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THE NON-LINEAR SCHRODINGER EQUATIONAND SOLITONS
James P. Gordon
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Soliton
John Scott Russell ( Solitary Water Wave –1834)
Zabusky and Kruskal (Solitons -- 1965)
Korteweg and DeVries (KdeV equation – 1895)
Zhakharov and Shabat (NLSE – 1971)
Hasegawa and Tappert (Lightwave Solitons – 1973)
Mollenauer et. al. (Observation in fiber – 1980)
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THE NONLINEAR SCHRODINGER EQUATION
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PHASE AND GROUP VELOCITIESZ=VT
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INVERSE GROUP VELOCITYTIME DELAY = Z/V
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FIBER NONLINEARITY
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DERIVATION OF THE NLS EQUATION
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DERIVATION OF THE NLS EQUATIONStep II: Shift to central frequency and retarded time
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PULSE REPRESENTATION IN ORDINARY VS RETARDED TIME
Pulses at 0 km: Pulses at 5000 km:
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)
Pulses separated by δλ = 0.4 nm in fiber with D = 0.1 ps/nm-km
Pulse at ω0
Pulse at 0.4 nm longerwavelength
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DERIVATION OF THE NLS EQUATIONStep III: Rescale the independent variables
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FOURIER TRANSFORMS
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THE NLS EQN: ACTION OF THE DISPERSIVE TERM
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DISPERSIVE BROADENING OF A GAUSSIAN PULSE VS Z(Minimum temporal width at origin)
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THE NLS EQN: ACTION OF THE NON-LINEAR TERM
t
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SPECTRAL BROADENING OF A GAUSSIAN PULSE ATZERO DISPERSION
(Peak non-linear phase shift indicated next to each spectrum.)
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ORIGIN OF THE SOLITON
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THE DISPERSIVE AND NON-LINEAR PHASE SHIFTS OF A SOLITON
Note that the dispersive and non-linear phase shifts sum to a constant.
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PATH-AVERAGE SOLITONS
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PULSE ENERGY AND FIBER DISPERSIONIN SAMPLE OF TRANSMISSION LINE USED FOR
TEST OF “PATH-AVERAGE” SOLITONS
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SIMULATED TRANSMISSION THRU SYSTEM WITHLUMPED AMPS AND PERIODICALLY VARYING DISPERSION
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DISPERSION RELATION FOR SOLITONS AND LINEAR WAVESSOLITON SPECTRAL DENSITY ALSO SHOWN
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NUMERICAL SOLUTION OF THE NLS EQN:THE SPLIT-STEP FOURIER TRANSFORM METHOD
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CONSTANTS OF THE NLS
uuuu
zui 2
2
2
||21
+∂∂
=∂∂
−
uuW ∫= 2||
)|||(|21 42 u
tudtH −∂∂
∫=
(Pulse Energy)
(Hamiltonian)
(More)
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GAUSSIAN PULSE APPROXIMATION
⎥⎦⎤
⎢⎣⎡ ++−⎟
⎠⎞
⎜⎝⎛= φβηπη itiWu 24
1
)(21exp
φβη and,, may depend on z but not on t
⎥⎦
⎤⎢⎣
⎡−
+=
πη
ηβη
2221 22
WWH
022
141
2
2
=⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛−−=
dzd
dzdWW
dzdH β
ηβη
πηηβ
This result gives a required relation between and dzdβ
dzdη
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βηη 2=dzd
From the NLS:
From the equation for H:5.122
2η
πηββ W
dzd
+−=
To make a soliton: 0=β πη2=W
These equations are very useful in picturing how solitons behave