Abstract

Dual-comb spectroscopy is a rapidly developing spectroscopic technique that does not require any opto-mechanical moving parts and enables broadband and high-resolution measurements with microsecond time resolution. However, for high sensitivity measurements and extended averaging times, high mutual coherence of the comb-sources is essential. To date, most dual-comb systems employ coherent averaging schemes that require additional electro-optical components, which increase system complexity and cost. More recently, computational phase correction approaches that enables coherent averaging of spectra generated by free-running systems have gained increasing interest. Here, we propose such an all-computational solution that is compatible with real-time data acquisition architectures for free-running systems. The efficacy of our coherent averaging algorithm is demonstrated using dual-comb spectrometers based on quantum cascade lasers, interband cascade lasers, mode-locked lasers, and optically-pumped microresonators.

© 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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2019 (1)

2018 (6)

M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, “Passively mode-locked interband cascade optical frequency combs,” Sci. Reports 8, 3322 (2018).
[Crossref]

J. Westberg, L. A. Sterczewski, F. Kapsalidis, Y. Bidaux, J. Wolf, M. Beck, J. Faist, and G. Wysocki, “Dual-comb spectroscopy using plasmon-enhanced waveguide dispersion compensated quantum cascade lasers,” Opt. Lett. 434522–4525 (2018).
[Crossref]

M. Yu, Y. Okawachi, A. G. Griffith, N. Picqué, M. Lipson, and A. L. Gaeta, “Silicon-chip-based mid-infrared dual-comb spectroscopy,” Nat. communications 9, 1869 (2018).
[Crossref]

N. B. Hébert, D. G. Lancaster, V. Michaud-Belleau, G. Y. Chen, and J. Genest, “Highly coherent free-running dual-comb chip platform,” Opt. Lett. 43, 1814–1817 (2018).
[Crossref] [PubMed]

A. V. Muraviev, V. O. Smolski, Z. E. Loparo, and K. L. Vodopyanov, “Massively parallel sensing of trace molecules and their isotopologues with broadband subharmonic mid-infrared frequency combs,” Nat. Photonics 12, 209–214 (2018).
[Crossref]

Z. Zhu, K. Ni, Q. Zhou, and G. Wu, “Digital correction method for realizing a phase-stable dual-comb interferometer,” Opt. Express 26, 16813–16823 (2018).
[Crossref] [PubMed]

2017 (9)

Q. Lu, D. Wu, S. Slivken, and M. Razeghi, “High efficiency quantum cascade laser frequency comb,” Sci. Reports 7, 43806 (2017).
[Crossref]

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 (2017).
[Crossref] [PubMed]

Y. Yang, D. Burghoff, J. Reno, and Q. Hu, “Achieving comb formation over the entire lasing range of quantum cascade lasers,” Opt. Lett. 42, 3888 (2017).
[Crossref] [PubMed]

L. A. Sterczewski, J. Westberg, L. Patrick, C. Soo Kim, M. Kim, C. L. Canedy, W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 75, 011014(2017).

J. Westberg, L. A. Sterczewski, and G. Wysocki, “Mid-infrared multiheterodyne spectroscopy with phase-locked quantum cascade lasers,” Appl. Phys. Lett. 110, 141108 (2017).
[Crossref]

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

O. Kara, L. Maidment, T. Gardiner, P. G. Schunemann, and D. T. Reid, “Dual-comb spectroscopy in the spectral fingerprint region using OPGaP optical parametric oscillators,” Opt. Express 25, 32713 (2017).
[Crossref]

N. B. Hébert, J. Genest, J.-D. Deschênes, H. Bergeron, G. Y. Chen, C. Khurmi, and D. G. Lancaster, “Self-corrected chip-based dual-comb spectrometer,” Opt. Express 25, 8168 (2017).
[Crossref] [PubMed]

L. A. Sterczewski, J. Westberg, and G. Wysocki, “Molecular dispersion spectroscopy based on Fabry–Perot quantum cascade lasers,” Opt. Lett. 42, 243 (2017).
[Crossref] [PubMed]

2016 (4)

