Abstract

We report the results of experimental observation and theoretical analysis of the generation of subharmonic and chaotic signals in a harmonically driven all-fiber phase modulator (PM). When a PM constructed with optical fiber wrapped around a piezoelectric cylinder is driven at high amplitude, we identified that the fiber itself moves with subharmonic frequencies while the piezoelectric cylinder maintains harmonic motion. A theoretical model is presented that is a modification of the model for a bouncing ball on an oscillating table. Some key physical parameters for the model are identified. Potential origins for the discrepancy between the experimental and theoretical analyses are discussed. Ways to suppress the nonlinear effect are also discussed.

© 2013 Optical Society of America

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References

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  1. D. E. N. Davies and S. A. Kingsley, “Method of phase-modulating signals in optical fibers: application to optical-telemetry systems,” Electron. Lett. 10, 21–22 (1974).
    [CrossRef]
  2. S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
    [CrossRef]
  3. D. A. Jackson, R. Priest, A. Dandridge, and A. B. Tveten, “Elimination of drift in a single-mode optical fiber interferometer using a piezoelectrically stretched coiled fiber,” Appl. Opt. 19, 2926–2929 (1980).
    [CrossRef]
  4. W. Jin, D. Uttamchandani, and B. Culshaw, “Direct readout of dynamic phase changes in a fiber-optic homodyne interferometer,” Appl. Opt. 31, 7253–7258 (1992).
    [CrossRef]
  5. Y. N. Ning, B. C. B. Chu, and D. A. Jackson, “Interrogation of a conventional current transformer by a fiber-optic interferometer,” Opt. Lett. 16, 1448–1450 (1991).
    [CrossRef]
  6. H. J. Jeong, J. H. Kim, H. W. Lee, and B. Y. Kim, “Birefringence modulation in fiber-optic phase modulators,” Opt. Lett. 19, 1421–1423 (1994).
    [CrossRef]
  7. Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.
  8. M. N. Zervas and R. C. Youngquist, “Subharmonics, chaos, and hysteresis in piezoelectric fiber-optic phase modulators,” in Proceedings of 4th International Conference on Optical Fiber Sensors 19 (IECE, 1986), pp. 343–346.
  9. M. N. Zervas and I. P. Giles, “Optical-fiber phase modulator with enhanced modulation efficiency,” Opt. Lett. 13, 404–406 (1988).
    [CrossRef]
  10. I. Takuro, Fundamentals of Piezoelectricity (Oxford University, 1996).
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).
  12. M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
    [CrossRef]
  13. G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” Opt. Quantum Electron. 19, 179–190 (1987).
    [CrossRef]
  14. N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, 1992).
  15. F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
    [CrossRef]
  16. P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
    [CrossRef]

2008

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

1994

1992

W. Jin, D. Uttamchandani, and B. Culshaw, “Direct readout of dynamic phase changes in a fiber-optic homodyne interferometer,” Appl. Opt. 31, 7253–7258 (1992).
[CrossRef]

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

1991

1988

1987

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” Opt. Quantum Electron. 19, 179–190 (1987).
[CrossRef]

1980

1975

S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
[CrossRef]

1974

D. E. N. Davies and S. A. Kingsley, “Method of phase-modulating signals in optical fibers: application to optical-telemetry systems,” Electron. Lett. 10, 21–22 (1974).
[CrossRef]

Abbott, T.

N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, 1992).

Andre, P. S.

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

Antunes, P.

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

Chu, B. C. B.

Culshaw, B.

Dandridge, A.

Davies, D. E. N.

D. E. N. Davies and S. A. Kingsley, “Method of phase-modulating signals in optical fibers: application to optical-telemetry systems,” Electron. Lett. 10, 21–22 (1974).
[CrossRef]

Gagnaire, H.

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

Giles, I. P.

Imai, M.

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

Jackson, D. A.

Jeong, H. J.

H. J. Jeong, J. H. Kim, H. W. Lee, and B. Y. Kim, “Birefringence modulation in fiber-optic phase modulators,” Opt. Lett. 19, 1421–1423 (1994).
[CrossRef]

Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.

Jin, W.

Kim, B. Y.

H. J. Jeong, J. H. Kim, H. W. Lee, and B. Y. Kim, “Birefringence modulation in fiber-optic phase modulators,” Opt. Lett. 19, 1421–1423 (1994).
[CrossRef]

Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.

