CHAPTER 1 SIGNAL PROPAGATION IN DWDM OPTICAL SYSTEMS

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1 1 CHAPTER 1 SIGNAL PROPAGATION IN DWDM OPTICAL SYSTEMS 1.1 INTRODUCTION TO DWDM SYSTEMS A Wavelength Division Multiplexing (WDM) optical communication system supports multiple signals multiplexed on to many wavelengths that are spaced to avoid the inter-channel interference and transmitted through a single fiber channel. Figure 1.1 demonstrates the concept of WDM. WDM is broadly classified as Coarse WDM (CWDM) and Dense WDM (DWDM) which are ITU standardized. CWDM (ITU G.695) specifies 18 channels of 20 nm spacing in the 1270 nm nm band and DWDM (ITU-T G.694.1) systems use channel spacing of the order of 200 GHz, 100 GHz, 50 GHz,25 GHz to provide more number of channels and hence improve the spectral efficiency. Figure 1.1 Wavelength division multiplexing DWDM technology is a cost effective solution to current and future demands on bandwidth. Another important aspect of WDM is that each optical channel can carry signals with any transmission format, (Agarwal

2 2 1997). In order to realize such high capacity, systems with spectral efficiency beyond 0.4b/s/Hz and data rate higher than 40 Gb/s need to be designed using the available bandwidth, (Fukuchi 2002, Bigo et al 2001) Operational Principles of DWDM A general setup for an optically amplified DWDM communication system is shown in Figure 1.2. Figure 1.2 An optically amplified DWDM system ( Binh 2003) The transmitter consists of a laser source, modulation device and a suitable line coding. Modulation can be either direct or external, direct modulation causes change in the refractive index of the material for higher input power and results in chirping effect (Binh 2003). In the case of external modulation, the laser is operated with a constant bias current and modulation is carried out using an external modulating device (Kaiser 2000). The external modulator can be an electro-optic Lithium Niobate (LiNbO 3 ) modulator used in a Mach-Zehnder Interferometer (MZIM) configuration. Figure 1.3 illustrates this modulator.

3 3 Figure 1.3 LiNbO 3 modulator in Mach-Zehnder configuration (Hodzic 2004) This modulator operates on the principle of electro optic effect where the refractive index of the modulator arm is changed by varying the applied electric field. This change in the refractive index induces a change of material propagation constant β, resulting in different phase shifts in both modulator arms. In the absence of the applied electric field the optical signal E 0 applied at the input is divided by a 3-dB coupler into two equal parts E 1, E 2, which propagate in lower and upper arm of the MZM and constructively interfere on arrival at the output, (Hodzic 2004). In the presence of the electric field the signals are shifted in phase relative to each other in both the arms. The signals can interfere constructively or destructively depending on the phase difference Δφ between the arms resulting in an amplitude modulation of the input signal (Wenke and Klimnek 1996). Thus, a 1 signal is obtained at the output when the signals interfere constructively and a 0 signal is obtained for destructive interference, (Kondo et al 2001). Selection of an appropriate line coding format plays a significant role in spectral efficiency improvement. Figure 1.4 shows the NRZ and RZ formats. In the case of NRZ format the signal level is held low for a 0 bit and high for a 1 bit. In the RZ format, for a 1 bit the first half of the bit

4 4 period is high followed by a low for the second half. For a 0 bit, the signal will be held low for the whole bit period. The bandwidth required for RZ is higher compared to NRZ but it is less susceptible to Inter Symbol Interference (ISI), which is the corruption of isolated 0 bits by their neighboring 1 bits (Wiley 2003) NRZ RZ Figure 1.4 NRZ and RZ line coding formats In DWDM systems we require a multiplexer to combine modulated outputs and couple it into a fiber and a de-multiplexer at the receiving end to separate the optical signals for detection and further processing, (Tan 2004). An Erbium Doped Fiber Amplifier (EDFA) provides gain in the optical domain and has the advantage of providing high gain, low noise, wide bandwidth and polarization independent operation, (Binh 2003).

5 5 1.2 NON-LINEAR PHENOMENA IN OPTICAL FIBERS Non-linearities in optical fibers can be classified as: (1) nonlinearities that arise from stimulated in-elastic scattering namely Stimulated Raman Scattering (SRS) and Stimulated Brillouin Scattering (SBS), and (2) non-linearities that arise from optically induced changes in the refractive index which result either in phase modulation namely Self Phase Modulation (SPM) and Cross Phase Modulation (XPM) or in the mixing of several waves and generation of new frequencies namely Modulation Instability and Four- Wave Mixing (FWM ), (Agarwal 1995) Self Phase Modulation (SPM) Increase in the input intensity changes the refractive index of the medium which in turn results in a phase modulation of the optical beam. This rate of phase modulation produces a frequency chirping resulting in spectral broadening. This non-linear spectral broadening adds to the chromatic dispersion of the fiber in the normal dispersion region of λ < λ ZDWL.. In the anomalous dispersion region of λ > λ ZDWL the non-linear dispersion to some extent cancels chromatic dispersion leading to pulse compression or when exactly balanced results in the formation of solitons ( Agarwal 1995) Cross Phase Modulation (XPM) This effect arises when a change in the intensity of one beam modulates the phase of a co-propagating wave. The chirp induced by the XPM effect can combine with the different parts of the signal pulse traveling with different GVD and result in a multi-peak temporal structure, (Sano et al 2000).

