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Coherent control of spectra effects with chirped femtosecond laser pulse
Optics Communications 236 (2004) 151–157 www.elsevier.com/locate/optcom
Coherent control of spectra effects with chirped femtosecond laser pulse Xiao-hong Song *, Shang-qing Gong, Wei-feng Yang, Shi-qi Jin, Xun-li Feng, Zhi-zhan Xu Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Shanghai 201800, China Received 27 October 2003; received in revised form 2 January 2004; accepted 9 March 2004
Abstract We investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagation in a twolevel-atom medium. It is found that the spectral feature depends crucially on the sign and magnitude of the initial chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. Ó 2004 Elsevier B.V. All rights reserved. PACS: 32.80.Qk; 42.62.Fi; 42.50.Md Keywords: Coherent control; Spectra; Chirped femtosecond laser pulse
1. Introduction The interaction of intense laser light with a collection of two-level atoms has been an area of active research for many years [1]. For example, the famous area theorem [2,3] is one of the most interesting features, which can predict and explain many fascinating effects, such as self-induced transparency and pulse compression. Moreover, the influence of various aspects, such as atomic
*
Corresponding author. Tel. +86-021-699-18266; fax: +86021-699-18000. E-mail address: [email protected] (X.-h. Song).
relaxation, detuning on the temporal dynamic of the optical pulse and the atomic system, have also been investigated in many literatures [4–6]. All these were derived within the standing slowly varying envelope approximation (SVEA) [7] and the rotating-wave approximation (RWA) [8]. However, these approximations clearly fail if the pulse duration approaches the duration of several optical cycles [9–11]. Recently, Xiao et al. [12] investigated the propagation of 5 fs pulse in a two-level-atom medium. It was found for small pulse area, that the variation of the few-cycle pulse area is caused by the pulse splitting but not by the pulse broadening or the pulse compression as in the case of
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.03.029
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X.-h. Song et al. / Optics Communications 236 (2004) 151–157
long-pulse [2]. Moreover, Hughes proposed for large pulse areas, that the area under individual carriers might themselves cause Rabi flopping (RF), which can lead to carrier-wave reshaping and significantly higher spectral components [11] and even generation of soft-X-ray [13]. Whereas, for small pulse areas, the self-induced transparency results are still recovered with minor modifications, i.e., no higher spectral components occur. In our previous paper, we investigated the spectra effects for two-color ultrashort pulses propagating in a two-level-atom medium [14]. It showed that higher spectral components can be produced even for small area pulses due to the interference effect of the two laser pulses. Moreover, these higher spectral components depend crucially on the relative phase u of the two pulses. Continuum and distinct peaks spectra can be achieved by adjusting the relative phase to equal to 0 and p. Furthermore, Kalosha et al. [15] found that due to intrapulse four-wave mixing, a large blueshift in the transmitted pulse and a large redshift in the reflected pulse can occur in dense media of two-level systems. The advances in generation and characterization techniques of ultrashort laser pulses made it practical to prepare precisely defined laser pulses [16,17], these open up new possibilities for studying and controlling dynamical processes in femtosecond time scale. Hitherto, frequency-synthesized or chirped laser pulses have been widely applied to high-order harmonic generation (HHG) [18–21], population transfer [22–24], multiphoton transitions between Rydberg states [25], etc. Both the interference of two-color pulses and the nontrivial time dependence of chirped pulse laser field may lead to very interesting results. Moreover, recent studies have manifested that chirping can also have much impacts on the laser pulses as it propagates through different medium. The propagation properties of chirped soliton pulses in optical nonlinear Kerr media were investigated by Desaix et al. [26]. The result indicates that the properties of asymptotically emerging solitons depend not only on the magnitude, but also on the form of the initial chirp. In particular, the splitting of an initial pulse into separating soliton pulse pairs only occurs for certain classes of initial chirp functions. The influence
of pulse shape and frequency chirp on stability of optical solitons was studied by Klaus and Shaw [27]. They found that destabilization occurs when eigenvalues of an associated Zakharov–Shabat system, which move along the positive imaginary axis with increasing chirp parameter, either are absorbed into the lower half plane or collide with a another eigenvalue. Furthermore, the signal and group velocities for chirped pulses propagating through a GaAs cavity were measured by Centini et al. [28]. They showed that the chirp substantially modifies the group velocity of the pulse, but leaves the signal velocity unaltered. In this paper, we investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagating in a two-level-atom medium. By solving the full Maxwell–Bloch (M–B) equations which avoid the limitations of SVEA and RWA as ultrashort pulses are considered [9,11,12,29], we find that the spectral feature depends crucially on the sign and magnitude of the initial chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp rate. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. This paper is organized as follows: in Section 2, we present a theoretical model of the interaction of few-cycle pulse with a two-level-atom medium. In Section 3, we investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagating in a two-level-atom medium. Finally, we offer some conclusions in Section 4.
