lengths in space and time-central-difference method 7.3.3 Numerical stability 7.3.4 Gain and phase 7.4 Coupled reflections 7.4.1 Kappa coupling but no gain or phase changes 7.4.2 Matrix formulation 7.4.3 Phase jumps replacing scattering 7.4.4 Fourier checks 7.5 A uniform Bragg laser: finite difference in time and space 7.5.1 Full coupled-wave equations 7.5.2. MATLAB code 7.5.3 Analytic against numeric solutions 7.6 Spontaneous emission and random fields 226 7.6.1 Spontaneous noise and travelling fields 226 7.6.2 Null correlation for different times, positions and directions 7.6.3 Spontaneous magnitude 7.6.4 Tutorial programs 7.7 Physical effects of discretisation in the frequency domain 7.7.1 Discretisation process-integrals to sums 7.7.2 Fast Fourier transform (FFT) 232 7.8 Finite-element strategies for a spectral filter 233 7.8.1 Lorentzian filter 233 7.8.2 Numerical implementation 235 7.9 Application of the filter theory to gain filtering 237 7.9.1 General 237 7.9.2 Filtering the gain in the travelling-wave equations 7.9.3 Numerical implementation 7.10 Basic DFB laser excited by spontaneous emission 7.10.1 Introduction and normalisation 7.10.2 Field equations 243 7.10.3 Charge-carrier rate equation 243 7.10.4 Numerical programs 246 7.11 Summary 248 7.12 References 249 x Contents 8 Future devices, modelling and systems analysis 8.

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