6 Solitons Solitons are generated by the combined influence of SPM and chromatic dispersion. This pulse retains its temporal and spectral shape as it propagates and can travel undistorted over long distances Agarwal (1995). The major type of soliton pulse that occurs in a non-linear fiber is the Dispersion Managed Solitons (DMS), where the intrinsic dispersion of the fiber can be overcome by periodically alternating the dispersion of the fiber so that local GVD remains relatively large compared to average GVD and can compensate for the non-linear dispersion, (Suzuki et al 1995, Jacob and Carter 1997). Hasegawa (2002) has developed the models for the optical solitons in the higher data rate of transmission Parametric Processes and FWM Parametric Processes in non-linear optics involve the energy transfer between the waves leading to χ (3) non-linearity. When the phase matching occurs where they have the same phase velocity this parametric process is enhanced. In such conditions when three waves interact it produces a fourth one and the interaction is between the fields and not between the powers or intensities. The frequencies and wave vectors must be conserved and wave vector conservation is called as phase matching. These conditions are shown in Equations (1.1) and (1.2), (Agarwal 1995). ±ω 1 ± ω 2 ± ω 3 = ± ω 4 (1.1) ± κ 1 ± κ 2 ± κ 3 = ± κ 4 (1.2) Four wave mixing (FWM) is a parametric process where three different frequencies interact to produce a cross talk spectral component that may fall in the desired signal frequency band, (Hill et al 1978).

7 7 1.3 LITERATURE REVIEW The main challenges in the high power and the high data rate optical transmissions especially in DWDM systems are the simultaneous interaction of SPM and XPM, XPM and FWM, SRS and FWM which changes the temporal and spectral characteristics of the pulses. Research on fiber design, transmission modulation format, dispersion mapping, receiver configuration, channel spacing technique, pulse propagation characteristics, Filtering technique etc contribute towards reducing the linear and non-linear distortions and improve system performance. In this thesis a detailed study has been done to investigate the impact of FWM on the performance of DWDM systems with reference to improvement in the spectral efficiency of the optical system. Impact of non-linear effects on pulse propagation and pulse evolution in a fiber are usually analyzed using the classical Non-Linear Schrödinger Equation (NLSE), (Agarwal 2001), which is a non-linear partial differential equation, with a closed form analytical solution existing, only for some specific cases. Several numerical methods were proposed for the investigation of non-linear effects in optical fibers (Taha and Ablowitz 1984). The most frequently used numerical method is the Split-Step Fourier Method (SSFM) (Harvey et al 2002), which belongs to a group of pseudo-spectral models characterized by small computing time with sufficient accuracy. The principle of SSFM is based on separately considering the non-linear and linear propagation effects in a single fiber segment. The SSFM accuracy can be increased by choosing the step width appropriately. A variety of step size selection criteria have been proposed for optimizing the split step method. A constant step size through out the transmission path is considered by Agrawal in The logarithmic step size

8 8 method is designed to efficiently suppress FWM products by Bosco et al (2000). In the walk off method, the step size is chosen to be inversely proportional to the product of the absolute value of dispersion and the spectral bandwidth of the signal (Sinkin et al 2003). This method is efficient for low power multi-channel systems. The proper step size has to be recognized considering the system parameters such as channel spacing, power per channel and number of channels. In this thesis the NLSE is analyzed by keeping constant and variable step sizes, the upper bound being set by the maximum non-linear phase shift, using iterative and non-iterative techniques. Limitations on light wave communications imposed by fiber nonlinearities like SPM, XPM, Stimulated Inelastic Scattering (SRS) effects and FWM effects were studied for the first time by Chraplyvy in 1990, in which the system power limitations has been plotted as a function of optical channels. Each of these non-linearities will affect specific light wave systems in different ways. The effect of fiber non-linearity on long distance communication in which the losses are compensated by the fiber amplifiers was studied by Marcuse (1991). Chraplyvy in 1990, has shown that if two WDM channels are sent in a uniform fiber without dispersion fluctuations, catastrophic build up of four wave mixing occurs, if one of the channel wavelength is at the zero dispersion wavelength. The location of any channel on the zero dispersion wave length is to be avoided to reduce FWM effects (Marcuse 1991). Light modulated in ASK format appears feasible at 2.5 Gb/s if two wavelengths in the normal dispersion wavelength region having 2 nm to 3 nm spacing and show less impact of non-linear effects (Tkach et al 1995).

