PHYSICAL REVIEW E 90, 013202 (2014)
Self-similar propagation and asymptotic optical waves in nonlinear waveguides Jun-Rong He,1 Lin Yi,1 and Hua-Mei Li2 1
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China (Received 26 February 2014; published 31 July 2014)
The properties of self-similar optical waves propagating in a tapered cubic-quintic nonlinear waveguide are investigated. Using a lens-type transformation we obtain the exact analytical self-similar solutions which describe the propagation of bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikinkshaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Beyond the exact analytical solutions, asymptotic optical waves are also found by employing a direct ansatz. These waves possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by numerical simulations. Finally, we investigate the generation and propagation properties of self-similar optical waves in a quintic nonlinear medium. DOI: 10.1103/PhysRevE.90.013202
PACS number(s): 41.20.−q, 42.65.Tg, 42.81.Dp, 42.65.Wi
I. INTRODUCTION
The study of self-similar solutions (similaritons) has been intensively carried out in various fields of physics, such as nonlinear optics [1], Bose-Einstein condensates (BECs) [2,3], plasmas [4], fluid dynamics, and condensed matter physics. In nonlinear optics, optical similaritons in nonautonomous systems and asymptotic similaritons in gain amplifier systems have been studied extensively due to their potential applications in nonlinearity and dispersion management systems. For instance, in the past decades, several remarkable results have been obtained concerning self-similar regime of collapse for spiral laser beams [5], stimulated Raman scattering [6], evolution of self-written waveguides [7], nonlinear propagation of pulses with parabolic intensity [8–10], and nonlinear compression of chirped solitary waves [11]. These self-similar waves possess many attractive features that make them potentially useful for various applications in fiber-optic telecommunications and photonics, since they can maintain their overall shapes but allow their amplitudes and widths to change, following the modulation of the system’s parameters, such as the dispersion, nonlinearity, gain, inhomogeneity, and so on. The dynamics of optical selfsimilar wave solutions is usually governed by the nonlinear Schr¨odinger equation (NLSE), where the self-similarity of the solutions has been allowed by reducing the governing equations to ordinary differential equations or even algebraic equations. Generally speaking, optical similaritons can be divided into two categories. The first category is the exact optical similaritons, which are mainly described by the exact solutions, including the bright and dark soliton solutions, the quasisoliton solutions, the solitary nonlinear Bloch waves, and the solitons on the continuous-wave background [12–19]. However, the existence of these similaritons requires a delicate balance between the system parameters such as dispersion, nonlinearity, gain, and inhomogeneity. These requirements are, sometimes, difficult to realize in real applications. The second category is the asymptotic optical similaritons, which are mainly described by the parabolic, Hermite-Gaussian, and hybrid functions [20–25] and exist under a wide range of system parameters. 1539-3755/2014/90(1)/013202(9)
Note that most of the previous theoretical and experimental studies are focused on optical pulses propagating in gradedindex waveguide amplifiers with cubic Kerr nonlinearity, i.e., the refractive index is n(z,x) = n0 + n1 F (z)x 2 + n2 I (z,x), where z is the propagation distance, x is the spatial coordinate, and I is the pulse intensity. The first two terms describe the linear part of the refractive index and the last term represents the Kerr-type nonlinearity with n2 being the cubic coefficient (positive n2 for self-focusing nonlinearity and negative n2 for self-defocusing nonlinearity). Here we assume n1 > 0 and the dimensionless tapering function F (z) can be negative or positive, corresponding to the graded-index waveguide acting as a focusing or defocusing lens. However, when the intensity of the optical pulse exceeds a certain value, a higher-order nonlinear effect such as the quintic nonlinearity should be taken into account [1]. In this situation, the refractive index is given by [26] n(z,x) = n0 + n1 F (z)x 2 + n2 γ (z)I (z,x) − n4 δ(z)I (z,x)2 , (1) where n4 is the quintic coefficient of the Kerr-type nonlinearity which may assume positive or negative values. The dimensionless functions γ (z) and δ(z) represent inhomogeneity of Kerr nonlinearities along medium. In experiment, the cubic-quintic (CQ) nonlinearities can be obtained by doping a fiber with two appropriate semiconductor materials [27]. Promising candidates for optical CQ materials include chalcogenide glasses [28], colloids [29], polydiacetylene para-toluene sulfonate [30], and some organic polymers [31]. To our knowledge, although exact self-similar solutions in CQ optical fibers and nonlinear waveguides have been reported [26,32], much less attention has been paid to investigating their behaviors in the tapered CQ sech2 -profile waveguide (this type of waveguide is experimentally realizable [33]). Another important feature is that we obtain the asymptotic compact optical waves in the CQ nonlinear waveguides, which are first in this paper. The above two types of self-similar solutions can be achieved by using a lens-type transformation. First, we reduce the governing equation to the CQ-NLSE with constant coefficients and obtain the exact analytical self-similar solutions which describe the propagation of
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©2014 American Physical Society
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PHYSICAL REVIEW E 90, 013202 (2014)
bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Second, we reduce the governing equation to a simple algebraic equation and obtain the asymptotic optical waves which possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by our numerical simulations. Finally, we consider the generation and propagation of self-similar optical waves (including the exact and asymptotic analytical waves) in a pure quintic nonlinear medium. This type of nonlinearity can be engineered in centrosymmetric nonlinear media doped with resonant impurities whose resonant frequencies lie sufficiently far away from the beam carrier frequency [34]. II. THE REDUCTION
Under the paraxial and the slowly varying envelope approximations, the nonlinear wave equation governing pulse propagation in a tapered CQ inhomogeneous nonlinear waveguide given by Eq. (1) can be written as 2
i
1 ∂ u k0 n1 k0 n2 ∂u + + F (z)x 2 u + γ (z)|u|2 u 2 ∂z 2k0 ∂x n0 n0 k0 n4 ig(z) u, (2) − δ(z)|u|4 u = n0 2
where k0 = 2π n0 /λ, with λ being the wavelength of the optical source, and g(z) is the gain (loss) coefficient. Introducing the normalized variables U = (k0 |n2 |LD /n0 )1/2 u,G(Z) = g(z)LD ,τ = n0 n4 /(k02 w02 n22 ), X = x/w0 , and Z = z/LD , where w0 = (2k02 n1 /n0 )−1/4 and LD = k0 w02 represent the characteristic transverse scale and the diffraction length, respectively, Eq. (2) then can be rewritten in a dimensionless form, 1 ∂ 2U 1 iG ∂U + U, + F X2 U + σ γ |U |2 U − τ δ|U |4 U = i 2 ∂Z 2 ∂X 2 2 (3) where σ = n2 /|n2 | = ±1 corresponds to self-focusing (+) and self-defocusing (−) nonlinearity of the waveguide, respectively, and F (Z), γ (Z), δ(Z), as well as G(Z) are the functions of the normalized distance Z. When δ(z) = 0, Eq. (3) is reduced to the cubic NLSE (CNLSE) with quadratic term and gain, which has been intensively investigated in Refs. [35–37]. Equation (3) also finds the applications in BECs. In this case, Z and X, respectively, represent the time and spatial coordinate, U (Z,X) is the wave function, the term 12 F X2 denotes the harmonic potential, γ (Z) and δ(Z) describe the two- and three-body interactions, and G is the gain or loss coefficient, which is phenomenologically incorporated to account for the interaction of atomic or thermal clouds. It should be noted that the self-similar pulse may possess a quadratic phase φ = Z X2 /(2 ), which indicates that the pulse could be linearly chirped ( Z = 0) or unchirped ( Z = 0) for the self-similar evolution [37]. In addition, by considering the evolution properties of the self-similar optical pulse, one can express the pulse’s intensity as |U (Z,X)|2 = Z exp[ 0 G(Z )dZ ]|Q(ζ,ξ )|2 / , with ξ ≡ X/ , where is a
positive function of the propagation distance Z that characterizes the change in the pulse’s width and ζ is a function of Z referring to the transformed propagation distance. Thus, by introducing the following lens-type transformation:
Z 2 E Q(ζ,ξ ) exp i X , (4) U (Z,X) =
2
Z with E ≡ exp[ 0 G(Z )dZ ], we can transform Eq. (3) into the form of dζ
1 E Qζ + Qξ ξ + σ γ |Q|2 Q − τ δ |Q|4 Q i dZ E 2 E
2
(5) + (F − ZZ )ξ 2 Q = 0. 2E It is seen that by appropriately choosing the relations among the functions F (Z), γ (Z), δ(Z), and G(Z), one can obtain various types of self-similar solutions for Eq. (3). We first construct exact self-similar solutions for Eq. (3) by reducing Eq. (5) to the following CQNLSE with constant coefficients: iQζ + 12 Qξ ξ + σ |Q|2 Q − τ |Q|4 Q = 0,
(6)
when the following relations are satisfied: dζ = −2 , γ = ( E)−1 , δ = E −2 , F = ZZ , (7) dZ Therefore, when the system parameters are given by Eq. (7), one can construct exact self-similar optical wave solutions to Eq. (3) by using the exact solutions of Eq. (6). In fact, there are various types of exact solutions to Eq. (6) [38], depending on the sign of σ and the values of τ . More general forms of exact solutions to the CQNLSE have been reported in Refs. [39,40]. When σ = 1, corresponding to the self-focusing cubic nonlinearity, the bright soliton solution reads Mb exp(i b ), (8) Qb (ξ,ζ ) = √ 1 + Nb cosh[pb (ξ − vζ )] √ √ where Mb = 2 ωb , pb = 2 2ωb , b = vξ + (ωb − v 2 /2)ζ , ωb = 3(1 − Nb2 )/(16τ ), and v is the velocity of the soliton. When σ = −1, corresponding to the self-defocusing cubic nonlinearity, the dark soliton solution is given by Qd (ξ,ζ ) =
Md sinh[pd (ξ − vζ )]
exp(i d ), (9) 1 + Nd sinh2 [pd (ξ − vζ )] √ pd = where Md = pd (3Nd − 2)Nd , 3(Nd − 1)/[2τ (3Nd − 2)2 ],
d = vξ − (ωd + v 2 /2)ζ , ωd = (3Nd − 1)pd2 /2, and Nd > 1,τ = 0. If we choose τ = 3/(8ωk2 ), we get the following kink soliton solution for σ = 1: Mk exp(i k ), Qk (ξ,ζ ) = (10) 2 + exp[pk (ξ − vζ )] where Mk = 2ωk , pk = 2ak ωk , and k = vξ − (v 2 − ωk2 )ζ /2 with ak = ±1 and ωk being an arbitrary real constant. This solution appears to be a kink soliton when ak = −1 and an antikink soliton when ak = 1. Therefore, as long as one chooses the exact solutions in Eqs. (8)–(10) and substitutes the appropriate Q(ξ,ζ ) into Eq. (4), one obtains the exact self-similar wave solutions to
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1
G
Magnitude
0.5
γ
W 0
F
Chirp
δ
−0.5
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
Z
FIG. 1. (Color online) Profiles of the tapering function, width, CQ nonlinearities, gain, and chirp function for n = G = 1.
