Abstract

The theory of nondegenerate four-wave mixing (NDFWM) in semiconductor lasers and amplifiers is presented with particular emphasis on the physical processes that lead to population pulsations. In the case of nearly degenerate four-wave mixing, modulation of the carrier density at the beat frequency Ω of the pump and probe waves creates a dynamic population grating whose effectiveness is governed by the spontaneous carrier lifetime τs. Such a grating affects both the gain and the refractive index of the probe wave. In particular, the probe gain exhibits features analogous to those observed in a detuned atomic system arising from the optical Stark effect. Both the gain grating and the index grating contribute to NDFWM, with the dominant contribution coming from the index grating. For detunings such that Ωτs ≫ 1, population pulsations correspond to modulation of the intraband population arising from spectral hole burning. Our results show that NDFWM is then limited by the phase-mismatch effects governed by the transit time τ rather than by the intraband population-relaxation time T1. Significant NDFWM is expected to occur for detunings up to about 300 GHz for typical transit-time values of 3 psec in semiconductor lasers.

© 1988 Optical Society of America

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  1. See various chapters in R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  2. Special issue on dynamic gratings and four-wave mixing, IEEE J. Quantum Electron. QE-22, 1194–1542 (1986).
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 14.
  4. E. E. Bergmann, I. J. Bigio, B. J. Feldman, and R. A. Fisher, “High-efficiency pulsed 10.6-μ m phase-conjugate reflection via degenerate four-wave mixing,” Opt. Lett. 3, 82–84 (1978).
    [CrossRef]
  5. G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-17, 374–380 (1981).
    [CrossRef]
  6. G. P. Agrawal, “Intracavity resonant degenerate four-wave mixing: bistability in phase conjugation,” J. Opt. Soc. Am. 73, 654–660 (1983).
    [CrossRef]
  7. A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463–464 (1979).
    [CrossRef]
  8. R. A. Fisher and B. J. Feldman, “On resonant phase conjugate reflection and amplification at 10.6 μ m in inverted CO2,” Opt. Lett. 4, 140–152 (1979).
    [CrossRef] [PubMed]
  9. J. Reintjes and L. J. Palumbo, “Phase conjugation in saturable amplifiers by degenerate frequency mixing,” IEEE J. Quantum Electron. QE-18, 1934–1940 (1982).
    [CrossRef]
  10. H. Nakajima and R. Frey, “Observation of bistable reflectivity of a phase-conjugated signal through intracavity nearly degenerate four-ware mixing,” Phys. Rev. Lett. 54, 1798–1801 (1985).
    [CrossRef] [PubMed]
  11. H. Nakajima and R. Frey, “Intracavity nearly degenerate four-wave mixing in a (GaAl)As semiconductor laser,” Appl. Phys. Lett. 47, 769–771 (1985).
    [CrossRef]
  12. H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. QE-22, 1349–1354 (1986).
    [CrossRef]
  13. R. Frey, “On-axis intracavity nearly degenerate four-wave mixing in semiconductor lasers,” Opt. Lett. 11, 91–93 (1986).
    [CrossRef]
  14. N. C. Kothari and R. Frey, “Bistable behavior of pump, probe, and conjugate signals through collinear intracavity nearly degenerate four-wave mixing,” Phys. Rev. A 34, 2013–2025 (1986).
    [CrossRef] [PubMed]
  15. G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. 12, 260–262 (1987).
    [CrossRef] [PubMed]
  16. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429–1450 (1964).
  17. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 9.
  18. S. T. Hendow and M. Sargent, “Theory of single-mode laser instabilities,” J. Opt. Soc. Am. B 2, 84–101 (1985).
    [CrossRef]
  19. A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
    [CrossRef]
  20. G. P. Agrawal, “Highly nondegenerate four-wave mixing in semiconductor lasers due to spectral hole-burning,” Appl. Phys. Lett. 51, 302–304 (1987).
    [CrossRef]
  21. Y. Nishimura and Y. Nishimura, “Spectral hole-burning and nonlinear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron. QE-9, 1011–1019 (1973).
    [CrossRef]
  22. B. Zee, “Broadening mechanism in semiconductor (GaAs) lasers: limitations to single mode power emission,” IEEE J. Quantum Electron. QE-14, 727–736 (1978).
    [CrossRef]
  23. M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653–2664 (1981).
    [CrossRef]
  24. R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
    [CrossRef]
  25. M. Asada and Y. Suematsu, “Density-matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
    [CrossRef]
  26. G. P. Agrawal, “Gain nonlinearities in semiconductor lasers: theory and application to distributed feedback lasers,” IEEE J. Quantum Electron. QE-23, 860–868 (1987).
    [CrossRef]
  27. G. P. Agrawal and N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
    [CrossRef]
  28. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
    [CrossRef]
  29. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
    [CrossRef]
  30. The expression for ΔN in Ref. 15 is in error. However, this does not affect the conclusions reported therein.
  31. B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1982).
    [CrossRef]
  32. F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
    [CrossRef]
  33. M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rep. 43, 223–265 (1978).
    [CrossRef]
  34. R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
    [CrossRef]
  35. T. Fu and M. Sargent, “Effect of signal detuning on phase conjugation,” Opt. Lett. 4, 366–368 (1979).
    [CrossRef] [PubMed]
  36. D. J. Harter and R. W. Boyd, “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980).
    [CrossRef]
  37. K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
    [CrossRef]
  38. J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a homogeneously broadened two-level-system with saturating pump waves,” IEEE J. Quantum Electron. QE-18, 1947–1952 (1982).
    [CrossRef]
  39. G. P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
    [CrossRef] [PubMed]
  40. K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
    [CrossRef]
  41. G. P. Agrawal, “Amplifier-induced crosstalk in multichannel coherent lightwave systems,” Electron. Lett. 23,Oct23 (1987).
    [CrossRef]

