Abstract

A method of phase locking two CO2 lasers by radiation exchange is presented. This phase-locking was achieved by use of a copper prism as a beam folding device in the resonators and extraction of the output radiation by a common output coupler. Energy exchange led to a phase-locked state if several locking conditions were fulfilled. The amount of radiation injected from one resonator to the second cavity could be adjusted by movement of the prism. The influence of the strength of coupling on the locking range was studied. The beat signal between the two unlocked lasers could be measured, whereas in the case of phase-locked operation twice the intensity was detected. Despite the inclusion of several assumptions, a simplified mathematical model delivered good agreement between calculated and experimental results.

© 2001 Optical Society of America

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  1. H. L. Stover, W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. 8, 91–93 (1966).
    [CrossRef]
  2. M. B. Spencer, W. E. Lamb, “Theory of two coupled laser,” Phys. Rev. A 2, 893–898 (1972).
    [CrossRef]
  3. W. J. Fader, “Theory of two coupled lasers,” IEEE J. Quantum Electron. QE-21, 1838–1844 (1985).
    [CrossRef]
  4. V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
    [CrossRef]
  5. V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
    [CrossRef]
  6. D. G. Youmans, “Phase locking of adjacent channel leaky waveguide CO2 lasers,” J. Appl. Phys. Lett. 44, 365–367 (1984).
    [CrossRef]
  7. R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
    [CrossRef]
  8. K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
    [CrossRef]
  9. K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
    [CrossRef]
  10. K. Schröder, A. Müller, D. Schuöcker, “Phase locking of lasers through diffraction effects,” Appl. Opt. 34, 8252–8259 (1995).
    [CrossRef] [PubMed]
  11. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
    [CrossRef]
  12. L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).
  13. V. V. Likhanskii, A. P. Napartovich, “Radiation emitted by optically coupled lasers,” Sov. Phys. Usp. 33, 228–252 (1990).
    [CrossRef]
  14. A. E. Siegman, Lasers (University Science, Sausalito, Calif. (1986), p. 1129.
  15. K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
    [CrossRef]
  16. C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
    [CrossRef]
  17. W. Weingartner, “Phase locking of CO2 lasers,” Ph.D. dissertation (Vienna University of Technology, Vienna, Austria, 2000).
  18. W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
    [CrossRef]
  19. W. J. Witteman, The CO2 Laser, Vol. 53 of Springer Series on Optical Sciences (Springer-Verlag, Berlin, 1987).
    [CrossRef]
  20. G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
    [CrossRef]
  21. MathWorks “Optimization Toolbox for Mathlab,” handbook MathWorks, 3 Apple Hill Drive, Natick, Mass. 01760, 1998), Sec. 3–3.
  22. R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
    [CrossRef]

2000 (1)

W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
[CrossRef]

1999 (1)

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

1998 (1)

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

1995 (1)

1992 (4)

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

1990 (1)

V. V. Likhanskii, A. P. Napartovich, “Radiation emitted by optically coupled lasers,” Sov. Phys. Usp. 33, 228–252 (1990).
[CrossRef]

1988 (1)

R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
[CrossRef]

1987 (2)

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

1985 (1)

W. J. Fader, “Theory of two coupled lasers,” IEEE J. Quantum Electron. QE-21, 1838–1844 (1985).
[CrossRef]

1984 (1)

D. G. Youmans, “Phase locking of adjacent channel leaky waveguide CO2 lasers,” J. Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

1973 (2)

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[CrossRef]

C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[CrossRef]

1972 (1)

M. B. Spencer, W. E. Lamb, “Theory of two coupled laser,” Phys. Rev. A 2, 893–898 (1972).
[CrossRef]

1966 (1)

H. L. Stover, W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. 8, 91–93 (1966).
[CrossRef]

Abramski, K. M.

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

Antyukhov, V. V.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
[CrossRef]

Apollonov, V. V.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Baker, H. J.

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

Bartelt-Berger, L.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Becker, U.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Bourdet, G. L.

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

Brauch, U.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Buczek, C. J.

C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[CrossRef]

Byrd, R. H.

R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
[CrossRef]

Cantor, A. J.

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

Colletto, C.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Colley, A. D.

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

Dan’shchikov, E. V.

V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
[CrossRef]

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Derzhavin, S. I.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Elkin, N. N.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Fader, W. J.

W. J. Fader, “Theory of two coupled lasers,” IEEE J. Quantum Electron. QE-21, 1838–1844 (1985).
[CrossRef]

Fleig, C.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Freiberg, R. J.

C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[CrossRef]

Giesen, A.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Hall, D. R.

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

Hart, R. A.

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

Hergenhan, G.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Hofflinger, B.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Hoppe, J.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Huegel, H.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Kazakov, A. A.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Kennedy, J. T.

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

Kislov, V. I.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Korotkov, V. A.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Koval, Y. P.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Kurokawa, K.

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[CrossRef]

Kuzminov, V. V.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Lamb, W. E.

