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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 21754–21761
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Dependence of dynamic Lorentz frequency shift on carrier-envelope phase and including local field effects

Chaojin Zhang, Weifeng Yang, Xiaohong Song, and Zhizhan Xu  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 21754-21761 (2009)
http://dx.doi.org/10.1364/OE.17.021754


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Abstract

We investigate the local field effects in a ZnO dense medium. Our results show due to the local-field effects, the Lorentz shifts can be found in the reflected spectra driven by the few-cycle laser pulse. Moreover, the dynamic Lorentz shifts depend sensitively on the carrier-envelope phase (CEP) of the few-cycle laser pulse, which provides a useful means to obtain the CEP information by the frequency shifts.

© 2009 OSA

1. Introduction

In recent years, due to the remarkable advancement of ultrashort laser pulses, it is feasible to generate light pulses with durations comparable to the carrier oscillation cycle [1

1. T. Brabec and F. Krausz, “Intense few-cycle laser field: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 ( 2000). [CrossRef]

]. For few-cycle laser pulses, many new phenomena have attracted extensive attention in resonant extreme nonlinear optics [2

2. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

9

9. C. Zhang, X. Song, W. Yang, and Z. Xu, “Carrier-envelope phase control of carrier-wave Rabi flopping in asymmetric semiparabolic quantum well,” Opt. Express 16(3), 1487–1496 ( 2008). [CrossRef] [PubMed]

]. For example, the phenomenon of the carrier-wave Rabi flopping was firstly predicted in two-level system by Hughes [2

2. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

], and it then was proved in experiment [3

3. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, “Signatures of carrier-wave Rabi flopping in GaAs,” Phys. Rev. Lett. 87(5), 057401 ( 2001). [CrossRef] [PubMed]

]. When few-cycle laser pulses propagate in a resonant two-level medium, a large red-shift in the reflected laser pulse is found [4

4. 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(3), 544–547 ( 1999). [CrossRef]

].

Many works have considered the LFE when laser interacts with the medium [22

22. C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47(2), 1247–1251 ( 1993). [CrossRef] [PubMed]

32

32. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A 40(5), 2446–2451 ( 1989). [CrossRef] [PubMed]

]. The generalized Maxwell-Bloch equations were presented with the LFE [22

22. C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47(2), 1247–1251 ( 1993). [CrossRef] [PubMed]

]. The reflection and transmission of ultrashort light pulses have been investigated [23

23. M. G. Benedict, V. A. Malyshev, E. D. Trifonov, and A. I. Zaitsev, “Reflection and transmission of ultrashort light pulses through a thin resonant medium: Local-field effects,” Phys. Rev. A 43(7), 3845–3853 ( 1991). [CrossRef] [PubMed]

]. In particular, the LFE plays an important role on generation of the Lorentz shift [26

26. J. T. Manassah and B. Gross, “The dynamical lorentz shift in an extended optically dense superradiant amplifier,” Opt. Express 1(6), 141–151 ( 1997). [CrossRef] [PubMed]

32

32. R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A 40(5), 2446–2451 ( 1989). [CrossRef] [PubMed]

]. The dynamic Lorentz shift in dense superradiant amplifier has been found [26

26. J. T. Manassah and B. Gross, “The dynamical lorentz shift in an extended optically dense superradiant amplifier,” Opt. Express 1(6), 141–151 ( 1997). [CrossRef] [PubMed]

]. A signature of near dipole-dipole interaction may provide a useful method for measuring the strength of the interaction [27

27. 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(7), 911–914 ( 1992). [CrossRef] [PubMed]

]. Moreover, in an experiment, the excitation dependence of the Lorentz local-field shift can be measured [29

29. H. Van Kampen, V. A. Sautenkov, C. J. C. Smeets, E. R. Eliel, and J. P. Woerdman, “Measurement of the excitation dependence of the Lorentz local-field shift,” Phys. Rev. A 59(1), 271–274 ( 1999). [CrossRef]

]. However, almost of these works are based on the slowly-varying-envelope approximation (SVEA) and the rotating-wave approximation (RWA) to solve the Maxwell-Bloch equations. In few-cycle regime, the SVEA and the RWA will break down [2

2. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

,4

4. 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(3), 544–547 ( 1999). [CrossRef]

,8

8. C. Van Vlack and S. Hughes, “Third-harmonic generation in disguise of second-harmonic generation revisited: role of thin-film thickness and carrier-envelope phase,” Opt. Lett. 32(2), 187–189 ( 2007). [CrossRef] [PubMed]

]. In our work, we investigate the new effects of the Lorentz shift driven by few-cycle laser pulses and solve numerically the full-wave Maxwell-Bloch equations without adopting the SVEA and the RWA. It is shown that the dynamic Lorentz shift can occur in the reflected spectra due to the LFE of the ZnO dense medium. Importantly, the Lorentz shift depends strongly on the CEP of the few-cycle laser pulse.