2. Theoretical model Consider the propagation of a few-cycle pulse laser along the z-axis in vacuum to an input interface of a two-level-atom medium at z ¼ 0, the propagation property can be modeled using the Maxwell–Bloch equations: ot Hy ¼
1 o z Ex ; l0
ð1Þ
X.-h. Song et al. / Optics Communications 236 (2004) 151–157
ot Ex ¼ e10 oz Hy e10 ot Px ;
ð2Þ
ot u ¼ c2 u x0 v;
ð3Þ
ot v ¼ c2 v þ x0 u þ 2Xw;
ð4Þ
ot w ¼ c1 ðw w0 Þ 2Xv;
ð5Þ
maximum of the pulse intensity envelope and v is the chirp rate. In the numerical analysis, the medium is initialized with u ¼ v ¼ 0; w0 ¼ 1 at t ¼ 0. The choice of z0 ensures that the pulse penetrates negligibly into the medium at t ¼ 0. We employ a standard finite-difference time-domain approach [9] for solving the full-wave Maxwell equations, and predictor–corrector method to solve the Bloch equations. The material and laser parameters are consistent with those in [12]: sp ¼ 5 fs, x0 ¼ xp ¼ 2:3 fs1 , z0 ¼ 15 lm, d ¼ 2 1029 1 A s m, N ¼ 4 1018 cm3 , c1 1 ¼ c2 ¼ 1 ns. The corresponding pulse areas is A ¼ Xm sp p=1:76, and Xm ¼ 1 fs1 corresponds to the electric field of Ex ¼ 5 109 V/m or an intensity of I ¼ 6:6 1012 W/cm2 . The time and space increments Dt and Dz are chosen to ensure cDt 6 Dz [30]. The results to follow can of course be scaled to various laser and material parameters.
where Ex and Hy are the electric and magnetic fields, c1 and c2 are the population and polarization relaxation constants, respectively. x0 is the transition frequency of the two-level-atom medium, and X ¼ dEx =h is the Rabi frequency. The macroscopic nonlinear polarization Px ¼ Ndu is related to the off-diagonal density matrix element q12 ¼ ðu þ ivÞ=2 and the population difference w ¼ q22 q11 between the upper and lower states, N is the density of the medium, d is the dipole moment. The refractive index is determined by the real part of q12 and the gain coefficient is proportional to the imaginary part of q12 . We consider a hyperbolic secant functional form for the initial chirped electric field, which can be written as
3. Numerical results and analysis In this section we present representative numerical solutions of the coupled M–B equations given by Eqs. (1)–(5). Fig. 1(a) presents the propagation of 3.1p optical pulse of 5 fs time duration with different propagation distances when no initial chirp is present. By virtue of the area theorem,
Xðz ¼ 0; tÞ ¼ Xm sec h½1:76ðt z0 =cÞ=sp cosðxp t þ 12vt2 Þ;
ð6Þ
where xp is the carrier frequency, Xm is the maximum Rabi frequency, sp is the full width at half
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z=0µm 0.5
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2.5
ω/ω 0
Fig. 1. (a) 3.1p chirp free pulse propagation through the two-level-atom medium at the respective distances of 0, 60 and 84 lm, (b) the corresponding spectra of 3.1p pulse at the same propagation distances.
