Abstract

The components of dissipative weakly coupled states, originating due to collinear Bragg light scattering by the acoustic wave of finite amplitude, are investigated in the square-law nonlinear medium with the linear optical losses. A novel theoretical model is developed for three-wave weakly coupled states of various pulse profiles, propagating in the quasi-stationary regime with the phase mismatch. The availability of both compact and infinite support is analyzed and compared with one another. Two limiting cases for cnoidal profiles are considered in detail and estimated. Their optical components are observed during the acousto-optical experiments in an α-quartz crystalline cell with calibrated optical losses.

© 2014 Optical Society of America

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  1. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).
  2. A. P. Sukhorukov, Nonlinear Wave Processes (Nauka, 1988).
  3. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  4. I. G. Mikhailov and V. A. Shutilov, “Diffraction of light on the ultra sound waves of large amplitude,” Akusticheskij Zhurnal 4, 174–183 (1958), in Russian.
  5. M. A. Breazcale and E. A. Hiedemann, “Diffraction patterns produced by finite amplitude waves,” J. Acoust. Soc. Am. 33, 700 (1961).
    [CrossRef]
  6. R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.
  7. T. H. Neighbors and W. G. Mayer, “Asymmetric light diffraction by pulsed ultrasonic waves,” J. Acoust. Soc Am. 74, 146–152 (1983).
    [CrossRef]
  8. K. Van Den Abeele and O. Leroy, “Light diffraction by ultrasonic pulses: analytical and numerical solutions of the extended Raman–Nath equations,” J. Acoust. Soc. Am. 88, 2298–2315 (1990).
    [CrossRef]
  9. F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
    [CrossRef]
  10. V. N. Parygin and A. V. Vershoubskiy, “Collinear diffraction of Gaussian optical beams by an acoustic pulse,” Pure Appl. Opt. 7, 733–746 (1998).
    [CrossRef]
  11. L. Adler, W. T. Yost, and J. H. Cantrell, “Subharmonics, chaos, and beyond,” in International Congress on Ultrasonics: Gdańsk 2011, AIP Conference Proceedings (2012), Vol. 1433, pp. 527–530.
  12. A. S. Shcherbakov and A. Aguirre Lopez, “Shaping the optical components of solitary three-wave weakly coupled states in a two-mode waveguide,” Opt. Express, 11, 1643–1649 (2003).
    [CrossRef]
  13. A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
    [CrossRef]
  14. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
  15. A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
    [CrossRef]
  16. A. D. Polyanin and V. F. Zaistev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (CRC, 2003).
  17. E. H. Neville, Elliptic Functions: A Primer (Pergamon, 1971).
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1964).
  19. R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
    [CrossRef]
  20. W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason 14, 123–134 (1967).
    [CrossRef]
  21. A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).
  22. A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
    [CrossRef]

2010 (1)

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

2008 (2)

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
[CrossRef]

2003 (1)

2000 (1)

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
[CrossRef]

1998 (1)

V. N. Parygin and A. V. Vershoubskiy, “Collinear diffraction of Gaussian optical beams by an acoustic pulse,” Pure Appl. Opt. 7, 733–746 (1998).
[CrossRef]

1990 (1)

K. Van Den Abeele and O. Leroy, “Light diffraction by ultrasonic pulses: analytical and numerical solutions of the extended Raman–Nath equations,” J. Acoust. Soc. Am. 88, 2298–2315 (1990).
[CrossRef]

1983 (1)

T. H. Neighbors and W. G. Mayer, “Asymmetric light diffraction by pulsed ultrasonic waves,” J. Acoust. Soc Am. 74, 146–152 (1983).
[CrossRef]

1967 (2)

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason 14, 123–134 (1967).
[CrossRef]

1961 (1)

M. A. Breazcale and E. A. Hiedemann, “Diffraction patterns produced by finite amplitude waves,” J. Acoust. Soc. Am. 33, 700 (1961).
[CrossRef]

1958 (1)

I. G. Mikhailov and V. A. Shutilov, “Diffraction of light on the ultra sound waves of large amplitude,” Akusticheskij Zhurnal 4, 174–183 (1958), in Russian.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Adler, L.

L. Adler, W. T. Yost, and J. H. Cantrell, “Subharmonics, chaos, and beyond,” in International Congress on Ultrasonics: Gdańsk 2011, AIP Conference Proceedings (2012), Vol. 1433, pp. 527–530.

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Aguirre Lopez, A.

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov and A. Aguirre Lopez, “Shaping the optical components of solitary three-wave weakly coupled states in a two-mode waveguide,” Opt. Express, 11, 1643–1649 (2003).
[CrossRef]

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

Balderas Mata, S. E.

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
[CrossRef]

Belokurova, O. I.

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

Blistanov, A. A.

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

Bondarenko, V. S.

