Abstract

A theory of four-wave mixing in semiconductor laser media is developed by considering the contributions of both the gain and index gratings created by the carrier-density modulation occurring at the beat frequency of the pump and the probe waves. The general formalism can be applied to semiconductor lasers operating below or above threshold. As an illustration, we consider the case in which the semiconductor laser is operated as a traveling-wave amplifier. The results show that the dominant contribution to the four-wave mixing process comes from the index grating. Further, the index grating makes the probe transmission asymmetric with respect to the pump–probe detuning.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. For a review, see various chapters in R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  2. H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985); IEEE J. Quantum Electron. QE-22, 1349 (1986).
    [CrossRef]
  3. R. Frey, Opt. Lett. 11, 91 (1986).
    [CrossRef] [PubMed]
  4. G. P. Agrawal, N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), Chap. 2.
    [CrossRef]
  5. T. Fu, M. Sargent, Opt. Lett. 4, 366 (1979).
    [CrossRef] [PubMed]
  6. D. J. Harter, R. W. Boyd, IEEE J. Quantum Electron. QE-16, 1126 (1980).
    [CrossRef]
  7. A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
    [CrossRef]
  8. G. P. Agrawal, C. Flytzanis, IEEE J. Quantum Electron. QE-17, 374 (1981).
    [CrossRef]
  9. G. P. Agrawal, J. Opt. Soc. Am. 73, 654 (1983).
    [CrossRef]
  10. H. Nakajima, R. Frey, Phys. Rev. Lett. 54, 1798 (1985).
    [CrossRef] [PubMed]
  11. N. C. Kothari, R. Frey, Phys. Rev. A 34, 2013 (1986).
    [CrossRef] [PubMed]

1986

N. C. Kothari, R. Frey, Phys. Rev. A 34, 2013 (1986).
[CrossRef] [PubMed]

R. Frey, Opt. Lett. 11, 91 (1986).
[CrossRef] [PubMed]

1985

H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985); IEEE J. Quantum Electron. QE-22, 1349 (1986).
[CrossRef]

H. Nakajima, R. Frey, Phys. Rev. Lett. 54, 1798 (1985).
[CrossRef] [PubMed]

1983

1981

G. P. Agrawal, C. Flytzanis, IEEE J. Quantum Electron. QE-17, 374 (1981).
[CrossRef]

1980

D. J. Harter, R. W. Boyd, IEEE J. Quantum Electron. QE-16, 1126 (1980).
[CrossRef]

1979

1975

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, J. Opt. Soc. Am. 73, 654 (1983).
[CrossRef]

G. P. Agrawal, C. Flytzanis, IEEE J. Quantum Electron. QE-17, 374 (1981).
[CrossRef]

G. P. Agrawal, N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), Chap. 2.
[CrossRef]

Bogatov, A. P.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
[CrossRef]

Boyd, R. W.

D. J. Harter, R. W. Boyd, IEEE J. Quantum Electron. QE-16, 1126 (1980).
[CrossRef]

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), Chap. 2.
[CrossRef]

Eliseev, P. G.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
[CrossRef]

Flytzanis, C.

G. P. Agrawal, C. Flytzanis, IEEE J. Quantum Electron. QE-17, 374 (1981).
[CrossRef]

Frey, R.

N. C. Kothari, R. Frey, Phys. Rev. A 34, 2013 (1986).
[CrossRef] [PubMed]

R. Frey, Opt. Lett. 11, 91 (1986).
[CrossRef] [PubMed]

H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985); IEEE J. Quantum Electron. QE-22, 1349 (1986).
[CrossRef]

H. Nakajima, R. Frey, Phys. Rev. Lett. 54, 1798 (1985).
[CrossRef] [PubMed]

Fu, T.

Harter, D. J.

D. J. Harter, R. W. Boyd, IEEE J. Quantum Electron. QE-16, 1126 (1980).
[CrossRef]

Kothari, N. C.

N. C. Kothari, R. Frey, Phys. Rev. A 34, 2013 (1986).
[CrossRef] [PubMed]

Nakajima, H.

