Abstract

We demonstrate, theoretically and experimentally, a new method to measure small changes in the cavity length of oscillators. The method is based on the high sensitivity of the phase of forced delay-line oscillators to changes in their cavity length. The oscillator phase is directly detected by mixing the oscillator output with the injected signal. We describe a comprehensive theoretical model for studying the signal and the noise at the output of a general forced delay-line oscillator with an instantaneous gain saturation and an amplitude-to-phase conversion. The results indicate that the magnitude and the bandwidth of the oscillator response to a small perturbation can be controlled by adjusting the injection ratio and the injected frequency. For signals with a frequency that is smaller than the device bandwidth, the oscillator noise is dominated by the noise of the injected signal. This noise is highly suppressed by mixing the oscillator output with the injected signal. Hence, the device sensitivity at frequencies below its bandwidth is limited only by the internal noise that is added in a single roundtrip in the oscillator cavity. We demonstrate the use of a forced oscillator as an acoustic fiber sensor in an optoelectronic oscillator. A good agreement is obtained between theory and experiments. The magnitude of the output signal can be controlled by adjusting the injection ratio while the noise power at low frequencies is not enhanced as in sensors that are based on a free-running oscillator.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  26. A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
    [Crossref]
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    [Crossref]
  30. K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
    [Crossref] [PubMed]
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  36. M. Fleyer and M. Horowitz, “Longitudinal mode selection in a delay-line homogeneously broadened oscillator with a fast saturable amplifier,” Opt. Express 25, 10632–10650 (2017).
    [Crossref] [PubMed]
  37. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
    [Crossref] [PubMed]
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    [Crossref]
  39. J.-J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

2017 (2)

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

M. Fleyer and M. Horowitz, “Longitudinal mode selection in a delay-line homogeneously broadened oscillator with a fast saturable amplifier,” Opt. Express 25, 10632–10650 (2017).
[Crossref] [PubMed]

2016 (3)

2015 (3)

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

A. Pikovsky, “Maximizing coherence of oscillations by external locking,” Phys. Rev. Lett. 115, 070602 (2015).
[Crossref] [PubMed]

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

2014 (1)

2013 (2)

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371: 20120464 (2013).
[Crossref] [PubMed]

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

2012 (1)

S. Usacheva and N. Ryskin, “Forced synchronization of a delayed-feedback oscillator,” Phys. D: Nonlinear Phenomena 241, 372–381 (2012).
[Crossref]

2010 (2)

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

2008 (3)

2005 (1)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

2004 (3)

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circ. 39, 1415–1424 (2004).
[Crossref]

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Physics 37, R197 (2004).
[Crossref]

2003 (1)

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Tech. 9, 57–79 (2003).
[Crossref]

1996 (1)

1993 (1)

R. York, “Nonlinear analysis of phase relationships in quasi-optical oscillator arrays,” IEEE Trans. Microwave Theory Tech. 41, 1799–1809 (1993).
[Crossref]

1981 (1)

A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun. 29, 1715–1720 (1981).
[Crossref]

1973 (1)

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[Crossref]

1966 (1)

D. Lesson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
[Crossref]

1965 (1)

L. Paciorek, “Injection locking of oscillators,” Proc. IEEE 53, 1723–1727 (1965).
[Crossref]

1946 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

1927 (1)

B. van der Pol, “Forced oscillations in a circuit with non-linear resistance (reception with reactive triode),” Philosophical Magazine and Journal of Science 7, 65–80 (1927).
[Crossref]

Acebrón, J. A.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
[Crossref]

Berman, M.

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Bonilla, L. L.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

Cahill, J.

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Cahill, J. P.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

Carter, G. M.

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Chembo, Y.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Chen, C. C.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Chen, J.

K. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems (Springer Science & Business Media, 2003).
[Crossref]

Chi, H.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Y. Zhu, X. Jin, H. Chi, S. Zheng, and X. Zhang, “High-sensitivity temperature sensor based on an optoelectronic oscillator,” Appl. Opt. 53, 5084–5087 (2014).
[Crossref] [PubMed]

Cho, D.

Cogdell, G.

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

Coillet, A.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Cole, J. H.

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

Dandridge, A.

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Physics 37, R197 (2004).
[Crossref]

Docherty, A.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

Eliyahu, D.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microwave Theory Tech. 56, 449–456 (2008).
[Crossref]

Erneux, T.

