## Near dipole-dipole effects on the propagation of few-cycle pulse in a dense two-level medium

Optics Express, Vol. 13, Issue 16, pp. 5913-5924 (2005)

http://dx.doi.org/10.1364/OPEX.13.005913

Acrobat PDF (391 KB)

### Abstract

The propagation behaviors, which include the carrier-envelope phase, the area evolution and the solitary pulse number of few-cycle pulses in a dense two-level medium, are investigated based on full-wave Maxwell-Bloch equations by taking Lorentz local field correction (LFC) into account. Several novel features are found: the difference of the carrier-envelope phase between the cases with and without LFC can go up to π at some location; although the area of ultrashort solitary pulses is lager than 2π, the area of the effective Rabi frequency, which equals to that the Rabi frequency pluses the product of the strength of the near dipole-dipole (NDD) interaction and the polarization, is consistent with the standard area theorem and keeps 2π; the large area pulse penetrating into the medium produces several solitary pulses as usual, but the number of solitary pulses changes at certain condition.

© 2005 Optical Society of America

## 1. Introduction

1. A. Baltuška, Z.Y. Wei, M. S. Pshenichnikov, and D. A. Wiersma, “Optical pulse compression to 5 fs at a 1-MHz repetition rate,” Opt. Lett. **22**, 102 (1997). [CrossRef] [PubMed]

2. A. Stingel, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti: sapphire laser,” Opt. Lett. **20**, 602 (1995). [CrossRef]

6. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. **81**, 3363–3366 (1998). [CrossRef]

6. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. **81**, 3363–3366 (1998). [CrossRef]

6. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. **81**, 3363–3366 (1998). [CrossRef]

*et al*. have found the area theorem invalid for an attosecond pulse propagating in a dense two-level medium [8

8. X.H. Song, S.Q. Gong, W.F. Yang, and Z.Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A **70**, 1 (2004). [CrossRef]

12. V.P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. **83**, 544–547 (1999). [CrossRef]

13. T. Serényi, C. Benedek, and M.G. Benedict, “Femtosecond light pulses in GaAs, third harmonic generation by carrier wave Rabi flopping,” Fortschr. Phys. **51**, 226–229 (2003). [CrossRef]

14. O.D. Mücke, T. Tritschler, and M. Wegener, “Signatures of carrier-wave Rabi flopping in GaAs,” Phys. Rev. Lett. **87**, 057401 (2001). [CrossRef] [PubMed]

*Φ*

_{CE}(defined as the phase of the carrier wave with respect to the pulse peak [15–16

15. G.G. Paulus, F. Grashon, H. Walther, P. Vllloresl, M. Nlsoll, S. Staglra, E. Prlorl, and S. De Sllvestrl, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature **414**, 182–184 (2001). [CrossRef] [PubMed]

15. G.G. Paulus, F. Grashon, H. Walther, P. Vllloresl, M. Nlsoll, S. Staglra, E. Prlorl, and S. De Sllvestrl, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature **414**, 182–184 (2001). [CrossRef] [PubMed]

18. A. de Bohan, P. Antoine, D. B. Miloševic, and B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. **81**, 1837–1840 (1998). [CrossRef]

19. R. M. Potvliege, N.J. Kylstra, and C.J. Joachain, “Photon emission by He^{+} in intense ultrashort laser pulses,” J. Phys. B **33**, L743–L748 (2000). [CrossRef]

20. 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]

*et al*. [20

20. 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]

22. M.E. Crenshaw, M. Scalora, and C.M. Bowden, “Ultrafast intrinsic optical switching in a dense medium of two-level atoms,” Phys. Rev. Lett. **68**, 911–914 (1992). [CrossRef] [PubMed]

22. M.E. Crenshaw, M. Scalora, and C.M. Bowden, “Ultrafast intrinsic optical switching in a dense medium of two-level atoms,” Phys. Rev. Lett. **68**, 911–914 (1992). [CrossRef] [PubMed]

