Redalyc.Soliton Management for Ultra-High Speed Telecommunications

Red de Revistas Científicas de América Latina, el Caribe, España y Portugal Sistema de Información Científica Vladimir N. Serkin, Akira Hasegawa Soliton Management for Ultra-high Speed Telecommunications Ciencia Ergo Sum, vol. 8, núm. 3, noviembre, 2001 Universidad Autónoma del Estado de México México Available in: http://www.redalyc.org/articulo.oa?id=10402208 Ciencia Ergo Sum, ISSN (Printed Version): 1405-0269 [email protected] Universidad Autónoma del Estado de México México How to cite Complete issue More information about this article Journal's homepage www.redalyc.org Non-Profit Academic Project, developed under the Open Acces Initiative n ú m e r o e s p e c i a l Soliton Management for Ultra-high Speed Telecommunications VLADIMIR N. SERKIN* AND AKIRA HASEGAWA** Manejo de solitones para telecomunicaciones de alta velocidad Resumen. La metodología desarrollada provee un método sistemático de encontrar un número infinito de las novedosas islas solitónicas (‘‘soliton islands’’) estables, brillantes y obscuras en un mar de olas solitarias, para la ecuación no lineal de Schrödinger con dispersión y no linealidad variables y con ganancia o absorción. Se muestra que los solitones existen sólo bajo ciertas condiciones y las funciones paramétricas que describen la dispersión, la no linealidad, la ganacia o absorción no homógenea, no pueden ser electas independientemente. Se han descubierto los regímenes de manejo fundamental solitónico para comunicaciones a velocidades ultra- rápidas a través de fibras ópticas. Palabras clave: telecomunicaciones en fibras ópticas, solitones. Abstract. The methodology developed provides for a systematic way to find an infinite number of the novel stable bright and dark ‘‘soliton islands’’ in a ‘’sea of solitary waves’’ of the non linear Schrödinger equation model with varying dispersion, non linearity and gain or absorption. It is shown that solitons exist only under certain conditions and the parameter functions describing dispersion, non linearity and gain or absorption inhomogeneities cannot be chosen independently. Fundamental soliton management regimes for ultra-high speed fiber optics telecommunications are discovered. Keywords: fiber optics telecommunications, solitons. Recepción: 8 de noviembre de 2000 interaction with other solitons, giving them a ‘‘particle-like’’ Aceptación: 8 de mayo de 2001 quality. The theory of NLSE solitons was developed for the first time in 1971 by Zakharov and Shabad [3]. Over the Introduction years, there have been many significant contributions to the development of the NLSE soliton theory (see, for example, The soliton –a solitary wave with the properties of a moving [1-9] and references therein). After predictions of the particle– is a fundamental object of nature. Solitons of the possibility of the existence [10] and experimental discovery non linear Schrödinger equation model (NLSE solitons) appear by Mollenauer, Stolen and Gordon [11], today, NLSE optical in many branches of modern science including physics and solitons are regarded as the natural data bits and as an important applied mathematics, non linear quantum field theory, condensed matter and plasma physics, nonlinear optics and * Benemérita Universidad Autónoma de Puebla, Instituto de Ciencias. Apartado. Postal quantum electronics, fluid mechanics, theory of turbulence 502, C.P. 72001, Puebla, Pue., Mexico. Email address: [email protected] and phase transitions, biophysics, and star formation. ** Soliton Communications, #403, 19-1 Awataguchi Sanjobocho, Higashiyama-ku, The current state-of-the-art in this very active field is Kyoto, 605-0035, Japan. Authors would like to express special gratitude to Prof. V. E. Zakharov for reading and reviewed, for instance [1, 2]. Characteristic properties of NLSE commenting on the entire manuscript and fruitful suggestions. Special thanks are due to solitons include a localized wave form that is retained upon Prof. T. H. Tieman for carefully checking the manuscript. V OL. 8 NÚMERO TRES, NOVIEMBRE 2001-FEBRERO 2002 CIENCIA ERGO SUM 305 n ú m e r o e s p e c i a l alternative for the next generation of ultra-high speed optical both temporal and spatial solitons are described by (1). In telecommunication systems [12-17]. the case of temporal solitons T is the non dimentional time Ultra-high speed optical communication is attracting word- in the retarded frame associated with the group velocity of wide interests as the 21st century infra-structure for wave packets at a particular optical carrier frequency. In computer based information age. Most agrees that optical the case of two-dimensional spatial solitons T=X represents soliton will play major role as the means of transmission, a transverse coordinate. however, it has not yet demonstrated decisive merit over linear transmission scheme because of it’s intrinsic problems. Theorem 