Abstract

Spectral phase conjugation with short pump pulses in a third-order nonlinear material is analyzed in depth. It is shown that if signal amplification is considered, the conversion efficiency can be significantly higher than previously considered, while the spectral phase conjugation operation remains accurate. A novel method of compensating for cross-phase modulation, the main parasitic effect, is also proposed. The validity of our theory and the performance of the spectral phase conjugation scheme are studied numerically.

© 2004 Optical Society of America

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References

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Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Rebane, J. Aaviksoo, and J. Kuhl, �??Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography,�?? Appl. Phys. Lett. 54, 93 (1989).
[CrossRef]

J. Appl. Phys. (1)

R. Quintero-Torres and M. Thakur, �??Measurement of the nonlinear refractive index of polydiacetylene using Michelson interferometry and z-scan,�?? J. Appl. Phys. 85, 401 (1999).
[CrossRef]

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

Opt. Lett. (6)

Quantum Electron. (1)

A. M. Weiner, D. E. Leaird, D. H. Reitze, and Eung Gi Paek, �??Femtosecond spectral holography,�?? IEEE J. Quantum Electron. 28, 2251 (1992).
[CrossRef]

Rev. Sci. Instrum. (1)

A. M. Weiner, �??Femtosecond pulse shaping using spatial light modulators,�?? Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Supplementary Material (2)

» Media 1: MPG (260 KB)     
» Media 2: MPG (285 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Setup of SPC by four-wave mixing. As (t) is the signal pulse, Ap (t) and Aq (t) are the pump pulses, and Ai (t) is the backward-propagating idler pulse (After Ref. [1]).

Fig. 2.
Fig. 2.

(a) Amplitude and (b) phase of output idler (solid lines) A i ( L 2 , t ) compared with input signal (dash lines) A s ( L 2 , t ) . XPM is neglected in this example. As predicted, the output idler is time-reversed and phase-conjugated with respect to the input signal. Parameters used are n 2 = 1×10-11 cm2/W, n 0 = 1.7, λ 0 = 800 nm, L = 2 mm, d = 5 μm, Ts = 1 ps, Tp = 100 fs, Ep = 12.8 nJ, pump fluence = E p Ld . Conversion efficiency is 100%.

Fig. 3.
Fig. 3.

Conversion efficiencies from simulations compared with predictions from first-order analysis and coupled-mode theory. Simulation results agree well with coupled-mode theory. See caption of Fig. 2 for parameters used.

Fig. 4.
Fig. 4.

(a) and (b) plot the normalized amplitude and phase of the output idler A i ( L 2 , t ) compared to the SPC of the input signal A s * ( L 2 , t ) respectively when XPM is present. The amplitude plots are normalized with respect to their peaks. The output idler is distorted and the conversion efficiency is only 34%, much lower than the theoretical efficiency 100%. (c) and (d) plot the same data, but with XPM compensation. The efficiency is back to 100% and the accuracy is restored.

Fig. 5.
Fig. 5.

Plots of amplitude and phase of one pump pulse with ideal phase adjustment according to Eq. (57) in time and frequency domain. Top-left: temporal envelope; Bottom-left: temporal phase; Top-right: envelope spectrum; Bottom-right: spectral phase. The simple pulse shape should be easily produced by many pulse shaping methods.

Fig. 6.
Fig. 6.

A movie that shows the evolution of the two pumps, signal and idler, with a signal energy below the pump depletion limit. See caption of Fig. 2 for parameters used. The signal energy is 1 pJ. [Media 1]

Fig. 7.
Fig. 7.

Similar to Fig. 6, but with a signal energy of 5 nJ, much above the pump depletion limit. [Media 2]

Fig. 8.
Fig. 8.

Amplitude and phase of output idler and input signal for the wave mixing process shown in Fig. 7 to demonstrate the effect of pump depletion.

Equations (61)