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

Breazcale, M. A.

M. A. Breazcale and E. A. Hiedemann, “Diffraction patterns produced by finite amplitude waves,” J. Acoust. Soc. Am. 33, 700 (1961).
[CrossRef]

Bridoux, E.

R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.

Cantrell, J. H.

L. Adler, W. T. Yost, and J. H. Cantrell, “Subharmonics, chaos, and beyond,” in International Congress on Ultrasonics: Gdańsk 2011, AIP Conference Proceedings (2012), Vol. 1433, pp. 527–530.

Carles, C.

R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.

Chkalova, V. V.

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

Cook, B. D.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason 14, 123–134 (1967).
[CrossRef]

Dixon, R. W.

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

Dodd, K.

K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).

Eilbeck, J. C.

K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).

Gibbon, J. D.

K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).

Hiedemann, E. A.

M. A. Breazcale and E. A. Hiedemann, “Diffraction patterns produced by finite amplitude waves,” J. Acoust. Soc. Am. 33, 700 (1961).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Klein, W. R.

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason 14, 123–134 (1967).
[CrossRef]

Leroy, O.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
[CrossRef]

K. Van Den Abeele and O. Leroy, “Light diffraction by ultrasonic pulses: analytical and numerical solutions of the extended Raman–Nath equations,” J. Acoust. Soc. Am. 88, 2298–2315 (1990).
[CrossRef]

Luna Castellanos, A.

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

Maximov, Je.

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
[CrossRef]

Mayer, W. G.

T. H. Neighbors and W. G. Mayer, “Asymmetric light diffraction by pulsed ultrasonic waves,” J. Acoust. Soc Am. 74, 146–152 (1983).
[CrossRef]

Mikhailov, I. G.

I. G. Mikhailov and V. A. Shutilov, “Diffraction of light on the ultra sound waves of large amplitude,” Akusticheskij Zhurnal 4, 174–183 (1958), in Russian.

Moriamez, M.

R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.

Morris, H.

K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).

Neighbors, T. H.

T. H. Neighbors and W. G. Mayer, “Asymmetric light diffraction by pulsed ultrasonic waves,” J. Acoust. Soc Am. 74, 146–152 (1983).
[CrossRef]

Neville, E. H.

E. H. Neville, Elliptic Functions: A Primer (Pergamon, 1971).

Parygin, V. N.

V. N. Parygin and A. V. Vershoubskiy, “Collinear diffraction of Gaussian optical beams by an acoustic pulse,” Pure Appl. Opt. 7, 733–746 (1998).
[CrossRef]

Perelomova, N. V.

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

Polyanin, A. D.

A. D. Polyanin and V. F. Zaistev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (CRC, 2003).

Pustovoit, V. I.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
[CrossRef]

Sánchez Lucero, D.

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

Shcherbakov, A. S.

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
[CrossRef]

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov and A. Aguirre Lopez, “Shaping the optical components of solitary three-wave weakly coupled states in a two-mode waveguide,” Opt. Express, 11, 1643–1649 (2003).
[CrossRef]

Shutilov, V. A.

I. G. Mikhailov and V. A. Shutilov, “Diffraction of light on the ultra sound waves of large amplitude,” Akusticheskij Zhurnal 4, 174–183 (1958), in Russian.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1964).

Strizhevskaya, N. S.

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

Sukhorukov, A. P.

A. P. Sukhorukov, Nonlinear Wave Processes (Nauka, 1988).

Torguet, R.

R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.

Van Den Abeele, K.

K. Van Den Abeele and O. Leroy, “Light diffraction by ultrasonic pulses: analytical and numerical solutions of the extended Raman–Nath equations,” J. Acoust. Soc. Am. 88, 2298–2315 (1990).
[CrossRef]

Vershoubskiy, A. V.

V. N. Parygin and A. V. Vershoubskiy, “Collinear diffraction of Gaussian optical beams by an acoustic pulse,” Pure Appl. Opt. 7, 733–746 (1998).
[CrossRef]

Windels, F. W.

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
[CrossRef]

Yost, W. T.

L. Adler, W. T. Yost, and J. H. Cantrell, “Subharmonics, chaos, and beyond,” in International Congress on Ultrasonics: Gdańsk 2011, AIP Conference Proceedings (2012), Vol. 1433, pp. 527–530.

Zaistev, V. F.

A. D. Polyanin and V. F. Zaistev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (CRC, 2003).

Akusticheskij Zhurnal (1)

I. G. Mikhailov and V. A. Shutilov, “Diffraction of light on the ultra sound waves of large amplitude,” Akusticheskij Zhurnal 4, 174–183 (1958), in Russian.

