Abstract

Spectral self-imaging (SI) is an efficient technique for controlling the line spacing (LS) of optical frequency combs (OFC). However, the degree of control is relatively limited, since the LS of the output comb must be set to be an integer sub-multiple of the input one. This technique can be extended to achieve arbitrary control of the comb LS by pre-conditioning the input comb with a properly designed spectral phase mask. This way, the output LS can be set to be any desired integer or fractional multiple of the input one. This generalized spectral SI process is intrinsically energy-preserving, which enables passive amplification of individual spectral lines of the comb when the scheme is designed for LS increase. Here we demonstrate the unique capabilities of generalized spectral SI in a simple dedicated fiber-optics platform, based on a frequency-shifting recirculating loop. When seeded with an external CW laser, the loop delivers a frequency comb with an arbitrary and reconfigurable quadratic spectral phase. We report lossless arbitrary control of the LS of the generated OFCs over six orders of magnitude, from the kHz to the GHz range, including passive amplification of individual comb lines by factors as high as 17 dB. The LS control is produced without modifying the features of the frequency comb. Practical applications of this LS control method are discussed.

© 2018 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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2018 (2)

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

B. C. Smith, B. Lomsadze, and S. T. Cundiff, “Optimum repetition rates for dual-comb spectroscopy,” Opt. Express 26(9), 12049–12056 (2018).
[Crossref] [PubMed]

2017 (4)

2016 (3)

2015 (6)

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

N. B. Hébert, V. Michaud-Belleau, J. D. Anstie, J.-D. Deschênes, A. N. Luiten, and J. Genest, “Self-heterodyne interference spectroscopy using a comb generated by pseudo-random modulation,” Opt. Express 23(21), 27806–27818 (2015).
[Crossref] [PubMed]

V. Durán, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” Opt. Express 23(23), 30557–30569 (2015).
[Crossref] [PubMed]

L. Romero Cortés, R. Maram, and J. Azaña, “Fractional averaging of repetitive waveforms induced by self-imaging effects,” Phys. Rev. A 92(4), 041804 (2015).
[Crossref]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Lossless fractional repetition-rate multiplication of optical pulse trains,” Opt. Lett. 40(3), 375–378 (2015).
[Crossref] [PubMed]

2014 (4)

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

J. Zhang and J. Yao, “Time-stretched sampling of a fast microwave waveform based on the repetitive use of a linearly chirped fiber Bragg grating in a dispersive loop,” Optica 1(2), 64–69 (2014).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser Photon. Rev. 8(3), 368–393 (2014).
[Crossref]

2013 (4)

2012 (1)

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

2011 (5)

N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nat. Photonics 5(4), 186–188 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
[Crossref] [PubMed]

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2006 (1)

2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

2001 (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

1981 (1)

Abramski, K. M.

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

Anstie, J. D.

Araki, T.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Azaña, J.

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
[Crossref] [PubMed]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Lossless fractional repetition-rate multiplication of optical pulse trains,” Opt. Lett. 40(3), 375–378 (2015).
[Crossref] [PubMed]

L. Romero Cortés, R. Maram, and J. Azaña, “Fractional averaging of repetitive waveforms induced by self-imaging effects,” Phys. Rev. A 92(4), 041804 (2015).
[Crossref]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

R. Maram and J. Azaña, “Spectral self-imaging of time-periodic coherent frequency combs by parabolic cross-phase modulation,” Opt. Express 21(23), 28824–28835 (2013).
[Crossref] [PubMed]

A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
[Crossref] [PubMed]

J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express 14(10), 4270–4279 (2006).
[Crossref] [PubMed]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

Bartels, A.

Beha, K.

Beltrán, M.

Bendahmane, A.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Berizzi, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Bogoni, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Bohn, B. J.

Braje, D. A.

Capria, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Caraquitena, J.

Charan, K.

Chen, L.

Coillet, A.

Cole, D. C.

Cortés, L. R.

Cundiff, S. T.

Curto, G. L.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Del’Haye, P.

Deschênes, J.-D.

Diddams, S. A.

Durán, V.

Fernández-Pousa, C. R.

Fortier, T. M.