23. M. Scalora and C.M. Bowden, “Propagation effects and ultrafast optical switching in dense media,” Phys. Rev. A **51**, 4048–4056 (1995). [CrossRef] [PubMed]

*w*has a nearly step-function response to the peak Rabi frequency Ω

_{0}after the pulse has passed. The system, initially in the ground state,

*w*=-1, is always returned to the ground state when the peak Rabi frequency is less than the strength of the NDD interaction ∈. However, the final state of the system is the fully excited state,

*w*=1, when Ω

_{0}is nearly equal to ∈. Thus, the nearly step-function response of the system constitutes a new and unique optical switch, which does not have hysteresis and is independent of dissipation from incoherent effects [22

22. M.E. Crenshaw, M. Scalora, and C.M. Bowden, “Ultrafast intrinsic optical switching in a dense medium of two-level atoms,” Phys. Rev. Lett. **68**, 911–914 (1992). [CrossRef] [PubMed]

*Φ*

_{CE}is sensitive to the NDD interaction. The effect of NDD arouses a decreasing in the center frequency and slightly increasing in the group velocity, which, combining with the self-phase modulation, lead to the change of the absolute phase

*Φ*

_{CE}. The difference of

*Φ*

_{CE}between the two cases with and without LFC can go up to π at some location, where Rabi frequency oscillates in opposite polarity. To be surprising, the area of solitary pulse, no matter LFC is considered or not, is evidently larger than 2π. However, the area of the effective Rabi frequency, which equals to that Rabi frequency Ω pluses the product of the strength of NDD interaction ∈ and the polarization

*u*, i.e. Ω +∈

*u*, keeps invariant area 2π in a relative large density range. Moreover, the polarization

*u*modifies the light-matter interaction. The interaction becomes stronger when the polarization following the Rabi frequency with identical direction, and becomes weaker when the direction of the polarization is opposite. When the area of input pulse is near a certain threshold, the large area pulse penetrating into the medium produces several solitary pulses as usual, however, the number of solitary pulses changes due to the light-matter interaction being enhanced or depressed.

## 2. Theoretical model

*ε*

_{0}is the electric permittivity in the vacuum.

35. V.P. Kalosha, M. Müller, and J. Herrmann, “Theory of solid-state laser mode locking by coherent semiconductor quantum-well absorbers,” J. Opt. Soc. Am. B **16**, 323 (1999). [CrossRef]

*μ*

_{0}is the magnetic permeability in the vacuum. In Eq. (2b) the macroscopic nonlinear polarization

*P*

_{x}

*=Ndu*is connected with the off-diagonal density matrix element

*ρ*

_{12}= (

*u*+

*iv*)/2 , the population difference

*w*=

*ρ*

_{22}-

*ρ*

_{11}between the upper and lower states, which are determined by the Bloch equations with LFC,

*N*is the density of the two-level medium and

*d*is the dipole moment. Substituting the local field

*E*

_{L}in Eq. (1) for the electric field in the interaction Hamiltonian operator and Eq. (12) in Ref [7

7. R.W. Ziolkowski, J.M. Arnold, and D.M. Gogny, “Ultrafast pulse interaction with tow-level atoms,” Phys. Rev. A **52**, 3082–3094 (1995). [CrossRef] [PubMed]

*d*E

_{x}/

*h*is the Rabi frequency,

*h*is the Planck’s constant divided by 2π,

*γ*

_{1}and

*γ*

_{2}are the polarization and population relaxation constant, respectively, ω

_{0}is the transition frequency of the two-level medium, and

*w*

_{0}is the initial population difference of the system.