M. Rösch, G. Scalari, G. Villares, L. Bosco, M. Beck, and J. Faist, “On-chip, self-detected terahertz dual-comb source,” Appl. Phys. Lett. 108, 171104 (2016).
[Crossref]

D. Burghoff, Y. Yang, and Q. Hu, “Computational multiheterodyne spectroscopy,” Sci. Adv. 2, e1601227 (2016).
[Crossref] [PubMed]

Y. Yang, D. Burghoff, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz multiheterodyne spectroscopy using laser frequency combs,” Optica 3, 499 (2016).
[Crossref]

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3, 414 (2016).
[Crossref]

2014 (4)

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] [PubMed]

T. Ideguchi, A. Poisson, G. Guelachvili, N. Picqué, and T. W. Hänsch, “Adaptive real-time dual-comb spectroscopy,” Nat. Commun. 5, 3375 (2014).
[Crossref] [PubMed]

J.-D. Deschênes and J. Genest, “Frequency-noise removal and on-line calibration for accurate frequency comb interference spectroscopy of acetylene,” Appl. Opt. 53, 731–735 (2014).
[Crossref] [PubMed]

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]

2012 (2)

J. Roy, J.-D. Deschênes, S. Potvin, and J. Genest, “Continuous real-time correction and averaging for frequency comb interferometry,” Opt. Express 20, 21932 (2012).
[Crossref] [PubMed]

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] [PubMed]

2009 (1)

L. Tan and J. Jiang, “Novel adaptive IIR filter for frequency estimation and tracking [DSP Tips Tricks],” IEEE Signal Process. Mag. 26, 186–189 (2009).
[Crossref]

2008 (1)

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent Multiheterodyne Spectroscopy Using Stabilized Optical Frequency Combs,” Phys. Rev. Lett. 100, 013902 (2008).
[Crossref] [PubMed]

2007 (1)

V. F. Pisarenko, “The Retrieval of Harmonics from a Covariance Function,” Geophys. J. Royal Astron. Soc. 33, 347–366 (2007).
[Crossref]

2004 (1)

R. M. Aarts, “Low-complexity tracking and estimation of frequency and amplitude of sinusoids,” Digit. Signal Process. 14, 372–378 (2004).
[Crossref]

2002 (1)

S. Schiller, “Spectrometry with frequency combs,” Opt. letters 27, 766–768 (2002).
[Crossref]

1997 (2)

P. Tichavsky and P. Handel, “Recursive estimation of linearly or harmonically modulated frequencies of multiple cisoids in noise,” IEEE 3, pp. 1925–1928 (1997).
[Crossref]

P. Tichavsky and P. Handel, “Recursive estimation of frequencies and frequency rates of multiple cisoids in noise,” Signal Process. 58, 117–129 (1997).
[Crossref]

1993 (1)

P. Werle, R. Mücke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131–139 (1993).
[Crossref]

Aarts, R. M.

R. M. Aarts, “Low-complexity tracking and estimation of frequency and amplitude of sinusoids,” Digit. Signal Process. 14, 372–378 (2004).
[Crossref]

Allmendinger, P.

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

Bagheri, M.

L. A. Sterczewski, J. Westberg, M. Bagheri, C. Frez, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, J. R. Meyer, and G. Wysocki, “Mid-infrared dual-comb spectroscopy with interband cascade lasers,” Opt. Lett. 44, 2113 (2019).
[Crossref] [PubMed]

M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, “Passively mode-locked interband cascade optical frequency combs,” Sci. Reports 8, 3322 (2018).
[Crossref]

Beck, M.

J. Westberg, L. A. Sterczewski, F. Kapsalidis, Y. Bidaux, J. Wolf, M. Beck, J. Faist, and G. Wysocki, “Dual-comb spectroscopy using plasmon-enhanced waveguide dispersion compensated quantum cascade lasers,” Opt. Lett. 434522–4525 (2018).
[Crossref]

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

M. Rösch, G. Scalari, G. Villares, L. Bosco, M. Beck, and J. Faist, “On-chip, self-detected terahertz dual-comb source,” Appl. Phys. Lett. 108, 171104 (2016).
[Crossref]

Bergeron, H.

Bewley, W.

L. A. Sterczewski, J. Westberg, L. Patrick, C. Soo Kim, M. Kim, C. L. Canedy, W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 75, 011014(2017).