M. B. Spencer, W. E. Lamb, “Theory of two coupled laser,” Phys. Rev. A 2, 893–898 (1972).
[CrossRef]

Lebedev, F. V.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Likhanskii, V. V.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

V. V. Likhanskii, A. P. Napartovich, “Radiation emitted by optically coupled lasers,” Sov. Phys. Usp. 33, 228–252 (1990).
[CrossRef]

Lucke, B.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Mashkovskii, D. A.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Masyukov, I. V.

V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
[CrossRef]

Menschig, A.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Muller, R. A.

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

Müller, A.

Mullot, G. M.

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

Napartovich, A. P.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

V. V. Likhanskii, A. P. Napartovich, “Radiation emitted by optically coupled lasers,” Sov. Phys. Usp. 33, 228–252 (1990).
[CrossRef]

Newman, L. A.

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

Opower, H.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Pis’mennyi, V. D.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Prokhorov, A. M.

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Scharl, S.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Schnabel, R. B.

R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
[CrossRef]

Schomburg, C.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Schröder, K.

W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
[CrossRef]

K. Schröder, A. Müller, D. Schuöcker, “Phase locking of lasers through diffraction effects,” Appl. Opt. 34, 8252–8259 (1995).
[CrossRef] [PubMed]

Schubert, M.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Schuöcker, D.

W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
[CrossRef]

K. Schröder, A. Müller, D. Schuöcker, “Phase locking of lasers through diffraction effects,” Appl. Opt. 34, 8252–8259 (1995).
[CrossRef] [PubMed]

Shultz, G. A.

R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Sausalito, Calif. (1986), p. 1129.

Skolnick, M. L.

C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[CrossRef]

Spencer, M. B.

M. B. Spencer, W. E. Lamb, “Theory of two coupled laser,” Phys. Rev. A 2, 893–898 (1972).
[CrossRef]

Springer, R.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Steier, W. H.

H. L. Stover, W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. 8, 91–93 (1966).
[CrossRef]

Steudle, D.

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Stover, H. L.

H. L. Stover, W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. 8, 91–93 (1966).
[CrossRef]

Troshchiev, V. E.

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

Vinet, J. Y.

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

Weingartner, W.

W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
[CrossRef]

W. Weingartner, “Phase locking of CO2 lasers,” Ph.D. dissertation (Vienna University of Technology, Vienna, Austria, 2000).

Witteman, W. J.

W. J. Witteman, The CO2 Laser, Vol. 53 of Springer Series on Optical Sciences (Springer-Verlag, Berlin, 1987).
[CrossRef]

Youmans, D. G.

D. G. Youmans, “Phase locking of adjacent channel leaky waveguide CO2 lasers,” J. Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

G. L. Bourdet, R. A. Muller, G. M. Mullot, J. Y. Vinet, “Cw injection phase locking in homogeneously broadened media. II. Phase locking and tunability of CO2 waveguide lasers,” Appl. Phys. B 43, 273–279 (1987).
[CrossRef]

Appl. Phys. Lett. (3)

H. L. Stover, W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. 8, 91–93 (1966).
[CrossRef]

R. A. Hart, L. A. Newman, A. J. Cantor, J. T. Kennedy, “Staggered hollow-bore CO2 waveguide laser array,” Appl. Phys. Lett. 51, 1057–1059 (1987).
[CrossRef]

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. J. Fader, “Theory of two coupled lasers,” IEEE J. Quantum Electron. QE-21, 1838–1844 (1985).
[CrossRef]

J. Appl. Phys. Lett. (1)

D. G. Youmans, “Phase locking of adjacent channel leaky waveguide CO2 lasers,” J. Appl. Phys. Lett. 44, 365–367 (1984).
[CrossRef]

Laser Optoelectron. (1)

L. Bartelt-Berger, U. Becker, U. Brauch, C. Colletto, C. Fleig, A. Giesen, G. Hergenhan, B. Hofflinger, J. Hoppe, H. Huegel, B. Lucke, A. Menschig, H. Opower, S. Scharl, C. Schomburg, M. Schubert, R. Springer, D. Steudle, “Single-mode fiber-coupled semiconductor laser systems for direct applications,” Laser Optoelectron. 31(1), 58–65 (1999).

Math. Progr. (1)

R. H. Byrd, R. B. Schnabel, G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Progr. 40, 247–263 (1988).
[CrossRef]

Opt. Commun. (1)

K. M. Abramski, H. J. Baker, A. D. Colley, D. R. Hall, “Phase-locked operation of intersecting CO2 waveguide lasers by four-wave mixing,” Opt. Commun. 90, 61–64 (1992).
[CrossRef]

Phys. Rev. A (1)

M. B. Spencer, W. E. Lamb, “Theory of two coupled laser,” Phys. Rev. A 2, 893–898 (1972).
[CrossRef]

Proc. IEEE (2)

K. Kurokawa, “Injection locking of microwave solid-state oscillators,” Proc. IEEE 61, 1386–1410 (1973).
[CrossRef]

C. J. Buczek, R. J. Freiberg, M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61, 1411–1431 (1973).
[CrossRef]

Quantum Electron. (1)

V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Y. P. Koval, V. V. Kuzminov, D. A. Mashkovskii, A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum Electron. 28, 344–346 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