This paper is organized as follows. The interaction of few-cycle pulses with a dense medium when considering the LFE is described in Sec. 2. The characteristics of the dynamic Lorentz shift with adjusting the CEP are presented in Sec. 3. We summarize the results in Sec. 4.

2. Theory

As done in Refs [2

2. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

,4

4. 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(3), 544–547 ( 1999). [CrossRef]

,30

30. M. E. Crenshaw, K. U. Sullivan, and C. M. Bowden, “Local field effects in multicomponent media,” Opt. Express 1(6), 152–159 ( 1997). [CrossRef] [PubMed]

], we consider the propagation of the few-cycle pulse in a two-level medium. The pulse initially propagates in the free-space region, then it partially enters the ZnO dense medium at z = 12 μm, and partially reflects backward. The backward reflected pulse is detected at z = 3 μm. Taking the initial laser field propagating along z direction, and polarized along x direction, the Maxwell equations are the following form:
Hyt=1μ0Exz,Ext=1ε0εsHyz1ε0εsPxt,
(1)
where μ0 is the magnetic permeability in the vacuum. Px=2NdRe[ρ12] is the macroscopic nonlinear polarization which connects with the off-diagonal element of the density matrix in the medium. The number density of medium is taken to be N=4.0×1020cm3 and the dipole matrix element d = 0.19 e•nm. εs is the relevant dielectric constant of the medium and εs=4 [8

8. C. Van Vlack and S. Hughes, “Third-harmonic generation in disguise of second-harmonic generation revisited: role of thin-film thickness and carrier-envelope phase,” Opt. Lett. 32(2), 187–189 ( 2007). [CrossRef] [PubMed]

], while within the vacuum, Px=0 and εs=1. The Bloch equations including the LFE take the forms
ρ12t=ω12ρ12+idElocn1τ2ρ12,
(2)
nt=i2dEloc(ρ12ρ12*)1τ1n,
(3)
where ρ12 is the off-diagonal element of the density matrix. n=ρ22ρ11 is the population difference between the excited and ground states. In the dense medium, the local-field is Eloc=Ex+P3ε0=Ex+Nd3ε0(ρ12+ρ12*) which shows the LFE is relevant to the macroscopic coherent polarization [22

22. C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47(2), 1247–1251 ( 1993). [CrossRef] [PubMed]

,23

23. M. G. Benedict, V. A. Malyshev, E. D. Trifonov, and A. I. Zaitsev, “Reflection and transmission of ultrashort light pulses through a thin resonant medium: Local-field effects,” Phys. Rev. A 43(7), 3845–3853 ( 1991). [CrossRef] [PubMed]

], while without the LFE, Eloc=Ex. The transition energy ω12 is close to the ZnO band gap (Es=3.3 eV). The excited-state lifetime and the dephasing time are set to τ1= and τ1=50 fs, respectively [7

7. T. Tritschler, O. D. Mücke, M. Wegener, and F. X. Kärtner, “Evidence for third-harmonic generation in disguise of second-harmonic generation in extreme nonlinear optics,” Phys. Rev. Lett. 90(21), 217404 ( 2003). [CrossRef] [PubMed]

,8

8. C. Van Vlack and S. Hughes, “Third-harmonic generation in disguise of second-harmonic generation revisited: role of thin-film thickness and carrier-envelope phase,” Opt. Lett. 32(2), 187–189 ( 2007). [CrossRef] [PubMed]

]. The full-wave Maxwell-Bloch equations are solved by employing Yee's finite-difference time-domain (FDTD) discretization scheme [33

33. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 ( 1966). [CrossRef]

38

38. J. Xiao, Z. Wang, and Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3), 031402 ( 2002). [CrossRef]

]. Mur absorbing boundary conditions [39

39. G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23(4), 377–382 ( 1981). [CrossRef]

] are incorporated with FDTD discretization in order to avoid the influence of the finite-space computational domain. The temporal and spatial increments Δt and Δz are chosen to ensure cΔtΔz [40

40. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23(8), 623–630 ( 1975). [CrossRef]

], i.e., Δt=2.5×1018s and Δz=1.5×109m.