Kim, J. H.

Kingsley, S. A.

S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
[CrossRef]

D. E. N. Davies and S. A. Kingsley, “Method of phase-modulating signals in optical fibers: application to optical-telemetry systems,” Electron. Lett. 10, 21–22 (1974).
[CrossRef]

Koh, Y. W.

Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.

Lee, H. W.

Lima, H.

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

Martini, G.

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” Opt. Quantum Electron. 19, 179–190 (1987).
[CrossRef]

Monteiro, J.

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

Mure-Ravaud, A.

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

Ning, Y. N.

Odajima, A.

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

Ohtsuka, Y.

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

Pelissier, S.

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

Pigeon, F.

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

Priest, R.

Reilly, J.

N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, 1992).

Shimizu, T.

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

Takuro, I.

I. Takuro, Fundamentals of Piezoelectricity (Oxford University, 1996).

Tufillaro, N. B.

N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, 1992).

Tveten, A. B.

Uttamchandani, D.

Veillas, C.

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

Yeo, Y. B.

Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.

Youngquist, R. C.

M. N. Zervas and R. C. Youngquist, “Subharmonics, chaos, and hysteresis in piezoelectric fiber-optic phase modulators,” in Proceedings of 4th International Conference on Optical Fiber Sensors 19 (IECE, 1986), pp. 343–346.

Zervas, M. N.

M. N. Zervas and I. P. Giles, “Optical-fiber phase modulator with enhanced modulation efficiency,” Opt. Lett. 13, 404–406 (1988).
[CrossRef]

M. N. Zervas and R. C. Youngquist, “Subharmonics, chaos, and hysteresis in piezoelectric fiber-optic phase modulators,” in Proceedings of 4th International Conference on Optical Fiber Sensors 19 (IECE, 1986), pp. 343–346.

Appl. Opt.

Electron. Lett.

D. E. N. Davies and S. A. Kingsley, “Method of phase-modulating signals in optical fibers: application to optical-telemetry systems,” Electron. Lett. 10, 21–22 (1974).
[CrossRef]

S. A. Kingsley, “Optical-fiber phase modulator,” Electron. Lett. 11, 453–454 (1975).
[CrossRef]

F. Pigeon, S. Pelissier, A. Mure-Ravaud, H. Gagnaire, and C. Veillas, “Optical fiber Young modulus measurement using an optical method,” Electron. Lett. 28, 1034–1035 (1992).
[CrossRef]

J. Lightwave Technol.

M. Imai, T. Shimizu, Y. Ohtsuka, and A. Odajima, “An electric-field sensitive fiber with coaxial electrodes for optical phase modulation,” J. Lightwave Technol. 5, 926–931 (1987).
[CrossRef]

Microw. Opt. Technol. Lett.

P. Antunes, H. Lima, J. Monteiro, and P. S. Andre, “Elastic constant measurement for standard and photosensitive single mode optical fibers,” Microw. Opt. Technol. Lett. 50, 2467–2469 (2008).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

G. Martini, “Analysis of a single-mode optical fiber piezoceramic phase modulator,” Opt. Quantum Electron. 19, 179–190 (1987).
[CrossRef]

Other

N. B. Tufillaro, T. Abbott, and J. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, 1992).

Y. B. Yeo, H. J. Jeong, Y. W. Koh, and B. Y. Kim, “All fiber-optic polarization scrambler,” in Proceedings of 2nd Optoelectronics & Communications Conference (GIST, 1997), pp. 278–279.

M. N. Zervas and R. C. Youngquist, “Subharmonics, chaos, and hysteresis in piezoelectric fiber-optic phase modulators,” in Proceedings of 4th International Conference on Optical Fiber Sensors 19 (IECE, 1986), pp. 343–346.

I. Takuro, Fundamentals of Piezoelectricity (Oxford University, 1996).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

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Figures (12)

Fig. 1.
Fig. 1.

(a) Schematic of PZT PM and (b) coordinate axes of the coiled fiber (R0, PZT’s outer radius; a, fiber’s radius; 3, fiber’s axial direction).

Fig. 2.
Fig. 2.

(a) Experiment setup (MZ interferometer). (b) Phase modulation amplitude versus applied frequency. (c) Phase modulation amplitude versus applied voltage.

Fig. 3.
Fig. 3.