T. Erneux, Applied Delay Differential Equations, Vol. 3 (Springer Science & Business Media, 2009).

Ferre-Pikal, E. S.

F. L. Walls and E. S. Ferre-Pikal, “Measurement of frequency, phase noise and amplitude noise,” in Wiley Encyclopedia of Electrical and Electronics Engineering (1999).
[Crossref]

Fleyer, M.

Giallorenzi, T.

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

Goune Chengui, G.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Gu, K.

K. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems (Springer Science & Business Media, 2003).
[Crossref]

Horng, T. S.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Horowitz, M.

Hsiao, C. H.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Jau, J. K.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Jin, X.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Y. Zhu, X. Jin, H. Chi, S. Zheng, and X. Zhang, “High-sensitivity temperature sensor based on an optoelectronic oscillator,” Appl. Opt. 53, 5084–5087 (2014).
[Crossref] [PubMed]

Kharitonov, V. L.

K. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems (Springer Science & Business Media, 2003).
[Crossref]

Kirkendall, C.

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

Kirkendall, C. K.

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Physics 37, R197 (2004).
[Crossref]

Kittipute, K.

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

Kurokawa, K.

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[Crossref]

Kurths, J.

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge University, 2003).

Larger, L.

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371: 20120464 (2013).
[Crossref] [PubMed]

Lee, B.

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Tech. 9, 57–79 (2003).
[Crossref]

Lee, J.

Lesson, D.

D. Lesson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
[Crossref]

Levy, E. C.

Li, C. J.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Li, J. Y.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Li, P.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Li, W.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

J.-J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

Lin, G.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Lin, J.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Liu, X.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Maleki, L.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microwave Theory Tech. 56, 449–456 (2008).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

E. Salik, N. Yu, L. Maleki, and E. Rubiola, “Dual photonic-delay line cross correlation method for phase noise measurement,” in Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 2004 (2004), pp. 303–306.
[Crossref]

Martinenghi, R.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

McDowell, E. J.

Menyuk, C. R.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Noise distribution in the radio frequency spectrum of optoelectronic oscillators,” Opt. Lett. 33, 2883–2885 (2008).
[Crossref] [PubMed]

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Namer, M.

Okusaga, O.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Paciorek, L.

L. Paciorek, “Injection locking of oscillators,” Proc. IEEE 53, 1723–1727 (1965).
[Crossref]

Pan, W.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Park, S.

Peng, K. C.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Pikovsky, A.

A. Pikovsky, “Maximizing coherence of oscillations by external locking,” Phys. Rev. Lett. 115, 070602 (2015).
[Crossref] [PubMed]

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge University, 2003).

Pritchett, J.

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Razavi, B.

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circ. 39, 1415–1424 (2004).
[Crossref]

B. Razavi, RF microelectronics (Prentice Hall, 2012, Vol. 1).

Ren, J.

Ritort, F.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

Rosenblum, M.

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge University, 2003).

Rubiola, E.

E. Rubiola, Phase Noise and Frequency Stability in Oscillators (Cambridge University, 2009).

E. Salik, N. Yu, L. Maleki, and E. Rubiola, “Dual photonic-delay line cross correlation method for phase noise measurement,” in Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 2004 (2004), pp. 303–306.
[Crossref]

Ryskin, N.

S. Usacheva and N. Ryskin, “Forced synchronization of a delayed-feedback oscillator,” Phys. D: Nonlinear Phenomena 241, 372–381 (2012).
[Crossref]

Saleh, A. A. M.

A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun. 29, 1715–1720 (1981).
[Crossref]

Saleh, K.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Salik, E.

E. Salik, N. Yu, L. Maleki, and E. Rubiola, “Dual photonic-delay line cross correlation method for phase noise measurement,” in Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 2004 (2004), pp. 303–306.
[Crossref]

Saratayon, P.

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

Seidel, D.

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microwave Theory Tech. 56, 449–456 (2008).
[Crossref]

Seo, D.

Shao, L.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Sherman, A.

Slotine, J.-J. E.

J.-J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

Sorenson, R.

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Spigler, R.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

Srisook, S.

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

Talla, A.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Talla Mbe, J.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Usacheva, S.

