp. (c) Cartoon representations of the electric field E of the fundamental optical TE fields of the pump and Stokes waves (upper panel), and the displacement field u of the lowest-order symmetry-allowed acoustic mode [26], associated with lateral stretching of the waveguide (lower panel). Exact profiles of the two modes are discussed in the following figures.
Fig. 2
Fig. 2 Optical waveguiding inside the central bar. (a,b) Spatial distribution of the dominant Ex component of the electric field (upper panel) and the P z component of the Poynting vector (lower panel) in the planes marked in the panels on the right. (c) Dispersion relation of the rectangular silicon waveguide with dimensions (1.2, 0.3) μm, unsupported (dashed blue line) and supported by the unpatterned ribs spaced by Λ = 1.2 μm (solid orange line). The horizontal line denotes the λ0 = 4 μm waveguiding wavelength.
Fig. 3
Fig. 3 Engineering acoustic stopband by patterning ribs. (a) Schematic of a unit cell of the phononic crystal. (b) Acoustic modes of the one-dimensional phononic crystal, calculated at positions marked with colored dots in (c). (c) Dispersion diagram of the one-dimensional phononic crystals for periodicity L = 2.25 μm and (dx, dy, dz) = (340, 120, 120) nm along the axis, patterned in the rib with square cross-section (wy, wz) = (180, 180) nm, used throughout the rest of the work. Green, blue and red lines denote the longitudinal (compressional), flexural and torsional modes, marked on the dispersion diagrams with circles. Hybridization is marked by intermediate colors. Lines are guides only. The complete acoustic stopband is centered at 3.35 GHz, and can be tuned to overlap with the mechanical resonance of the central waveguide by changing the length of the unit cell L, as shown in (d), where we mark the stopbands as shaded areas.
Fig. 4
Fig. 4 (a) Displacement field distribution of a lateral stretching mode of an unsupported waveguide. (b) Frequency and (c) mechanical quality factor Qm of the waveguide suspended by patterned ribs as a function of the core width ax. Stopband of the phononic crystal is marked as the gray area. The ribs comprise 4 unit cells of the phononic crystal on each side of the waveguide. (d,e) Schematics of the displacement fields in the unit cell, associated with modes marked in (a) as A and B. Plots in (e) demonstrate the displacement field calculated along the x axis, passing through the centres of the ribs. For the mode A with resonance within the stopband the displacement field decays exponentially over the length of the unit cells of the phononic crystal.
Fig. 5
Fig. 5 Acoustic frequencies (a), mechanical quality factors Qm(b) and Brillouin gain coefficients Γ (c) of suspended waveguide systems, calculated for the ribs comprised of an increasing number of unit cells of the phononic crystal. Orange crosses and green dots represent systems with patterned, and unpatterned ribs, respectively. Inset in (a) traces two anti-crossing mechanical modes, while the inset in (b) shows the calculated radiative mechanical quality factor Qm,rad of patterned waveguide, as a function of the ribs length, with an exponential fit. The dashed blue lines in (b) and (c) denote the values obtained for unsupported waveguides.

Equations (28)

