Abstract

A frequency tunable optoelectronic oscillator based on a directly modulated distributed-feedback (DFB) semiconductor laser under optical injection is proposed and experimentally demonstrated. Through optical injection, the relaxation oscillation frequency of the DFB laser is enhanced and its high modulation efficiency can enable the loop oscillation with a RF threshold gain of less than 20 dB. The DFB laser is a commercial semiconductor laser with a package of 10 GHz, and its packaging limitation can be overcome by optical injection. In our scheme, neither a high-speed external modulator nor an electrical bandpass filter is required, making the system simple and low-cost. Microwave signals with a frequency tuning range from 5.98 to 15.22 GHz are generated by adjusting the injection ratio and frequency detuning between the master and slave lasers. The phase noise of the generated 9.75 GHz microwave signal is measured to be −104.8 dBc/Hz @ 10 kHz frequency offset.

© 2015 Optical Society of America

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    [Crossref]
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    [Crossref]
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2014 (1)

2013 (2)

2012 (2)

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (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]

2010 (2)

2009 (2)

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

E. Shumakher, S. Ó. Dúill, and G. Eisenstein, “Optoelectronic oscillator tunable by an SOA based slow light element,” J. Lightwave Technol. 27(18), 4063–4068 (2009).
[Crossref]

2008 (1)

E. K. Lau, H. K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[Crossref]

2007 (3)

H. K. Sung, E. K. Lau, and M. C. Wu, “Optical single sideband modulation using strong optical injection-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 19(13), 1005–1007 (2007).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Scaling of resonance frequency for strong injection-locked lasers,” Opt. Lett. 32(23), 3373–3375 (2007).
[Crossref] [PubMed]

J. Yang, J. Yu, Y. Wang, L. Zhang, and E. Yang, “An optical domain combined dual-loop optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19(11), 807–809 (2007).
[Crossref]

2005 (1)

H. Tsuchida and M. Suzuki, “40-Gb/s optical clock recovery using an injection-locked optoelectronic oscillator,” IEEE Photon. Technol. Lett. 17(1), 211–213 (2005).
[Crossref]

2003 (1)

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

2002 (1)

2000 (1)

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

1997 (1)

X. S. Yao and L. Maleki, “Opto-electronic oscillator and its applications,” Proc. SPIE 3038, 97–107 (1997).

1996 (1)

1985 (2)

C. Henry, N. Olsson, and N. Dutta, “Locking range and stability of injection locked 1.54 µm InGaAsP semiconductor lasers,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985).
[Crossref]

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Atsuki, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Balakireva, I. V.

Belkin, L.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Belkin, M. E.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Ben, D.

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (2012).
[Crossref]

Boussert, B.

Caliman, A.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Chang-Hasnain, C. J.

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

Chembo, Y. K.

Chen, D.

Chen, Z.

Coillet, A.

Diallo, S.

Dúill, S. Ó.

Dutta, N.

C. Henry, N. Olsson, and N. Dutta, “Locking range and stability of injection locked 1.54 µm InGaAsP semiconductor lasers,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985).
[Crossref]

Eisenstein, G.

Ellafi, D.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Fang, T.

Goedgebuer, J.-P.

Guo, P.

Guo, R.

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (2012).
[Crossref]

Henriet, R.

Henry, C.

C. Henry, N. Olsson, and N. Dutta, “Locking range and stability of injection locked 1.54 µm InGaAsP semiconductor lasers,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985).
[Crossref]

Hu, W.

Iakovlev, V.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Jacobsen, G.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Jung, T.

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Kapon, E.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Kawashima, K.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Larger, L.

Lau, E. K.

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[Crossref]

H. K. Sung, E. K. Lau, and M. C. Wu, “Optical single sideband modulation using strong optical injection-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 19(13), 1005–1007 (2007).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Scaling of resonance frequency for strong injection-locked lasers,” Opt. Lett. 32(23), 3373–3375 (2007).
[Crossref] [PubMed]

Li, W.

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]

Lin, G.

Liou, K. Y.

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Loparev, A.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Lu, L.

Maleki, L.

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

X. S. Yao and L. Maleki, “Opto-electronic oscillator and its applications,” Proc. SPIE 3038, 97–107 (1997).

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13(8), 1725–1735 (1996).
[Crossref]

Martinenghi, R.

Mereuta, A.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Mogensen, F.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Murakami, A.

A. Murakami, K. Kawashima, and K. Atsuki, “Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection,” IEEE J. Quantum Electron. 39(10), 1196–1204 (2003).
[Crossref]

Olesen, H.

F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985).
[Crossref]

Olsson, N.

C. Henry, N. Olsson, and N. Dutta, “Locking range and stability of injection locked 1.54 µm InGaAsP semiconductor lasers,” IEEE J. Quantum Electron. 21(8), 1152–1156 (1985).
[Crossref]

Pan, M.

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (2012).
[Crossref]

Pan, S.

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (2012).
[Crossref]

S. Pan and J. Yao, “Wideband and frequency-tunable microwave generation using an optoelectronic oscillator incorporating a Fabry-Perot laser diode with external optical injection,” Opt. Lett. 35(11), 1911–1913 (2010).
[Crossref] [PubMed]

Parekh, D.

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

Pogurmirskiy, M.

Poinsot, S.

Porte, H.

Pu, T.

Rhodes, W. T.

Saleh, K.

Salzenstein, P.

Shumakher, E.

Sigov, A. S.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Sirbu, A.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Sun, T.

Sung, H. K.