Eq. (3). In the following section, we will discuss the generation and propagation properties of these exact self-similar optical waves in the tapered CQ nonlinear waveguide. We also study the stability of the analytical solutions by numerical simulations. III. COMPRESSIONS OF THE CHIRPED SELF-SIMILAR SOLUTIONS
From Eq. (7) it is found that once the pulse’s width (Z) is determined by the parameter F (Z), under the properly choice of the gain G(Z), we can directly present solutions for Eq. (3) by solving the rest of equations. Mathematically, from the last condition of Eq. (7), one can identify a soliton configuration corresponding to each F (Z). Although a host of F (Z) can be addressed, we will only demonstrate in the text an experimentally realizable example. According to the theory of sech2 -profile waveguides [33], the form of tapering corresponds to F (Z) = n2 − n(n + 1)sech2 (Z),
(11)
where n is a positive integer. This choice leads to the width
= sechn (Z) from Eq. (7). For present case, we restrict ourself to the lowest-order mode in the waveguide n = 1. With this, the chirp function is − tanh(Z)/2. From the first condition in Eq. (7) we find that the transformed propagation distance ζ is sinh(2Z)/4 + Z/2 + ζ0 , where ζ0 is an arbitrary constant.
Figure 1 shows the profiles of the tapering function F (Z), width (Z), CQ nonlinearities γ (Z) and δ(Z), gain G(Z), and chirp function with n = 1. The gain G(Z) is chosen as a positive constant, which is required for wavelength division multiplexing in an erbium-doped waveguide, and it is obtained by using optimized glass hosts. Here the cubic nonlinearity and quintic nonlinearity evolutions are γ (Z) = cosh(Z) exp(−GZ) and δ(Z) = exp(−2GZ), respectively, and E = exp(GZ). It is observed from Fig. 1 that the tapering function F (Z) changes from −1 to 1, which implies that the linear inhomogeneity of the waveguide should change from the focusing to the defocusing type. The cubic nonlinearity γ (Z) decreases from 1 to 0.5 asymptotically as the distance Z increases. Interestingly, the quintic nonlinearity δ(Z) and the width of the pulse are both decreasing from 1 to 0 in the limit Z → ∞. This indicates that Eq. (3) reduces to the CNLSE with the quadratic term and constant gain for large propagation distance in the sech2 -profile waveguides. A zero width of soliton implies the high-power and ultrashort pulse. When the gain G(Z) varies exponentially with the propagation distance [21], different behaviors of the CQ nonlinearities would occur. We emphasize that the other forms of exact solutions to Eq. (3) can also be constructed when the gain G is Z dependence. In this case, the system parameters can be derived by solving Eq. (7). In Figs. 2 and 3 we plot the evolutions of the self-similar soliton solutions for constant G. Figures 2(a) and 2(b) demonstrate the intensity profiles of the bright and dark self-similar solitons, respectively. Figures 3(a) and 3(b) demonstrate the intensity profiles of the kink and antikink self-similar solitons, respectively. It is observed that the solitons display a moving-compression behavior with the increase of distance. Thus, a key consequence of our exact solution for constant loss is that the pulse can be compressed to any required degree as Z → ∞ while maintaining its sech shape and linear chirp in the tapered CQ sech2 -profile waveguides. These exact solutions are particularly useful in the design of amplifying or attenuating pulse compressors for chirped solitary waves. Note that the center position of the soliton is v [sinh(2Z)/4 + Z/2 + ζ0 ], which is significantly dependent on the velocity v and the width of the solution. From Eq. (4) it follows that there are three cases for the peak amplitude by appropriately choosing E: (i) amplitude grows for increasing Z (as we discussed above), (ii) amplitude
FIG. 2. (Color online) Compressions of the bright and dark self-similar solitons in the sech2 -profile CQ nonlinear waveguide. (a) Intensity profile of the bright soliton given by Eqs. (4) and (8) for Nb = 0.5 and (b) intensity profile of the dark soliton given by Eqs. (4) and (9) for Nd = 2. The other parameters are ζ0 = 0,τ = G = 1,v = 3. 013202-3
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FIG. 3. (Color online) Compressions of the kink and antikink self-similar solitons in the sech2 -profile CQ nonlinear waveguide. Intensity profiles of (a) the kink soliton for ak = −1 and (b) the antikink soliton for ak = 1 given by Eqs. (4) and (10). The other parameters are ζ0 = 0,G = v = ωk = 1.
decreases for increasing Z [for instance, choosing E = sech2 (Z)(Z > 0)], and (iii) constant amplitude (E = E0 ). In cases (i) and (ii), the solutions correspond either to compressing or spreading solitary pulses which maintain a linear chirp. In case (iii), one can obtain the self-similar solution with constant amplitude (which would not go to infinity as Z → ∞). This type of solution may be useful for the stabilization of pulse in the waveguide. A very important aspect of the present issue is the stability of the obtained exact solutions; that is, how do they evolve when disturbed from their analytically given forms? This aspect of the problem can be addressed numerically. We ran simulations by means of the split-step Fourier-transform method with the initial condition U (0,X) in the framework of
2
Eq. (3). In Fig. 4(a), we have added a white noise in the initial pulse U (0,X), i.e., Upert (0,X) = U (0,X)[1 + 0.1R(X)] was taken as initial condition, where R(X) is a random number taking values in (0,1). It is found that the bright soliton can propagate in a stable way against the perturbation with an initial white noise. This may make the possibility for observing the compression of bright soliton in experiments more likely in a graded-index tapered CQ nonlinear waveguide. However, the situation differs markedly for the evolutions of the dark and kink solitons. Figures 4(b) and 4(c) show the direct numerical simulations of the dark and kink solitons in the framework of Eq. (3). It is seen that they are unstable, even without any of perturbation. This implies that the exact soliton solutions are not always stable. Note that the stability of
(a)
3 2
|U|
1 0.5 0 −10
1 0
0 −10
1 0
Z 10
0.5
X
Z
0
10
0
(d)
(c)
1.5
|U|2
2
4
|U|
1
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X
6
(b)
2
|U|2
1.5
2
0 −10
1 0
1 0.5
0 −10
1
0.5
0
Z
X 10
0.5
X
0
Z 10
0
FIG. 4. (Color online) (a) The stable evolution of the bright self-similar soliton given by Eqs. (4) and (8), with the same parameters as in Fig. 2(a) except for v = 0. Panels (b) and (c) show the unstable evolutions of the dark and kink self-similar solitons given by Eqs. (4), (9), and (10), respectively, with the same parameters as in Figs. 2(b) and 3(a) except for v = 0. (d) Evolution of a stable secant pulse in the CQ nonlinear waveguide within the framework of Eq. (3) when σ γ = τ δ = 1,G = 0. 013202-4
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bright soliton is limited to the condition (7), as mentioned. When the parameters in Eq. (7) gradually move from that value, the system becomes more and more unstable. Numerical simulations reveal that the bright soliton could still be stable if the deviation is within 10%. When the system parameters largely deviate from the idealized situation, other forms of solutions would be required to check the structural stability of the waveguides. In this situation, we introduce the secant function U (0,X) = sech(X) as the input pulse and make direct simulations in the framework of Eq. (3) with constant distributed coefficients. Note that a small initial white noise is included in the simulation. The result shows that the pulse can propagate stably with Z, see Fig. 4(d). IV. ASYMPTOTIC COMPACT SELF-SIMILAR SOLUTIONS
In this section, we shall put forward the asymptotic compact self-similar solutions to Eq. (3) when the relative strength of the dispersion is much less than those of the CQ nonlinearities. Within the framework of the CNLSE, we know that the asymptotic compact pulses can be generated in waveguide with the defocusing Kerr nonlinearity [37]. Therefore, we can assume that γ (Z) and δ(Z) are positive functions of Z satisfying σ γ > τ δ|U |2 with σ < 0 such that the effective nonlinearity is self-defocusing. To do this, we use the transformation Q(ζ,ξ ) = Q(Z,ξ )
= A(ξ ) exp iσ μ
Z 0
γ (Z )E(Z ) dZ ,
(Z )
(12)
and then reduce Eq. (5) to c0 A4 − A2 + μ − Kξ 2 = 0,
√
(13)
at |ξ | μ/K, and A(ξ ) = 0 otherwise, when the following relations are satisfied: 1 1, σ γ E
c0 =
2 τ δE , K= (F − ZZ ), σ γ 2σ γ E (14)
where c0 , μ, and K are assumed to be positive constants. Thus, we have τ < 0. Note that the reduction of Eq. (5) to Eq. (13) is essentially the same as that produced by the Thomas-Fermi approximation for the ground-state solution of the one-dimensional Gross-Pitaevskii equation with the harmonic potential [41]. Solving the algebraic Eq. (13), we obtain the following physical solution for A(ξ ): 1 − 1 − 4c0 (μ − Kξ 2 ) 2 A = , (15) 2c0 where μ is determined by the input power Pin =
∞
−∞
|U (0,X)| dX = 2
√ μ/K √ − μ/K
A2 dξ.