*w*

_{0}=-1 at t=0 means all the atoms initially should be in their lower states. The NDD parameter ∈ =

*Nd*

^{2}/3

*ε*

_{0}

*ħ*has unit of frequency, which presents the strength of the NDD interaction. For ∈ = 0, i.e. the NDD effect is not considered, Eq. (3) is consist with Eq. (2) in Ref. [12

12. V.P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. **83**, 544–547 (1999). [CrossRef]

36. K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302–307 (1966). [CrossRef]

7. R.W. Ziolkowski, J.M. Arnold, and D.M. Gogny, “Ultrafast pulse interaction with tow-level atoms,” Phys. Rev. A **52**, 3082–3094 (1995). [CrossRef] [PubMed]

37. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electronmagn. Compat. **EMC-23**, 377–382 (1981). [CrossRef]

^{-17}s and Δz=7.5×10

^{-9}m. The initial condition is

_{0}=25μ m, E

_{0}is the peak amplitude of the incident pulse, τ

_{p}is the full wide at half-maximum (FWHM) of the pulse intensity envelope, and c is the light velocity in the vacuum. The choice of z

_{0}ensures that the pulse penetrates negligibly into the medium at t=0. The medium is initialized with

*u*=

*v*=0, and the population difference

*w*

_{0}= -1 at t=0. In the following numerical analysis, all the material parameters we adopt are based on Ref. [12

12. V.P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. **83**, 544–547 (1999). [CrossRef]

_{p}=5 fs, ω

_{p}=ω

_{0}=2.3 fs

^{-1}(λ=830 nm), d=2×10

^{-29}Asm,

^{20}cm

^{-3}gives ∈ =0.0667 fs

^{-1}(Corresponding to ω

_{c}=0.2 fs

^{-1}in Ref. [12

**83**, 544–547 (1999). [CrossRef]

_{0}=

*d*E

_{0}/

*h*=1 fs

^{-1}corresponds to the electric field of E

_{x}=5×10

^{9}V/m or an intensity of I=6.6×10

^{12}W/cm

^{2}. The input envelope area can be get by A=Ω

_{0}τ

_{p}π/1.76.

## 3. Numerical results

*u, v, w*) but remain the 2ω part, and find it still valid that the effect of NDD arises a dynamic shift Δ

_{L}=∈

*w*in the transition frequency as stated in Refs.[23

23. M. Scalora and C.M. Bowden, “Propagation effects and ultrafast optical switching in dense media,” Phys. Rev. A **51**, 4048–4056 (1995). [CrossRef] [PubMed]

30. V. Malyshev and E.C. Jarque, “Spatial effects in nonlinear resonant reflection from the boundary of a dense semi-infinite two-level medium: normal incidence,” J. Opt. Soc. Am. B **14**, 1167–1178 (1997). [CrossRef]

39. R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two-level atoms,” Phys. Rep. **C7**, 101–179 (1973). [CrossRef]

*u*for a given area due to ultrashort duration and medium can be nearly completely inversed, whose average is near to zero, so the effect of NDD is negligible. For dense medium, which is weakly excited and ∈

*u*is comparable with the Rabi frequency, the NDD interaction makes some essential differences. Firstly, the reflected, penetrating, and transmitting pulses are investigated for different medium densities and different input pulse magnitude Ω

_{0}. It is found that the spectra of these fields are similar to those in Ref [12

**83**, 544–547 (1999). [CrossRef]

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

*t*) = Ω

_{m}sec

*h*(1.76(

*t*-

*t*

_{0})/

*)cos(*τ ^

_{p}*ω*

_{c}(

*t*-

*t*

_{0}) +

*ϕ*(

*t*)), where Ω

_{m}is the predicted maximum of the envelope, ω

_{c}is the central carrier frequency,

*t*

_{0}is the delay,

*is the predicted duration and*τ ^

_{p}*ϕ*(

*t*) the temporal pulse phase, we can determine the area, the center frequency and the absolute phase

*Φ*

_{CE}, i.e.

*ϕ*(

*t*

_{0}).

