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Novel Numerical Method to Model Multichannel Erbium-Doped Fiber Amplifier

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Novel Numerical Method to Model Multichannel Erbium-Doped Fiber Amplifier
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   PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [INFLIBNET India Order]  On: 7 July 2010  Access details: Access Details: [subscription number 920455929]  Publisher Taylor & Francis  Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Fiber and Integrated Optics Publication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713771194 Novel Numerical Method to Model Multichannel Erbium Doped FiberAmplifier Vishnu Priye a ; Dharmendra K. Singh b ; Subhash C. Arya aa  Electronics & Instrumentation Department, Indian School of Mines, Dhanbad, India b  Electronics &Communication Department, BIT Sindri, Dhanbad, India To cite this Article  Priye, Vishnu , Singh, Dharmendra K. and Arya, Subhash C.(2006) 'Novel Numerical Method to ModelMultichannel Erbium-Doped Fiber Amplifier', Fiber and Integrated Optics, 25: 5, 375 — 385 To link to this Article: DOI: 10.1080/01468030600817134 URL: http://dx.doi.org/10.1080/01468030600817134 Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdfThis article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.  Fiber and Integrated Optics , 25:375–385, 2006Copyright © Taylor & Francis Group, LLCISSN: 0146-8030 print/1096-4681 onlineDOI: 10.1080/01468030600817134 Novel Numerical Method to Model MultichannelErbium-Doped Fiber Amplifier VISHNU PRIYE Electronics & Instrumentation DepartmentIndian School of MinesDhanbad, India DHARMENDRA K. SINGH Electronics & Communication DepartmentBIT SindriDhanbad, India SUBHASH C. ARYA Electronics & Instrumentation DepartmentIndian School of MinesDhanbad, India We propose a novel numerical method based on genetic algorithm (GA) to solve theevolution equations that are the basis of modeling multichannel erbium-doped fiber amplifiers (EDFAs). The evolution equations are first transformed from an iteration problem to an optimization problem. The unknown parameter related to backward amplified spontaneous emission (BASE) is coded into a string of bits based on their  probable lower and upper limits. By defining proper random variation operators and cost function, the coupled evolution equations are optimized using GA for different values of erbium-doped fiber parameters. It is observed that GA is an efficient method  for analyzing the EDFA characteristics in the entire C-band. Keywords  genetic algorithm, evolution algorithm, EDFA modeling, optical ampli-fiers Introduction The fundamental and intrinsic advantages of the erbium-doped fiber amplifiers (EDFAs)like low noise, polarization insensitivity, and cross-talk immunity have made their deploy-ment in the actual communication system a reality [1, 2]. Due to the large bandwidth of amplification ∼ 40 nm each in C-band and L-band [3–5], terabits/s data transmission has been achieved using dense wavelength division multiplexed (DWDM) systems. In the Received 13 September 2005; accepted 17 April 2006.Address correspondence to Vishnu Priye, Electronics & Instrumentation Department, IndianSchool of Mines, Dhanbad, 826 004, India. E-mail: vpriye@hotmail.com 375  D o w nl o ad ed  B y : [ I N F L I B N E T  I ndi a  O rd e r]  A t : 04 :44 7  J ul y 2010  376 V. Priye et al. present research scenario, the forward and backward amplified spontaneous emissions(ASE) have assumed new importance not because they decide the achievable noise figure[6–9] of the multichannel amplifiers, but because they also can be used to pump the EDF along with 980-nm lasers to achieve L-band amplifiers [10–12]. The complete evolution equations that describe the length dependence of pump,signal, FASE (forward ASE), and BASE (backward ASE) represents a two-boundaryvalue problem [7–9] that requires iterative adjustment to the solution as initial conditions for the BASE is not known. The exact numerical method to solve the evolution equationsis known as the relaxation method [7] and henceforth will be referred to in this article asthe conventional method. In the relaxation method as discussed in Desurvire [7], the firstintegration over the entire EDF length is done without considering BASE by setting bothFASE and BASE as zero at the input end ( z = 0) of the EDF. The second integration isdone in the reverse direction starting from the output end ( z = l ) to the input end to obtaina set of quasi solutions for pump signal, FASE, and BASE. These quasi solutions are usedto perform back-and-forth integration until a steady state solution is reached. This methodis rigorous but poses a convergence problem and consumes a lot of computation time,when the total C-band spectrum (1525–1565 nm) is subdivided into wavelength strips of 1 nm to incorporate multichannel transmission. To overcome the problem, large numbersof analytical and approximate methods [5, 12–15] have been reported. Recently, black- box models have been proposed in which gain and noise figures of EDFA are representedby empirical formulae based on some experimentally determined parameters [15, 16]. In this article, we report a novel numerical method based on genetic algorithm (GA)to solve the evolution equations characterizing the propagation of pump, signal, FASE,and BASE in EDF [17, 18]. GA, also known as evolutionary algorithm, is basically asearch algorithm based on the mechanics of natural selection and