1987 (6)

G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. 12, 260–262 (1987).
[CrossRef] [PubMed]

G. P. Agrawal, “Highly nondegenerate four-wave mixing in semiconductor lasers due to spectral hole-burning,” Appl. Phys. Lett. 51, 302–304 (1987).
[CrossRef]

G. P. Agrawal, “Gain nonlinearities in semiconductor lasers: theory and application to distributed feedback lasers,” IEEE J. Quantum Electron. QE-23, 860–868 (1987).
[CrossRef]

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
[CrossRef]

G. P. Agrawal, “Amplifier-induced crosstalk in multichannel coherent lightwave systems,” Electron. Lett. 23,Oct23 (1987).
[CrossRef]

1986 (5)

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. QE-22, 1349–1354 (1986).
[CrossRef]

R. Frey, “On-axis intracavity nearly degenerate four-wave mixing in semiconductor lasers,” Opt. Lett. 11, 91–93 (1986).
[CrossRef]

N. C. Kothari and R. Frey, “Bistable behavior of pump, probe, and conjugate signals through collinear intracavity nearly degenerate four-wave mixing,” Phys. Rev. A 34, 2013–2025 (1986).
[CrossRef] [PubMed]

Special issue on dynamic gratings and four-wave mixing, IEEE J. Quantum Electron. QE-22, 1194–1542 (1986).

1985 (4)

H. Nakajima and R. Frey, “Observation of bistable reflectivity of a phase-conjugated signal through intracavity nearly degenerate four-ware mixing,” Phys. Rev. Lett. 54, 1798–1801 (1985).
[CrossRef] [PubMed]

H. Nakajima and R. Frey, “Intracavity nearly degenerate four-wave mixing in a (GaAl)As semiconductor laser,” Appl. Phys. Lett. 47, 769–771 (1985).
[CrossRef]

S. T. Hendow and M. Sargent, “Theory of single-mode laser instabilities,” J. Opt. Soc. Am. B 2, 84–101 (1985).
[CrossRef]

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

1983 (1)

1982 (6)

