Wideband tunable optoelectronic oscillator based on the deamplification of stimulated Brillouin scattering

Huanfa Peng; Yongchi Xu; Xiaofeng Peng; Xiaoqi Zhu; Rui Guo; Feiya Chen; Huayang Du; Yuanxiang Chen; Cheng Zhang; Lixin Zhu; Weiwei Hu; Zhangyuan Chen

doi:10.1364/OE.25.010287

Wideband tunable optoelectronic oscillator based on the deamplification of stimulated Brillouin scattering

Huanfa Peng, Yongchi Xu, Xiaofeng Peng, Xiaoqi Zhu, Rui Guo, Feiya Chen, Huayang Du, Yuanxiang Chen, Cheng Zhang, Lixin Zhu, Weiwei Hu, and Zhangyuan Chen

Optics Express
Vol. 25,
Issue 9,
pp. 10287-10305
(2017)
•https://doi.org/10.1364/OE.25.010287

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Huanfa Peng, Yongchi Xu, Xiaofeng Peng, Xiaoqi Zhu, Rui Guo, Feiya Chen, Huayang Du, Yuanxiang Chen, Cheng Zhang, Lixin Zhu, Weiwei Hu, and Zhangyuan Chen, "Wideband tunable optoelectronic oscillator based on the deamplification of stimulated Brillouin scattering," Opt. Express 25, 10287-10305 (2017)

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Related Topics
Table of Contents Category
- RF and Microwave Photonics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Radio frequency photonics (060.5625)
- Oscillators (230.4910)
- Scattering, stimulated Brillouin (290.5900)

About this Article
History
- Original Manuscript: March 2, 2017
- Revised Manuscript: April 11, 2017
- Manuscript Accepted: April 12, 2017
- Published: April 25, 2017

Accessible

Open Access

Abstract

A wideband tunable optoelectronic oscillator (OEO) based on the deamplification of stimulated Brillouin scattering (SBS) is proposed and experimentally demonstrated. A tunable single passband microwave photonic filter (MPF) utilizing phase modulation and SBS deamplification is used to realize the tunability of the OEO. Theoretical analysis of the MPF and phase noise performance of the OEO are presented. The frequency response of the MPF is determined by the + 1st sideband attenuation due to SBS deamplification and phase shift difference between the two sidebands due to chromatic dispersion and SBS. The close-in (< 1 MHz) phase noise of the proposed OEO is shown to be dominated by the laser frequency noise via phase shift of SBS. The conversion of the laser frequency noise to the close-in phase noise of the proposed OEO is effectively reduced compared with the OEO based on amplification by SBS. Tunable 7 to 40 GHz signals are experimentally obtained. The single-sideband (SSB) phase noise at 10 kHz offset is −128 dBc/Hz for 10.30 GHz signal. Compared with the OEO based on SBS amplification, the proposed OEO can achieve a phase noise performance improvement beyond 20 dB at 10 kHz offset. The maximum frequency and power drifts at 10.69 GHz are within 1 ppm and 1.4 dB during 1000 seconds, respectively. To achieve better close-in phase noise performance, lower frequency noise laser and higher pump power are preferred. The experimental results agree well with the theoretical models.

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References

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X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hansel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P. A. Tremblin, G. Santarelli, R. Holzwarth, and Y. L. Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017).
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J. P. Cahill, W. Zhou, and C. R. Menyuk, “Additive phase noise of fiber-optic links used in photonic microwave-generation systems,” Appl. Opt. 56(3), B18–B25 (2017).
[Crossref] [PubMed]

2016 (2)

M. Merklein, B. Stiller, I. V. Kabakova, U. S. Mutugala, K. Vu, S. J. Madden, B. J. Eggleton, and R. Slavík, “Widely tunable, low phase noise microwave source based on a photonic chip,” Opt. Lett. 41(20), 4633–4636 (2016).
[Crossref] [PubMed]

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[Crossref]

2015 (4)

