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58 ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 2, NOVEMBER 2010 TABLE I TAILBITING CODES OVER RING Z4 WITH CODE RATE R=1/2 FOR 16-QAM MODULATION(NATURAL LABELING AS IN [7]). S 16 TBR 64 TBR 256 TBR N f, g de_min f, g de_min f, g de_min 4 130,100 12,000 1300,1000 12,000 11100,10000 12,000 5 113,210 14,128 1130,2100 14,128 10132,10000 14,128 6 102,111 14,141 1121,1100 14,828 12330,11000 14,828 7 111,123 16,944 1312,1313 16,970 11331,20110 16,970 8 111,221 16,970 1121,120 18,129 - - [6] I. Bocharova, R. Johannesson, B. Kudryashov, and P. Ståhl, “Tail-biting codes: Bounds and search results,” IEEE Trans. Inf. Theory, vol. 48, pp. 597–610, Apr. 2000. [7] R. Carrasco and P. Farrell, “Ring-TCM for fixed and fading channels: land-mobile satellite fading channels with QAM,” IEE Proceedings- Communications, vol. 143, no. 5, pp. 281–288, Oct. 1996. [8] C. B. Weiß, “Code construction and decoding of parallel concatenated tail-biting codes,” IEEE Trans. Inf. Theory, vol. 47, pp. 366–386, Jan. 2001. [9] P.Remlein, “Theencoders with the feedback for thepacked transmission without tail symbols,” in VIII-th Poznan Workshop on Telecommunication, PWT ‘03, Poznan, Dec. 2003, pp. 165–169, (in polish). [10] J. Anderson, T. Aulin, and C. Sundberg, Digital PhaseModulation. NY Plenum Press, 1986. Piotr Remlein received the M. Sc. and Ph. D. degrees in telecommunications from the Poznan University of Technology, Poland, in 1991 and 2002, respectively. Since 1992 he has been working at the Faculty of Electronics and Telecommunications, Poznan University of Technology, where he currently is an Assistant Professor. His scientific interests cover wireless networks, communication theory, error control coding, cryptography, digital modulation, trellis coded modulation, continuous phase modulation, mobile communications, digital circuits design. He is author and co-author of over 50 publications and unpublished reports. He is a member of IEEE. Dawid Szłapka received the M. Sc. degree from the Faculty of Electronics and Telecommunications, Poznan University of Technology, in 2005.
ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 2, NOVEMBER 2010 59 Modeling Step Index Fiber to Soliton Propagation Abstract—Step index fiber modeling process is carried out through numerical solving of eigenvalue equation to calculate propagation constant for fundamental mod. Input data in the process is only index of refraction calculated from Sellmeier dispersive formula for appropriate mol percentage doping of germanium dioxide in silica glass fiber. Output data in the modeling process is optimal value of the normalized frequency, which guarantees that single mode operation region is equal to brightsolitonpropagation region.Finalverificationof theprocess is soliton generation up to sixth-order inside such modeled fiber. In this end nonlinear Schödinger equation is solved numerically for initial condition of hyperbolic secant form. Maximization of single mode operation and bright soliton propagation region is essential in wavelength division multiplexing technique. IndexTerms—eigenvalue equation,nonlinearSchödingerequation, solitons Tomasz Kaczmarek I. INTRODUCTION THE word soliton refers to special kinds of wave packets that can propagate undistorted over long distances. In the context of optical fibers solitons have found practical applications in the field of fiber-optic communications. Solitons results from a balance between group-velocity dispersion and self-phase modulation, both of which can be calculated in effect of step index fiber modeling process. Propagation of soliton in single-mode optical fiber is described by the nonlinear Schrödinger equation [1]–[4] j ∂A ∂z − β2 2 ∂ 2 A ∂T 2 + γ |A|2 A = 0, (1) where A is the slowly varying envelope of the pulse, γ is nonlinear parameter of the fiber, β2 is group velocity dispersion, z and T are spatial and time variable, respectively. Group velocity dispersion expressed in ps 2 /km is defined as the second derivative of mode propagation constant β with respect to frequency ω i.e. β2 = d 2 β/dω 2 , and is related to dispersion parameter D expressed in ps/(km · nm) through the relation D = −2πcβ2/λ 2 where c is the speed of light in vacuum. Nonlinear parameter is defined as follows [1], [4] γ = nNLk , (2) Aeff where nNL is nonlinear refractive index, Aeff is known as effective core area. For pulses as short as 1 ps and in case of single mode fiber, which core is made of silica glass doped by germanium dioxide, value of nNL is approximately equal to nNL = 2.2 · 10 −20 m 2 /W [1]. Effective core area is related to the transverse component of electric field vector E0 and T. Kaczmarek is with the Institute of Telecommunications, Photonics and Nanomaterials, Kielce University of Technology, Al. 1000-lecia P.P.7, 25-314 Kielce, Poland (e-mail: tkaczmar@tu.kielce.pl). effective core radius ωeff through the relations [1], [4] � ∞� 2π |E0 (r)| 0 Aeff = 2 �2 rdr ∞� |E0 (r)| 4 = πω rdr 2 eff , (3) 0 where r is radial coordinate in the cylindrical coordinate system. Absolute value of E0 is related to the transverse components of electric field vector Er and Eφ through well known formula |E0| = (|Er| 2 + |Eφ| 2 ) 1/2 . The transverse components are determined by the use of axial component of electric Ez and magnetic Hz field vectors through the following relations [3], [5], [6] Er1 = −j χ 2 Hr1 = −j χ 2 � β ∂Ez1 ∂r Eφ1 = −j χ2 � β r � β ∂Hz1 Hφ1 = −j χ 2 ∂Ez1 ∂φ + ωµ0 r − ωµ0 ∂r − ωε0n2 1 r � β ∂Hz1 r ∂φ + ωε0n 2 1 � ∂Hz1 , (4) ∂φ ∂Hz1 ∂r ∂Ez1 ∂φ ∂Ez1 ∂r � , (5) � , (6) � , (7) for the core. In case of claddingsubscript 1 should be changed to 2 and, moreover, variable χ 2 should be replaced with – σ 2 . Equations from (4) to (7) are essential for computing an average power curried by the core [5], [6] � P1 = π and cladding [5], [6] 0 a � P2 = π � Er1H ∗ φ1 − Eφ1H ∗ � r1 rdr, (8) +∞ � Er2H ∗ φ2 − Eφ2H ∗ r2 a � rdr, (9) where for example H∗ φ1 means complex conjugate to Hφ1. Averagepowerpropagatedinsidethe core P1 canbe expressed as percentage through the relation P1% = [P1/(P1 + P2)] · 100%. The expressions for Ez and Hz are given by [3], [5], [6] Ez1 = AEJm (χr) exp [j (mφ + ωt − βz)] , (10) Hz1 = AHJm (χr) exp [j (mφ + ωt − βz)] , (11) for the core and [3], [5], [6] Ez2 = BEKm (σr) exp [j (mφ + ωt − βz)] , (12) Hz2 = BHKm (σr) exp [j (mφ + ωt − βz)] , (13) for the cladding of the step index fiber, where AE, AH, BE and BH are arbitrary constants, Jm(χr) is the Bessel function
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