Abstract

Chalcogenide nanostructures are an interesting platform for ultra-fast nonlinear signal processing. However, stimulated Brillouin scattering (SBS) can represent a major obstacle to achieving this goal for continuous wave pumps, as it depletes the high-power pump wave. In this paper, we assess the SBS gain in an As2Se3 nanowire, which is found to be much larger than the SBS gain in As2Se3 fibers. Such a large SBS gain poses a severe problem, and standard concepts for fibers, such as Bragg gratings and tapers are found not to work for nanowires. However, through the introduction of an amorphous polycarbonate cladding, the elastic modes and the optical mode can be separated, and SBS can thus be efficiently suppressed by three and a half-orders of magnitude. At the same time, flexibility for other design goals, such as phase matching, is maintained.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).
  5. P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).
  6. A. W. Snyder, J. Love, Optical Waveguide Theory (Springer, 1983).
  7. D. Royer, E. Dieulesaint, Elastic Waves in Solids II: Generation, Acousto-optic Interaction, Applications (Springer, 2000).
  8. B. A. Auld, Acoustic Fields and Waves in Solids (Wiley, 1990).
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    [CrossRef]
  10. R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
    [CrossRef]
  11. Y. Ohmachi, N. Uchida, “Vitreous As2Se3: investigation of acousto-optical properties and application to infrared modulator,” J. Appl. Phys. 43, 1709–1712 (1972).
    [CrossRef]
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    [CrossRef]
  13. Z. Cimpl, F. Kosek, “Refractive index of As2–xSbxS3 and As2–xSbxSe3 systems,” Phys. Status Solidi A 93, K55–K58 (1986).
    [CrossRef]
  14. K. S. Abedin, “Observation of strong stimulated Brillouin scattering in single-mode As2Se3 chalcogenide fiber,” Opt. Express 13, 10266–10271 (2005).
    [CrossRef]
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    [CrossRef]
  17. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
    [CrossRef]
  18. X. Gai, S. Madden, D.-Y. Choi, D. Bulla, B. Luther-Davies, “Dispersion engineered Ge11.5As24Se64.5 nanowires with a nonlinear parameter of 136  W−1  m−1 at 1550  nm,” Opt. Express 18, 18866–18874 (2010).
    [CrossRef]
  19. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
    [CrossRef]
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  22. R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
    [CrossRef]
  23. H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
    [CrossRef]
  24. L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, 1986).

2012

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

T. Cheng, M. Liao, W. Gao, Z. Duan, T. Suzuki, Y. Ohishi, “Suppression of stimulated Brillouin scattering in all-solid chalcogenide-tellurite photonic bandgap fiber,” Opt. Express 20, 28846–28854 (2012).
[CrossRef]

2010

2008

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

2005

2002

1990

R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

1987

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

1986

Z. Cimpl, F. Kosek, “Refractive index of As2–xSbxS3 and As2–xSbxSe3 systems,” Phys. Status Solidi A 93, K55–K58 (1986).
[CrossRef]

1979

L. N. Durvasula, R. W. Gammon, “Brillouin scattering from shear waves in amorphous polycarbonate,” J. Appl. Phys. 50, 4339–4344 (1979).
[CrossRef]

1972

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[CrossRef]

Y. Ohmachi, N. Uchida, “Vitreous As2Se3: investigation of acousto-optical properties and application to infrared modulator,” J. Appl. Phys. 43, 1709–1712 (1972).
[CrossRef]

1967

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

1965

E. H. Bogardus, “Third-order elastic constants of Ge, MgO, and fused SiO2,” J. Appl. Phys. 36, 2504–2513 (1965).
[CrossRef]

I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965).
[CrossRef]

Abedin, K. S.

Aggarwal, I. D.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

Auld, B. A.

B. A. Auld, Acoustic Fields and Waves in Solids (Wiley, 1990).

Bogardus, E. H.

E. H. Bogardus, “Third-order elastic constants of Ge, MgO, and fused SiO2,” J. Appl. Phys. 36, 2504–2513 (1965).
[CrossRef]

Boyd, R. W.

