Abstract

In standing-wave laser oscillators the field energy density periodically varies along the resonator axis and creates a complex refractive-index grating in the gain medium. This grating couples the originally independent counterpropagating waves of the individual axial cavity modes. The coupling induces a mode frequency shift that is a nonlinear function of the unperturbed mode frequency. The uneven shifts can give rise to a substantial broadening of the beat-note linewidth of the multi-axial-mode free-running laser and to a corresponding increase in the threshold intracavity power for self-starting passive mode locking. The theoretical results are in good qualitative and quantitative agreement with previously reported experimental observations and measurements.

© 1993 Optical Society of America

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References

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  1. E. P. Ippen, in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical Physics (Springer-Verlag, New York, 1992).
  2. For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
    [CrossRef]
  3. See, e.g.,J. A. Fleck, Phys. Rev. B 1, 84 (1970).
    [CrossRef]
  4. F. Krausz, T. Brabec, Ch. Spielmann, Opt. Lett. 16, 235 (1991).
    [CrossRef] [PubMed]
  5. H. A. Haus, E. P. Ippen, Opt. Lett. 16, 1331 (1991).
    [CrossRef] [PubMed]
  6. C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
    [CrossRef]
  7. H. Kogelnik, C. V. Shank, J. Appl. Phys. 43,
  8. The connection between the real and imaginary parts of the atomic susceptibility can be found, for example, inA. Yariv, Quantum Electronics (Wiley, New York, 1989).
  9. The explicit quadratic dependence may be somewhat modified by N = N(gs), which makes ρ1/ρ dependent on gs.
  10. M. H. Ober, M. Hofer, M. E. Fermann, Opt. Lett. 18, 367 (1993).
    [CrossRef] [PubMed]
  11. D. E. Spence, P. N. Kean, W. Sibbett, Opt. Lett. 16, 42 (1991).
    [CrossRef] [PubMed]
  12. Y. M. Liu, K. W. Sun, P. R. Pruncal, S. A. Lyon, Opt. Lett. 17, 1219 (1992).
    [CrossRef] [PubMed]
  13. C. J. Flood, G. Giuliani, H. M. van Driel, Opt. Lett. 15, 218 (1990).
    [CrossRef] [PubMed]
  14. K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, J. G. Fujimoto, Opt. Lett. 18, 220 (1993).
    [CrossRef] [PubMed]

1993 (2)

1992 (2)

Y. M. Liu, K. W. Sun, P. R. Pruncal, S. A. Lyon, Opt. Lett. 17, 1219 (1992).
[CrossRef] [PubMed]

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

1991 (3)

1990 (1)

1970 (1)

See, e.g.,J. A. Fleck, Phys. Rev. B 1, 84 (1970).
[CrossRef]

1963 (1)

C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Brabec, T.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

F. Krausz, T. Brabec, Ch. Spielmann, Opt. Lett. 16, 235 (1991).
[CrossRef] [PubMed]

Curley, P. F.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

deMars, G.

C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Fermann, M. E.

M. H. Ober, M. Hofer, M. E. Fermann, Opt. Lett. 18, 367 (1993).
[CrossRef] [PubMed]

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

Fleck, J. A.

See, e.g.,J. A. Fleck, Phys. Rev. B 1, 84 (1970).
[CrossRef]

Flood, C. J.

Fujimoto, J. G.

Giuliani, G.

Haus, H. A.

Hofer, M.

M. H. Ober, M. Hofer, M. E. Fermann, Opt. Lett. 18, 367 (1993).
[CrossRef] [PubMed]

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

Ippen, E. P.

K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, J. G. Fujimoto, Opt. Lett. 18, 220 (1993).
[CrossRef] [PubMed]

H. A. Haus, E. P. Ippen, Opt. Lett. 16, 1331 (1991).
[CrossRef] [PubMed]

E. P. Ippen, in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical Physics (Springer-Verlag, New York, 1992).

Jacobson, J.

Kean, P. N.

Kogelnik, H.

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43,

Krausz, F.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

F. Krausz, T. Brabec, Ch. Spielmann, Opt. Lett. 16, 235 (1991).
[CrossRef] [PubMed]

Liu, Y. M.

Lyon, S. A.

