Modeling Propagation in Optical Fiber using Split- Step Wavelet in Linear Media

Transcription

1 International Journal of Electronic and Electrical Engineering. ISSN Volume 3, Number 3 (2010), pp International Research Publication House Modeling Propagation in Optical Fiber using Split- Step Wavelet in Linear Media Chakresh kumar 1, Devendra chack 2, Sandeep kumar 1 and Chandra Shekhar 3 1 Department of Electronics and communication, M.R.I.U Faridabad, INDIA 2 Department of Electronics and communication, College of Science and Engineering, Jhansi INDIA 3 Department of Electronics and communication,dronacharya College of Engineering, Gurgaon chakreshk@gmail.com, devendra.chack@gmail.com, chandu9890@indiatimes.com Abstract The propagation of optical pulses in a single mode fiber is analyzed by solving nonlinear Schrödinger (NLS) equation numerically. The propagation is modeled using a split-step wavelet method, analogous to the split-step Fourier method, while retaining the advantages of the wavelet transform. Pulses and pulse sequences are accommodated with equal ease, as is frequency chirping of the light source. The technique has been used for obtaining the propagation features of a pulse in an optical fiber. The effect of chirp and pulse power on the propagation characteristics has been studied. In this paper, we have analyzed pulse propagation in optical fibers using split-step wavelet method. The pulse propagation has been studied in linear regime. The effect of input power on the pulse shape has also been obtained. The wavelet transform method is a very powerful technique for analysis of propagation characteristics, both in terms of speed and the physical insight that it provides. Keywords:Wavelets, Propagation using, Fiber Optic Introduction Propagation in optical fibers is governed by the Non-Linear Schrödinger Equation (NLSE). This equation has to be solved numerically. The split-step Fourier Methods can be used to solve the NLSE numerically. This method is employed mainly because of its computational efficiency. But the solutions obtained by the Fourier method have a certain periodicity. Wavelets are an ideal tool for analyzing this. Wavelets have a

2 120 Chakresh Kumar et al zooming in property which allows us to express the higher order components, by a fewer number of concentrated wavelets. This means lowers number of coefficients.the Other major advantage is the computational efficiency. The fast wavelet transform is an O(N) computation as compared to the fast-fourier transform which is an O(N log N) computation. We shall discuss a method for the propagation of optical waveforms in a single mode fiber. Background Theory We will be using a system model which will give us the basis theory to implement wavelet analysis and will further explain how the wavelet transform is implemented. Step by step we shall build up the tools necessary for showing how any arbitrary waveform can be expressed using wavelets Wavelet Transform The wavelet analysis described in the introduction is known as the continuous wavelet transform or CWT. More formally it is written as: Where * denotes complex conjugation. This equation shows how a function ƒ(t) is decomposed into a set of basis functions ψ s,τ (t), called the wavelets. The variables s and τ are the new dimensions, scale and translation, after the wavelet transform. For completeness sake equation (2) gives the inverse wavelet transform. I will not expand on this since we are not going to use it: (1) The wavelets are generated from a single basic wavelet ψ (t), the so-called mother wavelet, by scaling and translation: (2) In (3) s is the scale factor, τ is the translation factor and the factor s -1/2 is for energy normalization across the different scales. It is important to note that in (1), (2) and (3) the wavelet basis functions are not specified. This is a difference between the wavelet transform and the Fourier transform, or other transforms. The theory of wavelet transforms deals with the general properties of the wavelets and wavelet transforms only. It defines a framework within one can design wavelets to taste and wishes. Using a mother wavelet h(t). (3)

3 Modeling Pulse Propagation in Optical Fiber 121 Propagation in Optical Fiber To understand the propagation of light in optical fibers, one needs to solve Maxwell s equations taking into account the nonlinear component of the polarization. After using a slowly varying approximation for the envelope of the pulses, one gets an equation similar to the nonlinear Schrödinger equation, Here, E is the slowly varying amplitude of the pulse envelope mentioned above. β 2 is the Group Velocity Dispersion parameter, while ϒis the Self Phase Modulation parameter related to the Kerr nonlinearity in the optical fiber. α is the fiber attenuation. This equation must be solved numerically. However, we can obtain an approximate solution by considering the linear and nonlinear parts of the above equation separately. Ignoring the nonlinear terms to begin with, and assuming a lossless fiber, we get; (4) (5) Here Ê (0, w) is the Fourier transform of the incident field at z = 0. It is necessary to choose a family of wavelet h mn (t) (6) Where h mn in the family of wavelet. We choose as our mother wavelet function (7) With the help of equ. (4) And equ. (5) We generate a family of wavelets. The propagated wavelet h mn (z, t) is given as. The complex envelope of the input electric field can be written as (8) (9)