R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Brinkmeyer, E.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

Bulla, D.

Camacho, R.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

Cheng, T.

Choi, D.-Y.

Cimpl, Z.

Z. Cimpl, F. Kosek, “Refractive index of As2–xSbxS3 and As2–xSbxSe3 systems,” Phys. Status Solidi A 93, K55–K58 (1986).
[CrossRef]

Cole, H. S.

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

Davids, P.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

Dieulesaint, E.

D. Royer, E. Dieulesaint, Elastic Waves in Solids II: Generation, Acousto-optic Interaction, Applications (Springer, 2000).

Dixon, R. W.

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

Duan, Z.

Durvasula, L. N.

L. N. Durvasula, R. W. Gammon, “Brillouin scattering from shear waves in amorphous polycarbonate,” J. Appl. Phys. 50, 4339–4344 (1979).
[CrossRef]

Fathpour, S.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

Gai, X.

Gammon, R. W.

L. N. Durvasula, R. W. Gammon, “Brillouin scattering from shear waves in amorphous polycarbonate,” J. Appl. Phys. 50, 4339–4344 (1979).
[CrossRef]

Gao, W.

Harbold, J. M.

Ilday, F. Ö.

Jalali, B.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Kosek, F.

Z. Cimpl, F. Kosek, “Refractive index of As2–xSbxS3 and As2–xSbxSe3 systems,” Phys. Status Solidi A 93, K55–K58 (1986).
[CrossRef]

Krause, M.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, 1986).

Legrand, D. G.

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

Liao, M.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, 1986).

Liu, Y. S.

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

Love, J.

A. W. Snyder, J. Love, Optical Waveguide Theory (Springer, 1983).

Luther-Davies, B.

Madden, S.

Malitson, I. H.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Narum, P.

R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Nguyen, V. Q.

Ohishi, Y.

Ohmachi, Y.

Y. Ohmachi, N. Uchida, “Vitreous As2Se3: investigation of acousto-optical properties and application to infrared modulator,” J. Appl. Phys. 43, 1709–1712 (1972).
[CrossRef]

Philipp, H. R.

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

Qiu, W.

W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).

Rakich, P. T.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).

Reinke, C.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

Renner, H.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

Royer, D.

D. Royer, E. Dieulesaint, Elastic Waves in Solids II: Generation, Acousto-optic Interaction, Applications (Springer, 2000).

Rza¸zewski, K.

R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Sanghera, J. S.

Shaw, L. B.

Smith, R. G.

Snyder, A. W.

A. W. Snyder, J. Love, Optical Waveguide Theory (Springer, 1983).

Soljacic, M.

W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).

Suzuki, T.

Uchida, N.

Y. Ohmachi, N. Uchida, “Vitreous As2Se3: investigation of acousto-optical properties and application to infrared modulator,” J. Appl. Phys. 43, 1709–1712 (1972).
[CrossRef]

Wang, Z.

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Wise, F. W.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Appl. Opt.

IEEE J. Quantum Electron.

M. Krause, H. Renner, S. Fathpour, B. Jalali, E. Brinkmeyer, “Gain enhancement in cladding-pumped silicon Raman amplifiers,” IEEE J. Quantum Electron. 44, 692–704 (2008).
[CrossRef]

J. Appl. Phys.

E. H. Bogardus, “Third-order elastic constants of Ge, MgO, and fused SiO2,” J. Appl. Phys. 36, 2504–2513 (1965).
[CrossRef]

R. W. Dixon, “Photoelastic properties of selected materials and their relevance for applications to acoustic light modulators and scanners,” J. Appl. Phys. 38, 5149–5153 (1967).
[CrossRef]

Y. Ohmachi, N. Uchida, “Vitreous As2Se3: investigation of acousto-optical properties and application to infrared modulator,” J. Appl. Phys. 43, 1709–1712 (1972).
[CrossRef]

L. N. Durvasula, R. W. Gammon, “Brillouin scattering from shear waves in amorphous polycarbonate,” J. Appl. Phys. 50, 4339–4344 (1979).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

Phys. Rev. A

R. W. Boyd, K. Rza¸zewski, P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Phys. Rev. X

P. T. Rakich, C. Reinke, R. Camacho, P. Davids, Z. Wang, “Giant enhancement of stimulated Brillouin scattering in the subwavelength limit,” Phys. Rev. X 2, 11008–11023 (2012).

