Abstract

We introduce a simple scheme for mode-density multiplication of an optical frequency comb (OFC) by a factor of square of an arbitrary integer N using phase modulation. This scheme is employed to multiply the mode density of an erbium-doped fiber laser OFC (repetition rate of 66.87 MHz) by factors of 42, 82, and 3 · 42 using an electro-optic phase modulator. The OFC multiplied by 42 is applied to direct-comb spectroscopy of methane with a spectral resolution of 4.18 MHz.

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

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    [Crossref]

2018 (4)

2017 (2)

2016 (5)

2015 (1)

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 µ m,” Appl. Phys. Express 8, 082402 (2015).
[Crossref]

2014 (2)

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]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8, 375–380 (2014).
[Crossref] [PubMed]

2013 (2)

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, 4139–4144 (2013).
[Crossref] [PubMed]

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

2012 (2)

K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and single-wavelength laser,” Appl. Phys. Lett. 101, 051105 (2012).
[Crossref]

P. Del’Haye, S. B. Papp, and S. A. Diddams, “Hybrid electro-optically modulated microcombs,” Phys. Rev. Lett. 109, 263901 (2012).
[Crossref]

2011 (1)

T. Shioda, K. Fujii, K. Kashiwagi, and T. Kurokawa, “High-resolution spectroscopy using interleaved 100 GHz optical frequency comb scanned by phase modulator,” Opt. Commun. 284, 5180–5184 (2011).
[Crossref]

2010 (2)

D. Mandridis, I. Ozdur, F. Quinlan, M. Akbulut, J. Plant, P. Juodawlkis, and P. Delfyett, “Low-noise, low repetition rate, semiconductor-based mode-locked laser source suitable for high bandwidth photonic analog-digital conversion,” Appl. Opt. 49, 2850–2857 (2010).
[Crossref] [PubMed]

A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, S. Aozasa, A. Mori, H. Nakano, A. Takada, and M. Koga, “Octave-spanning frequency comb generated by 250 fs pulse train emitted from 25 GHz externally phase-modulated laser diode for carrier-envelope-offset-locking,” Electron. Lett. 46, 1343–1344 (2010).
[Crossref]

2009 (2)

2008 (4)

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321, 1335–1337 (2008).
[Crossref] [PubMed]

I. Coddington, W. C. Swann, and N. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100, 013902 (2008).
[Crossref] [PubMed]

A. Bartels, D. Heinecke, and S. A. Diddams, “Passively mode-locked 10 GHz femtosecond Ti:sapphire laser,” Opt. Lett. 33, 1905–1907 (2008).
[Crossref] [PubMed]

S. Kobtsev, S. Kukarin, and Y. Fedotov, “Ultra-low repetition rate mode-locked fiber laser with high-energy pulses,” Opt. Express 16, 21936–21941 (2008).
[Crossref] [PubMed]

2007 (1)

S. A. Diddams, L. Hollberg, and V. Mbele, “Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb,” Nature 445, 627–630 (2007).
[Crossref] [PubMed]

2006 (1)

C.-B. Huang, Z. Jiang, D.E. Leaird, and A.M. Weiner, “High-rate femtosecond pulse generation via line-by-line processing of phase-modulated CW laser frequency comb,” Electron. Lett. 19, 1114–1115 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

A. Baltuška, M. Uiberacker, E. Goulielmakis, R. Kienberger, V. S. Yakovlev, T. Udem, T. W. Hänsch, and F. Krausz, “Phase-controlled amplification of few-cycle laser pulses,” IEEE J. Sel. Top. Quantum Electron. 9, 972–989 (2003).
[Crossref]

1988 (1)

M. Ohtsu, “Frequency stabilization in semiconductor lasers,” Opt. Quantum Electron. 20, 283–300 (1988).
[Crossref]

Akbulut, M.

Almaiman, A.

Aozasa, S.

A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, S. Aozasa, A. Mori, H. Nakano, A. Takada, and M. Koga, “Octave-spanning frequency comb generated by 250 fs pulse train emitted from 25 GHz externally phase-modulated laser diode for carrier-envelope-offset-locking,” Electron. Lett. 46, 1343–1344 (2010).
[Crossref]

Apolonski, A.

Araujo-Hauck, C.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321, 1335–1337 (2008).
[Crossref] [PubMed]

Azaña, J.

Babikov, Y.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Balling, P.

Baltuška, A.

A. Baltuška, M. Uiberacker, E. Goulielmakis, R. Kienberger, V. S. Yakovlev, T. Udem, T. W. Hänsch, and F. Krausz, “Phase-controlled amplification of few-cycle laser pulses,” IEEE J. Sel. Top. Quantum Electron. 9, 972–989 (2003).
[Crossref]

Bao, C.