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the area of 3.1p pulse will increase to 4p through pulse broadening. However, in our simulation, the increases of the pulse area is not caused by the pulse broadening, but by the pulse splitting, this is agreement with that in [12]. Fig. 1(b) shows the corresponding spectra of the pulses. It can be seen for relative large values of the propagation distance, that the resulting spectral feature surrounding the resonance frequency becomes quite complex and exhibits an oscillatory structure. Furthermore, the spectra are broadened during the course of propagation. The origin of oscillations and broadenings can be traced to the temporally oscillating tail [5] that appears in the transmitted pulse. This tail arises as a result of the pulse reshaping that occurs as the pulse propagates through the system [31]. Considering the initial input pulse, it is known that chirping a pulse can lead to a shift of the carrier frequency, and a reduction of the peak intensity [32]. The goal now is to investigate the impact of initial chirp on the propagation and spectral features during the propagation in the two-level-atom medium. The chirp rates we considered are in the range of values which a pulse with a fixed bandwidth can sustain. For relative small chirp rate, which are shown in Fig. 2 (positive chirp) and Fig. 3 (negative chirp), respectively. It can be seen that the propagation and spectral features are similar to those of chirp free resonant
case (see Fig. 1). However, there are some discrepancies between them: first, the splits of chirped pulses (see Figs. 2(a) and 3(a)) are slower than that of chirp free pulse (see Fig. 1(a)). Second, contrary to the chirp free case (see Fig. 1(b)), the spectra of initial chirped pulses (see Figs. 2(b) and 3(b)) exhibit obvious carrier frequency shifts during the propagation in the medium depending on the sign of the initial chirp: blueshift for positive chirp and redshift for negative chirp case. Moreover, the shift of negative chirped pulse is smaller than that of positive chirped pulse. With the increasing of the initial chirp rate, however, the frequency shift in the medium becomes smaller and smaller. Furthermore, the oscillations around the resonant frequency and the broadenings also become weaker and weaker. Fig. 4(a) shows the spectra of 3.1p pulse with chirp rate equal to 0.025 fs2 . It can be seen that though there is a large blueshift of the carrier frequency and a reduction of the peak intensity for the initial input chirped pulse (solid line) compared with chirp free case (dash dot line), spectral changes can hardly be observed in the course of propagation in the medium. Fig. 4(b) depicts the electric field profile at different propagation distances. It shows that the medium becomes approximately transparent to the pulse. These phenomena can be qualitatively interpreted by the intrinsic chirp which is produced in
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(b) Fig. 2. As in Fig. 1 but for v ¼ 0:005 fs2 .
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X.-h. Song et al. / Optics Communications 236 (2004) 151–157
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Fig. 3. As in Fig. 1 but for v ¼ 0:005 fs2 .
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Ω (ω)
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Fig. 4. (a) spectra of 3.1p chirped pulse at the respective distances of 0, 60 and 84 lm for v ¼ 0:025 fs2 , the dash line represents the spectrum of initial input pulse with non-chirp exists, (b) the corresponding electric field profile at the same propagation distances.
the course of propagation. Similar to that nearresonant case [4,5], an initial chirped pulse can also bring an intrinsic chirp during the propagation in the medium, which pulls the carrier frequency toward the resonant frequency. For relative small chirp rate, the initial chirp rate is larger than the intrinsic chirp, hence the spectra exhibit blueshift or redshift in the medium depending on the sign of initial chirp rate. Also because of this intrinsic chirp, the splits of chirped pulse are slower than
that of chirp free pulse, which results in the weakening of the oscillations and the broadenings. Whereas, with increasing the initial chirp rate to appropriate value, the intrinsic chirp will become approximately equal to the initial chirp, spectral changes can hardly be observed, and the medium seem to be approximately transparent to the pulse. However, for negative chirp case, transparency can not be realized. The spectra feature of 5 fs pulse with chirp rate equal to )0.025 fs2 is present
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0.8
0.4
z= 84 µ m
Ω (ω)
0.0 0. 0 0.8
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Ω (fs-1)
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Fig. 5. (a) As in Fig. 4 but for v ¼ 0:025 fs2 , (b) the corresponding electric field profile at the respective distances of 0 and 84 lm.
in Fig. 5. It can be seen that though there is no frequency shift during the propagation in the medium for the carrier frequency, higher spectral components can be observed on the propagation pulse. It is because that negative chirp makes the frequency of initial input pulse redshift, which increases the carrier wave length, and at the same time decreases the number of cycles in the envelope. For such pulse, the carrier-wave RF can occur, that manifests in local carrier reshaping and subsequently to the production of significantly
higher spectral components on the propagating pulse [11]. The corresponding electric field profiles are shown in Fig. 5(b), carrier-wave reshaping can be found, this is consistent with our qualitative analysis.