S. Usacheva and N. Ryskin, “Forced synchronization of a delayed-feedback oscillator,” Phys. D: Nonlinear Phenomena 241, 372–381 (2012).
[Crossref]

van der Pol, B.

B. van der Pol, “Forced oscillations in a circuit with non-linear resistance (reception with reactive triode),” Philosophical Magazine and Journal of Science 7, 65–80 (1927).
[Crossref]

Vicente, C. J. P.

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

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F. L. Walls and E. S. Ferre-Pikal, “Measurement of frequency, phase noise and amplitude noise,” in Wiley Encyclopedia of Electrical and Electronics Engineering (1999).
[Crossref]

Wang, F. K.

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

Wardkein, P.

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

Woafo, P.

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

Yan, L.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Yang, C.

Yao, X. S.

Yim, S.

Yoon, S.

York, R.

R. York, “Nonlinear analysis of phase relationships in quasi-optical oscillator arrays,” IEEE Trans. Microwave Theory Tech. 41, 1799–1809 (1993).
[Crossref]

Yu, N.

E. Salik, N. Yu, L. Maleki, and E. Rubiola, “Dual photonic-delay line cross correlation method for phase noise measurement,” in Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 2004 (2004), pp. 303–306.
[Crossref]

Zhang, X.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Y. Zhu, X. Jin, H. Chi, S. Zheng, and X. Zhang, “High-sensitivity temperature sensor based on an optoelectronic oscillator,” Appl. Opt. 53, 5084–5087 (2014).
[Crossref] [PubMed]

Zheng, S.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Y. Zhu, X. Jin, H. Chi, S. Zheng, and X. Zhang, “High-sensitivity temperature sensor based on an optoelectronic oscillator,” Appl. Opt. 53, 5084–5087 (2014).
[Crossref] [PubMed]

Zhou, J.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Zhou, W.

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010).
[Crossref] [PubMed]

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

Zhu, Y.

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Y. Zhu, X. Jin, H. Chi, S. Zheng, and X. Zhang, “High-sensitivity temperature sensor based on an optoelectronic oscillator,” Appl. Opt. 53, 5084–5087 (2014).
[Crossref] [PubMed]

Zou, X.

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52, 1–16 (2016).
[Crossref]

A. Talla, R. Martinenghi, G. Goune Chengui, J. Talla Mbe, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51, 1–8 (2015).
[Crossref]

IEEE J. Solid-State Circ. (1)

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circ. 39, 1415–1424 (2004).
[Crossref]

IEEE Photon. J. (1)

A. Docherty, C. R. Menyuk, J. P. Cahill, O. Okusaga, and W. Zhou, “Rayleigh-scattering-induced RIN and amplitude-to-phase conversion as a source of length-dependent phase noise in OEOs,” IEEE Photon. J. 5, 5500514 (2013).
[Crossref]

IEEE Trans. Commun. (1)

A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun. 29, 1715–1720 (1981).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microwave Theory Tech. 56, 449–456 (2008).
[Crossref]

F. K. Wang, C. J. Li, C. H. Hsiao, T. S. Horng, J. Lin, K. C. Peng, J. K. Jau, J. Y. Li, and C. C. Chen, “A novel vital-sign sensor based on a self-injection-locked oscillator,” IEEE Trans. Microwave Theory Tech. 58, 4112–4120 (2010).
[Crossref]

R. York, “Nonlinear analysis of phase relationships in quasi-optical oscillator arrays,” IEEE Trans. Microwave Theory Tech. 41, 1799–1809 (1993).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. D: Appl. Physics (1)

C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D: Appl. Physics 37, R197 (2004).
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J. Washington Academy of Sciences (1)

J. H. Cole, C. Kirkendall, A. Dandridge, G. Cogdell, and T. Giallorenzi, “Twenty-five years of interferometric fiber optic acoustic sensors at the naval research laboratory,” J. Washington Academy of Sciences 90, 40–57 (2004).

