Abstract

SWIFT spectroscopy (Shifted Wave Interference Fourier Transform Spectroscopy) is a coherent beatnote technique that can be used to measure the temporal profiles of periodic optical signals. While it has been essential in understanding the physics of various mid-infrared and terahertz frequency combs, its ultimate limits have not been discussed. We show that the envelope of a SWIFTS interferogram is physically meaningful and is directly related to autocorrelation. We derive analytical expressions for the SWIFTS signals of two prototypical cases—chirped pulses from a mode-locked laser and a frequency-modulated comb—and derive scaling laws for the noise of these measurements, showing how it can be mitigated. Finally, we confirm this analysis by performing the first SWIFTS measurements of near-infrared pulses from femtosecond lasers, establishing the validity of the technique for highly-dispersed sub-picojoule pulses.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2020 (2)

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

2019 (8)

J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13(2), 101–104 (2019).
[Crossref]

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

J. Hillbrand, M. Beiser, A. M. Andrews, H. Detz, R. Weih, A. Schade, S. Höfling, G. Strasser, and B. Schwarz, “Picosecond pulses from a mid-infrared interband cascade laser,” Optica 6(10), 1334 (2019).
[Crossref]

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

D. Burghoff, N. Han, and J. H. Shin, “Generalized method for the computational phase correction of arbitrary dual comb signals,” Opt. Lett. 44(12), 2966 (2019).
[Crossref]

D. Burghoff, N. Han, F. Kapsalidis, N. Henry, M. Beck, J. Khurgin, J. Faist, and Q. Hu, “Microelectromechanical control of the state of quantum cascade laser frequency combs,” Appl. Phys. Lett. 115(2), 021105 (2019).
[Crossref]

M. Singleton, M. Beck, and J. Faist, “Pulses from a mid-infrared quantum cascade laser frequency comb using an external compressor,” J. Opt. Soc. Am. B 36(6), 1676 (2019).
[Crossref]

2018 (1)

2017 (4)

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(1), 1 (2017).
[Crossref]

P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Analysis of operating regimes of terahertz quantum cascade laser frequency combs,” IEEE Trans. Terahertz Sci. Technol. 7(4), 351–359 (2017).
[Crossref]

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

2016 (2)

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(5), 499 (2016).
[Crossref]

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

2015 (1)

2014 (3)

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(6), 462–467 (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(8), 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(1), 5192 (2014).
[Crossref]

2012 (1)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[Crossref]

2011 (1)

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

2009 (2)

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[Crossref]

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol., A 116(4), 585–587 (2009).
[Crossref]

1996 (1)

S. Prein, S. Diddams, and J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123(4-6), 567–573 (1996).
[Crossref]

1994 (1)

1969 (1)

E. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[Crossref]

1958 (1)

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle ; Facteur de mérite,” J. Phys. Radium 19(3), 223–229 (1958).
[Crossref]

Amanti, M.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Andrews, A. M.

J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13(2), 101–104 (2019).
[Crossref]

J. Hillbrand, M. Beiser, A. M. Andrews, H. Detz, R. Weih, A. Schade, S. Höfling, G. Strasser, and B. Schwarz, “Picosecond pulses from a mid-infrared interband cascade laser,” Optica 6(10), 1334 (2019).
[Crossref]

Auth, D.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

Barbieri, S.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

Bartalini, S.

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

Beck, M.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

D. Burghoff, N. Han, F. Kapsalidis, N. Henry, M. Beck, J. Khurgin, J. Faist, and Q. Hu, “Microelectromechanical control of the state of quantum cascade laser frequency combs,” Appl. Phys. Lett. 115(2), 021105 (2019).
[Crossref]

M. Singleton, M. Beck, and J. Faist, “Pulses from a mid-infrared quantum cascade laser frequency comb using an external compressor,” J. Opt. Soc. Am. B 36(6), 1676 (2019).
[Crossref]

M. Singleton, P. Jouy, M. Beck, and J. Faist, “Evidence of linear chirp in mid-infrared quantum cascade lasers,” Optica 5(8), 948 (2018).
[Crossref]

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

Becker, S.

