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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 3 — Jan. 31, 2011
  • pp: 1963–1974
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High-order nonlinear Schrödinger equation and weak-light superluminal solitons in active Raman gain media with two control fields

Chengjie Zhu and Guoxiang Huang  »View Author Affiliations


Optics Express, Vol. 19, Issue 3, pp. 1963-1974 (2011)
http://dx.doi.org/10.1364/OE.19.001963


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Abstract

We propose a scheme to generate superluminal optical solitons in a four-level atomic system with two control fields via an active Raman gain. We derive a modified nonlinear Schrödinger equation with high-order corrections contributed from linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. We predict various optical solitons in different regimes of system parameters, and show that these optical solitons have superluminal propagating velocity and very low generation power.

© 2011 Optical Society of America

1. Introduction

Optical solitons, i.e. special types of optical wavepackets formed by the balance between nonlinearity and dispersion (and/or diffraction), have attracted much attention for decades due to their important applications for optical information processing and transmission [1

1. A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).

3

3. G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).

]. However, up yo now most optical solitons are produced in far-off resonant optical media such as glass-based optical fibers. The nonlinear effect in such media is very weak and hence to form an optical soliton a very high-intensity is needed.

In recent years, interest has focused on wave propagation in resonant optical media via electromagnetically induced transparency (EIT) [4

4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein. [CrossRef]

]. Propagation of a weak probe optical field in such media displays many new features due to the quantum interference effect induced by a strong control optical field, including large suppression of optical absorption, significant reduction of group velocity, and giant enhancement of Kerr nonlinearity [4

4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein. [CrossRef]

]. Based on these features, it has been shown that it is possible to produce weak-light ultraslow optical solitons in various EIT media [5

5. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

8

8. W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010). [CrossRef]

]. However, there are many drawbacks for the optical solitons obtained in EIT-based schemes, including pulse spreading at room temperature and very long response time due to ultraslow propagation, as indicated in Refs. [9

9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

, 10

10. K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008). [CrossRef]

].

Contrary to the EIT-based scheme which is absorptive in nature, active Raman gain (ARG) scheme, which is based on a gain-assisted configuration, is able to support a superluminal propagation for a probe field. In recent years, parallel to the study on EIT, works based on ARG schemes have also attracted considerable attention both theoretically and experimentally [9

9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

29

29. K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007). [CrossRef]

]. Especially, Agarwal and Dasgupta investigate a linear propagation of probe field in a four-level N-type atomic system and showed that a gain doublet appears due to a quantum interference induced by an additional control field [30

30. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]

]. Recently, we have studied nonlinear effect of an ARG-based four-level Λ-type atomic system and demonstrated that it is possible to obtain giant Kerr nonlinearity in such system [31

31. C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010). [CrossRef]

]. It is natural to consider the possibility of producing a superluminal optical soliton in ARG-based atomic ensemble. It is just this topic that will be addressed in this work. We shall show that, based on the ARG scheme and working on the gain minimum of the gain doublet, we can obtain a stable propagation of superluminal optical solitons. However, due to the resonant character of the system, the property of the superluminal optical solitons is very sensitive to probe pulse length. In general, solitons in such system must be described by a modified nonlinear Schrödinger (NLS) equation, which has high-order correction terms contributed from effects of linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. We shall show that superluminal optical solitons obtained in such system have many interesting features.

2. Model and solution in linear regime

The model under study is shown in Fig. 1, where a lifetime-broadened four-state atom with bare energy states |j〉 (j = 1, 2, 3, 4) interacts with three laser fields. The weak, pulsed probe filed of angular frequency ωp (half Rabi frequency Ωp) couples to the transition |3〉 ↔ |4〉, the strong, continuous-wave control field of angular frequency ωc (half Rabi frequency Ωc) couples to the transition |1〉 ↔ |4〉, and the strong, continuous-wave control field of angular frequency ωb (half Rabi frequency Ωb) couples to the transition |2〉 ↔ |3〉. States |1〉, |3〉, |4〉, and half Rabi frequencies Ωp and Ωc consists of an ARG core [9

9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

, 10

10. K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008). [CrossRef]

]. States |1〉, |2〉 and |3〉 can be chosen as hyperfine ground states. The transition |3〉 ↔ |2〉 is generally electric-dipole forbidden, and hence the control field of half Rabi frequency Ωb is a microwave field in our system.