4. Conclusions In conclusion, we investigate propagation and spectral effect of an initial chirped few-cycle pulse
X.-h. Song et al. / Optics Communications 236 (2004) 151–157
in the two-level-atom medium by solving the full Maxwell–Bloch equations. It is demonstrated that the initial chirp can substantially modify the behavior of the pulse and spectra: the spectral feature depend crucially on the sign and magnitude of the chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. Similar results can be achieved for other pulse area. The work is supported by the National Key Basic Research Special Foundation of China (NKBRSFC) (Grant No. G1999075200), the National Natural Science Key Foundation of China (Grant No. 10234030), and the Natural Science Foundation of Shanghai (Grant No. 03ZR14102). References [1] L. Allen, J.H. Eberly, Optical Resonance and Two-level Atoms, Dover, New York, 1987. [2] S.L. McCall, E.L. Hahn, Phys. Rev. 183 (1969) 457. [3] G.L. Lamb Jr., Rev. Mod. Phys. 43 (1971) 99. [4] A.M. Alhasan, J. Fiutak, W. Miklaszewski, Z. Phys. B 88 (1992) 349. [5] J.C. Diels, Phys. Lett. A 31 (1970) 111. [6] W. Miklaszewski, J. Opt. Soc. Am. B 12 (1995) 1909. [7] M. Born, E. Wolf, Principles of Optics, 5th ed., Pergamon, Oxford, 1975, Section 10.4. [8] W.E. Lamb Jr., Phys. Rev. A 134 (1964) 1429. [9] R.W. Ziolkowski, J.M. Arnold, D.M. Gogny, Phys. Rev. A 52 (1995) 3082.
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[10] M. M€ uller, V.P. Kalosha, J. Herrmann, Phys. Rev. A 58 (1998) 1372. [11] S. Hughes, Phys. Rev. Lett. 81 (1998) 3363. [12] J. Xiao, Z.Y. Wang, Z.Z. Xu, Phys. Rev. A 65 (2002) 031402(R). [13] S. Hughes, Phys. Rev. A 62 (2000) 055401. [14] X.H. Song, S.Q. Gong, S.Q. Jin, Z.Z. Xu, Phys. Rev. A 69 (2004) 015801. [15] V.P. Kalosha, J. Herrmann, Phys. Rev. Lett. 83 (1999) 544. [16] A. Baltuska, Z. Wei, M.S. Pshenichnikov, D.A. Wiersma, Opt. Lett. 22 (1997) 102. [17] M. Nisoli et al., Opt. Lett. 22 (1997) 522. [18] M. Protopapas, C.H. Keitel, P.L. Knight, Rep. Prog. Phys. 60 (1997) 389. [19] M. Protopapas, A. Sanpera, P.L. Knight, K. Burnett, Phys. Rev. A 52 (1995) R2527. [20] D.G. Lee, J.H. Kim, K.H. Hong, C.H. Nam, Phys. Rev. Lett. 87 (2001) 243902. [21] H.T. Kim, D.G. Lee, K.H. Hong, J.H. Kim, I.W. Choi, C.H. Nam, Phys. Rev. A 67 (2003) 051801(R). [22] J. Cheng, J.Y. Zhou, Phys. Rev. A 64 (2001) 065402. [23] S. Guerin, S. Thomas, H.R. Jauslin, Phys. Rev. A 65 (2002) 023409. [24] R. Netz, A. Nazarkin, R. Sauerbrey, Phys. Rev. Lett. 90 (2003) 063001. [25] C.W.S. Conover, M.C. Doogue, F.J. Struwe, Phys. Rev. A 65 (2002) 033414. [26] M. Desaix, L. Helczynski, D. Anderson, M. Lisak, Phys. Rev. E 65 (2002) 056602. [27] M. Klaus, J.K. Shaw, Opt. Commun. 197 (2001) 491. [28] M. Centini, M. Bloemer, K. Myneni, M. Scalora, C. Sibilia, M. Bertolotti, G.D. Aguanno, Phys. Rev. E 68 (2003) 016602. [29] V.P. Kalosha, J. Herrmann, Phys. Rev. Lett. 83 (1999) 544. [30] A. Taflove, M.E. Brodwin, IEEE Trans. Microwave Theory Tech. 23 (1975) 623. [31] J.K. Ranka, R.W. Schirmer, A.L. Gaeta, Phys. Rev. A 57 (1998) R36. [32] V.S. Malinovsky, J.L. Krause, Eur. Phys. J. D 14 (2001) 147.
