Abstract

A model to evaluate frequency locking by means of a power injection near a side mode of the free-running frequency is presented. Semiconductor optical nonlinearities govern the locking problem; they are represented by nonlinear coefficients that take into account both carrier fluctuations and all relevant fast phenomena. Field equations are derived from coupled-mode theory, accounting for longitudinal variations by means of appropriate mean values. Examples of results show both the single locked frequency-stable operation region and the case when the injected signal and the free-running field coexist.

© 1996 Optical Society of America

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References

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  1. S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
    [CrossRef]
  2. L. Li, K. Petermann, IEEE J. Quantum Electron. 29, 2988 (1993).
    [CrossRef]
  3. G. P. Bava, P. Debernardi, G. Osella, Opt. Lett. 20, 1643 (1995).
    [CrossRef] [PubMed]
  4. R. Lang, IEEE J. Quantum Electron. QE-18, 976 (1982).
    [CrossRef]
  5. L. Li, IEEE J. Quantum Electron. 30, 1701 (1994).
    [CrossRef]
  6. G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).
  7. A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
    [CrossRef]
  8. V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
    [CrossRef]

1995 (2)

G. P. Bava, P. Debernardi, G. Osella, Opt. Lett. 20, 1643 (1995).
[CrossRef] [PubMed]

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

1994 (3)

V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
[CrossRef]

L. Li, IEEE J. Quantum Electron. 30, 1701 (1994).
[CrossRef]

G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).

1993 (1)

L. Li, K. Petermann, IEEE J. Quantum Electron. 29, 2988 (1993).
[CrossRef]

1991 (1)

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

1982 (1)

R. Lang, IEEE J. Quantum Electron. QE-18, 976 (1982).
[CrossRef]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
[CrossRef]

Bava, G. P.

G. P. Bava, P. Debernardi, G. Osella, Opt. Lett. 20, 1643 (1995).
[CrossRef] [PubMed]

G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).

Debernardi, P.

G. P. Bava, P. Debernardi, G. Osella, Opt. Lett. 20, 1643 (1995).
[CrossRef] [PubMed]

G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).

Donati, S.

V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
[CrossRef]

Iannone, E.

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

Lang, R.

R. Lang, IEEE J. Quantum Electron. QE-18, 976 (1982).
[CrossRef]

Li, L.

L. Li, IEEE J. Quantum Electron. 30, 1701 (1994).
[CrossRef]

L. Li, K. Petermann, IEEE J. Quantum Electron. 29, 2988 (1993).
[CrossRef]

Manna, M.

V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
[CrossRef]

Mecozzi, A.

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

Murata, S.

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

Osella, G.

G. P. Bava, P. Debernardi, G. Osella, Opt. Lett. 20, 1643 (1995).
[CrossRef] [PubMed]

G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).

Petermann, K.

L. Li, K. Petermann, IEEE J. Quantum Electron. 29, 2988 (1993).
[CrossRef]

Scotti, S.

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

Shimizu, J.

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

Spano, P.

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

Suzuki, A.

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

Tomita, A.

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

IEEE J. Quantum Electron. (5)

L. Li, K. Petermann, IEEE J. Quantum Electron. 29, 2988 (1993).
[CrossRef]

R. Lang, IEEE J. Quantum Electron. QE-18, 976 (1982).
[CrossRef]

L. Li, IEEE J. Quantum Electron. 30, 1701 (1994).
[CrossRef]

A. Mecozzi, S. Scotti, E. Iannone, P. Spano, IEEE J. Quantum Electron. 31, 689 (1995).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30, 1537 (1994).
[CrossRef]

IEEE Trans. Photon. Technol. Lett. (1)

S. Murata, A. Tomita, J. Shimizu, A. Suzuki, IEEE Trans. Photon. Technol. Lett. 3, 1021 (1991).
[CrossRef]

Opt. Lett. (1)

Proc. Inst. Electr. Eng. Part J (1)

G. P. Bava, P. Debernardi, G. Osella, Proc. Inst. Electr. Eng. Part J 141, 119 (1994).

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Figures (5)

Fig. 1
Fig. 1

Frequency scheme: the vertical lines below the abscissa represent the possible oscillating modes without injection locking, Ω0 is the free spectral range, and p defines the number of resonances away from the lasing frequency (p > 0 if ω1 > ω2).

Fig. 2
Fig. 2

Stable locking range (black areas) versus normalized current for two different power injections displaced +10 Ω0 from the free-running pulsation: The gray areas correspond to stable two-signal operation; the white ones are unstable regions.

Fig. 3
Fig. 3

Output powers for signals at frequencies 1 (curves) and 2 (curves with squares) along the two-signal stability borderlines of Fig. 2. Continuous curves, upper boundary; dashed curves, lower borderline; thick solid lines, free-running power.

Fig. 4
Fig. 4

Stable locking range (black areas) versus injected power displaced +10 Ω0 from the free-running pulsation for two different currents: The gray areas correspond to stable two-signal operation; the white ones are unstable regions.

Fig. 5
Fig. 5

As in Fig. 4 but for two different frequencies of the locking signal and for I /Ith = 2.

Tables (1)

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Table 1 Notation Used

Equations (5)

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d n d τ = - n + Δ N i - ( n + Δ N s ) ( P 1 + P 2 ) + n g P 1 , d c 1 d τ = c 1 ( G n - H - σ P 1 - μ ˜ P 2 ) + K c in , d c 2 d τ = c 2 [ G n - μ P 1 - σ ( P 2 - P 20 ) ] ,
K = τ s v g / ( 2 L ) [ ( R 1 R 2 - 1 ) ( R 1 / R 2 + 1 ) × ( R 1 log R 1 R 2 ) - 1 ] 1 / 2 ,
σ = γ Δ N s h , μ ˜ = γ Δ N s [ h + 3 / 2 m ( - Ω ) ] , μ = γ Δ N s [ h + 3 / 2 m ( Ω ) ] .
m ( Ω ) = 1 + j α 1 + j Ω τ s + t 1 + j α t 1 + j Ω τ t + h 1 + j α h 1 + j Ω τ h ,
- n + Δ N i - ( n + Δ N s ) ( P 1 + P 2 ) + n g P 1 = 0 , [ G n - j x 2 - μ P 1 - σ ( P 2 - P 20 ) ] = 0 , P 1 G n - j x in - H - σ P 1 - μ ˜ P 2 2 = K 2 P in ,

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