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

Beiser, M.

Belyanin, A.

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

Benea-Chelmus, I.-C.

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

Bewley, W. W.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Blaser, S.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5(1), 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(7428), 229–233 (2012).
[Crossref]

Bonzon, C.

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

Breuer, S.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

Burghoff, D.

D. Burghoff, N. Han, and J. H. Shin, “Generalized method for the computational phase correction of arbitrary dual comb signals,” Opt. Lett. 44(12), 2966 (2019).
[Crossref]

D. Burghoff, N. Han, F. Kapsalidis, N. Henry, M. Beck, J. Khurgin, J. Faist, and Q. Hu, “Microelectromechanical control of the state of quantum cascade laser frequency combs,” Appl. Phys. Lett. 115(2), 021105 (2019).
[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(1), 1 (2017).
[Crossref]

P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Analysis of operating regimes of terahertz quantum cascade laser frequency combs,” IEEE Trans. Terahertz Sci. Technol. 7(4), 351–359 (2017).
[Crossref]

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(5), 499 (2016).
[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(2), 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(6), 462–467 (2014).
[Crossref]

Burghoff, D. P.

D. P. Burghoff, “Broadband terahertz photonics,” Ph.D. thesis, Massachusetts Institute of Technology (2014).

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(6), 462–467 (2014).
[Crossref]

Campa, A.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

Campo, G.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

Canedy, C. L.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Capasso, F.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

Cappelli, F.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

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[Crossref]

Nafa, M.

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

Natale, P. D.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

Nikolov, I.

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol., A 116(4), 585–587 (2009).
[Crossref]

Nong, H.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Opacak, N.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

Pastor, P. C.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

Patrick, C. L.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Piccardo, M.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

Picqué, N.

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[Crossref]

Pistore, V.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Prein, S.

S. Prein, S. Diddams, and J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123(4-6), 567–573 (1996).
[Crossref]

Ravaro, M.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

Reno, J. L.

Rösch, M.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

Rößbner, K.

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

Santarelli, G.

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

Scalari, G.

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[Crossref]

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

Schade, A.

Scheuermann, J.

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

Schwarz, B.

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

J. Hillbrand, M. Beiser, A. M. Andrews, H. Detz, R. Weih, A. Schade, S. Höfling, G. Strasser, and B. Schwarz, “Picosecond pulses from a mid-infrared interband cascade laser,” Optica 6(10), 1334 (2019).
[Crossref]

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13(2), 101–104 (2019).
[Crossref]

Shin, J. H.

Singleton, M.

Sirtori, C.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

Sterczewski, L. A.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Strasser, G.

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

J. Hillbrand, M. Beiser, A. M. Andrews, H. Detz, R. Weih, A. Schade, S. Höfling, G. Strasser, and B. Schwarz, “Picosecond pulses from a mid-infrared interband cascade laser,” Optica 6(10), 1334 (2019).
[Crossref]

J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13(2), 101–104 (2019).
[Crossref]

Strutz, T.

T. Strutz, Data Fitting and Uncertainty: A practical introduction to weighted least squares and beyond (Springer Vieweg, 2016), 2nd ed.

Sultanova, N.

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol., A 116(4), 585–587 (2009).
[Crossref]

Tignon, J.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Treacy, E.

E. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[Crossref]

Tzenov, P.

P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Analysis of operating regimes of terahertz quantum cascade laser frequency combs,” IEEE Trans. Terahertz Sci. Technol. 7(4), 351–359 (2017).
[Crossref]

Villares, G.

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5(1), 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(7428), 229–233 (2012).
[Crossref]

Vitiello, M. S.

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

Vurgaftman, I.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Walmsley, I. A.

Wang, F.

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Wang, Y.

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

Weih, R.