(16)
To check the analytical predictions, we consider the simplest case where F , γ , and G are constants. In this case,
by solving Eq. (14) we have G 3 18Kσ γ Z exp
= 9F − G2 3 and δ = c0 γ
σ τ
3
18Kσ γ 2G exp − Z , 9F − G2 3
(17)
(18)
Note that Eq. (17) implies σ (9F − G2 ) > 0, which means that for the defocusing Kerr nonlinearity it is F < G2 /9. The defocusing Kerr nonlinearity can be realized in planar silica semiconductor waveguides, such as ZnSe [42]. The approximate self-similar compact solution for Eq. (3) can be obtained by substituting Eqs. (12), (15), and (17) into Eq. (4). After some analysis, we find that the effective width and amplitude of the compact pulse increase exponentially as exp(GZ/3) and depend on the tapering function F . When the gain and propagation distance are given, the effective width at the asymptotic limit is increasing as F varies from negative to positive. Such a tapering function can be realized, as stated above, by the linear focusing or defocusing lens of the graded-index nonlinear waveguide. When F < 0, the larger the |F |, the smaller the effective width and the larger the pulse’s amplitude; when F > 0, the effective width becomes larger when F increases, while the pulse’s amplitude becomes smaller. In addition, it is possible to control the shape of the output compact pulses by choosing an appropriate tapering parameter, since (0) can vary from zero to infinity as F changes from −∞ to G2 /9. Thus, we may infer that with a strong-enough negative tapering parameter, one can generate a high-power, ultrashort pulse in the CQ nonlinear waveguide. Note that the above compact solution exists when the conditions σ γ > τ δ|U |2 and γ E 1 are valid. From the selfsimilar transformation (4), the second condition in Eq. (14), and Eq. (15), we find that σ γ > τ δ|U |2 is automatically satisfied. At the same time, the condition γ E 1 can be easily satisfied after a short propagation distance due to the exponentially increasing of E and . The analytical predictions are confirmed by direct numerical resolutions of Eq. (3) by use of the standard split-step Fourier method, as shown in Fig. 5. The input pulse is chosen as U (0,X) = exp(−X2 /2)/π 1/4 . It is observed that after about five propagation distance units, the results of numerical simulations and analytical predictions for the pulse’s intensity agree well with each other. From Fig. 5(b) one finds that the general forms of the width and amplitude of the pulse are in good agreement with the analytical results as well. Moreover, one can see that at the central region, the pulse intensity without the quintic nonlinearity is smaller than that with the quintic nonlinearity. Note that the chirp of the asymptotic compact solution is G/6, which is also confirmed by numerical simulations [see Fig. 5(c)] using a fast phase unwrapping algorithm [43]. Next, we consider additional simulations that involve pulses with the same input power but with different input profiles. We choose a hyperbolic secant pulse, U (0,X) = sech(X)/21/2 . The results of numerical simulations and analytical predictions for the general form of the pulse agree well with each other as well, see Fig. 6. However, we find that this agreement is not
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equation for the paraxial optical pulse in such a medium is
2
10
0
1
10
Intensity
0
i
10
Width
10
Amplitude
(a)
0
2 Z
4
6
(b) 0
10
0
−5
2
4
Z
10
6
20 Phase
0
−20
−10
10
(c)
−40 −20
−10
0 X
10
20
−20
−10
0 X
10
20
FIG. 5. (Color online) (a) The self-similar evolution of the compact pulse in the CQ nonlinear waveguide, starting with the Gaussian input pulse U (0,X) = exp(−X2 /2)/π 1/4 . From top to bottom, the propagation distance is Z = 6,5.6, and 5, respectively. (b) The pulse’s width and amplitude as functions of Z. (c) The phase (phase offset is ignored) of the pulse at propagation distance Z = 6. Here, and in the following figures, the solid lines and circles represent results of the direct numerical simulations of Eq. (3) and the analytical predictions, respectively. The dotted lines in (a) show the pulse intensity at Z = 6 when the quintic nonlinearity is absent. The parameters are γ = G = K = 1,F = G2 /60, and c0 = 0.05 such that μ = 0.8069.
as good as that in Fig. 5 after the same units of propagation distance. Therefore, we may infer that the initial Gaussian input pulse leads to the generation of asymptotic compact similariton more convergence with the analytical profile in the CQ nonlinear waveguide.
V. SELF-SIMILAR SOLUTIONS IN THE QUINTIC MODEL
In this section, we consider the self-similar solutions propagating in a pure quintic nonlinear model. The governing
1 ∂ 2U 1 iG ∂U + U. + F X2 U − τ δ|U |4 U = 2 ∂Z 2 ∂X 2 2
(19)
Equation (19) can also describe optical pulse propagating in nonlinear optical media with power-law nonlinearity [44]. It is known that physically, various materials, including semiconductors, exhibit power-law nonlinearity. Recently, a generic model for the quintic nonlinearity has been realized in a centrosymmetric nonlinear medium doped with resonant impurities in the limit of a large light carrier frequency detuning from the impurity resonance [34]. In BECs, the pure quintic nonlinearity can be derived by setting the s-wave scattering length as (t) to zero via the Feschbach resonance technique [41,45,46]. The pure quintic nonlinearity also appears in general NLSE-type systems near the transition from supercritical to subcritical bifurcations [47], pattern formation [48], and dissipative solitons [49]. Substituting the profile (4) back into Eq. (19) and requiring the first, third, and last conditions in Eq. (7) to be satisfied, one obtains a partial differential equation: iQζ + 12 Qξ ξ − τ |Q|4 Q = 0, which has the following exact bright self-similar solution for τ < 0 [44]:
√ 3 sech[ −τ (ξ − vζ )] exp(i ), (20) Q(ξ,ζ ) = 8 where = vξ − (v 2 + τ/4)ζ /2. When the tapering function F (Z) is given by Eq. (11), one again has = sechn (Z). The solution in this situation is plotted in Fig. 7(a), which also displays a moving-compression behavior with the increase of distance. The corresponding numerical evolution of the bright similariton profile is shown in Fig. 7(b), where a 10% withe noise is included. It is observed that stable bright similariton can be supported by the quintic nonlinear media. Asymptotic compact self-similar solutions to Eq. (19) can also be found when considering that the relative strength of the dispersion is much less than that of the quintic nonlinearity. We still assume that δ(Z) is a positive function of Z and employ the transformation (4) with Q(ζ,ξ ) = Z ) A(ξ ) exp[−iμ 0 E(Z dZ ], and then Eq. (19) reduces to
(Z )
2
Amplitude
10
Width
(a) 0
10
c1 A4 − μ − Kξ 2 = 0,
(21) √ at |ξ | −μ/K, and A(ξ ) = 0 otherwise, when the following relations are satisfied:
0
10
1
10
0
2
Z
4
6
Intensity
(b) 0
10
0
2
−5
10
Z
4
Phase
20 0
−20
−10
(c)
10
−20
−10
0 X
10
20
−40 −20
−10
0 X
1 1, τ δE 2
6
10
20
FIG. 6. (Color online) The same as in Fig. 5 except that the input pulse is U (0,X) = sech(X)/21/2 , i.e., the hyperbolic secant input pulse.
c1 =
τ δE ,
K=
2 (F − ZZ ), 2E
(22)
where c1 and μ are positive constants. Thus, we have K < 0. We obtain the following physical solution for A(ξ ) by solving Eq. (21): c1 (μ + Kξ 2 ) 2 A = , (23) c1 with μ being determined by the input power. To check the analytical predictions, we consider the simplest case where F and G are constants. In this case, by
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(b) 3
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Z
0
FIG. 7. (Color online) (a) Compressions of the bright self-similar soliton given by Eqs. (4) and (20) in the sech2 -profile nonlinear media. (b) The stable evolution of the bright self-similar soliton with initial zero velocity. The parameters are ζ0 = 0, − τ = G = 1,v = 3.
