Anharmonic Oscillator Derivation of Second Order Susceptibilities
INTRODUCTION TO SINGULAR NONLINEAR OPTICS...Group-velocity dispersion, diffraction and third-order...
Transcript of INTRODUCTION TO SINGULAR NONLINEAR OPTICS...Group-velocity dispersion, diffraction and third-order...
INTRODUCTION TO SINGULAR NONLINEAR OPTICS
FSU-Jena, Abbe School of Photonics’2011
LECTURE 1: Linear vs. nonlinear optics. Optical solitons.
LECTURE 2: Singular optical beams. Dark optical solitons– physics and applications.
LECTURE 3: Interactions between optical solitons.
LECTURE 4: Polychromatic spatial solitons.
INTRODUCTION TO SINGULAR NONLINEAR OPTICS
Linear vs. nonlinear optics. Optical solitons.
FSU-Jena, Abbe School of Photonics’2011
Linear vs. nonlinear optics. Optical solitons. Singular optical beams.
1. GVD, diffraction and - susceptibilities.2. Physical mechanisms of some third-order nonlinearities.3. Feynman diagrams and z-scanner for calculating and measuring
third-order nonlinear susceptibilities.4. Heuristic derivation of the nonlinear Schrödinger equation (NLSE).5. Numerical and approximate analytical procedure for solving the
NLSE.6. Optical solitons (exact analytical results, bright and dark solitons).
)3(χ
FSU-Jena, Abbe School of Photonics’2011
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
WikipediA: “In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or, alternatively, when the group velocity depends on the frequency”.
Note that:There is no source which emits spectrum containing only one frequency.
The single-frequency continuous-wave (cw) lasers emit radiation with a finite bandwidth.
In the lasers emitting transform-limited pulses their sensitivity to higher order dispersion is very strong pronounced.
Since the refractive index has dispersion, the wavenumber has dispersion too:
)()(2)(2)( ωωωπνωλπω n
cn
cnk ===
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
)()(2)(2)( ωωωπνωλπω n
cn
cnk ===
...)(61)(
21)()( 3
03
32
02
2
00
0
00
+−⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛+= ωω
ωωω
ωωω
ωω
ωωd
kdd
kdddkkk
0 0 0 0
100 0
( ) 1 ( ) grgr
nndk d dn dnn n Vd d c c c d c d cω ω ω ω
ωω ω ω ωω ω ω ω
−⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = + = + = =⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
/phV c n= /gr grV c n=0
0 0( )grdnn nd ω
ω ωω
⎛ ⎞= + ⎜ ⎟⎝ ⎠
phase velocity group velocity group refractive index
HIERARCHY : :
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
...)(61
)(21
)()( 303
32
02
2
00
0
00
+−⎟⎠
⎞⎜⎝
⎛+−⎟
⎠
⎞⎜⎝
⎛ ∂+−⎟
⎠
⎞⎜⎝
⎛+= ωω
ωωω
ωωω
ωω
ωωd
kdd
kddk
kk
/phV c n= /gr grV c n=
phase velocity group velocity group velocity dispersion
HIERARCHY :
0 0 0 0
2 2 2 3 2
2 2 2 2 2
1 22
d k dn d n d n d nd c d d c d c dω ω ω λ
ω λωω ω ω ω π λ
⎛ ⎞ ⎡ ⎤ ⎛ ⎞= + ≈ =⎜ ⎟ ⎜ ⎟⎢ ⎥
⎝ ⎠ ⎣ ⎦ ⎝ ⎠
0 00
2
2 2 2
1 1
gr
d k dd d Vω
gr
gr
dVV d ωω
βω ω
⎛ ⎞⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ω
⎛ ⎞− ⎜ ⎟
⎝ ⎠
2β=GVD
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
Example: SiO2 ; Sellmeier formula: ∑= −
+=m
j j
jjBn
122
22 1)(
ωωω
ω
500 700 900 1100 1300 15001,44
1,45
1,46
1,47
1,48
1,49
n
ngr
n(λ)
, ngr
(λ)
Wavelength (nm)500 700 900 1100 1300 1500
-12-8-4048
12162024
Wavelength (nm)d2 k/
dω2 (p
s2 /km
)
2 2( / 0) 1.27D d k d mλ ω μ≈ ≅
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
Engineering definition of DISPERSION:
Influence of the GVD:
Does not enrich the pulse spectrum.
Can create certain frequency distribution within the pulse envelope , i.e. chirp.
