?? ??a

1
?? ??aµµ??? ?pt??? ??µata se ?µ??e?? ?a?
pe???d??? µ?sa

• ???ast???? ???sµat?? ??e?t??????? ??sµ?? ?a? ??
  G?aµµ???? ?pt????
• ???

????? ??s?????, G?????? ??µ????, ?a?a??t??
?apa??????, ???????? ??t?a??d??
2
T?µata e? s??t?µ?a

1. Ge???? pe?? µ? ??aµµ???? ??µ?t??, µ? ??aµµ????
   ?pt???? pa?µ?? ?a? solitons
2. ?? ??aµµ???? ?pt???? pa?µ?? se ?µ??e?? µ?sa 21D
3. ?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
   µ?sa-??e?µat??? s???t???a

3
Ge???? pe?? µ? ??aµµ???? ??µ?t??, µ? ??aµµ????
?pt???? pa?µ?? ?a? solitons

• S???t???a, p??t? pa?at???s? ?? ?e?? st? a?????,
  1834

??apa??stas? pa?at???s?? Scott Russell,
Heriot-Watt University 1995

• S???t???a Ge????? ?a?ad????
• ??t?p?sµ??a ??µat?pa??ta µe µ??f? a?a?????t? ?
  pe???d??? µetaßa???µe??
• ??µ????????ta? ap? t?? aµ??ßa?a e??s????p?s?
  fa???µ????
• ?e? µetaß????? t? p??t??, t?? e????e?a, ? t??
  ta??t?ta t??? µet? ap? µeta?? t??? s??????se??,
  pa?aµ????? a?a?????ta sa? s?µat?d?a, e??? ?a? ?
  ???? soliton (1965)
• ???a? µ? ??aµµ???? ??t?t?te?

• ?a?at???se?? Russell
• ? ??µat?sµ?? e??a? e?t?p?sµ????, sta?e??? se
  p??t?? ?a? d?at??e?ta? ??a µe???e? ap?st?se??
• ? ta??t?ta e?a?t?ta? ap? t? p??t?? ?a? t? ß????
  t?? ?e???
• ?? ??µat?sµ?? a?t?? t?? t?p?? de?
  s?ss?µat????ta?, a?t??eta ap? ta s????? ??µata

N.J.Zabusky, M.D.Kruskal, Phys.Rev.Let. 15, 240,
1965 N.N. Akmediev, A.A. Ankiewicz, Solitons
Nonlinear pulses and beams (Chapman and Hall,
1997) E.Infeld, G.Rowlands, Nonlinear waves,
solitons and chaos (Cambridge university press,
1990) R.W.Boyd, Nonlinear Optics (Academic Press,
1992)
4
Ge???? pe?? µ? ??aµµ???? ??µ?t??, µ? ??aµµ????
?pt???? pa?µ?? ?a? solitons
?a fa???µe?a p?? e??s????p???ta? e??a? ? a??µa??
d?asp??? ?a? ? µ?-??aµµ??? ap????s? t?? µ?s??
?a?µ?? e?s?d??
?p????s? µ?s??
?a?µ?? e??d??

• S???t???? s?µa??e? ?s????p?a, s???? ?a?????
  asta???
• ??µata ?a? d?ata?a???, a??µa ?a? se ?s????
  µ?-??aµµ??? µ?s? de? s??µat????? apa?t?t?
  s???t???a
• ?? ??aµµ??? fa???µe?a ?at? t? d??d?s? d?ata?a???
  se ?µ??e?? µ?sa ?a? ? ?pa??? s???t?????
  µe?et??ta? e?tetaµ??a se t?µe?? ?p??
• ??-??aµµ??? ?pt???
• ???d?s? d?a????? st? p??sµa
• ?e?st?µ??a???? ??a t?? s??µat?sµ? ts????µ?? ?a?
  freak waves

d?asp??? (? pe????as?)
a?t?-est?as?
soliton
t
z
f?te???
s??te???
collision
5
Ge???? pe?? µ? ??aµµ???? ??µ?t??, µ? ??aµµ????
?pt???? pa?µ?? ?a? solitons
???te??p???s? µ? ??aµµ???? ??µ?t??
PDEs, ?? ??aµµ???? (f?s???)
??e? ????? a?a?????te? s???t?????? ??se??