J. Reintjes and L. J. Palumbo, “Phase conjugation in saturable amplifiers by degenerate frequency mixing,” IEEE J. Quantum Electron. QE-18, 1934–1940 (1982).
[CrossRef]

R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1982).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a homogeneously broadened two-level-system with saturating pump waves,” IEEE J. Quantum Electron. QE-18, 1947–1952 (1982).
[CrossRef]

G. P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
[CrossRef] [PubMed]

1981 (3)

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653–2664 (1981).
[CrossRef]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-17, 374–380 (1981).
[CrossRef]

1980 (1)

D. J. Harter and R. W. Boyd, “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980).
[CrossRef]

1979 (3)

1978 (3)

E. E. Bergmann, I. J. Bigio, B. J. Feldman, and R. A. Fisher, “High-efficiency pulsed 10.6-μ m phase-conjugate reflection via degenerate four-wave mixing,” Opt. Lett. 3, 82–84 (1978).
[CrossRef]

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rep. 43, 223–265 (1978).
[CrossRef]

B. Zee, “Broadening mechanism in semiconductor (GaAs) lasers: limitations to single mode power emission,” IEEE J. Quantum Electron. QE-14, 727–736 (1978).
[CrossRef]

1977 (1)

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

1975 (1)

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
[CrossRef]

1973 (1)

Y. Nishimura and Y. Nishimura, “Spectral hole-burning and nonlinear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron. QE-9, 1011–1019 (1973).
[CrossRef]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429–1450 (1964).

Agrawal, G. P.

G. P. Agrawal, “Four-wave mixing and phase conjugation in semiconductor laser media,” Opt. Lett. 12, 260–262 (1987).
[CrossRef] [PubMed]

G. P. Agrawal, “Highly nondegenerate four-wave mixing in semiconductor lasers due to spectral hole-burning,” Appl. Phys. Lett. 51, 302–304 (1987).
[CrossRef]

G. P. Agrawal, “Gain nonlinearities in semiconductor lasers: theory and application to distributed feedback lasers,” IEEE J. Quantum Electron. QE-23, 860–868 (1987).
[CrossRef]

G. P. Agrawal, “Amplifier-induced crosstalk in multichannel coherent lightwave systems,” Electron. Lett. 23,Oct23 (1987).
[CrossRef]

G. P. Agrawal, “Intracavity resonant degenerate four-wave mixing: bistability in phase conjugation,” J. Opt. Soc. Am. 73, 654–660 (1983).
[CrossRef]

G. P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
[CrossRef] [PubMed]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-17, 374–380 (1981).
[CrossRef]

G. P. Agrawal and N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Asada, M.

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

Aubourg, P.

Bergmann, E. E.

Bigio, I. J.

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
[CrossRef]

Boulnois, J. L.

Boyd, R. W.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

D. J. Harter and R. W. Boyd, “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980).
[CrossRef]

Buus, J.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

Cronin-Golomb, M.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Ducloy, M.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
[CrossRef]

Ezekiel, S.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Feldman, B. J.

Fisher, R. A.

Flytzanis, C.

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-17, 374–380 (1981).
[CrossRef]

Frey, R.

N. C. Kothari and R. Frey, “Bistable behavior of pump, probe, and conjugate signals through collinear intracavity nearly degenerate four-wave mixing,” Phys. Rev. A 34, 2013–2025 (1986).
[CrossRef] [PubMed]

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. QE-22, 1349–1354 (1986).
[CrossRef]

R. Frey, “On-axis intracavity nearly degenerate four-wave mixing in semiconductor lasers,” Opt. Lett. 11, 91–93 (1986).
[CrossRef]

H. Nakajima and R. Frey, “Intracavity nearly degenerate four-wave mixing in a (GaAl)As semiconductor laser,” Appl. Phys. Lett. 47, 769–771 (1985).
[CrossRef]

H. Nakajima and R. Frey, “Observation of bistable reflectivity of a phase-conjugated signal through intracavity nearly degenerate four-ware mixing,” Phys. Rev. Lett. 54, 1798–1801 (1985).
[CrossRef] [PubMed]

Fu, T.