Fig. 1 (Color online) Energy-level diagram and excitation scheme of Λ-type four state atomic system with active Raman gain. |j〉 (j = 1, 2, 3, 4) are bare atomic states. Δ4 is one-photon detuning. Ωp is the half Rabi frequency of weak, pulsed probe field. Ωc and Ωb are half Rabi frequencies of two strong, continuous control fields.

In order to achieve shape-preserving propagation of the probe field, we first examine linear dispersion property of the system. When the probe field is absent, the state of the system reads A1 = a1, A2 = A3 = 0, and A4 = −Ωca1/d4, where a1=a1=|d4|/|d4|2+|Ωc|2 is a constant. Such steady state can be obtained by using the condition j=14|Aj|2=1. Note that this approximation can only be used when the detuning Δ4 is very large or γjt ≪ 1, which result in the population in the energy levels |2〉 and |3〉 close to zero. For a very weak probe field Ωp = F exp(), here θ = Kzωt and F is a very small amplitude, one can easily get the linear dispersion of the system
K(ω)=ωcκ34(ω+d2)|Ωc|2|d4|2D|a1|2,
(2)
where D = |Ωb|2 – (ω + d2)(ω + d3). Notice that the angular frequency ω introduced here is the frequency shift (relative to the central frequency ωp) of the probe field. Thus when the system is excited the angular frequency of the probe field is given ωp + ω.

Shown in Fig. 2 are −Im(K(ω)) (Fig. 2(a)) and Re(K(ω)) (Fig. 2(b)), representing the negative imaginary part and real part of K(ω), respectively. System parameters are those given by typical alkali atomic (87Rb) gas, with [9

9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

] γ2 = 1.5 × 104s−1, γ3 = 5.5 × 104s−1 and γ4 = 1.8 × 108s−1. Other parameters are taken as Ωc = 1.5 × 107s−1, Δ2 = Δ3 = 0s−1, Δ4 = −1.0 × 109s−1, κ34 = 1 × 109cm−1s−1. From Fig. 2(a), we see that for small microwave control field (Ωb = 1.0 × 104 s−1) the gain spectrum has only a single peak (dashed line). In this case, near the central frequency (i.e. ω = 0) the probe field is highly amplified and hence unstable during propagation; however, for large microwave control field (Ωb = 1.0 × 106 s−1) the gain spectrum displays two peaks (solid line), i.e. a gain spectrum hole appears. In this case, within the spectrum hole between two gain peaks the probe field has a very small gain and thus can keep its shape during propagation if the dispersion of the system can be arrested by some other effect of the system. The appearance of such gain spectrum hole (gain doublet) comes from the quantum interference effect induced by the microwave control field [30

30. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]

, 31

31. C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010). [CrossRef]

]. From Fig. 2(b) we see that for small microwave control field (dashed line) the slope of ReK(ω) near ω = 0 is positive, and thus the probe field in this case has a subluminal (but unstable) propagation. For large microwave control field (solid line) the slope of ReK(ω) corresponding to the region of the gain spectrum hole is negative, and hence in this region the probe field has a superluminal propagation.

Fig. 2 (Color online) (a) Negative imaginary part (gain) −Im(K) and (b) real part Re(K) as functions of ω. Solid and dashed lines correspond to Ωb = 1.0 × 104 s−1 and Ωb = 1.0 × 106 s−1, respectively.

3. Asymptotic expansion and envelope equation

To obtain a divergence-free expansion, all quantities on the right hand side of Eq. (3) are considered as function of multiscale variables zl = ɛlz (l = 0,1, 2, 3), and tl = ɛlt (l =0,1). Then Eqs. (1a)(1d) become
(it0+d2)a2(l)+Ωb*a3(l)=M(l),
(4a)
(it0+d3)a3(l)+Ωp(l)a4(0)+Ωba2(l)=N(l)
(4b)
(it0+d4*)a4(l)*+Ωs(l)*a1(l)*=Q(l),
(4c)
i(z0+1ct0)Ωp(l)+κ34a3(l)a4(1)*=R(l).
(4d)
Explicit expression of M(l), N(l), Q(l) and R(l) can be systematically and analytically obtained, which are omitted here for saving space.

