Abstract

We investigate theoretically and experimentally an all-fibered frequency-shifting loop which includes an electro-optic amplitude modulator (EOM) and an optical amplifier, and is seeded by a continuous-wave laser. At variance with frequency-shifted feedback lasers, or Talbot lasers, that contain an acousto-optic frequency shifter, the EOM creates at each round-trip two side-bands that recirculate inside the loop. Benefiting from the high modulation frequency of the EOM, a wide optical frequency comb up to 40 GHz is generated. We demonstrate an original double-pulse regime when the loop length is a multiple of the RF modulation wavelength applied to the modulator. The inter-pulse interval is governed by both the bias voltage and modulation depth of the EOM. Besides, some typical waveforms such as saw-tooth and rectangle are experimentally obtained by properly setting operating frequency, bias voltage and the RF power. The system is modeled by a linear interference model that takes the amplitude modulation function and loop delay into account. The model explains the formation of pulse doublets and reproduces well all the experimental waveforms. Furthermore, the un-seeded loop driven above threshold also generates mode-locked picosecond pulse doublets with a continuously adjustable delay up to the modulation period.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  15. H. Guillet de Chatellus, L. R. Cortés, and J. Azaña, “Optical real-time Fourier transformation with kilohertz resolutions,” Optica 3(1), 1–8 (2016).
    [Crossref]
  16. C. Schnébelin and H. Guillet de Chatellus, “Optical spectral shaping with MHz resolution for arbitrary RF waveform generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM1B.7.
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2019 (1)

2018 (1)

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

2017 (1)

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

2016 (2)

2013 (3)

2011 (2)

2010 (1)

2009 (2)

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
[Crossref]

2007 (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

2005 (1)

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

1996 (1)

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

1994 (1)

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12(8), 1360–1368 (1994).
[Crossref]

1990 (1)

A. Takada and H. Miyazawa, “30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser,” Electron. Lett. 26(3), 216–217 (1990).
[Crossref]

1989 (1)

1988 (1)

F. V. Kowalski, S. J. Shattil, and P. D. Hale, “Optical pulse generation with a frequency shifted feedback laser,” Appl. Phys. Lett. 53(9), 734–736 (1988).
[Crossref]

1969 (1)

J. Hirano and T. Kimura, “Multiple mode-locking of lasers,” IEEE J. Quantum Electron. 5(5), 219–225 (1969).
[Crossref]

Azaña, J.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

H. Guillet de Chatellus, L. R. Cortés, and J. Azaña, “Optical real-time Fourier transformation with kilohertz resolutions,” Optica 3(1), 1–8 (2016).
[Crossref]

Baer, T.

Brinkmeyer, E.

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12(8), 1360–1368 (1994).
[Crossref]

Brunel, M.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

H. Zhang, M. Brunel, M. Romanelli, and M. Vallet, “Green pulsed lidar-radar emitter based on a multipass frequency-shifting external cavity,” Appl. Opt. 55(10), 2467–2473 (2016).
[Crossref] [PubMed]

Burla, M.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

Capmany, J.

Chuang, H.-P.

Clement, J.

Cortés, L. R.

Cundiff, S. T.

de Chatellus, H. G.

Fernández-Pousa, C. R.

Gasulla, I.

Glastre, W.

Guillet de Chatellus, H.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

H. Guillet de Chatellus, L. R. Cortés, and J. Azaña, “Optical real-time Fourier transformation with kilohertz resolutions,” Optica 3(1), 1–8 (2016).
[Crossref]

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88(3), 033828 (2013).
[Crossref]

H. Guillet de Chatellus, O. Jacquin, O. Hugon, W. Glastre, E. Lacot, and J. Marklof, “Generation of ultrahigh and tunable repetition rates in CW injection-seeded frequency-shifted feedback lasers,” Opt. Express 21(13), 15065–15074 (2013).
[Crossref] [PubMed]

Hale, P. D.

F. V. Kowalski, S. J. Shattil, and P. D. Hale, “Optical pulse generation with a frequency shifted feedback laser,” Appl. Phys. Lett. 53(9), 734–736 (1988).
[Crossref]

Hall, D. W.

Hirano, J.

J. Hirano and T. Kimura, “Multiple mode-locking of lasers,” IEEE J. Quantum Electron. 5(5), 219–225 (1969).
[Crossref]

Holman, K. W.

Hosako, I.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Huang, C.-B.

Hudson, D. D.

Hugon, O.

Jacquin, O.

Jones, D. J.

Jones, R. J.

Kafka, J. D.

Kanno, A.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Kawanishi, T.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Kimura, T.

J. Hirano and T. Kimura, “Multiple mode-locking of lasers,” IEEE J. Quantum Electron. 5(5), 219–225 (1969).
[Crossref]

Kitayama, K.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Kowalski, F. V.