solving Eq. (22) we have 18K G 3 Z exp
= 9F − G2 3 and δ=
c1 τ
3
(24)
18K 2G Z , exp − 9F − G2 3
(25)
From Eq. (24) we find that F < G2 /9; from Eq. (25) we get τ δ > 0. This implies that the asymptotic compact pulses can be generated in nonlinear media with the defocusing quintic nonlinearity, which is confirmed by the direct numerical simulations of Eq. (19). Since the self-defocusing nonlinearity is possible in semiconductor waveguides, Eq. (19) may be found in semiconductor waveguides doped with quantum dots by considering the theory in Ref. [34]. Note that the effective width and amplitude of the compact pulse increase exponentially as exp(GZ/3) and depend on the tapering function F . A high-power, ultrashort pulse may also be generated in the quintic nonlinear media with a strong-enough negative tapering parameter F . Further, the term δE 2 varies 2
(a) Width
10
10 Intensity
1
10
0
0
10
0
2 Z
4
(b)
0
−1
10
10
−2
0
1
2 Z
3
4
6 4 2 0 −2 −4
Phase
10
−3
10
−4
10
Amplitude
10 1
−5
0 X
5
(c) −5
0 X
5
FIG. 8. (Color online) (a) The evolution of the compact selfsimilar solution in the nonlinear media, starting with the Gaussian input pulse U (0,X) = exp(−X2 /2)/π 1/4 . From top to bottom, the propagation distance is Z = 4,3.8,3.6, and 3.4, respectively. (b) The solution’s width and amplitude as functions of Z. (c) The phase (phase offset is ignored) of the pulse at propagation distance Z = 4. The parameters are G = −K = 1,F = G2 /60, and c1 = 0.5 such √ that μ = 2/π .
as ∼exp(4GZ/3); thus, the condition τ δE 2 1 can be easily satisfied after a short propagation distance. The analytical predictions are confirmed by direct numerical simulations of Eq. (19), as shown in Fig. 8, where the input pulse is U (0,X) = exp(−X2 /2)/π 1/4 . It is observed that the general forms of the intensity profile, width, amplitude, and phase are in good agreement with the analytical results.
VI. CONCLUSIONS AND DISCUSSION
In conclusion, with the help of a lens-type transformation, we have mainly investigated the propagation properties of self-similar optical waves in a tapered CQ nonlinear waveguides. Various types of self-similar solutions were found by appropriately choosing the relations between the distributed coefficients. First, we reduced Eq. (3) to the CQNLSE with constant coefficients and obtained the exact self-similar solutions which describe the propagation of brightshaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions was examined by the numerical simulations such that stable bright solitons were found. Second, we reduced Eq. (3) to a simple algebraic equation and obtained the asymptotic optical waves by employing a direct ansatz. These waves possessed linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves was demonstrated. The analytical results were confirmed by numerical simulations. Finally, we studied the generation and propagation properties of self-similar solutions in the quintic nonlinear media, which may be found in semiconductor waveguides (such as ZnSe) doped with quantum dots. We now give a suitable experimental protocol to generate the obtained asymptotic compact pulses in realistic waveguides. We consider a 10-μm-thick planar silica waveguide, such as ZnSe [42], in which the input pulse is confined in the y direction. The refractive index in Eq. (1) can be graded with n0 ≈ 2.7,n1 ≈ 0.1 cm−2 , and n2 ≈ −7 × 10−14 cm2 /W near 532 nm [16,35,42,50], which lead to w0 ≈ 34 μm and the diffraction length LD ≈ 3.7 cm. In this situation, the required peak intensity is about 550 MW/cm2 (corresponding to the six propagation distance units in Figs. 5 and 6), translating into input power levels ∼100 W. Such power levels can be realized by employing a frequency doubled Nd:YAG laser near 532 nm [42,50].
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Besides nonlinear optics, amplification of matter waves is also found in BECs. The demonstration in Ref. [51] that matter waves can be amplified and maintain their phase is an exciting development in the rapidly growing field of matter-wave optics. Amplification of light and atoms in a BEC has been reported in Ref. [52]. Therefore, the results obtained in this paper could also be applied to BECs.
[1] G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, UK, 1996); G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, New York, 2007). [2] P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000); V. I. Kruglov, M. K. Olsen, and M. J. Collett, ibid. 72, 033604 (2005). [3] L. Wu, J. F. Zhang, and L. Li, New J. Phys. 9, 69 (2007); L. Wu, L. Li, and J.-F. Zhang, Phys. Rev. A 80, 013617 (2009). [4] T. Xu, B. Tian, L. L. Li, X. Lu, and C. Zhang, Phys. Plasmas 15, 102307 (2008). [5] V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992). [6] C. R. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992). [7] T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, Opt. Lett. 23, 268 (1998). [8] Christophe Finot, Guy Millot, Cyril Billet, and John Dudley, Opt. Express 11, 1547 (2003). ¨ Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. [9] F. O. Rev. Lett. 92, 213902 (2004). [10] S. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009). [11] J. D. Moores, Opt. Lett. 21, 555 (1996). [12] V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007). [13] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000); ,JETP Lett. 72, 89 (2000); ,IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002). [14] V. N. Serkin and T. L. Belyaeva, JETP Lett. 74, 573 (2001); ,Quantum Electron. 31, 1007 (2001). [15] V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003). [16] S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006); ,Opt. Express 15, 2963 (2007). [17] J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, Opt. Commun. 263, 328 (2006). [18] S. Kumar and A. Hasegawa, Opt. Lett. 22, 372 (1997). [19] Y. Ozeki and T. Inoue, Opt. Lett. 31, 1606 (2006). [20] M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000). [21] V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000). [22] V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, J. Opt. Soc. Am. B 19, 461 (2002); V. I. Kruglov and J. D. Harvey, ibid. 23, 2541 (2006). [23] J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007). [24] C. Finot and G. Millot, Opt. Express 13, 5825 (2005); ,13, 7653 (2005).
ACKNOWLEDGMENTS
The authors thank Professor Lu Li at Shanxi University for his useful suggestions. This work was supported by the National Natural Science Foundation of China under Grants No. U1230108, No. 11374266, and No. 11175158.
[25] S. Chen, L. Yi, D. S. Guo, and P. Lu, Phys. Rev. E 72, 016622 (2005). [26] C. Q. Dai, Y. Y. Wang, and X. G. Wang, J. Phys. A: Math. Theor. 44, 155203 (2011); J. D. He and J. F. Zhang, ibid. 44, 205203 (2011). [27] J. Herrmann, Opt. Commun. 87, 161 (1992); C. De Angelis, IEEE J. Quantum Electron. 30, 818 (1994). [28] F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, J. Non-Cryst. Solids 274, 232 (2000); M. J. Paz-Alonso and H. Michinel, Phys. Rev. Lett. 94, 093901 (2005). [29] M. Brunner, J. Dobnikar, H. H. vonGrunberg, and C. Bechinger, Phys. Rev. Lett. 92, 078301 (2004); E. L. Falcao-Filho, C. B. de Araujo, and J. J. Rodrigues, Jr., J. Opt. Soc. Am. B 24, 2948 (2007). [30] B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, Phys. Rev. Lett. 73, 597 (1994). [31] C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, J. Opt. Soc. Am. B 19, 369 (2002). [32] K. Senthilnathan, Qian Li, K. Nakkeeran, and P. K. A. Wai, Phys. Rev. A 78, 033835 (2008); J. F. Zhang, Q. Tian, Y. Y. Wang, C. Q. Dai, and L. Wu, ibid. 81, 023832 (2010). [33] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1973), Table 12-7 and p. 266; G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980). [34] S. A. Ponomarenko and S. Haghgoo, Phys. Rev. A 81, 051801(R) (2010). [35] L. Li, X. Zhao, and Z. Xu, Phys. Rev. A 78, 063833 (2008). [36] S. A. Ponomarenko and G. P. Agrawal, Opt. Lett. 32, 1659 (2007). [37] L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, Opt. Express 16, 6352 (2008); L. Wu, L. Li, and J. F. Zhang, Phys. Rev. A 78, 013838 (2008); L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, ibid. 78, 053807 (2008). [38] Kh. I. Pushkarov et al., Opt. Quantum Electron. 11, 471 (1979); Kh. I. Pushkarov and D. I. Pushkarov, Rep. Math. Phys. 17, 37 (1980); D. I. Pushkarov and S. Tanev, Opt. Commun. 124, 354 (1996); S. Tanev and D. I. Pushkarov, ibid. 141, 322 (1997); Pavel Honzatko, ibid. 127, 363 (1996); R. Y. Hao, L. Li, Z. H. Li, R. C. Yang, and G. S. Zhou, ibid. 245, 383 (2005). [39] Zhenya Yan and V. V. Konotop, Phys. Rev. E 80, 036607 (2009). [40] Y. M. Chen, S. H. Ma, and Z. Y. Ma, Chin. Phys. B 21, 050510 (2012). [41] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). [42] S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991). [43] M. A. Schofield and Y. M. Zhu, Opt. Lett. 28, 1194 (2003). [44] A. W. Snyder and D. J. Mitchell, Opt. Lett. 18, 101 (1993).
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[45] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003). [46] V. A. Brazhnyi, V. V. Konotop, and L. P. Pitaevskii, Phys. Rev. A 73, 053601 (2006). [47] E. A. Kuznetsov, J. Exp. Theor. Phys. 89, 163 (1999); D. Agafontsev, F. Dias, and E. A. Kuznetsov, JETP Lett. 87, 667 (2008). [48] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[49] J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000). [50] G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, Opt. Lett. 16, 156 (1991). [51] S. Inouye, T. Pfau, S. Gupta, A. P. Chikkatur, A. G¨orlitz, D. E. Pritchard, and W. Ketterle, Nature 402, 641 (1999). [52] S. Inouye, R. F. L¨ow, S. Gupta, T. Pfau, A. G¨orlitz, T. L. Gustavson, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 85, 4225 (2000).