1 The quasi-soliton concept is developed to overcome these Consider the NLSE model Eq. (1) with varying dispersion, difficulties and to demonstrate decisively its merit as the nonlinearity and gain or absorption. Suppose that the mean of information carrier of expected terabit/second Wronskian W{N2, D2} of the functions N2(Z) and D2(Z) is ultra-high speed network. nonvanishing, the two functions N (Z) and D (Z) are thus The problem of soliton management in the non linear 2 2 linearly independent. There are then an infinite number of systems described by the NLSE model with varying a solitary wave solutions for the Eq. (1) written in the coefficients is a new and important one (see, for example, the review of optical soliton dispersion management following form: principles and research as it currently stands in [18-23] and references therein). Soliton interaction in optical ± D (Z) ± P(Z) Z ± q (Z,T) = 2 P(Z)Q [P(Z)T ]exp± i T 2 + i ∫ K (Z’)dZ’ (2) telecommunication lines has attracted considerable attention N2(Z) 2 0 in view of its effect on achievable bit rates. Dispersion managed soliton technique now is the most + powerful technique in the fiber optics communication where the real function Q (S) describes canonical functional + η η systems. The 40 Gbit/s - 1000 km soliton transmission test form of bright (sign= +1, Q (S)= sech( P(Z)T)) or dark - η η was realized recently [23] in which Nakazawa group from (sign= -1, Q (S)= tanh( P(Z)T)) NLSE solitons [1,2,3], and Γ NTT Network Innovation Laboratories, Japan, used the part the real functions D2(Z), N2(Z), (Z) and P(Z) satisfy the of the Tokyo metropolitan optical loop network. equation system: In this Report we show that methodology based on the quasi-soliton concept, provides for a systematic way to discover ∂P(Z) W{N (Z), D (Z)} + P2 (Z)D (Z) = 0; 2 2 − D (Z)P(Z) = 2Γ(Z) ∂ 2 2 (3) a novel stable soliton management regimes for the non linear Z D2(Z), N2 (Z) Schrödinger equation (NLSE) model with varying dispersion, non linearity and gain or absorption. Quasi-soliton solutions for this model must be of rather general character than Theorem 2 canonical solitons of standard NLSE, because the generalized Consider the NLSE model (1) with varying dispersion, model takes into account arbitrary variations of group velocity nonlinearity and gain or absorption. Suppose that the dispersion, non linearity and gain or absorption. Wronskian W{N2, D2} is vanishing, the two functions N2(Z) and D2(Z) are thus linearly dependent. There are then an I. Novel soliton solutions for non linear Schrödinger infinite number conserving the pulse area solitary wave equation model solutions for the Eq. (1): Our starting point is the NLSE model with varying coefficients: ± ± P(Z) Z ± q (Z,T ) = CP(Z)Q [P(Z)T ]exp± i T 2 + i ∫ K (Z’)dZ’ ± 2 ± (4) ∂q 1 ∂ q ± 2 ± 2 0 i ± D (Z) + N (Z) q q = iΓ(Z)q (1) ∂Z 2 2 ∂T 2 2 where the real functions Q±(S) describe a canonical form NLSE (1) is written here in standard soliton units, as they of bright (Q+(S)) or dark (Q-(S)) NLSE solitons, and the real are commonly known. There it is assumed that the Γ functions P(Z), D2(Z), N2(Z) and (Z) satisfy the equation perturbations to the dispersion parameter D2(Z), non linearity system: Γ N2(Z) and to the amplification or absorption coefficient (Z) are not limited to the regime where they are smooth and 1 ∂ P (Z ) 1 ∂ P (Z ) 2Γ (Z ) = ; C 2 N ( Z ) = D (Z ) = − small. Due to the well known spatial-temporal analogy [3] P ∂ Z 2 2 P 2 ( Z ) ∂ Z (5) 306 CIENCIA ERGO SUM V OL. 8 NÚMERO TRES, NOVIEMBRE 2001-FEBRERO 2002 S OLITON MANAGEMENT FOR ULTRA-HIGH SPEED TELECOMMUNICATIONS The explicit solutions for the travelling solitary waves can given. The function E(Z) is required only to be a once- easily be constructed by applying the Galilei transformation differentiable and once integrable, but otherwise arbitrary and by using the equation for the “soliton center’’ Tsol(Z) function, there are no restrictions. There are then an infinite given by number of solutions for the Eq. (1) of the form of bright and dark solitons represented by the Eq. (2), where the main functions D (Z), P(Z) and Γ(Z) are given by: ∂T (Z) 2 sol = −VD (Z) (6) ∂Z 2 E(Z)N (Z) ∂ D (Z) = 2 ; 2Γ(Z) = ln(E(Z) / 2) where V is a soliton group velocity (in the case of spatial 2 2P(Z) ∂Z (9) soliton V=tanθ, and θ is the angle of propagation in the X-Z plane). 1 + P(Z) = ± exp− ∫ E(Z)N (Z)dZ + C The phase function K (Z) for the bright soliton solution 2 2 (10) of Eq. (1) is given by Case 3. Soliton intensity management. In this case the + 1 K (Z) = η2D (Z)P2(Z) soliton pulse intensity (peak power) is assumed to be 2 2 Θ 2 controlled by the function (Z)=D2(Z)P (Z)/N2(Z), where and the correspondent phase function K –(Z) for the dark the control function Θ(Z) is required only to be a once- soliton solution is represented by differentiable and once integrable.

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