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

M. S. Kirchner, D. A. Braje, T. M. Fortier, A. M. Weiner, L. Hollberg, and S. A. Diddams, “Generation of 20 GHz, sub-40 fs pulses at 960 nm via repetition-rate multiplication,” Opt. Lett. 34(7), 872–874 (2009).
[Crossref] [PubMed]

Genest, J.

Ghelfi, P.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Glastre, W.

González Hernández, J. I.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Guillet de Chatellus, H.

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
[Crossref] [PubMed]

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]

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]

Gupta, S.

Hänsch, T. W.

K. J. Mohler, B. J. Bohn, M. Yan, G. Mélen, T. W. Hänsch, and N. Picqué, “Dual-comb coherent Raman spectroscopy with lasers of 1-GHz pulse repetition frequency,” Opt. Lett. 42(2), 318–321 (2017).
[Crossref] [PubMed]

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Hariki, T.

Hébert, N. B.

Heinecke, D.

Hindle, F.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Hollberg, L.

Holzwarth, R.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Hovhannisyan, T.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Hsieh, Y.-D.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Hugon, O.

Inaba, H.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Iyonaga, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Jacquin, O.

Jannson, J.

Jannson, T.

Kaczmarek, P.

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

Kippenberg, T. J.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

Kirchner, M. S.

Krzempek, K.

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

Lacot, E.

Laghezza, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Lazzeri, E.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Li, B.

Li, M.

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Lossless fractional repetition-rate multiplication of optical pulse trains,” Opt. Lett. 40(3), 375–378 (2015).
[Crossref] [PubMed]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

Llorente, R.

Lomsadze, B.

Luiten, A. N.

Malacarne, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
[Crossref] [PubMed]

Manescau, A.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Maram, R.

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

L. Romero Cortés, R. Maram, and J. Azaña, “Fractional averaging of repetitive waveforms induced by self-imaging effects,” Phys. Rev. A 92(4), 041804 (2015).
[Crossref]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Lossless fractional repetition-rate multiplication of optical pulse trains,” Opt. Lett. 40(3), 375–378 (2015).
[Crossref] [PubMed]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

R. Maram and J. Azaña, “Spectral self-imaging of time-periodic coherent frequency combs by parabolic cross-phase modulation,” Opt. Express 21(23), 28824–28835 (2013).
[Crossref] [PubMed]

Marklof, J.

Martí, J.

Mélen, G.

Metcalf, A. J.

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

Michaud-Belleau, V.

Millot, G.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Minoshima, K.

A. Nishiyama, S. Yoshida, T. Hariki, Y. Nakajima, and K. Minoshima, “Sensitivity improvement of dual-comb spectroscopy using mode-filtering technique,” Opt. Express 25(25), 31730–31738 (2017).
[Crossref] [PubMed]

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Mohler, K. J.

Muriel, M. A.

J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
[Crossref] [PubMed]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

Nakajima, Y.

Newbury, N. R.

N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nat. Photonics 5(4), 186–188 (2011).
[Crossref]

Nikodem, M.

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

Nishiyama, A.

Onori, D.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Papp, S. B.

Pasquini, L.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Picqué, N.

K. J. Mohler, B. J. Bohn, M. Yan, G. Mélen, T. W. Hänsch, and N. Picqué, “Dual-comb coherent Raman spectroscopy with lasers of 1-GHz pulse repetition frequency,” Opt. Lett. 42(2), 318–321 (2017).
[Crossref] [PubMed]

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Pinna, S.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Pitois, S.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Porzi, C.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Probst, R. A.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Pudo, D.

Quinlan, F.

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

Rebolo, R.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Rojo, T.

Romero Cortés, L.

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

L. Romero Cortés, R. Maram, and J. Azaña, “Fractional averaging of repetitive waveforms induced by self-imaging effects,” Phys. Rev. A 92(4), 041804 (2015).
[Crossref]

Sakaguchi, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Scaffardi, M.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Scotti, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Serafino, G.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Sinefeld, D.

Smith, B. C.

Sobon, G.

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

Steinmetz, T.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Tainta, S.

Torres-Company, V.

V. Durán, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” Opt. Express 23(23), 30557–30569 (2015).
[Crossref] [PubMed]

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser Photon. Rev. 8(3), 368–393 (2014).
[Crossref]

Udem, T.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Van Howe, J.