natural genetics [19, 20]and recently has been successfully applied in other areas of optical communication tooptimize the synthesis of fiber gratings with advance characteristics [21], long-periodgrating filters [22], multi-layer coatings at oblique light incidence [23], and spectral equalization of EDFA [24]. We have first coded the values of BASE at the input of thefiber that fall within the expected range. Next we have defined a cost function basedon the behavior of BASE and developed our algorithm based on the evolution equationsto minimize the cost function. The present method yields accurate estimation of pump,FASE, and BASE in the C-band of EDF and can be applied to estimate the BASE andmultichannel amplification in EDFA. Formulation The general evolution equations based on the three-level atomic rate equations that de-scribe the evolution of co-directional pump power at pump wavelength, forward andbackward ASE, and signals at discrete wavelengths within the bandwidth of the EDFAare described in detail in Desurvire [7]. For the sake of continuity they are reproducedbelow: dqdz =− qρ 0 σ  pa    b 0 ρ(r)ρ 0 ψ p (r)AB 2 πrdr  (1) dp ± k dz =± ρ 0 σ  a (λ sk )    b 0 ρ(r)ρ 0 ψ sk (r)  C − DB  2 πrdr  (2)  D o w nl o ad ed  B y : [ I N F L I B N E T  I ndi a  O rd e r]  A t : 04 :44 7  J ul y 2010   Multichannel Erbium-Doped Fiber Amplifier 377  where A = 1 +  j  η sj  1 + η sj  (p + j   + p − j   )ψ sj  (r)  (3) B  = 1 + qψ p (r) +  j  (p + j   + p − j   )ψ sj  (r)  (4) C  = η sk  qψ p (r) 1 + η p +  j  (p + j   + p − j  1 + η sj  ψ sj  (r)  [ p ± k  + 2 p o k ]  (5) D  =  1 +  j  η sj  1 + η sj  (p + j   + p − j   )ψ sj  (r) + η p 1 + η p qψ p (r)  p ± k  (6)The different notations are the following: q  =  P  p (z)/P  sat  (λ p ) =  normalized pump power; p ± k  =  P  s (z)/P  sat  (λ s ) =  normalized forward and backward propagating signal and ASE power; P  p (z)  =  pump power; P  s (z)  =  signal or ASE power; P  sat   ( λ s  or  λ s )  =  saturation power at pump or signal wavelength; p ok  =  equivalent input noise power; ρ(r)  =  radial distribution of Erbium ions; σ  a  =  absorption cross-section at pump and signal wavelength; ψ p  or  ψ s  =  modal field at pump or signal wavelength; η p  or  η j   =  excitation efficiency at pump wavelength or signal at a wavelengthcorresponding to  j  .The terms  A ,  B ,  C , and  D  come into the evolution equation by considering the rateequations of pump power and signal powers in the total emission and absorption band of EDFA. The expressions of   A  and  B  given in Eqs. (3) and (4) contain terms that representthe evolution of pump and forward and backward signal powers. In the summation,the value of   j   corresponds to the number of individual wavelengths considered. Forexample,  j   = 25 implies that the C-band is divided into 25 equally spaced wavelengths.The expression  C  depicted in Eq. (5) incorporates equivalent input noise power andnormalized forward and backward propagating signal and ASE power besides pumppower and other constants. Similarly, Eq. (6) for  D  shows the multiplication of normalizedforward and backward propagating signal and ASE power and summation of pump andforward as well as backward signal power. When the transition is degenerate, there exist  D o w nl o ad ed  B y : [ I N F L I B N E T  I ndi a  O rd e r]  A t : 04 :44 7  J ul y 2010  378 V. Priye et al. Table 1 EDF parameters used in this articleErbiumconcentrationNumerical aperture(NA)CoreradiusErbium-dopedradius1 . 15 × 10 25 ions/m 3 0.61 1.64  µ m 1.64  µ m Table 2 Data required in solving the rate and propagation equationsScanningwavelengthrangesScanningwavelength step( λ )Central signalwavelength( λ c )Er 3 + metastablelifetime( τ  )Pumpwavelength1525–1565 nm 1 nm 1550 nm 10 ms 980 nm g 1  and  g 2  sublevel in the lower and upper states; then we have relation g 1 σ  a (λ) = g 2 σ  e (λ) or σ  e (λ)σ  a (λ) = g 1 g 2 = η(λ)η s  = σ  es σ  as >σ  ep σ  ap = η p This expression is used in gain coefficient for multilevel laser systems. The parameter η(λ)  is of central importance in the modeling of EDFAs. Evaluation equations based onthe three-level atomic rate equation that requires 980-nm pump wavelength has  σ  ap  = 2 . 17 × 10 − 25 and  σ  ep  =  0 . 0. This relationship shows that if the ratio of the stimulatedemission to absorption at the signal wavelength is not greater than that at the pumpingwavelength, a positive signal gain coefficient can never be achieved. For Er 3 + glasspumped as a three-level pumping scheme we have  η ep  =  0; hence, the condition isalways verified.The multichannel wavelengths are chosen by the value of   j  . For example, if   j   = 40,the entire C-band (1525–1565 nm) is divided into 40 channels separated by 1 nm. Inthe conventional method, termed “relaxation method” in Desurvire [7], the differentialequations are solved using the fourth-order Runge Kutta method and making iterativeforward and backward integration with the initial condition that BASE and FASE arezero at fiber input. The parameters and data required for solving rates and propagationequations used in the present article are given in Tables 1 and 2, respectively. Genetic Algorithm Modeling Genetic algorithm (GA) is adaptive heuristic search algorithm based on the evolutionaryideas of natural selection and genetics. As such, they represent an intelligent exploitation  D o w nl o ad ed  B y : [ I N F L I B N E T  I ndi a  O rd e r]  A t : 04 :44 7  J ul y 2010
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