W. Weingartner, K. Schröder, D. Schuöcker, “Active length control of two phase locked CO2 lasers with a digital signal processor,” Rev. Sci. Instrum. 71, 3298–3305 (2000).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. V. Antyukhov, E. V. Dan’shchikov, N. N. Elkin, V. A. Korotkov, F. V. Lebedev, V. V. Likhanskii, A. P. Napartovich, V. D. Pis’mennyi, V. E. Troshchiev, “Conditions for stable coherent operation of two CO2 lasers with unstable resonators,” Sov. J. Quantum Electron. 19, 1582–1586 (1992).
[CrossRef]

V. V. Antyukhov, E. V. Dan’shchikov, I. V. Masyukov, “Stimulated emission of collective modes in two optically coupled lasers,” Sov. J. Quantum Electron. 22, 203–209 (1992).
[CrossRef]

Sov. Phys. Usp. (1)

V. V. Likhanskii, A. P. Napartovich, “Radiation emitted by optically coupled lasers,” Sov. Phys. Usp. 33, 228–252 (1990).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (University Science, Sausalito, Calif. (1986), p. 1129.

W. Weingartner, “Phase locking of CO2 lasers,” Ph.D. dissertation (Vienna University of Technology, Vienna, Austria, 2000).

W. J. Witteman, The CO2 Laser, Vol. 53 of Springer Series on Optical Sciences (Springer-Verlag, Berlin, 1987).
[CrossRef]

MathWorks “Optimization Toolbox for Mathlab,” handbook MathWorks, 3 Apple Hill Drive, Natick, Mass. 01760, 1998), Sec. 3–3.

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Figures (8)

Fig. 1
Fig. 1

Experimental setup for phase locking two CO2 lasers. DSP, digital signal processor for control of the locked operation; DSO, digital storage oscilloscope.

Fig. 2
Fig. 2

Coupled laser scheme for the calculations. The outcoupling of only one resonator arm is depicted. Notation defined in text.

Fig. 3
Fig. 3

Measured intensity for various values of resonator mismatch ΔL0, showing a growing locking range for larger coupling factors M.

Fig. 4
Fig. 4

Calculated intensity for various coupling factors M, for several values of resonator mismatch ΔL0.

Fig. 5
Fig. 5

Calculated and experimental locking ranges for various calculated and experimental values of coupling factor M.

Fig. 6
Fig. 6

Lasing frequencies on the resonator mismatch for four values of coupling factor M. Lighter lines, ω1 - ω0; tuned resonator length, L 1. Thicker lines, ω2 - ω0; constant resonator length, L 2. Both sets of lines are normalized to 2L/ c.

Fig. 7
Fig. 7

Calculated values (solid curves) and measured values (crosses) of the phase shift between the output fields I i for M = 1.5%.

Fig. 8
Fig. 8

Measured intensity profile at a distance z = 4.5 m from the output mirror for locked and unlocked lasers.

Equations (20)

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Ilocked=cε02|E1|2+|E2|2+2E1E2 cos Φ,
Ilocked=2cε0|E|2,
unlocked: IE12+E22++EN2=NE2,
locked: IE1+E2++EN2=N2E2.
M=PcPi=Pc1-RPout+Pc1-R,
Pi=11-R1-M Pout.
r=R1/2 expjΦr, m=M1/2 expjΦm.
Gi=expln G0i21+|Ei|2.
Φ2=Φ0+ΔΦ2=2πN+2L2cω2-ω0.
Φ1=Φ0+ΔΦ1+ΔΦL=2πN+2L2cω1-ω0+2ω1c ΔL.
E1=E1r1-m2+E2αmG1 expjΦ1,  E2=E2r1-m2+E1αmG2 expjΦ2.
0=r1-m2 G1E1 cosΦ1+Φ+αmG1E2 cosΦ1-E1 cosΦ, 0=r1-m2 G1E1 sinΦ1+Φ+αmG1E2 sinΦ1-E1 sinΦ, 0=αmG2E1 cosΦ2+Φ+r1-m2 G2E2 cosΦ2-E2, 0=αmG2E1 sinΦ2+Φ+r1-m2 G2E2 sinΦ2.
exp-jΦ1r1-m2 G1-1-αmr1-m2-αmr1-m2exp-jΦ2r1-m2 G2-1E1E2=AE1E2=0.
0=exp-jΔΦ2r1-m22G1G2exp-jΔΦL-exp-jΔΦLr1-m2 G1+1r1-m2 G2×exp-jΔΦ+1-αmr1-t22.
exp-jΔΦ1,2=r1-m2G1G2×coshlnG1/G2+jΔΦL2×expjΔΦL21±×1-1-α2m2r21-m2×cosh-2lnG1/G2+jΔΦL21/2.
1-1-α2m2r21-m2cosh-2lnG1/G2+jΔΦL2=0.
exp-jΔΦ=1=r1-m2G1G2×exp-j ΔΦL21-α2m2r21-m21/2.
G1G2r21-m2-α2m2=1
IΦ=I1+I2+2I1I2 cosΦ.
IΦ=I1+I2+2I1I21T0Tcosω1-ω2tdt.

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