P. Wang, J. Xiong, T. Zhang, D. Chen, P. Xiang, J. Zheng, Y. Zhang, R. Li, L. Huang, T. Pu, and X. Chen, “Frequency tunable optoelectronic oscillator based on a directly modulated DFB semiconductor laser under optical injection,” Opt. Express 23(16), 20450–20458 (2015).
[Crossref] [PubMed]

L. Liu, J. Dong, D. Gao, A. Zheng, and X. Zhang, “On-chip passive three-port circuit of all-optical ordered-route transmission,” Sci. Rep. 5(1), 10190 (2015).
[Crossref] [PubMed]

H. Peng, C. Zhang, X. Xie, T. Sun, P. Guo, X. Zhu, W. Hu, and Z. Chen, “Tunable DC-60GHz RF generation utilizing a dual-loop optoelectronic oscillator based on stimulated Brillouin scattering,” J. Lightwave Technol. 33(13), 2707–2715 (2015).
[Crossref]

K. Xu, Z. Wu, J. Zheng, J. Dai, Y. Dai, F. Yin, J. Li, Y. Zhou, and J. Lin, “Long-term stability improvement of tunable optoelectronic oscillator using dynamic feedback compensation,” Opt. Express 23(10), 12935–12941 (2015).
[Crossref] [PubMed]

2014 (1)

J. Zhang, L. Gao, and J. P. Yao, “Tunable optoelectronic oscillator incorporating a single passband microwave photonic filter,” IEEE Photonics Technol. Lett. 26(4), 326–329 (2014).
[Crossref]

2013 (6)

X. Xie, C. Zhang, T. Sun, P. Guo, X. Zhu, L. Zhu, W. Hu, and Z. Chen, “Wideband tunable optoelectronic oscillator based on a phase modulator and a tunable optical filter,” Opt. Lett. 38(5), 655–657 (2013).
[Crossref] [PubMed]

D. Eliyahu, W. Liang, E. Dale, A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “Resonant widely tunable opto-electronic oscillator,” IEEE Photonics Technol. Lett. 25(15), 1535–1538 (2013).
[Crossref]

R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express 21(19), 22918–22936 (2013).
[Crossref] [PubMed]

G. J. Schneider, J. A. Murakowski, C. A. Schuetz, S. Shi, and D. W. Prather, “Radiofrequency signal-generation system with over seven octaves of continuous tuning,” Nat. Photonics 7(2), 118–122 (2013).
[Crossref]

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4(4), 2097 (2013).
[PubMed]

O. Okusaga, J. P. Cahill, A. Docherty, C. R. Menyuk, W. Zhou, and G. M. Carter, “Suppression of Rayleigh-scattering-induced noise in OEOs,” Opt. Express 21(19), 22255–22262 (2013).
[Crossref] [PubMed]

2012 (3)

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Tech. 60(6), 1735–1742 (2012).
[Crossref]

R. Tao, X. Feng, Y. Cao, Z. Li, and B. Guan, “Widely tunable single bandpass microwave photonic filter based on phase modulation and stimulated Brillouin scattering,” IEEE Photonics Technol. Lett. 24(13), 1097–1099 (2012).
[Crossref]

2011 (2)

L. Maleki, “Sources: The optoelectronic oscillator,” Nat. Photonics 5(12), 728–730 (2011).
[Crossref]

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

2010 (3)

A. A. Savchenkov, V. S. Ilchenko, W. Liang, D. Eliyahu, A. B. Matsko, D. Seidel, and L. Maleki, “Voltage-controlled photonic oscillator,” Opt. Lett. 35(10), 1572–1574 (2010).
[Crossref] [PubMed]

K. Volyanskiy, Y. K. Chembo, L. Larger, and E. Rubiola, “Contribution of laser frequency and power fluctuations to the microwave phase noise of optoelectronic oscillators,” J. Lightwave Technol. 28(18), 2730–2735 (2010).
[Crossref]