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[Crossref]

J. Hillbrand, M. Beiser, A. M. Andrews, H. Detz, R. Weih, A. Schade, S. Höfling, G. Strasser, and B. Schwarz, “Picosecond pulses from a mid-infrared interband cascade laser,” Optica 6(10), 1334 (2019).
[Crossref]

Westberg, J.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Wong, V.

Wysocki, G.

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (2017).
[Crossref]

Yang, Y.

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(1), 1 (2017).
[Crossref]

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(5), 499 (2016).
[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(2), 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(6), 462–467 (2014).
[Crossref]

Acta Phys. Pol., A (1)

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol., A 116(4), 585–587 (2009).
[Crossref]

Appl. Phys. Lett. (3)

D. Burghoff, N. Han, F. Kapsalidis, N. Henry, M. Beck, J. Khurgin, J. Faist, and Q. Hu, “Microelectromechanical control of the state of quantum cascade laser frequency combs,” Appl. Phys. Lett. 115(2), 021105 (2019).
[Crossref]

S. Becker, J. Scheuermann, R. Weih, K. Rößbner, C. Kistner, J. Koeth, J. Hillbrand, B. Schwarz, and M. Kamp, “Picosecond pulses from a monolithic gasb-based passive mode-locked laser,” Appl. Phys. Lett. 116(2), 022102 (2020).
[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(8), 081118 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

E. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[Crossref]

IEEE Trans. Terahertz Sci. Technol. (1)

P. Tzenov, D. Burghoff, Q. Hu, and C. Jirauschek, “Analysis of operating regimes of terahertz quantum cascade laser frequency combs,” IEEE Trans. Terahertz Sci. Technol. 7(4), 351–359 (2017).
[Crossref]

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

J. Phys. Radium (1)

P. Jacquinot, “Caractères communs aux nouvelles méthodes de spectroscopie interférentielle ; Facteur de mérite,” J. Phys. Radium 19(3), 223–229 (1958).
[Crossref]

Laser Photonics Rev. (1)

F. Wang, H. Nong, T. Fobbe, V. Pistore, S. Houver, S. Markmann, N. Jukam, M. Amanti, C. Sirtori, S. Moumdji, R. Colombelli, L. Li, E. Linfield, G. Davies, J. Mangeney, J. Tignon, and S. Dhillon, “Short terahertz pulse generation from a dispersion compensated modelocked semiconductor laser,” Laser Photonics Rev. 11(4), 1700013 (2017).
[Crossref]

Nat. Commun. (2)

L. Consolino, M. Nafa, F. Cappelli, K. Garrasi, F. P. Mezzapesa, L. Li, A. G. Davies, E. H. Linfield, M. S. Vitiello, P. D. Natale, and S. Bartalini, “Fully phase-stabilized quantum cascade laser frequency comb,” Nat. Commun. 10(1), 2938 (2019).
[Crossref]

G. Villares, A. Hugi, S. Blaser, and J. Faist, “Dual-comb spectroscopy based on quantum-cascade-laser frequency combs,” Nat. Commun. 5(1), 5192 (2014).
[Crossref]

Nat. Photonics (5)

S. Barbieri, M. Ravaro, P. Gellie, G. Santarelli, C. Manquest, C. Sirtori, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Coherent sampling of active mode-locked terahertz quantum cascade lasers and frequency synthesis,” Nat. Photonics 5(5), 306–313 (2011).
[Crossref]

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[Crossref]

F. Cappelli, L. Consolino, G. Campo, I. Galli, D. Mazzotti, A. Campa, M. S. D. Cumis, P. C. Pastor, R. Eramo, M. Rösch, M. Beck, G. Scalari, J. Faist, P. D. Natale, and S. Bartalini, “Retrieval of phase relation and emission profile of quantum cascade laser frequency combs,” Nat. Photonics 13(8), 562–568 (2019).
[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(6), 462–467 (2014).
[Crossref]

J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, “Coherent injection locking of quantum cascade laser frequency combs,” Nat. Photonics 13(2), 101–104 (2019).
[Crossref]