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Scaling of resonance frequency for strong injection-locked lasers,” Opt. Lett. 32(23), 3373–3375 (2007).
[Crossref] [PubMed]

H. K. Sung, E. K. Lau, and M. C. Wu, “Optical single sideband modulation using strong optical injection-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 19(13), 1005–1007 (2007).
[Crossref]

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Suruceanu, G.

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

Suzuki, M.

H. Tsuchida and M. Suzuki, “40-Gb/s optical clock recovery using an injection-locked optoelectronic oscillator,” IEEE Photon. Technol. Lett. 17(1), 211–213 (2005).
[Crossref]

Tanbun-Ek, T.

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Tang, Z.

Z. Tang, S. Pan, D. Zhu, R. Guo, Y. Zhao, M. Pan, D. Ben, and J. Yao, “Tunable optoelectronic oscillator based on a polarization modulator and a chirped FBG,” IEEE Photon. Technol. Lett. 24(17), 1487–1489 (2012).
[Crossref]

Tavernier, H.

Tishinin, D.

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Tsang, W. T.

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

Tsuchida, H.

H. Tsuchida and M. Suzuki, “40-Gb/s optical clock recovery using an injection-locked optoelectronic oscillator,” IEEE Photon. Technol. Lett. 17(1), 211–213 (2005).
[Crossref]

Volyanskiy, K.

Wang, R.

Wang, Y.

J. Yang, J. Yu, Y. Wang, L. Zhang, and E. Yang, “An optical domain combined dual-loop optoelectronic oscillator,” IEEE Photon. Technol. Lett. 19(11), 807–809 (2007).
[Crossref]

Wu, M. C.

H. K. Sung, X. X. Zhao, E. K. Lau, D. Parekh, C. J. Chang-Hasnain, and M. C. Wu, “Optoelectronic oscillators using direct-modulated semiconductor lasers under strong optical injection,” IEEE J. Sel. Top. Quantum Electron. 15(3), 572–577 (2009).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Frequency response enhancement of optical injection-locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[Crossref]

H. K. Sung, E. K. Lau, and M. C. Wu, “Optical single sideband modulation using strong optical injection-locked semiconductor lasers,” IEEE Photon. Technol. Lett. 19(13), 1005–1007 (2007).
[Crossref]

E. K. Lau, H. K. Sung, and M. C. Wu, “Scaling of resonance frequency for strong injection-locked lasers,” Opt. Lett. 32(23), 3373–3375 (2007).
[Crossref] [PubMed]

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Other (3)

M. E. Belkin, L. Belkin, A. Loparev, A. S. Sigov, G. Suruceanu, A. Sirbu, A. Mereuta, A. Caliman, D. Ellafi, V. Iakovlev, and E. Kapon, “VCSEL-based processing of microwave signals,” in Proceedings of International Topical Meeting on Microwave Photonics and the Asia-Pacific Microwave Photonics Conference (IEEE, 2014), pp. 284–287.

P. Wang, J. Xiong, T. Zhang, J. Zheng, T. Pu, and X. Chen, “Widely Tunable OEO Based on a Directly Modulated DFB Laser under Optical Injection,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2015), paper M3E.1.
[Crossref]

H. K. Sung, T. Jung, M. C. Wu, D. Tishinin, T. Tanbun-Ek, K. Y. Liou, and W. T. Tsang, “Modulation bandwidth enhancement and nonlinear distortion suppression in directly modulated monolithic injection-locked DFB lasers,” in Proceedings of International Topical Meeting on Microwave Photonics (IEEE, 2003), pp. 27–30.
[Crossref]

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of the proposed OEO. (TLS: tunable laser source, PC: polarization controller, OC: optical circulator, PBS: polarization beam splitter, OSA: optical spectrum analyzer, PD: photo-detector, LNA: low noise amplifier, ESA: electrical spectrum analyzer).
Fig. 2
Fig. 2 Measured frequency response of the free-running DFB laser under various bias current and a constant temperature of 23 °C.
Fig. 3
Fig. 3 Illustration of the frequency response of the DFB laser under optical injection. (a), (b), (c) are the optical spectra of the free-running laser, the injected laser without modulation, and the injected laser which is modulated by a RF signal with a modulation frequency of fm, respectively. (d) Frequency response of the injected laser.
Fig. 4
Fig. 4 Injection locking diagram as a function of injection ratio and frequency detuning.
Fig. 5
Fig. 5 Calculated frequency response for (a) various injection ratios and a fixed frequency detuning of 0.04 GHz; (b) various frequency detuning and a fixed injection ratio of 0 dB.
Fig. 6
Fig. 6 Measured frequency response of the injected DFB laser under (a) various injection ratio and a fixed frequency detuning of 0 GHz; (b) various frequency detuning and a fixed injection ratio of −0.01 dB.
Fig. 7
Fig. 7 Measured frequency response of the DFB laser with optimized injection parameters.
Fig. 8
Fig. 8 (a) Optical spectra of the OEO loop with and without the feedback. (b) Corresponding RF spectrum of the generated 11.72 GHz microwave signal with feedback on; RBW = 1 MHz.
Fig. 9
Fig. 9 Spectra of the generated electrical signals with the frequency tuned from 5.98 to 15.22 GHz; RBW = 1 MHz.
Fig. 10
Fig. 10 Frequency response of the (a) low noise amplifier; (b) power splitter.
Fig. 11
Fig. 11 Phase noise of the generated 9.75 GHz microwave signal. The phase noise at 10 kHz frequency offset is −104.8 dBc/Hz.
Fig. 12
Fig. 12 Phase noise performance at different microwave frequencies.

Equations (3)

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ω R 0 = g γ p P 0
ω R 2 = ω R0 2 +Δ ω R 2
Δ ω R =κ R 1r r sin φ 0

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