Bewley, W. W.

L. A. Sterczewski, J. Westberg, M. Bagheri, C. Frez, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, J. R. Meyer, and G. Wysocki, “Mid-infrared dual-comb spectroscopy with interband cascade lasers,” Opt. Lett. 44, 2113 (2019).
[Crossref] [PubMed]

M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, “Passively mode-locked interband cascade optical frequency combs,” Sci. Reports 8, 3322 (2018).
[Crossref]

Bidaux, Y.

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 (2017).
[Crossref] [PubMed]

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] [PubMed]

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] [PubMed]

Bosco, L.

M. Rösch, G. Scalari, G. Villares, L. Bosco, M. Beck, and J. Faist, “On-chip, self-detected terahertz dual-comb source,” Appl. Phys. Lett. 108, 171104 (2016).
[Crossref]

Burghoff, D.

Y. Yang, D. Burghoff, J. Reno, and Q. Hu, “Achieving comb formation over the entire lasing range of quantum cascade lasers,” Opt. Lett. 42, 3888 (2017).
[Crossref] [PubMed]

D. Burghoff, Y. Yang, and Q. Hu, “Computational multiheterodyne spectroscopy,” Sci. Adv. 2, e1601227 (2016).
[Crossref] [PubMed]

Y. Yang, D. Burghoff, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, “Terahertz multiheterodyne spectroscopy using laser frequency combs,” Optica 3, 499 (2016).
[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]

Canedy, C. L.

L. A. Sterczewski, J. Westberg, M. Bagheri, C. Frez, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, J. R. Meyer, and G. Wysocki, “Mid-infrared dual-comb spectroscopy with interband cascade lasers,” Opt. Lett. 44, 2113 (2019).
[Crossref] [PubMed]

M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, “Passively mode-locked interband cascade optical frequency combs,” Sci. Reports 8, 3322 (2018).
[Crossref]

L. A. Sterczewski, J. Westberg, L. Patrick, C. Soo Kim, M. Kim, C. L. Canedy, W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 75, 011014(2017).

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]

Chen, G. Y.

Coddington, I.

I. Coddington, N. Newbury, and W. Swann, “Dual-comb spectroscopy,” Optica 3, 414 (2016).
[Crossref]

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent Multiheterodyne Spectroscopy Using Stabilized Optical Frequency Combs,” Phys. Rev. Lett. 100, 013902 (2008).
[Crossref] [PubMed]

Deschênes, J.-D.

Faist, J.

J. Westberg, L. A. Sterczewski, F. Kapsalidis, Y. Bidaux, J. Wolf, M. Beck, J. Faist, and G. Wysocki, “Dual-comb spectroscopy using plasmon-enhanced waveguide dispersion compensated quantum cascade lasers,” Opt. Lett. 434522–4525 (2018).
[Crossref]

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

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 (2017).
[Crossref] [PubMed]

M. Rösch, G. Scalari, G. Villares, L. Bosco, M. Beck, and J. Faist, “On-chip, self-detected terahertz dual-comb source,” Appl. Phys. Lett. 108, 171104 (2016).
[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] [PubMed]

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] [PubMed]

Fradet, M.

M. Bagheri, C. Frez, L. A. Sterczewski, I. Gruidin, M. Fradet, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, and J. R. Meyer, “Passively mode-locked interband cascade optical frequency combs,” Sci. Reports 8, 3322 (2018).
[Crossref]

Frez, C.

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L. A. Sterczewski, J. Westberg, M. Bagheri, C. Frez, I. Vurgaftman, C. L. Canedy, W. W. Bewley, C. D. Merritt, C. S. Kim, M. Kim, J. R. Meyer, and G. Wysocki, “Mid-infrared dual-comb spectroscopy with interband cascade lasers,” Opt. Lett. 44, 2113 (2019).
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Appl. Opt. (1)

Appl. Phys. B (1)

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Appl. Phys. Lett. (3)

M. Rösch, G. Scalari, G. Villares, L. Bosco, M. Beck, and J. Faist, “On-chip, self-detected terahertz dual-comb source,” Appl. Phys. Lett. 108, 171104 (2016).
[Crossref]

