Abstract

We investigate the areas of extremely short, intense (area π) resonant optical pulses on propagation through an inhomogeneously broadened atomic medium. Experimentally, there is no apparent change in the shape of such pulses on propagation, in contradiction to the area theorem. Here we show that, although the main part of the pulse is unchanged for short propagation distances (αl1), it is followed by a long weak tail formed by free induction decay from the excited atoms. The tail lengthens and oscillates on propagation through the medium and permits the area theorem to be obeyed. These oscillations, which depend critically on the Doppler broadening, are reflected in the spectral analysis of the propagated pulse. We also introduce another mechanism for pulse reshaping, which is a generalization of the McCall–Hahn pulse breakup and operates at long propagation lengths (αl1), where the number of absorbing atoms encountered by the pulse is comparable with the number of photons in the pulse.

© 1999 Optical Society of America

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References

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  1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975); K. Shimoda, Introduction to Laser Physics (Springer-Verlag, Berlin, 1991); G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992).
  2. S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
    [CrossRef]
  3. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [CrossRef]
  4. S. L. McCall and E. L. Hahn, “Pulse-area–pulse-energy description of a traveling-wave laser,” Phys. Rev. A 2, 861–870 (1970).
    [CrossRef]
  5. H. M. Gibbs and R. E. Slusher, “Sharp line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
    [CrossRef]
  6. M. Matusovski, B. Vaynberg, and M. Rosenbluh, “0π pulse propagation in the extreme sharp line limit,” J. Opt. Soc. Am. B 13, 1994–1998 (1996).
    [CrossRef]
  7. M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
    [CrossRef] [PubMed]
  8. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
    [CrossRef]
  9. J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [CrossRef]
  10. J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
    [CrossRef]
  11. A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
    [CrossRef]
  12. W. Miklasewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1993).
    [CrossRef]
  13. W. Miklasewski, “Near resonant propagation of the light pulse in a homogeneously broadened two-level medium,” J. Opt. Soc. Am. B 12, 1909–1917 (1995).
    [CrossRef]
  14. R. K. Bullough and P. J. Caurdey, eds., Solitons (Springer-Verlag, Berlin, 1980); R. K. Dodd, J. C. Eilbeck, and J. D. Gibbon, eds., Solitons and Nonlinear Wave Equations (Academic, London, 1982); G. L. Lamb, Jr., ed., Elements of Soliton Theory (Wiley, New York, 1980).

1998 (1)

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
[CrossRef]

1996 (2)

M. Matusovski, B. Vaynberg, and M. Rosenbluh, “0π pulse propagation in the extreme sharp line limit,” J. Opt. Soc. Am. B 13, 1994–1998 (1996).
[CrossRef]

M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

W. Miklasewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1993).
[CrossRef]

1992 (1)

A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[CrossRef]

1984 (1)

J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

1972 (1)

H. M. Gibbs and R. E. Slusher, “Sharp line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

1970 (2)

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

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

1967 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Alhasan, A. M.

A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[CrossRef]

Balant, A. C.

J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Fiutak, J.

W. Miklasewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1993).
[CrossRef]

A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[CrossRef]

Gaeta, A. L.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs and R. E. Slusher, “Sharp line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Grichkowsky, D.

J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Hahn, E. L.

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

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Matusovski, M.

Matusovsky, M.

M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
[CrossRef] [PubMed]

McCall, S. L.

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

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Miklasewski, W.

W. Miklasewski, “Near resonant propagation of the light pulse in a homogeneously broadened two-level medium,” J. Opt. Soc. Am. B 12, 1909–1917 (1995).
[CrossRef]

W. Miklasewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1993).
[CrossRef]

A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[CrossRef]

Ranka, J. K.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
[CrossRef]

Rosenbluh, M.

M. Matusovski, B. Vaynberg, and M. Rosenbluh, “0π pulse propagation in the extreme sharp line limit,” J. Opt. Soc. Am. B 13, 1994–1998 (1996).
[CrossRef]

M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
[CrossRef] [PubMed]

Rothenberg, J. E.

J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Schirmer, R. W.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
[CrossRef]

Slusher, R. E.

H. M. Gibbs and R. E. Slusher, “Sharp line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

Vaynberg, B.

M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
[CrossRef] [PubMed]

M. Matusovski, B. Vaynberg, and M. Rosenbluh, “0π pulse propagation in the extreme sharp line limit,” J. Opt. Soc. Am. B 13, 1994–1998 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Phys. Rev. (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Phys. Rev. A (4)

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

H. M. Gibbs and R. E. Slusher, “Sharp line self-induced transparency,” Phys. Rev. A 6, 2326–2334 (1972).
[CrossRef]

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57, R36–R39 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

J. E. Rothenberg, D. Grichkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

M. Matusovsky, B. Vaynberg, and M. Rosenbluh, “High-intensity pulse propagation in the extreme sharp line limit,” Phys. Rev. Lett. 77, 5198–5201 (1996).
[CrossRef] [PubMed]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967).
[CrossRef]

Z. Phys. B (2)

A. M. Alhasan, J. Fiutak, and W. Miklasewski, “The influence of atomic relaxation on the resonant propagation of short light pulses,” Z. Phys. B 88, 349–358 (1992).
[CrossRef]

W. Miklasewski and J. Fiutak, “The effect of the homogeneous broadening on the propagation of the light pulses,” Z. Phys. B 93, 491–499 (1993).
[CrossRef]

Other (2)

R. K. Bullough and P. J. Caurdey, eds., Solitons (Springer-Verlag, Berlin, 1980); R. K. Dodd, J. C. Eilbeck, and J. D. Gibbon, eds., Solitons and Nonlinear Wave Equations (Academic, London, 1982); G. L. Lamb, Jr., ed., Elements of Soliton Theory (Wiley, New York, 1980).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975); K. Shimoda, Introduction to Laser Physics (Springer-Verlag, Berlin, 1991); G. P. Agrawal and R. W. Boyd, Contemporary Nonlinear Optics (Academic, San Diego, Calif., 1992).