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Self-similar propagation and asymptotic optical waves in nonlinear waveguides Jun-Rong He,1 Lin Yi,1 and Hua-Mei Li2 1
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China (Received 26 February 2014; published 31 July 2014)
The properties of self-similar optical waves propagating in a tapered cubic-quintic nonlinear waveguide are investigated. Using a lens-type transformation we obtain the exact analytical self-similar solutions which describe the propagation of bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikinkshaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Beyond the exact analytical solutions, asymptotic optical waves are also found by employing a direct ansatz. These waves possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by numerical simulations. Finally, we investigate the generation and propagation properties of self-similar optical waves in a quintic nonlinear medium. DOI: 10.1103/PhysRevE.90.013202
PACS number(s): 41.20.−q, 42.65.Tg, 42.81.Dp, 42.65.Wi
I. INTRODUCTION
The study of self-similar solutions (similaritons) has been intensively carried out in various fields of physics, such as nonlinear optics [1], Bose-Einstein condensates (BECs) [2,3], plasmas [4], fluid dynamics, and condensed matter physics. In nonlinear optics, optical similaritons in nonautonomous systems and asymptotic similaritons in gain amplifier systems have been studied extensively due to their potential applications in nonlinearity and dispersion management systems. For instance, in the past decades, several remarkable results have been obtained concerning self-similar regime of collapse for spiral laser beams [5], stimulated Raman scattering [6], evolution of self-written waveguides [7], nonlinear propagation of pulses with parabolic intensity [8–10], and nonlinear compression of chirped solitary waves [11]. These self-similar waves possess many attractive features that make them potentially useful for various applications in fiber-optic telecommunications and photonics, since they can maintain their overall shapes but allow their amplitudes and widths to change, following the modulation of the system’s parameters, such as the dispersion, nonlinearity, gain, inhomogeneity, and so on. The dynamics of optical selfsimilar wave solutions is usually governed by the nonlinear Schr¨odinger equation (NLSE), where the self-similarity of the solutions has been allowed by reducing the governing equations to ordinary differential equations or even algebraic equations. Generally speaking, optical similaritons can be divided into two categories. The first category is the exact optical similaritons, which are mainly described by the exact solutions, including the bright and dark soliton solutions, the quasisoliton solutions, the solitary nonlinear Bloch waves, and the solitons on the continuous-wave background [12–19]. However, the existence of these similaritons requires a delicate balance between the system parameters such as dispersion, nonlinearity, gain, and inhomogeneity. These requirements are, sometimes, difficult to realize in real applications. The second category is the asymptotic optical similaritons, which are mainly described by the parabolic, Hermite-Gaussian, and hybrid functions [20–25] and exist under a wide range of system parameters. 1539-3755/2014/90(1)/013202(9)
Note that most of the previous theoretical and experimental studies are focused on optical pulses propagating in gradedindex waveguide amplifiers with cubic Kerr nonlinearity, i.e., the refractive index is n(z,x) = n0 + n1 F (z)x 2 + n2 I (z,x), where z is the propagation distance, x is the spatial coordinate, and I is the pulse intensity. The first two terms describe the linear part of the refractive index and the last term represents the Kerr-type nonlinearity with n2 being the cubic coefficient (positive n2 for self-focusing nonlinearity and negative n2 for self-defocusing nonlinearity). Here we assume n1 > 0 and the dimensionless tapering function F (z) can be negative or positive, corresponding to the graded-index waveguide acting as a focusing or defocusing lens. However, when the intensity of the optical pulse exceeds a certain value, a higher-order nonlinear effect such as the quintic nonlinearity should be taken into account [1]. In this situation, the refractive index is given by [26] n(z,x) = n0 + n1 F (z)x 2 + n2 γ (z)I (z,x) − n4 δ(z)I (z,x)2 , (1) where n4 is the quintic coefficient of the Kerr-type nonlinearity which may assume positive or negative values. The dimensionless functions γ (z) and δ(z) represent inhomogeneity of Kerr nonlinearities along medium. In experiment, the cubic-quintic (CQ) nonlinearities can be obtained by doping a fiber with two appropriate semiconductor materials [27]. Promising candidates for optical CQ materials include chalcogenide glasses [28], colloids [29], polydiacetylene para-toluene sulfonate [30], and some organic polymers [31]. To our knowledge, although exact self-similar solutions in CQ optical fibers and nonlinear waveguides have been reported [26,32], much less attention has been paid to investigating their behaviors in the tapered CQ sech2 -profile waveguide (this type of waveguide is experimentally realizable [33]). Another important feature is that we obtain the asymptotic compact optical waves in the CQ nonlinear waveguides, which are first in this paper. The above two types of self-similar solutions can be achieved by using a lens-type transformation. First, we reduce the governing equation to the CQ-NLSE with constant coefficients and obtain the exact analytical self-similar solutions which describe the propagation of
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bright-shaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions is examined by numerical simulations such that stable bright solitons are found. Second, we reduce the governing equation to a simple algebraic equation and obtain the asymptotic optical waves which possess linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves is demonstrated. The analytical results are confirmed by our numerical simulations. Finally, we consider the generation and propagation of self-similar optical waves (including the exact and asymptotic analytical waves) in a pure quintic nonlinear medium. This type of nonlinearity can be engineered in centrosymmetric nonlinear media doped with resonant impurities whose resonant frequencies lie sufficiently far away from the beam carrier frequency [34]. II. THE REDUCTION
Under the paraxial and the slowly varying envelope approximations, the nonlinear wave equation governing pulse propagation in a tapered CQ inhomogeneous nonlinear waveguide given by Eq. (1) can be written as 2
i
1 ∂ u k0 n1 k0 n2 ∂u + + F (z)x 2 u + γ (z)|u|2 u 2 ∂z 2k0 ∂x n0 n0 k0 n4 ig(z) u, (2) − δ(z)|u|4 u = n0 2
where k0 = 2π n0 /λ, with λ being the wavelength of the optical source, and g(z) is the gain (loss) coefficient. Introducing the normalized variables U = (k0 |n2 |LD /n0 )1/2 u,G(Z) = g(z)LD ,τ = n0 n4 /(k02 w02 n22 ), X = x/w0 , and Z = z/LD , where w0 = (2k02 n1 /n0 )−1/4 and LD = k0 w02 represent the characteristic transverse scale and the diffraction length, respectively, Eq. (2) then can be rewritten in a dimensionless form, 1 ∂ 2U 1 iG ∂U + U, + F X2 U + σ γ |U |2 U − τ δ|U |4 U = i 2 ∂Z 2 ∂X 2 2 (3) where σ = n2 /|n2 | = ±1 corresponds to self-focusing (+) and self-defocusing (−) nonlinearity of the waveguide, respectively, and F (Z), γ (Z), δ(Z), as well as G(Z) are the functions of the normalized distance Z. When δ(z) = 0, Eq. (3) is reduced to the cubic NLSE (CNLSE) with quadratic term and gain, which has been intensively investigated in Refs. [35–37]. Equation (3) also finds the applications in BECs. In this case, Z and X, respectively, represent the time and spatial coordinate, U (Z,X) is the wave function, the term 12 F X2 denotes the harmonic potential, γ (Z) and δ(Z) describe the two- and three-body interactions, and G is the gain or loss coefficient, which is phenomenologically incorporated to account for the interaction of atomic or thermal clouds. It should be noted that the self-similar pulse may possess a quadratic phase φ = Z X2 /(2 ), which indicates that the pulse could be linearly chirped ( Z = 0) or unchirped ( Z = 0) for the self-similar evolution [37]. In addition, by considering the evolution properties of the self-similar optical pulse, one can express the pulse’s intensity as |U (Z,X)|2 = Z exp[ 0 G(Z )dZ ]|Q(ζ,ξ )|2 / , with ξ ≡ X/ , where is a
positive function of the propagation distance Z that characterizes the change in the pulse’s width and ζ is a function of Z referring to the transformed propagation distance. Thus, by introducing the following lens-type transformation:
Z 2 E Q(ζ,ξ ) exp i X , (4) U (Z,X) =
2
Z with E ≡ exp[ 0 G(Z )dZ ], we can transform Eq. (3) into the form of dζ
1 E Qζ + Qξ ξ + σ γ |Q|2 Q − τ δ |Q|4 Q i dZ E 2 E
2
(5) + (F − ZZ )ξ 2 Q = 0. 2E It is seen that by appropriately choosing the relations among the functions F (Z), γ (Z), δ(Z), and G(Z), one can obtain various types of self-similar solutions for Eq. (3). We first construct exact self-similar solutions for Eq. (3) by reducing Eq. (5) to the following CQNLSE with constant coefficients: iQζ + 12 Qξ ξ + σ |Q|2 Q − τ |Q|4 Q = 0,
(6)
when the following relations are satisfied: dζ = −2 , γ = ( E)−1 , δ = E −2 , F = ZZ , (7) dZ Therefore, when the system parameters are given by Eq. (7), one can construct exact self-similar optical wave solutions to Eq. (3) by using the exact solutions of Eq. (6). In fact, there are various types of exact solutions to Eq. (6) [38], depending on the sign of σ and the values of τ . More general forms of exact solutions to the CQNLSE have been reported in Refs. [39,40]. When σ = 1, corresponding to the self-focusing cubic nonlinearity, the bright soliton solution reads Mb exp(i b ), (8) Qb (ξ,ζ ) = √ 1 + Nb cosh[pb (ξ − vζ )] √ √ where Mb = 2 ωb , pb = 2 2ωb , b = vξ + (ωb − v 2 /2)ζ , ωb = 3(1 − Nb2 )/(16τ ), and v is the velocity of the soliton. When σ = −1, corresponding to the self-defocusing cubic nonlinearity, the dark soliton solution is given by Qd (ξ,ζ ) =
Md sinh[pd (ξ − vζ )]
exp(i d ), (9) 1 + Nd sinh2 [pd (ξ − vζ )] √ pd = where Md = pd (3Nd − 2)Nd , 3(Nd − 1)/[2τ (3Nd − 2)2 ],
d = vξ − (ωd + v 2 /2)ζ , ωd = (3Nd − 1)pd2 /2, and Nd > 1,τ = 0. If we choose τ = 3/(8ωk2 ), we get the following kink soliton solution for σ = 1: Mk exp(i k ), Qk (ξ,ζ ) = (10) 2 + exp[pk (ξ − vζ )] where Mk = 2ωk , pk = 2ak ωk , and k = vξ − (v 2 − ωk2 )ζ /2 with ak = ±1 and ωk being an arbitrary real constant. This solution appears to be a kink soliton when ak = −1 and an antikink soliton when ak = 1. Therefore, as long as one chooses the exact solutions in Eqs. (8)–(10) and substitutes the appropriate Q(ξ,ζ ) into Eq. (4), one obtains the exact self-similar wave solutions to
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1
G
Magnitude
0.5
γ
W 0
F
Chirp
δ
−0.5
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
Z
FIG. 1. (Color online) Profiles of the tapering function, width, CQ nonlinearities, gain, and chirp function for n = G = 1.