F. V. Kowalski, S. J. Shattil, and P. D. Hale, “Optical pulse generation with a frequency shifted feedback laser,” Appl. Phys. Lett. 53(9), 734–736 (1988).
[Crossref]

Kuri, T.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Lacot, E.

Li, J.

Lloret, J.

Maleki, L.

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

Marklof, J.

Miyazawa, H.

A. Takada and H. Miyazawa, “30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser,” Electron. Lett. 26(3), 216–217 (1990).
[Crossref]

Mora, J.

Morohashi, I.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Pan, C.-L.

J.-W. Shi, C.-B. Huang, and C.-L. Pan, “Millimeter-wave photonic wireless links for very high data rate communication,” NPG Asia Mater. 3(4), 41–48 (2011).
[Crossref]

Romanelli, M.

Romero Cortés, L.

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

Sabert, H.

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12(8), 1360–1368 (1994).
[Crossref]

Sales, S.

Sancho, J.

Schnébelin, C.

J. Clement, C. Schnébelin, H. G. de Chatellus, and C. R. Fernández-Pousa, “Laser ranging using coherent pulse compression with frequency shifting loops,” Opt. Express 27(9), 12000–12010 (2019).
[Crossref] [PubMed]

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

Shattil, S. J.

F. V. Kowalski, S. J. Shattil, and P. D. Hale, “Optical pulse generation with a frequency shifted feedback laser,” Appl. Phys. Lett. 53(9), 734–736 (1988).
[Crossref]

Shi, J.-W.

J.-W. Shi, C.-B. Huang, and C.-L. Pan, “Millimeter-wave photonic wireless links for very high data rate communication,” NPG Asia Mater. 3(4), 41–48 (2011).
[Crossref]

Takada, A.

A. Takada and H. Miyazawa, “30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser,” Electron. Lett. 26(3), 216–217 (1990).
[Crossref]

Tian, F.

Vallet, M.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

H. Zhang, M. Brunel, M. Romanelli, and M. Vallet, “Green pulsed lidar-radar emitter based on a multipass frequency-shifting external cavity,” Appl. Opt. 55(10), 2467–2473 (2016).
[Crossref] [PubMed]

Wells, J.

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

Xi, L.

Yang, H.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

Yang, S.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

Yao, J. P.

Yao, X. S.

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

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

Yasumura, Y.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Ye, J.

Yoshida, Y.

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

Zhang, H.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

H. Zhang, M. Brunel, M. Romanelli, and M. Vallet, “Green pulsed lidar-radar emitter based on a multipass frequency-shifting external cavity,” Appl. Opt. 55(10), 2467–2473 (2016).
[Crossref] [PubMed]

Zhang, X.

Zhao, C.

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. V. Kowalski, S. J. Shattil, and P. D. Hale, “Optical pulse generation with a frequency shifted feedback laser,” Appl. Phys. Lett. 53(9), 734–736 (1988).
[Crossref]

Electron. Lett. (2)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

A. Takada and H. Miyazawa, “30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser,” Electron. Lett. 26(3), 216–217 (1990).
[Crossref]

IEEE J. Quantum Electron. (2)

J. Hirano and T. Kimura, “Multiple mode-locking of lasers,” IEEE J. Quantum Electron. 5(5), 219–225 (1969).
[Crossref]

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

IEEE Microw. Mag. (1)

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

IEEE Photonics J. (1)

H. Yang, M. Brunel, H. Zhang, M. Vallet, C. Zhao, and S. Yang, “RF up-conversion and waveform generation using a frequency-shifting amplifying fiber loop, application to Doppler velocimetry,” IEEE Photonics J. 9(6), 7106609 (2017).
[Crossref]

J. Lightwave Technol. (4)

Nat. Commun. (1)

H. Guillet de Chatellus, L. Romero Cortés, C. Schnébelin, M. Burla, and J. Azaña, “Reconfigurable photonic generation of broadband chirped waveforms using a single CW laser and low-frequency electronics,” Nat. Commun. 9(1), 2438 (2018).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

NPG Asia Mater. (1)

J.-W. Shi, C.-B. Huang, and C.-L. Pan, “Millimeter-wave photonic wireless links for very high data rate communication,” NPG Asia Mater. 3(4), 41–48 (2011).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Optica (1)

Phys. Rev. A (1)

H. Guillet de Chatellus, E. Lacot, W. Glastre, O. Jacquin, and O. Hugon, “Theory of Talbot lasers,” Phys. Rev. A 88(3), 033828 (2013).
[Crossref]

Other (5)

A. Kanno, I. Morohashi, T. Kuri, I. Hosako, T. Kawanishi, Y. Yasumura, Y. Yoshida, and K. Kitayama, “16-Gbaud QPSK Radio Transmission using Optical Frequency Comb with Recirculating Frequency Shifter for 300-GHz RoF Signal,” inIEEE International Topical Meeting on Microwave Photonics, 2012, 298–301.
[Crossref]

