linear vs. nonlinear optics. optical solitons. - abbe school of

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.


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



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 ===



)()(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 : :



...)(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



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≈ ≅



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=



-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



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.



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



( )∇ = +L na

k a2 2

02Δ

I0

I0/4x

I



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).



( ) 2/2 aLDiff λπ=

diffraction length



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


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).


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 ).



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



N. B. Delone, V. P. Krainov, Fundamentals of nonlinear optics of atomic gases (Wiley, 1988 ).

( ) ( )( )( )[ ] ( )( )( )[ ]{( )( )( )[ ] ( )( )( )[ ] }11

113

,,

)3(

232

232/1

),,;3(

−−

−−

++++++−

++−−+−−−

×= ∑

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

+ + +



( ) ( )( )( )[ ] ( )( )( )[ ]{( )( )( )[ ] ( )( )( )[ ] }11

113

,,

)3(

232

232/1

),,;3(

−−

−−

++++++−

++−−+−−−

×= ∑



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).



R. B. Miles, S. E. Harris, IEEE JQE 9, 470 (1973).


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


3.2. Z-scanner for measuring third-order nonlinear susceptibilities.

M. Sheik-Bahae et al., JOSA B 11, 1009-1017 (1994).


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


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

∂ ∂ = − − ⇒ ∂ ∂ ↔ − −

∂ ∂ = − ⇒ ∂ ∂ ↔ −

∂ ∂ = − − ⇒ ∂ ∂ ↔ − −

.

:


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


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


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


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


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:



∂∂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.



-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).


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).


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



⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−= 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




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


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

+=⇒−±=



( ) ( )iKzKTKTzA exp2sech2),( =Therefore

( ) 0/lim and 0lim == ±∞→±∞→ dTdBB TT



( ) ( )iKzKTKTzA exp2sech2),( =Therefore

T

A



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



Case A: sign(β2)=-1 and n2>0 ( ) ( )iKzKTKTzA exp2sech2),( =

Case B: sign(β2)=+1 and n2<0 ( ) ( )ziUTUUTzA 2000 exptanh),( =


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!


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).


6. Optical solitons – The history in brief


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.






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)






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





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!

linear vs. nonlinear optics. optical solitons. - abbe school of

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