J. Westberg, L. A. Sterczewski, and G. Wysocki, “Mid-infrared multiheterodyne spectroscopy with phase-locked quantum cascade lasers,” Appl. Phys. Lett. 110, 141108 (2017).
[Crossref]

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

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IEEE Signal Process. Mag. (1)

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Nat. Commun. (2)

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M. Yu, Y. Okawachi, A. G. Griffith, N. Picqué, M. Lipson, and A. L. Gaeta, “Silicon-chip-based mid-infrared dual-comb spectroscopy,” Nat. communications 9, 1869 (2018).
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Figures (11)

Fig. 1
Fig. 1 Effects of frequency instabilities in frequency combs on the radio frequency spectrum. (a) CEO fluctuations. (b) Repetition rate fluctuations.
Fig. 2
Fig. 2 Demonstration of the efficacy of the coherent averaging algorithm. (a) Raw dual-comb rf spectrum acquired using two free-running terahertz quantum cascade lasers. (b) rf spectrum after timing and phase correction.
Fig. 3
Fig. 3 Effect of acquisition time on the shape of the DCS spectrum. (a) rf spectra at 1 μs, 10 μs, and 100 μs together with a zoom on the strongest beat note. Only the shortest time acquisition yields a nearly pure, acquisition time limited tone. Longer time acquisitions reveal the multi-peak complex shape of the beat notes. (b) Spectrogram of a 1 ms acquisition of two QCLs suffering from cryostat vibrations and optical feedback.
Fig. 4
Fig. 4 Block diagram of the proposed timing and phase correction. (t) - complex dual-comb signal, Δfrep(t) - repetition rate difference, Δf0(t) - difference in offset frequency, c(t) -corrected complex dual-comb signal.
Fig. 5
Fig. 5 Flowchart of signals in the CoCoA algorithm. (a) The IQ-demodulated DCS rf signal. (b) DDFG self-mixing spectrum. (c) The retrieved instantaneous differential repetition rate. (d) The resampled DDFG self-mixing spectrum. (e) The IQ-demodulated DCS rf signal after adaptive resampling. (f) The instantaneous difference offset frequency obtained from the fast frequency tracker. (g) the IQ-demodulated rf spectrum after correction. (h) The effect of solely correcting for difference frequency offset.
Fig. 6
Fig. 6 (a) Dual-comb spectrum of two ICL combs acquired over 1 ms, where one is operated in the high phase-noise regime. As a result, no resolvable rf beat notes are observed. (b) the DDFG self-mixing spectrum with no indication of Δfrep-harmonics, which is a signature of non-comb operation of at least one of the devices. (c) The intermode beat note of the device operating in comb-mode. (d) The intermode beat note of the device operating in high phase-noise regime. (e) DDFG self-mixing spectrum after resampling.
Fig. 7
Fig. 7 (a) Dual-comb spectrum of two ICL combs acquired over 1 ms, where both are operated in the comb-regime. (b) The DDFG self-mixing spectrum with several Δfrep-harmonics that can be used for the adaptive sampling procedure. (c) The rf spectrum after correction. (d) The resampled DDFG self-mixing spectrum. (e), (f) Zoom of a rf beat note showing the effects of the CoCoA algorithm. (g) Zoom of a DDFG self-mixing harmonic showing the effects of the adaptive sampling.
Fig. 8
Fig. 8 (a) DCS rf spectrum using two LWIR QCL combs, phase-locked via active feedback control of the injection current. (b) DCS rf spectrum from the same sources using the CoCoA algorithm described in this work.
Fig. 9
Fig. 9 Correction of the microresonator-based DCS data. (a) Temporal structure of the DCS signal without clearly identifiable bursts. (b) Radio frequency spectrum acquired over 2 ms. (c) The rf spectrum after correction. Aliased beat notes previously buried in noise appear above 1 GHz. (d), (e), (f) Zoom of selected beat notes showing a narrowing of the rf comb lines. (g) Allan deviation analysis [35] of beat note amplitude performed on the weak beat note located at 1039 MHz. (h) Allan deviation of beat note amplitude of the strong beat note located at 415 MHz. It is evident, that for very short time scales (single microseconds) phase correction introduces weak amplitude modulation, which slightly affects the uncertainty of amplitude estimate. Nevertheless, above 10 μs, the correction allows for reaching higher precision (lower uncertainty) of the peak amplitude down to the per mille range.
Fig. 10
Fig. 10 Correction of the passively mode-locked waveguide lasers DCS data. (a) Temporal structure of the DCS signal with clearly identifiable bursts. (b) Radio frequency spectrum mapping ∼2.3 THz acquired over 25.5 ms before and after CoCoA correction. An improvement of approximately 5 dB in SNR can be observed. (c) Zoom of selected beat notes in the rf spectrum before correction. (d) Zoom of selected beat notes in the rf spectrum after correction.
Fig. 11
Fig. 11 Spectral comparison of the different nonlinear operations applied to the acquired complex multiheterodyne signal in (a). Panel (b) shows the result of squaring the complex signal, which yields SHG and SFG without any DFG products. Panel (c) shows the spectrum of the squared real part, which creates a DC component together with SHG, SFG, and DFG. Note the spectral symmetry, and overlap of the components. (d) Spurious-free DFG signal obtained by adding the independently squared real and complex part of the signal. Slight aliasing is also visible for the highest order repetition rate harmonics.