H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985); IEEE J. Quantum Electron. QE-22, 1349 (1986).
[CrossRef]

H. Nakajima, R. Frey, Phys. Rev. Lett. 54, 1798 (1985).
[CrossRef] [PubMed]

Sargent, M.

Sverdlov, B. N.

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
[CrossRef]

Appl. Phys. Lett.

H. Nakajima, R. Frey, Appl. Phys. Lett. 47, 769 (1985); IEEE J. Quantum Electron. QE-22, 1349 (1986).
[CrossRef]

IEEE J. Quantum Electron.

D. J. Harter, R. W. Boyd, IEEE J. Quantum Electron. QE-16, 1126 (1980).
[CrossRef]

A. P. Bogatov, P. G. Eliseev, B. N. Sverdlov, IEEE J. Quantum Electron. QE-11, 510 (1975).
[CrossRef]

G. P. Agrawal, C. Flytzanis, IEEE J. Quantum Electron. QE-17, 374 (1981).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Phys. Rev. A

N. C. Kothari, R. Frey, Phys. Rev. A 34, 2013 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett.

H. Nakajima, R. Frey, Phys. Rev. Lett. 54, 1798 (1985).
[CrossRef] [PubMed]

Other

For a review, see various chapters in R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

G. P. Agrawal, N. K. Dutta, Long- Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), Chap. 2.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Variation of the conjugate reflectivity R with the pump–probe detuning (in normalized units) for three values of the linewidth-enhancement factor β.

Fig. 2
Fig. 2

Variation of the probe transmittivity T with the pump–probe detuning (in normalized units) for three values of β. Note the asymmetric enhancement of T for negative values of Ωτs.

Equations (28)

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

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 ) ,
P ( x , y , z , t ) = U ( x , y ) j P j ( z ) exp ( - i ω j t ) ,
P = 0 χ ( N ) E ,
χ ( N ) = - n c ω ( 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 ,
Ω = ω 1 - ω 0 = ω 0 - ω 2 .
N ( t ) = N ¯ + [ Δ N exp ( - i Ω t ) + c . c . ] ,
Δ N = Γ ( N ¯ - N 0 ) ( E 0 * E 1 + E 0 E 2 * ) ( 1 + E ¯ 0 2 / P s + i Ω τ s ) P s ,
P s = ω 0 / ( Γ a τ s )
P j = - 0 n c ω j ( β + i ) Γ g ( N ¯ ) × [ E j + Γ ( E 0 2 E j + E 0 2 E 3 - j * ) ( 1 + E ¯ 0 2 / P s + i Ω τ s ) P s ] .
Γ g ( N ¯ ) = g 0 1 + E ¯ 0 2 / P s ,
g 0 = Γ a [ ( I τ s / q V ) - N 0 ]
E ¯ 0 2 P s = g 0 g th - 1 = I - I th I th - I 0 ,
E 0 = P s [ A f exp ( i k 0 z ) + A b exp ( - i k 0 z ) ] , E 1 = P s A 1 exp ( i k 1 z ) ,             E 2 = P s A 2 exp ( - i k 2 z ) ,
P s d d z A j = ± i ω j 2 0 n c P j exp ( i k j z ) .
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 j = i 2 ( β + i ) g 0 1 + P 0 ( 1 + Γ P 0 1 + P 0 ± i Ω τ s ) ,
κ j = - 1 2 ( β + i ) g 0 1 + P 0 ( Γ P 0 1 + P 0 ± i Ω τ s ) .
P 0 = 2 P in { [ exp ( g 0 L ) - 1 ] / g 0 L } .
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 = ½ ( α 1 + α 2 * - i Δ k ) ,
α ¯ = ½ ( α 1 - α 2 * + i Δ k ) .
Δ ν = ( 1 + P 0 ) / ( π τ s ) .
G = - 2 Re ( α 1 ) = g 0 1 + P 0 [ 1 + Γ P 0 ( 1 + P 0 - β Ω τ s ) ( 1 + P 0 ) 2 + ( Ω τ s ) 2 ] .

Metrics