Coherent control of spectra effects with chirped femtosecond laser pulse Xiao-hong Song *, Shang-qing Gong, Wei-feng Yang, Shi-qi Jin, Xun-li Feng, Zhi-zhan Xu Laboratory for High Intensity Optics, Shanghai Institute of Optics and Fine Mechanics, Shanghai 201800, China Received 27 October 2003; received in revised form 2 January 2004; accepted 9 March 2004
Abstract We investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagation in a twolevel-atom medium. It is found that the spectral feature depends crucially on the sign and magnitude of the initial chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. Ó 2004 Elsevier B.V. All rights reserved. PACS: 32.80.Qk; 42.62.Fi; 42.50.Md Keywords: Coherent control; Spectra; Chirped femtosecond laser pulse
1. Introduction The interaction of intense laser light with a collection of two-level atoms has been an area of active research for many years [1]. For example, the famous area theorem [2,3] is one of the most interesting features, which can predict and explain many fascinating effects, such as self-induced transparency and pulse compression. Moreover, the influence of various aspects, such as atomic
*
Corresponding author. Tel. +86-021-699-18266; fax: +86021-699-18000. E-mail address: [email protected] (X.-h. Song).
relaxation, detuning on the temporal dynamic of the optical pulse and the atomic system, have also been investigated in many literatures [4–6]. All these were derived within the standing slowly varying envelope approximation (SVEA) [7] and the rotating-wave approximation (RWA) [8]. However, these approximations clearly fail if the pulse duration approaches the duration of several optical cycles [9–11]. Recently, Xiao et al. [12] investigated the propagation of 5 fs pulse in a two-level-atom medium. It was found for small pulse area, that the variation of the few-cycle pulse area is caused by the pulse splitting but not by the pulse broadening or the pulse compression as in the case of
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.03.029
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X.-h. Song et al. / Optics Communications 236 (2004) 151–157
long-pulse [2]. Moreover, Hughes proposed for large pulse areas, that the area under individual carriers might themselves cause Rabi flopping (RF), which can lead to carrier-wave reshaping and significantly higher spectral components [11] and even generation of soft-X-ray [13]. Whereas, for small pulse areas, the self-induced transparency results are still recovered with minor modifications, i.e., no higher spectral components occur. In our previous paper, we investigated the spectra effects for two-color ultrashort pulses propagating in a two-level-atom medium [14]. It showed that higher spectral components can be produced even for small area pulses due to the interference effect of the two laser pulses. Moreover, these higher spectral components depend crucially on the relative phase u of the two pulses. Continuum and distinct peaks spectra can be achieved by adjusting the relative phase to equal to 0 and p. Furthermore, Kalosha et al. [15] found that due to intrapulse four-wave mixing, a large blueshift in the transmitted pulse and a large redshift in the reflected pulse can occur in dense media of two-level systems. The advances in generation and characterization techniques of ultrashort laser pulses made it practical to prepare precisely defined laser pulses [16,17], these open up new possibilities for studying and controlling dynamical processes in femtosecond time scale. Hitherto, frequency-synthesized or chirped laser pulses have been widely applied to high-order harmonic generation (HHG) [18–21], population transfer [22–24], multiphoton transitions between Rydberg states [25], etc. Both the interference of two-color pulses and the nontrivial time dependence of chirped pulse laser field may lead to very interesting results. Moreover, recent studies have manifested that chirping can also have much impacts on the laser pulses as it propagates through different medium. The propagation properties of chirped soliton pulses in optical nonlinear Kerr media were investigated by Desaix et al. [26]. The result indicates that the properties of asymptotically emerging solitons depend not only on the magnitude, but also on the form of the initial chirp. In particular, the splitting of an initial pulse into separating soliton pulse pairs only occurs for certain classes of initial chirp functions. The influence
of pulse shape and frequency chirp on stability of optical solitons was studied by Klaus and Shaw [27]. They found that destabilization occurs when eigenvalues of an associated Zakharov–Shabat system, which move along the positive imaginary axis with increasing chirp parameter, either are absorbed into the lower half plane or collide with a another eigenvalue. Furthermore, the signal and group velocities for chirped pulses propagating through a GaAs cavity were measured by Centini et al. [28]. They showed that the chirp substantially modifies the group velocity of the pulse, but leaves the signal velocity unaltered. In this paper, we investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagating in a two-level-atom medium. By solving the full Maxwell–Bloch (M–B) equations which avoid the limitations of SVEA and RWA as ultrashort pulses are considered [9,11,12,29], we find that the spectral feature depends crucially on the sign and magnitude of the initial chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp rate. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. This paper is organized as follows: in Section 2, we present a theoretical model of the interaction of few-cycle pulse with a two-level-atom medium. In Section 3, we investigate the coherent control of spectra effects with chirped femtosecond laser pulse propagating in a two-level-atom medium. Finally, we offer some conclusions in Section 4.