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

2008 (1)

D. Eliyahu, D. Seidel, and L. Maleki, “RF amplitude and phase-noise reduction of an optical link and an opto-electronic oscillator,” IEEE Trans. Microw. Theory Tech. 56(2), 449–456 (2008).
[Crossref]

2006 (2)

M. Kaba, H.-W. Li, A. S. Daryoush, J.-P. Vilcot, D. Decoster, J. Chazelas, G. Bouwmans, Y. Quiquempois, and F. Deborgies, “Improving thermal stability of opto-electronic oscillators,” IEEE Microw. Mag. 7(4), 38–47 (2006).
[Crossref]

A. P. S. Khanna, “Microwave oscillators: The state of the technology,” Microwave J. 49(22), 22–44 (2006).

2005 (1)

M. G. Herraez, K. Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87(8), 081113 (2005).
[Crossref]

2002 (1)

T. Tanemura, Y. Takushima, and K. Kikuchi, “Narrowband optical filter, with a variable transmission spectrum, using stimulated Brillouin scattering in optical fiber,” Opt. Lett. 27(17), 1552–1554 (2002).
[Crossref] [PubMed]

2000 (1)

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36(1), 79–84 (2000).
[Crossref]

1998 (1)

W. Shieh and L. Maleki, “Phase noise of optical interference in photonic RF systems,” IEEE Photonics Technol. Lett. 10(11), 1617–1619 (1998).
[Crossref]

1996 (1)

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996).
[Crossref]

Alexandre, C.

Bergquist, J. C.

Bouchand, R.

Bouwmans, G.

Bowers, J. E.

D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4(1), 23–30 (2017).
[Crossref]

Buenrostro, J.

A. P. S. Khanna and J. Buenrostro, “2 to 22 GHz low phase noise sillicon bipolar YIG tuned oscillator using composite feedback,” in Proceedings of IEEE MTI-S Int. Microw. Symp. Dig., 1297–1299 (1992).

Cahill, J.

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

Cahill, J. P.

J. P. Cahill, W. Zhou, and C. R. Menyuk, “Additive phase noise of fiber-optic links used in photonic microwave-generation systems,” Appl. Opt. 56(3), B18–B25 (2017).
[Crossref] [PubMed]

Cao, Y.

Carter, G. M.

Cater, G.

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

Chan, E. H. W.

R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express 21(19), 22918–22936 (2013).
[Crossref] [PubMed]

Chazelas, J.

Chembo, Y. K.

Chen, D.

Chen, X.

Chen, Z.

H. Peng, X. Peng, Y. Xu, C. Zhang, L. Zhu, W. Hu, and Z. Chen, “Suppression of phase noise induced by optical interference in optoelectronic oscillators,” in Proceedings of Conference on Optoelectronics and Communications (OECC 2016).

Chowdhury, D.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Coq, Y. L.

Dai, J.

Dai, Y.

Dale, E.

Daryoush, A. S.

Datta, S.

Deborgies, F.

Decoster, D.

Diddams, S. A.

Docherty, A.

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

Dong, J.

L. Liu, J. Dong, D. Gao, A. Zheng, and X. Zhang, “On-chip passive three-port circuit of all-optical ordered-route transmission,” Sci. Rep. 5(1), 10190 (2015).
[Crossref] [PubMed]

Eggleton, B. J.

Eliyahu, D.

D. Eliyahu and L. Maleki, “Tunable, ultra-low phase noise YIG based optoelectronic oscillator,” in Proceedings of IEEE MTT-S Int. Microw. Symp. Dig., 2185–2187 (2003).

Feng, X.

Fortier, T. M.

Gao, D.

L. Liu, J. Dong, D. Gao, A. Zheng, and X. Zhang, “On-chip passive three-port circuit of all-optical ordered-route transmission,” Sci. Rep. 5(1), 10190 (2015).
[Crossref] [PubMed]

Gao, L.

