Abstract

Since the first demonstration of the quantum cascade laser (QCL) frequency comb in 2012, there have been open questions concerning the reproducibility of comb states, and, critically for metrological applications, the phase stability. To address these important issues, we measure the intermodal phase relation of a QCL comb operating at 8 μm using a coherent beatnote spectroscopy. We find these intermodal phase differences to be constant within experimental uncertainty over several repeat measurements, and the comb states reproducible to an average of 21 mrad after cycling the power of the device. These phases describe a comb state exhibiting a simple, linear chirp, which in fact corresponds to the lowest state of chirp to minimize the amplitude modulation, as required for combs driven by four-wave mixing in a gain medium with a short gain recovery lifetime. All together, these findings could pave the way for pulse shaping in the QCL platform.

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

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References

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2018 (1)

2017 (4)

Y. Bidaux, I. Sergachev, W. Wuester, R. Maulini, T. Gresch, A. Bismuto, S. Blaser, A. Muller, and J. Faist, “Plasmon-enhanced waveguide for dispersion compensation in mid-infrared quantum cascade laser frequency combs,” Opt. Lett. 42, 1604–1607 (2017).
[Crossref]

N. Henry, D. Burghoff, Y. Yang, Q. Hu, and J. B. Khurgin, “Pseudorandom dynamics of frequency combs in free-running quantum cascade lasers,” Opt. Eng. 57, 011009 (2017).
[Crossref]

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

M. Dong, N. M. Mangan, J. N. Kutz, S. T. Cundiff, and H. G. Winful, “Traveling wave model for frequency comb generation in single-section quantum well diode lasers,” IEEE J. Quantum Electron. 53, 1–11 (2017).
[Crossref]

2016 (2)

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Süess, P. Jouy, M. Beck, and J. Faist, “Dispersion engineering of quantum cascade laser frequency combs,” Optica 3, 252–258 (2016).
[Crossref]

2015 (4)

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
[Crossref]

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

2014 (5)

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

P. Del’Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

2012 (2)

2009 (1)

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

2008 (3)

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

2001 (1)

K. Sato, “100  GHz optical pulse generation using Fabry–Perot laser under continuous wave operation,” Electron. Lett. 37, 763–764 (2001).
[Crossref]

1998 (1)

1997 (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

1996 (1)

Accard, A.

Allmendinger, P.

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

Austerer, M.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Bacon, D. R.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Barry, L. P.

Beck, M.

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Süess, P. Jouy, M. Beck, and J. Faist, “Dispersion engineering of quantum cascade laser frequency combs,” Optica 3, 252–258 (2016).
[Crossref]

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
[Crossref]

Beere, H. E.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Beha, K.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

P. Del’Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

Belyanin, A.

M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Bidaux, Y.

Y. Bidaux, I. Sergachev, W. Wuester, R. Maulini, T. Gresch, A. Bismuto, S. Blaser, A. Muller, and J. Faist, “Plasmon-enhanced waveguide for dispersion compensation in mid-infrared quantum cascade laser frequency combs,” Opt. Lett. 42, 1604–1607 (2017).
[Crossref]

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

Bismuto, A.

Blaser, S.

Y. Bidaux, I. Sergachev, W. Wuester, R. Maulini, T. Gresch, A. Bismuto, S. Blaser, A. Muller, and J. Faist, “Plasmon-enhanced waveguide for dispersion compensation in mid-infrared quantum cascade laser frequency combs,” Opt. Lett. 42, 1604–1607 (2017).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

Boittin, R.

Bour, D.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Buchanan, M.

H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “QWIP performance and polarization selection rule,” in Intersubband Transitions in Quantum Wells: Physics and Devices (Springer, 1998), pp. 50–59.

Burghoff, D.

N. Henry, D. Burghoff, Y. Yang, Q. Hu, and J. B. Khurgin, “Pseudorandom dynamics of frequency combs in free-running quantum cascade lasers,” Opt. Eng. 57, 011009 (2017).
[Crossref]

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Cai, X.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Capasso, F.

M. Piccardo, D. Kazakov, N. A. Rubin, P. Chevalier, Y. Wang, F. Xie, K. Lascola, A. Belyanin, and F. Capasso, “Time-dependent population inversion gratings in laser frequency combs,” Optica 5, 475–478 (2018).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

Chan, C. W. I.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Chevalier, P.

