Abstract

A detailed examination of the nonlinear dynamical behavior of an acousto-optic Bragg cell in the near-Bragg regime of operation for the case of four scattered orders under intensity feedback is carried out. This problem is an extension of the standard ideal-Bragg feedback model whereby traditionally bistability, hysteresis, and chaotic oscillations are observed under zeroth- or first-order feedback of the scattered light. For the present case, the closed-loop equations are developed from a priori knowledge of the open-loop analytical solutions for four-order near-Bragg scattering. The results, obtained via computer simulation, reveal a variety of interesting dynamics, including bistability, bifurcation, hysteresis, chaotic oscillations (including in this case the relatively uncommon period-three behavior, in addition to the more usual period-doubling phenomenon en route to chaos), and potentially useful parametric dependence of these features. The observed results are interpreted in terms of system behavior for varying feedback gain and bias, the so-called Klein-Cook parameter Q, and time delay, and are compared with earlier work based on the ideal Bragg regime.

© 2002 Optical Society of America

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References

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  1. J. Chrostowski, C. Delisle, “Bistable optical switching based on Bragg diffraction,” Opt. Commun. 41, 71–74 (1982).
    [CrossRef]
  2. J. Chrostowski, “Noisy bifurcations in acousto-optic bistability,” Phys. Rev. A 26, 3023–3025 (1982).
    [CrossRef]
  3. J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
    [CrossRef]
  4. J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
    [CrossRef]
  5. P. P. Banerjee, T.-C. Poon, “Simulation of bistability and chaos in hybrid devices,” in Proceedings of the 30th Midwest Symposium on Circuits and Systems, K. Jabbour, ed., (North-Holland, Amsterdam, 1987), pp. 820–823.
  6. T.-C. Poon, S. K. Cheung, “Performance of a hybrid bistable device using an acoustooptic modulator,” Appl. Opt. 28, 4787–4791 (1989).
    [CrossRef] [PubMed]
  7. M. R. Chatterjee, J.-J. Huang, “Demonstration of A-O bistability and chaos by direct nonlinear circuit modeling,” Appl. Opt. 31, 2506–2517 (1992).
    [CrossRef] [PubMed]
  8. S.-T. Chen, M. R. Chatterjee, “A dual-input hybrid A-O set-reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147–3154 (1997).
    [CrossRef] [PubMed]
  9. V. I. Balakshy, A. V. Kazaryan, V. Y. Molchanov, “Deflectors with a feedback: new possibilities for image processing,” in International Conference on Optical Information Processing, Y. V. Gulyaev, D. R. Pape, eds., Proc. SPIE2051, 672–677 (1994).
    [CrossRef]
  10. M. R. Chatterjee, S. Ramchandran, “Feedback correction of angular error in grating readout,” in Photonic Devices and Algorithms for Computing III, K. M. Iftekharuddin, A. A. S. Awwal, eds., Proc. SPIE4470, 127–137 (2001).
    [CrossRef]
  11. T.-C. Poon, “A Feynman Diagram Approach to Multiple Plane Wave Scattering in Acousto-Optic Interactions,” Ph. D. dissertation, (University of Iowa, Iowa City, Iowa, 1982).
  12. R. C. Hilborn, Chaos and Nonlinear Dynamics, an Introduction for Scientists and Engineers (Oxford University, London, 1994).
  13. M. R. Chatterjee, E. Sonmez, “An overview of acousto-optic bistability, chaos and logical applications,” in Acousto-Optics and Applications IV, B. B. J. Linde, A. Sikorska, eds., Proc. SPIE4514, 41–60 (2001).
    [CrossRef]
  14. E. Sonmez, “Development and examination of acousto-optic logic devices” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).
  15. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, 1994).
  16. V. I. Balakshy, A. V. Kazaryan, “Laser beam direction stabilization by means of Bragg diffraction,” Opt. Eng. 38, 1154–1159 (1999).
    [CrossRef]
  17. S. Ramchandran, “Investigation of feedback in acoustooptic gratings and related applications” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).
  18. R. P. Feynman, R. B. Leighton, M. Sands, Feynman Lectures on Physics, The Commemorative Issue, Vols. 1–3 (Addison-Wesley, New York, 1989).
  19. P. P. Banerjee, U. Banerjee, H. Kaplan, “Response of an acousto-optic device with feedback to time-varying inputs,” Appl. Opt. 31, 1842–1852 (1992).
    [CrossRef] [PubMed]

1999 (1)

V. I. Balakshy, A. V. Kazaryan, “Laser beam direction stabilization by means of Bragg diffraction,” Opt. Eng. 38, 1154–1159 (1999).
[CrossRef]

1997 (1)

1992 (2)

1989 (1)

1983 (2)

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
[CrossRef]

1982 (2)

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

J. Chrostowski, “Noisy bifurcations in acousto-optic bistability,” Phys. Rev. A 26, 3023–3025 (1982).
[CrossRef]

Balakshy, V. I.