Phys. Status Solidi A

Z. Cimpl, F. Kosek, “Refractive index of As2–xSbxS3 and As2–xSbxSe3 systems,” Phys. Status Solidi A 93, K55–K58 (1986).
[CrossRef]

Polym. Eng. Sci.

H. R. Philipp, D. G. Legrand, H. S. Cole, Y. S. Liu, “The optical properties of bisphenol-A polycarbonate,” Polym. Eng. Sci. 27, 1148–1155 (1987).
[CrossRef]

Other

L. D. Landau, E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, 1986).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

COMSOL AB, “COMSOL Multiphysics,” (COMSOL AB, 2012).

A. W. Snyder, J. Love, Optical Waveguide Theory (Springer, 1983).

D. Royer, E. Dieulesaint, Elastic Waves in Solids II: Generation, Acousto-optic Interaction, Applications (Springer, 2000).

B. A. Auld, Acoustic Fields and Waves in Solids (Wiley, 1990).

W. Qiu, P. T. Rakich, M. Soljacic, Z. Wang, “Stimulated Brillouin scattering in slow light waveguides,” http://arxiv.org/abs/1210.0738v1 (2012).

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Figures (10)

Fig. 1.
Fig. 1.

Inverse effective area of As 2 Se 3 wire waveguide surrounded by silica cladding.

Fig. 2.
Fig. 2.

Group velocity dispersion in an As 2 Se 3 wire waveguide with silica cladding (blue), and with an amorphous polycarbonate (APC) cladding (red).

Fig. 3.
Fig. 3.

SBS gain spectrum for backward SBS in As 2 Se 3 wire waveguide with silica cladding. The phonon-mode profiles contributing to the two peaks are shown in Fig. 4. As shown, the largest contribution to the BSBS gain (a), comes from a longitudinal mode, shown in Fig. 4(a).

Fig. 4.
Fig. 4.

Displacement of elastic waveguide modes contributing to the BSBS gain shown in Fig. 3. The color scale shows the normalized total displacement | u | ; and the cones show the local direction, as well as the magnitude of the displacement. Modes (a) and (b) are longitudinal modes, pointed in the z -direction.

Fig. 5.
Fig. 5.

Pump power penalty in As 2 Se 3 wire waveguide with silica cladding for different pump input powers due to BSBS. The BSBS gain G B bwd = 2.84 · 10 4 ( W · m ) 1 , and the signal input power at z = L was S 0 = 1 pW . The black line marks the pump power threshold computed via Eq. (12).

Fig. 6.
Fig. 6.

Shift of SBS gain spectrum and effective area versus waveguide width for an As 2 Se 3 waveguide embedded in silica. The waveguide height is kept fixed at 500 nm. The waveguide mode is calculated at the pump wavelength λ p = 1.55 μm .

Fig. 7.
Fig. 7.

Pump power penalty due to BSBS in As 2 Se 3 wire with silica cladding and a taper at position z int . The suppression due to the shift of the SBS gain spectrum is 40 dB . The total waveguide length is 19.8 mm.

Fig. 8.
Fig. 8.

BSBS gain spectrum of As 2 Se 3 wire waveguide with amorphous polycarbonate (APC) cladding of height h = 2 μm . The pump wavelength is λ p = 1.55 μm . Shown are all lines originating from Love and Lamb modes in the APC cladding.

Fig. 9.
Fig. 9.

Maximum BSBS gain in As 2 Se 3 wire with APC cladding versus pump wavelength. The APC cladding has a height of h = 2 μm .

Fig. 10.
Fig. 10.

Lamb modes of APC film on silica substrate. The color scale shows the total displacement | u | ; and the arrows show the local direction of the displacement. (a) Lamb modes yielding the maximum BSBS gain in Fig. 8, at a pump wavelength of 1550 nm. (b) Mode 2, now computed at a different wavenumber q , which corresponds to a pump wavelength of 1500 nm. (c) Lamb mode yielding the maximum BSBS gain at a pump wavelength of 1500 nm.

