Abstract

Homogeneously broadened delay-line oscillators such as lasers or optoelectronic oscillators (OEOs) can potentially oscillate in a large number of cavity modes that are supported by their amplifier bandwidth. In a continuous wave operating mode, the oscillating mode is selected between one or few cavity modes that experience the highest small-signal gain. In this manuscript, we show that the oscillation mode of a homogeneously broadened oscillator can be selected from a large number of modes in a frequency region that can be broader than the full width at half maximum of the effective cavity filter. The mode is selected by a short-time injection of an external signal into the oscillator. After the external signal is turned off, the oscillation is maintained in the selected mode even if this mode has a significantly lower small-signal gain than that of other cavity modes. The stability of the oscillation is obtained due to nonlinear saturation effect in the oscillator amplifier. We demonstrate, experimentally and theoretically, mode selection in a long cavity OEO. We could select any desired mode between 400 cavity modes while maintaining ultra-low phase noise in the selected mode and in the non-oscillating modes. No mode-hopping was observed during our maximum measurement duration of about 24 hours.

© 2017 Optical Society of America

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References

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2016 (2)

2015 (2)

M. Fleyer, J. P. Cahill, M. Horowitz, C. R. Menyuk, and O. Okusaga, “Comprehensive model for studying noise induced by self-homodyne detection of backward Rayleigh scattering in optical fibers,” Opt. Express 23(20), 25635–25652 (2015).
[Crossref] [PubMed]

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

2014 (1)

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

2013 (2)

Y. Jiang, G. Bai, L. Hu, H. Li, Z. Zhou, J. Xu, and S. Wang, “Frequency locked single-mode optoelectronic oscillator by using low frequency RF signal injection,” IEEE Photon. Technol. Lett. 25(4), 382–384 (2013).
[Crossref]

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(2), 5500514 (2013).
[Crossref]

2010 (1)

2009 (2)

E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26(1), 148–159 (2009).
[Crossref]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

2008 (1)

K. H. Lee, J. Y. Kim, and W. Y. Choi, “Injection-locked hybrid optoelectronic oscillators for single-mode oscillation,” IEEE Photon. Technol. Lett. 20(19), 1645–1647 (2008).
[Crossref]

2004 (1)

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39(9), 1415–1424 (2004).
[Crossref]

2002 (1)

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

1999 (1)

W. Lewandowski, J. Azoubib, and W. J. Klepczynski, “GPS: primary tool for time transfer,” Proc. IEEE 87(1), 163–172 (1999).
[Crossref]

1996 (1)

1995 (1)

C. R. Doerr, M. Zirngibl, and C. H. Joyner, “Single longitudinal-mode stability via wave mixing in long-cavity semiconductor lasers,” IEEE Photon. Technol. Lett. 7(9), 962–964 (1995).
[Crossref]

1988 (1)

1985 (1)

1983 (1)

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

1981 (1)

A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun. 29(11), 1715–1720 (1981).
[Crossref]

1978 (1)

1977 (1)

1973 (1)

R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE 61(10), 1380–1385 (1973).
[Crossref]

1972 (1)

P. W. Smith, “Mode selection in lasers,” Proc. IEEE 60(4), 422–440 (1972).
[Crossref]

1955 (1)

W. A. Edson, “Frequency memory in multi-mode oscillators,” IRE Trans. Circuit Theory 2(1), 58–66 (1955).
[Crossref]

Abrams, R. L.

Adler, R.

R. Adler, “A study of locking phenomena in oscillators,” Proc. IEEE 61(10), 1380–1385 (1973).
[Crossref]

Agrawal, G. P.

Azoubib, J.

W. Lewandowski, J. Azoubib, and W. J. Klepczynski, “GPS: primary tool for time transfer,” Proc. IEEE 87(1), 163–172 (1999).
[Crossref]

Bai, G.

Y. Jiang, G. Bai, L. Hu, H. Li, Z. Zhou, J. Xu, and S. Wang, “Frequency locked single-mode optoelectronic oscillator by using low frequency RF signal injection,” IEEE Photon. Technol. Lett. 25(4), 382–384 (2013).
[Crossref]

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

Browning, I.

I. Browning and M. F. Lewis, “Theory of Multimoding in SAW Oscillators,” in Proceedings of IEEE Ultrasonics Symposium (IEEE, 1976), pp. 256–259.

Cahill, J. P.

M. Fleyer, J. P. Cahill, M. Horowitz, C. R. Menyuk, and O. Okusaga, “Comprehensive model for studying noise induced by self-homodyne detection of backward Rayleigh scattering in optical fibers,” Opt. Express 23(20), 25635–25652 (2015).
[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(2), 5500514 (2013).
[Crossref]

Carter, G. M.

