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Self-similar soliton-like beam generation and propagation in inhomogeneous coupled optical fiber media system
Optik 124 (2013) 7040–7043
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Optik journal homepage: www.elsevier.de/ijleo
Self-similar soliton-like beam generation and propagation in inhomogeneous coupled optical fiber media system Hongjuan Li b , Jinping Tian a,b,∗ , Lijun Song b a b
Computer Center of Shanxi University, Taiyuan, Shanxi 030006, PR China College of Physics and Electronics Engineering of Shanxi University, Taiyuan, Shanxi 030006, PR China
a r t i c l e
i n f o
Article history: Received 18 January 2013 Accepted 28 May 2013
PACS: 42.81.Dp 42.65.Jx 42.65.Tg 42.79.Ry
a b s t r a c t In this paper, we present a theoretical model of coupled nonlinear Schrödinger equation with variable coefficients describing the self-similar soliton-like optical pulse propagating in the coupled inhomogeneous optical fiber system. The exact self-similar bright soliton-like solution and its propagation characteristics under different perturbation conditions are studied in detail. It is found that, within the range of appropriate parameters, stable propagation of the calculated self-similar bright soliton-like optical pulse can be obtained. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Optical communication Self-similar solution Optical soliton Chirp
1. Introduction The all-optical communication link is going to play a vital role in the rapidly growing information technology areas, in which optical communication using optical soliton has become the subject of intense study. In recent years, the study of nonlinear fiber optics has attracted much attention and has played an important role in the development of optical communication systems and optical fiber lasers. Among these wide researches, one of the attentions has been focused on the self-similar propagation of optical pulses in an optical fiber amplifier, a dispersion-decreasing optical fiber, and a laser resonator [1], because self-similarity is a fundamental physical property that has been studied extensively in many areas of physics. In particular, the self-similar evolution of a nonlinear wave implies that the wave profile remains unchanged and its amplitude and width simply scale with time or propagation distance. Thus the study of self-similar solutions of the relevant nonlinear differential equations has become a topic of growing issue because of their ubiquity in the description of complicated phenomena, including the scaling properties of turbulent flow, the formation of fractals in nonlinear systems, and the wave collapse in hydrodynamics
∗ Corresponding author at: Computer Center of Shanxi University, Taiyuan, Shanxi 030006, PR China. Tel.: +86 0351 7010755; fax: +86 0351 7010755. E-mail address: [email protected] (J. Tian). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.137
[see, for example, Ref. [2] and references therein]. Also self-similar soliton-like or parabolic pulses have been observed experimentally in rare-earth-doped fiber [3–5] and Raman amplifiers [3,6]. In [3], the authors present an all-optical regeneration technique based on spectral filtering of self-similar pulses. We have known that self-similar pulses represent a new class of solution to the nonlinear Schrödinger (NLS) equation with gain [7]. Recently, coupled NLS equations have been extensively studied since the self-trapping of light beams in a slow Kerr-type nonlinear medium is well characterized by them. Meanwhile, coupled NLS equations also describe many interesting and important physical phenomena, including multi-component Bose–Einstein condensates (BECs), temporal incoherent solitons, and nonlinear interactions of optical waves with different polarizations or wavelength [9–20]. It is also of interest to note that the system descried by the general NLS equation with varying coefficients can lead to exact sech- and tanh-shaped self-similar pulses [1,8] in describing inhomogeneous optical fiber system and dispersion management system. In addition, when two polarization components of a wave interact nonlinearly at some central frequency or for the transmission of several channels by using of wavelength division multiplexing (WDM), at least two optical fields are to be transmitted and the system is governed by coupled NLS (CNLS) equations. When considering inhomogeneous media, we must discuss the CNLS equations with varying coefficients [21].
H. Li et al. / Optik 124 (2013) 7040–7043
In this letter, we present exact analytical self-similar solitonlike solutions of CNLS equations with variable coefficients. And the paper is organized as follows. In Section 2 the model and its solution are given analytically at first, and then the detailed numerical evolution is carried out. Section 3 is the conclusions.
