Abstract

The spatiotemporal dynamics of a light pulse injected into a dense, resonant medium under near-dipole–dipole interaction conditions is considered. Two regimes for soliton formation are investigated numerically and analytically. The first one occurs if the pulses are much shorter than both relaxation times. In the second, the pulse duration falls between those of the relaxation times and may be regarded as an incoherent one. In the latter case there are conditions in which the dephasing process is suppressed by dipole–dipole interatomic interaction. Not only are one-soliton solutions found but also a powerful pulse splitting into separate solitons is revealed. The regions in which solitons are found have been estimated. The possibility of finding incoherent solitons has been analyzed, with group-velocity dispersion and diffraction of tilted pulses, which are shown to make soliton formation easier, taken into account.

© 2002 Optical Society of America

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References

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  1. S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser amplifier,” Phys. Rev. A 2, 861–870 (1970).
    [CrossRef]
  2. C. M. Bowden and J. P. Dowling, “Near dipole–dipole effects in dense media: generalized Maxwell–Bloch equations,” Phys. Rev. A 47, 1247–1251 (1993).
    [CrossRef] [PubMed]
  3. Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
    [CrossRef] [PubMed]
  4. M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
    [CrossRef] [PubMed]
  5. A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
    [CrossRef]
  6. Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
    [CrossRef] [PubMed]
  7. M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
    [CrossRef] [PubMed]
  8. M. Segev and D. N. Christodoulides, “Incoherent solitons: self-trapping of weakly-correlated wave-packets,” in Optical Spatial Solitons, S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001).
  9. C. M. Bowden, A. Postan, and R. Inguva, “Invariant pulse propagation and self-phase modulation in dense media,” J. Opt. Soc. Am. B 8, 1081–1084 (1991).
    [CrossRef]
  10. A. M. Agranovich and V. I. Rupasov, “Self-induced transparency in a medium with space dispersion,” Sov. Phys. Solid State 18, 801–807 (1976).
  11. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley-Interscience, New York, 1975).
  12. J. H. Eberly, “Area theorem rederived,” Opt. Express 2, 173–176 (1998), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  13. I. S. Gradstein and I. M. Reazhik, Tablitsy Integralov, Summ, Ryadov i Proizvedenii (Nauka, Moscow, 1971).
  14. V. D. Gora, “Soliton light propagation under conditions of adiabatic following under a single-photon resonance,” Sov. J. Quantum Electron. 17, 763–766 (1990).
  15. R. A. Vlasov, “Tilted femtosecond solitons in cubic media,” Phys. Lett. 211, 82–86 (1996).
    [CrossRef]

2000 (1)

A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
[CrossRef]

1998 (2)

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

J. H. Eberly, “Area theorem rederived,” Opt. Express 2, 173–176 (1998), http://www.opticsexpress.org.
[CrossRef] [PubMed]

1996 (2)

R. A. Vlasov, “Tilted femtosecond solitons in cubic media,” Phys. Lett. 211, 82–86 (1996).
[CrossRef]

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

1994 (1)

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

1993 (1)

C. M. Bowden and J. P. Dowling, “Near dipole–dipole effects in dense media: generalized Maxwell–Bloch equations,” Phys. Rev. A 47, 1247–1251 (1993).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

V. D. Gora, “Soliton light propagation under conditions of adiabatic following under a single-photon resonance,” Sov. J. Quantum Electron. 17, 763–766 (1990).

1986 (1)

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
[CrossRef] [PubMed]

1976 (1)

A. M. Agranovich and V. I. Rupasov, “Self-induced transparency in a medium with space dispersion,” Sov. Phys. Solid State 18, 801–807 (1976).

1970 (1)

S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser amplifier,” Phys. Rev. A 2, 861–870 (1970).
[CrossRef]

Afanas’ev, A. A.

A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
[CrossRef]

Agranovich, A. M.

A. M. Agranovich and V. I. Rupasov, “Self-induced transparency in a medium with space dispersion,” Sov. Phys. Solid State 18, 801–807 (1976).

Ben-Aryeh, Y.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
[CrossRef] [PubMed]

Bowden, C. M.

C. M. Bowden and J. P. Dowling, “Near dipole–dipole effects in dense media: generalized Maxwell–Bloch equations,” Phys. Rev. A 47, 1247–1251 (1993).
[CrossRef] [PubMed]

C. M. Bowden, A. Postan, and R. Inguva, “Invariant pulse propagation and self-phase modulation in dense media,” J. Opt. Soc. Am. B 8, 1081–1084 (1991).
[CrossRef]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
[CrossRef] [PubMed]

Chen, Zh.