Pulses without any initial chirp always broaden in time.
In the same medium the pulse broadening depends on the initial pulse shape.
( ) 2
22 2 2
2 /1 1 1 2 2d cd d d d c d k cDL d L d d L d d d
π λτ τ ω τ π π βλ ω λ ω λ λ ω λ
⎛ ⎞= = = = − = −⎜ ⎟
⎝ ⎠
2( ) ( )sign D sign β= − [ ] 22 /fs cmβ = [ ] ( )2 / .D fs cm nm=
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
-8 -6 -4 -2 0 2 4 6 80,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4 z/LD
0 2 4
Inte
nsity
(arb
. uni
ts)
Time (arb. units)
-6 -4 -2 0 2 4 60,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4m=3 z/LD
0 1 2
Inte
nsity
(arb
. uni
ts)
Time (arb. units)
20 2/DL t β= - dispersion length
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
0,0 0,5 1,0 1,5 2,00,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0
3
1
2
t/t0
z/LD
Change of the duration of an input Gaussian pulse with no initial phase modulation (1), (2) with an initial phase modulation with the same sign as this caused by the group velocity dispersion, and (3) with an initial phase modulation opposite in sign to this caused by the group velocity dispersion.
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
By definition diffraction is each deviation of the straight light propagation, which is not due to reflection or refraction (even in gradient refractive index media).
When the refractive index changes at a distance of the order of , we use the term scattering.
( )∇ = +L na
k a2 2
02Δ
λ
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
( )∇ = +L na
k a2 2
02Δ
I0
I0/4x
I
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
Discrete diffraction - diffraction of light in the course of propagation light along periodic structure of strongly-coupled waveguides.
PRL 81 (16), pp. 3383-3386 (1988).
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
( ) 2/2 aLDiff λπ=
diffraction length
FSU-Jena, Abbe School of Photonics’2011
1. Group-velocity dispersion, diffraction and third-order nonlinear susceptibilities.
Generally, the polarization can be considered as a sum of a linear and nonlinear component:NLL PPPrrr
+=i.e. as expanded in a series with respect to the electric field amplitude
...:. )3()2( +++= EEEEEEPrrr
Mrrrr
χχκ
In the general case
...)3()2( +++= lkjijklkjijkjiji EEEEEEP χχκ
ijκ(2) (3), ,...,ijk ijklχ χ
- linear susceptibility
- n-th order nonlinear susceptibilities
,
.
,
where
FSU-Jena, Abbe School of Photonics’2011
2. Physical mechanisms of some third-order nonlinearities.
Local
Nonlocal
2224)3(
12)2(
)1(
/10~/10~
1~
VmVm
−
−
χ
χ
χ
R. W. Boyd, Nonlinear Optics (Academic, 2008).
FSU-Jena, Abbe School of Photonics’2011
3.1. Feynman diagrams for calculating third-order nonlinear susceptibilities.
g
g
m
n
ν
ω
ω
ω
3ω
Energy-level diagram of sodium.
Third-harmonic
Generation process.
Double-sided
Feynman diagram.
R. B. Miles, S. E. Harris, IEEE JQE 9, 470 (1973). N. B. Delone, V. P. Krainov, Fundamentals of nonlinear optics of atomic gases (Wiley, 1988 ).