7041
Analysis reveals that under this self-consistent condition and by setting = 1, the equations can be readily solved to obtain the following two types of results. For the case of c = / 0, we have W () = W0 exp(c),
() =
B0 2c
0 W0 −
2. Models and analysis
Pj () = Pj0 exp(−c), exp(c) +
Bj () = B0 exp(−c)
(10)
B0 sinh(c) c
and for the case of c = 0, we have The generally proposed two-coupled NLS equation with variable coefficient can be written as:
D(Z) Aj,TT + (Z)Aj |Ak |2 + i (Z)Aj = 0, 2 n
iAj,Z +
(j = 1, 2), (1)
k=1
where j = 1, 2 and A(Z, T ) is the electric field envelope. D(Z) > 0, (Z) > 0 and (Z) are functions of distance Z, and represent dispersion or diffraction, Kerr nonlinearity and gain (loss) respectively. = ±1, with upper (lower) sign corresponds to a self-focusing (self-defocusing) nonlinearity of the waveguide and in this paper we only consider the case of = 1. The variable T represents the local retarded time in the case of pulse propagation. To simplify the analytical process, we begin our analysis by making the following transformations
Z
t = a(Z)T ≡ T exp −
D(x)C(x)dx ,
0
Aj (Z, T ) = a(Z)
|Uk |2 + M()t 2 Uj + iF()Uj = 0,
F() =
2a4 (Z)D(Z)
(Z) −
˚j (, t) =
(j = 1, 2),
(4)
,
(5)
(Z)DZ (Z) − Z (Z)D(Z) 1 D(Z)C(Z) + 2 2(Z)D(Z)
.
(6)
t − () W ()
exp{i[˚j (, t)]}
1 2 ct + Bj ()t + j () 2
(7)
(8)
where the chirp effect can be expressed as −d˚j /dt = −ct + Bj () and the coefficient c is related to the wave front curvature, it is also a measure of the linear chirp imposed on the self-similar wave. Bj () and j () represent phase shift and phase, respectively, and Pj () is the pulse amplitude, W () is the pulse width. Substituting Eq. (7) with (8) into Eq. (4), collecting the similar terms, and separating the real and imaginary parts, we can obtain some lengthy but straightforward first-order differential equations for the pulse amplitude, width, phase shift and phase. The equations are self-consistent only if the following conditions are satisfied: M() =
1 2 c , 2
1 2
1 W02
exp
1 i
2
ct 2 + Bj ()t + j ()
exp(−2c) + j0 ,
t − () W ()
(12)
exp{i[Bj ()t + j ()]}
− B02
+ j0 ,
(13)
where
According to Ref. [21], the symmetry analysis of Eq. (4) indicates that a self-similar solution to this equation ought to be sought in the form Uj (, t) = Pj ()j
(11)
2
2 Pj0 = W0−2
(14)
j=1
(Z) + D(Z)C 2 (Z)
1 a2 (Z)D(Z)
t − () W ()
and when c=0, we have
j () =
where C(Z) is a frequency chirp parameter, and CZ
() = B0 + 0
4c
(3)
k=1
M() = −
j () =
B02 − W0−2
n
1 Uj,tt + Uj 2
Uj (, t) = Pj () sec h
Then Eq. (1) can be written as iUj, +
Uj (, t) = Pj () sec h
D(Z) i U (, t) exp C(Z)T 2 . 2 (Z) j
Bj () = B0 ,
where W0 , Pj0 , B0 , and 0 are integral constants describing the initial values of the corresponding parameters. At last, by writing the self-similar wave profile as j (, t) = j ( ) with similarity variable = (t − ())/W (), we can further obtain self-similar soliton-like / 0, we have solutions to Eq. (4) for the two cases. When c =
(2)
0
Pj () = Pj0 ,
Z
D(x)a2 (x)dx,
=
W () = W0 ,
F() =
1 c. 2
(9)
Here the pulse width, pulse amplitude, pulse phase shift and the center of the self-similar solutions are specified by Eqs. (10) and (11). The initial phase can be set as j0 = 0 for the aim of simplicity. Meanwhile, one can find that the self-similar waves are completely determined by the pulse amplitude Pj0 , the width W0 , the phase shift coefficient B0 , the beam center 0 , and the linear phase chirp c. If we define the pulse energy as Ej = Pj ()W () = Pj0 W0 , we can find that it is a constant. This means that the pulse energy is always conservative, namely, the system itself is a conservative one. In addition, from Eq. (10), one can find that the amplitudes of the selfsimilar pulses will increase exponentially along the propagating distance when c < 0, because of the exponentially increasing nature of exp(−c) and the widths will be compressed. When c > 0, because of the exponentially decreasing nature of exp(−c), the amplitudes of the self-similar waves will decrease exponentially and their widths will be broadened. From Eq. (11) one can also find that when c = 0, the self-similar waves will propagate steadily in the optical media with their profiles not changed. Consequently, combining the equations obtained, one can further get the exact self-similar soliton-like solution to Eq. (1). To the best of our knowledge, the above self-similar results for Eq. (1) have not been discussed before. These characteristics might have potential applications, e.g., it may provide an effective approach to compress or amplify optical pulse. To investigate the evolution properties of the exact self-similar soliton-like solution propagating in the system described by Eq. (1), in the following part, we will design the optimal system by appropriately choosing the different forms of the distributed parameters according to the specific problem to obtain proper parameter condition under which the self-similar soliton-like solution can propagate steadily in the proposed system. At first, from
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H. Li et al. / Optik 124 (2013) 7040–7043
Fig. 1. Evolution when = 1, = 0.005, C0 = 1, d0 = 1, 0 = 1, B0 = 0.2, and (a) c = 0.005, W0 = 1, P10 = 0.7; (b) c = −0.005, W0 = 2, P10 = 0.35; (c) c = 0, W0 = 1, P10 = 0.7.