Eq. (3). In the following section, we will discuss the generation and propagation properties of these exact self-similar optical waves in the tapered CQ nonlinear waveguide. We also study the stability of the analytical solutions by numerical simulations. III. COMPRESSIONS OF THE CHIRPED SELF-SIMILAR SOLUTIONS
From Eq. (7) it is found that once the pulse’s width (Z) is determined by the parameter F (Z), under the properly choice of the gain G(Z), we can directly present solutions for Eq. (3) by solving the rest of equations. Mathematically, from the last condition of Eq. (7), one can identify a soliton configuration corresponding to each F (Z). Although a host of F (Z) can be addressed, we will only demonstrate in the text an experimentally realizable example. According to the theory of sech2 -profile waveguides [33], the form of tapering corresponds to F (Z) = n2 − n(n + 1)sech2 (Z),
(11)
where n is a positive integer. This choice leads to the width
= sechn (Z) from Eq. (7). For present case, we restrict ourself to the lowest-order mode in the waveguide n = 1. With this, the chirp function is − tanh(Z)/2. From the first condition in Eq. (7) we find that the transformed propagation distance ζ is sinh(2Z)/4 + Z/2 + ζ0 , where ζ0 is an arbitrary constant.
Figure 1 shows the profiles of the tapering function F (Z), width (Z), CQ nonlinearities γ (Z) and δ(Z), gain G(Z), and chirp function with n = 1. The gain G(Z) is chosen as a positive constant, which is required for wavelength division multiplexing in an erbium-doped waveguide, and it is obtained by using optimized glass hosts. Here the cubic nonlinearity and quintic nonlinearity evolutions are γ (Z) = cosh(Z) exp(−GZ) and δ(Z) = exp(−2GZ), respectively, and E = exp(GZ). It is observed from Fig. 1 that the tapering function F (Z) changes from −1 to 1, which implies that the linear inhomogeneity of the waveguide should change from the focusing to the defocusing type. The cubic nonlinearity γ (Z) decreases from 1 to 0.5 asymptotically as the distance Z increases. Interestingly, the quintic nonlinearity δ(Z) and the width of the pulse are both decreasing from 1 to 0 in the limit Z → ∞. This indicates that Eq. (3) reduces to the CNLSE with the quadratic term and constant gain for large propagation distance in the sech2 -profile waveguides. A zero width of soliton implies the high-power and ultrashort pulse. When the gain G(Z) varies exponentially with the propagation distance [21], different behaviors of the CQ nonlinearities would occur. We emphasize that the other forms of exact solutions to Eq. (3) can also be constructed when the gain G is Z dependence. In this case, the system parameters can be derived by solving Eq. (7). In Figs. 2 and 3 we plot the evolutions of the self-similar soliton solutions for constant G. Figures 2(a) and 2(b) demonstrate the intensity profiles of the bright and dark self-similar solitons, respectively. Figures 3(a) and 3(b) demonstrate the intensity profiles of the kink and antikink self-similar solitons, respectively. It is observed that the solitons display a moving-compression behavior with the increase of distance. Thus, a key consequence of our exact solution for constant loss is that the pulse can be compressed to any required degree as Z → ∞ while maintaining its sech shape and linear chirp in the tapered CQ sech2 -profile waveguides. These exact solutions are particularly useful in the design of amplifying or attenuating pulse compressors for chirped solitary waves. Note that the center position of the soliton is v [sinh(2Z)/4 + Z/2 + ζ0 ], which is significantly dependent on the velocity v and the width of the solution. From Eq. (4) it follows that there are three cases for the peak amplitude by appropriately choosing E: (i) amplitude grows for increasing Z (as we discussed above), (ii) amplitude
FIG. 2. (Color online) Compressions of the bright and dark self-similar solitons in the sech2 -profile CQ nonlinear waveguide. (a) Intensity profile of the bright soliton given by Eqs. (4) and (8) for Nb = 0.5 and (b) intensity profile of the dark soliton given by Eqs. (4) and (9) for Nd = 2. The other parameters are ζ0 = 0,τ = G = 1,v = 3. 013202-3
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FIG. 3. (Color online) Compressions of the kink and antikink self-similar solitons in the sech2 -profile CQ nonlinear waveguide. Intensity profiles of (a) the kink soliton for ak = −1 and (b) the antikink soliton for ak = 1 given by Eqs. (4) and (10). The other parameters are ζ0 = 0,G = v = ωk = 1.
decreases for increasing Z [for instance, choosing E = sech2 (Z)(Z > 0)], and (iii) constant amplitude (E = E0 ). In cases (i) and (ii), the solutions correspond either to compressing or spreading solitary pulses which maintain a linear chirp. In case (iii), one can obtain the self-similar solution with constant amplitude (which would not go to infinity as Z → ∞). This type of solution may be useful for the stabilization of pulse in the waveguide. A very important aspect of the present issue is the stability of the obtained exact solutions; that is, how do they evolve when disturbed from their analytically given forms? This aspect of the problem can be addressed numerically. We ran simulations by means of the split-step Fourier-transform method with the initial condition U (0,X) in the framework of
2
Eq. (3). In Fig. 4(a), we have added a white noise in the initial pulse U (0,X), i.e., Upert (0,X) = U (0,X)[1 + 0.1R(X)] was taken as initial condition, where R(X) is a random number taking values in (0,1). It is found that the bright soliton can propagate in a stable way against the perturbation with an initial white noise. This may make the possibility for observing the compression of bright soliton in experiments more likely in a graded-index tapered CQ nonlinear waveguide. However, the situation differs markedly for the evolutions of the dark and kink solitons. Figures 4(b) and 4(c) show the direct numerical simulations of the dark and kink solitons in the framework of Eq. (3). It is seen that they are unstable, even without any of perturbation. This implies that the exact soliton solutions are not always stable. Note that the stability of
(a)
3 2
|U|
1 0.5 0 −10
1 0
0 −10
1 0
Z 10
0.5
X
Z
0
10
0
(d)
(c)
1.5
|U|2
2
4
|U|
1
0.5
X
6
(b)
2
|U|2
1.5
2
0 −10
1 0
1 0.5
0 −10
1
0.5
0
Z
X 10
0.5
X
0
Z 10
0
FIG. 4. (Color online) (a) The stable evolution of the bright self-similar soliton given by Eqs. (4) and (8), with the same parameters as in Fig. 2(a) except for v = 0. Panels (b) and (c) show the unstable evolutions of the dark and kink self-similar solitons given by Eqs. (4), (9), and (10), respectively, with the same parameters as in Figs. 2(b) and 3(a) except for v = 0. (d) Evolution of a stable secant pulse in the CQ nonlinear waveguide within the framework of Eq. (3) when σ γ = τ δ = 1,G = 0. 013202-4
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bright soliton is limited to the condition (7), as mentioned. When the parameters in Eq. (7) gradually move from that value, the system becomes more and more unstable. Numerical simulations reveal that the bright soliton could still be stable if the deviation is within 10%. When the system parameters largely deviate from the idealized situation, other forms of solutions would be required to check the structural stability of the waveguides. In this situation, we introduce the secant function U (0,X) = sech(X) as the input pulse and make direct simulations in the framework of Eq. (3) with constant distributed coefficients. Note that a small initial white noise is included in the simulation. The result shows that the pulse can propagate stably with Z, see Fig. 4(d). IV. ASYMPTOTIC COMPACT SELF-SIMILAR SOLUTIONS
In this section, we shall put forward the asymptotic compact self-similar solutions to Eq. (3) when the relative strength of the dispersion is much less than those of the CQ nonlinearities. Within the framework of the CNLSE, we know that the asymptotic compact pulses can be generated in waveguide with the defocusing Kerr nonlinearity [37]. Therefore, we can assume that γ (Z) and δ(Z) are positive functions of Z satisfying σ γ > τ δ|U |2 with σ < 0 such that the effective nonlinearity is self-defocusing. To do this, we use the transformation Q(ζ,ξ ) = Q(Z,ξ )
= A(ξ ) exp iσ μ
Z 0
γ (Z )E(Z ) dZ ,
(Z )
(12)
and then reduce Eq. (5) to c0 A4 − A2 + μ − Kξ 2 = 0,
√
(13)
at |ξ | μ/K, and A(ξ ) = 0 otherwise, when the following relations are satisfied: 1 1, σ γ E
c0 =
2 τ δE , K= (F − ZZ ), σ γ 2σ γ E (14)
where c0 , μ, and K are assumed to be positive constants. Thus, we have τ < 0. Note that the reduction of Eq. (5) to Eq. (13) is essentially the same as that produced by the Thomas-Fermi approximation for the ground-state solution of the one-dimensional Gross-Pitaevskii equation with the harmonic potential [41]. Solving the algebraic Eq. (13), we obtain the following physical solution for A(ξ ): 1 − 1 − 4c0 (μ − Kξ 2 ) 2 A = , (15) 2c0 where μ is determined by the input power Pin =
∞
−∞
|U (0,X)| dX = 2
√ μ/K √ − μ/K
A2 dξ.