C. Schnébelin and H. Guillet de Chatellus, “Optical spectral shaping with MHz resolution for arbitrary RF waveform generation,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM1B.7.
[Crossref]

L. Wang and S. LaRochelle, “Talbot Laser with Tunable GHz Repetition Rate using an Electro-Optic Frequency Shifter,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper JW2A.66.
[Crossref]

C. H. Lee, Microwave Photonics, 2nd ed. (CRC, 2013).

A. E. Siegman, Lasers (Mill Valley, 1986).

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Figures (9)

Fig. 1
Fig. 1 Sketch of the dual side-band FS loop. CW-SFL: continuous-wave single-frequency laser; PD: photodiode; TBPF: tunable bandpass filter; PC: polarization controller; EOM: Mach-Zehnder intensity modulator driven at frequency fm (SYN) and bias voltage Vb (DC); EDFA: erbium-doped Optical Fiber Amplifier.
Fig. 2
Fig. 2 Sketches of output signal vs time, with (a) different Γ and (b) different Γm. Simulation output power with (c) Γ = π/3 (blue), 5π/12 (green), π/2 (red), 7π/12 (light blue), and 2π/3 (purple), and (d) Γm = 0.5 (black), 0.55 (grey), 0.6 (yellow), 0.65 (brown), and 0.7 (green).
Fig. 3
Fig. 3 Experimental pulse doublet regime when fm = nfc: influence of the modulation frequency fm and the bias voltage Vb. Pdc = 25 dBm; (a) n = 1; (b) n = 10; (c) n = 100; (d) n = 500. In (a)-(b)-(c), Vb = 0 V (blue lines), 2 V (green lines), 4 V (red lines), 6 V (light blue lines), 8 V (purple lines). In (d), Vb = 2 V (blue line), 3 V (green line), 4 V (red line), 5 V (light blue line), 6 V (purple line). Pulse FWHM τp measured when Vb = 4 V: (a) τp = 13 ns, (b) 1.3 ns, (c) 130 ps, (d) 80 ps (detection limit).
Fig. 4
Fig. 4 Optical spectrum of the dual side-band FS loop. (a) fm = 673.7 MHz, the two arrows show the ± 19th harmonics. The total comb width is 39 × fm = 26.3 GHz. (b). fm = 3.369 GHz and the two arrows show the –5th to + 6th harmonics. The total comb width is 12 × fm = 40.4 GHz.
Fig. 5
Fig. 5 Influence of RF power on the pulse train. (a),(d) Pdc = 25 dBm; (b),(e) 23 dBm and (c), (f) 21 dBm. Upper row (blue curves): Vb = 2 V; lower row (red curves): Vb = 4 V.
Fig. 6
Fig. 6 Rectangle waveform generation; influence of the bias voltage and the RF power on the duty cycle. (a)-(c): Experiment results with (a) Vb = 1.5 V, Pdc = 21 dBm; (b) Vb = 2.6 V, Pdc = 20 dBm and (c) Vb = 3.3 V, Pdc = 19 dBm. (d)-(f): Simulation results with (d) Γ = 1 rad, Γm = 0.6; (e) Γ = 1.2 rad, Γm = 0.5 and (f) Γ = 1.4 rad, Γm = 0.3.
Fig. 7
Fig. 7 Saw-tooth waveforms with (a) fm = 1.0028 GHz, (b) fm = 0.99835 GHz. (red line: experiment; dotted blue line: simulation).
Fig. 8
Fig. 8 Resonant mode-locked double-pulse operation with fm = 7.611MHz. (a) Vb = 4 V. (b) Vb = 0 V.
Fig. 9
Fig. 9 Harmonic mode-locked double-pulse operation. Experimental laser output when (a) fm = 76.11 MHz, τp = 210 ps (b) fm = 761.1 MHz, τp = 70 ps.

Equations (9)

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[ E out1 E out2 ]=[ t 11 t 12 t 21 t 22 ][ E in1 E in2 ].
E out2 (t)= t 21 E in1 + t 22 E in2 ,
E out2 (t)= t 21 E in1 (t)+ t 22 γ E out2 (tτ) ϒ (1) (t).
E out2 (tτ)= t 21 E in1 (tτ)+ t 22 γ E out2 (t2τ) ϒ (2) (t).
E out2 (t)= t 21 E in1 (t)+ p=1 N t 21 t 22 p γ p l=1 p ϒ (l) (t) E in1 (tpτ) .
E out1 (t)= t 11 P in + t 21 t 12 p=1 N t 22 p1 γ p l=1 p ϒ (l) (t) P in ,
P out (t)= | t 11 + t 21 t 12 p=1 N t 22 p1 γ p l=1 p ϒ (l) (t) | 2 P in .
P out (t)= | t 11 + t 21 t 12 γsinθ(t) 1 t 22 γsinθ(t) | 2 P in ,
Δt= 1 2 f m [ 1 2 π sin 1 ( π/2Γ Γ m ) ].

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