The initial laser field propagating along z direction is
Ex(t=0,z)=E0sech[1.76(z/cz0/c)/τ0]cos[ω0(zz0)/c+φ],
(4)
where E0 is the peak field strength of the envelope. φ is the initial CEP, and τ0 is the full width at half maximum (FWHM) of the pulse intensity envelope. The choice of z0 ensures that the pulse penetrates negligibly into the medium at t = 0. The excitation pulse has a central frequency corresponding to a photon energy of ω0 = 1.4 eV and a FWHM intensity of τ0=5 fs. In our work, the Rabi frequency is comparable to the band gap frequency, so it is relevant to carrier-wave Rabi flopping. The envelope area of the input laser field we used is about 5π within the medium (the pulse area is defined as A=dεsE(t)dt, where E(t)is the envelope of the laser pulse), which is in the regime of the carrier-wave Rabi flopping (>4π) [2

2. S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

,8

8. C. Van Vlack and S. Hughes, “Third-harmonic generation in disguise of second-harmonic generation revisited: role of thin-film thickness and carrier-envelope phase,” Opt. Lett. 32(2), 187–189 ( 2007). [CrossRef] [PubMed]

].

3. Results and discussions

Then we discuss the relation between the dynamic Lorentz shift and the CEP of the reflected few-cycle laser pulse with the LFE. As shown in Fig. 2
Fig. 2 (Color online) The dynamic Lorentz shift of the normalized reflected spectra of the few-cycle laser pulse with the LFE.
, the dynamic Lorentz shifts of the reflected spectra become evident with adjusting the CEP. Forϕ=0, the dynamic Lorentz shift of the reflected spectrum is δ1=-19 .85 THZ (see Fig. 2, solid line), while for ϕ=0.5π, the dynamic Lorentz shift significantly increases (δ2=-29 .24THZ). The reason can be explained by the population differences near the front face of the medium in Fig. 3(a)
Fig. 3 (Color online) (a) The population difference n near the front face with the LFE. (b) The reemitted spectra [Eem(ω)] near the front face of the medium.
. The transitions between the ground and excited levels represented by the population distributions can be reversed because of the two-photon absorption. As expressed in Eq. (2), the LFE will obviously affect the population distribution, which directly influence the Lorentz shifts near the resonant frequency. Interestingly, the population distributions change with the CEP of the few-cycle laser pulse. Importantly, the population differences are same for the cases of the CEPs φ=0 and φ=π [see Fig. 3(a)]. This means that the period of the CEP-dependent spectral signal is π. The view can be further proved through the reemitted field Eem(t), i.e., single photon emission [see Fig. 3(b)]. When the resonant excitation is considered, the macroscopic coherent polarization Px(t) will build up gradually near the front face of the medium driven by the incident laser field, which acts as a source of a reemitted field Eem(t), i.e., Eem(ω)FFT(Pxt) [15

15. W. Yang, X. Song, S. Gong, Y. Cheng, and Z. Xu, “Carrier-envelope phase dependence of few-cycle ultrashort laser pulse propagation in a polar molecule medium,” Phys. Rev. Lett. 99(13), 133602 ( 2007). [CrossRef] [PubMed]

,42

42. C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 ( 2004). [CrossRef] [PubMed]

].

The CEP-dependent Lorentz shift will be more significant when the inversion symmetry of the system is broken, such as the presence of a static electric field. When a static electric field is added (Es=2%E0), the period of the CEP-dependent spectral signal becomes 2π as shown in Fig. 4
Fig. 4 (Color online) Same as in Fig. 2 but for the presence of a static electric field.
. Moreover, due to the LFE, the dynamic Lorentz shifts δ depend crucially on the CEP of few-cycle laser pulses. By analysis, the underlying physical mechanism for the phenomenon can be explained with the field strength. As presented in Ref [43

43. A. Brown and W. J. Meath, “On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules,” J. Chem. Phys. 109(21), 9351–9365 ( 1998). [CrossRef]

], the populations of the states can be strongly phase dependent if the field strength is increased. As a result, a 2% variation in the field amplitude can affect the population distributions near the front face of the medium as shown in Fig. 5(a)
Fig. 5 (Color online) Same as in Fig. 3 but the presence of a static electric field.
, which directly control the Lorentz shift decreasing or increasing. In order to demonstrate the validity of our analysis, the spectra of the reemitted field for different CEPs are performed to help interpret the CEP-dependent phenomena [see Fig. 5(b)].