Nature (1)

A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist, “Mid-infrared frequency comb based on a quantum cascade laser,” Nature 492(7428), 229–233 (2012).
[Crossref]

Opt. Commun. (1)

S. Prein, S. Diddams, and J.-C. Diels, “Complete characterization of femtosecond pulses using an all-electronic detector,” Opt. Commun. 123(4-6), 567–573 (1996).
[Crossref]

Opt. Eng. (2)

L. A. Sterczewski, J. Westberg, C. L. Patrick, C. S. Kim, M. Kim, C. L. Canedy, W. W. Bewley, C. D. Merritt, I. Vurgaftman, J. R. Meyer, and G. Wysocki, “Multiheterodyne spectroscopy using interband cascade lasers,” Opt. Eng. 57, 011014 (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(1), 1 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Optica (3)

Phys. Rev. A (1)

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93(4), 043812 (2016).
[Crossref]

Phys. Rev. Lett. (2)

J. Hillbrand, D. Auth, M. Piccardo, N. Opacak, E. Gornik, G. Strasser, F. Capasso, S. Breuer, and B. Schwarz, “In-phase and anti-phase synchronization in a laser frequency comb,” Phys. Rev. Lett. 124(2), 023901 (2020).
[Crossref]

M. Piccardo, P. Chevalier, B. Schwarz, D. Kazakov, Y. Wang, A. Belyanin, and F. Capasso, “Frequency-modulated combs obey a variational principle,” Phys. Rev. Lett. 122(25), 253901 (2019).
[Crossref]

Other (3)

P. Fellgett, “The multiplex advantage,” Ph.D. thesis, Ph. D. dissertation (University of Cambridge, Cambridge, UK, 1951) (1951).

D. P. Burghoff, “Broadband terahertz photonics,” Ph.D. thesis, Massachusetts Institute of Technology (2014).

T. Strutz, Data Fitting and Uncertainty: A practical introduction to weighted least squares and beyond (Springer Vieweg, 2016), 2nd ed.

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Figures (5)

Fig. 1.
Fig. 1. Interferograms of a chirped Gaussian pulse. a. I and Q interferograms measured from a pulse with $\tau _0$ =29 fs, along with an analytical fit for $D_2$ = -0.89 ps $^2$ . b. Corresponding SWIFTS intensity and frequency envelopes, along with an analytical fit. c. Calculated envelopes for different $D_2$ parameters.
Fig. 2.
Fig. 2. Interferograms of an FM comb. a. I and Q interferograms measured for a mid-IR QCL operating in a linearly-chirped regime, along with an analytical fit assuming $\Delta f$ =-0.96 THz (indicating negative chirp) [27]. b. Corresponding SWIFTS intensity and frequency envelopes, along with the analytical fit. c. Calculated envelopes for different $\Delta f$ parameters.
Fig. 3.
Fig. 3. Ten SWIFTS measurements of a chirped Gaussian pulse, with $\tau _0$ =30 fs, $f_{\textrm {res}}$ =0.51 THz, and $f_{\textrm {LO}}$ =2.03 GHz. The calculated $D_2$ standard deviation of 0.067 ps $^2$ closely matches the measured standard deviation of 0.058 ps $^2$ , and the measured value agrees with the value expected from the compressor’s grating separation (see Section 5).
Fig. 4.
Fig. 4. a. Raw intensity and frequency waveforms extracted directly from the measurement of a chirped pulse. Both functions possess unphysical spikes due to high-order coherence noise. b. Result of using the measured noise to calculate the median of the distribution, which regularizes the result and smooths out the spikes.
Fig. 5.
Fig. 5. a. Experimental setup used in the measurements of a Ti:Sapphire laser. The self-referenced LO scheme is used here. b. Measured group delay dispersion of the pulses, compared with the value expected from the grating distance. c. Chirped and nominally-unchirped pulse intensity and frequency waveforms (along with measured dispersions). d. Comparison of the intensity autocorrelation measured by an autocorrelator to that extracted by SWIFTS (for a pulse dispersed by 0.042 ps $^2$ ).