J. Zhang, L. Gao, and J. P. Yao, “Tunable optoelectronic oscillator incorporating a single passband microwave photonic filter,” IEEE Photonics Technol. Lett. 26(4), 326–329 (2014).
[Crossref]

Giunta, M.

Guan, B.

Guo, P.

Hansel, W.

Herraez, M. G.

M. G. Herraez, K. Y. Song, and L. Thevenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87(8), 081113 (2005).
[Crossref]

Holzwarth, R.

Horowitz, M.

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

Hu, W.

Huang, D.

D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4(1), 23–30 (2017).
[Crossref]

Huang, L.

Ilchenko, V. S.

Jiang, Y.

Joshi, A.

Kaba, M.

Kabakova, I. V.

Khanna, A. P. S.

A. P. S. Khanna, “Microwave oscillators: The state of the technology,” Microwave J. 49(22), 22–44 (2006).

Kikuchi, K.

Kirchner, M. S.

Kobyakov, A.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Larger, L.

Lee, H.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4(4), 2097 (2013).
[PubMed]

Lemke, N.

Levy, E.

W. Zhou, O. Okusaga, E. Levy, J. Cahill, A. Docherty, C. Menyuk, G. Cater, and M. Horowitz, “Potentials and challenges for optoelectronic oscillator,” Proc. SPIE 8255, 37 (2012).
[Crossref]

Lezius, M.

Li, H.-W.

Li, J.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4(4), 2097 (2013).
[PubMed]

Li, R.

Li, W.

Li, Z.

Liang, W.

Lin, J.

Liu, L.

L. Liu, J. Dong, D. Gao, A. Zheng, and X. Zhang, “On-chip passive three-port circuit of all-optical ordered-route transmission,” Sci. Rep. 5(1), 10190 (2015).
[Crossref] [PubMed]

Lours, M.

Ludlow, A.

Madden, S. J.

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Adv. Opt. Photonics (1)

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photonics 2(1), 1–59 (2010).
[Crossref]

Appl. Opt. (1)

J. P. Cahill, W. Zhou, and C. R. Menyuk, “Additive phase noise of fiber-optic links used in photonic microwave-generation systems,” Appl. Opt. 56(3), B18–B25 (2017).
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Appl. Phys. Lett. (1)

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IEEE J. Quantum Electron. (2)

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36(1), 79–84 (2000).
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IEEE Microw. Mag. (1)

IEEE Photonics Technol. Lett. (4)

W. Shieh and L. Maleki, “Phase noise of optical interference in photonic RF systems,” IEEE Photonics Technol. Lett. 10(11), 1617–1619 (1998).
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J. Zhang, L. Gao, and J. P. Yao, “Tunable optoelectronic oscillator incorporating a single passband microwave photonic filter,” IEEE Photonics Technol. Lett. 26(4), 326–329 (2014).
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Optica (1)

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Fig. 1 Left part: schematic diagram of the proposed OEO. PM: phase modulator; ISO: optical isolator; NZ-DSF: none-zero dispersion shifted fiber; OC: optical coupler; SMF1, SMF2: single-mode fibers; PD1, PD2: photodiodes; EC1, EC2: electrical couplers; LNA: low noise amplifier; ELPF: electrical lowpass filter; PA: power amplifier; PC: polarization controller; ESA: electrical spectrum analyzer. Right part: optical spectra of the nodes (A, B, and C). λ₀, λ_p: wavelength of the signal pump lasers; ν_B: Brillouin frequency shift.

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Fig. 2 (a) Setup for frequency response measurement of the MPF utilizing the sideband deamplification of SBS. (b) Schematic diagram of the amplitude and phase responses induced by the narrow-band optical gain and loss spectral areas of SBS to the + 1st phase modulated sideband.

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Fig. 3 Simulation results of the frequency response of the MPF. (a) The + 1st sideband amplitude response and phase shift difference compared with the −1st sideband induced by the chromatic dispersion and SBS. (b) Frequency responses of the MPF. (c) Comparison of passband responses with different center frequencies. (d) Calculated tunable single passband responses of the MPF.