9 9 In many multi-channel applications low dispersion fibers are utilized which necessitates consideration of XPM effects. Several experimental and analytical results have been presented in the literature highlighting the influence of XPM on WDM networks (Eiselt et al 1999, Hui et al 1999 and Cartaxo 1999). Furst et al (2001) has determined the maximum XPM limited distance on a SSMF and a NZDSF. For both fibers all the fiber spans are fully post compensated by inline DCF where the DCF is optimized to minimize the XPM induced Q penalty for 100 GHz channel spacing. Belloti et al (1998) discussed about the intensity distortion induced by cross phase modulation and dispersion. The effect of Cross phase modulation in multi span WDM optical system has been demonstrated by (Rongquing 2000). In this thesis the emphasis is on the effect of FWM than XPM, since, the influence of FWM quadruples when halving the channel spacing, while XPM effects are approximately inversely proportional to the channel spacing (Shtaif et al 1999). Furthermore signal distortion by XPM can be mitigated by optimal dispersion compensation schemes (Saunders et al 1996), whereas FWM distortions are resistant to dispersion compensation. Hence FWM ultimately limits the channel density and capacity of a DWDM system. An expression was formulated by Marcuse in 1991, for the power P i j k (L) existing in a fiber at length L denoting FWM effect generated due to frequencies f i, f j and f k and also the effect of the channel spacing was studied and concluded that the impact of FWM is low for Unequally Spaced channels (USC). The power loss P i in one channel can be computed by adding the power that has been transferred to frequency f i j k at the end of the fiber with the power lost due to FWM during propagation (Tkach 1995). The effect of FWM can be minimized by reducing the pulse overlapping in the WDM systems (Matera et al 2000).

10 10 The value of P i j k is dependent on the relation between the propagation constants of interacting channels. The difference between propagation constants Δβ for different channels governs the phase matching and the efficiency of FWM (Forgheiri et al 1995) Forgheri(1997). The larger the phase mismatch the smaller is the FWM efficiency and hence SMF performs better than NZDSF considering FWM products. The impact of FWM on the system performance depends on the wavelength at which new spectral components are generated (Shibata et al 1987). Forghieri in 1997 has adapted a design methodology for unequal spacing in which the frequency separation of any two channels is different from that of any other pair of channels. The experimental and theoretical results showed that P i j k depends on the number of channels and becomes constant if the number of channels exceeds a certain value, e.g. number of channels > 24 (Inoue 1991). The FWM effect can also be reduced by using a novel hybrid wavelength and Time division multiplexing techniques (Okada et al 1998). In this thesis the influence of FWM cross talk power in SSMF, NZDSF and DSF is analyzed and compared when the DWDM channels are equally and unequally spaced. Intra-channel FWM (IFWM) is due to the correlation between the spectral components of a single channel. The larger the time separation (bit period) between the pulses, the smaller is the energy transfer (Kumar 2001). This effect will be dominating in the equally spaced pulses (constant phase) where the non-linear interaction between optical pulses is resonant. Therefore it is essential that IFWM be suppressed by alternate phase modulation between adjacent pulses. IFWM represents a dominant performance limitation in systems with a short bit slot e.g. 40 Gb/s systems (Konrad and Petermann

11 ) and a sequential dispersion compensation scheme in which the dispersion compensation is done partly at the transmitter and partly at the receiver side. Hence the optimization of dispersion compensation for example by the use of a certain amount of a pre-compensation (Killey et al 2000) is crucial for their efficient suppression. By pre-chirping of optical pulses at the transmitter side (say by phase modulation), different spacing between pulses and a reduction of the non-linear resonance can be accomplished. In this thesis the effect of varying the DCF length for a fixed length of SSMF fiber is studied under three dispersion compensating schemes namely pre-, post-, and symmetrical compensation and the Q factor is estimated to compare the performances. Generally, In a DWDM system, the spectral efficiency is defined as the ratio of the capacity per channel to the channel spacing. To decrease the channel spacing in order to exploit the entire fiber bandwidth and to improve the spectral efficiency of the system, a modulation format that has the highest tolerance to non-linear effects need to be identified. Bergano et al (1999) studied the wavelength division multiplexing for the long haul system at 900 km with 45 km spacing for the NRZ format at 5 Gb/s for 20 channels. Vareille et al (2001) simulated 50 GHz spacing with 32 channels with 50 km amplifier spacing and 1000 km and improved the Q factor performances using RZ signal format. Since the terabit capacity per fiber has already become commercially viable higher bit rates of either 20 Gb/s or 40 Gb/s is the way forward. Yano et al in 1998 showed that duo binary format together with Polarization Division Multiplexing (PDM) can increase the spectral efficiency up to 1 b/s/hz for 100 km but resistance to non-linear effects like SPM is not so large at 10 Gb/s. NRZ signal modulation format showed a spectral

