Abstract

The time evolution of the nondegenerate four-wave mixing polarization in a semiconductor optical amplifier (SOA) is investigated theoretically by solution of the semiconductor Bloch equations, including all the relevant scattering processes. The results are in direct agreement with recent experimental measurements, showing that a treatment of coherent and incoherent effects in SOA’s on the level of a microscopic theory is highly desirable.

© 1998 Optical Society of America

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  1. See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
    [CrossRef]
  2. R. Indik, A. Knorr, R. Binder, J. V. Moloney, and S. W. Koch, Opt. Lett. 19, 966 (1994); S. Hughes, A. Knorr, and S. W. Koch, Opt. Lett. 21, 1052 (1996).
    [CrossRef] [PubMed]
  3. K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, Opt. Lett. 17, 874 (1992).
    [CrossRef]
  4. R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
    [CrossRef]
  5. A. Mecozzi, J. Mørk, and M. Hofmann, Opt. Lett. 21, 1017 (1996).
    [CrossRef] [PubMed]
  6. M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
    [CrossRef]
  7. T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
    [CrossRef]
  8. M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
    [CrossRef]
  9. A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
    [CrossRef]
  10. F. Rossi, S. Haas, and T. Kuhn, Phys. Rev. Lett. 72, 152 (1994); Phys. Rev. B 53, 15,855 (1996).
    [CrossRef] [PubMed]
  11. Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
    [CrossRef] [PubMed]
  12. S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
    [CrossRef]
  13. S. Hughes, Solid State Commun. 104, 107 (1997).
    [CrossRef]

1997 (1)

S. Hughes, Solid State Commun. 104, 107 (1997).
[CrossRef]

1996 (4)

S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
[CrossRef]

A. Mecozzi, J. Mørk, and M. Hofmann, Opt. Lett. 21, 1017 (1996).
[CrossRef] [PubMed]

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

1995 (1)

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (1)

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

1992 (2)

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, Opt. Lett. 17, 874 (1992).
[CrossRef]

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

1988 (1)

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

1986 (1)

See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
[CrossRef]

Binder, R.

Brorson, S. D.

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

Chow, W. W.

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

Haas, S.

F. Rossi, S. Haas, and T. Kuhn, Phys. Rev. Lett. 72, 152 (1994); Phys. Rev. B 53, 15,855 (1996).
[CrossRef] [PubMed]

Hall, K. L.

Henneberger, K.

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

Hofmann, M.

A. Mecozzi, J. Mørk, and M. Hofmann, Opt. Lett. 21, 1017 (1996).
[CrossRef] [PubMed]

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

Hughes, S.

S. Hughes, Solid State Commun. 104, 107 (1997).
[CrossRef]

S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
[CrossRef]

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

Indik, R.

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

R. Indik, A. Knorr, R. Binder, J. V. Moloney, and S. W. Koch, Opt. Lett. 19, 966 (1994); S. Hughes, A. Knorr, and S. W. Koch, Opt. Lett. 21, 1052 (1996).
[CrossRef] [PubMed]

Ippen, E. P.

K. L. Hall, G. Lenz, E. P. Ippen, and G. Raybon, Opt. Lett. 17, 874 (1992).
[CrossRef]

See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
[CrossRef]

Kesler, M. P.

See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
[CrossRef]

Knorr, A.

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
[CrossRef]

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

R. Indik, A. Knorr, R. Binder, J. V. Moloney, and S. W. Koch, Opt. Lett. 19, 966 (1994); S. Hughes, A. Knorr, and S. W. Koch, Opt. Lett. 21, 1052 (1996).
[CrossRef] [PubMed]

Koch, S. W.

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
[CrossRef]

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

R. Indik, A. Knorr, R. Binder, J. V. Moloney, and S. W. Koch, Opt. Lett. 19, 966 (1994); S. Hughes, A. Knorr, and S. W. Koch, Opt. Lett. 21, 1052 (1996).
[CrossRef] [PubMed]

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Kuhn, T.

F. Rossi, S. Haas, and T. Kuhn, Phys. Rev. Lett. 72, 152 (1994); Phys. Rev. B 53, 15,855 (1996).
[CrossRef] [PubMed]

Lenz, G.

Lindberg, M.

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Mecozzi, A.

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

A. Mecozzi, J. Mørk, and M. Hofmann, Opt. Lett. 21, 1017 (1996).
[CrossRef] [PubMed]

Moloney, J. V.

