Abstract

Frequency fluctuations of lasers cause a broadening of their line shapes. Although the relation between the frequency noise spectrum and the laser line shape has been studied extensively, no simple expression exists to evaluate the laser linewidth for frequency noise spectra that does not follow a power law. We present a simple approach to this relation with an approximate formula for evaluation of the laser linewidth that can be applied to arbitrary noise spectral densities.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Zhu and J. L. Hall, “Stabilization of optical phase/frequency of a laser system: application to a commercial dye laser with an external stabilizer,” J. Opt. Soc. Am. B 10, 802–816 (1993).
    [CrossRef]
  2. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
    [CrossRef]
  3. L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the AURIGA detector optical readout,” J. Opt. Soc. Am. B 20, 462–468 (2003).
    [CrossRef]
  4. M. Heurs, V. M. Quetschke, B. Willke, K. Danzmann, and I. Freitag, “Simultaneously suppressing frequency and intensity noise in a Nd:YAG nonplanar ring oscillator by means of the current-lock technique,” Opt. Lett. 29, 2148–2150 (2004).
    [CrossRef] [PubMed]
  5. J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
    [CrossRef]
  6. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
    [CrossRef]
  7. F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
    [CrossRef]
  8. I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
    [CrossRef] [PubMed]
  9. S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
    [CrossRef] [PubMed]
  10. D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
    [CrossRef]
  11. P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).
    [CrossRef]
  12. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
    [CrossRef]
  13. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  14. G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
    [CrossRef]
  15. J.-P. Tourrenc, “Caractérisation et modélisation du bruit d’amplitude optique, du bruit de fréquence et de la largeur de raie de VCSELs monomode,” Ph.D. dissertation (Université de Montpellier II, 2005).
  16. L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9, 485–493 (1991).
    [CrossRef]
  17. K. Kikuchi, “Effect of llf-type fm noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25, 684–688 (1989).
    [CrossRef]
  18. A. Godone and F. Levi, “About the radiofrequency spectrum of a phase noise-modulated carrier,” in Proceedings of the 12th European Frequency and Time Forum (1998), pp. 392–396.
  19. A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
    [CrossRef]
  20. S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
    [CrossRef]
  21. The modulation index β is defined as the ratio of the frequency deviation Δf over the modulation frequency fm, i.e., β=Δf/fm.

2010

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

2009

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

2008

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
[CrossRef]

2005

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

J.-P. Tourrenc, “Caractérisation et modélisation du bruit d’amplitude optique, du bruit de fréquence et de la largeur de raie de VCSELs monomode,” Ph.D. dissertation (Université de Montpellier II, 2005).

2004

2003

2002

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

1999

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

1998

A. Godone and F. Levi, “About the radiofrequency spectrum of a phase noise-modulated carrier,” in Proceedings of the 12th European Frequency and Time Forum (1998), pp. 392–396.

1993

1991

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9, 485–493 (1991).
[CrossRef]

1989

K. Kikuchi, “Effect of llf-type fm noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25, 684–688 (1989).
[CrossRef]

1984

P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).
[CrossRef]

1982

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[CrossRef]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

1958

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Acernese, F.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Akikusa, N.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

Alnis, J.

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Alshourbagy, M.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Antonucci, F.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Aoudia, S.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Arun, K. G.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Astone, P.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Ballardin, G.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Barone, F.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Barsotti, L.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Barsuglia, M.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bartalini, S.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

Bauer, T. S.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bergquist, J. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

Besnard, P.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

Bigotta, S.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Birindelli, S.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bizouard, M. A.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Blin, S.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

Boccara, C.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bondu, F.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bonelli, L.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Borri, S.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Bosi, L.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Braccini, S.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bradaschia, C.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Brillet, A.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Brisson, V.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Bulten, H. J.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Buskulic, D.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Cagnoli, G.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Calloni, E.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Campagna, E.

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

Cancio, P.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

Castrillo, A.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Conti, L.

Cruz, F. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

Danzmann, K.

De Natale, P.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Debarge, G.

P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).
[CrossRef]

di Sopra, F. M.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Edamura, T.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

Elliott, D. S.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[CrossRef]

Freitag, I.

Fujita, K.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

Gabrysch, M.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Galli, I.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

Gallion, P. B.

P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).
[CrossRef]

Gianfrani, L.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Giusfredi, G.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

Godone, A.

A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
[CrossRef]

A. Godone and F. Levi, “About the radiofrequency spectrum of a phase noise-modulated carrier,” in Proceedings of the 12th European Frequency and Time Forum (1998), pp. 392–396.

Gulden, K. H.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Hall, J. L.

Hänsch, T. W.

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Henry, C.

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Heurs, M.

Itano, W. M.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

Kan, H.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

Kikuchi, K.

K. Kikuchi, “Effect of llf-type fm noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25, 684–688 (1989).
[CrossRef]

Kolachevsky, N.

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Levi, F.