FSU-Jena, Abbe School of Photonics’2011
3.1. Feynman diagrams for calculating third-order nonlinear susceptibilities.
l
l
m
n
ν
ω
ω
ω
3ω
N. B. Delone, V. P. Krainov, Fundamentals of nonlinear optics of atomic gases (Wiley, 1988 ).
( )( )( )ωωωωωωωωωωχ
ν
νν
32~),,;3( 3
)3(
−−− lnlml
ll
kn
jmn
ilm
ijklrrrr
h
FSU-Jena, Abbe School of Photonics’2011
3.1. Feynman diagrams for calculating third-order nonlinear susceptibilities.
N. B. Delone, V. P. Krainov, Fundamentals of nonlinear optics of atomic gases (Wiley, 1988 ).
( ) ( )( )( )[ ] ( )( )( )[ ]{( )( )( )[ ] ( )( )( )[ ] }11
113
,,
)3(
232
232/1
),,;3(
−−
−−
++++++−
++−−+−−−
×= ∑
ωωωωωωωωωωωω
ωωωωωωωωωωωω
ωωωωχ
νν
νν
ννν
lnlmllnlml
lnlmllnlml
nm
ll
kn
jmn
ilmijkl rrrr
h
l
l
m
n
ν
ω
ω
ω
3ω
l
l
m
n
ν
ω
ω
ω
3ω
l
l
m
n
ν
ω
ω
ω
3ω
l
l
m
n
νω
ω
ω
3ω
+ + +
FSU-Jena, Abbe School of Photonics’2011
3.1. Feynman diagrams for calculating third-order nonlinear susceptibilities.
( ) ( )( )( )[ ] ( )( )( )[ ]{( )( )( )[ ] ( )( )( )[ ] }11
113
,,
)3(
232
232/1
),,;3(
−−
−−
++++++−
++−−+−−−
×= ∑
ωωωωωωωωωωωω
ωωωωωωωωωωωω
ωωωωχ
νν
νν
ννν
lnlmllnlml
lnlmllnlml
nm
ll
kn
jmn
ilmijkl rrrr
h
nli
Near two-photon resonance:
Γ+ nliΓ+
Y. Prior, IEEE J. Quant. Electron., QE-20, 1, 37 (1984).
Bates, Damgaard, Proc. R. Soc. London, 242, 101 (1949).
Bebb, Phys. Rev. 149, 1, 25 (1966).
FSU-Jena, Abbe School of Photonics’2011
3.1. Feynman diagrams for calculating third-order nonlinear susceptibilities.
R. B. Miles, S. E. Harris, IEEE JQE 9, 470 (1973).
FSU-Jena, Abbe School of Photonics’2011
3.2. Z-scanner for measuring third-order nonlinear susceptibilities.
M. Sheik-Bahae et al., Optics Letters 14, 955-957 (1989).
025.0)1(406.0 ΔΦ−=Δ − ST vp
( )22 /2exp1 aa wrS −−=
αα )exp(1)()( 00
Ltnkt −−Δ=ΔΦ
FSU-Jena, Abbe School of Photonics’2011
3.2. Z-scanner for measuring third-order nonlinear susceptibilities.
M. Sheik-Bahae et al., JOSA B 11, 1009-1017 (1994).
FSU-Jena, Abbe School of Photonics’2011
SI vs. ESU units
[ ] ( ) 2/)1(3)( / −=
nn ergcmχ [ ] ( ) )1()( / −= nn Vmχ
[ ] [ ] [ ] [ ]esun
esunnsmc
Wm )3(20
20
2 0395.0/
40/ χπγ ==
ESU SI
n=1 n=2 n=3 n=4
12.6 4.19.10-4 1.4.10-8 1.56.10-17)()( / nESU
nSI χχ
FSU-Jena, Abbe School of Photonics’2011
4. Heuristic derivation of the nonlinear Schrödinger equation (1-D NLSE).
In the nonlinear optics, in media with cubic (e.g. Kerr) nonlinearity,
2 2( , ) ; ( , )n n E k k Eω ω= =
Taylor series expansion
0 0
222
0 0 0 22
1( ) ( ) ....2
k k kk k EEω ω
ω ω ω ωω ω∂ ∂ ∂
− = − + − + +∂ ∂ ∂
Electromagnetic (optical) field amplitude :
( ) ( )0 0 0exp{ [ ]}E E i t k k zω ω= − − −
Correspondence between the differential operators and the multipliers in the Taylor series :
( ) ( )( ) ( )
( ) ( )
0 0
0 0
2 22 2 2 20 0
/ /
/ /
/ /
E z i k k E z i k k
E t i E t i
E t E t
ω ω ω ω
ω ω ω ω
∂ ∂ = − − ⇒ ∂ ∂ ↔ − −
∂ ∂ = − ⇒ ∂ ∂ ↔ −
∂ ∂ = − − ⇒ ∂ ∂ ↔ − −
.
:
FSU-Jena, Abbe School of Photonics’2011
2 22
22 2
12
E k E k E ki E Ez t t Eω ω
∂ ∂ ∂ ∂ ∂ ∂= − − +
∂ ∂ ∂ ∂ ∂ ∂
The formal substitution in the Taylor series yields
Recalling that
and
.
0 00
2
2 2 2
1 1 gr
gr gr
dVd k dd d V V dω ωω
βω ω ω
⎛ ⎞ ⎛ ⎞⎛ ⎞= = = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠grVd
dk 1
0
1 =⎟⎠⎞
⎜⎝⎛=
ωωβ
we get (2) 2
222
1 02gr
E E E ki E Ez V t t E
β∂ ∂ ∂ ∂+ − + =
∂ ∂ ∂ ∂.
