Abstract

The analytical theory of four-wave mixing in semiconductor amplifiers is developed. The theory applies to arbitrary saturation conditions. If carrier depletion and spectral hole burning are considered, the conversion efficiency is the maximum at a total input power that produces a gain compression of e−2 (approximately −9 dB), independently of the physical parameters of the amplifier.

© 1994 Optical Society of America

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  1. J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
    [CrossRef]
  2. G. P. Agrawal, J. Opt. Soc. Am. B 5, 147 (1988).
    [CrossRef]
  3. K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
    [CrossRef]
  4. G. P. Agrawal, IEEE J. Quantum Electron. QE-23, 860 (1987).
    [CrossRef]
  5. A. Uskov, J. Mørk, J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral hole buring,”submitted toIEEE J. Quantum Electron.
  6. K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
    [CrossRef]
  7. K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
    [CrossRef]

1993 (1)

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

1992 (2)

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

1990 (1)

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

1988 (1)

1987 (1)

G. P. Agrawal, IEEE J. Quantum Electron. QE-23, 860 (1987).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, J. Opt. Soc. Am. B 5, 147 (1988).
[CrossRef]

G. P. Agrawal, IEEE J. Quantum Electron. QE-23, 860 (1987).
[CrossRef]

Dawson, J. W.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Eisenstein, G.

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

Hall, K. J.

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

Hall, K. L.

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

Ippen, E. P.

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

Kakui, M.

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

Kikuchi, K.

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

Koren, U.

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

Lee, T. P.

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

Lenz, G.

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

Mark, J.

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

A. Uskov, J. Mørk, J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral hole buring,”submitted toIEEE J. Quantum Electron.

Miller, B. I.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Mørk, J.

A. Uskov, J. Mørk, J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral hole buring,”submitted toIEEE J. Quantum Electron.

Newkirk, M. A.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Park, N.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Reybon, G.

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

Uskov, A.

A. Uskov, J. Mørk, J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral hole buring,”submitted toIEEE J. Quantum Electron.

Vahala, K. J.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Zah, C. E.

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

Zhou, J.

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

Appl. Phys. Lett. (3)

J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, B. I. Miller, Appl. Phys. Lett. 63, 1179 (1993).
[CrossRef]

K. J. Hall, J. Mark, E. P. Ippen, G. Eisenstein, Appl. Phys. Lett. 56, 1740 (1990).
[CrossRef]

K. L. Hall, G. Lenz, E. P. Ippen, U. Koren, G. Reybon, Appl. Phys. Lett. 61, 2512 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Kikuchi, M. Kakui, C. E. Zah, T. P. Lee, IEEE J. Quantum Electron. 28, 151 (1992).
[CrossRef]

G. P. Agrawal, IEEE J. Quantum Electron. QE-23, 860 (1987).
[CrossRef]

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

Other (1)

A. Uskov, J. Mørk, J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral hole buring,”submitted toIEEE J. Quantum Electron.

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Figures (1)

Fig. 1
Fig. 1

Four-wave-mixing efficiency η versus the sum of the pump and the probe input powers S(0) for the values of the physical constants reported in the text.

Equations (24)

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d E 0 d z = { ½ [ g ( z ) γ sc ] + i Δ k ( z ) } E 0 ,
g ( z ) = g ¯ 1 + S ( z ) / P s [ 1 S ( z ) ] ,
Δ k ( z ) = [ α β S ( z ) ] g ¯ 1 + S ( z ) / P s ,
d E 2 d z = { ½ g ( z ) γ sc ] + i Δ k ( z ) } E 2 g ¯ 1 + S ( z ) / P s × E 0 2 E 1 * P s { ( 1 i α ) 2 [ 1 + S ( z ) / P s + i Ω τ s ] + ( 1 i β ) P s 2 ( 1 + i Ω τ 1 ) } .
d E 1 d z = { ½ ( z ) γ sc ] + i Δ k ( z ) } E 1 ,
E j ( z ) = E j ( 0 ) exp 0 z d z { ½ g ( z ) γ sc ] + i Δ k ( z ) } , j = 0 , 1 .
E 2 ( z ) = R ( Ω ) exp 0 z d z × { ½ [ g ( z ) γ sc ] + i Δ k ( z ) } E ( z ) ,
E ( z ) = 0 z d z g ¯ 1 + S ( z ) / P s E 0 2 ( z ) E 1 * ( z ) P s × exp 0 z d z { ½ [ g ( z ) γ sc ] + i Δ k ( z ) } ,
R ( Ω ) = ( 1 i α ) 2 ( 1 + i Ω τ s ) ( 1 i β ) P s 2 ( 1 + i Ω τ 1 ) .
E ( z ) = E 0 2 ( 0 ) E 1 * ( 0 ) P s 0 z d z g ¯ 1 + S ( z ) / P s × exp 0 z d z [ g ( z ) γ sc ] .
0 z d z g ¯ 1 + S ( z ) / P s exp 0 z d z [ g ( z ) γ sc ] = P s S ( 0 ) 1 1 + P s { 0 z d z [ g ( z ) γ sc ] + ( g ¯ γ sc ) z } .
exp 0 z d z [ g ( z ) γ sc ] = S ( z ) S ( 0 ) ,
E ( z ) = E 0 2 ( 0 ) E 1 * ( 0 ) S ( 0 ) 1 1 + P s ln G 0 ( z ) G ( z ) ,
G ( z ) = exp 0 z d z [ g ( z ) γ sc ] ,
G 0 ( z ) = exp [ ( g ¯ γ sc ) z ] .
η = | E 2 ( L ) | 2 | E 1 ( 0 ) | 2 = [ | E 0 ( 0 ) | 2 | E 0 ( 0 ) | 2 + | E 1 ( 0 ) | 2 ] 2 | R ( Ω ) | 2 ( 1 + P s ) 2 ( ln G 0 G ) 2 G ,
G = G ( L ) = | E 0 ( L ) | 2 | E 0 ( 0 ) | 2 = | E 1 ( L ) | 2 | E 1 ( 0 ) | 2 = exp 0 L d z [ g ( z ) γ sc ]
( G G 0 ) max = e 2 ,
d S d z = [ g ¯ 1 + S / P s ( 1 S ) γ sc ] S ,
S ( 0 ) P s = ( g ¯ γ sc ) ( 1 χ ) ( g ¯ P s + γ sc ) ( G χ ) ,
χ = ( G G 0 ) ξ , ξ = g ¯ P s + γ sc g ¯ ( 1 + P s ) .
S ( L ) P s = GS ( 0 ) P s G G 1 ln ( G 0 G ) .
g ( z ) = g ¯ 1 + S ( z ) / P s g ¯ S ( z )
1 4 [ G + 1 G 1 ( ln G 0 G ) 2 + 2 ln G 0 G ] 2

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