(16)
To check the analytical predictions, we consider the simplest case where F , γ , and G are constants. In this case,
by solving Eq. (14) we have G 3 18Kσ γ Z exp
= 9F − G2 3 and δ = c0 γ
σ τ
3
18Kσ γ 2G exp − Z , 9F − G2 3
(17)
(18)
Note that Eq. (17) implies σ (9F − G2 ) > 0, which means that for the defocusing Kerr nonlinearity it is F < G2 /9. The defocusing Kerr nonlinearity can be realized in planar silica semiconductor waveguides, such as ZnSe [42]. The approximate self-similar compact solution for Eq. (3) can be obtained by substituting Eqs. (12), (15), and (17) into Eq. (4). After some analysis, we find that the effective width and amplitude of the compact pulse increase exponentially as exp(GZ/3) and depend on the tapering function F . When the gain and propagation distance are given, the effective width at the asymptotic limit is increasing as F varies from negative to positive. Such a tapering function can be realized, as stated above, by the linear focusing or defocusing lens of the graded-index nonlinear waveguide. When F < 0, the larger the |F |, the smaller the effective width and the larger the pulse’s amplitude; when F > 0, the effective width becomes larger when F increases, while the pulse’s amplitude becomes smaller. In addition, it is possible to control the shape of the output compact pulses by choosing an appropriate tapering parameter, since (0) can vary from zero to infinity as F changes from −∞ to G2 /9. Thus, we may infer that with a strong-enough negative tapering parameter, one can generate a high-power, ultrashort pulse in the CQ nonlinear waveguide. Note that the above compact solution exists when the conditions σ γ > τ δ|U |2 and γ E 1 are valid. From the selfsimilar transformation (4), the second condition in Eq. (14), and Eq. (15), we find that σ γ > τ δ|U |2 is automatically satisfied. At the same time, the condition γ E 1 can be easily satisfied after a short propagation distance due to the exponentially increasing of E and . The analytical predictions are confirmed by direct numerical resolutions of Eq. (3) by use of the standard split-step Fourier method, as shown in Fig. 5. The input pulse is chosen as U (0,X) = exp(−X2 /2)/π 1/4 . It is observed that after about five propagation distance units, the results of numerical simulations and analytical predictions for the pulse’s intensity agree well with each other. From Fig. 5(b) one finds that the general forms of the width and amplitude of the pulse are in good agreement with the analytical results as well. Moreover, one can see that at the central region, the pulse intensity without the quintic nonlinearity is smaller than that with the quintic nonlinearity. Note that the chirp of the asymptotic compact solution is G/6, which is also confirmed by numerical simulations [see Fig. 5(c)] using a fast phase unwrapping algorithm [43]. Next, we consider additional simulations that involve pulses with the same input power but with different input profiles. We choose a hyperbolic secant pulse, U (0,X) = sech(X)/21/2 . The results of numerical simulations and analytical predictions for the general form of the pulse agree well with each other as well, see Fig. 6. However, we find that this agreement is not
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equation for the paraxial optical pulse in such a medium is
2
10
0
1
10
Intensity
0
i
10
Width
10
Amplitude
(a)
0
2 Z
4
6
(b) 0
10
0
−5
2
4
Z
10
6
20 Phase
0
−20
−10
10
(c)
−40 −20
−10
0 X
10
20
−20
−10
0 X
10
20
FIG. 5. (Color online) (a) The self-similar evolution of the compact pulse in the CQ nonlinear waveguide, starting with the Gaussian input pulse U (0,X) = exp(−X2 /2)/π 1/4 . From top to bottom, the propagation distance is Z = 6,5.6, and 5, respectively. (b) The pulse’s width and amplitude as functions of Z. (c) The phase (phase offset is ignored) of the pulse at propagation distance Z = 6. Here, and in the following figures, the solid lines and circles represent results of the direct numerical simulations of Eq. (3) and the analytical predictions, respectively. The dotted lines in (a) show the pulse intensity at Z = 6 when the quintic nonlinearity is absent. The parameters are γ = G = K = 1,F = G2 /60, and c0 = 0.05 such that μ = 0.8069.
as good as that in Fig. 5 after the same units of propagation distance. Therefore, we may infer that the initial Gaussian input pulse leads to the generation of asymptotic compact similariton more convergence with the analytical profile in the CQ nonlinear waveguide.
V. SELF-SIMILAR SOLUTIONS IN THE QUINTIC MODEL
In this section, we consider the self-similar solutions propagating in a pure quintic nonlinear model. The governing
1 ∂ 2U 1 iG ∂U + U. + F X2 U − τ δ|U |4 U = 2 ∂Z 2 ∂X 2 2
(19)
Equation (19) can also describe optical pulse propagating in nonlinear optical media with power-law nonlinearity [44]. It is known that physically, various materials, including semiconductors, exhibit power-law nonlinearity. Recently, a generic model for the quintic nonlinearity has been realized in a centrosymmetric nonlinear medium doped with resonant impurities in the limit of a large light carrier frequency detuning from the impurity resonance [34]. In BECs, the pure quintic nonlinearity can be derived by setting the s-wave scattering length as (t) to zero via the Feschbach resonance technique [41,45,46]. The pure quintic nonlinearity also appears in general NLSE-type systems near the transition from supercritical to subcritical bifurcations [47], pattern formation [48], and dissipative solitons [49]. Substituting the profile (4) back into Eq. (19) and requiring the first, third, and last conditions in Eq. (7) to be satisfied, one obtains a partial differential equation: iQζ + 12 Qξ ξ − τ |Q|4 Q = 0, which has the following exact bright self-similar solution for τ < 0 [44]:
√ 3 sech[ −τ (ξ − vζ )] exp(i ), (20) Q(ξ,ζ ) = 8 where = vξ − (v 2 + τ/4)ζ /2. When the tapering function F (Z) is given by Eq. (11), one again has = sechn (Z). The solution in this situation is plotted in Fig. 7(a), which also displays a moving-compression behavior with the increase of distance. The corresponding numerical evolution of the bright similariton profile is shown in Fig. 7(b), where a 10% withe noise is included. It is observed that stable bright similariton can be supported by the quintic nonlinear media. Asymptotic compact self-similar solutions to Eq. (19) can also be found when considering that the relative strength of the dispersion is much less than that of the quintic nonlinearity. We still assume that δ(Z) is a positive function of Z and employ the transformation (4) with Q(ζ,ξ ) = Z ) A(ξ ) exp[−iμ 0 E(Z dZ ], and then Eq. (19) reduces to
(Z )
2
Amplitude
10
Width
(a) 0
10
c1 A4 − μ − Kξ 2 = 0,
(21) √ at |ξ | −μ/K, and A(ξ ) = 0 otherwise, when the following relations are satisfied:
0
10
1
10
0
2
Z
4
6
Intensity
(b) 0
10
0
2
−5
10
Z
4
Phase
20 0
−20
−10
(c)
10
−20
−10
0 X
10
20
−40 −20
−10
0 X
1 1, τ δE 2
6
10
20
FIG. 6. (Color online) The same as in Fig. 5 except that the input pulse is U (0,X) = sech(X)/21/2 , i.e., the hyperbolic secant input pulse.
c1 =
τ δE ,
K=
2 (F − ZZ ), 2E
(22)
where c1 and μ are positive constants. Thus, we have K < 0. We obtain the following physical solution for A(ξ ) by solving Eq. (21): c1 (μ + Kξ 2 ) 2 A = , (23) c1 with μ being determined by the input power. To check the analytical predictions, we consider the simplest case where F and G are constants. In this case, by
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(b) 3
|U|2
2 1
0 −10
1 0
0.5
X 10
Z
0
FIG. 7. (Color online) (a) Compressions of the bright self-similar soliton given by Eqs. (4) and (20) in the sech2 -profile nonlinear media. (b) The stable evolution of the bright self-similar soliton with initial zero velocity. The parameters are ζ0 = 0, − τ = G = 1,v = 3.
solving Eq. (22) we have 18K G 3 Z exp
= 9F − G2 3 and δ=
c1 τ
3
(24)
18K 2G Z , exp − 9F − G2 3
(25)
From Eq. (24) we find that F < G2 /9; from Eq. (25) we get τ δ > 0. This implies that the asymptotic compact pulses can be generated in nonlinear media with the defocusing quintic nonlinearity, which is confirmed by the direct numerical simulations of Eq. (19). Since the self-defocusing nonlinearity is possible in semiconductor waveguides, Eq. (19) may be found in semiconductor waveguides doped with quantum dots by considering the theory in Ref. [34]. Note that the effective width and amplitude of the compact pulse increase exponentially as exp(GZ/3) and depend on the tapering function F . A high-power, ultrashort pulse may also be generated in the quintic nonlinear media with a strong-enough negative tapering parameter F . Further, the term δE 2 varies 2
(a) Width
10
10 Intensity
1
10
0
0
10
0
2 Z
4
(b)
0
−1
10
10
−2
0
1
2 Z
3
4
6 4 2 0 −2 −4
Phase
10
−3
10
−4
10
Amplitude
10 1
−5
0 X
5
(c) −5
0 X
5
FIG. 8. (Color online) (a) The evolution of the compact selfsimilar solution in the nonlinear media, starting with the Gaussian input pulse U (0,X) = exp(−X2 /2)/π 1/4 . From top to bottom, the propagation distance is Z = 4,3.8,3.6, and 3.4, respectively. (b) The solution’s width and amplitude as functions of Z. (c) The phase (phase offset is ignored) of the pulse at propagation distance Z = 4. The parameters are G = −K = 1,F = G2 /60, and c1 = 0.5 such √ that μ = 2/π .
as ∼exp(4GZ/3); thus, the condition τ δE 2 1 can be easily satisfied after a short propagation distance. The analytical predictions are confirmed by direct numerical simulations of Eq. (19), as shown in Fig. 8, where the input pulse is U (0,X) = exp(−X2 /2)/π 1/4 . It is observed that the general forms of the intensity profile, width, amplitude, and phase are in good agreement with the analytical results.