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Fig. 4 (a) Phase noise model of the proposed dual-loop OEO based on the deamplification of SBS. (b) Phasor representation of a noisy sinusoid RF carrier.

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Fig. 5 Simulation results of the phase noise performance for the proposed OEO. (a) A comparison of phase noise induced by different noise sources in the proposed OEO. (b) A comparison of additive phase noises contributed by the conversion of laser frequency noise via SBS deamplification and amplification. (c) Simulated SSB phase noise of the OEOs based on the deamplification and amplification of SBS. (d) A comparison of phase noise at 10 kHz offset between the OEOs based on the deamplification and amplification of SBS.

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Fig. 6 Measured frequency response of the MPF. (a) Frequency responses induced by the sideband deamplification and amplification of SBS. (b) Comparison of frequency responses at different passband frequencies.

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Fig. 7 Measured frequency responses of the tunable single passband MPF by changing the wavelength of the signal laser. (a) Passband responses with center frequency tuned from 7 GHz to 20 GHz. (b) Passband responses with center frequency tuned from 21 GHz to 36 GHz.

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Fig. 8 Optical and electrical spectra of the generated 10.3 GHz signal. (a) Optical spectrum. (b) Electrical spectrum.

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Fig. 9 (a) Electrical spectra for the tunable range from 7 to 12 GHz. (b) Electrical spectra for the tunable range from 13 to 20 GHz. (c) Electrical spectra for the tunable range from 21 to 28 GHz. (d) Electrical spectra for the tunable range from 29 to 40 GHz.

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Fig. 10 Measured SSB phase noises of the microwave signals generated by the proposed OEO. (a) SSB phase noises of the generated 8.28 GHz with dual-loop fiber lengths of 1.1 km-1.2 km and 2.1 km-2.2 km. (b) Measured SSB phase noise of the generated signal with different oscillation frequencies and dual-loop fiber lengths of 2.1 km-2.2 km.

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Fig. 11 Measured frequency and power fluctuations of the proposed OEO. (a), (b): Frequency and power fluctuations of the generated 10.69 GHz signal, respectively.

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Fig. 12 Experimental measurements for the tested laser frequency noises and the corresponding SSB phase noises of the OEOs. (a) Measured frequency noises of the lasers. (b) A comparison between the calculated and measured SSB phase noises. The olive curve (1), blue curve (3), and red curve (5) stand for the measured SSB phase noises. The wine curve (2), magenta curve (4), and purple curve (6) stand for the calculated SSB phase noises through model in Eq. (17), but multiplied the transfer function of the dual-loop OEO. Dark yellow curve (7) stands for the phase noise induced by the flicker phase noise of the cascaded amplifiers.

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Fig. 13 (a) Comparison of SSB phase noise of the OEOs based on SBS amplification and deamplification. (b) Comparison of phase noise at 10 kHz offset for the two OEOs.

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Tables (2)

Table 1 Simulation parameters for the response of MPF

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Table 2 Simulation parameters for the phase noise of the proposed OEO

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Equations (24)

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E (t) \approx \sqrt{P_{0}} e^{j 2 π f_{0} t} [J_{- 1} (β) e^{- j 2 π f_{m} t} + J_{0} (β) + J_{1} (β) e^{j 2 π f_{m} t}]

E^{'} (t) \approx \sqrt{P_{0}} e^{j (2 π f_{0} t - β_{0} L)} [α \cdot J_{- 1} (β) e^{- j 2 π f_{m} (t - \frac{n_{g} L}{c})} e^{j \frac{2 π D L λ_{0}^{2} f_{m}^{2}}{c}} e^{- j δ} + J_{0} (β) + J_{1} (β) e^{j 2 π f_{m} (t - \frac{n_{g} L}{c})} e^{j \frac{2 π D L λ_{0}^{2} f_{m}^{2}}{c}}]