The heuristic element - Kerr type nonlinearity:2
0 2n n n E= +
(2) 22
22
1 02gr
E E Ei n Ez V t t c
β ω∂ ∂ ∂+ − + =
∂ ∂ ∂E 1-D NLSE
FSU-Jena, Abbe School of Photonics’2011
4. Heuristic derivation of the nonlinear Schrödinger equation (3-D NLSE).
For a spatially confined optical beam
In this case
Electromagnetic (optical) field amplitude :
20
2220 zyx kkkk ++=
The paraxial approximation means that 20
22 kkk yx <<+
z
yxz
z
yxz
z
yxz k
kkk
kkk
kk
kkkk
0
22
020
22
020
22
00 2211
++=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ++≈
++=
Taylor series expansion
( ) ( ) 22
202
2
00
22
00
0021
2E
Ekkk
kkk
kkkkz
yxz
∂
∂+−
∂∂
+−∂∂
≈+
−−≈− ωωω
ωωω ωω
[ ]{ }ykxkzkktiEE yxz −−−−−= )()(exp 000 ωω
FSU-Jena, Abbe School of Photonics’2011
Correspondence between the differential operators and the multipliers in the Taylor series :
20
2220
22
00
00
222222
222222
)(/)(/
)(/)(/
)(/)(/
//
//
ωωωω
ωωωω
−−↔∂∂⇒−−=∂∂
−−↔∂∂⇒−−=∂∂
−−↔∂∂⇒−−=∂∂
−↔∂∂⇒−=∂∂
−↔∂∂⇒−=∂∂
tEtE
itEitE
kkizEkkizE
kyEkyE
kxEkxE
zz
yy
xx
The formal substitution in the Taylor series yields
.022
11 222
2)2(
2
2
2
2
0
=∂
∂+
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂ EE
Ek
tEE
yxkE
tVzi
zgr
β
FSU-Jena, Abbe School of Photonics’2011
For Kerr nonlinearities
In this way we get the (3+1)-D NLSE in its most simple form
.022
11 222
2)2(2
0
=+∂∂
−∇+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
⊥ EEnct
EEk
EtVz
izgr
ωβ
In the paraxial approximation
22 ncE
k ω=
∂
∂
zkk 00 ≈
(2) 2 2 22
2 2 20
1 ( )2 2
i fz t x yψ β ψ ψ ψ ψ
β⎛ ⎞∂ ∂ ∂ ∂
− + + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
Comment: Space-time analogy
FSU-Jena, Abbe School of Photonics’2011
5. Numerical and approximate analytical procedure for solving the NLSE.
G. P. Agrawal, Nonlinear fiber optics (Academic, Boston, 1989).
( Split-step Fourier method ≡ Beam propagation method )
02
2202
22 =+− AAnkA
tA
zi
∂∂β
∂∂
00
,,IAu
Lz
Tt
Disp
=== ςτ
0202
20 1,
InkLTL NLDisp ==
β 22
00202 βTInkLLN NLDisp ==
02
)( 222
22 =+∂∂
−∂∂ uuNusignui
τβ
ς
1-D NLSE :
Normalized
coordinates :
NLSE in
“soliton”
variables:
FSU-Jena, Abbe School of Photonics’2011
G. P. Agrawal, Nonlinear fiber optics (Academic, Boston, 1989).
∂∂ςu
D N u= +( )) )
)D i
sign= −
( )β ∂
∂τ2
2
22
2uiN =)
The method : Dispersion only Nonlinearity only
u(ζ, τ)
h
u h hD hN u( , ) exp( ) exp( ) ( , )ς τ ς τ+ ≅) )
[ ] }{exp( ) ( , ) exp ( ) ( , )hD u F hD i F u) )
ς τ ς τ= −1 Ω
Error due to the splitting of the operators - of the order of h2.
Can be enhanced to the order of h3 by an iterative procedure.
FSU-Jena, Abbe School of Photonics’2011
G. P. Agrawal, Nonlinear fiber optics (Academic, Boston, 1989).
-9 -6 -3 0 3 6 9
t/T0
z/LDisp0
0.5
1
4
3
2
1
0
|u|2
Nice test for checking the accuracy: Propagation of a bright temporal soliton with N=3.