VI. CONCLUSIONS AND DISCUSSION
In conclusion, with the help of a lens-type transformation, we have mainly investigated the propagation properties of self-similar optical waves in a tapered CQ nonlinear waveguides. Various types of self-similar solutions were found by appropriately choosing the relations between the distributed coefficients. First, we reduced Eq. (3) to the CQNLSE with constant coefficients and obtained the exact self-similar solutions which describe the propagation of brightshaped solitons, dark-shaped solitons, kink-shaped solitons, and antikink-shaped solitons. The stability of the solutions was examined by the numerical simulations such that stable bright solitons were found. Second, we reduced Eq. (3) to a simple algebraic equation and obtained the asymptotic optical waves by employing a direct ansatz. These waves possessed linear chirps and can propagate self-similarly. The possibility of controlling the shape of output asymptotic optical waves was demonstrated. The analytical results were confirmed by numerical simulations. Finally, we studied the generation and propagation properties of self-similar solutions in the quintic nonlinear media, which may be found in semiconductor waveguides (such as ZnSe) doped with quantum dots. We now give a suitable experimental protocol to generate the obtained asymptotic compact pulses in realistic waveguides. We consider a 10-μm-thick planar silica waveguide, such as ZnSe [42], in which the input pulse is confined in the y direction. The refractive index in Eq. (1) can be graded with n0 ≈ 2.7,n1 ≈ 0.1 cm−2 , and n2 ≈ −7 × 10−14 cm2 /W near 532 nm [16,35,42,50], which lead to w0 ≈ 34 μm and the diffraction length LD ≈ 3.7 cm. In this situation, the required peak intensity is about 550 MW/cm2 (corresponding to the six propagation distance units in Figs. 5 and 6), translating into input power levels ∼100 W. Such power levels can be realized by employing a frequency doubled Nd:YAG laser near 532 nm [42,50].
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Besides nonlinear optics, amplification of matter waves is also found in BECs. The demonstration in Ref. [51] that matter waves can be amplified and maintain their phase is an exciting development in the rapidly growing field of matter-wave optics. Amplification of light and atoms in a BEC has been reported in Ref. [52]. Therefore, the results obtained in this paper could also be applied to BECs.
[1] G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, UK, 1996); G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, New York, 2007). [2] P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000); V. I. Kruglov, M. K. Olsen, and M. J. Collett, ibid. 72, 033604 (2005). [3] L. Wu, J. F. Zhang, and L. Li, New J. Phys. 9, 69 (2007); L. Wu, L. Li, and J.-F. Zhang, Phys. Rev. A 80, 013617 (2009). [4] T. Xu, B. Tian, L. L. Li, X. Lu, and C. Zhang, Phys. Plasmas 15, 102307 (2008). [5] V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992). [6] C. R. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992). [7] T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, Opt. Lett. 23, 268 (1998). [8] Christophe Finot, Guy Millot, Cyril Billet, and John Dudley, Opt. Express 11, 1547 (2003). ¨ Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. [9] F. O. Rev. Lett. 92, 213902 (2004). [10] S. Chen and J. M. Dudley, Phys. Rev. Lett. 102, 233903 (2009). [11] J. D. Moores, Opt. Lett. 21, 555 (1996). [12] V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007). [13] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85, 4502 (2000); ,JETP Lett. 72, 89 (2000); ,IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002). [14] V. N. Serkin and T. L. Belyaeva, JETP Lett. 74, 573 (2001); ,Quantum Electron. 31, 1007 (2001). [15] V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003). [16] S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006); ,Opt. Express 15, 2963 (2007). [17] J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, Opt. Commun. 263, 328 (2006). [18] S. Kumar and A. Hasegawa, Opt. Lett. 22, 372 (1997). [19] Y. Ozeki and T. Inoue, Opt. Lett. 31, 1606 (2006). [20] M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000). [21] V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000). [22] V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, J. Opt. Soc. Am. B 19, 461 (2002); V. I. Kruglov and J. D. Harvey, ibid. 23, 2541 (2006). [23] J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007). [24] C. Finot and G. Millot, Opt. Express 13, 5825 (2005); ,13, 7653 (2005).
ACKNOWLEDGMENTS
The authors thank Professor Lu Li at Shanxi University for his useful suggestions. This work was supported by the National Natural Science Foundation of China under Grants No. U1230108, No. 11374266, and No. 11175158.
[25] S. Chen, L. Yi, D. S. Guo, and P. Lu, Phys. Rev. E 72, 016622 (2005). [26] C. Q. Dai, Y. Y. Wang, and X. G. Wang, J. Phys. A: Math. Theor. 44, 155203 (2011); J. D. He and J. F. Zhang, ibid. 44, 205203 (2011). [27] J. Herrmann, Opt. Commun. 87, 161 (1992); C. De Angelis, IEEE J. Quantum Electron. 30, 818 (1994). [28] F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, J. Non-Cryst. Solids 274, 232 (2000); M. J. Paz-Alonso and H. Michinel, Phys. Rev. Lett. 94, 093901 (2005). [29] M. Brunner, J. Dobnikar, H. H. vonGrunberg, and C. Bechinger, Phys. Rev. Lett. 92, 078301 (2004); E. L. Falcao-Filho, C. B. de Araujo, and J. J. Rodrigues, Jr., J. Opt. Soc. Am. B 24, 2948 (2007). [30] B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, Phys. Rev. Lett. 73, 597 (1994). [31] C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, J. Opt. Soc. Am. B 19, 369 (2002). [32] K. Senthilnathan, Qian Li, K. Nakkeeran, and P. K. A. Wai, Phys. Rev. A 78, 033835 (2008); J. F. Zhang, Q. Tian, Y. Y. Wang, C. Q. Dai, and L. Wu, ibid. 81, 023832 (2010). [33] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1973), Table 12-7 and p. 266; G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980). [34] S. A. Ponomarenko and S. Haghgoo, Phys. Rev. A 81, 051801(R) (2010). [35] L. Li, X. Zhao, and Z. Xu, Phys. Rev. A 78, 063833 (2008). [36] S. A. Ponomarenko and G. P. Agrawal, Opt. Lett. 32, 1659 (2007). [37] L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, Opt. Express 16, 6352 (2008); L. Wu, L. Li, and J. F. Zhang, Phys. Rev. A 78, 013838 (2008); L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, ibid. 78, 053807 (2008). [38] Kh. I. Pushkarov et al., Opt. Quantum Electron. 11, 471 (1979); Kh. I. Pushkarov and D. I. Pushkarov, Rep. Math. Phys. 17, 37 (1980); D. I. Pushkarov and S. Tanev, Opt. Commun. 124, 354 (1996); S. Tanev and D. I. Pushkarov, ibid. 141, 322 (1997); Pavel Honzatko, ibid. 127, 363 (1996); R. Y. Hao, L. Li, Z. H. Li, R. C. Yang, and G. S. Zhou, ibid. 245, 383 (2005). [39] Zhenya Yan and V. V. Konotop, Phys. Rev. E 80, 036607 (2009). [40] Y. M. Chen, S. H. Ma, and Z. Y. Ma, Chin. Phys. B 21, 050510 (2012). [41] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). [42] S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991). [43] M. A. Schofield and Y. M. Zhu, Opt. Lett. 28, 1194 (2003). [44] A. W. Snyder and D. J. Mitchell, Opt. Lett. 18, 101 (1993).
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[45] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003). [46] V. A. Brazhnyi, V. V. Konotop, and L. P. Pitaevskii, Phys. Rev. A 73, 053601 (2006). [47] E. A. Kuznetsov, J. Exp. Theor. Phys. 89, 163 (1999); D. Agafontsev, F. Dias, and E. A. Kuznetsov, JETP Lett. 87, 667 (2008). [48] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
[49] J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000). [50] G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, Opt. Lett. 16, 156 (1991). [51] S. Inouye, T. Pfau, S. Gupta, A. P. Chikkatur, A. G¨orlitz, D. E. Pritchard, and W. Ketterle, Nature 402, 641 (1999). [52] S. Inouye, R. F. L¨ow, S. Gupta, T. Pfau, A. G¨orlitz, T. L. Gustavson, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 85, 4225 (2000).
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