A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
[CrossRef]

A. Godone and F. Levi, “About the radiofrequency spectrum of a phase noise-modulated carrier,” in Proceedings of the 12th European Frequency and Time Forum (1998), pp. 392–396.

Marin, F.

L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the AURIGA detector optical readout,” J. Opt. Soc. Am. B 20, 462–468 (2003).
[CrossRef]

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Matveev, A.

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Mazzotti, D.

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

I. Galli, S. Bartalini, P. Cancio, G. Giusfredi, D. Mazzotti, and P. D. Natale, “Ultra-stable, widely tunable and absolutely linked mid-IR coherent source,” Opt. Express 17, 9582–9587(2009).
[CrossRef] [PubMed]

Mercer, L. B.

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9, 485–493 (1991).
[CrossRef]

Micalizio, S.

A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
[CrossRef]

Moser, M.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Natale, P. D.

Quetschke, V. M.

Rosa, M. D.

Roy, R.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[CrossRef]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Smith, S. J.

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[CrossRef]

Stéphan, G. M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

Tam, T. T.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

Têtu, M.

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

Tourrenc, J.-P.

J.-P. Tourrenc, “Caractérisation et modélisation du bruit d’amplitude optique, du bruit de fréquence et de la largeur de raie de VCSELs monomode,” Ph.D. dissertation (Université de Montpellier II, 2005).

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Udem, T.

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Viciani, S.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Willke, B.

Yamanishi, M.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

Young, B. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

Zhu, M.

IEEE J. Quantum Electron.

P. B. Gallion and G. Debarge, “Quantum phase noise and field correlation in single frequency semiconductor laser systems,” IEEE J. Quantum Electron. 20, 343–350 (1984).
[CrossRef]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

K. Kikuchi, “Effect of llf-type fm noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25, 684–688 (1989).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: Hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron. 44, 12(2008).
[CrossRef]

J. Lightwave Technol.

L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9, 485–493 (1991).
[CrossRef]

J. Opt. Soc. Am. B

Metrologia

A. Godone, S. Micalizio, and F. Levi, “Rf spectrum of a carrier with a random phase modulation of arbitrary slope,” Metrologia 45, 313–324 (2008).
[CrossRef]

Opt. Commun.

S. Viciani, M. Gabrysch, F. Marin, F. M. di Sopra, M. Moser, and K. H. Gulden, “Line shape of a vertical cavity surface emitting laser,” Opt. Commun. 206, 89–97 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112, 1940–1949 (1958).
[CrossRef]

Phys. Rev. A

D. S. Elliott, R. Roy, and S. J. Smith, “Extracavity laser band shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[CrossRef]

G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71, 043809 (2005).
[CrossRef]

F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia, K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Barsotti, M. Barsuglia, T. S. Bauer, S. Bigotta, S. Birindelli, M. A. Bizouard, C. Boccara, F. Bondu, L. Bonelli, L. Bosi, S. Braccini, C. Bradaschia, A. Brillet, V. Brisson, H. J. Bulten, D. Buskulic, G. Cagnoli, E. Calloni, and E. Campagna, “Laser with an in-loop relative frequency stability of 1.0×10−21 on a 100 ms time scale for gravitational-wave detection,” Phys. Rev. A 79, 053824 (2009).
[CrossRef]

J. Alnis, A. Matveev, N. Kolachevsky, T. Udem, and T. W. Hänsch, “Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry–Pérot cavities,” Phys. Rev. A 77, 053809 (2008).
[CrossRef]

Phys. Rev. Lett.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82, 3799–3802 (1999).
[CrossRef]

S. Bartalini, S. Borri, P. Cancio, A. Castrillo, I. Galli, G. Giusfredi, D. Mazzotti, L. Gianfrani, and P. De Natale, “Observing the intrinsic linewidth of a quantum-cascade laser: beyond the Schawlow–Townes limit,” Phys. Rev. Lett. 104, 083904 (2010).
[CrossRef] [PubMed]

Other

J.-P. Tourrenc, “Caractérisation et modélisation du bruit d’amplitude optique, du bruit de fréquence et de la largeur de raie de VCSELs monomode,” Ph.D. dissertation (Université de Montpellier II, 2005).

A. Godone and F. Levi, “About the radiofrequency spectrum of a phase noise-modulated carrier,” in Proceedings of the 12th European Frequency and Time Forum (1998), pp. 392–396.

The modulation index β is defined as the ratio of the frequency deviation Δf over the modulation frequency fm, i.e., β=Δf/fm.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Numerical calculation of the laser line shape S E ( δ ν ) for a fixed frequency noise level h 0 = 1 Hz 2 / Hz and different values of the cutoff frequency: a, f c = 0.03 Hz ; b, f c = 0.3 Hz ; c, f c = 3 Hz ; and d, f c = 30 Hz . The line shapes are normalized to help the comparison of their full width at half- maximum (FWHM). The line shape evolves from a Gaussian when f c h 0 and to a Lorentzian when f c h 0 .

