2.3 Cross Phase Modulation (XPM)
2.3.1 XPM phase shift
The optical phase shift due to XPM can be derived from the nonlinear Schrodinger equation and is given by equation (2.18) as
N z
e
dÇ
(2.24)Unlike the expression for SPM, the phase shift depends on the power Pp=\Apf o f the neighbouring channels p , increasing with channel num ber N. Due to the different velocity o f each channel, the bit-pattem o f the interacting channels is shifted after the propagation as illustrated by Fig. 2.13. This time-dependent alignment o f the channels is characterised by the walk-off. The w alk-off param eter dsp is the most important parameter characterising XPM and is a result o f different group velocities Vg(5) and Vg(p) o f the channels s and p . Low channel w alk-off results in enhanced XPM due to the build-up o f nonlinear phase modulation. It is defined as
r U ^ T T
L
► ( ► (
Fig. 2.13 W alk-off Z)-(2^-2p) between 2 PRBS-m odulated channels at and Xp over SSMF,
dsp<Q\ the detected channel is faster (this example), dsp>0: the interfering channel propagates faster than the detected channel
In the case o f constant dispersion within the interval AÀ^p = expression (2.25) is given by D'AÀsp [CHI96]. For dsp=0 the phase shift reaches a maximum For channels with orthogonal polarisation states, is reduced to a third o f this value as described in section. 2.3.4.
Chapter 2: Theory o f optical fibre transmission 35
The chirp acquired during the interaction o f the two channels is due to transitions in the interfering channel. A rising edge in the interfering channel causes a red-shift towards longer wavelengths as a result of an increased refractive index for the signal under intensity, according to equation (2.12), whilst a falling edge results in blue-shift. However, the shape of the associated intensity distortion o f the detected channel at depends on the sign o f the dispersion as shown in Fig. 2.14.
(a) Generation o f XPM chirp intensity
chirp at À,
pump at Af,
(b) Generation o f distortion for given pulse at A,
D>0
i
(f>s XPMI
PM-IM conversionzxo
intensity Fig. 2.14 intensity timeImpact o f XPM on intensity modulated channels, (a): introduction o f chirp at due to pulse transitions in the neighbouring channel at Ap,(b): PM-to-IM conversion due to fibre dispersion resulting in pulse distortion. Red-shifted chirp is introduced by a rising edge and blue-shifted by a falling edge o f the pulses at Ap.
After the nonlinear interaction, the chirped frequency components o f the pulse propagate with different velocities in the dispersive fibre. The two consequences are timing jitter and amplitude distortion leading to horizontal and vertical eye-closure, respectively. Firstly, timing jitter occurs because o f the different arrival time o f pulse components. This is a result o f the relative delay the chirped components experience during propagation over the fibre following the nonlinear process. Secondly, amplitude distortion occurs due to PM-IM conversion in the fibre. It has been shown in simulations that the impact o f jitter on the BER
Chapter 2: Theory o f optical fibre transmission 36
is low for NRZ signals compared to amplitude distortion, although both processes occur simultaneously [EIS99b]. However, timing jitter is expected to be significant for RZ pulses due to the shorter pulse width. It was also confirmed in experiments which are presented in chapter 4 o f this thesis that intensity distortion dominates jitter for NRZ signals. XPM distortion due to PM-IM conversion, therefore, is the key topic o f this thesis work and is discussed in more detail in section 2.3.5.
The early quantitative analysis o f XPM in optical communication systems has only focused on the phase distortion in equation (2.24) taking into account GVD [CHI94, KAG94]. A model described in [CHI96] can be used to derive as a function o f fibre length, dispersion, AZ and modulation frequency without solving the NLSE. The calculation presented by the authors was restricted to a sinusoidal modulation co o f the interfering pump channel at Pp{t). The probe channel Ps{t) is initially CW, so that
I |2 / >. (2.26)
Pp( 0 , 0 = \A p( 0 ,0 | = Ppo + P p cos{(ot)
Equation (2.26) is substituted into the general expression for given by (2.24) and the XPM phase modulation o f the probe at is obtained as
( ^ , 0 = COS CO t ---i l + (p
v„ V. g J
(2.27)
where cp is a phase retardation factor dependent on co. It is o f importance when contributions o f several sinusoidal components add up to Pp(t). A^'™^ is the amplitude o f the XPM-induced phase shift and can be simplified in the case o f low walk-off a, « 1 /Z as
(2.28) This expression is constant in co and only dependent on the power in the m odulated pump channel. In a long fibre with strong walk-off dsp=D AÀ,, equation (2.28) can be approximated as
(2.29) o}D ^À
The resultant amount o f phase modulation is inversely proportional to the modulation frequency co or walk-off D-AZ since both parameters increase the averaging effect between the bit-pattem in both channels preventing a build-up o f phase modulation. Therefore, the XPM process can be understood to act as a low-pass filter with a \/co- or 1/c/sp-characteristic for the input intensity Pp(co) o f the co-propagating channel. In a similar way, decreases when increasing the channel spacing.