One thing that should be highlighted is that such alternative phenomena are not limited to the certain value but are valid for a range of the number density of medium N. As shown in Fig. 6
Fig. 6 (Color online) Same as in Fig. 2 but for the number density of medium N=3.0×1020cm3.
, due to the LFE, the characteristics of the dynamic Lorentz shift depending on the CEP of few-cycle laser pulses also occur. Our results may provide a potential method to gain the information about the CEP of few-cycle laser pulses by measuring the dynamic Lorentz shifts.

4. Conclusions

In conclusion, we have studied the LFE on the reflected spectra of few-cycle laser pulses in a dense medium. It has been found that when considering the LFE, the Lorentz shift can be obtained in the reflected spectrum. Interestingly, the dynamic Lorentz shifts vary significantly with adjusting the CEP of the few-cycle laser pulse. Moreover, when the inversion symmetry of the system is broken, the CEP-dependent spectra effects will be further enhanced. Our results suggest that the CEP-dependent effect can provides a useful route to obtain the CEP information.

Acknowledgments

References and links

1.

T. Brabec and F. Krausz, “Intense few-cycle laser field: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 ( 2000). [CrossRef]

2.

S. Hughes, “Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 ( 1998). [CrossRef]

3.

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, “Signatures of carrier-wave Rabi flopping in GaAs,” Phys. Rev. Lett. 87(5), 057401 ( 2001). [CrossRef] [PubMed]

4.

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(3), 544–547 ( 1999). [CrossRef]

5.

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, F. X. Kärtner, G. Khitrova, and H. M. Gibbs, “Carrier-wave Rabi flopping: role of the carrier-envelope phase,” Opt. Lett. 29(18), 2160–2162 ( 2004). [CrossRef] [PubMed]

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O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, “Role of the carrier-envelope offset phase of few-cycle pulses in nonperturbative resonant nonlinear optics,” Phys. Rev. Lett. 89(12), 127401 ( 2002). [CrossRef] [PubMed]

7.

T. Tritschler, O. D. Mücke, M. Wegener, and F. X. Kärtner, “Evidence for third-harmonic generation in disguise of second-harmonic generation in extreme nonlinear optics,” Phys. Rev. Lett. 90(21), 217404 ( 2003). [CrossRef] [PubMed]

8.

C. Van Vlack and S. Hughes, “Third-harmonic generation in disguise of second-harmonic generation revisited: role of thin-film thickness and carrier-envelope phase,” Opt. Lett. 32(2), 187–189 ( 2007). [CrossRef] [PubMed]

9.

C. Zhang, X. Song, W. Yang, and Z. Xu, “Carrier-envelope phase control of carrier-wave Rabi flopping in asymmetric semiparabolic quantum well,” Opt. Express 16(3), 1487–1496 ( 2008). [CrossRef] [PubMed]

10.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 ( 2000). [CrossRef] [PubMed]

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G. G. Paulus, F. Grasbon, H. Walther, P. Villoresi, M. Nisoli, S. Stagira, E. Priori, and S. De Silvestri, “Absolute-phase phenomena in photoionization with few-cycle laser pulses,” Nature 414(6860), 182–184 ( 2001). [CrossRef] [PubMed]

12.

W. Yang, X. Song, R. Li, and Z. Xu, “Generation of intense extreme supercontinuum radiation via resonant propagation effects,” Phys. Rev. A 78(2), 023836 ( 2008). [CrossRef]

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C. Lemell, X. M. Tong, F. Krausz, and J. Burgdörfer, “Electron emission from metal surfaces by ultrashort pulses: determination of the carrier-envelope phase,” Phys. Rev. Lett. 90(7), 076403 ( 2003). [CrossRef] [PubMed]

14.

C. Van Vlack and S. Hughes, “Carrier-envelope-offset phase control of ultrafast optical rectification in resonantly excited semiconductors,” Phys. Rev. Lett. 98(16), 167404 ( 2007). [CrossRef] [PubMed]

15.

W. Yang, X. Song, S. Gong, Y. Cheng, and Z. Xu, “Carrier-envelope phase dependence of few-cycle ultrashort laser pulse propagation in a polar molecule medium,” Phys. Rev. Lett. 99(13), 133602 ( 2007). [CrossRef] [PubMed]

16.

