Abstract

A transient analysis of pulse formation and mode locking for parametric traveling-wave modulation is considered. Steady-state cavity mode-locking conditions providing pulse width, and chirp taking mirror or facet dispersion into account, are discussed. A Gaussian-pulse parameter analysis for dispersive compensation resulting in intracavity pulse compression is introduced. Numerical estimates for semiconductor lasers are given.

© 1986 Optical Society of America

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References

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  1. D. Haas, J. Wurl, J. McLean, T. K. Gustafson, Opt. Lett. 9, 445 (1984).
    [CrossRef] [PubMed]
  2. T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
    [CrossRef]
  3. H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
    [CrossRef]
  4. J. P. van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983). For a laser with confinement factors of 0.42 and 0.24 a value for the real part of ∂2k/∂ω2 is found to be of the same order of magnitude as the value of αgT22 assumed here.
    [CrossRef]
  5. D. J. Kuizenga, A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
    [CrossRef]
  6. D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
    [CrossRef]
  7. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  8. P. Laporta, V. Magni, Appl. Opt. 24, 2014 (1985).
    [CrossRef] [PubMed]
  9. H. A. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
    [CrossRef]
  10. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, Toronto. 1961).

1985

1984

1983

J. P. van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983). For a laser with confinement factors of 0.42 and 0.24 a value for the real part of ∂2k/∂ω2 is found to be of the same order of magnitude as the value of αgT22 assumed here.
[CrossRef]

1981

H. A. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

1974

H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
[CrossRef]

1973

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

1970

D. J. Kuizenga, A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
[CrossRef]

1969

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Casey, H. C.

H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
[CrossRef]

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, Toronto. 1961).

Gustafson, T. K.

D. Haas, J. Wurl, J. McLean, T. K. Gustafson, Opt. Lett. 9, 445 (1984).
[CrossRef] [PubMed]

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Haas, D.

Haus, H. A.

H. A. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Kelley, P. L.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Kuizenga, D. J.

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

D. J. Kuizenga, A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
[CrossRef]

Laporta, P.

Lifsitz, J. R.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Logan, R. A.

J. P. van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983). For a laser with confinement factors of 0.42 and 0.24 a value for the real part of ∂2k/∂ω2 is found to be of the same order of magnitude as the value of αgT22 assumed here.
[CrossRef]

Lund, T.

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

Magni, V.

McLean, J.

Panish, M. B.

H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
[CrossRef]

Phillion, D. W.

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

Sell, D. D.

H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
[CrossRef]

Siegman, A. E.

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

D. J. Kuizenga, A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
[CrossRef]

Taran, J. P.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

van der Ziel, J. P.

J. P. van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983). For a laser with confinement factors of 0.42 and 0.24 a value for the real part of ∂2k/∂ω2 is found to be of the same order of magnitude as the value of αgT22 assumed here.
[CrossRef]

Wurl, J.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Appl. Opt.

Appl. Phys. Lett.

H. C. Casey, D. D. Sell, M. B. Panish, Appl. Phys. Lett. 24, 63 (1974). As pointed out in Ref. 1, background material dispersion in GaAs is estimated to be ≅ 104 psec nm−1 km−1 in contrast to αgT22, which gives ≅ 105 psec nm−1 km−1. This ignores the Van Hove cusp, which may be avoided at the lasing frequency by using a large optical cavity. We note that this also implies equal mode spacing for continuous-wave oscillation.
[CrossRef]

IEEE J. Quantum Electron.

J. P. van der Ziel, R. A. Logan, IEEE J. Quantum Electron. QE-19, 164 (1983). For a laser with confinement factors of 0.42 and 0.24 a value for the real part of ∂2k/∂ω2 is found to be of the same order of magnitude as the value of αgT22 assumed here.
[CrossRef]

D. J. Kuizenga, A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).
[CrossRef]

Jpn. J. Appl. Phys.

H. A. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

Opt. Commun.

D. J. Kuizenga, D. W. Phillion, T. Lund, A. E. Siegman, Opt. Commun. 9, 221 (1973).
[CrossRef]

Opt. Lett.

Phys. Rev.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, P. L. Kelley, Phys. Rev. 177, 306 (1969).
[CrossRef]

Other

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, Toronto. 1961).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

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Figures (2)

Fig. 1
Fig. 1

Plot showing the behavior of Q as a function of z. (a) γ + 1 and γ − 1 in the complex γ plane. The branch cut between γ = −1 and γ = +1 is indicated. For γ real a phase flip of π occurs in γ − 1 or γ + 1 as the end points of the branch cut are traversed. This results in the singular behavior of Q shown in (b), which is taken from Dwight.10 The normalized quantities are γ = Q/ωp2 and y = 2ωp2αgT22z = βz. For the parameters αm = −i1.4 cm−1, αg = 30 cm−1, and T2 = 5 × 10−14 sec, ωm = 1.3 × 1012 rad sec−1 (corresponding to L = 200 μm), ωp2 = 4.0 × 1024 sec−2 (corresponding to a full 1/e pulse-intensity width of 1 psec at γ = 1), and β = 0.60 cm−1.

Fig. 2
Fig. 2

The γ > 1 branch of Fig. 1(b) (loss or gain modulation) indicating the values of γ0 and γ for successively smaller values of , where is defined through b = bc(1 − ) and bc is the mirror dispersion coefficient of Eq. (11). The spacing between γ0 and γ is βL on the vertical axis. The parameters assumed are αm = −i1.4 cm−1, αl + 1/L ln(1/R0) = 30 cm−1, and L = 200 μm. For b = 0, ωp2 = 4 × 1024 sec−1, so that γ = 1 corresponds to a pulse width of 1 psec.

Equations (13)

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Δ E ξ + E z = α g T 2 2 2 E ξ 2 - i α m 2 ω m 2 ξ 2 E + ( α g - α l ) E .
E ( z , t ) = exp { i P ( z ) - ½ Q ( z ) [ ξ + η ( z ) ] 2 } .
- d Q d z = 2 α g T 2 2 Q 2 - i α m ω m 2 ,
d i P d z = - α g T 2 2 Q + i α m 2 ω m 2 η 2 + ( α g - α l ) ,
d η d z + i α m ω m 2 η Q = - Δ .
β z + c = ½ ln ( 1 + γ / 1 - γ ) ,
γ = tanh ( β z + c ) .
exp [ i P ( z ) ] = exp [ i P ( 0 ) + ( α g - α l ) z ] [ sech ( β z + c ) sech c ] 1 / 2 ,
γ = [ A γ 0 + B ] / C γ 0 + D ] ,
γ 1 = [ A m γ + B m ] / [ C m γ + D m ] ,
γ 0 2 + γ 0 B 1 + D m C - A m B 1 + D m C = 0.
( α g - α l ) L - ln ( 1 R 0 ) = 1 4 ln - 1 + γ 0 2 ( - 1 + γ 2 ) .
b c = - i / [ 2 ω p 2 tanh ( β L ) ] .

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