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 - 1610 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

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).

1.1.1 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 (LiNbO3) modulator used

in a Mach-Zehnder Interferometer (MZIM) configuration. Figure 1.3

illustrates this modulator.

3

Figure 1.3 LiNbO3 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

E0 applied at the input is divided by a 3-dB coupler into two equal parts E1,

E2, 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

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).

1 0 0 1

5

1.2 NON-LINEAR PHENOMENA IN OPTICAL FIBERS

Non-linearities in optical fibers can be classified as: (1) non-

linearities 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).

1.2.1 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 < λZDWL.. In the

anomalous dispersion region of λ > λZDWL 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).

1.2.2 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

1.2.3 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.

1.2.4 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).

±ω1 ± ω2 ± ω3 = ± ω4 (1.1)

± κ1 ± κ2 ± κ3 = ± 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

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 1995. The logarithmic step size

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 non-

linearities 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

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 Pi j k (L)

existing in a fiber at length L denoting FWM effect generated due to

frequencies fi ,fj and fk 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 Pi in one channel can be computed by adding the

power that has been transferred to frequency fi 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

The value of Pi 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 Pi 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

2001) 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

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 (DM-

soliton) 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

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 non-

linearity 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 pre-

amplification and interferometric detection is equivalent to that of the

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 inter-

modulation 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 non-

linearity 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

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

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

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spurious non-linear cross talk components bounded by non-

linear 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.

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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

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

2 3

3 02 0 2 3

( )E E EE i ( )Z t 6 t

2

2 2R1 2

0

Ei{ E E E E Et t

(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/vg = t-β1Z). 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

0n n

( )

(1.2)

Where n = 1, 2, 3 and β2 and β3 are the GVD parameters and ω0 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,

122

d 2 cDd

dDSd

(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

optical solitons Agarwal (1995). The non-linear coefficient ‘’ determines the

magnitude of the non-linear effect and is given by,

γ = ( 2π / λ )( n2 / A eff) (1.4)

where λ is the free space wavelength, A eff is the effective core area and n2 is

the non-linear index. Aeff for conventional single mode fibers is 80 µm2, for

dispersion shifted fibers it is 50 µm2 and for dispersion compensating fibers it

is 20 µm2. 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

m 1111R1

d ( ) 0d

I (1.5)

where Re(χ1111(ω = 0)) = 1

m 1122R 2

d ( ) 0d

I (1.6)

The ζR 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

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 non-

linear effects.

1.6.1 Split Step Fourier Method for Solving NLSE

Pulse propagation in optical fiber is analyzed by applying the split-

step 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),

222

I A i A A i A AZ 2 t 2

(1.7)

where A is the complex field envelope, Z is the distance, β2 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

A(Z, t) (L N)A(Z, t)Z

(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/2 -------------- > <--------------- h/2 ------------ >

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 non-

linear 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),

2NLMAX max E h (1.10)

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).

1.6.2 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 β2 (ps2/km) and β3 (ps3/km)

and in wavelength domain as dispersion D (ps/nm.km ) and dispersion slope S

(ps / nm2.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

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 non-

linear 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

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

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 193.1 THz, power =0.07325W fiber of length 780

km with β2 = -20 (ps2/km), effective area 80 m2, n2 = 2.6 -20 m2/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 non-

iterative 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

Figure 1.10 Output spectrum with ΦMAXNL = 20 mrad

Figure 1.11 Output spectrum with ΦMAX

NL = 2 mrad

29

1.6.3 Impact of SRS on Pulse Propagation

1.6.3.1 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.

1.6.3.2 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 33 0

2 0 2 3

21111

0

( )E E EE i ( )Z T 6 T

i (1 )E E h (s) E(T s) ds

(1.11)

The term h1111(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

signals with sampled signals, noise bins and parameterized signals and vice

versa ( Ramprasad and Meenakshi 2006).

1.6.3.3 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).

2 33 ii i 2 i i i

l i 1 0 i i 2 3z

( )E E i ( ) E E( ( ) ( )) ( )ET 2 T 6 T

Number of SS Number of PS

2 2k i

k 1 l 1

i iNumber of SS Number of PS

2(SS) (PS)ik k il l

k 1 l 1

(2 ) E (1 ) E (2 )

i E

R E p R P

(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

sampled signals present in the fiber the SRS effect is represented through

Inter-band Raman Scattering.

1.6.3.4 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 193.1 THz and 193.15 THz

with 50 GHz spacing. The input power is 20 dBm, group delay constant of the

fiber is 490000 ps/km, GVD constant is 4.5 ps/ nm /km, fiber length 300 km,

dispersion slope constant 0.11 ps/nm2 /km, effective area 60 m2, 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

Figure 1.12 Fiber output signal under inter-band Raman scattering

Figure 1.13 Spectrum under inter-band Raman scattering

33

Figure 1.14 Input spectrum considering two sources

Figure 1.15 Inter-band Raman scattering for two channels

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 non-

linear 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

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.