Fig. 2. Evolution when 10% of white noise is added, the parameters are respectively the same as those in Fig. 1.
the expressions of M() and F(), one can obtain a group of restric-
Z
−1
tive conditions as C(Z) = ca(Z)2 and a(Z)2 = (C0 + 2c 0 D(x)dx) , where C0 is a real integral constant. Because the solutions include two distributed functions D(Z) and (Z), thus by choosing the form most approximate to the real conditions, one can explain the various soliton control systems or dispersion management systems. Here as an example, we consider a system with variable group velocity dispersion and nonlinear Kerr effects respectively as: D(Z) = d0 cos2 ( Z) exp( Z),
(Z) = 0 cos2 ( Z) exp( Z),
(15)
where d0 , 0 , , and are arbitrary constants. Then the gain (loss) coefficient can be obtained as:
(Z) = cD(Z)a(Z)2 = cD(Z)(C0 + 2c
−1
Z
D(x)dx)
(16)
Fig. 3. Evolution when 10% amplitude perturbation is added, the parameters are respectively the same as those in Fig. 1.
0
From Eq. (16) we can see that if the linear chirp parameter c=0, we will have (Z) = 0. Therefore, we may think that the linear chirp results from linear gain (loss) and this means that we can compensate the linear gain (loss) by properly choosing the linear chirp in the real optical communication system. By using the splitstep Fourier method and the Runge–kutta method, we give the evolution plot in Fig. 1(a–c) respectively for the cases of c = 0.005, c = −0.005, and c = 0. The other parameters are given in the figure caption. It can be found that the pulses can propagate stably in the system described by Eq. (1) and the evolution properties are in accordance with the theoretical expectation very well. In this paper, we only give the figures for A1 (Z, T ), the figures for A2 (Z, T ) are similar. It is worth noting that the existence of the self-similar solitonlike solutions depends on the particular nonlinear and dispersive features of the medium. However, in real application, it is difficult to achieve exact balances between them. Therefore, the study of stability of the pulse propagation for various perturbations is very necessary. Here, we first add 10% white noise to the pulse to investigate its evolution properties, and the result is shown in Fig. 2. One can find that there are no clear changes compared to Fig. 1, except for some small oscillation attached to the solitons’ amplitude. To further study the pulse stability, secondly, we add 10% amplitude perturbation to the pulse and the result is shown in Fig. 3. It can be found that the pulse can also propagate stably without clear changes in the pulse shape. So we may infer that the evolution of the pulse is not sensitive to the perturbed conditions. This may make the soliton control technique more realistic and leave scope for more physical explanations and applications in the future. So far, we have investigated the cases of one single pulse propagating in the given system. However, another effect on the pulse propagation is the interaction between two adjacent solitons propagating simultaneously. For the purpose of understanding the effect
Fig. 4. Interaction of two equal amplitude pulses with initial pulse separation S = 10 when W0 = 1, P10 = 0.7and (a) c = 0.005, (b) c = 0.01, (c) c = 0.03. The other parameters are the same as in Fig. 1.
