Abstract

Electromagnetic (EM) bubbles (EMB's), unipolar, super-short, and intense nonoscillating solitary pulses of EM radiation, can be generated in a gas of nonlinear atoms by available half-cycle pulses (HCP's). We investigate how EMB's characteristics (amplitude, length, formation distance, and total number) are controlled by the amplitude and length of originating HCP's. We also predict shocklike wave fronts in the multibubble regime.

© 1997 Optical Society of America

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  1. P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
    [CrossRef]
  2. D. Grischkowsky, S. Keidin, M. van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990)R. A. Cheville and D. Grischkowsky, Opt. Lett. 20, 1646 (1995).
    [CrossRef] [PubMed]
  3. J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
    [CrossRef]
  4. B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995).
    [CrossRef]
  5. R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
    [CrossRef] [PubMed]
  6. A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
    [CrossRef] [PubMed]
  7. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. Non-Linear Opt. Phys. Mater. 4, 831 (1995).
    [CrossRef] [PubMed]
  8. R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971) ;J. E. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973); E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991).
    [CrossRef] [PubMed]
  9. Propagation is more sensitive to the precision of constitutive equations than to that of Maxwell equations.7,8 The former, however, can also be simplified within certain limits. For slow and weak fields  ( τ0-1,f0< <1) , the Maxwell  +  constitutive equations can be reduced to a modified Kordeweg–De Vries equation for both the quantum and classical cases:  ∂f/∂ ζ-A nl f2 ∂f/∂ τ-∂3f/∂ τ3=0 , where for a TLS,  A nl =3/2 . In the opposite limit of a very strong and fast field, the Maxwell–Bloch equations can be reduced to a sine Gordon equation  ∂2 ϕR/∂ ζ ∂ τ1=sin  ϕR  for a Rabi phase  ϕR≡∫ -∞  τ1  fd τ ( τ1=τ-ζ ) . Both of these equations are fully integrable and have single solitons of exactly the same profile as an EMB.
  10. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
    [PubMed]
  11. P. H. Bucksbaum, Department of Physics, University of Michigan, Ann Arbor, Mich., 48109, personal communication, 1996.

1995 (2)

B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995).
[CrossRef]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. Non-Linear Opt. Phys. Mater. 4, 831 (1995).
[CrossRef] [PubMed]

1994 (2)

A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[PubMed]

1993 (1)

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

1990 (2)

1988 (1)

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

1971 (1)

R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971) ;J. E. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973); E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991).
[CrossRef] [PubMed]

Ahmad, F.

R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971) ;J. E. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973); E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991).
[CrossRef] [PubMed]

Auston, D. H.

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Bucksbaum, P. H.

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

P. H. Bucksbaum, Department of Physics, University of Michigan, Ann Arbor, Mich., 48109, personal communication, 1996.

Bullough, R. K.

R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971) ;J. E. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973); E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991).
[CrossRef] [PubMed]

Fattinger, Ch.

Glownia, J. H.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

Grischkowsky, D.

Hu, B. B.

Jones, R. R.

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

Kaplan, A. E.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. Non-Linear Opt. Phys. Mater. 4, 831 (1995).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[PubMed]

A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef] [PubMed]

Keidin, S.

Misewich, J. A.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

Nuss, M. S.

B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995).
[CrossRef]

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Shkolnikov, P. L.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. Non-Linear Opt. Phys. Mater. 4, 831 (1995).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[PubMed]

Smith, P. R.

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Sorokin, P. P.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

van Exter, M.

You, D.

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

J. Chem. Phys. (1)

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. A (1)

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[PubMed]

Phys. Rev. Lett. (4)

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. Non-Linear Opt. Phys. Mater. 4, 831 (1995).
[CrossRef] [PubMed]

R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971) ;J. E. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973); E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991).
[CrossRef] [PubMed]

Other (2)

Propagation is more sensitive to the precision of constitutive equations than to that of Maxwell equations.7,8 The former, however, can also be simplified within certain limits. For slow and weak fields  ( τ0-1,f0< <1) , the Maxwell  +  constitutive equations can be reduced to a modified Kordeweg–De Vries equation for both the quantum and classical cases:  ∂f/∂ ζ-A nl f2 ∂f/∂ τ-∂3f/∂ τ3=0 , where for a TLS,  A nl =3/2 . In the opposite limit of a very strong and fast field, the Maxwell–Bloch equations can be reduced to a sine Gordon equation  ∂2 ϕR/∂ ζ ∂ τ1=sin  ϕR  for a Rabi phase  ϕR≡∫ -∞  τ1  fd τ ( τ1=τ-ζ ) . Both of these equations are fully integrable and have single solitons of exactly the same profile as an EMB.

P. H. Bucksbaum, Department of Physics, University of Michigan, Ann Arbor, Mich., 48109, personal communication, 1996.

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Figures (3)

Fig. 1
Fig. 1

Normalized formation distance ζEMBfthr3 of the EMB precursor versus normalized incident amplitude, N0=f0/fthr. Line 1, π5/N03 [expression  (9), below]; line 2, 3/N02 [expression  (10), below]; filled circles, first saddle point appearance in a field profile. Inset, double-EMB formation by HCP with N0=2.

Fig. 2
Fig. 2

EMB amplitude fEMB versus the amplitude f0 and length τ0 (inset) of the incident HCP. Solid curve, EMB precursor, n=1; broken curves, higher-order EMB's.

Fig. 3
Fig. 3

Formation of a shocklike wave front for N0=33 as the wave propagates in ζ. Inset, superimposition of the field profiles at different ζ, illustrating the front formation.

Equations (12)

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η·=-fp·, p̈+p=fη,
2f/ζ˜2-2f/τ˜2=Q2p/τ˜2.
βEMB1-Q/[2(1+fEMB2/4)].
f(τ)=f0/cosh(2τ/τ0).
fEMBaf0-(a-1)fthr, a=constant2.
q=Ω1+Q/ΩSTΩST2-Ω21/2.
(Ωfast/ΩST)2=2ΩST3+1-8ΩST3+11/22ΩST3
βfast1-Q1+8ΩST3+11/2.
ζEMBπ5f03.
ζshζEMB3f02fthr, f0>4fthr.
NEMB=L(N0), N0f0/fthr=f0τ0/4,
fn/fthr=2(N0-n)+1, nN0,

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