12 12 efficiency of 0.8 b/s/hz with PDM and 1.2 b/s/hz with vestigial sideband modulation when the de-multiplexer settings are adapted for suppressing PDM (Bigo et al 2001). The results obtained proved to be much worse in the case of long haul transmission systems where the impact of polarization mode dispersion and Cross phase modulation dominated while using the polarization division multiplexing technique. Sano (2000) reduced the impact of non-linear effects by Chirped RZ (CRZ) signal format. To further improve the performance Ohhira et al (2001) discussed about the alternate chirp RZ (AC-RZ) format which compensates the dispersion with alternate values and almost nullifies the residual dispersion and mitigates the effect of SPM. Menyuk and Grigorian (2002) studied the performance and the eye measurement diagram for CRZ systems. Morita et al (2000) worked on the dispersion managed soliton (DMsoliton) and found that in spite of larger resistance to SPM-GVD effect it requires wider bandwidth. Hence this degrades the spectral efficiency of the system at higher date rate and more number of channels. Cundiff et al (1999) has studied the propagation of highly chirped pulses in the presence of the non linear effects. Mingchia (2004) studied the fiber nonlinearity limitations in DWDM systems. Sullivan (1997) presented the impact of modulation instability in multi span optical amplified IMDD systems were pre Blow and Wood (1989) derived a single wave equation relating the transient Stimulated Raman effect in fiber, Also formula for small signal gain spectrum which includes the effect of SRS,FWM and modulation instabilities are derived. Menyuk et al (1991) derived the equations governing the Raman Effect in birefringent fibers both the parallel and perpendicular Raman effects are taken into considerations. Bigo et al (1999) related the effect of the power distribution over various types of fibers and Raman gain coefficient is estimated for the same. Stolen et al (1989) described the

13 13 response function of the Raman gain in time domain and also the effect of Raman gain in the region of anomalous and normal dispersion has been studied. Miyamoto et al (2000) introduced Carrier Suppressed Return to Zero (CS-RZ) format whose performance is further improved by using narrow band optical band pass filters (Miyakawa et al 2001). Vestigial Side Band-RZ (VSB-RZ) format is employed by Tsuritani et al (2002) which can provide large resistance to non-linear effect without increasing the spectral width. Regarding modulation formats for 40 Gb/s DWDM applications, Carrier Suppressed Return to Zero (CS-RZ) is a promising format, because it could be implemented by using conventional balanced receiver, encoder / decoder and so on (Rasmussen et al 2002). CS-RZ is more robust to nonlinearity and band limitation than RZ and NRZ formats. Therefore CSRZ with optical filtering would be one of the solutions for highly spectral efficient Ultra-long-haul (ULH) transmission. The improvement in spectral efficiency has been determined for asymmetrical and symmetrical pre-filtering CSRZ technique by Cai et al (2002) and Agata et al (2003). Non-binary modulation has longer symbol duration than binary and offers improved resistance to transmission impairments, including Chromatic and Polarization Mode Dispersions (PMD) (Wang and Kahn 2004). DPSK has been recently employed for fiber transmission with various formats including NRZ, RZ and CRZ. In the absence of fiber non-linearity all these formats achieve the same SNR performance. However when the input power and data rate is increased the non-linear effects deteriorate the performance and is usually assessed by changes in the Q factor and eye pattern. Tonguz and Wagner (2001) showed that for DPSK, the performance with optical preamplification and interferometric detection is equivalent to that of the

14 14 standard differentially coherent detection. Xu et al (2003) compared the performances of RZ PSK and on off keying in long haul dispersion managed transmissions. In this thesis, the interferometric detection is used for all the subsequent studies on PSK based systems. The impact of ASE and non-linear effects like SPM and XPM on the channel capacity has been studied by Green et al (2002). Mitra and Stark in 2001 argued that the capacity of the DWDM system is limited most fundamentally by XPM; as the signal propagates, chromatic dispersion converts XPM induced phase modulation to intensity noise. Capacity limitations caused by XPM was further studied by Green et al (2002). Using perturbation methods Narimanou and Mitra (2002) found the channel capacity of a single channel system. To quantify the SNR in the presence of XPM noise, The impact of non-linear effects can be reduced or cancelled using phase conjugation (Brener et al 2000). The noise introduced by intermodulation products is equivalent to an increase in the number of ASE photons (Desuvire et al 2002). The quantum limited spectral efficiency depends on SNR and the number of signal photons and the effect of fiber nonlinearity is equivalent to an increase in the number of ASE photons. The interaction of ASE noise with the non-linear phase noise induced by SPM and XPM will affect the signal phase, limiting the performance of systems using phase modulated signals such as DPSK (Kim and Gnauck 2003). The statistical properties of non-linear phase noise depend on the mean non-linear phase shift and SNR (Ho 2003). Xu and Brandt 2002 studied the behavior of asymmetric Gaussian and non Gaussian statistics of the optical communication systems. Song et al (1999) derived a phase matching factor considering the impact of SPM and XPM which shifts the peak FWM efficiency away from