Microwave and Opt. Tech. Lett. (1)

Y. Zhu, J. Zhou, X. Jin, H. Chi, X. Zhang, and S. Zheng, “An optoelectronic oscillator-based strain sensor with extended measurement range,” Microwave and Opt. Tech. Lett. 57, 2336–2339 (2015).
[Crossref]

Opt. Express (4)

Opt. Fiber Tech. (1)

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Tech. 9, 57–79 (2003).
[Crossref]

Opt. Lett. (2)

Phil. Trans. R. Soc. A (1)

L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Phil. Trans. R. Soc. A 371: 20120464 (2013).
[Crossref] [PubMed]

Philosophical Magazine and Journal of Science (1)

B. van der Pol, “Forced oscillations in a circuit with non-linear resistance (reception with reactive triode),” Philosophical Magazine and Journal of Science 7, 65–80 (1927).
[Crossref]

Phys. D: Nonlinear Phenomena (1)

S. Usacheva and N. Ryskin, “Forced synchronization of a delayed-feedback oscillator,” Phys. D: Nonlinear Phenomena 241, 372–381 (2012).
[Crossref]

Phys. Rev. Lett. (1)

A. Pikovsky, “Maximizing coherence of oscillations by external locking,” Phys. Rev. Lett. 115, 070602 (2015).
[Crossref] [PubMed]

Proc. IEEE (3)

L. Paciorek, “Injection locking of oscillators,” Proc. IEEE 53, 1723–1727 (1965).
[Crossref]

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[Crossref]

D. Lesson, “A simple model of feedback oscillator noise spectrum,” Proc. IEEE 54, 329–330 (1966).
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Proc. IRE (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).
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Rev. Modern Phys. (1)

J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Rev. Modern Phys. 77, 137 (2005).
[Crossref]

Sci. Rep. (1)

K. Kittipute, P. Saratayon, S. Srisook, and P. Wardkein, “Homodyne detection of short-range doppler radar using a forced oscillator model,” Sci. Rep. 7, 43680 (2017).
[Crossref] [PubMed]

Other (9)

B. Razavi, RF microelectronics (Prentice Hall, 2012, Vol. 1).

F. L. Walls and E. S. Ferre-Pikal, “Measurement of frequency, phase noise and amplitude noise,” in Wiley Encyclopedia of Electrical and Electronics Engineering (1999).
[Crossref]

K. Gu, J. Chen, and V. L. Kharitonov, Stability of Time-Delay Systems (Springer Science & Business Media, 2003).
[Crossref]

J.-J. E. Slotine and W. Li, Applied Nonlinear Control (Prentice Hall, 1991).

T. Erneux, Applied Delay Differential Equations, Vol. 3 (Springer Science & Business Media, 2009).

A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Vol. 12 (Cambridge University, 2003).

E. Rubiola, Phase Noise and Frequency Stability in Oscillators (Cambridge University, 2009).

E. Salik, N. Yu, L. Maleki, and E. Rubiola, “Dual photonic-delay line cross correlation method for phase noise measurement,” in Proceedings of the 2004 IEEE International Frequency Control Symposium and Exposition, 2004 (2004), pp. 303–306.
[Crossref]

O. Okusaga, J. Pritchett, R. Sorenson, W. Zhou, M. Berman, J. Cahill, G. M. Carter, and C. R. Menyuk, “The OEO as an acoustic sensor,” in 2013 Joint European Frequency and Time Forum International Frequency Control Symposium (EFTF/IFC) (2013), pp. 66–68.
[Crossref]

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Figures (12)
Fig. 1
Fig. 1 Schematic description of a delay-line injection-locked oscillator which is used for measurement of small variations in its cavity delay. G is a nonlinear element with instantaneous amplitude saturation and with amplitude-to-phase conversion, τ(t) is a time dependent delay, H(ω) is the frequency response of a bandpass filter that determines the operating bandwidth of the oscillator, ωinj is the frequency of an external signal that is injected into the oscillator, and x(t) denotes a phasor. The signal causes a small variation in the delay τ(t) that causes a phase fluctuation Δφ(t) in the oscillating signal with respect to the forcing signal. The oscillator phase variations are detected by mixing the oscillator output with the forcing signal.

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Fig. 2
Fig. 2 Transfer function |Ts(ωs)|2 as a function of the signal frequency ωs and the injection power ratio Γinj; τ0 is the average cavity delay. Different lines correspond to constant power enhancements in dB. Dashed red line corresponds to the frequency ωs = ωr. The phase lag Δφ0 equals zero.

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Fig. 3
Fig. 3 Transfer function |Ts(ωs)|2 of the signal at frequency ωs, for which ωsτ0 = 10−3 rad, versus the phase lag Δφ0 and the injection ratio Γinj. Different lines correspond to a constant enhancement in dB. The signal is significantly enhanced when the phase lag approaches 90°.

