Abstract

We derive a delay-differential equation model that describes continuous-wave (cw) or passively Q-switched (PQS) two-frequency solid-state lasers submitted to frequency-shifted feedback (FSF). The study focuses on the locking of the beat note between the two free-running laser frequencies to a reference external frequency. The locking domain is obtained analytically in the cw regime. The PQS regime is treated by adding a saturable absorber population in the model equations. In this case, numerical simulations permit us to evaluate a locking range that is smaller than in the cw case. We find good agreement between the theoretical predictions and experiments carried out with a cw diode-pumped dual-polarization Nd:YAG laser as well as with previously published experimental results obtained with cw Er:Yb:glass [Opt. Lett. 32, 1099 (2007)] and PQS Nd:YAG [Opt. Lett. 33, 2524 (2008)] lasers. Applications of the FSF locking technique include the lidar–radar technique, for which a highly coherent beat note is required.

© 2011 Optical Society of America

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References

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  1. K. Otsuka and H. Kawaguchi, “Period-doubling bifurcations in detuned lasers with injected signals,” Phys. Rev. A 29, 2953–2956 (1984).
    [CrossRef]
  2. K. Otsuka, “Ultrahigh sensitivity laser Doppler velocimetry with a microchip solid-state laser,” Appl. Opt. 33, 1111–1114(1994).
    [PubMed]
  3. E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
    [CrossRef]
  4. P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
    [CrossRef]
  5. L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
    [CrossRef]
  6. F. V. Kowalski, P. D. Hale, and S. J. Shattil, “Broadband continuous-wave laser,” Opt. Lett. 13, 622–624 (1988).
    [CrossRef] [PubMed]
  7. H. Guillet de Chatellus and J. P. Pique, “Statistical properties of frequency shifted feedback lasers,” Opt. Commun. 283, 71–77 (2010).
    [CrossRef]
  8. H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12, 1360–1368 (1994).
    [CrossRef]
  9. L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
    [CrossRef] [PubMed]
  10. M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
    [CrossRef] [PubMed]
  11. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science, 1986).
  13. M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
    [CrossRef]
  14. A. Le Floch and G. Stephan, “La condition de résonance dans les lasers anisotropes contenant des lames biréfringentes,” C. R. Acad. Sci. B 277, 265–268 (1973).
  15. M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
    [CrossRef]
  16. K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
    [CrossRef] [PubMed]
  17. N. D. Lai, F. Bretenaker, and M. Brunel, “Coherence of pulsed microwave signals carried by two-frequency solid-state lasers,” J. Lightwave Technol. 21, 3037–3042 (2003).
    [CrossRef]
  18. L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
    [CrossRef]
  19. L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712(2002).
    [CrossRef] [PubMed]
  20. R. Diaz, S.-C. Chan, and J.-M. Liu, “Lidar detection using a dual-frequency source,” Opt. Lett. 31, 3600–3602 (2006).
    [CrossRef] [PubMed]
  21. T. Erneux, Applied Delay Differential Equations (Springer,2009).

2010

H. Guillet de Chatellus and J. P. Pique, “Statistical properties of frequency shifted feedback lasers,” Opt. Commun. 283, 71–77 (2010).
[CrossRef]

2008

2007

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
[CrossRef] [PubMed]

2006

2003

2002

2001

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

1999

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

1997

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

1995

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

1994

K. Otsuka, “Ultrahigh sensitivity laser Doppler velocimetry with a microchip solid-state laser,” Appl. Opt. 33, 1111–1114(1994).
[PubMed]

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12, 1360–1368 (1994).
[CrossRef]

1992

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

1988

1984

K. Otsuka and H. Kawaguchi, “Period-doubling bifurcations in detuned lasers with injected signals,” Phys. Rev. A 29, 2953–2956 (1984).
[CrossRef]

1980

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

1973

A. Le Floch and G. Stephan, “La condition de résonance dans les lasers anisotropes contenant des lames biréfringentes,” C. R. Acad. Sci. B 277, 265–268 (1973).

Besesty, P.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

Bielawski, S.

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

Bretenaker, F.

N. D. Lai, F. Bretenaker, and M. Brunel, “Coherence of pulsed microwave signals carried by two-frequency solid-state lasers,” J. Lightwave Technol. 21, 3037–3042 (2003).
[CrossRef]

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712(2002).
[CrossRef] [PubMed]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Brinkmeyer, E.

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12, 1360–1368 (1994).
[CrossRef]

Brunel, M.

M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
[CrossRef] [PubMed]

N. D. Lai, F. Bretenaker, and M. Brunel, “Coherence of pulsed microwave signals carried by two-frequency solid-state lasers,” J. Lightwave Technol. 21, 3037–3042 (2003).
[CrossRef]

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712(2002).
[CrossRef] [PubMed]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Chan, S.-C.