Tables (1)

Tables Icon

Table 1. Properties of Materials Considered a

Equations (62)

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

× E ˜ ( r , t ) = μ 0 t H ˜ ( r , t ) , × H ˜ ( r , t ) = n ( x , y ) 2 ε 0 t E ˜ ( r , t ) , · ( n ( x , y ) 2 E ˜ ( r , t ) ) = 0 , · H ˜ ( r , t ) ,
σ i j es = 1 2 ε 0 ε 2 p i j k l E ˜ k E ˜ l ,
f i es = x j σ i j es .
p i j k l = ( p 11 p 12 p 44 ) δ i j k l + p 12 δ i j δ k l + p 44 δ i k δ j l .
p 44 = p 11 p 12 2 .
σ i j rp = ε 0 ε ( E ˜ i E ˜ j 1 2 δ i j | E ˜ | 2 ) μ 0 ( H ˜ i H ˜ j 1 2 δ i j | H ˜ | 2 ) ,
f i rp = x j σ i j rp .
q = k p k s , Ω = ω p ω s .
2 t 2 U ( r , t ) = c t ( x , y ) 2 × × U ( r , t ) + c l ( x , y ) 2 ( · U ( r , t ) ) .
d P d z = G B P S α p P ,
d S d z = + G B P S α s S .
G B = lim δ z 0 1 δ z ω s Ω 1 P p P s f ( r , t ) · u ˙ ( r , t ) d x d y d z ,
G B ( Ω ) = m G m ( Γ m / 2 ) 2 ( Ω Ω m ) 2 + ( Γ m / 2 ) 2
G m = ω s Q 2 Ω 2 | f * ( x , y ) · u m ( x , y ) d x d y | 2 N s N p N m ,
N p / s = 1 2 Re ( E p / s × H p / s * d x d y ) ,
N m = ρ ( x , y ) | u m ( x , y ) | 2 d x d y .
A eff = 12 Z 0 2 N 2 n eff 2 chg [ 2 | E ( x , y ) | 4 + | E ( x , y ) 2 | 2 ] d x d y ,
D λ = 2 π c 0 λ 2 d 2 β d ω 2 ,
n Silica = 1 + B 1 λ 2 λ 2 C 1 2 + B 2 λ 2 λ 2 C 2 2 + B 3 λ 2 λ 2 C 3 2 ,
n As 2 Se 3 = 1 + A 1 2 λ 2 λ 2 A 2 2 + A 3 2 λ 2 λ 2 19 2 + A 4 2 λ 2 λ 2 4 A 2 2 ,
G i = P i , f w m ( z = L ) P s , f w m ( z = 0 ) = exp ( α L ) ( sinh ( γ P L eff ) ) 2 , L eff = 1 exp ( α L ) α , α = α p = α s , f w m = α i , f w m , γ = 2 π n 2 A eff λ p .
P crit bwd = 21 G B bwd L eff .
Penalty = P ( z = L ) P ( z = 0 ) · exp ( α L ) .
Δ ω ω 2 Δ n n sin ( π d / a ) π .
Δ n n = Δ ω 2 ω π sin ( π d / a ) .
T dB = 2 · N · 10 · log 10 [ 1 ( n 1 n 2 n 1 + n 2 ) 2 ] N · 20 · log 10 [ 1 ( Δ n 2 n ) 2 ] = N · 1.5 · 10 14 dB unit cell .
Ω = q v A , eff = 2 2 π n eff , p λ p v A , eff ,
d P d z = G B , 1 P S 1 G B , 2 P S 2 α P ,
d S 1 d z = + G B , 1 P S 1 α S 1 ,
d S 2 d z = + G B , 2 P S 2 α S 2 .
sin ϑ l c l , co = sin ϑ l c l , cl , sin ϑ t c t , co = sin ϑ t c t , cl ,
n APC = D 0 + D 1 λ 2 + D 2 λ 2 + D 3 λ 4 + D 4 λ 6 + D 5 λ 8 ,