Choi, H.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

Cockburn, J. W.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Coillet, A.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

Corzine, S.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Cundiff, S. T.

M. Dong, N. M. Mangan, J. N. Kutz, S. T. Cundiff, and H. G. Winful, “Traveling wave model for frequency comb generation in single-section quantum well diode lasers,” IEEE J. Quantum Electron. 53, 1–11 (2017).
[Crossref]

Darmo, J.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Davies, A. G.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Dean, P.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Debeau, J.

Del’Haye, P.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

P. Del’Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Diddams, S. A.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

P. Del’Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

Diehl, L.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

Dikmelik, Y.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

Dong, M.

M. Dong, N. M. Mangan, J. N. Kutz, S. T. Cundiff, and H. G. Winful, “Traveling wave model for frequency comb generation in single-section quantum well diode lasers,” IEEE J. Quantum Electron. 53, 1–11 (2017).
[Crossref]

Elsaesser, T.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Faist, J.

Y. Bidaux, I. Sergachev, W. Wuester, R. Maulini, T. Gresch, A. Bismuto, S. Blaser, A. Muller, and J. Faist, “Plasmon-enhanced waveguide for dispersion compensation in mid-infrared quantum cascade laser frequency combs,” Opt. Lett. 42, 1604–1607 (2017).
[Crossref]

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Süess, P. Jouy, M. Beck, and J. Faist, “Dispersion engineering of quantum cascade laser frequency combs,” Optica 3, 252–258 (2016).
[Crossref]

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Freeman, J. R.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Gaal, P.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Gao, J.-R.

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Giovannini, M.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

Gordon, A.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Green, R. P.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Gresch, T.

Han, N.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Hänsch, T. W.

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

Hayton, D. J.

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Henry, N.

N. Henry, D. Burghoff, Y. Yang, Q. Hu, and J. B. Khurgin, “Pseudorandom dynamics of frequency combs in free-running quantum cascade lasers,” Opt. Eng. 57, 011009 (2017).
[Crossref]

Hipke, A.

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

Höfler, G.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Hu, Q.

N. Henry, D. Burghoff, Y. Yang, Q. Hu, and J. B. Khurgin, “Pseudorandom dynamics of frequency combs in free-running quantum cascade lasers,” Opt. Eng. 57, 011009 (2017).
[Crossref]

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Hugi, A.

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

Ideguchi, T.

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

Jouy, P.

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Süess, P. Jouy, M. Beck, and J. Faist, “Dispersion engineering of quantum cascade laser frequency combs,” Optica 3, 252–258 (2016).
[Crossref]

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Kao, T.-Y.

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Kärtner, F. X.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Kazakov, D.

Khurgin, J. B.

N. Henry, D. Burghoff, Y. Yang, Q. Hu, and J. B. Khurgin, “Pseudorandom dynamics of frequency combs in free-running quantum cascade lasers,” Opt. Eng. 57, 011009 (2017).
[Crossref]

J. B. Khurgin, Y. Dikmelik, A. Hugi, and J. Faist, “Coherent frequency combs produced by self frequency modulation in quantum cascade lasers,” Appl. Phys. Lett. 104, 081118 (2014).
[Crossref]

Kowalski, B.

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Krysa, A. B.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Kuehn, W.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Kutz, J. N.

M. Dong, N. M. Mangan, J. N. Kutz, S. T. Cundiff, and H. G. Winful, “Traveling wave model for frequency comb generation in single-section quantum well diode lasers,” IEEE J. Quantum Electron. 53, 1–11 (2017).
[Crossref]

Lascola, K.

Lelarge, F.

Li, L.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Linfield, E. H.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Liu, H. C.

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “QWIP performance and polarization selection rule,” in Intersubband Transitions in Quantum Wells: Physics and Devices (Springer, 1998), pp. 50–59.

Loh, W.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

Maier, T.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Mangan, N. M.

M. Dong, N. M. Mangan, J. N. Kutz, S. T. Cundiff, and H. G. Winful, “Traveling wave model for frequency comb generation in single-section quantum well diode lasers,” IEEE J. Quantum Electron. 53, 1–11 (2017).
[Crossref]

Mangold, M.

P. Jouy, J. M. Wolf, Y. Bidaux, P. Allmendinger, M. Mangold, M. Beck, and J. Faist, “Dual comb operation of 8.2  um quantum cascade laser frequency comb with 1  W optical power,” Appl. Phys. Lett. 111, 141102 (2017).
[Crossref]

Martinez, A.