V. I. Balakshy, A. V. Kazaryan, “Laser beam direction stabilization by means of Bragg diffraction,” Opt. Eng. 38, 1154–1159 (1999).
[CrossRef]

V. I. Balakshy, A. V. Kazaryan, V. Y. Molchanov, “Deflectors with a feedback: new possibilities for image processing,” in International Conference on Optical Information Processing, Y. V. Gulyaev, D. R. Pape, eds., Proc. SPIE2051, 672–677 (1994).
[CrossRef]

Banerjee, P. P.

P. P. Banerjee, U. Banerjee, H. Kaplan, “Response of an acousto-optic device with feedback to time-varying inputs,” Appl. Opt. 31, 1842–1852 (1992).
[CrossRef] [PubMed]

P. P. Banerjee, T.-C. Poon, “Simulation of bistability and chaos in hybrid devices,” in Proceedings of the 30th Midwest Symposium on Circuits and Systems, K. Jabbour, ed., (North-Holland, Amsterdam, 1987), pp. 820–823.

Banerjee, U.

Chatterjee, M. R.

S.-T. Chen, M. R. Chatterjee, “A dual-input hybrid A-O set-reset flip-flop and its nonlinear dynamics,” Appl. Opt. 36, 3147–3154 (1997).
[CrossRef] [PubMed]

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

M. R. Chatterjee, E. Sonmez, “An overview of acousto-optic bistability, chaos and logical applications,” in Acousto-Optics and Applications IV, B. B. J. Linde, A. Sikorska, eds., Proc. SPIE4514, 41–60 (2001).
[CrossRef]

M. R. Chatterjee, S. Ramchandran, “Feedback correction of angular error in grating readout,” in Photonic Devices and Algorithms for Computing III, K. M. Iftekharuddin, A. A. S. Awwal, eds., Proc. SPIE4470, 127–137 (2001).
[CrossRef]

Chen, S.-T.

Cheung, S. K.

Chrostowski, J.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
[CrossRef]

J. Chrostowski, “Noisy bifurcations in acousto-optic bistability,” Phys. Rev. A 26, 3023–3025 (1982).
[CrossRef]

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

Delisle, C.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
[CrossRef]

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

Feynman, R. P.

R. P. Feynman, R. B. Leighton, M. Sands, Feynman Lectures on Physics, The Commemorative Issue, Vols. 1–3 (Addison-Wesley, New York, 1989).

Hilborn, R. C.

R. C. Hilborn, Chaos and Nonlinear Dynamics, an Introduction for Scientists and Engineers (Oxford University, London, 1994).

Huang, J.-J.

Kaplan, H.

Kazaryan, A. V.

V. I. Balakshy, A. V. Kazaryan, “Laser beam direction stabilization by means of Bragg diffraction,” Opt. Eng. 38, 1154–1159 (1999).
[CrossRef]

V. I. Balakshy, A. V. Kazaryan, V. Y. Molchanov, “Deflectors with a feedback: new possibilities for image processing,” in International Conference on Optical Information Processing, Y. V. Gulyaev, D. R. Pape, eds., Proc. SPIE2051, 672–677 (1994).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, M. Sands, Feynman Lectures on Physics, The Commemorative Issue, Vols. 1–3 (Addison-Wesley, New York, 1989).

Molchanov, V. Y.

V. I. Balakshy, A. V. Kazaryan, V. Y. Molchanov, “Deflectors with a feedback: new possibilities for image processing,” in International Conference on Optical Information Processing, Y. V. Gulyaev, D. R. Pape, eds., Proc. SPIE2051, 672–677 (1994).
[CrossRef]

Poon, T.-C.

T.-C. Poon, S. K. Cheung, “Performance of a hybrid bistable device using an acoustooptic modulator,” Appl. Opt. 28, 4787–4791 (1989).
[CrossRef] [PubMed]

P. P. Banerjee, T.-C. Poon, “Simulation of bistability and chaos in hybrid devices,” in Proceedings of the 30th Midwest Symposium on Circuits and Systems, K. Jabbour, ed., (North-Holland, Amsterdam, 1987), pp. 820–823.

T.-C. Poon, “A Feynman Diagram Approach to Multiple Plane Wave Scattering in Acousto-Optic Interactions,” Ph. D. dissertation, (University of Iowa, Iowa City, Iowa, 1982).

Ramchandran, S.