Chembo, Y. K.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

Chengui, G. R. G.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

Choi, W. Y.

K. H. Lee, J. Y. Kim, and W. Y. Choi, “Injection-locked hybrid optoelectronic oscillators for single-mode oscillation,” IEEE Photon. Technol. Lett. 20(19), 1645–1647 (2008).
[Crossref]

Coillet, A.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[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(2), 5500514 (2013).
[Crossref]

Doerr, C. R.

C. R. Doerr, M. Zirngibl, and C. H. Joyner, “Single longitudinal-mode stability via wave mixing in long-cavity semiconductor lasers,” IEEE Photon. Technol. Lett. 7(9), 962–964 (1995).
[Crossref]

Edson, W. A.

W. A. Edson, “Frequency memory in multi-mode oscillators,” IRE Trans. Circuit Theory 2(1), 58–66 (1955).
[Crossref]

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

Erneux, T.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

Figueiredo, J. M. L.

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

Fleyer, M.

Godard, A.

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

Graindorge, P.

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

Hendow, S. T.

Horowitz, M.

Hosseini, S. E.

Hu, L.

Y. Jiang, G. Bai, L. Hu, H. Li, Z. Zhou, J. Xu, and S. Wang, “Frequency locked single-mode optoelectronic oscillator by using low frequency RF signal injection,” IEEE Photon. Technol. Lett. 25(4), 382–384 (2013).
[Crossref]

Jacquot, M.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

Jahanbakht, S.

Javaloyes, J.

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

Jiang, Y.

Y. Jiang, G. Bai, L. Hu, H. Li, Z. Zhou, J. Xu, and S. Wang, “Frequency locked single-mode optoelectronic oscillator by using low frequency RF signal injection,” IEEE Photon. Technol. Lett. 25(4), 382–384 (2013).
[Crossref]

Joyner, C. H.

C. R. Doerr, M. Zirngibl, and C. H. Joyner, “Single longitudinal-mode stability via wave mixing in long-cavity semiconductor lasers,” IEEE Photon. Technol. Lett. 7(9), 962–964 (1995).
[Crossref]

Kha?iretdinov, K. A.

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

Kim, J. Y.

K. H. Lee, J. Y. Kim, and W. Y. Choi, “Injection-locked hybrid optoelectronic oscillators for single-mode oscillation,” IEEE Photon. Technol. Lett. 20(19), 1645–1647 (2008).
[Crossref]

Klepczynski, W. J.

W. Lewandowski, J. Azoubib, and W. J. Klepczynski, “GPS: primary tool for time transfer,” Proc. IEEE 87(1), 163–172 (1999).
[Crossref]

Kong, F.

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

Kurths, J.

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

Larger, L.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

Lee, K. H.

K. H. Lee, J. Y. Kim, and W. Y. Choi, “Injection-locked hybrid optoelectronic oscillators for single-mode oscillation,” IEEE Photon. Technol. Lett. 20(19), 1645–1647 (2008).
[Crossref]

Levy, E. C.

Lewandowski, W.

W. Lewandowski, J. Azoubib, and W. J. Klepczynski, “GPS: primary tool for time transfer,” Proc. IEEE 87(1), 163–172 (1999).
[Crossref]

Lewis, M. F.

I. Browning and M. F. Lewis, “Theory of Multimoding in SAW Oscillators,” in Proceedings of IEEE Ultrasonics Symposium (IEEE, 1976), pp. 256–259.

Li, H.

Y. Jiang, G. Bai, L. Hu, H. Li, Z. Zhou, J. Xu, and S. Wang, “Frequency locked single-mode optoelectronic oscillator by using low frequency RF signal injection,” IEEE Photon. Technol. Lett. 25(4), 382–384 (2013).
[Crossref]

Li, W.

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

Lin, G.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

Lind, R. C.

Maleki, L.

Martin, P.

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

Martinenghi, R.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

Mbé, J. H. T.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

Menyuk, C. R.

Namer, M.

Okhotnikov, O. G.

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

Okusaga, O.

Pauliat, G.

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

Peil, M.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E 79(2), 026208 (2009).
[Crossref]

Pepper, D. M.

Pikovsky, A.

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

Rakhval’skii, M. P.

A. P. Bogatov, P. G. Eliseev, O. G. Okhotnikov, M. P. Rakhval’skiĭ, and K. A. KhaȈıretdinov, “Interaction of modes and self-stabilization of single-frequency emission from injection lasers,” Sov. J. Quantum Electron. 13(9), 1221–1229 (1983).
[Crossref]

Razavi, B.