2. Theoretical model Consider the propagation of a few-cycle pulse laser along the z-axis in vacuum to an input interface of a two-level-atom medium at z ¼ 0, the propagation property can be modeled using the Maxwell–Bloch equations: ot Hy ¼
1 o z Ex ; l0
ð1Þ
X.-h. Song et al. / Optics Communications 236 (2004) 151–157
ot Ex ¼ e10 oz Hy e10 ot Px ;
ð2Þ
ot u ¼ c2 u x0 v;
ð3Þ
ot v ¼ c2 v þ x0 u þ 2Xw;
ð4Þ
ot w ¼ c1 ðw w0 Þ 2Xv;
ð5Þ
maximum of the pulse intensity envelope and v is the chirp rate. In the numerical analysis, the medium is initialized with u ¼ v ¼ 0; w0 ¼ 1 at t ¼ 0. The choice of z0 ensures that the pulse penetrates negligibly into the medium at t ¼ 0. We employ a standard finite-difference time-domain approach [9] for solving the full-wave Maxwell equations, and predictor–corrector method to solve the Bloch equations. The material and laser parameters are consistent with those in [12]: sp ¼ 5 fs, x0 ¼ xp ¼ 2:3 fs1 , z0 ¼ 15 lm, d ¼ 2 1029 1 A s m, N ¼ 4 1018 cm3 , c1 1 ¼ c2 ¼ 1 ns. The corresponding pulse areas is A ¼ Xm sp p=1:76, and Xm ¼ 1 fs1 corresponds to the electric field of Ex ¼ 5 109 V/m or an intensity of I ¼ 6:6 1012 W/cm2 . The time and space increments Dt and Dz are chosen to ensure cDt 6 Dz [30]. The results to follow can of course be scaled to various laser and material parameters.
where Ex and Hy are the electric and magnetic fields, c1 and c2 are the population and polarization relaxation constants, respectively. x0 is the transition frequency of the two-level-atom medium, and X ¼ dEx =h is the Rabi frequency. The macroscopic nonlinear polarization Px ¼ Ndu is related to the off-diagonal density matrix element q12 ¼ ðu þ ivÞ=2 and the population difference w ¼ q22 q11 between the upper and lower states, N is the density of the medium, d is the dipole moment. The refractive index is determined by the real part of q12 and the gain coefficient is proportional to the imaginary part of q12 . We consider a hyperbolic secant functional form for the initial chirped electric field, which can be written as
3. Numerical results and analysis In this section we present representative numerical solutions of the coupled M–B equations given by Eqs. (1)–(5). Fig. 1(a) presents the propagation of 3.1p optical pulse of 5 fs time duration with different propagation distances when no initial chirp is present. By virtue of the area theorem,
Xðz ¼ 0; tÞ ¼ Xm sec h½1:76ðt z0 =cÞ=sp cosðxp t þ 12vt2 Þ;
ð6Þ
where xp is the carrier frequency, Xm is the maximum Rabi frequency, sp is the full width at half
z=84µm
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1.0
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z=0µm
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Ω (ω )
-1
Ω (fs )
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100
200
t (fs)
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2.5
ω/ω 0
Fig. 1. (a) 3.1p chirp free pulse propagation through the two-level-atom medium at the respective distances of 0, 60 and 84 lm, (b) the corresponding spectra of 3.1p pulse at the same propagation distances.