of frequency chirp parameter c on two neighboring solitons, we investigate their transmission properties. Here the spacing of beam center is defined as S and the amplitude ratio of the two pulses is defined as k. Fig. 4 gives the contour plot of two neighboring pulses for the case of c > 0when the spacing of beam center S = 10. We can see that, increasing of chirp parameter c will lead to the two pulses repelling each other except for the changes of pulse intensity. We have known that one can control the interaction of the two adjacent pulses through changing the spacing of beam center and the amplitude ratio of the two pulses. Therefore, using the parameters given in Fig. 1, we give the further numerical simulations. Results show that when the spacing of beam center changes from S = 4 to S = 7, the interaction between two pulses is reduced and further increase the spacing to S = 10, the interaction between two pulses will disappear within the propagation distance of more than one hundred normalized dispersion lengths. The results are shown in Fig. 5. However, this will limit the bit rate of the soliton communication system, so we must further investigate more suitable conditions which can not only prevent interaction between neighboring solitons but also ensure the bit rate of the soliton communication system. So in the simulation, we still set the spacing of beam center of two pulses as S = 7, but adjust the amplitude ratio
H. Li et al. / Optik 124 (2013) 7040–7043
7043
theoretical significance to the research and application of optical fiber communication system. Acknowledgment This work was supported in part by the Provincial Natural Foundation of Shanxi (Grant no. 2012011010-4). References
Fig. 5. Evolution of two pulses with same amplitudes (k = 1) and different spacing of beam center. (a) c = 0, S = 4, W0 = 1, P10 = 0.7; (b) c = 0, S = 5, W0 = 1, P10 = 0.7; (c) c = 0, S = 7, W0 = 1, P10 = 0.7; (d) c = 0, S = 10, W0 = 1, P10 = 0.7; (e) c = 0.005, S = 10, W0 = 1, P10 = 0.7; (f) c = –0.005, S = 10, W0 = 2, P10 = 0.35; the other parameters are the same as those in Fig. 1.
Fig. 6. Evolution of two pulses with spacing of beam center S = 7 and amplitude ratio k = 1.2 when (a) c = 0.005, W0 = 1, P10 = 0.7; (b) c = –0.005, W0 = 2, P10 = 0.35; (c) c = 0, W0 = 1, P10 = 0.7; the other parameters are the same as those in Fig. 1.
of two pulses as k = 1.2. Simulations show that the two pulses can also propagate stably without interaction or shape changing no matter the intensity of pulses is increasing or decreasing along the propagation direction. The results are shown in Fig. 6. Therefore, we may infer that changing the amplitude ratio of two pulses can reduce the interaction between two pulses with smaller spacing of the beam center of two pulses, and it is more suitable for reducing the interaction between the neighboring solitons. This result is advantageous to increase the information bit rate in optical soliton communications. 3. Conclusion In conclusion, firstly, we obtained the self-similarity solution of the variable coefficient coupling nonlinear Schrödinger equation. The stability analysis and numerical evolution are given to study the propagation properties of the self-similar soliton-like pulses for the given system. Results show that within the proper parameter conditions, the pulses can propagate stably and the outer white noises and amplitude perturbations have no clear effect on the pulses. Also, by choosing proper amplitude ratio and proper beam center spacing of two pulses, the interaction between two adjacent pulses can be well restrained. Our result may have
[1] S.H. Chen, L. Yi, D.S. Guo, P.X. Lu, Self-similar evolutions of parabolic Hermite–Gaussian, and hybrid optical pulses: universality and diversity, Phys. Rev. E 72 (2005) 016622–016626. [2] T.S. Raju, P.K. Panigrahi, K. Porsezian, Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain, Phys. Rev. E 72 (2005) 046612–046617. [3] C. Finot, S. Pitois, G. Millot, Regenerative 40 Gbit/s wavelength converter based on similariton generation, Opt. Lett. 30 (2005) 1776–1778. [4] M.E. Fermann, V.I. Kruglov, B.C. Thomsen, J.M. Dudley, J.D. Harvery, Self-similar propagation and amplification of parabolic pulses in optical fibers, Phys. Rev. Lett. 84 (2000) 6010–6013. [5] J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H.-J. Fuchs, E.-B. Bley, H. Zellmer, A. Tünnermann, High-power femtosecond Yb-doped fiber amplifier, Opt. Lett. 10 (2002) 628–638. [6] C. Finot, G. Millot, C. Billet, J.M. Dudley, Experimental generation of parabolic pulses via Raman amplification in optical fiber, Opt. Lett. 11 (2003) 1547–1552. [7] V.I. Kruglov, A.C. Peacock, J.M. Dudley, J.D. Harvey, Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers, Opt. Lett. 25 (2000) 1753–1755. [8] S.H. Chen, L. Yi, Chirped self-similar solutions of a generalized nonlinear Schrödinger equation model, Phys. Rev. E 71 (2005) 016606–016609. [9] G.Q. Chang, H.G. Winful, A. Galvanauskas, T.B. Norris, Incoherent selfsimilarities of the coupled amplified nonlinear Schrödinger equations, Phys. Rev. E 73 (2006) 016616–016619. [10] A. Ankiewicz, W. Krolikowski, M.N. Akhmediev, Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions, Phys. Rev. E 59 (1999) 6079–6087. [11] H. Buljan, M. Srgev, M. Soljacic, N.K. Efremidis, D.N. Christodoulides, Whitelight solitons, Opt. Lett. 28 (2003) 1239–1241. [12] M. Soljacic, K. Steiglitz, S.M. Sears, M. Segev, M.H. Jakubowski, R. Squier, Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 