Harter, D. J.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

D. J. Harter and R. W. Boyd, “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980).
[CrossRef]

Hendow, S. T.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
[CrossRef]

Inoue, K.

K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
[CrossRef]

Kazarinov, R. F.

R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
[CrossRef]

Kothari, N. C.

N. C. Kothari and R. Frey, “Bistable behavior of pump, probe, and conjugate signals through collinear intracavity nearly degenerate four-wave mixing,” Phys. Rev. A 34, 2013–2025 (1986).
[CrossRef] [PubMed]

Kwonk, S.-K.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Kyuma, K.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Lamb, W. E.

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429–1450 (1964).

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 9.

Lau, K. Y.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Logan, R. A.

R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
[CrossRef]

Mollow, B. R.

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1982).
[CrossRef]

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Mukai, T.

K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
[CrossRef]

Nakajima, H.

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. QE-22, 1349–1354 (1986).
[CrossRef]

H. Nakajima and R. Frey, “Intracavity nearly degenerate four-wave mixing in a (GaAl)As semiconductor laser,” Appl. Phys. Lett. 47, 769–771 (1985).
[CrossRef]

H. Nakajima and R. Frey, “Observation of bistable reflectivity of a phase-conjugated signal through intracavity nearly degenerate four-ware mixing,” Phys. Rev. Lett. 54, 1798–1801 (1985).
[CrossRef] [PubMed]

Narum, P.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Nilsen, J.

J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a homogeneously broadened two-level-system with saturating pump waves,” IEEE J. Quantum Electron. QE-18, 1947–1952 (1982).
[CrossRef]

Nishimura, Y.

Y. Nishimura and Y. Nishimura, “Spectral hole-burning and nonlinear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron. QE-9, 1011–1019 (1973).
[CrossRef]

Y. Nishimura and Y. Nishimura, “Spectral hole-burning and nonlinear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron. QE-9, 1011–1019 (1973).
[CrossRef]

Osinski, M.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

Palumbo, L. J.

J. Reintjes and L. J. Palumbo, “Phase conjugation in saturable amplifiers by degenerate frequency mixing,” IEEE J. Quantum Electron. QE-18, 1934–1940 (1982).
[CrossRef]

Raymer, M. G.

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

Reintjes, J.

J. Reintjes and L. J. Palumbo, “Phase conjugation in saturable amplifiers by degenerate frequency mixing,” IEEE J. Quantum Electron. QE-18, 1934–1940 (1982).
[CrossRef]

Saitoh, T.

K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
[CrossRef]

Sargent, M.

S. T. Hendow and M. Sargent, “Theory of single-mode laser instabilities,” J. Opt. Soc. Am. B 2, 84–101 (1985).
[CrossRef]

T. Fu and M. Sargent, “Effect of signal detuning on phase conjugation,” Opt. Lett. 4, 366–368 (1979).
[CrossRef] [PubMed]

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rep. 43, 223–265 (1978).
[CrossRef]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 9.

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 9.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 14.

Suematsu, Y.

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653–2664 (1981).
[CrossRef]

Sverdlov, B. N.

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
[CrossRef]

Tomita, A.

A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463–464 (1979).
[CrossRef]

Vahala, K.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Van Lerberghe, A.

Wu, F. Y.

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Yamada, M.

M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653–2664 (1981).
[CrossRef]

Yariv, A.

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a homogeneously broadened two-level-system with saturating pump waves,” IEEE J. Quantum Electron. QE-18, 1947–1952 (1982).
[CrossRef]

Zee, B.