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the area of 3.1p pulse will increase to 4p through pulse broadening. However, in our simulation, the increases of the pulse area is not caused by the pulse broadening, but by the pulse splitting, this is agreement with that in [12]. Fig. 1(b) shows the corresponding spectra of the pulses. It can be seen for relative large values of the propagation distance, that the resulting spectral feature surrounding the resonance frequency becomes quite complex and exhibits an oscillatory structure. Furthermore, the spectra are broadened during the course of propagation. The origin of oscillations and broadenings can be traced to the temporally oscillating tail [5] that appears in the transmitted pulse. This tail arises as a result of the pulse reshaping that occurs as the pulse propagates through the system [31]. Considering the initial input pulse, it is known that chirping a pulse can lead to a shift of the carrier frequency, and a reduction of the peak intensity [32]. The goal now is to investigate the impact of initial chirp on the propagation and spectral features during the propagation in the two-level-atom medium. The chirp rates we considered are in the range of values which a pulse with a fixed bandwidth can sustain. For relative small chirp rate, which are shown in Fig. 2 (positive chirp) and Fig. 3 (negative chirp), respectively. It can be seen that the propagation and spectral features are similar to those of chirp free resonant
case (see Fig. 1). However, there are some discrepancies between them: first, the splits of chirped pulses (see Figs. 2(a) and 3(a)) are slower than that of chirp free pulse (see Fig. 1(a)). Second, contrary to the chirp free case (see Fig. 1(b)), the spectra of initial chirped pulses (see Figs. 2(b) and 3(b)) exhibit obvious carrier frequency shifts during the propagation in the medium depending on the sign of the initial chirp: blueshift for positive chirp and redshift for negative chirp case. Moreover, the shift of negative chirped pulse is smaller than that of positive chirped pulse. With the increasing of the initial chirp rate, however, the frequency shift in the medium becomes smaller and smaller. Furthermore, the oscillations around the resonant frequency and the broadenings also become weaker and weaker. Fig. 4(a) shows the spectra of 3.1p pulse with chirp rate equal to 0.025 fs2 . It can be seen that though there is a large blueshift of the carrier frequency and a reduction of the peak intensity for the initial input chirped pulse (solid line) compared with chirp free case (dash dot line), spectral changes can hardly be observed in the course of propagation in the medium. Fig. 4(b) depicts the electric field profile at different propagation distances. It shows that the medium becomes approximately transparent to the pulse. These phenomena can be qualitatively interpreted by the intrinsic chirp which is produced in
1.2 z=84µm
1.0
z=60µm
0.8 0.0
z=0µm
0.6
z=84µm
0.5 0.0
Ω (ω)
Ω (fs-1)
1.0
-0.5
2.5
z=60µ m
0.4 0.0
2.5
0.2
z=0µm
-1.0 0
(a)
100
200
t (fs)
300
0.0 0.0
400
0.5
(b) Fig. 2. As in Fig. 1 but for v ¼ 0:005 fs2 .
1.0
ω /ω 0
1.5
2 .0
2.5
X.-h. Song et al. / Optics Communications 236 (2004) 151–157
155
1.2
z=84µm
1.0
Ω (ω)
z=60µm
0.5
Ω (fs-1)
1.0
z=0µm
0.0 -0.5
z=84µm
0.8 0.0 0.6
2.5
z=60µm
0.4 0.0 0.2
2.5
-1.0 0
100
(a)
200
300
0.0 0.0
400
z=0µm 0.5
1.0
2.0
2.5
ω /ω 0
(b)
t (fs)
1.5
Fig. 3. As in Fig. 1 but for v ¼ 0:005 fs2 .