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A p x + 1 v x A p t = [ 2 A s A i A q * + ( A p 2 + 2 A q 2 + 2 A s 2 + 2 A i 2 ) A p ] ,
A q x + 1 v x A q t = [ 2 A s A i A p * + ( 2 A p 2 + A q 2 + 2 A s 2 + 2 A i 2 ) A q ] ,
A s z + 1 v A s t = [ 2 A p A q A i * + ( 2 A p 2 + 2 A q 2 + A s 2 + 2 A i 2 ) A s ] ,
A i z + 1 v A i t = [ 2 A p A q A s * + ( 2 A p 2 + 2 A q 2 + 2 A s 2 + A i 2 ) A i ] ,
γ = 3 ω 0 χ ( 3 ) 8 c n 0 ,
A p ( 0 ) x t = A p ( t ) ,
A q ( 0 ) x t = A q ( t ) ,
A s ( 0 ) z t = F ( t z v )
A i ( 0 ) z t = 0 .
A i ( 1 ) ( L 2 , t ) = j F * ( t + L 2 v ) 2 γv A p ( t ) A q ( t ) dt ,
η ( 1 ) A i ( 1 ) ( L 2 , t ) 2 dt A s ( 1 ) ( L 2 , t ) 2 dt = [ 2 γv A p ( t ) A q ( t ) dt ] 2 ,
L v > > T s > > ( T p or T q ) > > d v x ,
v A s z t z + A s z t t = jg ( t ) A i * z t ,
v A i z t z + A i z t t = jg ( t ) A s * z t ,
where g ( t ) = 2 γv A p ( t ) A q ( t ) .
v A i * z t z + A i * z t t = j g * ( t ) A s z t ,
A ˜ s κ t = A s z t exp ( jκz ) dz ,
A ˜ i κ t = A i * z t exp ( jκz ) dz .
jκv A ˜ s + A ˜ s t = jg ( t ) A ˜ i ,
jκv A ˜ i + A ˜ i t = j g * ( t ) A ˜ s .
exp ( jκvt ) ( jκv A ˜ s + A ˜ s t ) = jg ( t ) exp ( jκvt ) A ˜ i ,
exp ( jκvt ) ( jκv A ˜ i + A ˜ i t ) = j g * ( t ) exp ( jκvt ) A ˜ s ,
t [ exp ( jκvt ) A ˜ s ] = jg ( t ) exp ( jκvt ) A ˜ i ,
t [ exp ( jκvt ) A ˜ i ] = j g * ( t ) exp ( jκvt ) A ˜ s .
A κ t = exp ( jκvt ) A ˜ s = exp ( jκvt ) A s z t exp ( jκz ) dz ,
B κ t = exp ( jκvt ) A ˜ i = exp ( jκvt ) A i * z t exp ( jκz ) dz ,
A t = jg ( t ) exp ( 2 jκvt ) B ,
B t = j g * ( t ) exp ( 2 jκvt ) A .
v A s z + A s t = 0 ,
v A i z + A i t = 0 .
κv = Ω ,
g ( t ) exp ( 2 jκvt ) g ( t ) .
A t = jg ( t ) B ,
B t = j g * ( t ) A .
A s z L 2 v = F ( L 2 v z v ) ,
A i ( z , L 2 v ) = 0 .
A κ t = A κ L 2 v cosh [ L 2 v t g ( t ) dt ] ,
B κ t = j A κ L 2 v exp ( ) sinh [ L 2 v t g ( t ) dt ] .
A s z t = F ( t z v ) cosh [ L 2 v t g ( t ) dt ] ,
A i z t = j F * ( t z v ) exp ( ) sinh [ L 2 v t g ( t ) dt ] .
A i ( L 2 , t ) = j F * ( t + L 2 v ) exp ( ) sinh [ g ( t ) dt ] .
η A i ( L 2 , t ) 2 dt A s ( L 2 , t ) 2 dt = sinh 2 [ 2 γv A p ( t ) A q ( t ) dt ] .
v A s z t z + A s z t t = jg ( t ) A i * z t + jc ( t ) A s z t ,
v A i z t z + A i z t t = jg ( t ) A s * z t + jc ( t ) A i z t ,
where g ( t ) = 2 γv A p ( t ) A q ( t ) ,
c ( t ) = 2 γv [ A p ( t ) 2 + A q ( t ) 2 ] .
A κ t = exp [ jκvt j t c ( t ) dt ] A s z t exp ( jκz ) dz ,
B κ t = exp [ jκvt + j t c ( t ) dt ] A i * z t exp ( jκz ) dz .
A t = jg ( t ) exp [ 2 j t c ( t ) dt ] B ,
B t = j g * ( t ) exp [ 2 j t c ( t ) dt ] A .
θ ( t ) = θ 0 + 2 t c ( t ) dt .
A t = j g ( t ) exp ( j θ 0 ) B ,
B t = j g ( t ) exp ( j θ 0 ) A .
A s z t = F ( t z v ) exp [ j t c ( t ) dt ] cosh [ t g ( t ) dt ] ,
A i z t = j F * ( t z v ) exp [ j θ 0 + j t c ( t ) dt ] sinh [ t g ( t ) dt ] ,
A i ( L 2 , t ) = j F * ( t + L 2 v ) exp [ j θ 0 + j c ( t ) dt ] sinh [ g ( t ) dt ] .
θ p ( t ) + θ q ( t ) = θ 0 + 4 γv t A p ( t ) 2 + A q ( t ) 2 dt .
A p ( t ) = A q ( t ) = exp ( t 2 2 T p 2 ) ,
F ( τ ) = A s 0 { exp [ 1 + j 2 ( τ + 2 T s T s ) 2 ] + 1 2 exp [ 1 2 ( τ 2 T s T s ) 2 ] } .
A p > > 2 γ A s A i A q d
E s < < n 0 d T s η 0 γ η .

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