Chapter 2: Theory o f optical fibre transmission 37
2.3.2 Channel w alk-off
The w alk-off d^p is a result o f fibre dispersion and leads to wavelength-dependent group velocities o f the two interacting channels as described by equation (2.25). The parameter dsp=D'AÀsp increases with channel spacing and dispersion resulting in a time-dependent channel alignment. The physical process o f XPM is directly influenced by the sign and the absolute value o f dsp. The sign o f dsp determines the relative shift between the bit-pattems o f the two channels. In this thesis, dsp>0 indicates that the interfering channel at Àp propagates faster than the detected channel at The absolute value o f dsp determines the degree o f averaging during the nonlinear interaction between the two channels. For dsp^ 0 the interfering channel is shifted with respect to the detected channel and pulse transitions, thus, induce both positive and negative chirp on a given pulse section o f the detected channel. As a result o f the interference between opposite chirp components, the build-up o f XPM phase m odulation can be prevented. This averaging effect is also evident in equation (2.24). For dsp=0 the phase modulation can be directly expressed as a function o f Ap(t). However, for dsp^ 0 the argument o f Ap(t) is time-shifted by a factor determined by dsp during the nonlinear interaction. In this case, an integration over the pulse sequence o f Ap(t) must be perform ed to determine
The w alk-off can be normalised to the bit period Tq=1/B when different bit-rates B are considered. In Fig. 2.15 the normalised w alk-off dsp/To is plotted for 2 channels at lOGbit/s as a function o f AT for 60km o f fibre with different dispersion values. The w alk-off increases linearly with AT reaching 10 bits at AT%Inm in the case o f SSMF (D= 16ps/(km 'nm )). In contrast, DSF exhibits a low w alk-off o f less than a bit over the entire interval o f 3nm suggesting a significant build-up o f XPM phase modulation. The slope dsp/(àÀ.’To) is determined by the fibre dispersion D, however it can also be reduced at lower bit-rates.
H igh walk-off and linear attenuation: The attenuation a during the w alk-off determines the build-up o f XPM since the amount o f nonlinear chirp is power dependent. For an idealised case o f CF=0 and dsp^O it can be shown that at some distances no phase modulation is generated as the chirp introduced by a rising edge in the pump channel is completely cancelled by the opposite sign o f chirp associated with a falling edge [EVA99]. For 1010- m odulation in the interfering channel, the nonlinear chirp o f the detected channel is compensated every multiple o f 2'Tq. However, in the case o f attenuation a ^ 0 the absolute value o f the chirp components generated by two subsequent pulse transitions o f opposite chirp is different and exact cancellation will no longer exist. A simple model was derived by
Chapter 2: Theory o f optical fibre transm ission 38
Marcuse [MAR94J to determine the minimum required channel spacing AT so that the walk- off along the effective length L^ff, necessary for averaging, is at least twice the pulse width Tq.
IT .
DL
(2.30)
eff
For SSMF (D=16ps/(kmmm)), Lejf= 20km, 7'o=100ps, the required spacing is at least AT„„„ =0.62nm. The minimum channel spacing AT,„,„ for SSMF, DSF (0.5ps/(kmmm)) and NZ-DSF (4ps/(kmmm)) according to equation (2.30) is included in Fig. 2.15. In the single-span experiments described in this thesis T„„„~ Inm for SSMF and >5nm for DSF. Equation (2.30) therefore only provides an estimation o f AT„„„. One limiting assumption is a constant channel power over However, according to (2.19) the power is reduced by the factor Me after resulting in incomplete compensation o f the chirp. In addition, this model does not take into account the additional chirp due to SPM and the PM-IM conversion.
15 10 5 DSF 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Fig. 2.15 AT [nm]
Normalised channel w alk-off d^/Toas a function o f AT over 60km fibre span at lOGbit/s. SSMF: Z)=16ps/(km-nm), NZ-DSF: Z)=4ps/(knTnm) and DSF: Z)=0.5ps/(km-nm), minimum channel spacing AT„„„ to suppress XPM calculated according to equation (2.30), grey area: low w alk-off <2-Tq within L^ff-
Low walk-off and linear attenuation: In [SHTOO] the effect of small walk-off 0 on XPM IS discussed. The walk-off length was introduced by Shtaif and is defined as = Az/r/^^ where r/^^=D'A/Lo and At denotes the 10-90% rise time o f the pulse transitions. For narrow channel spacing AT„.o, the walk-off length increases due to the small difference in velocity between the two channels. As experimentally shown by the authors and indicated by equation (2.28) the amount of XPM generated becomes independent of dsp for i.e. the amount of XPM does not vary with AT in low dispersion fibre such as DSF or NZDSF (moderate dispersion). In the experiments described in chapter 4 of this thesis this effect was observed for AT<0.5nm in DSF. Therefore, both L\X„,i„{2TçJD=Lcjj) and AT,,.^(T„.o=T^ are important
Chapter 2: Theory o f optical fibre transmission 39
parameters characterising the impact o f XPM in W DM systems in the case o f high and low channel walk-off.