IEEE J. Quantum Electron. (1)

R. W. Dixon, “Acoustic diffraction of light in anisotropic media,” IEEE J. Quantum Electron. QE-3, 85–93 (1967).
[CrossRef]

IEEE Trans. Sonics Ultrason (1)

W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason 14, 123–134 (1967).
[CrossRef]

J. Acoust. Soc Am. (1)

T. H. Neighbors and W. G. Mayer, “Asymmetric light diffraction by pulsed ultrasonic waves,” J. Acoust. Soc Am. 74, 146–152 (1983).
[CrossRef]

J. Acoust. Soc. Am. (2)

K. Van Den Abeele and O. Leroy, “Light diffraction by ultrasonic pulses: analytical and numerical solutions of the extended Raman–Nath equations,” J. Acoust. Soc. Am. 88, 2298–2315 (1990).
[CrossRef]

M. A. Breazcale and E. A. Hiedemann, “Diffraction patterns produced by finite amplitude waves,” J. Acoust. Soc. Am. 33, 700 (1961).
[CrossRef]

J. Opt. (1)

A. S. Shcherbakov, D. Sánchez Lucero, A. Luna Castellanos, and O. I. Belokurova, “Direct multi-channel optical spectrum analysis of radio-wave signals using collinear wave heterodyning in single acousto-optical cell,” J. Opt. 12, 045203 (2010).
[CrossRef]

J. Opt. A (2)

A. S. Shcherbakov, S. E. Balderas Mata, Je. Maximov, and A. Aguirre Lopez, “The existence of five-wave non-collinear acousto-optical weakly coupled states,” J. Opt. A 10, 085106 (2008).
[CrossRef]

A. S. Shcherbakov, Je. Maximov, and S. E. Balderas Mata, “Shaping the dissipative collinear three-wave coupled states in a two-mode medium with a square-law nonlinearity and linear non-optical losses,” J. Opt. A 10, 025001 (2008).
[CrossRef]

Opt. Express (1)

Pure Appl. Opt. (1)

V. N. Parygin and A. V. Vershoubskiy, “Collinear diffraction of Gaussian optical beams by an acoustic pulse,” Pure Appl. Opt. 7, 733–746 (1998).
[CrossRef]

Ultrasonics (1)

F. W. Windels, V. I. Pustovoit, and O. Leroy, “Collinear acousto-optic interaction using two nearby sound frequencies,” Ultrasonics 38, 586–589 (2000).
[CrossRef]

Other (10)

L. Adler, W. T. Yost, and J. H. Cantrell, “Subharmonics, chaos, and beyond,” in International Congress on Ultrasonics: Gdańsk 2011, AIP Conference Proceedings (2012), Vol. 1433, pp. 527–530.

R. Torguet, C. Carles, E. Bridoux, and M. Moriamez, “Diffraction of light by finite amplitude acoustic waves in nonlinear crystals,” in Ultrasonics Symposium (1972), pp. 147–150.

K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations (Academic, 1984).

A. P. Sukhorukov, Nonlinear Wave Processes (Nauka, 1988).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

A. D. Polyanin and V. F. Zaistev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed. (CRC, 2003).

E. H. Neville, Elliptic Functions: A Primer (Pergamon, 1971).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1964).

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

A. A. Blistanov, V. S. Bondarenko, N. V. Perelomova, N. S. Strizhevskaya, and V. V. Chkalova, in Acoustic Crystals, M. P. Shaskolskaya, ed. (Nauka, 1982).

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Figures (9)

Fig. 1.
Fig. 1.

Acoustic component. With growing m, initially a rectangular pulse of the finite width π, creating a compact support, is converted into a hyperbolic secant pulse giving infinite support for optical components of the coupled state.

Fig. 2.
Fig. 2.

Optical component. Initially trigonometric profile with compact support π is transformed into a hyperbolic profile with infinite support as the modulus m grows from zero to unity.

Fig. 3.
Fig. 3.

Evolution of the optical components |a0|2 and |a1|2 in the weakly coupled acousto-optical states. (a) Hyperbolic profiles with η=1.5, infinite support. (b) Trigonometric profiles with η=5, N=2, the compact support L00.562.

Fig. 4.
Fig. 4.

Localization distance L0 versus the mismatch η with σ as a parameter: (a) for trigonometric profile with N=1 and (b) for hyperbolic secant profile.

Fig. 5.
Fig. 5.

Localization distance L0 and relative efficiency of localization Ef on the plane {σ,η}. (a) Trigonometric profile with N=1 and (b) for hyperbolic secant profile.

Fig. 6.
Fig. 6.

Light intensity |a1|2 of three-wave weakly coupled state at J0=1 and ξ=0 versus the ratio (η/σ) and the product (αx): (a) is for σ/α=2.0 and (b) is for σ/α=4.0.

Fig. 7.
Fig. 7.

Schematic arrangement of the experimental setup.

Fig. 8.
Fig. 8.