B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39(9), 1415–1424 (2004).
[Crossref]

Romeira, B.

B. Romeira, F. Kong, W. Li, J. M. L. Figueiredo, J. Javaloyes, and J. Yao, “Broadband chaotic signals and breather oscillations in an optoelectronic oscillator incorporating a microwave photonic filter,” J. Lightw. Technol. 32(20), 3933–3942 (2014).
[Crossref]

Roosen, G.

A. Godard, G. Pauliat, G. Roosen, P. Graindorge, and P. Martin, “Side-mode gain in grating-tuned extended-cavity semiconductor lasers: investigation of stable single-mode operation conditions,” IEEE J. Quantum Electron. 38(4), 390–401 (2002).
[Crossref]

Rosenblum, M.

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

Rubiola, E.

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

Saleh, A. A. M.

A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun. 29(11), 1715–1720 (1981).
[Crossref]

Saleh, K.

A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
[Crossref]

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A. F. Talla, R. Martinenghi, G. R. G. Chengui, J. H. T. Mbé, K. Saleh, A. Coillet, G. Lin, P. Woafo, and Y. K. Chembo, “Analysis of phase-locking in narrow-band optoelectronic oscillators with intermediate frequency,” IEEE J. Quantum Electron. 51(6), 1–8 (2015).
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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of mode selection in OEOs. (a) A free-running OEO oscillates at frequency fj. An external signal with a frequency fsig, which is approximately equal to the frequency of the m-th cavity mode, is injected into OEO for a short time. (b) After the external signal is switched-off, the oscillation frequency equals fm. Dashed curve indicates the transfer function of the open-loop cavity gain.

Fig. 2
Fig. 2

Schematic description of the experiment setup. Laser is a CW laser source, MZM is a Mach-Zehnder modulator, L is a 7.5-km fiber, PD is a photo-detector, G1 and G2 are RF amplifiers, C1, C2, C3 and C4 are directional couplers with coupling ratios of −6 dB, −6 dB, −20 dB and −5 dB, respectively, BPF is a bandpass filter with a central frequency of 10 GHz and a bandwidth of 10 MHz, PS is an electronically controlled phase shifter, SSA is a signal source analyzer, and "Switch" is an RF switch that is controlled by the computer (PC). The outputs of a I/Q mixer are sampled and used to measure the relative phase and frequency between the OEO signal and the signal generator.

Fig. 3
Fig. 3

Power transmission (solid blue) and phase (solid back) of the intracavity OEO filter, measured by using network analyzer, around a central frequency of 10 GHz with a frequency resolution of 12.5 kHz.

Fig. 4
Fig. 4

(a) Beating frequency between the OEO signal and the signal generator after the external signal is switched off at t = 1 s, marked by a dashed vertical red curve. The beating frequency is calculated by performing a derivation to the measured outputs of the I/Q mixer that are sampled at a rate of 40 kSamples/sec. The frequency difference between the signal generator and the stand-alone oscillator is only about 5 Hz. The injection ratio was equal to −10 dB and it enabled to injection-lock the OEO before the injected signal was turned off. (b) Transient response of the OEO after the injected signal was turned off at t = 0 s, in case that the frequency detuning between the injected signal and the corresponding OEO cavity mode was intentionally increased to 130 Hz. The figure shows the voltage of one of the outputs of the I/Q mixer that was sampled at a rate of 250 kSamples/sec. A transient behavior to a stand-alone oscillation occurs over a period of 30 μs, as marked by the two dashed red vertical lines.

Fig. 5
Fig. 5

Frequency drift of the stand-alone OEO versus time for a carrier frequency of 9.997 GHz (blue), 10 GHz (red) and 10.005 GHz (green) calculated every 30 seconds from the measured beating signal between the OEO and the synthesizer that was used for injection. The oscillation frequency changes smoothly due to variations in the environmental conditions. No mode-hopping is observed during the measurement duration of 24 hours.

Fig. 6
Fig. 6

RF spectra of the stand-alone OEO after setting its frequency by a short-time injection of an external signal at twelve different frequencies in the region of 9.996–10.007 GHz. The difference between the oscillation frequencies was chosen to be 1 MHz. Different colors of the plots correspond to different oscillation frequencies. The spectra were measured by using an RF spectrum analyzer with a resolution bandwidth of 10 kHz.