Fig. 2
Fig. 2

Upper graph: Numerical computation showing the evolution of the linewidth (FWHM) with the cutoff frequency f c in the case of low-pass filtered white noise. The dots have been calculated by numerical integration of the exact relations Eqs. (1, 2). The continuous line is given by our approximate formula Eq. (7). Both horizontal and vertical scales have been normalized to the noise level h 0 . The behavior at low and high cutoff frequencies is indicated by the asymptotic response (red dashed lines). Lower graph: Relative error between the exact and approximate values.

Fig. 3
Fig. 3

A typical laser frequency noise spectral density composed of flicker noise at low frequencies and white noise at high frequencies. The dashed line given by S δ ν ( f ) = 8 ln ( 2 ) f / π 2 separates the spectrum into two regions whose contributions to the laser line shape is very different: the high modulation index area contributes to the linewidth, whereas the low modulation index area contributes only to the wings of the line shape (see the text for details).

Fig. 4
Fig. 4

Pure flicker frequency noise model of Eq. (11) with α = 1 , 1.2, 1.5, 1.7, and 2.0. The axes are normalized with respect to the frequency f m , at which S δ ν intersects the β-separation line.

Fig. 5
Fig. 5

Evolution of the laser linewidth with respect to the measurement time in the case of a frequency noise spectrum composed of flicker noise as shown in Fig. 4. The dots have been obtained by numerical integration of the exact relation between the frequency noise and the line shape given by Eqs. (1, 2). The red lines are the values given by the approximate formulas Eqs. (12, 13).

Fig. 6
Fig. 6

Frequency noise model used to study laser linewidth reduction using a servo loop. We assume that the free-running laser noise level h b = 1000 Hz 2 / Hz is reduced to h a = 100 Hz 2 / Hz with a servo loop having a bandwidth f b of a, 100 Hz ; b, 300 Hz ; c, 500 Hz ; and d, 1500 Hz . The dashed line represents the β-separation line. The minimum servo-loop bandwidth necessary to efficiently reduce the laser linewidth is f b min = π 2 h b / ( 8 ln ( 2 ) ) .

Fig. 7
Fig. 7

Evolution of the laser linewidth (FWHM) with the servo-loop bandwidth f b for the frequency noise model presented in Fig. 6. Special values of the servo bandwidth, for which the line shape is represented in Fig. 8, are indicated by the following points: a, f b = 100 Hz ; b, f b = 300 Hz ; c, f b = 500 Hz ; and d, f b = 1500 Hz . The continuous line has been obtained by numerical integration of the exact relation Eqs. (1, 2), and the dashed line has been obtained with our approximate formula Eqs. (9, 10).

Fig. 8
Fig. 8

Evolution of the laser line shape with the servo-loop bandwidth for the frequency noise model presented in Fig. 6. We chose the following values of the servo bandwidth: a, f b = 100 Hz ; b, f b = 300 Hz ; c, f b = 500 Hz ; and d, f b = 1500 Hz , which correspond to the points indicated in Fig. 7.

Equations (14)

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

Γ E ( τ ) = E 0 2 e i 2 π ν 0 τ e 2 0 S δ ν ( f ) sin 2 ( π f τ ) f 2 d f ,
S E ( ν ) = 2 e i 2 π ν τ Γ E ( τ ) d τ .
S δ ν ( f ) = { h 0 if     f f c 0 if    f > f c .
Γ E ( τ ) = E 0 2 e i 2 π ν 0 τ e 2 h 0 f c ( sin 2 ( π f c τ ) π f c τ Si ( 2 π f c τ ) ) ,
S E ( ν ) = E 0 2 h 0 ( ν ν 0 ) 2 + ( π h 0 / 2 ) 2 ,
S E ( ν ) = E 0 2 ( 2 π h 0 f c ) 1 / 2 e ( ν ν 0 ) 2 2 h 0 f c ,
FWHM = h 0 ( 8 ln ( 2 ) f c / h 0 ) 1 / 2 [ 1 + ( 8 ln ( 2 ) π 2 f c h 0 ) 2 ] 1 / 4 ,
f c * = π 2 8 ln ( 2 ) h 0 1.78 h 0 .
FWHM = ( 8 ln ( 2 ) A ) 1 / 2 ,
A = 1 / T o H ( S δ ν ( f ) 8 ln ( 2 ) f / π 2 ) S δ ν ( f ) d f ,
S δ ν ( f ) f m = 8 ln ( 2 ) π 2 ( f f m ) α ,
FWHM = f m 8 ln ( 2 ) π [ ln ( f m T o ) ] 1 / 2 ,
FWHM = f m 8 ln ( 2 ) π [ ( f m T o ) α 1 1 α 1 ] 1 / 2 .
Γ E ( τ ) = E 0 2 e i 2 π ν 0 τ e h b π 2 | τ | h a h b f b ( ω b τ Si ( ω b τ ) 2 sin 2 ( ω b τ 2 ) ) ,

Metrics