As highlighted earlier, in Section 3.2, the magnitude of the step-size parameter must be set small enough so that the errors associated with the numerical approximation are not too great. Too small a step-size however, although producing very accurate results would needlessly lengthen the computational effort and time.
In [9] simulation was performed for single channel transm ission and an upper bound to the step-size was set by the condition that the nonlinear phase-shift (proportional to
pow er) due to self-phase modulation (equation (2.27)) m ust not exceed n m rad, ensuring
the amplitude distortion as a result o f phase modulation in the nonlinear and dispersive system was dominated by the physical processes not numerical error. W hen more than one channel is present the magnitude of the step-size becom es even more important^. In multi-channel systems a stepsize too large can very easily over-estimate the effect o f the nonlinearity within the system, specifically in the generation o f FW M power.
The approximate analytical solution for pow er generated by four-w ave mixing (see Section 2.4.3) reveals its oscillatory nature whereby pow er is continuously extracted from the input frequencies to generate the FW M power and then transferred back to the original frequencies over a certain distance, or beat length, proportional to 1/Af5 w here Afi is the phase-mismatch parameter. W ith regard to the step-size, if it is much smaller than this beat-length then the transfer of pow er to and from the FW M products will be accurately calculated. W hereas, if the step-size is comparable to, or larger than, the beat length then the FW M pow er will be incorrect. Indeed, some resonance conditions may occur as the step-size matches the beat length, enhancing the FWM pow er at each step, i.e. w hen the
sin^ argument o f the FWM efficiency process (equation (2.45)) equals a multiple o f 7i
with L replaced by the step-size, [10]
Ês.pL = \p^A,(û^dz = n 2 n (3.9)
These unphysical resonances are analogous to the real situation where amplifier position can enhance FW M by appropriate location at multiples o f the FW M beat length [11] (see chapter 4).
To examine the effect of the step-size on the generation o f FW M pow er a simple transm ission system is simulated with the step-size varied: tw o C W channels spaced lOOGHz apart w ere propagated over fibre with dispersions o f 5, 20 and 100 ps/nm /km . The first-order generated frequencies appeared at 100 G Hz either side of the input frequencies, as expected. The FW M pow er at the end of the fibre for increasing step-size is plotted in Figure 3.2 (length given in the caption). It show s both the maximum step- size which gives accurate results and the over-estimation o f FW M pow er (by 2 to 5 orders of m agnitude). As the dispersion increases so the FW M pow er diminishes (the phase- mismatch parameter increases, reducing FWM efficiency) but the maximum permitted
^ B rought to the au th o r’s attention in discussion w ith C .G lingener, U niv eristat D ortm und, D -44227 D ortm und.
stepsize decreases. For 5ps/nm/km dispersion there is negligible effect in reducing the step-size below 0.1km whereas for lOOps/nm/km it m ust be reduced to 0.01km . As the FW M beat length is inversely proportional to the phase-mismatch param eter, larger dispersion creates a smaller beat length and hence, as given in equation (3.9), the step- size must be reduced to avoid resonance and maintain numerical accuracy.
The effect of step-size has also been shown to cause M odulation Instability (MI) [12]. As discussed in Section 2.4.4 a periodic nonlinear phase grating allows phase matching for the M I process in normal dispersion fibres. In this instance the phase-grating is induced through the computational step-size, producing instability sidebands w hose frequencies are inversely proportional to the step-size. The step size must be chosen such that the MI sideband frequencies occur outwith the spectral window of the computation.
le-I le-2 l e - 3 « le-4 le-5 X — 0.001 ower) F W M ( u p pe r) 0.1 S t e p - s iz e ( k m ) (a) (lower FWM (upper) l e - 6 - k 0.1 S t e p - s iz e ( k m ) ( b ) ie -l (lower) F W M 0.001 0.1 S t e p - s iz e ( k m ) ( c)
F ig u re 3 .2 U p p e r antd lo w e r first o r d e r F W M p o w e r ( a r b itr a r y u n its ) as a f u n c tio n o f th e n u m e ric a l ste p -s iz e , fo r tw o 5 m W CW c h a n n e ls s p a c e d lO O G H z a p a rt, a fte r fib r e w ith d is p e rs io n a n d le n g th o f (a) 5 p s /n m /k m a n d 5 0 k m (b ) 2 0 p s /n m /k m a n d 5 0 k m (c) lO O p s/n m /k m , 10km .