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

E ( t ) = n E n e i ω n t ,
S ( t , τ ) = 1 2 ( E ( t ) + E ( t τ ) ) 2
= 1 2 ( E 2 ( t ) + E 2 ( t τ ) + 2 E ( t ) E ( t τ ) )
= 1 2 n , m E n E m e i ω n m t ( 1 + e i ω n m τ + 2 e i ω m τ ) ,
S 0 ( τ ) = 1 2 n E n E n ( 1 + 1 + 2 e i ω n τ )
S + ( τ ) = 1 2 n E n + L E n e i ( ω n + L , n ω LO ) t ( 1 + e i ω n + L , n τ + 2 e i ω n τ )
S ^ 0 ( τ ) = n E n E n e i ω n τ
S ^ + ( τ ) = n E n + L E n e i ω n τ .
S ^ 0 ( τ ) = n E n ( τ ) E n ( τ ) e i ω n τ
S ^ + ( τ ) = n E n + L ( τ ) E n ( τ ) e i ( ω n + L , n ω LO ) c 2 v τ e i ω n τ .
S 0 ( n ) = 1 τ m a x E n ( τ ) E n ( τ ) d τ = | E n | 2
S + ( n ) = 1 τ m a x E n + L ( τ ) E n ( τ ) e i ( ω n + L , n ω LO ) c 2 v τ d τ .
| g n | 2 = | 1 τ m a x 0 τ m a x e i ( ω n + L , n ω LO ) c 2 v τ d τ | 2 = sinc 2 ( c 4 v ( ω n + L , n ω LO ) τ m a x )
| g n | 2 = sinc 2 ( π T ( f n + L , n f L O ) ) .
| g n | 2 = | 1 T 0 T e i ϕ n ( t ) d t | 2
| g n | 2 = 2 ( π T f FWHM ) 2 ( e π T f FWHM + π T f FWHM 1 )
| g n | 2 = 1 T 2 0 T d t 0 T d t e i ( ϕ n ( t ) ϕ n ( t ) 1 1 2 T 2 0 T d t 0 T d t ( ϕ n ( t ) ϕ n ( t ) ) 2
= 1 1 T 0 T ϕ n 2 ( t ) d t
I ( τ ) | S ^ + ( τ ) | 2
ω ( τ ) d d τ arg S ^ + ( τ ) = 1 | S ^ + ( τ ) | 2 Im ( S ^ + ( τ ) d S ^ + d τ ) .
A I ( τ ) L ( S ^ + ( τ ) S ^ + ( τ ) )
A ω I ( τ ) L ( Im ( S ^ + ( τ ) d S ^ + d τ ) )
A I ( τ ) = n , m > 0 ( E n + L E n E m + L E m + E n L E n E m L E m ) e i ω n m τ
A ω I ( τ ) = Re n , m > 0 ω n ( E n + L E n E m + L E m E n L E n E m L E m ) e i ω n m τ ,
I A C ( τ ) = l n , m > 0 E n + l E n E m + l E m e i ω n m τ
E a ( t ) = τ 0 2 τ 0 2 + i D 2 E 0 exp ( t 2 2 ( τ 0 2 + i D 2 ) ) e i ω 0 t
S + ^ ( τ ) = 1 T r T r / 2 T r / 2 ( E a ( t + τ / 2 ) E a ( t τ / 2 ) + E a ( t + τ / 2 ) E a ( t τ / 2 ) ) e i ω LO t d t
S 0 ^ ( τ ) = 1 T r T r / 2 T r / 2 ( E a ( t + τ / 2 ) E a ( t τ / 2 ) + E a ( t + τ / 2 ) E a ( t τ / 2 ) ) d t ,