90 (2003) 254102–254105. [13] T. Kanna, M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 86 (2001) 5043–5046. [14] T.S. Ku, M.F. Shih, A.A. Sukhorukov, Y.S. Kivshar, Coherence controlled soliton interactions, Phys. Rev. Lett. 94 (2005) 063904–063907. [15] A.A. Sukhorukov, N.N. Akhmediev, Coherent and incoherent contributions to multisoliton complexes, Phys. Rev. Lett. 83 (1999) 4736–4739. [16] T. Kanna, E.N. Tsoy, N. Akhmediev, On the solution of multicomponent nonlinear Schrodinger equations, Opt. Lett. 330 (2004) 224–229. [17] E.P. Bashkin, A.V. Vagov, Instability and stratification of a two-component Bose–Einstein condensate in a trapped ultra-cold gas, Phys. Rev. B 56 (1997) 6207–6212. [18] A. Hasegawa, Envelope soliton of random phase waves, Phys. Fluids 20 (1977) 2155–2156. [19] A. Hasegawa, Self-confinement of multimode optical pulse in a glass fiber, Opt. Lett. 5 (1980) 416–417. [20] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001. [21] S.A. Ponomarenko, G.P. Agrawal, Do solitonlike self-similar waves exist in nonlinear optical media? Phys. Rev. Lett. 97 (2006) 013901–013904.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Self-similar soliton-like beam generation and propagation in inhomogeneous coupled optical fiber media system Hongjuan Li b , Jinping Tian a,b,∗ , Lijun Song b a b
Computer Center of Shanxi University, Taiyuan, Shanxi 030006, PR China College of Physics and Electronics Engineering of Shanxi University, Taiyuan, Shanxi 030006, PR China
a r t i c l e
i n f o
Article history: Received 18 January 2013 Accepted 28 May 2013
PACS: 42.81.Dp 42.65.Jx 42.65.Tg 42.79.Ry
a b s t r a c t In this paper, we present a theoretical model of coupled nonlinear Schrödinger equation with variable coefficients describing the self-similar soliton-like optical pulse propagating in the coupled inhomogeneous optical fiber system. The exact self-similar bright soliton-like solution and its propagation characteristics under different perturbation conditions are studied in detail. It is found that, within the range of appropriate parameters, stable propagation of the calculated self-similar bright soliton-like optical pulse can be obtained. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Optical communication Self-similar solution Optical soliton Chirp
1. Introduction The all-optical communication link is going to play a vital role in the rapidly growing information technology areas, in which optical communication using optical soliton has become the subject of intense study. In recent years, the study of nonlinear fiber optics has attracted much attention and has played an important role in the development of optical communication systems and optical fiber lasers. Among these wide researches, one of the attentions has been focused on the self-similar propagation of optical pulses in an optical fiber amplifier, a dispersion-decreasing optical fiber, and a laser resonator [1], because self-similarity is a fundamental physical property that has been studied extensively in many areas of physics. In particular, the self-similar evolution of a nonlinear wave implies that the wave profile remains unchanged and its amplitude and width simply scale with time or propagation distance. Thus the study of self-similar solutions of the relevant nonlinear differential equations has become a topic of growing issue because of their ubiquity in the description of complicated phenomena, including the scaling properties of turbulent flow, the formation of fractals in nonlinear systems, and the wave collapse in hydrodynamics
∗ Corresponding author at: Computer Center of Shanxi University, Taiyuan, Shanxi 030006, PR China. Tel.: +86 0351 7010755; fax: +86 0351 7010755. E-mail address: [email protected] (J. Tian). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.137
[see, for example, Ref. [2] and references therein]. Also self-similar soliton-like or parabolic pulses have been observed experimentally in rare-earth-doped fiber [3–5] and Raman amplifiers [3,6]. In [3], the authors present an all-optical regeneration technique based on spectral filtering of self-similar pulses. We have known that self-similar pulses represent a new class of solution to the nonlinear Schrödinger (NLS) equation with gain [7]. Recently, coupled NLS equations have been extensively studied since the self-trapping of light beams in a slow Kerr-type nonlinear medium is well characterized by them. Meanwhile, coupled NLS equations also describe many interesting and important physical phenomena, including multi-component Bose–Einstein condensates (BECs), temporal incoherent solitons, and nonlinear interactions of optical waves with different polarizations or wavelength [9–20]. It is also of interest to note that the system descried by the general NLS equation with varying coefficients can lead to exact sech- and tanh-shaped self-similar pulses [1,8] in describing inhomogeneous optical fiber system and dispersion management system. In addition, when two polarization components of a wave interact nonlinearly at some central frequency or for the transmission of several channels by using of wavelength division multiplexing (WDM), at least two optical fields are to be transmitted and the system is governed by coupled NLS (CNLS) equations. When considering inhomogeneous media, we must discuss the CNLS equations with varying coefficients [21].