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Fig. 4
Fig. 4 Schematic description of the experiment setup. Laser is a CW laser, MZM is a Mach-Zehnder modulator, PFS is a piezo-electric fiber stretcher, L is a fiber with a length L, PD is a photo-detector, G1 – G3 are amplifiers, SSA is a signal source analyzer, BPF is a bandpass filter, C1 – C4 are directional couplers with coupling ratios of −6 dB, −6 dB, −20 dB and −5 dB, respectively. PC is a personal computer, ϕ is electro-mechanical phase shifter, and LPF is a low-pass filter with a cutoff frequency of 20 kHz.

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Fig. 5
Fig. 5 Amplitude-to-amplitude conversion coefficient γam, derived from the open-loop power transmission, measured between the input to amplifier G1 and the output of the photo-detector. At operating condition, the power at the entrance of G1 is −7.8 dBm and the system is deeply saturated with γam = 0.06.

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Fig. 6
Fig. 6 Phase noise of the free-running OEO (green curve) and the phase noise of the injected source (yellow curve), measured by using SSA. The internal phase noise of the OEO (bottom blue curve) was extracted from the noise of the free-running OEO by using Eq. (24). Empirical fits to the curves are shown by dashed curves.

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Fig. 7
Fig. 7 Phase noise (solid blue curve) and amplitude noise (solid yellow curve) spectra of the injected-locked OEO, measured by using SSA. The measured phase noise is compared with the PSD, calculated by using Eq. (23) (dashed red curve). The injection ratio was Γinj = −30 dB and the phase lag was Δφ0 = 0°. The device bandwidth equals fr = 4.8 kHz.

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Fig. 8
Fig. 8 Phase noise PSD at the output of the mixer SB,noise(f) (solid blue curve), calculated for an injection ratio Γinj = −40 dB and an average phase lag Δφ0 = 0. The result is compared to the calculated phase noise PSD of the injection-locked OEO, SN(ω) (solid red curve) and to the measured phase noise of the external signal (dashed black curve), where f in these two curves represents the frequency offset with respect to the carrier frequency finj. The solid yellow curve in the figure gives the calculated phase noise PSD at the mixer output when the internal noise source is set to zero. The vertical dashed blue line marks the device bandwidth of fr = 1.6 kHz. The results indicate that inside the device bandwidth, the effect of the injected noise is highly suppressed at the mixer output.

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Fig. 9
Fig. 9 Spectrum of a beating signal and the noise, measured by using A/D converter (blue solid curve), for a sinusoidal signal at 30 Hz that was supplied to the PFS and caused a stretching of the fiber with an amplitude of about 80 nm. The result is compared to the calculated noise PSD (solid red curve) and to the signal power (red dot). The resolution bandwidth equals RBW = 1 Hz, the injection power ratio equals Γinj = −40 dB, and the average phase lag equals Δφ0 = 0°.

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Fig. 10
Fig. 10 Transfer function of the signal, |Ts(ω)|2, calculated by using Eq. (11), for injection ratios of −20 dB, −30 dB and −40 dB and a phase lag Δφ0 = 0 (blue curves), which is compared to the measured results (red dots).

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Fig. 11
Fig. 11 (a) Measured temporal response of the phase at the mixer output when the fiber was stretched by a square voltage with a period of 0.2 s that was supplied to the PFS and caused a change of 0.96 μm in the fiber length. The response was measured for (i) Γinj = −30 dB, Δφ0 = 0°; (ii) Γinj = −30 dB, Δφ0 = 60°; (iii) Γinj = −40 dB, Δφ0 = 0°, and (iv) Γinj = −40 dB, Δφ0 = 60°. The corresponding power enhancement factors of experiments (i) −(iv) are 30, 36, 40 and 46 dB, respectively. Figure 11(b) shows a close-up on the initial response that is compared to the theoretical response function (dashed black curves), calculated by using Eq. (18). The mixer outputs were filtered by a low-pass filter with a FWHM of 40 kHz and was sampled at a rate of 100 kSamples/sec.

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Fig. 12
Fig. 12 Output signals that correspond to graphs (i) (blue curve) and (iv) (purple curve) in Fig. 11(a) that are obtained after normalizing the measured phase by the mean steady-state phase. The enhancement factor equals 30 dB for curve (i) and 46 dB for curve (iv). The normalized noise decreases for the higher enhancement factor case.