• Korteweb-de Vries
• Kadomstev-Petviashvili
• Sine-Gordon

6
?? ??aµµ???? ?pt???? pa?µ?? se ?µ??e?? µ?sa, d??
?a? t???? d?ast?se??
?p?ped?? ??µat?d???? 1 ? 2 e????s?e? d?ast?se??
3D µ?s? 2 ? 3 e????s?e? d?ast?se??
NLSE 31 d?ast?se??
7
?? ??aµµ???? ?pt???? pa?µ?? se ?µ??e?? µ?sa, d??
?a? t???? d?ast?se??
?µ??, se µ?sa µe ?a?????? d?asp??? de? ????µe
collapse (de? ????µe ?a? soliton ß?ßa?a...)
S??se?? ??asp????-pe?????? ast??e?a?
???at?t?ta d?µ??????a? ?a? e?????? pa?µ?? ?a?
a?t???? p?? ?a pa?aµ????? a?a?????ta ?
t??????st?? s???e?t??µ??a ??a ??p??e? ap?st?se??
Bidispersive ?a µ?sa p?? eµfa?????? a?t??eta
p??s?µa pe????as?? ?a? d?asp????

• ?????µ?t? ???es? ??µ?t?? X
• ?????ep?d?as? ?a? ??e???? pa?µ??-a?t????

L.W.Liou, X.D.Cao, C.J.McKinstrie, G.P.Agrawal
Phys.Rev.A 46, 4202, 1992
8
?? ??aµµ???? ?pt???? pa?µ?? se ?µ??e?? µ?sa, d??
?a? t???? d?ast?se??
?????µ?t? ???es? ??µ?t?? t?p?? X

• ??µata ?
• a?a?????te? ??se?? t?? ??aµµ???? ??µat????
  e??s?s?? µe ?pe??? e????e?a
• s??a?t?s?a?? e??a? ?????sµa s??a?t?se?? Bessel
• p??? d?s???? ?a a?apa?a?????

?p???e? d??at?t?ta ???es?? t??? ap? ??????
pa?µ??? (p.?. G?a??s?a????, sech ? CW)
????µ?t??? ep???s? NLSE 21D ??????? pa?µ??
Gaussian ( as?e??? CW)
P2Pc CW 0.1? ?fp/2 ?? ??aµµ???
P2Pc CW 0.1? ?f0 ?? ??aµµ???
P4Pc no CW
?? ??aµµ???
G?aµµ???
J.Salo, J.Fagerholm, A.T.Friberg, M.M.Saloma,
Phys.Rev. E 62, 4261, 2000
9
?? ??aµµ???? ?pt???? pa?µ?? se ?µ??e?? µ?sa, d??
?a? t???? d?ast?se??
?????ep?d?as? ?a? ??e???? pa?µ?? ?a? a?t????,
pa???s?a ???µ?st???? CW
??????????? µet??es? pa?µ??

• ?? ap?t??esµa t?? a????ep?d?as?? e?a?t?ta?
• T?s?, f?s?, ?s??, ????a a?????? pa?µ??
• F?s? ?a? ?s?? CW
• ????a??a f?s??? fa???µe?a
• ?????ep?d?as? pa?µ??
• Bidispersion
• ???s??s? p?e?????? fasµat???? pe?????? (µ?
  ??aµµ???t?ta)
• ?? pa?µ?? e??d??
• ??af??et??? e????s?a ?a? ??????? µetat?p?s?
• Fasµat??? µetat?p?s?

Input, CW 0.2A ?fp, f1p/2, f2-p/2
Input, CW 0.2A ?fp
Input, CW 0.2A
Input, CW 0.2A ?f0
10
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
?fa?µ???? st? µ? ??aµµ??? ?pt???
F???µata se ?pt???? ??e? (Fiber gratings)
S?st????e? ?? G?aµµ???? ??µat?d???? (a) 1D
AlGaAs, (b) 2D silica glass
?? G?aµµ???? F?t?????? ???sta???
(b) ?pt??? epa???? (s?µß??? 4 ep?ped?? ??µ?t??)
(a) e????s?? p??f??
Review papers D.N. Christodoulides et al,
Discretizing light behaviour in linear and
nonlinear waveguide lattices, Nature 424, 817
(2003) A.A. Sukhorukov et al, Spatial Optical
Solitons in Waveguide Arrays, IEEE J. Quant.
Electron. 39, 31 (2003) J.W. Fleischer et al,
Spatial photonics in nonlinear waveguide arrays
, Opt. Express 13, 1780 (2005)
11
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
?d??t?te? ??e?µat???? S???t?????