E_{+ 1 s t} (L) = E_{+ 1 s t} (0) \exp (- g_{B} I_{p} L)

g_{B} = \frac{1}{2} \frac{g_{p}}{1 - 2 j (\frac{v - v_{B}}{Δ v_{B}})}

E_{+ 1 s t} (L) = E_{+ 1 s t} (0) \cdot α \cdot e^{j δ}

v - v_{B} = f_{o s c} - f_{m}

α = \exp (Re (- g_{B} I_{p} L)) = \exp (- g_{p} I_{p} L \frac{1}{1 + 4 {(\frac{f_{o s c} - f_{m}}{Δ v_{B}})}^{2}})

δ = Im (- g_{B} I_{p} L) = g_{p} I_{p} L \frac{\frac{f_{m} - f_{o s c}}{Δ v_{B}}}{1 + 4 {(\frac{f_{m} - f_{o s c}}{Δ v_{B}})}^{2}}

I_{p h} \propto \sqrt{[1 + α^{2} - 2 α \cos (\frac{4 π D L λ_{0}^{2} f_{m}^{2}}{c} - δ)]} \cos (2 π f_{m} t + φ_{0})

| H (j f_{m}) | \propto \sqrt{[1 + α^{2} - 2 α \cos (\frac{4 π D L λ_{0}^{2} f_{m}^{2}}{c} - δ)]}

φ_{e} + φ_{o} = 2 k π \begin{matrix} (k = 0, \pm 1, \pm 2...) \end{matrix}

I_{p h}^{'} \propto [\cos (φ_{C D}) - α_{\min} \cdot \cos (φ_{C D} + δ)] \cos (2 π f_{o s c} t) - [\sin (φ_{C D}) + α_{_{\min}} \cdot \sin (φ_{C D} + δ)] \sin (2 π f_{o s c} t)

I_{p h}^{'} \propto (1 - α_{\min}) \cos (2 π f_{o s c} t) + (φ_{C D} + α_{\min} \cdot φ_{C D} + α_{\min} \cdot δ) \sin (2 π f_{o s c} t)

I_{p h_n o i s e d}^{'} \propto (1 - α_{\min}) \cos (2 π f_{o s c} t) + α_{\min} \cdot Δ δ \cdot \sin (2 π f_{o s c} t)

Δ φ_{D} \approx \tan (Δ φ_{D}) = \frac{α_{\min}}{1 - α_{\min}} \cdot Δ δ

Δ δ \approx \frac{g_{p} I_{p} L}{Δ v_{B}} \cdot (Δ v_{l a s e r_s} - Δ v_{l a s e r_p} - \frac{2 n_{p} v_{A}}{c} Δ v_{l a s e r_p})

S_{S B S_D} (f) = {[\frac{1}{\exp (g_{p} I_{p} L) - 1} \frac{g_{p} I_{p} L}{Δ v_{B}}]}^{2} \cdot [S_{v, l a s e r_s} (f) + S_{v, l a s e r_p} (f)]

Δ φ_{A} \approx \tan (Δ φ_{A}) = \frac{G_{\max}}{1 - G_{\max}} \cdot Δ δ

S_{S B S_A} (f) = {[\frac{\exp (g_{p} I_{p} L)}{1 - \exp (g_{p} I_{p} L)} \frac{g_{p} I_{p} L}{Δ v_{B}}]}^{2} \cdot [S_{v, l a s e r_s} (f) + S_{v, l a s e r_p} (f)]

S_{C D} (f) = \frac{1}{2} {(\frac{2 π f_{o s c} λ_{0}^{2} D L}{c})}^{2} \cdot S_{v, l a s e r_s} (f)

S_{I n t e r f e r e n c e} (f) = 2 r_{c} \cdot r_{P D} \cdot \sin^{2} (2 π f_{o s c} τ_{d}) \frac{\sin^{2} (π f τ_{d})}{f^{2}} \cdot S_{v, l a s e r_s} (f)