_{0}=1.8 fs

^{-1}corresponding to 5.1π and ∈ = 1/3

*fs*

^{-1}, in the case without LFC, the absolute phase

*Φ*

_{CE}is -0.48 rad at z=79 μ m. However, when the LFC is considered,

*Φ*

_{CE}changes into 2.75 rad at the same location, and then the difference of

*Φ*

_{CE}is about 1.0π, i.e. the Rabi frequency oscillates in opposite direction. When we observe the pulse at z=80 μ m, the absolute phase in the case without LFC is 2.12 rad and changes into -2.07 rad in the case with LFC. So, if we confine the difference of

*Φ*

_{CE}in the range (-π, π], the difference of

*Φ*

_{CE}between two cases becomes into 0.67π, i.e. the difference changes about π/3 during the length increasing 1 μ m. Although the difference keeps changing in the medium, it remains constant after transmitting into free space. As an example, the difference remains 1.0π if pulses exit into free space at z=79 μ m, i.e. the medium zone length L=39 μ m. Numerical results show that the absolute phases of pulses transmitting into free space are constant for both cases, but the difference varies linearly in the rate of -0.36π/μ m with the medium zone length increasing, see Fig. 2. The difference goes back to the same value every about 4.5 μ m and is near to zero for the medium zone length L=40.5 μ m. For other input pulse and medium density, consistent conclusion can also be drawn. It is clear that the carrier-envelope phase

*Φ*

_{CE}of the pulse is sensitive to the NDD effect.

*Φ*

_{CE}can be interpreted as a combination result of three factors: the decreasing in the center frequency, the increasing in the group velocity and the self-phase modulation. For the case without LFC, the center frequency of the pulse is 1.74 times the resonant frequency, i.e. 1.74ω

_{0}, but it decreases to 1.67ω

_{0}owing to the dynamic redshift in the resonance frequency when the LFC is considered, which means that the center frequency decreases by about 0.16 fs

^{-1}, as shown in Fig. 1. Numerical results show that the group velocity increases slightly and the self-phase modulations of Ω for both cases are negligible. The change in the absolute phase mainly attributes to the change in the center frequency and the group velocity.

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

*u*, the pulse interacts with dense medium as a pulse whose area is 2.0π, see Fig. 1. So we define an effective Rabi frequency: Ω

_{eff}= Ω+∈

*u*, i.e. an effective coupling of light-matter. As shown in Figs. 3(a-f), for smaller density of medium ∈ = 0.1

*fe*

^{-1}and weaker incident pulse Ω

_{0}=1.4 fs

^{-1}in comparision with the case in Fig. 1, the area of Ω, no matter the LFC is considered or not, is about 2.1π, but that of Ω

_{eff}is 2.0π. For larger density of medium ∈ = 0.4

*fs*

^{-1}and incident pulse Ω

_{0}=1.8 fs

^{-1}, the area of Ω in both cases is about 2.4π, while that of Ω

_{eff}is still 2.0π. In dense medium, the area of Rabi frequency Ω is obviously larger than the value predicted by the standard area theorem. However, numerical experiments show that, in spite the density of medium and the peak of envelope of Rabi frequency vary in a relative large range, the area of Ω

_{eff}keeps invariant area 2π.

*u*modifies the light-matter interaction: the polarization enhances the light-matter interaction when it follows the field with identical direction, and suppresses the interaction when it oscillates in opposite direction to the field. We find that there are some area thresholds of input pulse, where the NDD effect causes the number of solitary pulses, which are produced by the penetrating part, to change. As an example in Fig. 4, for larger medium density, ∈ =2/3 fs

^{-1}, and higher input pulse, Ω

_{0}=5 fs

^{-1}corresponding to 14π, in the case without LFC, the penetrating part produces four solitary pulses: two unipolar half-cycle solitons and two high-frequency oscillating solitary pulses. However, when the LFC is considered, the effect of NDD causes that the slowest oscillating pulse changes into a unipolar half-cycle pulse, which is well investigated in Refs.[12

**83**, 544–547 (1999). [CrossRef]

41. A.E. Kaplan and P.L. Shkolnikov, “Electromagnetic “Bubbles” and shock waves: unipolar, non oscillating EM solitions,” Phys. Rev. Lett. **75**, 2316–2319 (1995). [CrossRef] [PubMed]

42. R.K. Bullough and F. Ahmad, “Exact solutions of the self-induced transparency equations,” Phys. Rev. Lett. **27**, 330–333 (1971). [CrossRef]

