Abstract

In standard weak interaction theory, acousto-optic Bragg analysis typically assumes that the incident light and sound beams are uniform plane waves. Acousto-optic Bragg diffraction with nonuniform profiled input beams is numerically examined under open loop via a transfer function formalism. Unexpected deviations in the first-order diffracted beam from the standard theory are observed for high Q values. These deviations are significant because the corresponding closed-loop system is sensitive to input amplitudes and initial conditions, and the overall impact on the dynamical behavior has not been studied previously in standard analyses. To explore the effect of such nonuniform output profiles on the feedback system, the numerically generated scattered output is fed back to the acoustic driver, and the resulting nonlinear dynamics are manipulated to create novel monostable, bistable, multistable, and chaotic regimes. The effects of the nonuniform input on these regimes are examined using the techniques of Lyapunov exponents and bifurcation maps. The orbital behavior is characterized with quadratic maps, which are an intuitive method of predicting the parametric behavior of the system. The latter trajectory-based approach offers yet a third arm in the process of developing a fuller understanding of the profiled output beam under feedback. The results of this work indicate that the nonlinear dynamical thresholds of the hybrid cell are significantly different for the profiled propagation problem than for the uniform case. The mono and bistable regimes do not coincide with the well-known uniform plane wave results, and the chaotic thresholds, which are critical to understanding encryption applications, are altered noticeably.

© 2014 Optical Society of America

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References

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  1. A. Korpel, Acousto-Optics, 2nd ed. (Dekker, 1997).
  2. S.-T. Chen and M. R. Chatterjee, “A numerical analysis and expository interpretation of the diffraction of light by ultrasonic waves in the Bragg and Raman-Nath regimes using multiple scattering theory,” IEEE Trans. Ed. 39, 56–68 (1996).
    [CrossRef]
  3. A. Korpel and T.-C. Poon, “Explicit formalism for acousto-optic multiple plane-wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
    [CrossRef]
  4. M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).
  5. J. Chrostowski and C. Delisle, “Bistable piezoelectric Fabry–Perot interferometer,” Can. J. Phys. 57, 1376–1379 (1979).
    [CrossRef]
  6. J. Chrostowski and C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
    [CrossRef]
  7. P. P. Banerjee, U. Banerjee, and H. Kaplan, “Response of an acousto-optic device with feedback to time-varying inputs,” Appl. Opt. 31, 1842–1852 (1992).
    [CrossRef]
  8. M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
    [CrossRef]
  9. M. R. Chatterjee and F. S. Almehmadi, “Numerical analysis of first-order acousto-optic Bragg diffraction of profiled optical beams using open-loop transfer functions,” Opt. Eng (to be published).
  10. A. K. Ghosh and P. Verma, “Lyapunov exponent of chaos generated by acousto-optic modulators with feedback,” Opt. Eng. 50, 017005 (2011).
    [CrossRef]
  11. M. R. Chatterjee and J.-J. Huang, “Demonstration of acousto-optic bistability and chaos by direct nonlinear circuit modeling,” Appl. Opt. 31, 2506–2517 (1992).
    [CrossRef]
  12. J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
    [CrossRef]
  13. J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
    [CrossRef]
  14. G. VanWiggeren and R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
    [CrossRef]
  15. T. Bountis, “Fundamental concepts of the theory of chaos and fractals,” in Chaos Applications in Telecommunications, P. Stavroulakis, ed. (CRC Press, 2006), pp. 313–379.

2012

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

2011

A. K. Ghosh and P. Verma, “Lyapunov exponent of chaos generated by acousto-optic modulators with feedback,” Opt. Eng. 50, 017005 (2011).
[CrossRef]

1998

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

G. VanWiggeren and R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

1996

S.-T. Chen and M. R. Chatterjee, “A numerical analysis and expository interpretation of the diffraction of light by ultrasonic waves in the Bragg and Raman-Nath regimes using multiple scattering theory,” IEEE Trans. Ed. 39, 56–68 (1996).
[CrossRef]

1992

1990

M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).

1982

J. Chrostowski and C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

1981

J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
[CrossRef]

1980

1979

J. Chrostowski and C. Delisle, “Bistable piezoelectric Fabry–Perot interferometer,” Can. J. Phys. 57, 1376–1379 (1979).
[CrossRef]

Albert, J.

J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
[CrossRef]

Almehmadi, F. S.

M. R. Chatterjee and F. S. Almehmadi, “Numerical analysis of first-order acousto-optic Bragg diffraction of profiled optical beams using open-loop transfer functions,” Opt. Eng (to be published).