R. H. Stolen et al., Opt. Lett. 8, 186 (1983).
FSU-Jena, Abbe School of Photonics’2011
5. Numerical and approximate analytical procedure for solving the NLSE.
NLSE L
Oiler-Lagrange equation
Probefunctions
<LG> δ∫<LG>dx=0
System of ODEsfor the variational
parameters
D. Anderson, Phys. Rev. A27, 3135-3145 (1983).
FSU-Jena, Abbe School of Photonics’2011
NLSE L
Oiler-Lagrange equation
Trialfunctions
<LG> δ∫<LG>dx=0
System of ODEsfor the variational
parameters
22
2NLi k
x∂ ∂ψ α ψ ψ ψ∂ ∂τ
= + L 42
**
2)2/( ψ
∂τ∂ψα
∂∂ψψ
∂∂ψψ
NLkxx
i +−⎥⎦
⎤⎢⎣
⎡−=
* * *( / ) ( / )x xψ τ ψ τ ψ⎡ ⎤∂ ∂ ∂ ∂ ∂
+ −⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦L=0
D. Anderson, Phys. Rev. A27, 3135-3145 (1983).
FSU-Jena, Abbe School of Photonics’2011
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−= 22
2
)()(2
exp)(),( τττψ xibxa
xAx
∫∞
∞−= τdGG LL
⎭⎬⎫
⎩⎨⎧
+⎟⎠⎞
⎜⎝⎛ +−+⎥
⎦
⎤⎢⎣
⎡−= 4
422332*
*
2114
2Aak
abAa
dxdbaA
xAA
xAAia NL
G α∂∂
∂∂πL
0=∫ dxGLδ
aAk
aab
dxdba
abdxda
NL 2
32
224
4
+−=
−=
αα
α
Trial function:
Result: System of ODEs for the variational parameters
D. Anderson, Phys. Rev. A27, 3135-3145 (1983).
FSU-Jena, Abbe School of Photonics’2011
D. Anderson, Phys. Rev. A27, 3135-3145 (1983).
aA
kadx
ad NL2
3
2
2
2
24 αα+=
.22)(;0)(21 0
2
22
constaEk
aaa
dxda NL +−=Π=Π+⎟
⎠⎞
⎜⎝⎛ αα
aAk
aab
dxdba
abdxda
NL 2
32
224
4
+−=
−=
αα
α
Dsol
NL
IAka
dxdbxb
constxaxadxda
12
2 22
|| 0 and 0)(
.)0()( i.e. 0→=⇒
⎪⎪⎭
⎪⎪⎬
⎫
==⇒
====α
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons (exact analytical results, bright and dark solitons).
02
)( 22
22 =+∂∂
−∂∂ AA
TAsign
zAi βNLSE in “soliton”
Variables (N=1):
Case A: sign(β2)=-1 and n2>0
)exp()(),( iKzTBTzA =
B(T) – Real amplitude, which does not change along the propagation path length
The substitution in the NLSE leads to
021 3
2
2
=+− BKBdT
Bdi.e. to 0
4241 422
=⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−⎟⎠⎞
⎜⎝⎛ BKB
dTdB
dTd
( ) .2sech2)( 2 2 constKTKTBBKBdTdB
+=⇒−±=
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons (exact analytical results, bright and dark solitons).
( ) ( )iKzKTKTzA exp2sech2),( =Therefore
( ) 0/lim and 0lim == ±∞→±∞→ dTdBB TT
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons (exact analytical results, bright and dark solitons).
( ) ( )iKzKTKTzA exp2sech2),( =Therefore
T
A
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons (exact analytical results, bright and dark solitons).
Case A: sign(β2)=-1 and n2>0 ( ) ( )iKzKTKTzA exp2sech2),( =
Case B: sign(β2)=+1 and n2<0 ( ) ( )ziUTUUTzA 2000 exptanh),( =
xx
xx
xx eeeet
eet
t −
−
− +−
=−
== )tanh( ; 2)sech()cosh(
1
-4 -2 0 2 40,0
0,2
0,4
0,6
0,8
1,0
|A|2
T
0,0
0,5
1,0
1,5
2,0Ph
ase,
rad
-4 -2 0 2 40,0
0,2
0,4
0,6
0,8
1,0
|A|2
T
0,0
0,5
1,0
1,5
2,0
Phas
e, r
ad
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons (exact analytical results, bright and dark solitons).