Maulini, R.

Meek, S. A.

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

Merghem, K.

Mohandas, R. A.

D. R. Bacon, J. R. Freeman, R. A. Mohandas, L. Li, E. H. Linfield, A. G. Davies, and P. Dean, “Gain recovery time in a terahertz quantum cascade laser,” Appl. Phys. Lett. 108, 081104 (2016).
[Crossref]

Muller, A.

Müller, T.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Murdin, B.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Murdoch, S. G.

Norris, T. B.

H. Choi, L. Diehl, Z.-K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, “Gain recovery dynamics and photon-driven transport in quantum cascade lasers,” Phys. Rev. Lett. 100, 167401 (2008).
[Crossref]

Papp, S. B.

P. Del’Haye, A. Coillet, W. Loh, K. Beha, S. B. Papp, and S. A. Diddams, “Phase steps and resonator detuning measurements in microresonator frequency combs,” Nat. Commun. 6, 5668 (2015).
[Crossref]

P. Del’Haye, K. Beha, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014).
[Crossref]

Parz, W.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Piccardo, M.

Picqué, N.

A. Hipke, S. A. Meek, T. Ideguchi, T. W. Hänsch, and N. Picqué, “Broadband Doppler-limited two-photon and stepwise excitation spectroscopy with laser frequency combs,” Phys. Rev. A 90, 011805 (2014).
[Crossref]

Pidgeon, C.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Ramdane, A.

Reimann, K.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Reno, J. L.

D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Evaluating the coherence and time-domain profile of quantum cascade laser frequency combs,” Opt. Express 23, 1190–1202 (2015).
[Crossref]

D. Burghoff, T.-Y. Kao, N. Han, C. W. I. Chan, X. Cai, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz laser frequency combs,” Nat. Photonics 8, 462–467 (2014).
[Crossref]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Riedi, S.

Ritchie, D. A.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Roberts, J. S.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Rosales, R.

Rösch, M.

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
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Sato, K.

K. Sato, “100  GHz optical pulse generation using Fabry–Perot laser under continuous wave operation,” Electron. Lett. 37, 763–764 (2001).
[Crossref]

Scalari, G.

M. Rösch, G. Scalari, M. Beck, and J. Faist, “Octave-spanning semiconductor laser,” Nat. Photonics 9, 42–47 (2015).
[Crossref]

Schneider, H.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Sergachev, I.

Strasser, G.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Süess, M. J.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Trebino, R.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Tredicucci, A.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Troccoli, M.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Unterrainer, K.

W. Kuehn, W. Parz, P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, T. Müller, J. Darmo, K. Unterrainer, M. Austerer, G. Strasser, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Ultrafast phase-resolved pump-probe measurements on a quantum cascade laser,” Appl. Phys. Lett. 93, 151106 (2008).
[Crossref]

Villares, G.

G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Süess, P. Jouy, M. Beck, and J. Faist, “Dispersion engineering of quantum cascade laser frequency combs,” Optica 3, 252–258 (2016).
[Crossref]

G. Villares and J. Faist, “Quantum cascade laser combs: effects of modulation and dispersion,” Opt. Express 23, 1651–1669 (2015).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5, 5192 (2014).
[Crossref]

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492, 229–233 (2012).
[Crossref]

Vinh, N. Q.

R. P. Green, A. Tredicucci, N. Q. Vinh, B. Murdin, C. Pidgeon, H. E. Beere, and D. A. Ritchie, “Gain recovery dynamics of a terahertz quantum cascade laser,” Phys. Rev. B 80, 075303 (2009).
[Crossref]

Walmsley, I. A.

Wang, C. Y.

A. Gordon, C. Y. Wang, L. Diehl, F. X. Kärtner, A. Belyanin, D. Bour, S. Corzine, G. Höfler, H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, “Multimode regimes in quantum cascade lasers: from coherent instabilities to spatial hole burning,” Phys. Rev. A 77, 053804 (2008).
[Crossref]

Wang, Y.

Wasilewski, Z. R.

H. C. Liu, M. Buchanan, and Z. R. Wasilewski, “QWIP performance and polarization selection rule,” in Intersubband Transitions in Quantum Wells: Physics and Devices (Springer, 1998), pp. 50–59.