1.2 1.0
2.5
z=60µm
0.4 0.0 0.2 0.0 0.0
(a)
z=84µm
0.8 0.0 0.6
z=0µm 1.0
1.5
2.0
ω /ω 0
z=60µm z=84µm
0.0 -0.5
2.5
0.5
z=0µm
0.5
Ω (fs-1)
Ω (ω)
1.0
-1.0
2.5
0
(b)
100
200
300
400
t (fs)
Fig. 4. (a) spectra of 3.1p chirped pulse at the respective distances of 0, 60 and 84 lm for v ¼ 0:025 fs2 , the dash line represents the spectrum of initial input pulse with non-chirp exists, (b) the corresponding electric field profile at the same propagation distances.
the course of propagation. Similar to that nearresonant case [4,5], an initial chirped pulse can also bring an intrinsic chirp during the propagation in the medium, which pulls the carrier frequency toward the resonant frequency. For relative small chirp rate, the initial chirp rate is larger than the intrinsic chirp, hence the spectra exhibit blueshift or redshift in the medium depending on the sign of initial chirp rate. Also because of this intrinsic chirp, the splits of chirped pulse are slower than
that of chirp free pulse, which results in the weakening of the oscillations and the broadenings. Whereas, with increasing the initial chirp rate to appropriate value, the intrinsic chirp will become approximately equal to the initial chirp, spectral changes can hardly be observed, and the medium seem to be approximately transparent to the pulse. However, for negative chirp case, transparency can not be realized. The spectra feature of 5 fs pulse with chirp rate equal to )0.025 fs2 is present
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0.8
0.4
z= 84 µ m
Ω (ω)
0.0 0. 0 0.8
2.5
0.4
z= 6 0 µ m 0.0 0. 0 0.8
2.5
0.4
z= 0 µ m
0.0 0.0
0.5
1. 0
1. 5
ω /ω
(a)
1.0
z=0µm
-1
Ω (fs )
Ω (fs-1)
z=84µm
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
30
2 .5
0
1.0
0.5
(b)
2.0
40
50
60
70
310
320
t (fs)
330
340
350
t (fs)
Fig. 5. (a) As in Fig. 4 but for v ¼ 0:025 fs2 , (b) the corresponding electric field profile at the respective distances of 0 and 84 lm.
in Fig. 5. It can be seen that though there is no frequency shift during the propagation in the medium for the carrier frequency, higher spectral components can be observed on the propagation pulse. It is because that negative chirp makes the frequency of initial input pulse redshift, which increases the carrier wave length, and at the same time decreases the number of cycles in the envelope. For such pulse, the carrier-wave RF can occur, that manifests in local carrier reshaping and subsequently to the production of significantly
higher spectral components on the propagating pulse [11]. The corresponding electric field profiles are shown in Fig. 5(b), carrier-wave reshaping can be found, this is consistent with our qualitative analysis.
4. Conclusions In conclusion, we investigate propagation and spectral effect of an initial chirped few-cycle pulse
X.-h. Song et al. / Optics Communications 236 (2004) 151–157
in the two-level-atom medium by solving the full Maxwell–Bloch equations. It is demonstrated that the initial chirp can substantially modify the behavior of the pulse and spectra: the spectral feature depend crucially on the sign and magnitude of the chirp rate. For small chirp rate, blueshift or redshift of the spectrum can occur depending on the sign of the chirp. However, for relative large chirp rate, the medium can become transparent to positive chirped pulse, whereas in the case of negative chirped pulse, higher spectral components can be produced even for small area pulses. Similar results can be achieved for other pulse area. The work is supported by the National Key Basic Research Special Foundation of China (NKBRSFC) (Grant No. G1999075200), the National Natural Science Key Foundation of China (Grant No. 10234030), and the Natural Science Foundation of Shanghai (Grant No. 03ZR14102). References [1] L. Allen, J.H. Eberly, Optical Resonance and Two-level Atoms, Dover, New York, 1987. [2] S.L. McCall, E.L. Hahn, Phys. Rev. 183 (1969) 457. [3] G.L. Lamb Jr., Rev. Mod. Phys. 43 (1971) 99. [4] A.M. Alhasan, J. Fiutak, W. Miklaszewski, Z. Phys. B 88 (1992) 349. [5] J.C. Diels, Phys. Lett. A 31 (1970) 111. [6] W. Miklaszewski, J. Opt. Soc. Am. B 12 (1995) 1909. [7] M. Born, E. Wolf, Principles of Optics, 5th ed., Pergamon, Oxford, 1975, Section 10.4. [8] W.E. Lamb Jr., Phys. Rev. A 134 (1964) 1429. [9] R.W. Ziolkowski, J.M. Arnold, D.M. Gogny, Phys. Rev. A 52 (1995) 3082.
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