RF spectra of interference signals at applied voltages of (a) 0.5Vpp and (b) 5.0Vpp (fm=25.1kHz; vertical axis, 10dBm/div; horizontal axis, 10kHz/div; Res., 290 Hz).

Fig. 4.
Fig. 4.

Vibration detection on the side surface of PZT and fiber. (a) Schematic of Michelson interferometer (1, reflection from the fiber’s surface; 2, 4% internal reflection). (b) RF spectrum of interference signal from PZT surface. (c) RF spectrum of interference signal from fiber’s surface (fm=25.1kHz; Res., 150 Hz).

Fig. 5.
Fig. 5.

Experimental schematic for controlling the tension. W, weight.

Fig. 6.
Fig. 6.

Spectrum change versus tension on/off (5.0Vpp, 4.7 g weight). (a) Tension on, (b) reduced tension, (c) tension further reduced, and (d) minimum tension before losing contact (fm=25.1kHz; Res., 290 Hz).

Fig. 7.
Fig. 7.

RF spectrum of interference signal for tension on/off and oil. (a) Tension on without oil. (b) Tension off without oil. (c) Tension off with oil (fm=25.1kHz; Res., 290 Hz).

Fig. 8.
Fig. 8.

Schematic of the bouncing ball models: (a) bouncing ball and (b) modified bouncing ball.

Fig. 9.
Fig. 9.

(a) Axial elongation of fiber under tension (dot loop, neutral position; solid loop, stretched fiber). (b) Radial expansion of fiber loop.

Fig. 10.
Fig. 10.

Results of simulation at fspring=26kHz: time domain (left), frequency domain (right). (a) α=0.8, fspring=26kHz, Δxspring=1×107m, Atable=3.0×107m; (b) α=0.8, fspring=26kHz, Δxspring=1×1010m, Atable=3.0×107m.

Fig. 11.
Fig. 11.

Results of modeling at fspring=7.7kHz: time domain (left), frequency domain (right). (a) 1/2 subharmonics: α=0.8, fspring=7.7kHz, Δxspring=1×107m, Atable=1.0×107m; (b) 1/3 subharmonics: α=0.8, fspring=7.7kHz, Δxspring=1×107m, Atable=2.0×107m; (c) 1/4 subharmonics: α=0.8, fspring=7.7kHz, Δxspring=1×1010m, Atable=2.0×107m; (d) random motion: α=0.8, fspring=7.7kHz, Δxspring=1×107m, Atable=3.0×107m

Fig. 12.
Fig. 12.

Simulation results for ranges of values of α, Atable, and Δxspring at fspring=7.7kHz [Cross (red), 1/2 subharmonics; sphere (green), 1/3 subharmonics; box (black), 1/4 subharmonics; diamond (blue), random motion].

Equations (23)

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

ϕ=2πλnL,
Δϕ=2πλΔ(nL)=2πλ(LΔn+nΔL).
S1=S2=σS3=σΔLL,
Δni=n32jpijSj,
Δϕ=2πnλΔL{1n22[p12σ(p11+p12)]}.
Δϕ0.78(2πλnΔL).
ΔL=N2πΔR=2πNd33V,
Δϕ=4π2nd33NλV{1n22[p12σ(p11+p12)]},
Δϕ=ϕmsin(ωmt),
ϕm=4π2nd33λ{1n22[p12σ(p11+p12)]}NV0,
I(t)=I02[1+βcos(Δϕ+ϕ0)]=I02(1+β{[J0(ϕm)+2n=1J2n(ϕm)cos(2nωmt)]cos(ϕ0)[2n=1J2n1(ϕm)sin((2n1)ωmt)]sin(ϕ0)}).
ytable(t)=Atable[sin(2πftablet+ϕtable)+1],
yball(t)=yk+vk(ttk)12g(ttk)2,
(vkuk)=α(vkuk).
xtable(t)=Atablesin(2πftablet+ϕtable)+Δxspring,
xball(t)=Aball,ksin(2πfspringt+ϕball,k),
(vkuk)=α(vkuk).
T=k0ΔL=πa2YΔLL0.
k0=πa2L0Y=a22R0Y.
U=0ΔLFdL=0ΔLTdL=12k0(ΔL)2=12k0(2πΔR)2,
K=12Mv2.
vmax=2πk0MΔRmax=2πfspringΔRmax.
fspring=k0M=12πYρR02.

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