To make the procedure of obtaining iterative solutions of Eqs. (4a)(4d) more trackable, we express Eqs. (4a)(4d) in the following forms:
a2(l)=1Ωb[N(l)(it0+d3)a3(l)Ωp(l)a4(0)],
(5a)
a3(l)=1κ34a4(0)*[R(l)+i(z0+1ct0)Ωp(l)],
(5b)
L^Ωp(l)=S(l),
(5c)
where
L^=i(z0+1ct0)[|Ωb|2(it0+d2)(it0+d3)]κ34|a4(0)|2(it0+d2),
(6a)
S(l)=R(l)[(it0+d2)(it0+d3)|Ωb|2]+κ34a4(0)*[ΩbM(l)(it0+d2)N(l)]
(6b)
(l = 1, 2, 3). Equations (5a)(5c) can be solved order by order.

3.1. First-order approximation

The case for l = 1 is just the linear problem solved in the last section. we thus have the linear dispersion relation (2). The first-order approximation solution reads
Ωp(1)=Feiθ,
(7a)
a2(1)=a4(0)Ωb*DFeiθ,
(7b)
a3(1)=a4(0)(ω+d2)DFeiθ,
(7c)
where θ = K (ω)z0ωt0 = K (ω)zωt and F is yet to be determined envelope function depending on variables x1, y1, t1, z1 and z2. Note that the dispersion function K(ω) enters the expression of the probe field, this implies that all-order dispersion coefficients are appropriately introduced in the first-order approximation.

3.2. Second-order approximation

For l = 2 we can obtain M(2), N(2), Q(2) and R(2) by substituting the first-order solution Eqs. (7a)(7c) into Eqs. (4a)(4d), and also obtain the explicit expression of S(2). Equation (5c) becomes
L^Ωp(2)=iDeiθ(Fz1+1vgFt1).
(8)
Notice that since exp() is the eigensolution of the operator . Solvability condition of Eq. (8) requires
i(Fz1+1vgFt1)=0
(9)
where vg = 1/K1 (Kj ≡ [jK(ω)/∂ωj] (j = 1,2,...) ) is the group velocity of the probe pulse. The second-order approximation solution reads
Ωp(2)=0,
(10a)
a2(2)=ieiθΩbκ34a4(0)*[(Kωc)+(ω+d3)(1vg1c)]Ft1,
(10b)
a3(2)=ieiθκ34a4(0)*(1vg1c)Ft1,
(10c)
a4(0)a4(2)*+a4(0)*a4(2)=ieα¯z2|F|2|a4(0)|2|Ωc|2+|d4|2{|Ωc|2(|ω+d2D|2+|ΩbD|2)|d4|2[(ω+d2D)*d4+c.c.]},
(10d)
where c.c. represents complex conjugate and α/2 = Im(K(ω)) and = ɛ2α.

3.3. Third-order approximation

Similarly, for l = 3 we obtain
L^Ωp(3)=D[iFz2K222Ft12W|F|2F]eiθ
(11)
with
W=iκ34(ω+d2)eα¯z2|a4(0)|2|Ωc|2+|d4|2{|Ωc|2(|ω+d2D|2+|ΩbD|2)|d4|2[(ω+d2D)*d4+c.c.]}.
The solvability condition of the Eq. (11) gives rise to
iFz2K222Ft12W|F|2F=0,
(12)
which is the NLS equation describing evolution of the envelope function F. The third-order solution reads
a1(3)=i(α1F*Ft1α1*FF*t1)eα¯z2,
(13)
a2(3)=(α2(1)2Ft12+α2(2)|F|2F)eiθ,
(14)
a3(3)=(α3(1)2Ft12+α3(2)|F|2F)eiθ,
(15)
Ωp(3)=0,
(16)
where explicit coefficients of α1, α2(1), α2(2), α3(1) and α3(2) are omitted.

As indicated above, for pulse propagation in resonant systems, the dispersion and nonlinear effects are very sensitive to pulse length. For a shorter pulse, the third-order approximation is not enough and hence one must go to fourth-order. Using the solutions from the first- to third-orders we can solve the fourth-order approximation equations. After a detailed calculation we obtain the solvability condition
iFz3iK363Ft13iβ1eα¯z2t1(|F|2F)+iβ2eα¯z2Ft1(|F|2)=0,
(17)
where β1 and β2 are constants.