• ?a µa??µat??? µ??t??a p?? pe?????f??? t? d??d?s?
  ??e?µat???? S???t????? se µ?sa µe e????s?a
  a??µ??????e?a e??a? µ?-????????s?µa. ?e t??
  a?st??? µa??µat??? ?????a de? ?p?????? s???t???a!
  ?p?????? ?µ?? e???sta e?t?p?sµ??a µ?-??aµµ???
  ??µata.
• ? p?????a s???t?????? ??µ?t?? se µ?-??aµµ???
  p???µata ??e? p???t??? d?af??et??? ?a?a?t???st???
  ap? t?? pe??pt?s? µ?-??aµµ???? ?µ???µ??f?? µ?s??.
• ? e????s?a a??µ??????e?a t?? µ?s?? s??ep??eta?
  ap??e?a t?? ?d??t?ta? µetaf?????? s?µµet??a?
  (translational invariance) µe ap?t??esµa
• ?e?????sµ??? ????t???t?ta s???t?????
• S??µat?sµ? t?? s???t????? se s???e???µ??e? ??se??
  se s??s? µe t?? ?e?µet??a t?? p???µat??
• ?p? te????????? ?p??? ????? ?d?a?te?? e?d?af????
• ?pa?t??? s?µa?t??? µ????te?? ?s?? ??a t??
  eµf???s? µ?-??aµµ???? ?d??t?t?? ?a? t??
  s??µat?sµ? t???
• ?p????? ?a ???s?µ?p??????? se efa?µ????
  d??µ?????s?? ?a? µeta????? ?pt???? s?µ?t?? se
  aµ???? ?pt???? s?s?e???

?fa?µ???? - S?ed?as? / ?atas?e?? (engineering)
S???et?? F?t?????? ??µ?? µe ep???µ?t??
?d??t?te? - ???aµ???? ?pt???? ??e???? (d??aµ???
e?a?t?µe?? ap? t?? ?s??, ??e???? µe ?pt??? s?µata
(p.?. XPM)
12
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
T?s? ?a? ??st??e?a S???t????? se S???ete?
F?t?????? ??µ??
Te????µe µ?a f?t????? d?µ? ?p?? t?s? ?? ??aµµ????
?s? ?a? ?? µ?-??aµµ???? ?d??t?te? t?? µ?s?? e??a?
e????s?a a??µ????e?e??
n0(x), ??aµµ???? de??t?? d????as?? n2(x),
µ?-??aµµ???? de??t?? d????as?? e, d?ata?a?t???
pa??µet???
????? ??a St?s?µe? ??se??
???aµ??? s?st?µa
Hamiltonian 11/2 ßa?µ?? e?e??e??a?
13
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
?? ad?at??a?t? s?st?µa (e0) ??a ßgt0 ??e? µ?a
?µ???????? t????? p?? a?t?st???e? st?
st?s?µ? s???t???? t?? NLSE ??a ???e x0.

• ? ?µ???????? t?????
• s??µat??eta? ap? t?? ?e?a ???s? t?? e?sta????
  ?a? t?? asta???? p???ap??t?ta? t?? sa?µat????
  st?s?µ?? s?µe??? st? µ?d??
• e??a? ??e?st? ?aµp??? p?? ap?te?e?ta? ap? ?pe??a
  µ?-e????s?a (nontransverse) s?µe?a t?µ??
  (?µ???????? s?µe?a) t?? d?? p???ap??t?t??

14
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
? pa???s?a d?ata?a??? (e?0) ??e? sa? ap?t??esµa
t?? ?s???? t??p?p???s? (sp?s?µ?) a?t?? t??
e?a?s??t?? t?????? ?a? t?? eµf???s? s?µe???
e????s?a? t?µ?? t?? d?? p???ap??t?t??.
? s????t?s? Melnikov M(x0) e??a? a?????? t??
ap?stas?? d(x0) t?? d?? p???ap??t?t?? ?p?? a?t?
µet??ta? p??? se µ?a t?µ? Poincare.

• ?? µ?de??sµ?? t?? s????t?s?? Melnikov
• a?t?st?????? se ?µ???????? s?µe?a
• p??sd???????? ??a t? d?ata?a?µ??? s?st?µa ta
  d?a???t? µ??? t?? (a????? s??e????) ???????e?a?
  ??se?? µe pa??µet?? x0

S. Wiggins, Introduction to Applied Nonlinear
Dynamical Systems and Chaos, Springer (2003)
15
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
?e??pt?s? n0(x)cos(x), n2(x)0
G?a ??e? t?? pe??pt?se?? e0.1

• G?a ??a ta ß
• ??sta??? s???t???? e?t?p?sµ??? st? ??s? µe??st??
  t?? n0(x), x00
• ?sta??? s???t???? e?t?p?sµ??? st? ??s? e?a??st??
  t?? n0(x), x0p

16
?? ??aµµ???? ?pt???? pa?µ?? se pe???d???
µ?sa-??e?µat??? s???t???a
?e??pt?s? n0(x)cos(x), n2(x)-4.8cos(x)
?d??? a???µ?? s???t?????, st?? ?d?e? ??se??,
d?af??et???? t?p?? e?st??e?a? ??a ß 0.1, 1.
????t?s? t?? e?st??e?a? ap? ?s?? / ??????
e???? / sta?e?? d??d?s??
17
???ast???? ???sµat?? ??e?t??????? ??sµ?? ?a? ??
G?aµµ???? ?pt???? ??? nmoshon_at_central.ntua.gr ?a?.
???????? ??t?a??d?? kyriakos_at_central.ntua.gr