B. Zee, “Broadening mechanism in semiconductor (GaAs) lasers: limitations to single mode power emission,” IEEE J. Quantum Electron. QE-14, 727–736 (1978).
[CrossRef]

Appl. Phys. Lett. (5)

A. Tomita, “Phase conjugation using gain saturation of a Nd:YAG laser,” Appl. Phys. Lett. 34, 463–464 (1979).
[CrossRef]

H. Nakajima and R. Frey, “Intracavity nearly degenerate four-wave mixing in a (GaAl)As semiconductor laser,” Appl. Phys. Lett. 47, 769–771 (1985).
[CrossRef]

G. P. Agrawal, “Highly nondegenerate four-wave mixing in semiconductor lasers due to spectral hole-burning,” Appl. Phys. Lett. 51, 302–304 (1987).
[CrossRef]

K. Inoue, T. Mukai, and T. Saitoh, “Nearly degenerate four-wave mixing in a traveling-wave semiconductor laser amplifier,” Appl. Phys. Lett. 51, 1051–1053 (1987).
[CrossRef]

K. Vahala, K. Kyuma, A. Yariv, S.-K. Kwonk, M. Cronin-Golomb, and K. Y. Lau, “Narrow linewidth, single frequency semiconductor laser with a phase-conjugate external-cavity mirror,” Appl. Phys. Lett. 49, 1563–1565 (1986).
[CrossRef]

Electron. Lett. (1)

G. P. Agrawal, “Amplifier-induced crosstalk in multichannel coherent lightwave systems,” Electron. Lett. 23,Oct23 (1987).
[CrossRef]

IEEE J. Quantum Electron. (13)

J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a homogeneously broadened two-level-system with saturating pump waves,” IEEE J. Quantum Electron. QE-18, 1947–1952 (1982).
[CrossRef]

Y. Nishimura and Y. Nishimura, “Spectral hole-burning and nonlinear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron. QE-9, 1011–1019 (1973).
[CrossRef]

B. Zee, “Broadening mechanism in semiconductor (GaAs) lasers: limitations to single mode power emission,” IEEE J. Quantum Electron. QE-14, 727–736 (1978).
[CrossRef]

A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, “Anomalous interaction of spectral modes in a semiconductor laser,” IEEE J. Quantum Electron. QE-11, 510–515 (1975).
[CrossRef]

M. Asada and Y. Suematsu, “Density-matrix theory of semiconductor lasers with relaxation broadening model—gain and gain-suppression in semiconductor lasers,” IEEE J. Quantum Electron. QE-21, 434–442 (1985).
[CrossRef]

G. P. Agrawal, “Gain nonlinearities in semiconductor lasers: theory and application to distributed feedback lasers,” IEEE J. Quantum Electron. QE-23, 860–868 (1987).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

D. J. Harter and R. W. Boyd, “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980).
[CrossRef]

H. Nakajima and R. Frey, “Collinear nearly degenerate four-wave mixing in intracavity amplifying media,” IEEE J. Quantum Electron. QE-22, 1349–1354 (1986).
[CrossRef]

J. Reintjes and L. J. Palumbo, “Phase conjugation in saturable amplifiers by degenerate frequency mixing,” IEEE J. Quantum Electron. QE-18, 1934–1940 (1982).
[CrossRef]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-17, 374–380 (1981).
[CrossRef]

Special issue on dynamic gratings and four-wave mixing, IEEE J. Quantum Electron. QE-22, 1194–1542 (1986).

J. Appl. Phys. (2)

M. Yamada and Y. Suematsu, “Analysis of gain suppression in undoped injection lasers,” J. Appl. Phys. 52, 2653–2664 (1981).
[CrossRef]

R. F. Kazarinov, C. H. Henry, and R. A. Logan, “Longitudinal mode self-stabilization in semiconductor lasers,” J. Appl. Phys. 53, 4631–4644 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (6)

Phys. Rep. (1)

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rep. 43, 223–265 (1978).
[CrossRef]

Phys. Rev. A (4)

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interaction in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1982).
[CrossRef]

N. C. Kothari and R. Frey, “Bistable behavior of pump, probe, and conjugate signals through collinear intracavity nearly degenerate four-wave mixing,” Phys. Rev. A 34, 2013–2025 (1986).
[CrossRef] [PubMed]

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429–1450 (1964).