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Supplementary Material (1)

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» Supplement 1       Supplementary 1

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Figures (5)

Fig. 1.
Fig. 1. (a) Coherent beatnote spectroscopy setup [17], with the detected signal referenced to the QCL “clock.” We measure with a maximum path difference corresponding to a resolution of 0.125  cm1, or about 3.75 GHz. The fast RF correlation traces are measured on a quantum well infrared photodetector [20] (QWIP), biased through a bias tee using a source measurement unit (SMU, Keithley 2400). The reference and signal are amplified using a pair of low-noise, high-bandwidth RF amplifiers, and pre-downconverted to the lower frequency 25 MHz using Miteq mixers. The DC autocorrelation is collected on a separate, more sensitive mercury cadmium telluride (MCT) detector. (b) Example of measured DC and demodulated (SWIFTS) interferogram traces (left); zoom about centerburst (right), showing that, when the interferometer arms are balanced, the RF signal is at a minimum, characteristic of FM combs. (c) Computed spectrum product |E(ω)|2|E(ω+Δω)|2 [17] (blue) with the complex spectrum [Eq. (5), green] overlaid, both having been resampled at the repetition ωr. The similarity of the spectral envelopes demonstrates plainly the equidistance across the entire lasing bandwidth. (d) 95% confidence intervals for calculated ϕnϕn1 (left axis), with the corresponding group delay (right axis). Estimated as explained in Supplement 1, Section 1.
Fig. 2.
Fig. 2. (a) Amplitude spectrum and computed phase for one measurement set. The orange-dashed line indicates the quadratic fit to the phase over the range 11701270  cm1, marked by the black-dotted lines, yielding an estimated field GDD of 6.4  ps2. (b) Instantaneous intensity and frequency of the field, recovered by the inverse Fourier transform of the measurement state in (a). Spiking in the instantaneous frequency coincides with the intensity going to zero, where the phase at that instant is undefined.
Fig. 3.
Fig. 3. (Orange) GDD of the laser output field, as predicted by Eq. (7) for a maximally chirped pulse from the (15  dB) estimated spectral bandwidth (purple stars) and round trip frequency, and estimated by parabolic fit to unwrapped phase profile (blue).
Fig. 4.
Fig. 4. (a) Twelve amplitude and group delay traces, with the power having been cycled prior to measuring each. The slight vertical offsets observed between the group delay traces translate only to relative time shifts between the waveforms. These have hence been subtracted for visual clarity, without changing the result. (b) Standard deviation of the results. (c) Histogram, demonstrating that the phases are reproduced to an accuracy of about 21 mrad. In both (b) and (c), the red line indicates the mean, taken from the log-normal fit in orange.
Fig. 5.
Fig. 5. (a) Merit as a function of the chirp parameter. Parameters: τ21=0.5  ps, τ22=0.1  ps, τrt=133  ps. The red circles at about 0.062 indicate the first minimum, which coincides with the first minimum of amplitude modulation. The latter is taken to be the variance of the instantaneous intensity over one comb period. Inset: spectral amplitude as a function of mode index, chosen to be a sum of 3 Gaussians. σ=10 modes, A=[1,0.8,1], offset ±17, g0=1.12. (b) Intensity (top) and frequency (bottom) as a function of time for the state at the local optimum 0.062, showing a linear chirp and rapid amplitude modulations. (c) Beats present in the intracavity field for a given chirp (see Supplement 1, Section 4). The dashed line corresponds to our optimal chirp and shows the onset of strong high-frequency beat tones.

Equations (8)

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E(t)=nAnei(ω0+nωr)t.
E*(t)E(t+τ)=nAn[An1*eiωrt+An+1*eiωrt]ei(ω0+nωr)τ,
x(τ)=E(t+τ)E*(t)cos(ωrt)=12nAn[An1*+An+1*]ei(ω0+nωr)τ,
y(τ)=E(t+τ)E*(t)sin(ωrt)=12inAn[An+1*An1*]ei(ω0+nωr)τ.
X(ω)iY(ω)=n|An||An1|ei(ϕnϕn1)δ[ω(ω0+nωr)].
ϕn={n=0n(ϕnϕn1)n<00n=0n=0nϕnϕn1n>0.
GDD=Δ(ϕnϕn1)Δω=12πfrfBW.
dAndt=(Gn1)AniDnAnGnk,l=N/2N/2AmAkAl*BklCklκklmn