15 15 the zero dispersion wavelength. Distributed Raman amplifiers are used to reduce the ASE noise and improve the OSNR. Neilsen in 2000 studied the performance of an optical system using Raman amplifiers without considering the fiber non-linear effects. In this thesis, a Carrier Suppressed RZ (CSRZ) formatted DQPSK modulated 40-Gb/s DWDM transmission system using Hybrid Raman / Erbium-doped fiber Amplifiers (HREA) considering the non-linear effects, is investigated by simulation and the OSNR and the Q factor are estimated. The narrow line width modulation format like CSRZ-DQPSK is generated by the use of Fiber Bragg filtering in which different transfer functions can be devised by proper combinations of the apodization profile, intensity and pitch of the grating, (Kashyap 1999). Low dispersion filters (LDF) based on FBGs have attracted increased interest over the past few years (Ibsen et al 1998) due to their very high selectivity, flat top spectral response, steep spectral roll off, low insertion loss and flattened in-band group delay profile which permits their use in high speed networks. For these reasons, these filters have found widespread applications in low channel spacing (<50 GHz) Optical Add/Drop Multiplexers (OADM) and hence in this thesis FBG-based LDF DWDM filters are used in our studies for channel spacing of 20 GHz. Foresteri (2000) evaluated the error probability in light wave systems with chromatic dispersion, pre- and post- detection filtering. Menyuk (2003) used Multi-canonical Monte Carlo simulation to obtain accurate BER in optical communication systems.

16 16 Sinkin (2003) studied the effect of non-linearites on various modulation formats like RZ, NRZ and other ASK formats considering SPM, XPM and its impact on BER. Mizuochi et al (2003) made a comparative study of DPSK and OOK DWDM systems over transoceanic distances and their performance degradations due to non-linear noise. Mecozzi (2004) predicted the probability density function of the non-linear noise to be Gaussian distributed in PSK systems. When the fiber non-linearities such as the self phase modulation, cross phase modulation and four wave mixing are not considered, the ASE noise is said to have a Gaussian distribution and when the fiber non-linearities are taken into account the ASE noise is said to have Chi-squared distribution (Ronald Holzlohner 2002) due to the complex interactions between signal and noise components. In this thesis, BER analysis is carried out with both Gaussian and Chi-square noise distributions and comparisons made and the dependency of BER versus SNR on the modulation format is studied. 1.4 THESIS OBJECTIVES The major objectives of the research work presented in this thesis are, To study the impact of non-linear effects on pulse propagation by solving the Non-linear Schrödinger equation using Split Step Fourier method considering constant and varying step sizes and iterative and non-iterative procedures using MATLAB, the proposed variable step size being bounded by the non-linear phase shift. To study the impact of SRS effect in terms of Full Raman Response and Inter-Band Scattering, by modifying the NLSE and to study the effect of step size in the generation of

17 17 spurious non-linear cross talk components bounded by nonlinear phase shift. To explore the concept of unequal channel allocation for reducing the impact of FWM in DWDM systems, wherein the unequal channel allocation based on Optical Orthogonal Codes is studied and its performance with respect to FWM generation in SSMF, NZDSF and DSF based DWDM systems is investigated. To compare the performance of DWDM systems with equal and unequal channel spacing in terms of Q factor for various modulation formats like RZ, CSRZ-DPSK and CSRZ- DQPSK, so as to identify the best format suitable for high bit rate systems. To model the Phase shift keying modulations CSRZ-DQPSK after studying various formats like RZ /NRZ - DPSK and RZ /NRZ DQPSK using MATLAB SIMULINK, and also compares performances when using various filtering techniques like Super Gaussian Higher Order, Arrayed Waveguide Grating and Fiber Bragg Filter. To analyze the effect of varying the DCF length for a fixed length of SSMF under pre-, post-, and symmetrical dispersion compensation schemes and to compare Q factor performance for RZ, NRZ and DQPSK modulation formats so as to identify the best dispersion compensation scheme. To study the spectral efficiency performance of CSRZ DQPSK with hybrid Raman amplification, fiber Bragg for narrow bandwidth filtering and pre-dispersion compensation.

18 18 To compare the BER for various modulation formats, computed using saddle point approximation method using improved numerical techniques like secant method and golden section algorithm and thereby identify the best modulation format for long haul high bit rate DWDM systems, that is robust to dispersion and non-linear impairments. 1.5 THESIS ORGANIZATION The thesis is organized into the following chapters: Chapter 1 presents a brief introduction about DWDM systems and non-linear phenomena in optical fibers. The literature review is presented followed by the objectives of the research work. The impact of non-linear effects on signal/pulse propagation in an optical fiber is studied using the Split Step Fourier method for solving the Non-linear Schrödinger equation considering constant and varying step sizes and iterative and non-iterative procedures. In Chapter 2 the limitations imposed by FWM components on DWDM systems has been studied for equal channel spacing and unequal spacing channel scenarios and the performance towards mitigating the effect of FWM is analyzed for SSMF, NZDSF and DSF fibers. In Chapter 3, the system modeling of various phase shift keying modulation formats have been analyzed under the impact of various filtering techniques, dispersion compensating techniques, channel spacing, fiber type and non-linear effects, with the aim of identifying a suitable modulation format that is robust to linear and non-linear impairments.