Chartier, G.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

Contarino, V. M.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

Day, R.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

de Chatellus, H. Guillet

H. Guillet de Chatellus and J. P. Pique, “Statistical properties of frequency shifted feedback lasers,” Opt. Commun. 283, 71–77 (2010).
[CrossRef]

Derozier, D.

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

Diaz, R.

Dolfi, D.

Emile, O.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Erneux, T.

T. Erneux, Applied Delay Differential Equations (Springer,2009).

Ferrand, B.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Fulbert, L.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Gilles, H.

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
[CrossRef] [PubMed]

Girard, S.

L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
[CrossRef] [PubMed]

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

Glorieux, P.

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

Hale, P. D.

Herczfeld, P. R.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

Huignard, J.-P.

Kawaguchi, H.

K. Otsuka and H. Kawaguchi, “Period-doubling bifurcations in detuned lasers with injected signals,” Phys. Rev. A 29, 2953–2956 (1984).
[CrossRef]

Kervevan, L.

L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
[CrossRef] [PubMed]

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Kowalski, F. V.

Lacot, E.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

Lai, N. D.

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Laroche, M.

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

L. Kervevan, H. Gilles, S. Girard, and M. Laroche, “Beat-note jitter suppression in a dual-frequency laser using optical feedback,” Opt. Lett. 32, 1099–1101 (2007).
[CrossRef] [PubMed]

Le Floch, A.

L. Morvan, N. D. Lai, D. Dolfi, J.-P. Huignard, M. Brunel, F. Bretenaker, and A. Le Floch, “Building blocks for a two-frequency laser lidar-radar: a preliminary study,” Appl. Opt. 41, 5702–5712(2002).
[CrossRef] [PubMed]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

A. Le Floch and G. Stephan, “La condition de résonance dans les lasers anisotropes contenant des lames biréfringentes,” C. R. Acad. Sci. B 277, 265–268 (1973).

Leprince, P.

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

Liu, J.-M.

Mandel, P.

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

Marty, J.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Molva, E.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Morvan, L.

Mullen, L. J.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

Nerin, P.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

Otsuka, K.

K. Otsuka, “Ultrahigh sensitivity laser Doppler velocimetry with a microchip solid-state laser,” Appl. Opt. 33, 1111–1114(1994).
[PubMed]

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

K. Otsuka and H. Kawaguchi, “Period-doubling bifurcations in detuned lasers with injected signals,” Phys. Rev. A 29, 2953–2956 (1984).
[CrossRef]

Pique, J. P.

H. Guillet de Chatellus and J. P. Pique, “Statistical properties of frequency shifted feedback lasers,” Opt. Commun. 283, 71–77 (2010).
[CrossRef]

Puget, P.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

Sabert, H.

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12, 1360–1368 (1994).
[CrossRef]

Shattil, S. J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Stephan, G.

A. Le Floch and G. Stephan, “La condition de résonance dans les lasers anisotropes contenant des lames biréfringentes,” C. R. Acad. Sci. B 277, 265–268 (1973).

Stoeckel, F.

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

Vallet, M.

M. Brunel and M. Vallet, “Pulse-to-pulse coherent beat note generated by a passively Q-switched two-frequency laser,” Opt. Lett. 33, 2524–2526 (2008).
[CrossRef] [PubMed]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Vieira, A. J. C.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

Appl. Opt.

Appl. Phys. B

L. Kervevan, H. Gilles, S. Girard, M. Laroche, and P. Leprince, “Self-mixing laser Doppler velocimetry with a dual-polarization Yb:Er glass laser,” Appl. Phys. B 86, 169–176(2007).
[CrossRef]

C. R. Acad. Sci. B

A. Le Floch and G. Stephan, “La condition de résonance dans les lasers anisotropes contenant des lames biréfringentes,” C. R. Acad. Sci. B 277, 265–268 (1973).

Electron. Lett.

P. Nerin, P. Puget, P. Besesty, and G. Chartier, “Self-mixing using a dual-polarisation Nd:YAG microchip laser,” Electron. Lett. 33, 491–492 (1997).
[CrossRef]

IEEE J. Quantum Electron.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

L. J. Mullen, A. J. C. Vieira, P. R. Herczfeld, and V. M. Contarino, “Application of RADAR technology to aerial LIDAR systems for enhancement of shallow underwater target detection,” IEEE Trans. Microwave Theory Tech. 43, 2370–2377(1995).
[CrossRef]

J. Lightwave Technol.

H. Sabert and E. Brinkmeyer, “Pulse generation in fiber lasers with frequency shifted feedback,” J. Lightwave Technol. 12, 1360–1368 (1994).
[CrossRef]

N. D. Lai, F. Bretenaker, and M. Brunel, “Coherence of pulsed microwave signals carried by two-frequency solid-state lasers,” J. Lightwave Technol. 21, 3037–3042 (2003).
[CrossRef]

Opt. Commun.

H. Guillet de Chatellus and J. P. Pique, “Statistical properties of frequency shifted feedback lasers,” Opt. Commun. 283, 71–77 (2010).
[CrossRef]

Opt. Lett.