Digitized oscilloscope traces for the transmitted |C0|2 (upper traces) and scattered |C1|2 (lower ones) light intensities observed at the output of the above-described α-quartz collinear AOC under action of a rectangular acoustic pulse of finite amplitude and time width about 10.5 μs: (a) N=1, f(1)81.408MHz and (b) N=2, f(2)81.517MHz.

Fig. 9.
Fig. 9.

(a) Digitized oscilloscope traces for the transmitted light |C0|2 and (b) scattered light |C1|2 intensities observed at the output of the above-described α-quartz collinear AOC at the acoustic frequency fH=81.426MHz under action of a hyperbolic secant acoustic pulse of finite amplitude and time width about 10.5 μs at the acoustic intensity level 1%.

Tables (1)

Tables Icon

Table 1. Estimations Using α-Quartz Crystal with L0=6cm, σ=0.43cm1, and Obtaining the Needed Acoustic Power Density P=1.0W/mm2 for the Three Cases

Equations (39)

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

ux+1Vut0,
a0x+α0a0+1ca0t=q1a1uexp(2iηx),
a1x+α1a1+1ca1t=q0a0u*exp(2iηx),
a0x+αa0=q1a1U[t(x/V)]exp(2iηx),
a1x+αa1=q0a0U*[t(x/V)]exp(2iηx).
2f0x2[2iη+(1U)(Ux)]f0x+q0q1|U|2f0=0,
2f1x2[2iη+(1U*)(U*x)]f1x+q0q1|U|2f1=0
Ux+[2iη2f0x2(f0x)1]Uq2f0(f0x)1U|U|2=0,
U*x+[2iη2f1x2(f1x)1]U*q2f1(f1x)1U*|U|2=0,
|U|2=(f0x)(f0*x)×[C0q2f0f0*]1=(f1x)(f1*x)×[C1q2f1f1*]1.
U(x)=CU(f0x)×exp{2iηx+q2f0(f0*x)×[C0q2f0f0*]1dx},
|U|2=4b2(η2+b2x2)q2[(b2+4η2)exp(2bx2)b2],
U(x)=2BCU(iηbx)×exp{bx2+2(bx+iη)dx1exp[2bx2·{1(4η2/b2)}]}.
f0=Bcn(bx,m)·exp(ip0x),
|U|2=(b2q2)4η2cn2(bx,m)+b2dn2(bx,m)sn2(bx,m)4η2+b2sn2(bx,m).
|U|2=(b2q2)·4η2cos2(bx)+b2sin2(bx)4η2+b2sin2(bx),m=0;
|U|2=(b2q2)sech2(bx)[4η2+b2tanh2(bx)]4η2+b2tanh2(bx),m=1.
|U|2=(b2q2)=const,withm=0;
|U|2=(b2q2)·sech2(bx),withm=1.
2f0,1x2[±2iη+(1U0)(U0x)]f0,1x+q2U02f0,1=0.
2Y0x2+[2iη+σtanh(σx)]Y0x+q2H2sech2(σx)·Y0=0.
y0=Bsech(Bx)·exp(ip0x),
Y0=Z1(0)σsech(σx)·exp(2iηx)+Z2(0)·2iη+σtanh(σx)σ(σ2+4η2),
|Y0|2(x)=J02σ2σ2+4η2sech2(σx),
|Y1|2(x)=J02·4η2+σ2tanh2(Ax)σ2+4η2.
2f0,1x22iηf0,1x+σ2f0,1=0
f0,1=P0,1exp[ix(±ησ2+η2)]+Q0,1exp[ix(±η+σ2+η2)],
|F1|2=J02σ2σ2+η2·sin2(xσ2+η2),
|F0|2=J02[η2σ2+η2+σ2σ2+η2·cos2(xσ2+η2)].
sn(z,m)sin(z)m4[zsin(z)cos(z)]cos(z),
cn(z,m)cos(z)m4[zsin(z)cos(z)]sin(z)
|a0|2(x,ξ)=J02σ2σ2+4η2sech2[σ(xξ)]exp(2αx),
|a1|2(x,ξ)=J02exp(2αx)4η2+σ2tanh2[σ(xξ)]σ2+4η2.
|a0|2(x,ξ)=J02[η2σ2+η2+σ2σ2+η2·cos2[(xξ)σ2+η2]]exp(2αx),
|a1|2(x,ξ)=J02σ2σ2+η2sin2[(xξ)σ2+η2]exp(2αx).
Trigonometric profile:L0=πN/σ2+η2,
Hyperbolic profile:L0=2σ1arcsech{0.1·[1+(4η2/σ2)]1/2}.
Trigonometric profile:η=π2N2L02σ2,Ef=σ2(σ2+η2)1;
Hyperbolic profile:η=(σ/2)100sech2(σL0/2)1,Ef=σ2(σ2+4η2)1.

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