Combining Eqs. (9), (12) and (17) we obtain an envelope equation with complex coefficients of the form (after returning to original variables)
iUz+iα2UK222Uτ2iK363Uτ3W|U|2Uiβ1τ(|U|2U)+iβ2Uτ(|U|2)=0,
(18)
where τ = tz/Vg and U = ɛFeiαz/2.

4. Formation of superluminal optical solitons

Equation (18) derived in the preceding section is in fact a Ginzburg-Landau equation [33

33. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein. [CrossRef]

] with some high-order terms since its coefficients are complex. In general case, such equation does not allows soliton solutions. However, if a realistic set of system parameters can be found so that the imaginary part of these coefficients can be made much smaller than their corresponding real parts, it is possible to obtain a shape-preserving, localized probe pulse that can propagate for an extended distance without significant distortion. We show that this is indeed possible for the present ARG system. We take the parameters used in Ref. [9

9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

], which are the same as those in Fig. 2 Other parameters are chosen as Ωc = 1.5 × 107s−1, Ωb = 1.0 × 106s−12 = 0.1 × 106s−13 = 0.5 × 106s−14 = −1.0 × 109s−1,κ34 = 1 × 109cm−1s−1. With the above parameters, we obtain K0 = −(2.16 + i0.14)cm−1, K1 = −(14.97 + i0.04) × 10−7cm−1s, K2 = −(21.35 + i0.01) × 10−13cm−1s2, K3 = −(39.26 + i0.07) × 10−19cm−1s3, W = (20.52 + i0.008) × 10−16cm−1s2, β1 = (27.62 + i0.01) × 10−22cm−1s3, β2 = −(2.39 + i0.1) × 10−21cm−1s3. We see that the imaginary part of each coefficient of the high-order Ginzburg-Landau Eq. (17) is indeed much smaller than its real part. The physical reason of so small imaginary part is due to the quantum destructive interference effect induced by the control field Ωb, by which the transition passage |2〉 ↔ |3〉 superposes on the transition passage |3〉 ↔ |4〉 destructively to make the population in the energy level |3〉 vanish, and hence suppress the gain of the probe field largely.

When neglecting the small imaginary part of the coefficients, the Ginzburg-Landau Eq. (18) is converted into the dimensionless high-order NLS equation
ius+2uσ2+2|u|2u=i(g0u+g1(|u|2u)σ+g2u|u|2σ+g33uσ3)+g4uσ.
(19)
where we have introduced dimensionless quantities s = −z/(2LD), σ = τ/τ0, u = U/U0, and gj = 2LD/Lj (j = 0–4), with τ0 the characteristic time length of the probe pulse, LD=τ02/K˜2 the characteristic dispersion length, L0 = 2/α the characteristic absorption length, L1=τ0/(β˜1U02) the characteristic nonlinear dispersion length, L2=τ0/(β˜2U02) the characteristic delay length in nonlinear refractive index, L3=6τ03/K˜3 the characteristic third-order dispersion length, L4 = τ0/ImK1 the characteristic differential absorption length, and U0=(1/τ0)K˜2/W˜ the characteristic Rabi frequency of the probe field. Here, the quantity with tilde mean its real part, e.g. 2 ≡ ReK2. Notice that in order to form soliton the characteristic Rabi frequency U0 has been obtained by setting LD = LNL, where LNL is the characteristic nonlinearity length defined by LNL=1/(W˜U02). From the definition of the characteristic lengths Lj given above we see that for smaller τ0 the high-order dispersion and high-order nonlinear effect must be included in the envelope equation. The characteristic amplitude U0 can be taken as the expansion parameter ɛ used in Eq. (3).

Notice that each term in Eq. (19) has clear physical meaning. The second and the third terms on the left hand side describe respectively the second-order dispersion and Kerr nonlinearity of the system. The terms from the first to the fourth ones in the square bracket on the right hand side describe linear absorption (proportional to g0), nonlinear dispersion (proportional to g1), delay response of nonlinear refractive index (proportional to g2), and third-order dispersion (proportional to g3), respectively. The last term describes differential absorption (proportional to g4).