Fig. 7
Fig. 7

Phase noise spectra of a stand-alone OEO that was set to oscillate at frequencies of 9.997, 10, 10.003 and 10.006 GHz. Peaks at frequency offsets of 27.3 kHz, 54.6 kHz and 81.9 kHz correspond to the spurious modes of the OEO. Besides spurs power, the phase noise does not significantly depend on the oscillation frequency. The frequency resolution of the SSA was automatically set around the first three spurs to about 500 Hz, 1 kHz and 1.5 kHz respectively.

Fig. 8
Fig. 8

Phase noise power spectral density of the first (black diamond) and the second (blue circle) spurious modes located at frequency offsets of about 27.3 kHz and 54.5 kHz with respect to the carrier frequency, respectively. The power densities are shown as a function of the frequency offset of the carrier frequency with respect to the central frequency of the OEO filter (10 GHz). The powers were measured around each spur with a frequency resolution of approximately 1.8 Hz. Minimum powers of the spurs are obtained at frequencies that are different than the central frequency of the OEO filter. A sharp increase in the spurs power is obtained at the boundaries of the stable operating region.

Equations (22)

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a 0 e j φ 0 = f G ( a 0 ) e j φ 0 H ( ω c ) e j ( θ ω c τ ) .
x m ( t ) = { h ( t ) e j ω m t } * { f G [ | x m ( t τ ) | ] e j x m ( t τ ) } e j ( θ ω m τ ) + n m ( t ) ,
f G [ | x m ( t ) | ] f G ( a 0 , m ) + f G ( a 0 , m ) δ a m ( t ) + O [ δ 2 a m ( t ) ] .
f G [ | x m ( t ) | ] e j x m ( t ) f G ( a 0 , m ) + f G ( a 0 , m ) δ a m ( t ) + j f ( a 0 , m ) δ φ m ( t ) + O [ δ 2 a m ( t ) , δ 2 φ m ( t ) , δ a m ( t ) δ φ m ( t ) ] .
f G [ | x m ( t ) | ] e j x m ( t ) = f G ( a 0 , m ) + Δ + δ x m ( t ) + Δ + δ x m * ( t ) ,
δ x m ( t ) = { h ( t ) e j ω m ( t ) } * { Δ + δ x m ( t τ ) + Δ δ x m * ( t τ ) } e j ( θ ω m τ ) + n m ( t ) .
δ x m ( t ) = k = δ x m , k ( t ) e j ( ω m + k ω m ) t ,
n m ( t ) = k = n m , k ( t ) e j ( ω m + k ω m ) t ,
δ x m , k ( t ) = | H ( ω m + k ) | [ Δ + δ x m , k ( t τ ) + Δ + δ x m , k * ( t τ ) ] + n m , k ( t ) .
δ a m , k ( t ) = γ am ( a 0 , m ) β k ( ω m ) δ a m , k ( t τ ) + j a 0 , m η k ( ω m ) δ φ m , k ( t τ ) + n m , k a ( t ) ,
a 0 , m δ φ m , k ( t ) = j γ am ( a 0 , m ) η k ( ω m ) δ a m , k ( t τ ) + a 0 , m β k ( ω m ) δ φ m , k ( t τ ) + a 0 , m n m , k φ ( t ) ,
β k ( ω m ) = 1 2 | H ( ω m ) | [ | H ( ω m + k ) | + | H ( ω m k ) | ] ,
η k ( ω m ) = 1 2 | H ( ω m ) | [ | H ( ω m + k ) | | H ( ω m k ) | ]
δ φ m , k ( t ) = β k ( ω m ) δ φ m , k ( t τ ) + n m , k φ ( t ) ,
δ φ m , k ( t ) + ( g m , k / τ ) δ φ m , k ( t ) = n m , k φ ( t ) / [ τ β k ( ω m ) ] ,
g m , k = β k 1 ( ω m ) 1 .
δ φ m , k ( t ) = δ φ m , k ( 0 ) e g m , k t / τ + 0 t d t e g m , k ( t t ) / τ n m , k φ ( t ) / [ τ β k ( ω m ) ] .
δ φ m , 0 ( t ) = δ φ m , 0 ( 0 ) + 0 t d t n m , 0 φ ( t ) / τ .
β k ( ω m ) < 1 .
β k ( ω m ) 1 + ( ω m + k ω m ) 2 H ˜ m ,
S δ φ m , k ( Δ ω ) = N m , k φ ( Δ ω ) | 1 β k ( ω m ) exp ( j Δ ω τ ) | 2 ,
S δ φ m , k ( 0 ) = N m , k φ ( 0 ) ( ω m + k ω m ) 4 H ˜ m 2 .

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