19 19 In Chapter 4, a study on the spectral efficiency performance of CSRZ-DQPSK under different amplification mechanisms is undertaken. The ASE noise generated by optical amplifiers decreases the OSNR. Since the distributed Raman amplifiers can reduce the equivalent transmission fiber loss, they are used to improve the Q factor. In Chapter 5, the BER of a phase modulated optical communication system is computed by Saddle Point Approximation method using improved Numerical techniques and the BER curves are compared for various modulation formats to identify the best format. Two major numerical techniques, the Secant Method and the Golden Search Section Algorithm are used to compute the saddle points and the performance of all these numerical techniques is compared in terms of computation time and complexities of the system. 1.6 SIGNAL PROPAGATION IN A NON-LINEAR FIBER An understanding of the pulse propagation and pulse evolution in a non-linear optical fiber is realized by solving and analyzing the Non-linear Schrödinger Equation (NLSE) (Agarwal 1995). In our work the NLSE equation is analyzed with respect to the computation time and step size, related to the non-linear phase shift, and also the impact of SRS effect in terms of full Raman response and inter band scattering. The NLSE taking into consideration third order effects as well as the Raman self shift time is considered for the analysis and is given by (Agarwal 1995),

20 E E 3( 0 ) E E i 2( 0) 2 3 Z t 6 t 2 i 2 E { E E E E R1E 2 t t 0 2 (1.1) In Equation (1.1), E = E (Z, T) is the electric envelope of a frame moving with the group velocity (T = t Z/v g = t-β 1 Z). The first term in the equation takes accounts for the slow changes of the electric field along the fiber length. The second term accounts for the linear losses of optical fiber. The derivatives of the propagation constant of the fiber mode propagation constant with respect to the frequency are, n n 0 n ( ) (1.2) Where n = 1, 2, 3 and β 2 and β 3 are the GVD parameters and ω 0 is the reference frequency. The third term in Equation (1.1) represents the group velocity dispersion and is the effect responsible for pulse broadening. The fourth term is the second-order GVD, also known as third-order dispersion (TOD). The pulse shape becomes asymmetric due to the effect of third order dispersion. The wavelength domain expressions are, D S d d 2c dd d (1.3) The first term in the right-hand side in Equation (1.1) accounts for the self phase modulation effect. It is responsible for the broadening of the pulse spectra and, in the presence of anomalous GVD, for the formation of

21 21 optical solitons Agarwal (1995). The non-linear coefficient determines the magnitude of the non-linear effect and is given by, γ = ( 2π / λ )( n 2 / A eff ) (1.4) where λ is the free space wavelength, A eff is the effective core area and n 2 is the non-linear index. A eff for conventional single mode fibers is 80 µm 2, for dispersion shifted fibers it is 50 µm 2 and for dispersion compensating fibers it is 20 µm 2. The second term in the right-hand side of Equation (1.1) accounts for the self-steepening effect. It leads to an asymmetry in the SPM-broadened spectra of ultra short pulses and is responsible for the formation of optical shocks Agarwal (2002). The last term in Equation (1.1) accounts for the intra-pulse Raman scattering effect with the parameter ζ R1 being the parallel Raman self-shift time. The Raman self shift time ζ R1 and ζ R2 ( Fukuchi 2002) associated with the parallel SRS effect are defined as d I ( ) d m 1111 R1 0 (1.5) where Re(χ 1111 (ω = 0)) = 1 d I ( ) d m 1122 R 2 0 (1.6) The ζ R parameter is related to the slope of the imaginary part of the Raman susceptibility at zero frequency offset (Blow and Wood 1989). The parameter ρ in Equation (1.1) is the fractional contribution of the delayed response of the material to the total non-linearity (Tchofo et al 1998). The

22 22 intra-pulse Raman scattering effect is responsible for the self-frequency shift, that is, the energy transfer from higher to lower spectral components. It also leads to a decay of higher order solitons into its constituents. The intra -pulse Raman scattering plays the most important role among the higher order nonlinear effects Split Step Fourier Method for Solving NLSE Pulse propagation in optical fiber is analyzed by applying the splitstep Fourier method to numerically solve the Non-Linear Schrödinger Equation (NLSE), which includes the effect of first order and second order GVD, self phase modulation (SPM) and cross phase modulation (XPM) due to fiber non-linearity, and fiber attenuation. A simplified version of Equation (1.1) is depicted in Equation (1.7), (Agarwal 1995), IA i2 2 A A i A A 2 Z 2 t 2 (1.7) where A is the complex field envelope, Z is the distance, β 2 is the second order dispersion, and γ is the non-linear coefficient. This method applies the linear operator in the frequency domain and the non-linear operator in the time domain during each step and the field solution is evolved (Agarwal 1995). To understand the Split step method it is useful to rewrite Equation (1.7) as,