C. Zhang, W. Yang, X. Song, and Z. Xu, “Carrier-envelope phase dependence of the spectra of reflected few-cycle laser pulses in the presence of a static electric field,” Phys. Rev. A 79(4), 043823 ( 2009). [CrossRef]

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C. Zhang, W. Yang, X. Song, and Z. Xu, “Phase control of higher spectral components in the presence of a static electric field,” J. Phys. B 42(5), 055602 ( 2009). [CrossRef]

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R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Frequency shifts in emission and absorption by resonant systems of two-level atoms,” Phys. Rep. 7(3), 101–179 ( 1973). [CrossRef]

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M. E. Crenshaw, “Comparison of quantum and classical local-field effects on two-level atoms in a dielectric,” Phys. Rev. A 78(5), 053827 ( 2008). [CrossRef]

22.

C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: Generalized Maxwell-Bloch equations,” Phys. Rev. A 47(2), 1247–1251 ( 1993). [CrossRef] [PubMed]

23.

M. G. Benedict, V. A. Malyshev, E. D. Trifonov, and A. I. Zaitsev, “Reflection and transmission of ultrashort light pulses through a thin resonant medium: Local-field effects,” Phys. Rev. A 43(7), 3845–3853 ( 1991). [CrossRef] [PubMed]

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E. Paspalakis, A. Kalini, and A. F. Terzis, “Local field effects in excitonic population transfer in a driven quantum dot system,” Phys. Rev. B 73(7), 073305 ( 2006). [CrossRef]

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D. V. Novitsky, “Compression of an intensive light pulse in photonic-band-gap structures with a dense resonant medium,” Phys. Rev. A 79(2), 023828 ( 2009). [CrossRef]

26.

J. T. Manassah and B. Gross, “The dynamical lorentz shift in an extended optically dense superradiant amplifier,” Opt. Express 1(6), 141–151 ( 1997). [CrossRef] [PubMed]

27.

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(7), 911–914 ( 1992). [CrossRef] [PubMed]

28.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and nonlinear optical measurements of the Lorentz local field,” Phys. Rev. Lett. 67(8), 972–975 ( 1991). [CrossRef] [PubMed]

29.

H. Van Kampen, V. A. Sautenkov, C. J. C. Smeets, E. R. Eliel, and J. P. Woerdman, “Measurement of the excitation dependence of the Lorentz local-field shift,” Phys. Rev. A 59(1), 271–274 ( 1999). [CrossRef]

30.

M. E. Crenshaw, K. U. Sullivan, and C. M. Bowden, “Local field effects in multicomponent media,” Opt. Express 1(6), 152–159 ( 1997). [CrossRef] [PubMed]

31.

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effects of the dynamic Lorentz shift on four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 42(1), 494–497 ( 1990). [CrossRef] [PubMed]

32.

R. Friedberg, S. R. Hartmann, and J. T. Manassah, “Effect of local-field correction on a strongly pumped resonance,” Phys. Rev. A 40(5), 2446–2451 ( 1989). [CrossRef] [PubMed]

33.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antenn. Propag. 14(3), 302–307 ( 1966). [CrossRef]

34.

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37.

W. Yang, S. Gong, and Z. Xu, “Enhancement of ultrafast four-wave mixing in a polar molecule medium,” Opt. Express 14(16), 7216–7223 ( 2006). [CrossRef] [PubMed]

38.

J. Xiao, Z. Wang, and Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3), 031402 ( 2002). [CrossRef]

39.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23(4), 377–382 ( 1981). [CrossRef]

40.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23(8), 623–630 ( 1975). [CrossRef]

41.

K. Xia, S. Gong, C. Liu, X. Song, and Y. Niu, “Near dipole-dipole effects on the propagation of few-cycle pulse in a dense two-level medium,” Opt. Express 13(16), 5913–5924 ( 2005). [CrossRef] [PubMed]

42.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 ( 2004). [CrossRef] [PubMed]

43.

A. Brown and W. J. Meath, “On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules,” J. Chem. Phys. 109(21), 9351–9365 ( 1998). [CrossRef]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.2250) Ultrafast optics : Femtosecond phenomena
(320.7150) Ultrafast optics : Ultrafast spectroscopy

ToC Category:
Ultrafast Optics

History
Original Manuscript: September 21, 2009
Revised Manuscript: October 30, 2009
Manuscript Accepted: November 2, 2009
Published: November 12, 2009

Citation
Chaojin Zhang, Weifeng Yang, Xiaohong Song, and Zhizhan Xu, "Dependence of dynamic Lorentz frequency shift on carrier-envelope phase and including local field effects," Opt. Express 17, 21754-21761 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21754


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References

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