Opt. Rev.

M. Brunel, O. Emile, F. Bretenaker, A. Le Floch, B. Ferrand, and E. Molva, “Tunable two-frequency lasers for lifetime measurements,” Opt. Rev. 4, 550–552 (1997).
[CrossRef]

Phys. Rev. A

K. Otsuka, P. Mandel, S. Bielawski, D. Derozier, and P. Glorieux, “Alternate time scale in multimode lasers,” Phys. Rev. A 46, 1692–1695 (1992).
[CrossRef] [PubMed]

K. Otsuka and H. Kawaguchi, “Period-doubling bifurcations in detuned lasers with injected signals,” Phys. Rev. A 29, 2953–2956 (1984).
[CrossRef]

E. Lacot, R. Day, and F. Stoeckel, “Coherent laser detection by frequency-shifted optical feedback,” Phys. Rev. A 64, 043815(2001).
[CrossRef]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q switched by a Cr4+:yttrium aluminum garnet absorber,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science, 1986).

T. Erneux, Applied Delay Differential Equations (Springer,2009).

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Figures (7)

Fig. 1
Fig. 1

Schematic of the experimental setup. Two-frequency laser: M 1 and M 2 , cavity mirrors; AM, active medium; QWP 1 , 2 , quarter-wave plates; E x , y , cross-polarized intracavity fields; ν x , y , eigenfrequencies; SA, saturable absorber (pulsed regime only). Reinjection arm: AO, acousto-optic frequency shifter driven at frequency f AO ; PR, polarization rotator; M 3 , mirror; L m : matching lens; τ, feedback round-trip time. Output: P, polarizer, E out , output field.

Fig. 2
Fig. 2

(a) Schematic of a pulse train (repetition rate f rep ) carrying a beat note locked to 2 f AO . (b) Schematic of the output RF spectra corresponding to (left) an incoherent beat note at f b delivered by a free-running laser, or to (right) a fully coherent beat note locked to f b = 2 f AO .

Fig. 3
Fig. 3

Typical experimental spectral analysis of the output pulse train power. Resolution bandwidth ( RBW ) = 30 kHz . (a) Unlocked beat note, i.e., Δ ν > γ e / 2 π . (b) Locked beat note, i.e., Δ ν < γ e / 2 π . Inset, RBW = 1 Hz .

Fig. 4
Fig. 4

Experimental cw locking range as a function of the diffraction efficiency of the AO.

Fig. 5
Fig. 5

Output pulse intensity evolution versus time when f rep = 10 kHz and Δ ν = 0 . (a) Experimental pulse. (b) Calculated pulse with the parameters g = 0.3 , R 3 = 0.4 , and η = 3 .

Fig. 6
Fig. 6

Experimental spectral analysis of the output power for two different frequency spans. f AO = 92.5 MHz . (a), (b) Unlocked regime. (c), (d) Locked regime. (a), (c)  RBW = 1 MHz . (b), (d)  RBW = 300 Hz .

Fig. 7
Fig. 7

Calculated intensity and phase difference time evolution for three detunings. (a)  Δ ν = 10 MHz . (b)  Δ ν = 1 MHz . (c)  Δ ν = 100 kHz . The parameters are g = 1 , R 3 = 0.1 , η = 3 , leading to γ e / 2 π = 2.2 MHz .

Equations (11)

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n ˙ x , y = γ / / P x , y [ γ / / + ζ ( | E x , y | 2 + β | E y , x | 2 ) ] n x , y + n ˜ x , y ,
E ˙ x = { i 2 π ν x + [ Γ x + κ ( n x + β n y ) ] / 2 } E x + E ˜ x ,
E ˙ y = { i 2 π ν y + [ Γ y + κ ( n y + β n x ) ] / 2 } E y + E ˜ y + γ e E x ( t τ ) e i ( 4 π f AO t + ψ ) .
n ˙ x , y = γ / / η n 0 ( γ / / + | E x , y | 2 + β | E y , x | 2 ) n x , y + n ˜ x , y ,
E ˙ x = ( Γ + n x + β n y ) E x / 2 + E ˜ x ,
E ˙ y = ( Γ + n y + β n x ) E y / 2 + i 2 π Δ ν E y + E ˜ y + γ e E x ( t τ ) .
ϕ ˙ y = 2 π Δ ν γ e | E x ( t τ ) E y ( t ) | sin [ ϕ y ( t ) ϕ x ( t τ ) ] .
γ e / 2 π Δ ν γ e / 2 π .
E ˙ x = ( Γ + n x + β n y a x ) E x / 2 + E ˜ x ,
E ˙ y = ( Γ + n y + β n x a y ) E y / 2 + i 2 π Δ ν E y + E ˜ y + γ e E x ( t τ ) .
a ˙ = γ a a 0 [ γ a + μ 0 ( 1 + C a ) ( | E x | 2 + | E y | 2 ) / 2 ] a ,

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