Phys. Rev. Lett. (2)

H. Nakajima and R. Frey, “Observation of bistable reflectivity of a phase-conjugated signal through intracavity nearly degenerate four-ware mixing,” Phys. Rev. Lett. 54, 1798–1801 (1985).
[CrossRef] [PubMed]

F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, “Observation of amplification in a strongly driven two-level atomic system at optical frequencies,” Phys. Rev. Lett. 38, 1077–1080 (1977).
[CrossRef]

Other (5)

The expression for ΔN in Ref. 15 is in error. However, this does not affect the conclusions reported therein.

G. P. Agrawal and N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
[CrossRef]

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chap. 9.

See various chapters in R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 14.

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Figures (9)

Fig. 1
Fig. 1

Variation of the probe gain (normalized to its value expected in the absence of the pump wave) with the normalized pump–probe detuning Ωτs for several pump intensities P0 (normalized to the saturation intensity). For τs = 2 nsec, Ωτs = 1 corresponds to a detuning of about 80 MHz.

Fig. 2
Fig. 2

Variation of the index change (normalized to its value expected in the absence of the pump wave) with the normalized pump–probe detuning Ωτs for several pump intensities.

Fig. 3
Fig. 3

Effect of the linewidth enhancement factor β on the probe-gain spectrum at a fixed pump intensity equal to the saturation intensity (P0 = 1).

Fig. 4
Fig. 4

Variation of the conjugate reflectivity R and the probe transmittance T with the normalized pump–probe detuning Ωτs, for several incident pump intensities when the semiconductor laser operates as a traveling-wave amplifier with g0L = 4.

Fig. 5
Fig. 5

Same as for Fig. 4 except that the effect of linewidth enhancement factor β on the reflectivity and transmittance spectra is shown at a fixed pump intensity.

Fig. 6
Fig. 6

Same as for Fig. 4 except that the effect of small-signal gain on the reflectivity and transmittance spectra is shown for Pin = 0.1 and β = 5.

Fig. 7
Fig. 7

Reflectivity spectra under conditions identical to those of Fig. 5 except for the larger values of the linewidth enhancement factor β.

Fig. 8
Fig. 8

Reflectivity and transmittance spectra for the case of highly nondegenerate four-wave mixing resulting from spectral hole burning. Pump–probe detuning is normalized to the intraband population-relaxation time T1. For T1 = 0.3 psec, ΩT1 = 1 corresponds to a detuning of about 500 GHz. Other parameters are T2/T1 = 1/3 and τ/T1 = 10.

Fig. 9
Fig. 9

Same as for Fig. 8 except that the population-relaxation time T1 has been doubled. As a result, T2/T1 = 1/6 and τ/T1 = 5.

Equations (78)