M. R. Chatterjee, S. Ramchandran, “Feedback correction of angular error in grating readout,” in Photonic Devices and Algorithms for Computing III, K. M. Iftekharuddin, A. A. S. Awwal, eds., Proc. SPIE4470, 127–137 (2001).
[CrossRef]

S. Ramchandran, “Investigation of feedback in acoustooptic gratings and related applications” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).

Sands, M.

R. P. Feynman, R. B. Leighton, M. Sands, Feynman Lectures on Physics, The Commemorative Issue, Vols. 1–3 (Addison-Wesley, New York, 1989).

Sonmez, E.

M. R. Chatterjee, E. Sonmez, “An overview of acousto-optic bistability, chaos and logical applications,” in Acousto-Optics and Applications IV, B. B. J. Linde, A. Sikorska, eds., Proc. SPIE4514, 41–60 (2001).
[CrossRef]

E. Sonmez, “Development and examination of acousto-optic logic devices” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, 1994).

Tremblay, R.

J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
[CrossRef]

Vallee, R.

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

Appl. Opt. (4)

Can. J. Phys. (2)

J. Chrostowski, R. Vallee, C. Delisle, “Self-pulsing andchaos in acoustooptic bistability,” Can. J. Phys. 61, 1143–1148 (1983).
[CrossRef]

J. Chrostowski, C. Delisle, R. Tremblay, “Oscillation in an acoustooptic bistable device,” Can. J. Phys. 61, 188–191 (1983).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (1)

V. I. Balakshy, A. V. Kazaryan, “Laser beam direction stabilization by means of Bragg diffraction,” Opt. Eng. 38, 1154–1159 (1999).
[CrossRef]

Phys. Rev. A (1)

J. Chrostowski, “Noisy bifurcations in acousto-optic bistability,” Phys. Rev. A 26, 3023–3025 (1982).
[CrossRef]

Other (10)

P. P. Banerjee, T.-C. Poon, “Simulation of bistability and chaos in hybrid devices,” in Proceedings of the 30th Midwest Symposium on Circuits and Systems, K. Jabbour, ed., (North-Holland, Amsterdam, 1987), pp. 820–823.

S. Ramchandran, “Investigation of feedback in acoustooptic gratings and related applications” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).

R. P. Feynman, R. B. Leighton, M. Sands, Feynman Lectures on Physics, The Commemorative Issue, Vols. 1–3 (Addison-Wesley, New York, 1989).

V. I. Balakshy, A. V. Kazaryan, V. Y. Molchanov, “Deflectors with a feedback: new possibilities for image processing,” in International Conference on Optical Information Processing, Y. V. Gulyaev, D. R. Pape, eds., Proc. SPIE2051, 672–677 (1994).
[CrossRef]

M. R. Chatterjee, S. Ramchandran, “Feedback correction of angular error in grating readout,” in Photonic Devices and Algorithms for Computing III, K. M. Iftekharuddin, A. A. S. Awwal, eds., Proc. SPIE4470, 127–137 (2001).
[CrossRef]

T.-C. Poon, “A Feynman Diagram Approach to Multiple Plane Wave Scattering in Acousto-Optic Interactions,” Ph. D. dissertation, (University of Iowa, Iowa City, Iowa, 1982).

R. C. Hilborn, Chaos and Nonlinear Dynamics, an Introduction for Scientists and Engineers (Oxford University, London, 1994).

M. R. Chatterjee, E. Sonmez, “An overview of acousto-optic bistability, chaos and logical applications,” in Acousto-Optics and Applications IV, B. B. J. Linde, A. Sikorska, eds., Proc. SPIE4514, 41–60 (2001).
[CrossRef]

E. Sonmez, “Development and examination of acousto-optic logic devices” (M. S. thesis, Binghamton University, Binghamton, New York, 2001).

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley, Reading, 1994).

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Figures (6)

Fig. 1
Fig. 1

Schematic illustrating feedback equation.

Fig. 2
Fig. 2

(a) Plot of I 1 vs. n for Q = 3π, α0 = 2, β = 4.2; (b) Q = 8π, α0= 2, β = 5 (onset of chaos for high Q); (c) Q = 3π, α0 = 2, β = 5 (no chaos, still at period 4); (d) Q = 3π, α0 = 1, β = 4.7 (chaos sets in already for low Q); (e) Q = 8π, α0 = 1, β = 4.7); (f) Q = 3π, α0 = 1, β = 6 (low Q period 3 cycle within the chaotic regime. The transients show chaotic behavior, which reinforces our earlier statement about period three being a periodic window within chaos); (g) (on the next page) Q = 3π, α0 = 4, β = 5 (period 2 at high β).

Fig. 3
Fig. 3

(a) Plot of I 1 vs. α0 for Q = 3π, β = 4.5, TD ratio = 0.02; (b) Q = 8π, β = 4.5, TD ratio = 0.02 (high Q chaos); (c) Q = 3π, β = 5, TD ratio = 0.02 (onset of chaos for low Q).