Al-Saedi, M. A.

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

Banerjee, P. P.

Banerjee, U.

Bountis, T.

T. Bountis, “Fundamental concepts of the theory of chaos and fractals,” in Chaos Applications in Telecommunications, P. Stavroulakis, ed. (CRC Press, 2006), pp. 313–379.

Chatterjee, M. R.

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

S.-T. Chen and M. R. Chatterjee, “A numerical analysis and expository interpretation of the diffraction of light by ultrasonic waves in the Bragg and Raman-Nath regimes using multiple scattering theory,” IEEE Trans. Ed. 39, 56–68 (1996).
[CrossRef]

M. R. Chatterjee and J.-J. Huang, “Demonstration of acousto-optic bistability and chaos by direct nonlinear circuit modeling,” Appl. Opt. 31, 2506–2517 (1992).
[CrossRef]

M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).

M. R. Chatterjee and F. S. Almehmadi, “Numerical analysis of first-order acousto-optic Bragg diffraction of profiled optical beams using open-loop transfer functions,” Opt. Eng (to be published).

Chen, S.-T.

S.-T. Chen and M. R. Chatterjee, “A numerical analysis and expository interpretation of the diffraction of light by ultrasonic waves in the Bragg and Raman-Nath regimes using multiple scattering theory,” IEEE Trans. Ed. 39, 56–68 (1996).
[CrossRef]

Chrostowski, J.

J. Chrostowski and C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

J. Chrostowski and C. Delisle, “Bistable piezoelectric Fabry–Perot interferometer,” Can. J. Phys. 57, 1376–1379 (1979).
[CrossRef]

Delisle, C.

J. Chrostowski and C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

J. Chrostowski and C. Delisle, “Bistable piezoelectric Fabry–Perot interferometer,” Can. J. Phys. 57, 1376–1379 (1979).
[CrossRef]

Ghosh, A. K.

A. K. Ghosh and P. Verma, “Lyapunov exponent of chaos generated by acousto-optic modulators with feedback,” Opt. Eng. 50, 017005 (2011).
[CrossRef]

Goedgebuer, J.-P.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Huang, J.-J.

Kaplan, H.

Korpel, A.

Larger, L.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Poon, T.-C.

M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).

A. Korpel and T.-C. Poon, “Explicit formalism for acousto-optic multiple plane-wave scattering,” J. Opt. Soc. Am. 70, 817–820 (1980).
[CrossRef]

Porte, H.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Roy, R.

G. VanWiggeren and R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Sitter, D. N.

M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).

Tremblay, R.

J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
[CrossRef]

VanWiggeren, G.

G. VanWiggeren and R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Verma, P.

A. K. Ghosh and P. Verma, “Lyapunov exponent of chaos generated by acousto-optic modulators with feedback,” Opt. Eng. 50, 017005 (2011).
[CrossRef]

Vincent, D.

J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
[CrossRef]

Acustica

M. R. Chatterjee, T.-C. Poon, and D. N. Sitter, “Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles,” Acustica 71, 81–91 (1990).

Appl. Opt.

Can. J. Phys.

J. Chrostowski and C. Delisle, “Bistable piezoelectric Fabry–Perot interferometer,” Can. J. Phys. 57, 1376–1379 (1979).
[CrossRef]

J. Albert, R. Tremblay, and D. Vincent, “Hybrid bistable optical device using an acousto-optic waveguide modulator,” Can. J. Phys. 59, 1251–1253 (1981).
[CrossRef]

IEEE Trans. Ed.

S.-T. Chen and M. R. Chatterjee, “A numerical analysis and expository interpretation of the diffraction of light by ultrasonic waves in the Bragg and Raman-Nath regimes using multiple scattering theory,” IEEE Trans. Ed. 39, 56–68 (1996).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

J. Chrostowski and C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
[CrossRef]

Opt. Eng.

M. A. Al-Saedi and M. R. Chatterjee, “Examination of the nonlinear dynamics of a chaotic acousto-optic Bragg modulator with feedback under signal encryption and decryption,” Opt. Eng. 51, 018003 (2012).
[CrossRef]

A. K. Ghosh and P. Verma, “Lyapunov exponent of chaos generated by acousto-optic modulators with feedback,” Opt. Eng. 50, 017005 (2011).
[CrossRef]

Phys. Rev. Lett.

J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Science

G. VanWiggeren and R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Other

T. Bountis, “Fundamental concepts of the theory of chaos and fractals,” in Chaos Applications in Telecommunications, P. Stavroulakis, ed. (CRC Press, 2006), pp. 313–379.