The solution property of Eq. (19) is determined by coefficients gj (j = 0 – 4). Using the system parameters given above we have calculated values of gj as functions of the pulse length τ0 of the probe field, which is shown in Fig. 3. We see that g3 and g4 are not sensitive to τ0, but g0, g1 and g2 change rapidly as τ0 varies. Based on the result of Fig. 3(a) and Fig. 3(b) we can divide the Eq. (19) into several regimes and hence obtain different soliton solutions, which are given as follows:
  1. If τ0 ≥ 2.0 × 10−6 s, g1, g2, g3, and g4 are much smaller than g0 and hence can be neglected. In this case Eq. (19) is reduced to the perturbed NLS equation
    ius+2uσ2+2u|u|2=ig0u,
    (20)
    The single-soliton solution of this equation can be obtained approximately. The Rabi frequency of the probe field corresponding to such soliton, after returning to original variables, reads
    Ωp=eg0z/LDτ0K˜2W˜sech[eg0z/LDτ0(tzV˜g)]exp[iK˜0zi1e2g0z/LD4g0].
    (21)
    We see that the perturbations result in not only an increase of soliton amplitude but also a decrease of soliton width.
  2. If τ0 ≤ 2.0 ×10−6 s, g0 and g4 are much smaller than g1, g2 and g3. Thus Eq. (19) in this case is simplified into
    ius+2uσ2+2u|u|2=i[g1(|u|2u)σ+g2u(|u|2)σ+g33uσ3].
    (22)
    This equation admits exact soliton solutions [34

    34. M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation,” Phys. Rev. Lett. 78, 448 (1997). [CrossRef]

    36

    36. S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E. 60, R45 (1999). [CrossRef]

    ]. Hence we obtain the Rabi frequency of the probe field
    Ωp=U03(β+3q22q)g32(3c1+c2)sech[β+3q22qg3(tz/V˜gτ0+βz2g3LD)]×exp(i[q3q2+(β+3q22q)(13q)]z2g32LDiqg3tz/V˜gτ0+iK˜0z),
    (23)
    with the conditions β +3q2 – 2q > 0 and q ≠ 1/3, where β is a free real number, c1 = g1/(2g3), c2 = g2/(2g3), and q = (3c1 + 2c2 − 3)/[6(c1 + c2)].

Fig. 3 (Color online) Values of coefficients g0, g4 (panel (a)), and g1, g2, g3 (panel (b)) of Eq. (19) as function of the pulse length τ0. Both panels (a) and (b) are plotted by the parameters given by γ2 = 1.5 × 104s−1, γ3 = 5. 5 × 104s−1, γ4 = 1.8 × 108s−1, Ωc = 1. 5 × 107s−1, Ωb = 1.0 × 106s−1, Δ2 = 0.1 × 106s−1, Δ3 = 0.5 × 106s−1, Δ4 = −1.0 × 109s−1, and κ34 = 1 × 109cm−1s−1.

By considering small imaginary part of the coefficients and using the above data, we obtain U0 = 1.8×107s−1, LD = LNL = 1.5cm, α = −0.14cm−1. The propagating velocity of the soliton (21) for τ0 = 3.5 × 10−6 s is
V˜g=2.24×105c,
(24)
indicating a superluminal propagation.

In Fig. 4 we have shown the propagation of the superluminal optical solitons and their interaction. Shown in Fig. 4(a) is the waveshape of |Ωp/U0|2 versus time t and propagating z based on Eqs. (1a)(1d). The initial condition (black solid line) is given by Eq. (23) with the parameters τ0 = 1.5 × 10−6s, γ2 = 1.5 × 104s−1, γ3 = 5.5 × 104s−1, γ4 = 1.8 × 108s−1, Ωc = 1.5 × 107s−1, Ωb = 1.0 × 106s−1, Δ2 = 0.1 × 106s−1, Δ3 = 0.5 × 106s−1, Δ4 = −1.0 × 109s−1, and κ34 = 1 × 109cm−1s−1. The (red) dot-dashed line is the soliton profile at the propagation distance z = 2LD. The (green) dotted line is the result for τ0 = 3.5 × 10−6s, with the other parameters the same as above. The (blue) dashed line the soliton profile at the propagation distance z = 4LD. From these results we see that the superluminal optical soliton is fairly stable when propagates to 2LD. However, there exists a small radiation appearing on its right wing for τ0 = 3.5 × 10−6s and for the soliton propagating to 4LD. The physical reason for the appearance of such small radiation on the tail is due to that the approximation necessary to obtain the analytic solution (i.e. Eq. (23)) breaks down for a large pulse width for long propagation distance. We have also investigated two-soliton collisions based on the high-order NLS equation (19). The result is potted in Fig. 4(b)–4(d). Shown in Fig. 4(b) is the result of a collision between two solitons with the input condition u(0,σ) =1.0sech(σ – 3.0) exp(−i1.2σ + iπ)+1.0sech(σ +5.0) exp(i1.2σ). Figure. 4(c) reports the result of a collision between two solitons, which occurs for the initial condition u(0, σ) = 1.0sech(σ − 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ). We see the both solitons preserve their wave shapes after collisions. However, there exists many radiations when two large-amplitude solitons propagate to a long distance with the initial condition u(0, σ) = 1.5sech(σ − 3.0)exp(−i1.2σ)+1.0sech(σ+5.0) exp(i1.2σ), as shown in Fig. 4(d).