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Figures (15)

Fig. 1
Fig. 1

Experimental results: Normalized autocorrelation for a 6-ps intense pulse of approximate area 4π before propagation and after propagation through potassium vapor with αl=170.

Fig. 2
Fig. 2

Pulse shape as a function of time for four values of αl, calculated from Maxwell–Bloch equations. (a) The main part of the pulse: Note that it remains almost unchanged even for αl=20. (b) The long tail: For short optical paths the tail is a decaying signal of length T2*. Propagation for larger αl causes it to strengthen and oscillate between positive and negative values. The parameters used correspond to those of the experiment: μ=7.2D, T2*=0.2 ns, N=1013cm-3, ω=2.45×1015 Hz, T1=28 ns, T2=5 ns, corresponding to foreign gas broadening.

Fig. 3
Fig. 3

Pulse area as obtained numerically from Maxwell–Bloch equations and analytically from the area theorem as a function of αl. The initial pulse area input is 3.1π.

Fig. 4
Fig. 4

Shape of the main part of the pulse as a function of time for αl=100. The pulse begins to reshape according to the usual breakup mechanism into two pulses of 2π.

Fig. 5
Fig. 5

Pulse shape as a function of time for 4π and 10π input pulses of identical duration for the same optical depth αl1000. 4π pulses break up more relative to the main part of the pulse.

Fig. 6
Fig. 6

Pulse shape as a function of time for low area input of 0.5π. The tail intensity is of the same order of magnitude as the pulse intensity, and thus reshaping is observed.

Fig. 7
Fig. 7

Positions of the Bloch vector during pulse propagation: (a) before the pulse, (b) after 500 ps, which is a long time after the pulse for αl1, (c) after 500 ps for αl5, (d) after 500 ps for αl20. The tail changes the Bloch vector position. Note that in (d) the vector has turned to the other side of the Bloch sphere and therefore produces a negative field, as can be seen for the same αl and time in Fig. 2(b).

Fig. 8
Fig. 8

Pulse shape as a function of time for αl=50 for 3.1π and 5.1π pulses. (a) Main part of the pulse, (b) tail of the pulse, which is identical for both pulses because of the same excitation of the population.

Fig. 9
Fig. 9

Pulse shape as a function of time at αl=50 for 3.1π and 4.9π pulses. (a) Main part of the pulse, (b) tail of the pulse, which is of the same magnitude but opposite sign because of the opposite sign of the v components of the Bloch vector.

Fig. 10
Fig. 10

Pulse tails of 3.1π and 3.6π pulses. The average intensity of the tail formed for the same αl is larger for a 3.1π pulse.

Fig. 11
Fig. 11

Pulse shape as a function of time for different inhomogeneous broadening times T2*=0.2 ns and T2*=0.4 ns with the same propagation length (αl=50) and the same initial pulse area (3.1π). When T2* is smaller the tail has more oscillations, higher intensity, and a shorter time scale.

Fig. 12
Fig. 12

Tail of a 3.1π pulse when inhomogeneous broadening is neglected for three propagation lengths [which are larger than those calculated in Fig. 2(b) with inhomogeneous broadening]. The tail in this case is much less intense, has fewer oscillations, and does not lengthen on propagation.

Fig. 13
Fig. 13

Pulse spectra for (a) a 3.1π pulse and (b) a 4.9π pulse for three propagation lengths. The main shape of the spectrum does not change because there are no significant changes in the pulse shape. The tail is reflected by changes in the intensities of the central frequencies in the spectrum.

Fig. 14
Fig. 14

Central frequencies of the pulse spectrum in Fig. 12. The intensity at the resonance frequency becomes constant in accordance with the area theorem. Farther from resonance there are oscillations that correspond to the frequencies of oscillations in the tail.

Fig. 15
Fig. 15

Central frequencies of pulse spectrum for a 3.1π pulse for two Doppler broadening times, T2*=0.2 ns and T2*=0.4 ns, and with inhomogeneous broadening neglected (for the same propagation length αl=50). The range of frequencies of oscillations in the spectrum about the resonance in the last case is much smaller than for the inhomogeneous broadening case because there are oscillations in the tail with smaller frequencies.

Equations (11)

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dAdz=-½ αsinA
A(z)=k-(z,t)dt,
z+ηct(z, t)=-α0-v(z, t, Δω)g(Δω)d(Δω),
dudt=-Δωv-uT2,
dvdt=Δωu+kw-vT2,
dwdt=-kv-w-w0T1.
A(z)=k-(z, t)dt,
dA(z)dz=-αsinA(z),
α=4π2ωNμ2g(0)/ηc.
E(z, Δω)=12π-(z, t)exp(-iΔωt)dt.
E(z, 0)=12π-(z, t)dt=1k2πA(z).

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