23 23 A(Z,t) Z (L N)A(Z,t) (1.8) where L is the linear part and N is the non-linear part. When the electric field has propagated from Z to Z + h, the split step approximation is given by A(Z + h,t)=exp(hl)exp(hn)a(z,t) (1.9) A schematic illustration of SSM method is shown in Figure 1.5. First half Dispersion only (distance h/2) Non-linear Region Symmetrical Second half Dispersion only (distance h/2) < h > < h/ > < h/ > Z=0 Z+h/2 Z+h/2 Z+h Figure 1.5 Schematic illustration of SSFM method (Agarwal 1995) In our study the step size is bounded by the non-linear phase shift and pulse propagation is analyzed based on Equation (1.1) using MATLAB. The exponential propagation model is considered and the calculations are set to be iterative with fixed or variable step size which is bounded by the nonlinear phase shift. The step size determines whether a particular non-linear effect gets accounted for in the model. The step size h is determined through the value of the parameter (Sinkin et al 2003), NL 2 MAX max E h (1.10)

24 24 In the constant step size case, it is calculated once using the input signal s maximum intensity. In the variable step size case, h is calculated at each step ( Ramprasad and Meenakshi 2006) Fiber Simulation Results Using NLSE The study involves simulating the propagation of an optical field in a single mode fiber with the dispersive and non-linear effects taken into account by a direct numerical integration of the modified NLSE. In our simulation using OPT SIMULATION software 4.6 and MATLAB, the effects like attenuation, dispersion and non-linear effects are considered. Attenuation is defined as a fixed constant value. Generally in frequency domain the Group velocity dispersion (GVD) effects are included as β 2 (ps 2 /km) and β 3 (ps 3 /km) and in wavelength domain as dispersion D (ps/nm.km ) and dispersion slope S (ps / nm 2.km). The following two cases are considered for step size selection: (1) Constant Step size with iterative calculations and (2) Variable Step size bounded by the maximum non-linear phase shift with non-iterative calculations. The non-iterative approach is the fastest but the iterative approach with constant step size suppresses instability improving the quality of the results, (Ramprasad and Meenakshi 2006). A Single Gaussian pulse is assumed to be the input signal with 2 mw peak power and having zero chirp parameter. The Gaussian parameter order is set to one for the case of Gaussian and greater than one for Super Gaussian pulses. The initial pulse period is 50 pico-seconds and the wavelength is 1550 nm. The dispersion parameter is -2 ps/nm.km. The length of the fiber is 500 km and the maximum non-linear phase shift is 1 m radian. The optical amplifier is assumed with a flat frequency response and no noise.

25 25 This simulation is carried out for the variable step size bounded by the non linear phase shift using the MATLAB software and the received optical pulse at the fiber output is shown in Figure 1.6. The received power can also be improved by increasing the gain of the receiver amplifier. Figure 1.6 Received power for a length of 500 km Figure 1.7 shows the signal at the fiber output for a constant simulation step size of h = 20, considering the initial peak power of the signal source as 2 mw and the dispersion parameter as - 40 ps / nm.km.. The nonlinear phase shift is 2 m rad. The 'y axis is the power output in the unit of watts and the x axis is the time division in the unit of seconds. The output shows the side tones where the peak value is not converging and hence the step size has to be reduced to suppress these side tones. Peak convergence and falling of spurious components is observed from Figure 1.8 where the step size is reduced to 10. The generation of unwanted components will be reduced and the received output power level will be more in the case of reduced step size. Figure 1.9 shows the received output with an initial peak signal power of 2 mw and the initial pulse period of 50 pico seconds with the step size formulated by the non-linear phase shift bounded technique. The

26 26 higher order dispersion term is also included. The output profile shows asymmetry as expected. Figure 1.7 Received power with the step size h=20 Figure 1.8 Received power with the step size h=10

27 27 Figure 1.9 Received power with inclusion of higher order dispersion In order to compare the impact of step size on FWM and spurious component generation, at THz, power = W fiber of length 780 km with β 2 = -20 (ps 2 /km), effective area 80 m 2, n 2 = m 2 /W and 2 mrad phase shift. The spectrum of the single channel signal obtained at the fiber output for maximum non-linear phase shift of 20 m rad and 2 m rad are obtained as shown in Figures 1.10 and 1.11, respectively. For the first case of large step size iterative calculations are used with constant step size and in the second case the step size is variable and bounded by a smaller value and noniterative calculations are used. The spurious component generation is observed to be less for the variable step size non-iterative calculation and the accuracy is also improved in this case.

28 28 Figure 1.10 Output spectrum with Φ MAX NL = 20 mrad Figure 1.11 Output spectrum with Φ MAX NL = 2 mrad

29 Impact of SRS on Pulse Propagation Raman Scattering The Stimulated Raman Scattering is characterized by the interaction of the signal wave and the molecular oscillations in a non-linear medium Agrawal (2001). There in, certain part of the optical energy is absorbed by the medium and the other part is shifted in frequency due to reduced energy. Raman induced spectral asymmetry and its dependence on the fiber dispersion, in particular, the sign and slope of the dispersion has been studied by Grosz et al (2002) where the third order dispersion is neglected. Raman scattering is classified into Inter band Raman scattering and Full Raman response ( Bigo et al 2001). We have studied the impact of these two effects using the modified NLSE with our proposed varying step size method bounded by the non-linear phase shift. Raman scattering is simulated by the interaction of various signals Analytical Model of Full Raman Response Effect Considering the Full Raman Response effect and neglecting the self steepening effect, Agarwal (2001) Equation (1.1) gets modified as, 2 3 E E ( 30) E E i 2( 0) 2 3 Z T 6 T 2 i (1 )E E h 1111(s) E(T s) ds 0 (1.11) The term h 1111 (s) is the time domain Raman Response function and it is the Fourier transform of Raman susceptibility. For SRS effect the convolution integrals are calculated to represent the interaction of sampled