4 122 Chakresh Kumar et al where P in (t) is the power launched into the fiber and φ (t) is the optical power launched into the fiber. The propagating waveform can be then expressed in terms of wavelet coefficients as. Where (10) (11) We can find out the power out of an optical fiber, which is given the power launched into the fiber. Linear Condition The pulse propagation in an optical fiber using the split step wavelet method. We consider Gaussian pulse as an input. We use Eqns. (10) and (11) to analyze the propagation in linear regime. The parameters used for the analysis are given in Table 1. By using the linear equation we will propagate the pulse in optical fiber. The initial pulse is of the form Ein = P in (t) exp(iαt), Where P in (t)=p 0 exp(-t 2 /t 0 2 ) Where t 0 is the pulse width, and P 0 is its peak power.choosing the pulse width t 0 = 0.4ps, and peak power P o = 1mw, the pulse is propagated in the linear mode. Figure-1 shows the graph for pulse propagating in linear mode at different distances. Plot (a) gives the input pulse profile, (b) is at Z=0.5 km, (c) at Z=1Km, (d) at Z=1.5Km and (e) at Z=2km. We observe pulse broadening with increasing propagation distance which can be due to group velocity dispersion. Parameters of the optical fiber. the pulse propagation in an optical fiber using the split step wavelet method. We consider Gaussian pulse as an input. We use Eqns. (10) and (11) to analyze the propagation in linear regime. The parameters used for the analysis are given in Table 1. By using the linear equation we will propagate the pulse in optical fiber. The initial pulse is of the form Ein = P in (t) exp(iαt), Where

5 Modeling Pulse Propagation in Optical Fiber 123 P in (t)=p 0 exp(-t 2 /t 0 2 ) Where t 0 is the pulse width, and P 0 is its peak power Choosing the pulse width t 0 = 0.4ps, and peak power P o = 1mw, the pulse is propagated in the linear mode. Figure-1 shows the graph for pulse propagating in linear mode at different distances. Plot (a) gives the input pulse profile, (b) is at Z=0.5 km, (c) at Z=1Km, (d) at Z=1.5Km and (e) at Z=2km. We observe pulse broadening with increasing propagation distance which can be due to group velocity dispersion. Table 1. Parameters of the optical fiber Symbol Parameter Value Units β 2 GVD -0.1 ps 2 /m p Optical power mw Fiber attenuation 0.2 α Figure 1

6 124 Chakresh Kumar et al Conclusions In this paper we have described that when the pulse is propagated in the Liner media it is effect by the Dispersion. References [1] O. Audouin, L. Proingent, and J.-P. Hamaide, Limitations on transoceanic optical communications by dispersion, Kerr nonlin- earity and amplifier noise, in h o c. 17th Euro. Conf. Optic. Com- mun. (ECOC 91), Paris, France, 1991, pp [2] J. C. Cartledge and A. F. Elrefaie, Effect of chirping-induced waveform distortion on the performance of direct detection re- ceivers using travelingwave semiconductor optical amplifiers, J. Lightwaue Technol., vol. 9, pp , Feb [3] A. Naka and S. Saito, Fibre transmission distance determined by eye opening degradation due to selfphase modulation and group velocity dispersion, Electron. Lett., vol. 28, no. 24, pp , [4] T. R. Taha and M. J. Ablowitz, Analytic and numerical aspects of certain nonlinear evolution equations. 2. Numerical, nonlinear Schrodinger-equation, J. Comput. Phys., vol. 55, no. 2, pp , [5] G. P. Agrawal, Nonlinear Fiber Optics. San Diego, C A Academic, [6] I. Daubechies, The wavelet transform, time-frequency localiza- tion and signal analysis, IEEE Trans. Informat. Theov, vol. 36, pp , NOV [7] G. Strang, Wavelet transforms versus Fourier transform, Bull.Amer. Math. Soc., vol. 28, no. 2, pp , [8] L. J. Cimini, L. J. Greenstein, and A. A. M. Saleh, Optical equalization to combat the effects of laser chirp and fiber disper- sion, J. Lightwave Technol., vol. 8, pp , May [9] H. J. A. da Silva and J. J. OReilly, Optical pulse modeling with Hermite- Gaussian functions, Opt. Lett., vol. 14, no. 10, pp , [10] A. Djupsjobacka, Residual chirp in integrated-optic modulators, IEEE Photon. Technol. Lett., vol. 4, pp , Jan [11] M. A. Muschietti and B. Torresani, Pyramidal algorithms for Littlewood- Paley decompositions, S U M J. Math. Ann., to be published. [12] G. P. Agrawal, Nonlinear Fiber Optics, 3rd Ed. New York: Academic, 2001.