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

Cherstvyi, A. G.

A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
[CrossRef]

Christodoulides, D. H.

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

Coskun, T. H.

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

Dowling, J. P.

C. M. Bowden and J. P. Dowling, “Near dipole–dipole effects in dense media: generalized Maxwell–Bloch equations,” Phys. Rev. A 47, 1247–1251 (1993).
[CrossRef] [PubMed]

Eberly, J. H.

Englund, J. C.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
[CrossRef] [PubMed]

Gora, V. D.

V. D. Gora, “Soliton light propagation under conditions of adiabatic following under a single-photon resonance,” Sov. J. Quantum Electron. 17, 763–766 (1990).

Gudel, H. U.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser amplifier,” Phys. Rev. A 2, 861–870 (1970).
[CrossRef]

Hehlen, M. P.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Inguva, R.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser amplifier,” Phys. Rev. A 2, 861–870 (1970).
[CrossRef]

Mitchell, M.

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

Postan, A.

Rai, J.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Rai, S.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Rand, S. C.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Rupasov, V. I.

A. M. Agranovich and V. I. Rupasov, “Self-induced transparency in a medium with space dispersion,” Sov. Phys. Solid State 18, 801–807 (1976).

Segev, M.

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

Shih, M.

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

Shu, Q.

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

Vlasov, R. A.

A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
[CrossRef]

R. A. Vlasov, “Tilted femtosecond solitons in cubic media,” Phys. Lett. 211, 82–86 (1996).
[CrossRef]

J. Exp. Theor. Phys. (1)

A. A. Afanas’ev, R. A. Vlasov, and A. G. Cherstvyi, “Optical solitons in dense resonant media,” J. Exp. Theor. Phys. 90, 428–433 (2000).
[CrossRef]

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

Opt. Express (1)

Phys. Lett. (1)

R. A. Vlasov, “Tilted femtosecond solitons in cubic media,” Phys. Lett. 211, 82–86 (1996).
[CrossRef]

Phys. Rev. A (3)

S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser amplifier,” Phys. Rev. A 2, 861–870 (1970).
[CrossRef]

C. M. Bowden and J. P. Dowling, “Near dipole–dipole effects in dense media: generalized Maxwell–Bloch equations,” Phys. Rev. A 47, 1247–1251 (1993).
[CrossRef] [PubMed]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, “Intrinsic optical bistability in collections of spatially distributed two-level atoms,” Phys. Rev. A 34, 3917–3926 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

M. P. Hehlen, H. U. Gudel, Q. Shu, J. Rai, S. Rai, and S. C. Rand, “Cooperative bistability in dense, excited atomic systems,” Phys. Rev. Lett. 73, 1103–1106 (1994).
[CrossRef] [PubMed]

M. Mitchell, Zh. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[CrossRef] [PubMed]

Science (1)

Zh. Chen, M. Mitchell, M. Segev, T. H. Coskun, and D. H. Christodoulides, “Self-trapping of dark incoherent light beams,” Science 280, 889–892 (1998).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

V. D. Gora, “Soliton light propagation under conditions of adiabatic following under a single-photon resonance,” Sov. J. Quantum Electron. 17, 763–766 (1990).

Sov. Phys. Solid State (1)

A. M. Agranovich and V. I. Rupasov, “Self-induced transparency in a medium with space dispersion,” Sov. Phys. Solid State 18, 801–807 (1976).

Other (3)

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley-Interscience, New York, 1975).

I. S. Gradstein and I. M. Reazhik, Tablitsy Integralov, Summ, Ryadov i Proizvedenii (Nauka, Moscow, 1971).

M. Segev and D. N. Christodoulides, “Incoherent solitons: self-trapping of weakly-correlated wave-packets,” in Optical Spatial Solitons, S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001).

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Figures (7)

Fig. 1
Fig. 1

Modification of soliton shape under conditions of coherent interaction at z=1.5α-1: [ωL=0 (solid curve), ωL0 (dashed curve); inset, influence of inhomogeneous broadening (Γinh=6×1012 s-1)] on the transformation of soliton shape.

Fig. 2
Fig. 2

Pattern of 4π coherent pulse splitting in a dense medium for various propagation lengths L (in units of α-1).

Fig. 3
Fig. 3

Character of pulse absorption by a dense medium for the parameters σ=11, b=10, Δ=-11, and Wth=0.091 when the initial pulse energy (a) W0=0.055 is less than Wth and (b) W0=0.1 is greater than Wth.