A. Korpel, Acousto-Optics, 2nd ed. (Dekker, 1997).

M. R. Chatterjee and F. S. Almehmadi, “Numerical analysis of first-order acousto-optic Bragg diffraction of profiled optical beams using open-loop transfer functions,” Opt. Eng (to be published).

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Figures (21)

Fig. 1.
Fig. 1.

A-O closed-loop hybrid system with an arbitrary incident beam profile.

Fig. 2.
Fig. 2.

First-order intensity Bragg diffraction versus the optical phase shift for Q=20, 177, and 533.

Fig. 3.
Fig. 3.

Hysteresis loop when β˜=2.42 and Td=0.015s showing transition from bistable to multistable oscillations for uniform plane wave input.

Fig. 4.
Fig. 4.

Chaotic oscillations for uniform plane wave input when β˜; is increased to 3 and Td=0.025s.

Fig. 5.
Fig. 5.

Hysteresis loop with an arbitrary Gaussian incident beam profile for Q=20, Λ=0.1mm, Td=1μs, β˜=1.28, and I1(0)=0.

Fig. 6.
Fig. 6.

Hysteresis loop with an arbitrary Gaussian incident beam profile for Q=177, Λ=33.6μm, Td=1μs, β˜=1.7, and I1(0)=0.

Fig. 7.
Fig. 7.

Hysteresis loop with an arbitrary Gaussian incident beam profile for Q=533, Λ=19.4μm, Td=1μs, β˜=1.92, and I1(0)=0.

Fig. 8.
Fig. 8.

Nonlinear dynamics with three different values of the effective feedback gain for Q=20.

Fig. 9.
Fig. 9.

Nonlinear dynamics versus optical phase shift with a profiled Gaussian incident beam when β˜; is increased to 1.6 for Q=20.

Fig. 10.
Fig. 10.

Nonlinear dynamics versus optical phase shift with a profiled Gaussian incident beam when β˜ is increased to 2.2 for Q=20.

Fig. 11.
Fig. 11.

Lyapunov exponent and bifurcation maps versus the optical phase shift when β˜=1, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 12.
Fig. 12.

Lyapunov exponent and bifurcation maps versus the optical phase shift when β˜=1.2, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 13.
Fig. 13.

Lyapunov exponent and bifurcation maps versus the optical phase shift when β˜=1.5, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 14.
Fig. 14.

Lyapunov exponent and bifurcation maps versus the optical phase shift when β˜=2, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 15.
Fig. 15.

Lyapunov exponent and bifurcation maps versus the effective feedback gain when α^=1, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 16.
Fig. 16.

Lyapunov exponent and bifurcation maps versus the effective feedback gain when α^=1.2, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 17.
Fig. 17.

Lyapunov exponent and bifurcation maps versus the effective feedback gain when α^=2, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 18.
Fig. 18.

Lyapunov exponent and bifurcation maps versus the effective feedback gain when α^=3, Iinc=2, Λ=0.1mm, and I1(0)=0.

Fig. 19.
Fig. 19.

Lyapunov exponent and bifurcation maps versus the optical phase shift with a uniform plane wave input when β˜=2.

Fig. 20.
Fig. 20.

Dynamical analysis of the quadratic maps of nonuniform incident beams when Q=20, β˜=1, Λ=0.1mm, Iinc=1, and I1(0)=0.

Fig. 21.
Fig. 21.

Dynamical analysis of the quadratic maps of nonuniform incident beams when Q=20, β˜=2.3, Λ=0.1mm, Iinc=1, and I1(0)=0.

Equations (10)

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

H˜0(δ)=E˜0(ζ)ξ=1E˜inc=ejδQ4(δQ4)2+(α^2)2((δQ4)2+(α^2)2cos((δQ4)2+(α^2)2)+jδQ4sin((δQ4)2+(α^2)2)),
H˜1(δ)=E˜1(ζ)ξ=1E˜inc=j(α^2)ejδQ4(δQ4)2+(α^2)2(sin((δQ4)2+(α^2)2)).
Eout(r)=E˜inc(δ)H˜(δ)ej2πλδϕBr(ϕBλ)dδ.
Einc12πσer22σ2,Einc=1.
Eince2π2(σtanδϕBλ)2,Einc=1.
I1(t)=Iincsin2{12[α^0(t)+β˜I1(tTD)]}.
Iph(t)=|f(12[α^0(t)+β˜(Iph(tTD))])|2.
ΔI1εenλ,
Iph(n)=|f(12[α^0(n)+β˜(Iph(n1))])|2,
Xn+1=μXn(1Xn).

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