THE NON-LINEAR SCHRODINGER EQUATION AND SOLITONS James P. Gordon 1 Soliton John Scott Russell ( Solitary Water Wave –1834) Korteweg and DeVries (KdeV equation – 1895) Zabusky and Kruskal (Solitons -- 1965) Zhakharov and Shabat (NLSE – 1971) Hasegawa and Tappert (Lightwave Solitons – 1973) Mollenauer et. al. (Observation in fiber – 1980) 2 THE NONLINEAR SCHRODINGER EQUATION 3 PHASE AND GROUP VELOCITIES Z=VT 4 INVERSE GROUP VELOCITY TIME DELAY = Z/V 5 FIBER NONLINEARITY 6 DERIVATION OF THE NLS EQUATION / 7 DERIVATION OF THE NLS EQUATION Step II: Shift to central frequency and retarded time 8 PULSE REPRESENTATION IN ORDINARY VS RETARDED TIME Pulses separated by δλ = 0.4 nm in fiber with D = 0.1 ps/nm-km Pulses at 0 km: Pulses at 5000 km: Pulse at ω0 Pulse at 0.4 nm longer wavelength 0 100 200 300 400 25,000,000,100 25,000,000,350 Ordinary Time (ps) 0 100 200 300 400 0 100 200 300 400 Retarded Time (ps) 9 DERIVATION OF THE NLS EQUATION Step III: Rescale the independent variables 10 FOURIER TRANSFORMS 11 THE NLS EQN: ACTION OF THE DISPERSIVE TERM 12 DISPERSIVE BROADENING OF A GAUSSIAN PULSE VS Z (Minimum temporal width at origin) 13 THE NLS EQN: ACTION OF THE NON-LINEAR TERM t 14 SPECTRAL BROADENING OF A GAUSSIAN PULSE AT ZERO DISPERSION (Peak non-linear phase shift indicated next to each spectrum.) 15 ORIGIN OF THE SOLITON 16 THE DISPERSIVE AND NON-LINEAR PHASE SHIFTS OF A SOLITON Note that the dispersive and non-linear phase shifts sum to a constant. 17 PATH-AVERAGE SOLITONS 18 PULSE ENERGY AND FIBER DISPERSION IN SAMPLE OF TRANSMISSION LINE USED FOR TEST OF “PATH-AVERAGE” SOLITONS 19 SIMULATED TRANSMISSION THRU SYSTEM WITH LUMPED AMPS AND PERIODICALLY VARYING DISPERSION 20 DISPERSION RELATION FOR SOLITONS AND LINEAR WAVES SOLITON SPECTRAL DENSITY ALSO SHOWN 21 NUMERICAL SOLUTION OF THE NLS EQN: THE SPLIT-STEP FOURIER TRANSFORM METHOD 22 CONSTANTS OF THE NLS ∂u 1 ∂2u − i = + | u |2 u ∂z 2 ∂u 2 W = ∫| u |2u (Pulse Energy) 1 ∂u H = ∫ dt(| |2 − | u |4 ) (Hamiltonian) 2 ∂t (More) 23 GAUSSIAN PULSE APPROXIMATION 1 ⎛η ⎞4 ⎡ 1 2 ⎤ u = W ⎜ ⎟ exp − (η + iβ )t + iφ ⎝π ⎠ ⎣⎢ 2 ⎦⎥ η , β , and φ may depend on z but not on t 1 ⎡η 2 + β 2 η ⎤ H = W ⎢ −W ⎥ 2 ⎣ 2η 2π ⎦ dH 1 ⎡⎛ β 2 W ⎞ dη 2β dβ ⎤ W ⎜1 ⎟ 0 = ⎢⎜ − 2 − ⎟ + ⎥ = dz 4 ⎣⎝ η 2πη ⎠ dz η dz ⎦ dβ dη This result gives a required relation between and dz dz 24 dη From the NLS: = 2βη dz dβ 2 2 W 1.5 From the equation for H: = β −η + η dz 2π These equations are very useful in picturing how solitons behave To make a soliton: β = 0 W = 2πη 25.