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For most pulse sequences A P /i) cannot be analytically determined, due to the complex form o f the inverse Fourier transform to be applied to the expressions o f A P /cd) given in (2.37) and (2.39). Therefore, analytical calculations can be experimentally verified by measurements in the frequency domain [HUI99]. However, the investigation o f XPM in the time domain is preferred in most cases since the waveform distortion can be linked to eye closure and the impact o f system parameters on XPM is evident. The inverse Fourier transform o f equation (2.37) is known for only a few special cases such as sinusoidal m odulation in the pump channel. An expression for rux was derived based [SCH99] based on expression (2.37). The sinusoidal pump is assumed to be undistorted by SPM and GVD, e.g. in the case o f a single span. For o/DÀ^L/(4m:)<^\ and a * Z » l equation (2.37) can be simplified by sin{bL)^ olDX^LI{Akc) and cos{bL)^\. In the time frame moving with Vg it reduces to

àPXœ) =

(0)) •

(0, a>\e-’^ \- L

\ a y

An analytical solution o f (2.42) can be obtained for a pump channel given by

(2 42)

(2.43) The sinusoidal waveform approximates the 1010-bit pattern well in case of, for example, a bandwidth limitation due to the optical modulator and receiver. APs(L,t) is obtained by substituting the Fourier transform o f equation (2.43), Pp(0,co), into (2.42) and taking the

Chapter 2; Theory o f optical fibre transmission 51

inverse Fourier transform. The expression for nix is then given by the peak-to-peak amplitude

o f A P s (L ,t) normalised to the time-averaged power <Ps>

= yP ,pO

Kc^Ia"' + co^D^ •

(2.44)

In Fig. 2.24 the analytical expression (2.44) for is compared with the results o f a split-step Fourier simulation for Z=60km SSMF, lOdBm/channel, CK=0.21dB/km, T=1555nm,

16.5ps/(kmmm), ;^1.18/(W*km) and 1010...-m odulation at lOGbit/s. A good agreement with less than 15% error is achieved between AT=0.4nm and A/l=1.6nm demonstrating the accuracy o f equation (2.44). The small under-estimation o f the calculated distortion is due to the smoother transitions o f the sinusoidal pump approximation in comparison to the Bessel- filtered pulses used in the simulation.

In summary, several techniques have been proposed to enable fast and accurate estimation o f XPM-induced distortion. The XPM effect can be adequately described by the transfer function Hsp(co). Although the different analytical approaches provide simple expressions for

the intensity spectrum P s(co), this parameter is difficult to relate to waveform distortion and

eye closure measured in the experiments. The inverse Fourier transform required for obtaining P / t ) o f the probe channel has analytical solutions only for a limited number o f

waveforms such as sinusoidal pump approximation but it determines nix accurately for single span links. For longer distances the accuracy o f these techniques can be improved taking into account pump distortion due to SPM and GVD.

0 .4 ■ simulation - analytical 0 . 3 - 0.2- 0 .5 ^ ° A;i[nm ]

Fig.2.24 Comparison o f analytical solution and simulation for rrix vs.AÀ, pum p-probe configuration, Z.=60km SSMF, 10 1 0 ...-m odulation at lOGbit/s, lOdBm/channel

Chapter 2: Theory o f optical fibre transmission 52

2.4 Summary

In this chapter, linear and nonlinear effects affecting W DM transmission systems were discussed. Fibre attenuation and dispersion require periodic signal amplification and dispersion compensation to increase transmission distance. The Kerr-effect is a result o f the intensity dependence o f the refractive index at high channel powers and leads to the combination o f SPM and XPM, under study in this thesis. SPM and XPM result in a phase variation due to power fluctuation in the same and the neighbouring channels, respectively. The nonlinear phase modulation due to SPM and XPM is converted into signal distortion by the fibre dispersion leading to eye closure and increase o f the bit error-rate.

The channel walk-off is important in characterising XPM. For low walk-off, maximum XPM phase modulation is generated. Increasing fibre dispersion increases the PM-EM conversion efficiency but also reduces the initial XPM phase modulation. It was also shown that residual dispersion is important for the understanding o f XPM and dispersion. This relatively complex nature o f the XPM effect is the main motivation for a detailed study o f XPM as a function of typical system parameters in the following chapters o f this thesis. XPM intensity distortion can be calculated using analytical techniques which provide insight into the physical nature of XPM. In chapter 4 the XPM intensity distortion is measured as a function o f bit-rate in a 2- span link and compared with the analytically calculated transfer function H(co). In chapter 5 the cross-phase modulation index rrix is calculated as a function o f distance and channel spacing and the results are used to predict the impact o f XPM on the Q-factor in a 2-channel transmission experiment.

Chapter 3: Cross-phase modulation - literature review 53

Chapter 3

Cross-phase modulation -