H. Li et al. / Optik 124 (2013) 7040–7043
In this letter, we present exact analytical self-similar solitonlike solutions of CNLS equations with variable coefficients. And the paper is organized as follows. In Section 2 the model and its solution are given analytically at first, and then the detailed numerical evolution is carried out. Section 3 is the conclusions.
7041
Analysis reveals that under this self-consistent condition and by setting = 1, the equations can be readily solved to obtain the following two types of results. For the case of c = / 0, we have W () = W0 exp(c),
() =
B0 2c
0 W0 −
2. Models and analysis
Pj () = Pj0 exp(−c), exp(c) +
Bj () = B0 exp(−c)
(10)
B0 sinh(c) c
and for the case of c = 0, we have The generally proposed two-coupled NLS equation with variable coefficient can be written as:
D(Z) Aj,TT + (Z)Aj |Ak |2 + i (Z)Aj = 0, 2 n
iAj,Z +
(j = 1, 2), (1)
k=1
where j = 1, 2 and A(Z, T ) is the electric field envelope. D(Z) > 0, (Z) > 0 and (Z) are functions of distance Z, and represent dispersion or diffraction, Kerr nonlinearity and gain (loss) respectively. = ±1, with upper (lower) sign corresponds to a self-focusing (self-defocusing) nonlinearity of the waveguide and in this paper we only consider the case of = 1. The variable T represents the local retarded time in the case of pulse propagation. To simplify the analytical process, we begin our analysis by making the following transformations
Z
t = a(Z)T ≡ T exp −
D(x)C(x)dx ,
0
Aj (Z, T ) = a(Z)
|Uk |2 + M()t 2 Uj + iF()Uj = 0,
F() =
2a4 (Z)D(Z)
(Z) −
˚j (, t) =
(j = 1, 2),
(4)
,
(5)
(Z)DZ (Z) − Z (Z)D(Z) 1 D(Z)C(Z) + 2 2(Z)D(Z)
.
(6)
t − () W ()
exp{i[˚j (, t)]}
1 2 ct + Bj ()t + j () 2
(7)
(8)
where the chirp effect can be expressed as −d˚j /dt = −ct + Bj () and the coefficient c is related to the wave front curvature, it is also a measure of the linear chirp imposed on the self-similar wave. Bj () and j () represent phase shift and phase, respectively, and Pj () is the pulse amplitude, W () is the pulse width. Substituting Eq. (7) with (8) into Eq. (4), collecting the similar terms, and separating the real and imaginary parts, we can obtain some lengthy but straightforward first-order differential equations for the pulse amplitude, width, phase shift and phase. The equations are self-consistent only if the following conditions are satisfied: M() =
1 2 c , 2
1 2
1 W02
exp
1 i
2
ct 2 + Bj ()t + j ()
exp(−2c) + j0 ,
t − () W ()
(12)
exp{i[Bj ()t + j ()]}
− B02
+ j0 ,
(13)
where
According to Ref. [21], the symmetry analysis of Eq. (4) indicates that a self-similar solution to this equation ought to be sought in the form Uj (, t) = Pj ()j
(11)
2
2 Pj0 = W0−2
(14)
j=1
(Z) + D(Z)C 2 (Z)
1 a2 (Z)D(Z)
t − () W ()
and when c=0, we have
j () =
where C(Z) is a frequency chirp parameter, and CZ
() = B0 + 0
4c
(3)
k=1
M() = −
j () =
B02 − W0−2
n
1 Uj,tt + Uj 2
Uj (, t) = Pj () sec h
Then Eq. (1) can be written as iUj, +
Uj (, t) = Pj () sec h
D(Z) i U (, t) exp C(Z)T 2 . 2 (Z) j
Bj () = B0 ,
where W0 , Pj0 , B0 , and 0 are integral constants describing the initial values of the corresponding parameters. At last, by writing the self-similar wave profile as j (, t) = j ( ) with similarity variable = (t − ())/W (), we can further obtain self-similar soliton-like / 0, we have solutions to Eq. (4) for the two cases. When c =
