John and canal defined of
Солитоны в океане: The “Rogue” Wave
N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace”, Physics Letters, A 373 (2009) 675–678.
Оптические солитоны
Space: Broadening by Diffraction
Time: Broadening by Group Velocity Dispersion
Broadening +
Narrowing Via a Nonlinear Effect
= Soliton (Self-Trapped beam)
Optical Solitons
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Spatio-Temporal |
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Temporal |
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Spatial |
Homogeneous Media |
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Discrete Media |
1D, 2D
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Cavity Solitons |
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Propagating Solitons |
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Media |
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Kerr n=n2I |
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Liquid Crystals |
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Local Non-local |
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Kerr-like |
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Photorefractive |
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Quadratic |
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Gain Media |
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Пространственные солитоны
Spatial Solitons 2D
Уширение – дифракционная расходимость +
Фокусировка за счет Керровской нелинейности = Солитон
Временные солитоны
Уширение – дисперсия (аномальная) +
Сжатие за счет Керровской нелинейности (самомодуляция) = Солитон
Пространственно – временной солитон «оптическая пуля»
x
t
Electromagnetic pulses that do not spread in time and space
Characteristic Lengths
Temporal Dispersion : LD (T ) T02/ | k2 |
2
Spatial Diffraction : LD (r ) kw0 / 2
Nonlinear Length : LNL [kvacn2Ppeak / Aeff ] 1
Soliton : LNL LD (T ) LD (r )
Soliton period : z0 LD / 2
Require: dispersion length (time) diffraction length (space) nonlinear length
Нелинейное волновое уравнение
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2 E |
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E |
0 |
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{PL |
PNL} |
Kerr(3) |
EEE |
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depends on nonlinear |
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c2 t2 |
t2 |
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mechanism |
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0 (1) E |
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2 |
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E(r ) A(x, y) exp[i{kz |
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t}] |
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2 E n0 |
2 E 2 0PNL |
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c2 |
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spatial |
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2 |
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2 |
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NL |
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Slowly varying phase |
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2ik |
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E |
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P |
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and amplitude |
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z |
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0 |
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diffraction |
nonlinearity |
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approximation (SVEA,1st |
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temporal |
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2 |
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NL |
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order perturbation theory) 2ik |
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E kk |
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E |
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P |
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z |
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2 |
T 2 |
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0 |
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Group velocity dispersion
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Nonlinear Mode |
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| E | 0 |
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z |
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Spatial soliton |
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Солитоны высших порядков
A higher-order soliton is a soliton pulse the energy of which is higher than that of a fundamental soliton by a factor which is the square of an integer number (i.e. 4, 9, 16, etc.).
The temporal shape of such a pulse is not constant, but rather varies periodically during propagation
Когерентное взаимодействие: волна или частица
1 |
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=0 |
= |
Incoherent Soliton Interaction |
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= /2 |
=3 /2 |
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0 |
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0 |
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0 |
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0 |
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0 |
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500 |
0 |
500 |
0 |
500 |
50 |
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out particle-like behavior |
1. |
Number of solitons in = Number of solitons |
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For 0, also wave-like behavior - energy exchange occurs via nonlinear mixing |
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