Fig. 4
Fig. 4

Soliton shape under conditions of incoherent-light–medium interaction: 1, L=1α-1; 2, L=5α-1. Dashed curve, input envelope (z=0).

Fig. 5
Fig. 5

Range of stability of an incoherent soliton with regard to variables b and Δ (inset, the same under the influence group velocity dispersion).

Fig. 6
Fig. 6

Spatiotemporal dynamics of a 4π incoherent pulse in a dense resonant medium.

Fig. 7
Fig. 7

Conditions of incoherent interaction: modification of the input 2π pulse shape when τp>t2 (solid curve, Tp=10-10 s, T2=10-11 s) and τpt2 (dashed curve, Tp=1.1×10-11 s, T2=10-11 s) for two lengths: 1, L=5α-1 and 2, L=10α-1.

Equations (54)

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EZ+1V ET+i2 2kω2 2ET2=2iπkμn02N0P,
PT=iμEn+i(Ω+ΩLn)P-PT2ι,
nT=2iμ(E*P-P*E)ι.
E=μE/ωp,z=Zωp/V,t=ωpT,
δ=Ω/ωp,ωL=ΩL/ωp,t2=ωpT2,
ωp2=2πN0μ2ω/n02,D=ωpV2 2kω2.
z+tE+iD 2Et2=iP,
Pt=iEn+i(δ+ωLn)P-Pt2,
nt=2i(E*P-P*E).
Ghτ={(zj, tk),zj=(j-1)hz,j=1, 2 , 
L˜=[L/hz],tk=zj+(k-1)ht,
k=1, 2 ,, M˜=[T/ht]},
E(τ)=A(τ)exp[iφ(τ)].
1ξ dAdτ2=qA2-12A4-rA6,
dϕdτ=δ+ωL2-ωLγ2A2.
ξ=2-γωL(δ+ωL),
q=[4-γ(δ+ωL)2]/(4γξ),
r=(ωL2γ2)/(4ξ),γ=1/v0-1.
A(τ)=2q[1+1+16rq cosh(2ξqτ)]1/2,
φ(τ)=δ+ωL2τ-ωLγ2 -τ|A|2dτ.
1v0-1=τp21+[(δ+ωL)τP/2]2.
θ=2 - A(τ)dτ2π1-3ωL232[1+(δτP/2)2].
ΩR=[4|E|2+(δ+ωLn)2]1/2.
P=i t2nE1-i(Δ+bn),
dndt=-4t2n|E|21+(Δ+bn)2,
w(z, t)=-t|E(z, t)|2dt,
Fz+Ft=-t2nF1+(Δ+bn)2,
ψz+ψt=-t2n(Δ+bn)1+(Δ+bn)2.
Wz=-t2 - nF2dt1+(Δ+bn)2,
dWdz=-α¯W(1-σW),
W(z)=W0 exp(-α¯z)(1-σW0)+σW0 exp(-α¯z).
W(z)=W0σW0-(σW0-1)exp(α¯z).
F(z, τ)A(τ)=A0 sech(τ/τp),
ψ(z, τ)φ(τ)=(α2+β2τPA02 ln[cosh(τ/τP)],
τp=1/α1=γ|λ|2/t2,
A01/(τpβ1)=|λ|2/(2{[τpt2(1+Δ2-b2)]}1/2).
α=-iγχ0α1+iα2,β=iγχβ1+iβ2,χ0=it2/λ,χ=-i 4t2(1-iΔ)λ2|λ|2.
A02|λ|2/(8τpt2).
2[1+(Δ+b)2]1+Δ2-b21,
1+Δ2-b2>0.
1/v0-1=t2τp/|λ|2.
θ=2πβ1=π|λ|2τpt2(1+Δ2-b2)1/2.
Ez+Et+iD 2Et2=iχ0+χ -t|E|2dtE.
E=A(τ)exp[iφ(τ)+iηz].
A dAdτ+Dγ ddτ A2 dφdτ-α1A2+β1A2 -τ A2dτ=0,
A dφdτ-Dγ d2Adτ2+DγAdφdτ2+β2A -τ A2dτ=0.
φτ=-β2 -τ A2dτ.
dA2dτ-2Dγ ddτ A2β2 -τ A2dτ-2α1A2
+2β1A2 -τ A2dτ=0.
w¯(τ)=-τ A2dτ.
dw¯dτ=2α1w¯-β1-4α1Dβ2γw¯2.
w¯=α1β¯1[tanh(α1τ)+1],
β¯1=β1-4α1Dβ2γ.
dw¯dτ=A2=α12β¯1 1cosh2(α1τ),

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