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2 E - n 2 c 2 2 E t 2 = 1 0 c 2 2 P t 2 ,
E ( x , y , z , t ) = U ( x , y ) j E j ( z ) exp ( - i ω j t ) ,
Ω = ω 1 - ω 0 = ω 0 - ω 2
P = 0 χ ( N ) E ,
χ ( N ) = - n c ω 0 ( β + i ) g ( N )
g ( N ) = a ( N - N 0 ) .
d N d t = I q V - N τ s - g ( N ) ω 0 E 2 + D 2 N ,
d N d t = I q V - N τ s - g ( N ) ω 0 E 2 ,
N ( t ) = N ¯ + [ Δ N exp ( - i Ω t ) + c . c . ] ,
N ¯ = I ( τ s / q V ) + N 0 E ¯ 0 2 / P s 1 + E ¯ 0 2 / P s ,
Δ N = - C ( N ¯ - N 0 ) ( E 0 * E 1 + E 0 E 2 * ) / P s ( 1 + E ¯ 0 2 / P s - i Ω τ s ) ,
P s = ω 0 / ( Γ a τ s )
Γ = 0 w 0 d U ( x , y ) 2 d x d y - U ( x , y ) 2 d x d y .
P ( x , y , z , t ) = U ( x , y ) j P j ( z ) exp ( - i ω j t ) .
P 0 ( z ) = 0 A g ( N ¯ ) E 0 ( z ) ,
P 1 ( z ) = 0 A [ g ( N ¯ ) E 1 ( z ) + a Δ N E 0 ( z ) ] ,
P 2 ( z ) = 0 A [ g ( N ¯ ) E 2 ( z ) + a Δ N E 0 * ( z ) ] ,
A = - ( n c / ω 0 ) ( β + i )
g ( N ¯ ) = a ( N ¯ - N 0 ) = ( a τ s / q V ) I - a N 0 1 + E ¯ 0 2 / P s
Γ g ( N ¯ ) = g th = α m + α int ,
α m = 1 2 L ln ( 1 R 1 R 2 )
E ¯ 0 2 P s = g 0 g th - 1 = I - I th I th - I 0 ,
g 0 = Γ a N ¯ 0 ( I / I 0 - 1 ) ,
I th = I 0 + q V g th / ( a Γ τ s ) ,
I 0 = q V N 0 / τ s
P 1 = 0 ( χ p E 1 + χ FWM E 2 * ) ,
χ p = A g ( N ¯ ) ( 1 - C P 0 1 + P 0 - i Ω τ s ) ,
χ FWM = - A g ( N ¯ ) C ( E 0 2 / P s ) 1 + P 0 - i Ω τ s .
P 0 = E ¯ 0 2 / P s
χ p = Δ ɛ = 2 n [ Δ n - i g p ( c / 2 ω 0 ) ] .
Δ n = Δ n 0 [ 1 - C P 0 ( 1 + P 0 - Ω τ s / β ) ( 1 + P 0 ) 2 + ( Ω τ s ) 2 ] ,
g p = g ( N ¯ ) [ 1 - C P 0 ( 1 + P 0 + β Ω τ s ) ( 1 + P 0 ) 2 + ( Ω τ s ) 2 ] ,
Ω p = - 1 + P 0 β τ s [ ( 1 + β 2 ) 1 / 2 + 1 ] .
Ω 2 L = ( Δ 2 + Ω R 2 ) 1 / 2 ,
Ω 2 L Δ ( 1 + Ω R 2 2 Δ 2 ) .
Δ = - [ ( 1 + β 2 ) 1 / 2 + 1 ] / ( β τ s ) .
d 2 E j d z 2 + k j 2 E j = - ω j 2 Γ 0 c 2 P j ,
E 0 = P s 1 / 2 [ E f ( z ) exp ( i k 0 z ) + E b ( z ) exp ( - i k 0 z ) ] ,
E 1 = P s 1 / 2 [ A 1 ( z ) exp ( i k 1 z ) ] ,
E 2 = P s 1 / 2 [ A 2 ( z ) exp ( - i k 2 z ) ] .
d E f d z = - α 0 E f ,             d E b d z = α 0 E b ,
d A 1 d z = - α 1 A 1 + i κ 1 A 2 * exp ( i Δ k z ) ,
d A 2 * d z = α 2 * A 2 * + i κ 2 A 1 exp ( - i Δ k z ) ,
α 0 = - ( 1 - i β ) g 0 2 ( 1 + P 0 )