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Equations (58)

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

a ( t ) e j φ ( t ) = x inj ( t ) + h ( t ) * { f [ a ( t ′ ) + n a ( t ′ ) ] e j { φ ( t ′ ) + n φ ( t ′ ) } } | t ′ = t − τ ( t ) ,
a 0 = f G ( a 0 ) H 0 cos ( ζ 0 + ϕ 0 − ω inj τ 0 ) + b cos Δ ϕ 0 ,
0 = f G ( a 0 ) H 0 sin ( ζ 0 + ϕ 0 − ω inj τ 0 ) − b sin Δ φ 0 ,
sin ( ω inj τ 0 − ϕ 0 − ζ 0 ) = r inj sin ( ω inj τ 0 − ϕ 0 − ζ 0 − Δ φ 0 ) .
r inj ≥ | sin ( ω inj τ 0 − ω k τ 0 ) | .
Δ φ 0 = − sin − 1 { sin ( ω inj τ 0 − ω k τ 0 ) / r inj } + ( ω inj τ 0 − ω k τ 0 ) ,
a ( t ) = a 0 + Δ a sig ( t ) + Δ a N ( t ) ,
φ ( t ) = ω inj t + Δ φ 0 + Δ φ sig ( t ) + Δ φ N ( t ) ,
Δ φ sig ( ω ) = − ω inj δ τ ( ω ) T s ( ω ) ,
T s ( ω ) = 1 − r inj cos Δ φ 0 1 − ( 1 − r inj cos Δ φ 0 ) exp ( − j ω τ 0 )
T s ( ω ) ≈ G s 1 + j ω τ r ,
G s = ( 1 − r inj cos Δ φ 0 ) / ( r inj cos Δ φ 0 )
Δ φ sig ( t ) = ( 1 − r inj cos Δ φ 0 ) [ Δ φ sig ( t − τ 0 ) − ω inj δ τ ( t ) ] .
Δ φ sig ( t ) = − ω inj ∑ k = 0 ∞ ( 1 − r inj cos Δ φ 0 ) k + 1 δ τ ( t − k τ 0 ) .
Δ φ step ( t ) = − ω inj δ τ 0 ∑ k = 0 ⌊ t / τ 0 ⌋ ( 1 − r inj cos Δ φ 0 ) k + 1 .
Δ φ step ( t ) = − ω inj δ τ 0 G s [ 1 − ( 1 − r inj cos Δ φ 0 ) ⌊ t / τ 0 ⌋ ] ,
( 1 − r inj cos Δ φ 0 ) ⌊ t / τ 0 ⌋ = exp { ⌊ t / τ 0 ⌋ ln ( 1 − r inj cos Δ φ 0 ) } ≈ e − t / τ r ,
Δ φ step ( t ) ≈ − ω inj δ τ 0 G s ( 1 − e − t / τ r ) .
h s ( t ) ≈ G s τ r e − t / τ r u ( t ) ,
Δ φ sig ( t ) ≈ − ω inj G s τ r ∫ − ∞ t d t ′ e − ( t − t ′ ) / τ r δ τ ( t ′ ) .
Δ φ N ( ω ) = n φ , inj ( ω ) T s ( ω ) / G s + r inj n φ , inj ( ω ) γ pm sin Δ φ 0 e − j ω τ 0 T s ( ω ) + n ˜ φ ( ω ) e − j ω τ 0 T s ( ω ) ,
n ˜ φ ( ω ) = n φ ( ω ) + γ pm e − j ω τ 0 n a ( ω ) / a 0
S N ( ω ) = ( 1 / G s ) 2 | T s ( ω ) | 2 | 1 + e − j ω τ 0 q s ( ω ) γ pm tan Δ φ 0 | 2 S inj ( ω ) + | T s ( ω ) | 2 S φ ˜ ( ω ) .
S OEO – free ( ω ) = S φ ˜ ( ω ) / ( ω τ 0 ) 2 ,
I ( t ) = v ( t ) cos [ Δ φ 0 + Δ φ B ( t ) + ϕ P ] , Q ( t ) = v ( t ) sin [ Δ φ 0 + Δ φ B ( t ) + ϕ P ] ,
Δ φ B ( t ) = n φ , inj ( t ) − Δ φ N ( t ) − Δ φ sig ( t ) ,