(2)
0
Pj () = Pj0 ,
Z
D(x)a2 (x)dx,
=
W () = W0 ,
F() =
1 c. 2
(9)
Here the pulse width, pulse amplitude, pulse phase shift and the center of the self-similar solutions are specified by Eqs. (10) and (11). The initial phase can be set as j0 = 0 for the aim of simplicity. Meanwhile, one can find that the self-similar waves are completely determined by the pulse amplitude Pj0 , the width W0 , the phase shift coefficient B0 , the beam center 0 , and the linear phase chirp c. If we define the pulse energy as Ej = Pj ()W () = Pj0 W0 , we can find that it is a constant. This means that the pulse energy is always conservative, namely, the system itself is a conservative one. In addition, from Eq. (10), one can find that the amplitudes of the selfsimilar pulses will increase exponentially along the propagating distance when c < 0, because of the exponentially increasing nature of exp(−c) and the widths will be compressed. When c > 0, because of the exponentially decreasing nature of exp(−c), the amplitudes of the self-similar waves will decrease exponentially and their widths will be broadened. From Eq. (11) one can also find that when c = 0, the self-similar waves will propagate steadily in the optical media with their profiles not changed. Consequently, combining the equations obtained, one can further get the exact self-similar soliton-like solution to Eq. (1). To the best of our knowledge, the above self-similar results for Eq. (1) have not been discussed before. These characteristics might have potential applications, e.g., it may provide an effective approach to compress or amplify optical pulse. To investigate the evolution properties of the exact self-similar soliton-like solution propagating in the system described by Eq. (1), in the following part, we will design the optimal system by appropriately choosing the different forms of the distributed parameters according to the specific problem to obtain proper parameter condition under which the self-similar soliton-like solution can propagate steadily in the proposed system. At first, from
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Fig. 1. Evolution when = 1, = 0.005, C0 = 1, d0 = 1, 0 = 1, B0 = 0.2, and (a) c = 0.005, W0 = 1, P10 = 0.7; (b) c = −0.005, W0 = 2, P10 = 0.35; (c) c = 0, W0 = 1, P10 = 0.7.
Fig. 2. Evolution when 10% of white noise is added, the parameters are respectively the same as those in Fig. 1.
the expressions of M() and F(), one can obtain a group of restric-
Z
−1
tive conditions as C(Z) = ca(Z)2 and a(Z)2 = (C0 + 2c 0 D(x)dx) , where C0 is a real integral constant. Because the solutions include two distributed functions D(Z) and (Z), thus by choosing the form most approximate to the real conditions, one can explain the various soliton control systems or dispersion management systems. Here as an example, we consider a system with variable group velocity dispersion and nonlinear Kerr effects respectively as: D(Z) = d0 cos2 ( Z) exp( Z),
(Z) = 0 cos2 ( Z) exp( Z),
(15)
where d0 , 0 , , and are arbitrary constants. Then the gain (loss) coefficient can be obtained as:
(Z) = cD(Z)a(Z)2 = cD(Z)(C0 + 2c
−1
Z
D(x)dx)
(16)
Fig. 3. Evolution when 10% amplitude perturbation is added, the parameters are respectively the same as those in Fig. 1.