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

R. Indik, A. Knorr, R. Binder, J. V. Moloney, and S. W. Koch, Opt. Lett. 19, 966 (1994); S. Hughes, A. Knorr, and S. W. Koch, Opt. Lett. 21, 1052 (1996).
[CrossRef] [PubMed]

Mørk, J.

A. Mecozzi, J. Mørk, and M. Hofmann, Opt. Lett. 21, 1017 (1996).
[CrossRef] [PubMed]

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

Paul, A. E.

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

Petter, U.

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

Rappen, T.

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

Raybon, G.

Rossi, F.

F. Rossi, S. Haas, and T. Kuhn, Phys. Rev. Lett. 72, 152 (1994); Phys. Rev. B 53, 15,855 (1996).
[CrossRef] [PubMed]

Schäfer, W.

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

Scott, D. C.

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

Stix, M. S.

See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
[CrossRef]

Stroucken, T.

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

Wegener, M.

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

Appl. Phys. Lett. (2)

See, for example, M. S. Stix, M. P. Kesler, and E. P. Ippen, Appl. Phys. Lett. 48, 1722 (1986); J. Mark and J. Mørk, Appl. Phys. Lett. 61, 2281 (1992); A. D’Ottiva, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, J. Landreau, A. Ougazzaden, and J. C. Bouley, Appl. Phys. Lett. 64, 2492 (1994); K. Kikuchi, A. Amano, C. E. Zah, and T. P. Lee, Appl. Phys. Lett. 64, 548 (1994).
[CrossRef]

M. Hofmann, S. D. Brorson, J. Mørk, and A. Mecozzi, Appl. Phys. Lett. 68, 3236 (1996).
[CrossRef]

J. Chem. Phys. (1)

A. Knorr, S. Hughes, T. Stroucken, and S. W. Koch, J. Chem. Phys. 27, 210 (1996); S. Hughes, A. Knorr, and S. W. Koch, J. Opt. Soc. Am. B 14, 754 (1997).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

Pulse propagation and its influence on FWM were studied previously in the relaxation-time-approximation, which, however, does not imply a consistent calculation of the decay times:?W. W. Chow, R. Indik, A. Knorr, S. W. Koch, and J. V. Moloney, Phys. Rev. A 52, 2479 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (3)

R. Binder, D. C. Scott, A. E. Paul, M. Lindberg, K. Henneberger, and S. W. Koch, Phys. Rev. B 45, 1107 (1992).
[CrossRef]

T. Rappen, U. Petter, M. Wegener, and W. Schäfer, Phys. Rev. B 48, 4879 (1993); Phys. Rev. B 49, 774 (1994).
[CrossRef]

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Phys. Rev. Lett. (1)

F. Rossi, S. Haas, and T. Kuhn, Phys. Rev. Lett. 72, 152 (1994); Phys. Rev. B 53, 15,855 (1996).
[CrossRef] [PubMed]

Solid State Commun. (2)

S. Hughes, A. Knorr, and S. W. Koch, Solid State Commun. 100, 555 (1996).
[CrossRef]

S. Hughes, Solid State Commun. 104, 107 (1997).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

(a) Time-dependent FWM signal at T=30 and 300  K. The input irradiance in gigawatts per square centimeter is shown. (b) Distribution of electrons before and at the center of the pulse excitations at 30 and 300  K. The thicker curves, at time t=- are in quasi-equilibrium; the thinner lines are at a time near the center of the optical pulse.

Fig. 2
Fig. 2

Time-dependent FWM signal (solid curve) as in Fig.  1(a) that includes all the relevant scattering mechanisms. Also shown is the calculation when nondiagonal dephasing is not included (dotted curve) and for excitation near the bandgap (dashed curve, which includes nondiagonal dephasing).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Pkt=-iΔkPk-iΩkfke+fkh-1+Pkt|corr,
fke/ht=iPkΩk-iPkΩk+fke/ht|corr.
fkat|corr=Γinak, f1-fka-Γoutak, ffka+(PkΩk*|corr+c.c.),
Pkt|corr=-Γdpk, fPk+q Γndpk, q, fPk+q,
Γndpk,q, f=πkb, a=e,h2Vq21-fka1-fkbfk-qb+fkafkb1-fk-qb-Pk*Pk-q×δak+bk-ak+q-bk-q,
FWMτdτEpτEtt-τexpiΔτ2,

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