S + ^ ( τ ) = | E 0 | 2 π τ 0 T r e 1 4 τ 0 2 ( τ 2 + ( D 2 2 + τ 0 4 ) ω LO 2 ) ( e i ω 0 τ + D 2 ω LO 2 τ 0 2 τ + e i ω 0 τ D 2 ω LO 2 τ 0 2 τ )
S 0 ^ ( τ ) = | E 0 | 2 π τ 0 T r e 1 4 τ 0 2 τ 2 ( e i ω 0 τ + e i ω 0 τ ) .
A I ( τ ) = | E 0 | 4 π τ 0 2 T r 2 e 1 2 τ 0 2 ( τ 2 + ( D 2 2 + τ 0 4 ) ω LO 2 ) ( e 1 τ 0 2 D 2 ω LO τ + e 1 τ 0 2 D 2 ω LO τ )
A ω I ( τ ) = | E 0 | 4 π τ 0 2 T r 2 e 1 2 τ 0 2 ( τ 2 + ( D 2 2 + τ 0 4 ) ω LO 2 ) ( e 1 τ 0 2 D 2 ω LO τ e 1 τ 0 2 D 2 ω LO τ ) ω 0 .
ω ( t ) = ω 0 + 1 2 π Δ ω ω r [ t ]
ϕ ( t ) = ω 0 t + 1 4 π Δ ω ω r [ t ] 2
E a ( t ) = E 0 exp ( i 1 4 π Δ ω ω r [ t ] 2 ) e i ω 0 t
S + ^ ( τ ) = | E 0 | 2 4 ( 2 π L ) 2 ( Δ ω τ ) 2 ( 1 ) L + 1 sin ( Δ ω τ 2 ) ( Δ ω τ cos ( ω 0 τ ) + i 2 π L sin ( ω 0 τ ) )
S 0 ^ ( τ ) = | E 0 | 2 4 Δ ω τ ( 1 ) L + 1 sin ( Δ ω τ 2 ) cos ( ω 0 τ )
A I ( τ ) = 4 | E 0 | 4 ( 2 π L ) 2 + ( Δ ω τ ) 2 ( ( 2 π L ) 2 ( Δ ω τ ) 2 ) ( 1 cos ( Δ ω τ ) )
A ω I ( τ ) = 16 | E 0 | 4 π L Δ ω τ ( ( 2 π L ) 2 ( Δ ω τ ) 2 ) ( 1 cos ( Δ ω τ ) ) ω 0 .
Var ( ϕ n ) = σ 2 4 ( 1 | S + ( ω n ) | 2 + 1 | S + ( ω n L ) | 2 ) σ 2 2 | S + ( ω n ) | 2 = 1 4 SNR ( ω n )
Var ( τ g ( ω n ) ) 1 4 ω LO 2 SNR ( ω n )
w = 1 Var ϕ = 4 SNR ( ω 0 ) e 2 ( ω ω 0 ) 2 τ 0 2
Var ( D 2 ) = C 2 π 1 SNR ( ω 0 ) ω res τ 0 3 ω LO 2
Var ( t FWHM ) = C 2 π 1 SNR ( ω 0 ) ω res τ 0 ω LO 2 4 ln 2
Var ( D 2 ) = C 3 SNR ( ω 0 ) 1 ω LO Δ ω 3 .
I ( t ) = n , m > 0 E n E m e i ω n m t
f ( t ) = 1 I 2 ( t ) Re n , m > 0 f n E n E m e i ω n m t .
Var ( Φ n m ) = k = n k = m L Var ( ϕ ( ω k ) ) = k = n k = m L 1 4 SNR ( ω k ) .
E n E m = E n ( t ) E m ( t ) e i ϵ n m = E n ( t ) E m ( t ) e 1 2 Var ( Φ n m )
Var E n E m = | E n E m E n E m | 2 = | E n ( t ) E m ( t ) | 2 ( 1 e Var ( Φ n m ) )
SNR E n E m = | E n E m | 2 Var E n E m = 1 e Var ( Φ n m ) 1 .

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