Fig. 4
Fig. 4

(a) Plot of I 0 vs. α0 for Q = 3π, β = 5, TD ratio = 0.02 [note complementary nature relative to Fig. 3(c)]. (b) Plot of the sum of I 1 and I 0 vs. α0 for Q = 3π, β = 5, TD ratio = 0.02 {note reduction of chaos compared to I 0 [Fig. 4(a)] and I 1 [Fig. 3(c)]}.

Fig. 5
Fig. 5

(a) Plot of I 0 vs. α0 for Q = 8π, β = 4.5, TD ratio = 0.02; (b) Q = 8π, β = 4.5, TD ratio = 0.01125 [note higher chaos compared to Fig. 5(a), which has a higher TD ratio]; (c) Q = 3π, β = 5, TD ratio = 0.01125 (note extremely chaotic behavior toward the left-hand side, while the region to the right-hand side of the loop seems to be very regular for all gains for this value of Q).

Fig. 6
Fig. 6

(a) Plot of I 1 vs. α0 for Q = 8π, β = 4.5, TD ratio = 0.02; (b) Q = 3π, β = 4.5, TD ratio = 0.02.

Tables (1)

Tables Icon

Table 1 Summary of Dynamics of I1 When Bias Voltage is Not 0

Equations (21)

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

E1n+1=Einc sinαˆ0+βE12n2,
Xn+1=μXn1-Xn,
I1n+1=Iincfαn+1,
αn+1=α0+βI1n,
dE-1dξ=-j α exp-jQξE02,
dE0dξ=-j αexpjQξE-1+E12,
dE1dξ=-j αexpjQξE2+E02,
dE2dξ=-j α exp-jQξE12,
E0=-14expj2Q-α+5α2+4Q2+4αQ1/24×2Q+α5α2+4Q2+4αQ1/2-1-expj2Q-α-5α2+4Q2+4αQ1/24×2Q+α5α2+4Q2+4αQ1/2+1+expj2Q+α+5α2+4Q2-4αQ1/24×2Q-α5α2+4Q2-4αQ1/2-1-expj2Q+α-5α2+4Q2-4αQ1/24×2Q-α5α2+4Q2-4αQ1/2+1.
E-1=-jα2exp-j2Q+α-5α2+4Q2+4αQ1/24-exp-j2Q+α+5α2+4Q2+4αQ1/245α2+4Q2+4αQ1/2+expjα-2Q+5α2+4Q2-4αQ1/24-expjα-2Q-5α2+4Q2-4αQ1/245α2+4Q2-4αQ1/2.
L1=α216sin xx2+sin yy2+α2 sin x sin y cosα28xy,
I0=14cos2 x+cos2 y+2Q+α2 sin2 x64x2+2Q-α2 sin2 y64y2+2Q-αcos x sin y sinα28y+cos x cos y cosα22+22Q-α2Q+αsin x sin y cosα264xy-2Q+αcos y sin x sinα28x,
I1=14cos2 x+cos2 y+2Q+α2 sin2 x64x2+2Q-α2 sin2 y64y2-2Q-αcos x sin y sinα28y-cos x cos y cosα22-22Q-α2Q+αsin x sin y cosα264xy+2Q+αcos y sin x sinα28x,
I2=α216sin xx2+sin yy2-α2 sin x sin y cosα28xy,
x=5α2+4Q2+4αQ1/24, y=5α2+4Q2-4αQ1/24.
Xn+1=λXn1-Xn,
5α2+4Q2+4αQ=2Q+α2+2α2, 5α2+4Q2+4αQ1/2=2Q+α1+2α2Q+α21/2.
limQE014expjα22+exp-jα22=cosα2,
limQE1=-j sinα2,
With x=4Q2+4αQ+5α21/24, and y=4Q2-4αQ+5α21/24I-1+I0+I1+I2=214cos2x+cos2 y+2Q+α2 sin2 x64x2+2Q-α2 sin2 y64y2+α216sin xx2+sin yy2 =12cos2 x+cos2 y+sin2 xx22Q+α232+α28+sin2 yy22Q-α232+α28 =12cos2 x+cos2 y+sin2 xx22Q+α232+4α232+sin2 yy22Q-α232+4α232 =12cos2 x+cos2 y+sin2 xx24Q2+4αQ+5α232+sin2 yy24Q2-4αQ+5α232.
LHS=12cos2 x+cos2 y+16 sin2 x4Q2+4αQ+5α24Q2+4αQ+5α232+16 sin2 y4Q2-4αQ+5α24Q2-4αQ+5α232 =12cos2 x+cos2 y+12sin2 x+sin2 y=1,

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