Equations (13)

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s ˜ ( t ) = exp ( i 2 π Δ f 0 t ) n = 1 N exp ( i 2 π n Δ f rep t ) .
( e { s ˜ ( t ) } ) 2 + ( 𝕀 m { s ˜ ( t ) } ) 2 = N DC + 2 m = 1 N l = 1 m 1 cos ( 2 π ( m l ) Δ f rep t ) DFG .
t ( t ) = 0 t k Δ f rep k Δ f rep ( τ ) d τ ,
r ^ k = r ^ k 1 + x k 1 γ [ x k + x k 2 2 x k 1 r ^ k 1 ]
r k = cos ( 2 π Δ f 0 ( k ) T s ) ,
Δ f ^ 0 ( k ) = cos 1 ( r ^ k ) f s 2 π .
s ˜ c ( t ) = exp ( i 2 π 0 t Δ f 0 ( τ ) Δ f 0 d τ ) s ˜ ( t ) .
s ˜ ( t ) = n = 1 N cos ( 2 π ( Δ f 0 + n Δ f rep ) t ) + i n = 1 N sin ( 2 π ( Δ f 0 + n Δ f rep ) t ) = = n = 1 N exp ( i 2 π ( Δ f 0 + n Δ f rep ) t ) ,
( n = 1 N a n ) 2 + n = 1 N a n 2 + 2 m = 1 N l = 1 m 1 a l a m .
[ s ˜ ( t ) ] 2 = n = 1 N exp ( i 2 π ( 2 Δ f 0 + 2 n Δ f rep ) t ) SHG + 2 m = 1 N l = 1 m 1 exp ( i 2 π ( 2 Δ f 0 + ( l + m ) Δ f rep ) t ) SFG .
( e { s ˜ ( t ) } ) 2 = N 2 DC + 1 2 n = 1 N cos ( 2 π ( 2 Δ f 0 + 2 n Δ f rep ) t ) SHG + + m = 1 N l = 1 m 1 cos ( 2 π ( 2 Δ f 0 + ( l + m ) Δ f rep ) t ) SFG + + m = 1 N l = 1 m 1 cos ( 2 π ( m l ) Δ f rep t ) DFG .
( 𝕀 m { s ˜ ( t ) } ) 2 = N 2 DC 1 2 n = 1 N cos ( 2 π ( 2 Δ f 0 + 2 n Δ f rep ) t ) SHG + m = 1 N l = 1 m 1 cos ( 2 π ( 2 Δ f 0 + ( l + m ) Δ f rep ) t ) SFG + + m = 1 N l = 1 m 1 cos ( 2 π ( m l ) Δ f rep t ) DFG .
( e { s ˜ ( t ) } ) 2 + ( 𝕀 m { s ˜ ( t ) } ) 2 = N DC + 2 m = 1 N l = 1 m 1 cos ( 2 π ( m l ) Δ f rep t ) DFG .

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