**83**, 544–547 (1999). [CrossRef]

41. A.E. Kaplan and P.L. Shkolnikov, “Electromagnetic “Bubbles” and shock waves: unipolar, non oscillating EM solitions,” Phys. Rev. Lett. **75**, 2316–2319 (1995). [CrossRef] [PubMed]

^{-1}, for weaker input pulse, Ω

_{0}=4.3 fs

^{-1}, as shown in Fig. 5, when the LFC is not considered, the penetrating field produces three oscillating solitary pulses. Numerical investigation shows that the polarization

*u*well follows the pulse in time, but with opposite direction. So, the effect of NDD depresses the light-matter interaction and arouses that the penetrating part only produces two oscillating pulses. As shown by above analysis, when the area of input pulse is close to a certain threshold, the effect of NDD could cause the number of solitary pulses to change.

## 4. Conclusion

*Φ*

_{CE}of solitary pulse drastically changes as comparison with the case without LFC. At some location, the difference of

*Φ*

_{CE}between two cases with and without LFC can go up near to π, i.e. the Rabi frequency oscillates in opposite direction. The absolute phases of pulses exiting into free space are constant for both cases, but the difference varies linearly with the medium zone length. The absolute phase

*Φ*

_{CE}has essential influence on a large number of extreme nonlinear interactions owing to the extremely short duration of pulse, and then the NDD effect plays an important role. Secondly, in a dense medium, the area of the stable state pulse, which deviates from the value predicted by the standard area theorem, is larger than 2π. However, if we consider the effective coupling of light-matter, i.e. Ω+∈

*u*, the area keeps 2π in spite the medium density varies in a relative large range. Furthermore, the polarization

*u*can enhance or depress the strength of light-matter interaction. This effect accounts for the change in the number of solitary pulses at certain input pulse area.

## Acknowledgments

## References and links

1. | A. Baltuška, Z.Y. Wei, M. S. Pshenichnikov, and D. A. Wiersma, “Optical pulse compression to 5 fs at a 1-MHz repetition rate,” Opt. Lett. |

2. | A. Stingel, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, “Sub-10-fs mirror-dispersion-controlled Ti: sapphire laser,” Opt. Lett. |

3. | L. Allen and J.H. Eberly, “theorem” in |

4. | S.L. McCall and E.L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. |

5. | S.L. McCall and E.L. Hahn, “Self-induced transparency,” Phys. Rev. |

6. | S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. |

7. | R.W. Ziolkowski, J.M. Arnold, and D.M. Gogny, “Ultrafast pulse interaction with tow-level atoms,” Phys. Rev. A |

8. | X.H. Song, S.Q. Gong, W.F. Yang, and Z.Z. Xu, “Propagation of an attosecond pulse in a dense two-level medium,” Phys. Rev. A |

9. | X.H. Song, S.Q. Gong, S.Q. Jin, and Z.Z. Xu, “Formation of higher spectral components in a two-level medium driven by two-color ultrashort laser pulses,” Phys. Rev. A |

10. | X.H. Song, S.Q. Gong, W.F. Yang, S.Q. Jin, X.L. Feng, and Z.Z. Xu, “Coherent control of spectra effects with chirped femtosecond laser pulse,” Opt. Comm. |

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12. | V.P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. |

13. | T. Serényi, C. Benedek, and M.G. Benedict, “Femtosecond light pulses in GaAs, third harmonic generation by carrier wave Rabi flopping,” Fortschr. Phys. |

14. | O.D. Mücke, T. Tritschler, and M. Wegener, “Signatures of carrier-wave Rabi flopping in GaAs,” Phys. Rev. Lett. |

15. | G.G. Paulus, F. Grashon, H. Walther, P. Vllloresl, M. Nlsoll, S. Staglra, E. Prlorl, and S. De Sllvestrl, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature |

16. | A. Baltuška, Th. Udem, M. Ulberacker, M. Hentschel, E. Goullelmakls, C. Gohle, R. Holzwarth, V.S. Yakovlev, A. Scrlnzl, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature |

17. | X. Liu, “Nonsequential double ionization with few-cycle laser pulses,” Phys. Rev. Lett. |