Δ φ B , noise ( ω ) = n φ , inj ( ω ) j ω τ 0 + r inj γ pm sin Δ φ 0 r inj cos Δ φ 0 + j ω τ 0 − e − j ω τ 0 n ˜ φ ( ω ) r inj cos Δ φ 0 + j ω τ 0 ,
S B , noise ( ω ) = ( G s + 1 ) 2 S inj ( ω ) [ ( ω r τ 0 ) 2 ( γ pm tan Δ φ 0 ) 2 + ( ω τ 0 ) 2 ] + S φ ˜ ( ω ) 1 + ( ω / ω r ) 2 .
SNR = ( ω inj δ τ 0 ) 2 [ 1 2 π ∫ 0 ∞ d ω ( ω τ 0 ) 2 S inj ( ω ) + S φ ( ω ) 1 + ( ω / ω r ) 2 ] − 1 ,
SNR ^ = 〈 Δ φ B ( t ) 〉 2 〈 Δ φ B 2 ( t ) 〉 − 〈 Δ φ B ( t ) 〉 2 ,
f [ a 0 + Δ a ( t ) ] ≈ f G ( a 0 ) e i ζ ( a 0 ) + [ f G ′ ( a 0 ) + j ζ ′ ( a 0 ) f G ( a 0 ) ] e j ζ ( a 0 ) Δ a ( t ) ,
Δ a ( t ) = { γ am q s ( t ) − γ pm q c ( t ) } * [ Δ a 0 ( t − τ 0 ) + n a ( t − τ 0 ) ] − a 0 q c ( t ) * [ Δ φ ( t − τ 0 ) − ω inj δ τ ( t ) + n φ ( t − τ 0 ) ] + b n φ , inj ( t ) sin Δ φ 0 + n a , inj ( t ) cos Δ φ 0 ,
a 0 Δ φ ( t ) = { γ am q c ( t ) + γ pm q s ( t ) } * [ Δ a ( t − τ 0 ) + n a ( t − τ 0 ) ] + a 0 q s ( t ) * [ Δ φ ( t − τ 0 ) − ω inj δ τ ( t ) + n φ ( t − τ 0 ) ] + b n φ , inj ( t ) cos Δ φ 0 − n a , inj ( t ) sin Δ φ 0 ,
q c ( t ) = ( 1 − r inj cos Δ φ 0 ) h im ( t , ω inj ) − r inj sin Δ φ 0 h r ( t , ω inj ) , q s ( t ) = ( 1 − r inj cos Δ φ 0 ) h r ( t , ω inj ) + r inj sin Δ φ 0 h im ( t , ω inj ) ,
Δ a ( ω ) = { γ am q s ( ω ) − γ pm q c ( ω ) } [ Δ a ( ω ) + n a ( ω ) ] e − j ω τ 0 − a 0 q c ( ω ) e − j ω τ 0 [ Δ φ ( ω ) + n φ ( ω ) − e j ω τ 0 ω inj δ τ ( ω ) ] + b n φ , inj ( ω ) sin Δ φ 0 + n a , inj ( ω ) cos Δ φ 0 ,
a 0 Δ φ ( ω ) = { γ am q c ( ω ) + γ pm q s ( ω ) } e − j ω τ 0 [ Δ a ( ω ) + n a ( ω ) ] + a 0 q s ( ω ) e − j ω τ 0 [ Δ φ ( ω ) + n φ ( ω ) − e j ω τ 0 ω inj δ τ ( ω ) ] + b n φ , inj ( ω ) cos Δ φ 0 − n a , inj ( ω ) sin Δ φ 0 ,
q c ( ω ) = ( 1 − r inj cos Δ φ 0 ) h im ( ω , ω inj ) − r inj sin Δ φ 0 h r ( ω , ω inj ) , q s ( ω ) = ( 1 − r inj cos Δ φ 0 ) h r ( ω , ω inj ) + r inj sin Δ φ 0 h im ( ω , ω inj )
h r ( ω , ω inj ) = 1 2 H 0 [ H ( ω inj + ω ) e − j ϕ 0 + H * ( ω inj − ω ) e j ϕ 0 ] , h im ( ω , ω inj ) = − j 2 H 0 [ H ( ω inj + ω ) e − j ϕ 0 − H * ( ω inj − ω ) e j ϕ 0 ] .