Fig. 4 (Color online) (a) The wave shape of |Ωp/U0|2 versus time t and propagating distance z based on Eqs. (1a)(1d). The initial condition (black solid line) is given by Eq. (23) with τ0 = 1.5 × 10−6s. The (red) dot-dashed line is the soliton profile at the propagation distance z = 2LD. The (green) dotted line is the result for τ0 = 3.5 × 10−6s. The (blue) dashed line the soliton profile at the propagation distance z = 4LD. (b) Collision between two solitons with the initial condition u(0,σ) = 1.0sech(σ – 3.0) exp(–i1.2σ + iπ) + 1.0sech(σ + 5.0) exp(i1.2σ). (c) Collision between two solitons with the initial condition u(0, σ) = 1.0sech(σ − 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ). (d) Collision between two solitons with larger amplitude amplitudes. The initial condition is u(0,σ) = 1.5sech(σ – 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ).

The propagating velocity of the soliton (23) H is determined by 1/V˜gH=1/V˜gβτ0/(2LDg3). We obtain
V˜gH=2.379×106c
(25)
for τ0 = 1.8 × 10−6 s and β = −2.73. We see that the optical solitons predicted here has also a superluminal propagating velocity.

It is easy to calculate the threshold of optical power density required to generate the super-luminal optical solitons given above. Note that the energy flux of the probe filed is given by Poynting’s vector i.e. P = ∫dS(Ep × Hp) · ez, where ez is the unit vector in the propagation direction. Using Ep = (Ep, 0, 0), Hp = (0,ɛ0cnpEp, 0) (np = 1 + c Re(K/ωp) is refractive index), and Ep = (/|μ34|) Ωp exp[i(kpzωpt)] + c.c., we can get the expressions of P for different soliton solutions. Then the integration of P over a carrier-wave period gives the average energy flux of the probe field, i.e.
P¯=P¯max,1sech2[1τ0(tzV˜g)],(forthesoliton(21)),
(26a)
P¯=P¯max,2×sech2[β+3q22qd3τ0(tzV˜gH)],(forthesoliton(23))
(26b)
with corresponding peak powers
P¯max,12ɛ0cnpS0(h¯|μ34|)21τ02K˜2W˜,
(27a)
P¯max,2ɛ0cnpS0(h¯|μ34|)26(β+3q22q)g32τ02(3c1+2c2)K˜2W˜,
(27b)
where S0 = πR2 is the cross-section area of the probe laser beam. Using the above numerical values of system parameters and take |μ34| = 2.1 × 10−27 cm C and S0 = 1.0 × 10−4 cm2 we obtain max,1 = 1.4 μW and max,2 = 0.876 μW, respectively. Thus, to generate the superluminal optical solitons in the system, only very low input power is needed. This is drastically different from the optical soliton generation schemes in fiber-based passive media, where much higher input power is required in order to bring out the nonlinear effect required for the soliton formation.

5. Conclusion

We have proposed a scheme for generating superluminal optical solitons in a four-level ARG atomic system with two control fields. By using a standard method of multiple-scales, we have derived a Ginzburg-Landau equation with high-order corrections contributed from linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. Based on the quantum interference effect induced by the control field, the imaginary part of the coefficients of the Ginzburg-Landau equation can be made to be much smaller than their real part, and hence the Ginzburg-Landau equation is reduced to a high-order nonlinear Schrödinger equation. We have made predictions of various optical solitons in different regimes of system parameters, and shown that the optical solitons obtained display many interesting features, including superluminal propagating velocity and very low generation power. Such optical solitons may have potential applications in optical information processing and engineering.