U ¨ ( y , z , t ) = c t 2 × × U ( y , z , t ) + c l 2 ( · U ( y , z , t ) ) ,
σ i y ( y = h ) = 0 , i ,
U i 0 i , as y ,
σ i y ( 0 + ) = σ i y ( 0 ) , i ,
U i ( 0 + ) = U i ( 0 ) , i .
σ i j = E 1 + ν [ U i j + ν 1 2 ν l = x , y , z U l l δ i j ] ,
E = ρ c t 2 [ 3 c l 2 4 c t 2 c l 2 c t 2 ] , ν = 1 2 c l 2 2 c t 2 c l 2 c t 2 .
U i j = 1 2 ( U j x i + U i x j ) .
f · u ˙ d x d y d z = σ i j x j · U ˙ i d x d y d z = σ i j U ˙ i x j d x d y d z = 2 σ i j U ˙ i j d x d y d z ,
f · u ˙ d x d y d z = 2 Ω σ i j Im u i j d x d y d z
U ¨ x ( y , z , t ) = c t 2 ( 2 y 2 + 2 z 2 ) U x ( y , z , t ) .
U x = { B exp ( β y ) cos ( q z Ω t ) , y < 0 , [ A 1 cos ( k y ) + A 2 sin ( k y ) ] cos ( q z Ω t ) , 0 y h .
k 2 = ( Ω c t , f ) 2 q 2 ,
β 2 = q 2 ( Ω c s , s ) 2 .
tan ( k h ) = A 2 A 1 ,
E f 1 + ν f k A 2 = E s 1 + ν s β B , A 1 = B .
tan ( k h ) = E s E f 1 + ν f 1 + ν s β k ,
U x = { B exp ( β y ) cos ( q z Ω t ) , y < 0 , [ A 1 cosh ( α y ) + A 2 sinh ( α y ) ] cos ( q z Ω t ) , 0 y h ,
α 2 = q 2 ( Ω c t , f ) 2 .
α tanh ( α h ) = E s E f 1 + ν f 1 + ν s β ,
U = ϕ + × H ,
x ϕ = 0 , H y = H z = 0 , H x = H ,
H ¨ = c t 2 Δ H ,
ϕ ¨ = c l 2 Δ ϕ .
ϕ ( y , z , t ) = { A exp ( α y ) cos ( q z Ω t ) , y < 0 , ( C 1 cos ( k y ) + C 2 sin ( k y ) ) cos ( q z Ω t ) , 0 y h , H ( y , z , t ) = { B exp ( β y ) sin ( q z Ω t ) , 0 < y , ( D 1 cos ( l y ) + D 2 sin ( l y ) ) sin ( q z Ω t ) , 0 y h ,
l 2 = ( Ω c s , f ) 2 q 2 , k 2 = ( Ω c l , f ) 2 q 2 , α 2 = q 2 ( Ω c l , s ) 2 , β 2 = q 2 ( Ω c s , s ) 2 .
ϕ ( y , z , t ) = { A exp ( α y ) cos ( q z Ω t ) , y < 0 , ( C 1 cosh ( α y ) + C 2 sinh ( α y ) ) cos ( q z Ω t ) , 0 y h , H ( y , z , t ) = { B exp ( β y ) sin ( q z Ω t ) , 0 < y , ( D 1 cos ( l y ) + D 2 sin ( l y ) ) sin ( q z Ω t ) , 0 y h ,
α 2 = q 2 ( Ω c l , f ) 2 .
ϕ ( y , z , t ) = { A exp ( α y ) cos ( q z Ω t ) , y < 0 , ( C 1 cosh ( α y ) + C 2 sinh ( α y ) ) cos ( q z Ω t ) , 0 y h , H ( y , z , t ) = { B exp ( β y ) sin ( q z Ω t ) , 0 < y , ( D 1 cosh ( β y ) + D 2 sinh ( β y ) ) sin ( q z Ω t ) , 0 y h ,
β 2 = q 2 ( Ω c s , f ) 2 .

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