Barbe, A.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Bartels, A.

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, 27–30 (2016).
[Crossref]

Benner, D. Chris

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Bernath, P. F.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Birk, M.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Bizzocchi, L.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Boudon, V.

L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
[Crossref]

Braje, D. A.

Brasch, V.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8, 375–380 (2014).
[Crossref] [PubMed]

Brown, L. R.

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

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Schmidt, W.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321, 1335–1337 (2008).
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J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8, 375–380 (2014).
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L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
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L. S. Rothman, I. E. Gordon, Y. Babikov, A. Barbe, D. Chris Benner, P. F. Bernath, M. Birk, L. Bizzocchi, V. Boudon, L. R. Brown, A. Campargue, K. Chance, E. A. Cohen, L. H. Coudert, V. M. Devi, B. J. Drouin, A. Fayt, J.-M. Flaud, R. R. Gamache, J. J. Harrison, J.-M. Hartmann, C. Hill, J. T. Hodges, D. Jacquemart, A. Jolly, J. Lamouroux, R. J. Le Roy, G. Li, D. A. Long, O. M. Lyulin, C. J. Mackie, S. T. Massie, S. Mikhailenko, H. S. P. Müller, O. V. Naumenko, A. V. Nikitin, J. Orphal, V. Perevalov, A. Perrin, E. R. Polovtseva, C. Richard, M. A. H. Smith, E. Starikova, K. Sung, S. Tashkun, J. Tennyson, G. C. Toon, V. G. Tyuterev, and G. Wagner, “The HITRAN2012 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 130, 4–50 (2013).
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Wagner, G.

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J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nat. Photonics 8, 375–380 (2014).
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Yan, M.

G. Millot, S. Pitois, M. Yan, T. Hovhannisyan, A. Bendahmane, T. W. Hänsch, and N. Picqué, “Frequency-agile dual-comb spectroscopy,” Nat. Photonics 10, 27–30 (2016).
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Appl. Opt. (2)

Appl. Phys. Express (1)

S. Okubo, K. Iwakuni, H. Inaba, K. Hosaka, A. Onae, H. Sasada, and F-L. Hong, “Ultra-broadband dual-comb spectroscopy across 1.0–1.9 µ m,” Appl. Phys. Express 8, 082402 (2015).
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Appl. Phys. Lett. (1)

K. Urabe and O. Sakai, “Absorption spectroscopy using interference between optical frequency comb and single-wavelength laser,” Appl. Phys. Lett. 101, 051105 (2012).
[Crossref]

Electron. Lett. (2)

C.-B. Huang, Z. Jiang, D.E. Leaird, and A.M. Weiner, “High-rate femtosecond pulse generation via line-by-line processing of phase-modulated CW laser frequency comb,” Electron. Lett. 19, 1114–1115 (2006).
[Crossref]

A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, S. Aozasa, A. Mori, H. Nakano, A. Takada, and M. Koga, “Octave-spanning frequency comb generated by 250 fs pulse train emitted from 25 GHz externally phase-modulated laser diode for carrier-envelope-offset-locking,” Electron. Lett. 46, 1343–1344 (2010).
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IEEE J. Sel. Top. Quantum Electron. (1)

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram of pulse train in black and a phase-modulation sequence of Eq. (6) in orange. This phase modulation increases the mode density of the OFC by a factor of 42. (b) Pulses categorized into four groups according to the remainders of the pulse number divided by four. In the first group, the pulse numbers are the multiple of 4, and in the second (third, fourth) group, those are the multiple of 4 plus 1 (2,3). The carrier phase shift for the pulses is also given. (c) The mode spectra of the OFC without and with the phase modulation. The black, red, green, and blue modes correspond to the first-, second-, third-, and fourth-group pulses, respectively. (d) An alternative phase modulation sequence giving the same mode spectrum as (a).
Fig. 2
Fig. 2 (a) Experimental setup. A glass cell (GC) filled with methane is inserted for spectroscopy. (b) and (c) are beat-note spectral lines at 20 MHz recorded by the spectrum analyzer with RBW of 1 kHz and 1 Hz, respectively, when the ECLD frequency is stabilized with respect to one of the OFC modes. In (c), the linewidth is limited by RBW of the spectrum analyzer. Abbreviations: FA for an optical fiber amplifier, HNLF for a highly-nonlinear fiber, BS for a 50:50 beam splitter, AFG for an arbitrary function generator, FC for a fiber coupler, BPD for balanced photodetectors, SA for an rf spectrum analyzer, BPF for an rf bandpass filter at 20 MHz, RFA for an rf amplifier (15 dB), M for an rf mixer, PZT for a piezoelectric transducer to adjust the laser cavity length, and LO for a local oscillator.
Fig. 3
Fig. 3 Spectra of the beat notes of the ECLD with (a) the original OFC, and with the phase-modulated OFC for (b) P = 16, (d), 64, and (f) 48, respectively. The RBW of the spectrum analyzer is 300 kHz. The repetition frequency of the original OFC is indicated by dotted lines. In order to distinguish each OFC mode clearly, figures of (c), (e), and (g) are enlarged spectra of (b), (d), and (f), respectively and recorded with the RBW of 10 kHz.
Fig. 4
Fig. 4 Phase modulation for the mode-density multiplication of the OFC by factor of P1 × P2, where P1 and P2 are prime to each other. In (a), two electro-optic phase modulators (EOM1 and EOM2) are used in series. In (b), two modulation signals are added and applied to one EOM. AFG1 and AFG2 provide modulation signals for P1 and P2, respectively.
Fig. 5
Fig. 5 Absorption spectrum of the R(6) transitions of the 2ν3 band of methane at 1645.57 nm recorded by cw-comb heterodyne spectroscopy using (a) the OFC without the modulation and (b) the phase-modulated OFC for P = 16. The horizontal axis indicates the optical frequency of the phase-modulated OFC relative to the cw laser frequency. The blue dotted lines indicate the center frequencies and the intensity of the absorption lines. The red curve is the fitted spectrum assuming the Doppler broadening at 293 K. (c) and (d) show the residuals of the normalized and fitted spectra in (a) and (b), respectively.