Ober, M. H.

M. H. Ober, M. Hofer, M. E. Fermann, Opt. Lett. 18, 367 (1993).
[CrossRef] [PubMed]

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

Pruncal, P. R.

Schmidt, A. J.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43,

Sibbett, W.

Spence, D. E.

Spielmann, Ch.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

F. Krausz, T. Brabec, Ch. Spielmann, Opt. Lett. 16, 235 (1991).
[CrossRef] [PubMed]

Statz, H.

C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

Sun, K. W.

Tamura, K.

Tang, C. L.

C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

van Driel, H. M.

Wintner, E.

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

Yariv, A.

The connection between the real and imaginary parts of the atomic susceptibility can be found, for example, inA. Yariv, Quantum Electronics (Wiley, New York, 1989).

IEEE J. Quantum Electron. (1)

For a review see, e.g.,F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, Ch. Spielmann, E. Wintner, A. J. Schmidt, IEEE J. Quantum Electron. 28, 2097 (1992).
[CrossRef]

J. Appl. Phys. (2)

C. L. Tang, H. Statz, G. deMars, J. Appl. Phys. 34, 2289 (1963).
[CrossRef]

H. Kogelnik, C. V. Shank, J. Appl. Phys. 43,

Opt. Lett. (7)

Phys. Rev. B (1)

See, e.g.,J. A. Fleck, Phys. Rev. B 1, 84 (1970).
[CrossRef]

Other (3)

The connection between the real and imaginary parts of the atomic susceptibility can be found, for example, inA. Yariv, Quantum Electronics (Wiley, New York, 1989).

The explicit quadratic dependence may be somewhat modified by N = N(gs), which makes ρ1/ρ dependent on gs.

E. P. Ippen, in Ultrafast Phenomena VIII, Vol. 55 of Springer Series in Chemical Physics (Springer-Verlag, New York, 1992).

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Figures (2)

Fig. 1
Fig. 1

Circulating radiation in a linear laser resonator under steady-state conditions. The intracavity losses have been incorporated into the mirror reflectivities.

Fig. 2
Fig. 2

Effective modulation amplitude (ρ1) of the energy density relative to the average energy density (ρ) in the gain medium as a function of the number of oscillating modes (N) in a resonator with Lr = 1 m, m0 = 2 × 106 (λ0 = 1 μm), L1 = L2, nLg = 10 mm, and ρk = ρ/N. The effective modulation amplitude is defined by ρ 1 = 2 0 L g ρ 1 2 ( z ) d z , where the factor 2 ensures that ρ1/ρ = 1 for N = 1.

Equations (14)

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n c ( z ) = n + j c ω α + ( n 1 + j c ω α 1 ) cos ( 2 β 0 z + φ ) ,
b 1 = exp [ ( g s / 4 ) - j ϕ g ] ( K a 1 + a 2 ) ,
b 2 = exp [ ( g s / 4 ) - j ϕ g ] ( a 1 + K a 2 ) ,
K = - j L g sin Δ Δ ,
= 1 2 ( β 0 n 1 + j α 1 ) .
a i = exp [ - ( l i / 2 ) - j ϕ i ] b i ,             i = 1 , 2.
1 4 δ g s + j L r c δ ω = L g sin Δ Δ [ g s 4 sin ( ϕ 1 + ϕ g ) - j cos ( ϕ 1 + ϕ g ) ] ,
Δ ν 3 dB 2 Δ ν k 2 1 / 2 = Δ ν α 2 + Δ ν n 2 ,
Δ ν α = g s 2 π 2 α 1 L g Δ ν 0
Δ ν n = 2 β 0 π n 1 L g Δ ν 0
κ P > π ln N Δ ν 3 dB Δ ν 0 ,
ρ ( z ) = ρ + ρ 1 ( z ) = k = - ( N - 1 ) / 2 ( N - 1 ) / 2 ρ k - k = - ( N - 1 ) / 2 ( N - 1 ) / 2 ρ k cos 2 π m k L r ( n z - L 1 ) ,
α ( z ) - g s 4 L g α 1 cos ( 2 ω 0 c n z + φ ) ,
α 1 = g s 2 L g P P s ( 1 + 2 P P s ) - 1 ρ 1 ρ .

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