Acknowledgments

G. Huang thanks L. Deng and V. V. Konotop for fruitful discussions. This work was supported by NSF-China under Grant Nos. 10434060 and 10874043, by the Key Development Program for Basic Research of China under Grant Nos. 2005CB724508 and 2006CB921104.

References and links

1.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).

2.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

3.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).

4.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein. [CrossRef]

5.

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]

6.

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005). [CrossRef]

7.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008). [CrossRef]

8.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010). [CrossRef]

9.

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]

10.

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008). [CrossRef]

11.

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993). [CrossRef] [PubMed]

12.

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994). [CrossRef] [PubMed]

13.

R. Y. Chiao and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345. [CrossRef]

14.

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000). [CrossRef]

15.

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001). [CrossRef]

16.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001). [CrossRef] [PubMed]

17.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003). [CrossRef]

18.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003). [CrossRef] [PubMed]

19.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003). [CrossRef]

20.

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003). [CrossRef]

21.

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003). [CrossRef]

22.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003). [CrossRef]

23.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003). [CrossRef]

24.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004). [CrossRef]

25.

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]

26.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006). [CrossRef]

27.

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006). [CrossRef]

28.

J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006). [CrossRef] [PubMed]

29.

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007). [CrossRef]

30.

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]

31.

C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010). [CrossRef]

32.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

33.

S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein. [CrossRef]

34.

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation,” Phys. Rev. Lett. 78, 448 (1997). [CrossRef]

35.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998). [CrossRef]

36.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E. 60, R45 (1999). [CrossRef]

OCIS Codes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 21, 2010
Revised Manuscript: October 7, 2010
Manuscript Accepted: October 11, 2010
Published: January 19, 2011

Citation
Chengjie Zhu and Guoxiang Huang, "High-order nonlinear Schrödinger equation and weak-light superluminal solitons in active Raman gain media with two control fields," Opt. Express 19, 1963-1974 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1963


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References

  1. A. Hasegawa, and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).
  2. Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
  3. G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).
  4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005) (and references therein). [CrossRef]
  5. Y. Wu, and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004). [CrossRef] [PubMed]
  6. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72, 016617 (2005). [CrossRef]
  7. C. Hang, and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008). [CrossRef]
  8. W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010). [CrossRef]
  9. L. Deng, and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007). [CrossRef] [PubMed]
  10. K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Dopplerbroadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008). [CrossRef]
  11. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993). [CrossRef] [PubMed]
  12. A. M. Steinberg, and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994). [CrossRef] [PubMed]
  13. R. Y. Chiao, and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345. [CrossRef]
  14. L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature 406, 277 (2000). [CrossRef]
  15. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001). [CrossRef]
  16. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001). [CrossRef] [PubMed]
  17. . A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003). [CrossRef]
  18. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003). [CrossRef] [PubMed]
  19. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature 425, 695 (2003). [CrossRef]
  20. M. D. Stenner, and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003). [CrossRef]
  21. R. G. Ghulghazaryan, and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003). [CrossRef]
  22. K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003). [CrossRef]
  23. L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68, 066606 (2003). [CrossRef]
  24. E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004). [CrossRef]
  25. G. S. Agarwal, and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]
  26. A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006). [CrossRef]
  27. M. Janowicz, and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 046613 (2006). [CrossRef]
  28. J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006). [CrossRef] [PubMed]
  29. K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007). [CrossRef]
  30. G. S. Agarwal, and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004). [CrossRef]
  31. C. Zhu, and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” Eur. Phys. J. D 56, 231 (2010). [CrossRef]
  32. A. Jeffery, and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).
  33. S. Aranson, and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002) (and references therein). [CrossRef]
  34. M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schr¨odinger Equation,” Phys. Rev. Lett. 78, 448 (1997). [CrossRef]
  35. K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schr¨odinger Equations,” Phys. Rev. Lett. 80, 1425 (1998). [CrossRef]
  36. S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60, R45 (1999). [CrossRef]

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