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Lossless fractional repetition-rate multiplication of optical pulse trains,” Opt. Lett. 40(3), 375–378 (2015).
[Crossref] [PubMed]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

Vercesi, V.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

Wang, K.

Weiner, A. M.

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser Photon. Rev. 8(3), 368–393 (2014).
[Crossref]

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

M. S. Kirchner, D. A. Braje, T. M. Fortier, A. M. Weiner, L. Hollberg, and S. A. Diddams, “Generation of 20 GHz, sub-40 fs pulses at 960 nm via repetition-rate multiplication,” Opt. Lett. 34(7), 872–874 (2009).
[Crossref] [PubMed]

Wilken, T.

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

Xu, C.

Yan, M.

K. J. Mohler, B. J. Bohn, M. Yan, G. Mélen, T. W. Hänsch, and N. Picqué, “Dual-comb coherent Raman spectroscopy with lasers of 1-GHz pulse repetition frequency,” Opt. Lett. 42(2), 318–321 (2017).
[Crossref] [PubMed]

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Yao, J.

Yasui, T.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Yokoyama, S.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Yoshida, S.

Zhang, J.

Electron. Lett. (1)

A. M. Weiner, A. J. Metcalf, S. A. Diddams, T. M. Fortier, and F. Quinlan, “Broadly tunable, low timing jitter, high repetition rate optoelectronic comb generator,” Electron. Lett. 51(20), 1596–1598 (2015).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser Photon. Rev. 8(3), 368–393 (2014).
[Crossref]

Nat. Commun. (1)

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

Nat. Photonics (2)

N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nat. Photonics 5(4), 186–188 (2011).
[Crossref]

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10(1), 27–30 (2016).
[Crossref]

Nature (3)

T. Wilken, G. L. Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. González Hernández, R. Rebolo, T. W. Hänsch, T. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature 485(7400), 611–614 (2012).
[Crossref] [PubMed]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Opt. Commun. (2)

G. Sobon, K. Krzempek, P. Kaczmarek, K. M. Abramski, and M. Nikodem, “10 GHz passive harmonic mode-locking in Er-Yb double-clad fiber laser,” Opt. Commun. 284(18), 4203–4206 (2011).
[Crossref]

A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011).
[Crossref]

Opt. Express (10)

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]

L. Chen and D. Pudo, “Simple estimation of pulse amplitude noise and timing jitter evolution through the temporal Talbot effect,” Opt. Express 15(10), 6351–6357 (2007).
[Crossref] [PubMed]

J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express 14(10), 4270–4279 (2006).
[Crossref] [PubMed]

B. Li, K. Charan, K. Wang, T. Rojo, D. Sinefeld, and C. Xu, “Nonresonant background suppression for coherent anti-Stokes Raman scattering microscopy using a multi-wavelength time-lens source,” Opt. Express 24(23), 26687–26695 (2016).
[Crossref] [PubMed]

A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
[Crossref] [PubMed]

R. Maram and J. Azaña, “Spectral self-imaging of time-periodic coherent frequency combs by parabolic cross-phase modulation,” Opt. Express 21(23), 28824–28835 (2013).
[Crossref] [PubMed]

N. B. Hébert, V. Michaud-Belleau, J. D. Anstie, J.-D. Deschênes, A. N. Luiten, and J. Genest, “Self-heterodyne interference spectroscopy using a comb generated by pseudo-random modulation,” Opt. Express 23(21), 27806–27818 (2015).
[Crossref] [PubMed]

V. Durán, S. Tainta, and V. Torres-Company, “Ultrafast electrooptic dual-comb interferometry,” Opt. Express 23(23), 30557–30569 (2015).
[Crossref] [PubMed]

A. Nishiyama, S. Yoshida, T. Hariki, Y. Nakajima, and K. Minoshima, “Sensitivity improvement of dual-comb spectroscopy using mode-filtering technique,” Opt. Express 25(25), 31730–31738 (2017).
[Crossref] [PubMed]

B. C. Smith, B. Lomsadze, and S. T. Cundiff, “Optimum repetition rates for dual-comb spectroscopy,” Opt. Express 26(9), 12049–12056 (2018).
[Crossref] [PubMed]

Opt. Lett. (6)

Optica (2)

Phys. Rev. A (2)