Case A: sign(β2)=-1 and n2>0 ( ) ( )iKzKTKTzA exp2sech2),( =
Case B: sign(β2)=+1 and n2<0 ( ) ( )ziUTUUTzA 2000 exptanh),( =
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The early history in brief
“ I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped -not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along thechannel apparently without change of form or diminution of speed. ...Such, in the month of August 1834, was my first chance interview with that a singular and beautiful phenomenon which I have called the Wave of Translation.“
John Scott Russel, Reports on Waves
== The wave speed is proportional to the amplitude thereby showing, that higher waves travel faster.
== Russel determined the shape of the solitary wave to be that of a sech2-function.
== He discovered the solution to an as yet unknown equation!
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The early history in brief
1870 – Boussinesq and Rayleigh independently derive expressions for the shape and the speed of this wave
1875 – Korteweg and deVries derive the corresponding partial differential equation (KdV-equation).
Soliton at the Scott RusselAqueduct near the Heriot-Watt
University, Edinburgh (July 12, 1995).
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The history in brief
1870 – Boussinesq and Rayleigh independently derive expressions for the shape and the speed of this wave
1875 – Korteweg and deVries derive the corresponding partial differential equation (KdV-equation)
mid sixties - Zabusky and Kruskal (Princeton University) named these waves solitons
Exact definition : A soliton is a large amplitude coherent pulse of very stable solitary wave, the exact solution of a wave equation,
whose shape and speed are not altered by a collision with other solitary waves.(Only localized solutions of exactly integrable one-dimensional systems are called solitons. )
Physical definition : The solitary waves and solitons can be understood as a balance between the effect of dispersion and that of nonlinearity.
Localized excitations described by inexactly integrable nonlinear equations are termed solitary waves.
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The history in brief
1870 – Boussinesq and Rayleigh independently derive expressions for the shape and the speed of this wave
1875 – Korteweg and deVries derive the corresponding partial differential equation (KdV-equation)
mid sixties - Zabusky and Kruskal (Princeton University) named these waves solitons
03
3
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂ U
zzU
tβ
0),(2
22
22
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+∂∂ tzG
zmti ψψhh
Equation Field of validity History
Solitary waves on the water surface in narrow channels.
KortewegDe Vries(1895)
In a plasma placed in a strong magnetic field there may propagate solitons.
In the theory of superfluidity – waves in a Bose gas.
Phenomena in the nonlinear optics.
Sagdeev (1958)Kadomtsev andKarpmann (1973)
Ginzburg and Pitaevskii (1961)
Zakharov and Schabat (1972)
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The history in brief
1870 – Boussinesq and Rayleigh independently derive expressions for the shape and the speed of this wave
1875 – Korteweg and deVries derive the corresponding partial differential equation (KdV-equation)
mid sixties - Zabusky and Kruskal (Princeton University) named these waves solitons
1972 – Zakharov and Shabat develop the inverse scattering theory
1973 – Hasegawa and Tappert proposed that solitons could be used in optical communication systems
1977 – Operation of the first commercial optical communication system
1988 – Mollenauer and co-workers demonstrate soliton transmission over 6000 km without repeaters
1992 – Bell Labs research team transmitted solitons error-free at 5 Gb/s over more than 15.000 km
FSU-Jena, Abbe School of Photonics’2011
6. Optical solitons – The history in brief
1870 – Boussinesq and Rayleigh independently derive expressions for the shape and the speed of this wave
1875 – Korteweg and deVries derive the corresponding partial differential equation (KdV-equation)
mid sixties - Zabusky and Kruskal (Princeton University) named these waves solitons1972 – Zakharov and Shabat develop the inverse scattering theory
1973 – Hasegawa and Tappert proposed that solitons could be used in optical communication systems
1977 – Operation of the first commercial optical communication system
1988 – Mollenauer and co-workers demonstrate soliton transmission over 6000 km without repeaters
1992 – Bell Labs research team transmitted solitons error-free at 5 Gb/s over more than 15.000 kmFirst observation in the temporal domain:
1987 –P. Emplit et al. (Opt. Commun. 62, p. 374)1988 – A. Weiner et al. (Phys. Rev. Lett. 61, p. 2445)
1D- and quasi-2D experiments:1990 – D.- Andersen et al. (Opt. Lett. 15, p. 783)1991 – G. Swartzlander, Jr.(Phys. Rev. Lett. 66, p. 1583)
There are so many open questions …
Thank you for your attention!