Abstract

Four-wave mixing with optical pulses is considered analytically. It is shown that, whereas true time reversal of an amplitude pulse is not possible with continuous-wave pumping, this can be achieved by short-pulse pumping of a long, narrow, nonlinear medium, although time-dependent phase variations are not time reversed.

© 1980 Optical Society of America

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References

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  1. For a recent review see, for example, A Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14650 (1978);see also A. Yariv, IEEE J. Quantum Electron. QE-15524 (1979).
    [Crossref]
  2. J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372 (1978).
    [Crossref]
  3. P. D. Maker, R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
    [Crossref]

1978 (2)

For a recent review see, for example, A Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14650 (1978);see also A. Yariv, IEEE J. Quantum Electron. QE-15524 (1979).
[Crossref]

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

1965 (1)

P. D. Maker, R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Maker, P. D.

P. D. Maker, R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

Terhune, R. W.

P. D. Maker, R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

Yariv, A

For a recent review see, for example, A Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14650 (1978);see also A. Yariv, IEEE J. Quantum Electron. QE-15524 (1979).
[Crossref]

Appl. Phys. Lett. (1)

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372 (1978).
[Crossref]

IEEE J. Quantum Electron. (1)

For a recent review see, for example, A Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14650 (1978);see also A. Yariv, IEEE J. Quantum Electron. QE-15524 (1979).
[Crossref]

Phys. Rev. (1)

P. D. Maker, R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Configuration of incident fields E1, E2, and E3 and the nonlinear medium.

Fig. 2
Fig. 2

Comparison of incident and backward pulses for short-pulse pumping of the nonlinear medium. (a) An observer at O sees the incident and backward amplitude envelopes as being time reversed; points a–e on the pulse pass him in the order e, d, c, b, a for the incident pulse and in the order a′, b′, c′, d′, e′ for the corresponding points on the backward pulse. (b) An observer at O sees the wavefronts a′–e′ of the backward pulse pass in the order a′, b′, c′, d′, e′, compared to the order e, d, c, b, a for the corresponding phase fronts of the incident pulse; but additionally the phase leads δ and γ have been changed to phase lags with respect to z on the backward pulse (although they remain phase leads with respect to t). The pattern of spacings between phase fronts is therefore not time reversed. (The phase leads and lags have been exaggerated for clarity.) The phase envelopes giving rise to these phase-front patterns show also that true time reversal does not apply because of the inversion of the backward phase envelope.

Equations (21)

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E 1 ( x , t ) = ½ A 1 ( t ) e i ( ω t k x ) + c . c . ; A 1 ( t ) = a 1 ( ω 1 ) e i ω 1 t d ω 1 ,
E 2 ( x , t ) = ½ A 2 ( t ) e i ( ω t + k x ) + c . c . ; A 2 ( t ) = a 2 ( ω 2 ) e i ω 2 t d ω 2 ,
E 3 ( z , t ) = ½ A 3 ( t z υ ) e i ( ω t k z ) + c . c . ; A 3 ( t z υ ) = a 3 ( ω 3 ) e i ω 3 [ t ( z / υ ) ] d ω 3 ,
E 4 ( z , t ) = ½ A 4 ( r , t ) e i ( ω t + k z ) + c . c . ; A 4 ( r , t ) = a 4 ( r , ω 4 ) e i ω 4 t d ω 4 .
P ( NL ) ( r , t ) = ½ χ ( 3 ) ( ω + ω 1 + ω 2 ω 3 ; ω + ω 1 , ω + ω 2 , ω ω 3 ) a 1 ( ω 1 ) a 2 ( ω 2 ) × a 3 * ( ω 3 ) exp { i [ ( ω + ω 1 + ω 2 + ω 3 ) t + ( k + ω 3 υ ) z ] } × d ω 1 d ω 2 d ω 3 + c . c .
υ a 4 z i ω 4 a 4 = i 2 π ω n 2 χ ( 3 ) × ( ω + ω 4 ; + ω + ω 1 , ω + ω 2 , ω ω 1 ω 2 + ω 4 ) × a 1 ( ω 1 ) a 2 ( ω 2 ) a 3 * ( ω 1 + ω 2 ω 4 ) × e i ( ω 1 + ω 2 ω 4 ) ( z / υ ) d ω 1 d ω 2 .
a 4 ( z , ω 4 ) = i 2 π ω n 2 υ e i ω 4 z / υ χ ( 3 ) a 1 a 2 a 3 * × ( z / 2 ) L ( z / 2 ) L e i ( ω 1 + ω 2 2 ω 4 ) ( z / υ ) d z d ω 1 d ω 2 .
( z / 2 ) L ( z / 2 ) L e i ( ω 1 + ω 2 2 ω 4 ) ( z / υ ) d z = υ ( ω 1 + ω 2 2 ω 4 ) sin ( ω 1 + ω 2 2 ω 4 ) z L υ
ω 1 + ω 2 = 2 ω 4 .
a 4 ( z , ω 4 ) = D ( ω 4 ) a 3 * ( ω 4 ) e i ω 4 ( z / υ ) ,
D ( ω 4 ) = i ( 2 π ) 2 ω n 2 χ ( 3 ) ( ω + ω 4 ; ω + ω 1 , ω + 2 ω 4 ω 1 , ω ω 4 ) a 1 ( ω 1 ) a 2 ( 2 ω 4 ω 1 ) d ω 1 .
a 1 ( ω 1 ) a 2 ( 2 ω 4 ω 1 ) d ω 1 = 1 2 π A 1 ( t ) A 2 ( t ) e 2 i ω 4 t d t .
A 4 = i 8 π n 2 c ω [ 2 π χ ( 3 ) n ] F × a 3 * ( ω 4 ) e i ω 4 [ t + ( z / υ ) 2 t p ] d ω 4 ,
F = ( n c / 8 π ) A 1 ( t ) A 2 ( t ) d t .
A 4 ( t z υ ) = i 8 π ω n 2 c [ 2 π χ ( 3 ) n ] FA 3 * [ ( t 2 t p ) z υ ] .
A 3 ( τ ) = α 3 ( τ ) e i ϕ 3 ( τ ) ,
A 4 α 3 [ ( t 2 t p ) z υ ] e i ϕ 0 ,
A 4 α 3 [ ( t 2 t p ) z υ ] exp { i ϕ 3 [ ( t 2 t p ) z υ ] } .
E 3 ( z , t ) = α 3 ( t z υ ) cos [ ω t k z + ϕ 3 ( t z υ ) ] ,
E 4 ( z , t ) = 8 π ω n 2 c | 2 π χ ( 3 ) n | | F | α 3 [ ( t 2 t p ) z υ ] × cos { ω t + k z ϕ 3 [ ( t 2 t p ) z υ ] + ψ 0 } ,
I 4 ( t z υ ) = ( 8 π ω n 2 c ) 2 [ 2 π | χ ( 3 ) | n ] 2 × | F | 2 I 3 [ ( t 2 t p ) z υ ] ,

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