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]

L. Romero Cortés, R. Maram, and J. Azaña, “Fractional averaging of repetitive waveforms induced by self-imaging effects,” Phys. Rev. A 92(4), 041804 (2015).
[Crossref]

Phys. Rev. Appl. (1)

L. Romero Cortés, R. Maram, H. Guillet de Chatellus, and J. Azaña, “Subnoise detection and passive amplification of frequency combs through customized coherent spectral energy redistribution,” Phys. Rev. Appl. 9, 064017 (2018).
[Crossref]

Sci. Rep. (1)

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, and T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4(1), 3816 (2015).
[Crossref] [PubMed]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

Other (2)

V. Torres-Company, J. Lancis, and P. Andres, “Space-time analogies in optics”, Ed. E. Wolf, in Progress in Optics 56, 1–81 (2011).

H. G. de Chatellus, L. R. Cortés, M. Burla, C. Schnébelin, and J. Azaña, “Agile photonic generation of arbitrary RF chirped waveforms,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper SF2L.3.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Illustration of energy-preserving control of an input OFC by integer/fractional division/multiplication of the input comb LS.
Fig. 2
Fig. 2 Principle of arbitrary LS control of an input OFC. a, scheme of the generalized spectral SI concept (involving a combination of temporal and spectral SI) for user-defined LS control of the OFC, with the notation in the text. In the plots, t stands for time and f for frequency. The intensity of the train of pulses is plotted in red, and the power spectral density (PSD) is represented in blue. The temporal and spectral phases ( ψ and φ respectively) are plotted in green and pink, respectively. The parabola-shaped dotted curves illustrate the intrinsic quadratic profile of the involved phases through the three steps of the SI process. In the shown example, q   =   3 and m   =   4 .
Fig. 3
Fig. 3 Numerical simulations illustrating LS control of a Gaussian mode-locked input OFC (input LS = f S ) through generalized spectral self-imaging, involving a combination of temporal SI (quadratic spectral phase filtering, left column) followed by standard spectral SI (quadratic temporal phase modulation, middle column) of the original OFC. The output and input OFCs for each of the simulated cases are shown in the right column.
Fig. 4
Fig. 4 Simplified circuit diagram of a CW-seeded FSL with arbitrary LS control. The loop is seeded with a narrow-linewidth single-frequency (CW) laser and includes an Erbium Doped Fiber Amplifier (EDFA), an optical isolator, a tunable optical bandpass filter (OBPF) and an acousto-optic frequency shifter (AOFS) driven by an arbitrary function generator (AFG). The required temporal SI effect (step 1) is inherently realized by the FSL, while the desired spectral SI processes (steps 2 and 3) are achieved by temporal phase modulation of the pulse train from the FSL using a single external electro-optic phase modulator (EOPM) driven by an arbitrary waveform generator (AWG).
Fig. 5
Fig. 5 Spectral and temporal properties of the experimental CW injection-seeded FSL. a, experimental 100 GHz-bandwidth OFC generated from the implemented FSL. The AOFS operates at f s   ~80 MHz in a frequency down-shifting mode, i.e., the frequency of the CW seed laser ( f 0 ) corresponds to the rightmost line. b, generation of a built-in tunable quadratic phase, and the related temporal SI effects, in a 30 GHz-bandwidth seeded FSL ( f c = 13.13 MHz). The spectral phase is measured by heterodyning the output of the FSL with a fraction of the seed laser, and the temporal trace is detected with a fast photo-detector (see text). According to the theory of temporal SI, when f s / f c = p / q the repetition rate of the train of pulses at the FSL output is equal to q f s , as in b.1 (integer SI, q = 1) and b.3, b.4 (fractional SI, q = 8, and q = 5, respectively). The results in b.2 illustrate the influence of the quadratic spectral phase on the pulse shape (or equivalent GVD) in the vicinity of the integer SI condition, leading to the observed severe temporal broadening of the individual pulses.
Fig. 6