0
From Eq. (16) we can see that if the linear chirp parameter c=0, we will have (Z) = 0. Therefore, we may think that the linear chirp results from linear gain (loss) and this means that we can compensate the linear gain (loss) by properly choosing the linear chirp in the real optical communication system. By using the splitstep Fourier method and the Runge–kutta method, we give the evolution plot in Fig. 1(a–c) respectively for the cases of c = 0.005, c = −0.005, and c = 0. The other parameters are given in the figure caption. It can be found that the pulses can propagate stably in the system described by Eq. (1) and the evolution properties are in accordance with the theoretical expectation very well. In this paper, we only give the figures for A1 (Z, T ), the figures for A2 (Z, T ) are similar. It is worth noting that the existence of the self-similar solitonlike solutions depends on the particular nonlinear and dispersive features of the medium. However, in real application, it is difficult to achieve exact balances between them. Therefore, the study of stability of the pulse propagation for various perturbations is very necessary. Here, we first add 10% white noise to the pulse to investigate its evolution properties, and the result is shown in Fig. 2. One can find that there are no clear changes compared to Fig. 1, except for some small oscillation attached to the solitons’ amplitude. To further study the pulse stability, secondly, we add 10% amplitude perturbation to the pulse and the result is shown in Fig. 3. It can be found that the pulse can also propagate stably without clear changes in the pulse shape. So we may infer that the evolution of the pulse is not sensitive to the perturbed conditions. This may make the soliton control technique more realistic and leave scope for more physical explanations and applications in the future. So far, we have investigated the cases of one single pulse propagating in the given system. However, another effect on the pulse propagation is the interaction between two adjacent solitons propagating simultaneously. For the purpose of understanding the effect
Fig. 4. Interaction of two equal amplitude pulses with initial pulse separation S = 10 when W0 = 1, P10 = 0.7and (a) c = 0.005, (b) c = 0.01, (c) c = 0.03. The other parameters are the same as in Fig. 1.
of frequency chirp parameter c on two neighboring solitons, we investigate their transmission properties. Here the spacing of beam center is defined as S and the amplitude ratio of the two pulses is defined as k. Fig. 4 gives the contour plot of two neighboring pulses for the case of c > 0when the spacing of beam center S = 10. We can see that, increasing of chirp parameter c will lead to the two pulses repelling each other except for the changes of pulse intensity. We have known that one can control the interaction of the two adjacent pulses through changing the spacing of beam center and the amplitude ratio of the two pulses. Therefore, using the parameters given in Fig. 1, we give the further numerical simulations. Results show that when the spacing of beam center changes from S = 4 to S = 7, the interaction between two pulses is reduced and further increase the spacing to S = 10, the interaction between two pulses will disappear within the propagation distance of more than one hundred normalized dispersion lengths. The results are shown in Fig. 5. However, this will limit the bit rate of the soliton communication system, so we must further investigate more suitable conditions which can not only prevent interaction between neighboring solitons but also ensure the bit rate of the soliton communication system. So in the simulation, we still set the spacing of beam center of two pulses as S = 7, but adjust the amplitude ratio
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theoretical significance to the research and application of optical fiber communication system. Acknowledgment This work was supported in part by the Provincial Natural Foundation of Shanxi (Grant no. 2012011010-4). References
Fig. 5. Evolution of two pulses with same amplitudes (k = 1) and different spacing of beam center. (a) c = 0, S = 4, W0 = 1, P10 = 0.7; (b) c = 0, S = 5, W0 = 1, P10 = 0.7; (c) c = 0, S = 7, W0 = 1, P10 = 0.7; (d) c = 0, S = 10, W0 = 1, P10 = 0.7; (e) c = 0.005, S = 10, W0 = 1, P10 = 0.7; (f) c = –0.005, S = 10, W0 = 2, P10 = 0.35; the other parameters are the same as those in Fig. 1.
Fig. 6. Evolution of two pulses with spacing of beam center S = 7 and amplitude ratio k = 1.2 when (a) c = 0.005, W0 = 1, P10 = 0.7; (b) c = –0.005, W0 = 2, P10 = 0.35; (c) c = 0, W0 = 1, P10 = 0.7; the other parameters are the same as those in Fig. 1.
of two pulses as k = 1.2. Simulations show that the two pulses can also propagate stably without interaction or shape changing no matter the intensity of pulses is increasing or decreasing along the propagation direction. The results are shown in Fig. 6. Therefore, we may infer that changing the amplitude ratio of two pulses can reduce the interaction between two pulses with smaller spacing of the beam center of two pulses, and it is more suitable for reducing the interaction between the neighboring solitons. This result is advantageous to increase the information bit rate in optical soliton communications. 3. Conclusion In conclusion, firstly, we obtained the self-similarity solution of the variable coefficient coupling nonlinear Schrödinger equation. The stability analysis and numerical evolution are given to study the propagation properties of the self-similar soliton-like pulses for the given system. Results show that within the proper parameter conditions, the pulses can propagate stably and the outer white noises and amplitude perturbations have no clear effect on the pulses. Also, by choosing proper amplitude ratio and proper beam center spacing of two pulses, the interaction between two adjacent pulses can be well restrained. Our result may have
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