30 30 signals with sampled signals, noise bins and parameterized signals and vice versa ( Ramprasad and Meenakshi 2006) Analytical model of Inter band Raman Scattering The Equation (1.12) given below considers the effect of Inter-band Raman Scattering, (Agarwal 2001, Press 1992). E E i ( ) E ( ) E 2 3 i i 2 i i 3 i i ( l( i) 1( 0)) ( i)ei 2 3 z T 2 T 6 T Number of SS Number of PS 2 2 (2 ) E (1 ) E (2 ) k i k1 l1 i Number of SS Number of PS (SS) 2 (PS) Rik Ek p R il P l k1 l1 i E i (1.12) In this chapter this equation (1.12) is analyzed with the proposed variable step size bounded by the non linear phase shift and the effect of Full Raman and inter-band scattering is studied. The meaning of the terms on the left-hand side of the equation is the same as in the Full Raman approach fiber model. The first term in the right hand side gives the SPM contribution of the sampled signals, the second term gives XPM contributions of the sampled signals and the third term is the XPM contribution of the Parameterized signals. The fourth and the fifth terms describe the SRS induced interactions between the sampled signal and rest of the sampled signals and with the parameterized signals, respectively. In our analysis it is assumed that the power of noise bins remains much smaller compared to that of sampled and parametric samples of the signal. The parameterized signals also assume the FWM effect. With multiple

31 31 sampled signals present in the fiber the SRS effect is represented through Inter-band Raman Scattering Simulation results In this section, the propagation of an arbitrary configuration of optical signals in a single-mode fiber is simulated by taking into account of Stimulated Raman Scattering (SRS) effects. In order to analyze the stimulated effects, the NLSE equation is modified to represent the evolution of sampled signals, parameterized signals and noise bin signals. Figures 1.12 and 1.13 show the fiber output signal under Raman Scattering and it s spectrum obtained by solving the Equation (1.12). The step size is bounded by the phase shift 2 mrad. Figure 1.14 shows the simulation result for two input pulses. The two sources are at THz and THz with 50 GHz spacing. The input power is 20 dbm, group delay constant of the fiber is ps/km, GVD constant is 4.5 ps/ nm /km, fiber length 300 km, dispersion slope constant 0.11 ps/nm 2 /km, effective area 60 m 2, peak Raman gain coefficient 9.9 e -014 m/w, pump wavelength of peak Raman coefficient 1000 nm and Raman self shift time is 5 fsec. The other simulation parameters are same as that used to solve Equation (1.1) in the above section. Raman scattering leads to the interactions among all sampled signals, parameterized signals and noise bins. Inter-band Raman scattering is an approximation to the full expression of the Raman polarization provided that the frequency separation of the interacting signals is much larger than their individual spectral bandwidths. In Full Raman response, SRS effect will be represented through the convolution integral of time domain Raman response function and the field intensity resulting in a single frequency band confining all the sampled signals being formed. The Figures 1.15 and 1.16 show the Inter-band Raman Scattering and Full Raman Response.

32 32 Figure 1.12 Fiber output signal under inter-band Raman scattering Figure 1.13 Spectrum under inter-band Raman scattering

33 33 Figure 1.14 Input spectrum considering two sources Figure 1.15 Inter-band Raman scattering for two channels

34 34 Figure 1.16 Full Raman scattering for two channels The SRS cross talk impact is more pronounced in the Full Raman Scattering response and hence the impact of Full Response Raman scattering on generating spurious components is more when compared to Inter-band Raman scattering. 1.7 SUMMARY In this Chapter a brief introduction about DWDM systems and nonlinear phenomena in optical fibers is presented. The objectives of the research work are presented. The Split Step Fourier method is applied to solve the NLSE and a comparison of solutions for the constant step size and the variable step size cases bounded by the non-linear phase shift considering exponential propagation model is carried out. Iterative and non-iterative techniques for solving the NLSE using MATLAB are also explored. The generation of inter-modulation spurious components was found to be reduced

35 35 in the case of the variable step size than the case of fixed large step size, the step size being bounded by the non-linear phase shift in both cases. The results are also verified using the simulation package OPTI SIMULATION package. In order to analyze the impact of stimulated in-elastic effects, the NLSE considering the evolution of sampled signals, parameterized signals and noise bin signals, was modified with the proposed varying step size method. Raman scattering is simulated by the interaction of various signals. The effect of Inter-Band Raman scattering and Full Raman response was studied. In terms of the generation of inter-modulation components the Raman Effect due to Full Raman response is found to dominating compared to the Inter-band scattering.

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