Equations (22)

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ϕ j + P = ϕ j ,
E ( t ) = E 0 j = g ( t j t r ) exp [ i ω c ( t j t r ) ] exp [ i ( j ϕ o + ϕ j ) ] ,
E ˜ ( ω ) = E ( t ) exp ( i ω t ) d t = E 0 ω r P g ˜ ( ω ω c ) k = δ ( ω + ω o k ω r P ) Φ ^ ( k mod P ) ,
Φ ^ k = j = 0 P 1 exp ( i ϕ j ) exp ( 2 π i j k P ) .
ϕ N j + j = 2 π j j N ,
ϕ j = { 0 for j = 0 , 1 , 2 , 3 , 4 , 8 , 10 , 12 π / 2 for j = 5 , 15 π for j = 6 , 9 , 11 , 14 3 π / 2 for j = 7 , 13
Φ ^ k = j = 0 N 1 j = 0 N 1 exp ( 2 π i j j N ) exp [ 2 π i ( N j + j ) k N 2 ] .
Φ ^ k = j = 0 N 1 j = 0 N 1 exp [ 2 π i j ( j + k ) N ] exp ( 2 π i j k N 2 ) .
j = 0 N 1 exp ( 2 π i j k N ) = N δ k mod N , 0 ,
Φ ^ k = j = 0 N 1 N exp ( 2 π i j k N 2 ) δ ( j + k ) mod N , 0 = N exp ( 2 π i ( N k ) k N 2 ) .
ϕ j + P n ( n ) = ϕ j ( n ) ,
Φ ^ k n ( n ) = j = 0 P n exp [ i ϕ j ( n ) ] exp ( 2 π i j k n P n ) .
exp [ i ϕ j ( n ) ] = 1 P n k n = 0 P n Φ ^ k n ( n ) exp [ 2 π i j k n P n ] .
Φ ^ k j = 0 P 1 P 2 1 exp [ i ϕ j ( 1 ) ] exp [ i ϕ j ( 2 ) ] exp ( 2 π i j k P 1 P 2 ) .
Φ ^ k = 1 P 1 P 2 j = 0 P 1 P 2 1 k 1 = 0 P 1 1 k 2 = 0 P 2 1 Φ ^ k 1 ( 1 ) Φ ^ k 2 ( 2 ) exp ( 2 π i j P 2 k 1 + P 1 k 2 P 1 P 2 ) exp ( 2 π i j k P 1 P 2 ) ,
j = 0 P 1 P 2 1 exp ( 2 π i j P 2 k 1 + p 1 k 2 P 1 P 2 ) exp ( 2 π i j k P 1 P 2 ) = { P 1 P 2 for ( P 2 k 1 + P 1 k 2 ) mod P 1 P 2 = k 0  otherwise .
( P 2 k 1 + P 1 k 2 ) mod P 1 P 2 = k
( P 2 k 1 + P 1 k 2 ) mod P 1 P 2 = k .
P 2 k 1 + P 1 k 2 k = a P 1 P 2
P 2 k 1 + P 1 k 2 k = b P 1 P 2 ,
P 2 ( k 1 k 1 ) + P 1 [ k 2 k 2 + ( b a ) P 2 ] = 0 .
Φ ^ k = Φ ^ k 1 ( 1 ) Φ ^ k 2 ( 2 ) ,

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