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

U ( r ) = b ( z ) u q ( r ) = b ( z ) u ˜ q ( r ) e i q z ,
u ˜ q ( r + L z ^ ) = u ˜ q ( r ) ,
E ( i ) ( r ) = a ( i ) ( z ) e k ( i ) ( i ) ( r ) = a ( i ) ( z ) e ˜ k ( i ) ( i ) ( r ) e i k ( i ) z ,
Γ 4 ω Q m | 𝒬 1 ( PE ) + 𝒬 1 ( MB ) | 2 b 𝒫 ( 1 ) 𝒫 ( 2 ) ,
𝒬 1 ( PE ) ( z ) = ε 0 d 2 r ε a 2 i j k l [ e i ( 1 ) ] * e j ( 2 ) p i j k l k u l * ,
𝒬 1 ( MB ) ( z ) = A d r ( u * n ^ ) [ ( ε a ε b ) ε 0 ( n ^ × e ( 1 ) ) * ( n ^ × e ( 2 ) ) ( 1 ε a 1 ε b ) 1 ε 0 ( n ^ · d ( 1 ) ) * ( n ^ · d ( 2 ) ) ] .
Q m 1 = Q m , visc 1 + Q m , rad 1 .
( i ) ( z ) = 2 ε 0 d 2 r ε ( r ) [ e ( i ) ( r ) ] * e ( i ) ( r ) ,
𝒫 ( i ) ( z ) = 2 d 2 r z ^ ( [ e ( i ) ( r ) ] * × h ( i ) ( r ) ) .
v E = 𝒫 ( i ) ( i ) .
v E = 𝒫 ( i ) ( z ) ( i ) ( z ) .
i ω ( 1 ) [ 𝒫 ( 1 ) ( z ) z a ( 1 ) ( z ) + ( 1 ) ( z ) t a ( 1 ) ( z ) ] = a ( 2 ) ( z ) b * ( z ) ( i ω ( 1 ) ) 2 𝒬 1 ( z ) ,
𝒬 i ( z ) = d 2 r [ [ e ( i ) ( r ) ] * Δ d ( i ) ( r ) [ d ( i ) ( r ) ] * Δ e ( i ) ( r ) μ 0 [ h ( i ) ( r ) ] * Δ h ( i ) ( r ) ] .
z z + L d z [ z a ( 1 ) ( z ) ] z = z 𝒫 ( 1 ) ( z ) z a ( 1 ) ( z ) z z + L d z 𝒫 ( 1 ) ( z ) = z a ( 1 ) ( z ) 𝒫 ( 1 ) L ,
z a ( 1 ) ( z ) + 1 v ( 1 ) t a ( 1 ) ( z ) = i ω ( 1 ) a ( 2 ) ( z ) b * ( z ) 𝒬 1 𝒫 ( 1 ) ,
z a ( 2 ) ( z ) + 1 v ( 2 ) t a ( 2 ) ( z ) = i ω ( 2 ) a ( 1 ) ( z ) b ( z ) 𝒬 2 𝒫 ( 2 ) ,
i Ω j k l [ ( c i z k l k + j c i j z l ) u l ( r ) z b ( z ) 2 i Ω ρ ( r ) u i ( r ) t b ( z ) + ( j η i j k l k u l ( r ) ) b ( z ) + [ a ( 1 ) ( z ) ] * a ( 2 ) ( z ) f i ] + c . c . = 0 .
b ( z ) = 2 Ω 2 d 2 r ρ ( r ) | u ( r ) | 2 , 𝒫 b ( z ) = 2 i Ω d 2 r i k l c z i k l [ u i ( r ) ] * k u l ( r ) ,
𝒫 b ( z ) z b ( z ) + b ( z ) t b ( z ) + b ( z ) α 𝒫 b ( z ) = i Ω [ a ( 1 ) ( z ) ] * a ( 2 ) ( z ) 𝒬 b ( z ) ,
𝒬 b ( z ) = d 2 r [ u ( x , y ; z ) ] * f ( x , y ; z )
α 𝒫 b ( z ) = Ω 2 d 2 r i j k l [ j u i * ( r ) ] η i j k l k u l ( r ) ,
z b ( z ) + 1 v b t b ( z ) + α 𝒫 b 𝒫 b b ( z ) = i Ω [ a ( 1 ) ( z ) ] * a ( 2 ) ( z ) 𝒬 b 𝒫 b ,
b ( z ) i Ω [ a ( 1 ) ( z ) ] * a ( 2 ) ( z ) 𝒬 b α 𝒫 b ,
z a ( 1 ) ( z ) = ω ( 1 ) Ω a ( 1 ) ( z ) | a ( 2 ) ( z ) | 2 𝒬 1 * 𝒬 b 𝒫 ( 1 ) α 𝒫 b ,
z a ( 2 ) ( z ) = ω ( 2 ) Ω a ( 2 ) ( z ) | a ( 1 ) ( z ) | 2 𝒬 2 𝒬 b * 𝒫 ( 2 ) α 𝒫 b .
Γ = 2 ω Θ [ 𝒬 b * 𝒬 1 ] α 𝒫 b 𝒫 ( 1 ) 𝒫 ( 2 ) .
Q b = ( Ω ˜ ) 2 ( Ω ) α ( Ω ˜ ) v b = ( Ω ˜ ) 2 Q b v b = Ω 2 Q b v b ,
Γ = 2 ω Θ [ 𝒬 b * 𝒬 1 ] α 𝒫 b 𝒫 ( 1 ) 𝒫 ( 2 ) = 4 ω Q b v b [ 𝒬 b * 𝒬 1 ] 𝒫 b 𝒫 ( 1 ) 𝒫 ( 2 ) = 4 ω Q b | 𝒬 1 | 2 b 𝒫 ( 1 ) 𝒫 ( 2 ) .

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