Abstract

The copropagation of an intense pump pulse and a weak probe pulse in a lossy Kerr nonlinear medium has been studied. Specifically, the influence that the group-velocity dispersion, the attenuation, and the initial time delay between the pulses have on the amount of cross-phase modulation induced by the pump pulse are highlighted. Analytic expressions for the phase modulation and the chirp experienced by an initial unchirped Gaussian pulse are presented.

© 1995 Optical Society of America

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References

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  1. K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
    [CrossRef]
  2. R. R. Alfano and S. L. Shapiro, “Observation of self phase modulation and small scale filamentation in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
    [CrossRef]
  3. R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibres,” Phys. Rev. A 17, 1448–1453 (1978).
    [CrossRef]
  4. M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, “Cross-phase modulation in optical fibres,” Opt. Lett. 12, 625–627 (1987).
    [CrossRef] [PubMed]
  5. R. R. Alfano and P. P. Ho, “Self-, cross-, induced-phase modulation of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
    [CrossRef]
  6. T. Morioka and M. Saruwatari, “Demonstration of chirping manipulation of laser diode chirped optical pulses utilizing cross phase modulation,” Electron. Lett. 25, 646–648 (1989).
    [CrossRef]
  7. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1989), App. 5, p. 307.
  8. G. P. Agrawal, Nonlinear Fibre Optics (Academic, New York, 1989), Chap. 7, p. 176.
  9. M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
    [CrossRef]
  10. S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
    [CrossRef]
  11. Ref. 8, p. 200.
  12. J. T. Manassah, “Induced phase modulation of the stimulated Raman pulse in optical fibers,” Appl. Opt. 26, 3747–3749 (1987).
    [CrossRef] [PubMed]
  13. P. S. Spencer and K. A. Shore, Future Directions of Nonlinear Dynamics in Physical and Biological Systems, Proceedings of the NATO MIDIT Advanced Studies Institute Meeting, P. L. Christiansen, ed. (Plenum, New York, 1992), pp. 417–420.

1994 (1)

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

1993 (1)

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

1992 (1)

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

1989 (1)

T. Morioka and M. Saruwatari, “Demonstration of chirping manipulation of laser diode chirped optical pulses utilizing cross phase modulation,” Electron. Lett. 25, 646–648 (1989).
[CrossRef]

1988 (1)

R. R. Alfano and P. P. Ho, “Self-, cross-, induced-phase modulation of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

1987 (2)

1978 (1)

R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibres,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

1970 (1)

R. R. Alfano and S. L. Shapiro, “Observation of self phase modulation and small scale filamentation in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fibre Optics (Academic, New York, 1989), Chap. 7, p. 176.

Aitchison, J. S.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Alfano, R. R.

R. R. Alfano and P. P. Ho, “Self-, cross-, induced-phase modulation of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Observation of self phase modulation and small scale filamentation in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Al-hemyari, K.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Burrus, C. A.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1989), App. 5, p. 307.

Chang, Y. H.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1989), App. 5, p. 307.

Delfyett, P. J.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Dienes, A.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Heritage, J. P.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Ho, P. P.

R. R. Alfano and P. P. Ho, “Self-, cross-, induced-phase modulation of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

Hong, M. Y.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Ironside, C. N.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Islam, M. N.

Kang, J. U.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Lin, C.

R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibres,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Manassah, J. T.

Mollenauer, L. F.

Morioka, T.

T. Morioka and M. Saruwatari, “Demonstration of chirping manipulation of laser diode chirped optical pulses utilizing cross phase modulation,” Electron. Lett. 25, 646–648 (1989).
[CrossRef]

Raybon, G.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Saruwatari, M.

T. Morioka and M. Saruwatari, “Demonstration of chirping manipulation of laser diode chirped optical pulses utilizing cross phase modulation,” Electron. Lett. 25, 646–648 (1989).
[CrossRef]

Shang, H. T.

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Observation of self phase modulation and small scale filamentation in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Shore, K. A.

P. S. Spencer and K. A. Shore, Future Directions of Nonlinear Dynamics in Physical and Biological Systems, Proceedings of the NATO MIDIT Advanced Studies Institute Meeting, P. L. Christiansen, ed. (Plenum, New York, 1992), pp. 417–420.

Simpson, J. R.

Smith, J. S.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Spencer, P. S.

P. S. Spencer and K. A. Shore, Future Directions of Nonlinear Dynamics in Physical and Biological Systems, Proceedings of the NATO MIDIT Advanced Studies Institute Meeting, P. L. Christiansen, ed. (Plenum, New York, 1992), pp. 417–420.

Stegeman, G. I.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Stolen, R. H.

Villeneuve, A.

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Whinnery, J. R.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Wiesenfeld, J. M.

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. Al-hemyari, A. Villeneuve, J. U. Kang, J. S. Aitchison, C. N. Ironside, and G. I. Stegeman, “Ultrafast all-optical switching in GaAlAs directional couplers at 1.55 μ m without multiphoton absorption,” Appl. Phys. Lett. 63, 3562–3564 (1993).
[CrossRef]

Electron. Lett. (1)

T. Morioka and M. Saruwatari, “Demonstration of chirping manipulation of laser diode chirped optical pulses utilizing cross phase modulation,” Electron. Lett. 25, 646–648 (1989).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

S. P. Dijaili, J. M. Wiesenfeld, G. Raybon, C. A. Burrus, A. Dienes, J. S. Smith, and J. R. Whinnery, “Cross-phase modulation in a semiconductor laser amplifier determined by a dispersive technique,” IEEE J. Quantum Electron. 28, 141–149 (1992).
[CrossRef]

R. R. Alfano and P. P. Ho, “Self-, cross-, induced-phase modulation of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

R. H. Stolen and C. Lin, “Self-phase modulation in silica optical fibres,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Phys. Rev. Lett. (1)

R. R. Alfano and S. L. Shapiro, “Observation of self phase modulation and small scale filamentation in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Other (4)

Ref. 8, p. 200.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Cambridge, 1989), App. 5, p. 307.