ℒ (f) = (\frac{F k T + 2 q I_{p h} R + N_{r i n} I_{p h}^{2} R}{2 P_{R F}} + \frac{b_{- 1}}{f} + S_{S B S_D} (f) + S_{C D} (f) + S_{I n t e r f e r e n c e} (f)) \cdot | H_{d u a l - l o o p} (j f) |^{2}

S_{v, l a s e r} (f) = C_{1} + C_{2} / f

\frac{Δ f_{o s c}}{f_{o s c}} = \frac{Δ φ_{D}}{2 Q}

Name	Symbol	Value
Fiber length	$L$	$1 k m$
Dispersion coefficient	$D$	$5 p s / (n m \cdot k m)$
Effective mode area	$A_{e f f}$	$63.6 μ m^{2}$
Line center Brillouin gain coefficient	$g_{p}$	$2 \times 10^{- 11} m / W$
Loss bandwidth of SBS	$Δ v_{B}$	$50 M H z$
Wavelength of signal laser	$λ_{0}$	$1550 n m$
Pump power	$P_{p}$	$11.7 m W$
Brillouin frequency shift	$v_{B}$	$10.7 G H z$

Name	Symbol	Value
Oscillation frequency	$f_{o s c}$	$10 G H z$
ORL of fiber connector	$r_{c}$	$- 50 d B$
ORL of PD	$r_{P D}$	$- 30 d B$
Round trip time of double reflection process	$τ_{d}$	$1 μ s$
White frequency noise coefficient of pump	$C_{1}$	$1.0 \times 10^{3} H z^{2} / H z$
Flicker frequency noise coefficient of pump	$C_{2}$	$1.0 \times 10^{8} H z^{3} / H z$
Noise figure of amplifier	$F$	$7 d B$
Room temperature	$T$	$300 K$
Photo-current at the output of PD	$I_{p h}$	$4 m A$
Load resistance of PD	$R$	$50 Ω$
Relative intensity noise of signal laser	$N_{r i n}$	$- 158 d B / H z$
Flicker noise of the amplifier	$b_{- 1}$	$- 120 d B \cdot r a d^{2} / H z$
RF power from PD	$P_{R F}$	$400 μ W$

Name	Symbol	Value
Fiber length	$L$	$1 k m$
Dispersion coefficient	$D$	$5 p s / (n m \cdot k m)$
Effective mode area	$A_{e f f}$	$63.6 μ m^{2}$
Line center Brillouin gain coefficient	$g_{p}$	$2 \times 10^{- 11} m / W$
Loss bandwidth of SBS	$Δ v_{B}$	$50 M H z$
Wavelength of signal laser	$λ_{0}$	$1550 n m$
Pump power	$P_{p}$	$11.7 m W$
Brillouin frequency shift	$v_{B}$	$10.7 G H z$

Name	Symbol	Value
Oscillation frequency	$f_{o s c}$	$10 G H z$
ORL of fiber connector	$r_{c}$	$- 50 d B$
ORL of PD	$r_{P D}$	$- 30 d B$
Round trip time of double reflection process	$τ_{d}$	$1 μ s$
White frequency noise coefficient of pump	$C_{1}$	$1.0 \times 10^{3} H z^{2} / H z$
Flicker frequency noise coefficient of pump	$C_{2}$	$1.0 \times 10^{8} H z^{3} / H z$
Noise figure of amplifier	$F$	$7 d B$
Room temperature	$T$	$300 K$
Photo-current at the output of PD	$I_{p h}$	$4 m A$
Load resistance of PD	$R$	$50 Ω$
Relative intensity noise of signal laser	$N_{r i n}$	$- 158 d B / H z$
Flicker noise of the amplifier	$b_{- 1}$	$- 120 d B \cdot r a d^{2} / H z$
RF power from PD	$P_{R F}$	$400 μ W$