α j = α 0 ( 1 - C P 0 1 + P 0 ± i Ω τ s ) ,
κ j = - i α 0 ( 2 C E f ( z ) E b ( z ) 1 + P 0 ± i Ω τ s ) ,
Δ k = k 2 - k 1 = - 2 n ¯ Ω / c
P 0 = 1 L 0 L [ E f ( z ) 2 + E b ( z ) 2 ] d z ,
E f ( 0 ) 2 = E b ( L ) 2 = P in .
P in = g 0 L 2 P 0 1 + P 0 [ exp ( g 0 L 1 + P 0 ) - 1 ] - 1 .
E f ( z ) E b ( z ) = P in exp [ g 0 L 2 ( 1 + P 0 ) ]
R = | A 2 * ( 0 ) A 1 ( 0 ) | 2 = | κ 2 sin ( p L ) p cos ( p L ) + α sin ( p L ) | 2 ,
T = | A 1 ( L ) A 1 ( 0 ) | 2 = | p exp ( - α ¯ L ) p cos ( p L ) + α sin ( p L ) | 2 ,
p = ( κ 1 κ 2 * - α 2 ) 1 / 2 ,
α = ( α 1 + α 2 * + i Δ k ) / 2 ,
α ¯ = ( α 1 - α 2 * - i Δ k ) / 2.
Δ ν = 1 + P 0 π τ s .
T 0 = | A 1 ( L ) A 1 ( 0 ) | 2 = exp ( g 0 L 1 + P 0 ) .
d d t ρ 11 + ρ 11 - ρ ¯ 11 T 1 = μ i ( ρ 12 - ρ 21 ) E ,
d d t ρ 12 + ( i ω T + 1 T 2 ) ρ 12 = μ i ( ρ 11 - ρ ¯ 22 ) E ,
P = μ ( ω T ) D ( ω T ) ( ρ 12 + ρ 21 ) d ω T ,
ρ 11 = ρ ¯ 11 + Δ ρ 11 ( 0 ) + [ Δ ρ 11 ( Ω ) exp ( i Ω t ) + c . c . ] .
P 0 = 0 [ χ L ( ω 0 ) + χ 1 ( 0 ) ] E 0 ,
P 1 = 0 [ χ L ( ω 1 ) + χ 1 ( Ω ) + χ 2 ( Ω ) ] E 1 + 0 χ 3 ( Ω ) E 2 * ,
P 2 = 0 [ χ L ( ω 2 ) + χ 1 ( - Ω ) + χ 2 ( - Ω ) ] E 2 + 0 χ 3 ( - Ω ) E 1 * ,
χ L ( ω j ) = - n c ω j ( β + i ) g ( N ¯ , ω j )
g ( N ¯ , ω j ) = g ( N ¯ ) [ 1 - ( ω p - ω j ) 2 / Δ ω g 2 ] ,
χ 1 ( Ω ) = i n c ω 0 g ( N ¯ ) C ( 1 - i β ¯ ) ( 1 - i Ω T 2 / 2 ) E 0 2 P ¯ s ,
χ 2 ( Ω ) = i n c ω 0 g ( N ¯ ) C ( 1 - i β ¯ ) ( 1 - i Ω T 2 / 2 ) ( 1 - i Ω T 1 ) E 0 2 P ¯ s ,
χ 3 ( Ω ) = i n c ω 0 g ( N ¯ ) C [ 1 - i β ¯ ( 1 - i Ω T 2 ) ] ( 1 - i Ω T 2 ) ( 1 - i Ω T 1 ) E 0 2 P ¯ s ,
P ¯ s = 2 / ( μ 2 T 1 T 2 )
β ¯ = 1 g 0 T 2 ( d g d ω ) ω = ω 0
C = 1 Γ 0 w 0 d U ( x , y ) 4 d x d y
α 0 = - g 0 2 ( 1 + P 0 ) [ 1 - i β - ( 1 - i β ¯ ) C r P 0 ] ,
α 1 = - g 0 2 ( 1 + P 0 ) [ 1 - i β - Ω 2 Δ ω g 2 - ( 1 - i β ¯ ) C r P 0 1 - i Ω T 2 / 2 × ( 1 + 1 1 - i Ω T 1 ) ] ,
κ 1 = i g 0 2 ( 1 + P 0 ) C r [ 1 - i β ¯ ( 1 - i Ω T 2 ) ] ( 1 - i Ω T 2 ) ( 1 - i Ω T 1 ) 2 E f ( z ) E b ( z ) .
β ¯ = 2 ( ω 0 - ω p ) T 2 Δ ω g 2 2 ( ω 0 - ω p ) T 2 ,
g p = g 0 1 + P 0 { 1 - Ω 2 Δ ω g 2 - C r P 0 1 + ( Ω T 2 / 2 ) 2 × [ 1 + 1 - Ω 2 T 1 T 2 / 2 1 + ( Ω T 1 ) 2 ] } .

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