G. P. Agrawal, Nonlinear Fibre Optics (Academic, New York, 1989), Chap. 7, p. 176.

P. S. Spencer and K. A. Shore, Future Directions of Nonlinear Dynamics in Physical and Biological Systems, Proceedings of the NATO MIDIT Advanced Studies Institute Meeting, P. L. Christiansen, ed. (Plenum, New York, 1992), pp. 417–420.

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Figures (6)

Fig. 1
Fig. 1

Phase across the probe calculated for various ηz values, assuming T0 = −1.0 and (ωsn2α02)/c = 2π.

Fig. 2
Fig. 2

Chirp profiles for various ηz values associated with the phase profiles shown in Fig. 1.

Fig. 3
Fig. 3

Probe spectral profile for various time delays. The time delays are normalized with respect to the pulse width.

Fig. 4
Fig. 4

Probe spectrum assuming walk-off only. The two group-velocity mismatches used were ηz = 0.5 and ηz = 1.0.

Fig. 5
Fig. 5

Walk-off with positive and negative time delays. The normalized time delay was set at T0 = ±0.5, and the velocity mismatch was set at ηz = 1.0. In case (A) the faster of the two pulses is launched into the nonlinear medium after the slower pulse. In case (B) the faster of the two pulses enters the nonlinear medium first.

Fig. 6
Fig. 6

All three effects are included. The normalized time delay was set at T0 = ±0.5, the velocity mismatch was set at ηz = 1.0, and the loss was set at αz = 0.5. In case (A) either T0 > 0 and η > 0 or T0 < 0 and η < 0, while for case (B) either T0 > 0 and η < 0 or T0 < 0 and η > 0.

Equations (28)

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2 z 2 E x ( z , t ) 1 c 2 2 t 2 E x ( z , t ) μ 0 2 t 2 P x ( z , t ) = 0 ,
P x ( z , t ) = P L ( z , t ) + P NL ( z , t ) = 0 [ χ ( 1 ) + χ ( 3 ) | E x | 2 ] E x ( z , t ) .
E x ( z , t ) = A p ( z , t ) exp [ i ( β p z ω p t ) ] + A s ( z , t ) exp [ i ( β s z ω s t ) ] ,
ι ( A p z + 1 υ p A p t ) + ω p n 2 c [ | A p | 2 A p + 2 | A s | 2 A p ] + ι α p 2 A p = 0 ,
ι ( A s z + 1 υ s A s t ) + ω s n 2 c [ | A s | 2 A s + 2 | A p | 2 A s ] + ι α s 2 A s = 0 ,
A p ( z , t ) = a ( z , t ) exp [ ι ϕ ( z , t ) ] ,
A s ( z , t ) = b ( z , t ) exp [ ι ϕ ( z , t ) ] .
a z + 1 υ p a t = α a a 2 ,
b z + 1 υ s b t = α b b 2 ,
ϕ z + 1 υ p ϕ t = n 2 ω p a 2 c ,
θ z + 1 υ s θ t = 2 n 2 ω s a 2 c .
a ( 0 , t ) = a 0 F [ ( t t 0 ) / τ ] ,
b ( 0 , t ) = b 0 G [ t / τ ] ,
ϕ ( 0 , t ) = θ ( 0 , t ) = 0 ,
a ( z , t ) = a 0 F [ ( t t 0 z / υ p ) / τ ] exp ( α p z / 2 ) ,
b ( z , t ) = b 0 G [ ( t z / υ s ) / τ ] exp ( α s z / 2 ) ,
ϕ ( z , t ) = ω p n 2 a 0 2 c L eff ( z ) F 2 [ ( t t 0 z / υ p ) / τ ] ,
L eff ( z ) = 1 exp ( α p z ) α p .
θ z + 1 υ s θ t = 2 n 2 ω s a 0 2 c F 2 [ ( t t 0 z / υ p ) / τ ] exp ( α p z ) .
a ( U , Z ) = a 0 F [ ( U t 0 ) / τ η Z ] exp ( α p Z / 2 ) ,
Z = z + 1 υ s t ,
θ Z = 2 n 2 ω s a 0 2 c F 2 [ ( U t 0 ) / τ η Z ] exp ( α p Z ) ,
θ ( z , t ) = 2 ω s n 2 a 0 2 c 0 z exp ( α p z ) F 2 { [ t t 0 z / υ s z ( 1 / υ p 1 / υ s ) ] / τ } d z .
θ ( z , t ) = π ψ τ { erf [ α 2 η ( t η z ) 2 η ] erf ( α 2 t η 2 η ) } ,
ψ = ω s n 2 a 0 2 c η exp ( α α 4 t η 4 η 2 ) .
δ ω = θ ( z , t ) t .
δ ω = π ψ α τ η { erf [ α 2 η ( t η z ) 2 η ] erf ( α 2 t η 2 η ) } + 2 ψ τ ( exp { [ α 2 η ( t η z ) 2 η ] 2 } exp [ ( α 2 t η 2 η ) 2 ] ) .
S ( ω ) = | b ( z , t ) exp { ι [ θ ( z , t ) + ( ω ω s ) t ] } d t | 2 .

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