M ( ω ) [ Δ a ( ω ) a 0 Δ φ ( ω ) ] = V ( ω ) { a 0 ω inj δ τ ( ω ) [ 0 1 ] + e − j ω τ 0 [ n a ( ω ) a 0 n φ ( ω ) ] } + [ cos Δ φ 0 − sin Δ φ 0 ] n a , inj ( ω ) + [ sin Δ φ 0 cos Δ φ 0 ] b n φ , inj ( ω ) ,
M ( ω ) = I − e − j ω τ 0 V ( ω ) ,
V ( ω ) = [ γ am q s ( ω ) − γ pm q c ( ω ) − q c ( ω ) γ am q c ( ω ) + γ pm q s ( ω ) q s ( ω ) ] .
M − 1 ( ω ) d ( ω ) = I − e − j ω τ 0 [ q s ( ω ) q c ( ω ) − γ am q c ( ω ) − γ pm q s ( ω ) γ am q s ( ω ) − γ pm q c ( ω ) ] ,
d ( ω ) = e − 2 j ω τ 0 γ am [ q s 2 ( ω ) + q c 2 ( ω ) ] + 1 − ( 1 + γ am ) e − j ω τ 0 q s ( ω ) + e − j ω τ 0 γ pm q c ( ω ) .
Δ a sig ( ω ) = a 0 ω inj δ τ ( ω ) d ( ω ) q c ( ω ) ,
a 0 Δ φ sig ( ω ) = a 0 ω inj δ τ ( ω ) d ( ω ) [ − q s ( ω ) + e − j ω τ 0 γ am ( q c 2 ( ω ) + q s 2 ( ω ) ) ] ,
H ( ω inj + ω ) ≈ [ H 0 + H 0 ′ ω + 1 2 H 0 ″ ω 2 + O ( ω 3 ) ] × exp [ j ϕ 0 + j ϕ 0 ′ ω + j 1 2 ϕ 0 ″ ω 2 + O ( ω 3 ) ] ,
h r ( ω , ω inj ) ≈ 1 + O ( ω 2 ) , h im ( ω , ω inj ) ≈ O ( | ω | ) .
q c ( ω ) ≈ − r inj sin Δ φ 0 , q s ( ω ) ≈ 1 − r inj cos Δ φ 0 .
| q c ( ω ) | ≪ | q s ( ω ) | .
d ( ω ) ≈ e − 2 j ω τ 0 γ am q s 2 ( ω ) + 1 − ( 1 + γ am ) e − j ω τ 0 q s ( ω ) = [ 1 − e − j ω τ 0 q s ( ω ) ] [ 1 − γ am e − j ω τ 0 q s ( ω ) ] .
Δ φ sig ( ω ) = − ω inj δ τ ( ω ) q s ( ω ) 1 − q s ( ω ) exp ( − j ω τ 0 ) .
Δ φ N ( ω ) = r inj r φ , inj ( ω ) cos Δ φ 0 + γ pm sin Δ φ 0 e − j ω τ 0 q s ( ω ) 1 − q s ( ω ) exp ( − j ω τ 0 ) + n ˜ φ ( ω ) e − j ω τ 0 q s ( ω ) 1 − q s ( ω ) exp ( − j ω τ 0 ) ,
n ˜ φ ( ω ) = n φ ( ω ) + γ pm e − j ω τ 0 n a ( ω ) / a 0
Δ a N ( ω ) / a 0 ≈ r inj sin Δ φ 0 [ e − j ω τ 0 Δ φ N ( ω ) + n φ ( ω ) + n φ , inj ( ω ) ] + γ am e − j ω τ 0 n a ( ω ) .
σ 2 ( ω inj ) = M ( ω inj ) ( r inj 2 − 2 r inj cos Δ φ 0 + 1 ) ,
M ( ω inj ) = max ω ( | h r ( ω , ω inj ) | 2 + | h im ( ω , ω inj ) | 2 ) = 1 2 H 0 2 max ω [ | H ( ω inj + ω ) | 2 + | H ( ω inj − ω ) | 2 ] .
r inj ≥ max { | sin ( ω inj τ 0 − ω k τ 0 ) | , 1 − 1 M ( ω inj ) } ,
cos Δ φ 0 > 1 2 r inj [ r inj 2 + 1 − 1 M ( ω inj ) ] .

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