Fig. 6 Integer multiplication of the comb LS by a factor of 100. a, driving voltage signal applied to the EOPM (green) and theoretical phase sequence (grey). b, output optical power spectrum (dark blue, “PM on”). The input optical spectrum (generated in the FSL) is plotted in red (“PM off”) for comparison. The width of the individual frequency lines (~2 MHz at −3 dB) is unchanged by the SI process and is limited by the measurement integration time of the OSA (inset). c, input (red, “PM off”) and output (blue, “PM on”) baseband spectra acquired through self-heterodyne measurements. Inset: illustration of the possibility of selecting the specific set of output comb lines, within the original frequency grid, by temporal shift of the PM sequence. The relative temporal delay between the PM sequences corresponding to the blue and the purple curves in the plot is equal to 2 / ( 100 f s ) (twice the period of the rate-multiplied temporal pulse train). Such a delay in the PM profile produces a shift in the absolute frequency location of the generated OFC of twice the input LS (shifting two lines in the original frequency grid). The width of the input and output frequency lines (~1 MHz at −3dB) is unchanged, and limited by the measurement integration time.
Fig. 7
Fig. 7 Fractional multiplication/division of the comb LS. Experimental comparison of the baseband spectra of the input frequency comb directly generated from the FSL (red, “PM off”) and the output frequency comb after the SI process (blue/purple, “PM on”). All spectra are measured by heterodyning with the seed laser. a, fractional LS multiplication ( q = 25 , m = 2 ). Inset: the relative temporal shift between the PM sequences corresponding to the blue and the purple curves is 2 / ( 25 f s ) . b, fractional LS division ( q = 3 , m = 11 ) . Inset: the temporal shift between the PM sequences corresponding to the blue and the purple curves is 2 / ( 3 f s ) .
Fig. 8
Fig. 8 Integer division of the comb LS. Comparison of the baseband spectra of the OFC directly generated from the FSL (red, “PM off”) and at the output after the SI process (blue, “PM on”). All spectra are measured by heterodyning with the seed laser. In each case, the output OFC exhibits an LS that is decreased (divided) by a factor of m with respect to the original one with a, q = 1 , m = 10 ; b, q = 1 , m = 100 ; c, q = 1 , m = 1000 ; and d, q = 1 , m = 10000 . Notice that the noise floor level and width of the spectral lines (~2 kHz at −3dB, limited by the measurement integration time) remain unchanged.
Fig. 9
Fig. 9 Application of LS control for enhanced OFC detection and kHz frequency spacing spectroscopy. a.1-2, demonstration of LS control for improved OFC detection. The input comb (orange curve) generated by the FSL ( f s = 79.91 MHz) cannot be resolved using spectrometers having a larger frequency resolution (a1: 140 MHz and a.2: 1 GHz). Self-imaging (SI) of the input OFC produces OFCs (in blue) identical to the input one but with LS exceeding the resolution limit of the detection, revealing the detailed comb structure. The output LS is multiplied by 7 and 100 in a.1 and a.2 resp. b.1-4, adaptive frequency sampling rate by SI for spectroscopy with kHz frequency point spacing. The output OFC is sent through a Fabry-Perot (FP) resonator (LS: 16 GHz, finesse: 250) for characterization of the resonator’s spectral response. The FP theoretical spectral transmission function is plotted in red (dash). The transmitted optical spectrum is measured by heterodyning the output of the FP with the monochromatic (CW) seed laser (acquisition time: 100 µs) (b.1). b.2, the input comb generated by the FSL (LS = 80.93 MHz, orange curve) cannot resolve the transmission peaks of the FP (red curve) nor the additional FP transverse modes at ~1,4 and ~1,7 GHz, contrary to self-imaged OFCs displayed on b.3 (blue comb: LS = 40.46 MHz, and cyan comb: LS = 8.093 MHz). b.4, LS division through SI by factors of 100 (top, comb LS = 809.3 kHz) and 1000 (bottom, comb LS = 80.93 kHz) enables simultaneous, precise sampling of the FP transmission peaks with kHz point spacing.

Equations (6)

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φ k = π p q k 2
ψ n = π s m n 2
E ( t ) = E 0 e i ω 0 t k 0 g ( k ) e i k ( ω s t ω 0 τ c ) e i π k ( k + 1 ) f s f c
E ( t ) = E 0 e i ω 0 t n 0 g ( n ) e i n ω s t
p ' = 8 p [ 1 2 ] q ( [ 1 2 p ] q ) 2 ,
p ' = p ( [ 1 p ] q ) 2 ,

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