18. | A. de Bohan, P. Antoine, D. B. Miloševic, and B. Piraux, “Phase-dependent harmonic emission with ultrashort laser pulses,” Phys. Rev. Lett. |

19. | R. M. Potvliege, N.J. Kylstra, and C.J. Joachain, “Photon emission by He |

20. | C.M. Bowden and J.P. Dowling, “Near-dipole-dipole effects in dense media: generalized Maxwell-Bloch equations,” Phys. Rev. A |

21. | J.D. Jackson, “relation”, in |

22. | M.E. Crenshaw, M. Scalora, and C.M. Bowden, “Ultrafast intrinsic optical switching in a dense medium of two-level atoms,” Phys. Rev. Lett. |

23. | M. Scalora and C.M. Bowden, “Propagation effects and ultrafast optical switching in dense media,” Phys. Rev. A |

24. | O.G. Calderón, M.A. Antón, and F. Carreño, “Near dipole-dipole effects in a V-type medium with vacuum induced coherence,” Eur. Phys. J. D |

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27. | C.M. Bowden, A. Postan, and R. Inguva, “Invariant pulse propagation and self-phase modulation in dense media,” J. Opt. Soc. Am. B |

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29. | J.P. Dowling and C.M. Bowden, “Near dipole-dipole effects in lasing without inversion: an enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. |

30. | V. Malyshev and E.C. Jarque, “Spatial effects in nonlinear resonant reflection from the boundary of a dense semi-infinite two-level medium: normal incidence,” J. Opt. Soc. Am. B |

31. | 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 |

32. | M.G. Benedict, V.A. Malyshev, E.D. Trfonov, and A.I. Zaitsev, “Reflection and transmission of ultrashort light pulses through a thin resonant medium: local-field effects,” Phys. Rev. A |

33. | J.T. Manassah and B. Gross, “The dynamical Lorentz shift in an extended optically dense superradiant amplifier,” Opt. Express |

34. | F.A. Hopf, C.M. Bowden, and W.H. Louisell, “Mirrorless optical bistability with the use of the local-field correction,” Phys. Rev. A |

35. | V.P. Kalosha, M. Müller, and J. Herrmann, “Theory of solid-state laser mode locking by coherent semiconductor quantum-well absorbers,” J. Opt. Soc. Am. B |

36. | K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

37. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electronmagn. Compat. |

38. | A. Taflove, “cΔt≤ Δz”, in |

39. | R. Friedberg, S.R. Hartmann, and J.T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two-level atoms,” Phys. Rep. |

40. | W. Miklaszewski, “Near-resonant propagation of the light pulse in a homogenerously broadened two-level medium,” J. Opt. Soc. Am. B |

41. | A.E. Kaplan and P.L. Shkolnikov, “Electromagnetic “Bubbles” and shock waves: unipolar, non oscillating EM solitions,” Phys. Rev. Lett. |

42. | R.K. Bullough and F. Ahmad, “Exact solutions of the self-induced transparency equations,” Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.0320) Ultrafast optics : Ultrafast optics

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 7, 2005

Revised Manuscript: July 14, 2005

Published: August 8, 2005

**Citation**

Keyu Xia, Shangqing Gong, Chengpu Liu, Xiaohong Song, and Yueping Niu, "Near dipole-dipole effects on the propagation of few-cycle pulse in a dense two-level medium," Opt. Express **13**, 5913-5924 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-5913

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### References

- A. Baltuška, Z.Y. Wei, M. S. Pshenichnikov, and D. A. Wiersma, �??Optical pulse compression to 5 fs at a 1-MHz repetition rate,�?? Opt. Lett. 22, 102 (1997). [CrossRef] [PubMed]
- A. Stingel, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, �??Sub-10-fs mirror-dispersion-controlled Ti: sapphire laser,�?? Opt. Lett. 20, 602 (1995). [CrossRef]
- L. Allen and J.H. Eberly, �??Theorem�?? in Optical Resonance and Two-Level Atoms, (Wiley, New York, 1975)
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