falk lederer and oleg egorov - uni-dortmund.detdohnal/iciam_talks/lederer.pdf · falk lederer and...

Discrete Discrete Cavity Cavity SolitonsSolitons

Falk Lederer and Oleg Egorov

Institute of Condensed Matter Theory and Solid State Optics

Friedrich-Schiller-Universität Jena, Germany

mail: [email protected]

web: www.photonik.uni-jena.de

F. Lederer, FSU Jena, Discrete Cavity Solitons, Zürich 07/07

CreditsCredits

Most of the work was done by Oleg Egorov

- U. Peschel, O. Egorov, and F. Lederer, ‘Discrete cavity solitons’, Opt. Lett. 29 (2004) 1909

- O. Egorov, U. Peschel, and F. Lederer, 'Mobility of discrete cavity solitons', g , , , y y ,Phys. Rev. E 72 (2005) 066603

- O. Egorov, U. Peschel, and F. Lederer, 'Discrete quadratic cavity solitons’ Phys. Rev. E 71 (2005) 056612 y ( )

- O. Egorov and F. Lederer, and K. Staliunas, ‘Sub-diffractive discrete cavity solitons’, Opt. Lett. (to appear August 1)

O E d F L d d Y S Ki h ’ H d i li d- O. Egorov and F. Lederer, and Y. S. Kivshar,’ How does an inclined holding beam affect discrete modulational instability and soliton formation in arrays of nonlinear cavities? ‘ Opt. Exp. 15 (07) 4149

The work was supported by a grant of the Deutsche Forschungs-gemeinschaft, Forschergruppe 532.


CreditsCredits -- CollaborationsCollaborations

- U. Peschel, O. Egorov, and F. Lederer, ‘Discrete cavity solitons’, Opt. Lett. 29 (2004) 1909

- O. Egorov, U. Peschel, and F. Lederer, 'Mobility of discrete cavity solitons', g , , , y y ,Phys. Rev. E 72 (2005) 066603

- O. Egorov, U. Peschel, and F. Lederer, 'Discrete quadratic cavity solitons’ Phys. Rev. E 71 (2005) 056612 y ( )

- O. Egorov and F. Lederer, and K. Staliunas, ‘Sub-diffractive discrete cavity solitons’, Opt. Lett. (to appear August 1)

O E d F L d d Y S Ki h ’ H d i li d- O. Egorov and F. Lederer, and Y. S. Kivshar,’ How does an inclined holding beam affect discrete modulational instability and soliton formation in arrays of nonlinear cavities? ‘ Opt. Exp. 15 (07) 4149

The work was supported by a grant of the Deutsche Forschungs-gemeinschaft, Forschergruppe 532.


OutlineOutline

1 I d i1. Introduction

2 Resting Discrete Cavity Solitons (DCS)2. Resting Discrete Cavity Solitons (DCS)

3. Mobility of Resting DCS vs. Moving DCS

4. Sub-diffractive DCS


Cavity Solitons Cavity Solitons

INTRACAVITY FIELD

TRANSMITTED FIELDREFLECTED FIELD

FIELD

PUMP FIELD


Cavity Cavity Solitons Solitons

wide aperture passive resonators

2 22

NL2 2 in ( , )1 ( ) ( , )2

i u t xi u u t xt x y

⎡ ⎤⎛ ⎞∂ ∂ ∂+ + + + +α χ =⎢ ⎥⎜ ⎟ Δ∂ ∂ ∂⎝ ⎠⎣ ⎦

F. Lederer, et al.

Phys. Rev. A56, R3366 (1998)

L. Lugiato et al.

Phys. Rev. Lett. 79, 2042 (1998)

EU project FUNFACSEU project FUNFACSJ Tredicce M Giudici Univ Nice


J. Tredicce, M. Giudici, Univ. Nice

Discrete Solitons Discrete Solitons waveguide arrayg y

( ) 2u∂ ( )12

1 0n n nn nC u uui u uz + −

∂+ γ =

∂+ +

D N Christodoulides and R I Joseph

Y. Silberberg et al., Phys. Rev.Lett. 91(98) 3383

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13 (1988) 794


Discrete Cavity SolitonsDiscrete Cavity Solitons

wide pump field

Bragg mirrors

+ mirrors → feedback and radiation losses

array of nonlinear waveguides

+ pump field → energy supply


Array of Nonlinear Waveguide ResonatorsArray of Nonlinear Waveguide Resonators

PPLNPeriodically Poled Lithium Niobate

AlGaAs below half band gapLithium Niobate

Bragg mirrorsg p

W Sohler Paderborn

fast quadratic nonlinearity

I S Aitchison Toronto

fast cubic nonlinearity


W. Sohler, Paderborn I.S.Aitchison,Toronto

Array of Coupled Array of Coupled Defects Defects in in PCs PCs Lithium Niobate – quadratic NL

pump

H. Hartung, E.-B. Kley, A.Tünnermann, p pW. Wesch, FSU Jena

Semiconductor – cubic NL

A. Scherer et al.Appl Phys Lett 79 4289 (2001)


Appl. Phys. Lett 79, 4289 (2001)

Single Single Cavity ResponseCavity Response

mean-field model for the single cavity E0

20( )

ui i u u u ET

∂+ + Δ + γ =

∂

upumplosses

detuningKerr NL

bistability

|u|2← stable← unstable 3 solutions for 3Δ ⋅ γ < −

|E0|2← stable


Array of Array of Coupled CavitiesCoupled Cavities

evanescent coupling

( , )x yenu 1nu +1nu −



evanescent coupling


Mean-field and tight-binding approximations

( ) 2 01 1 (2 )n n nn n n nC u u uui i u u u ET + −

+∂

+ + Δ +∂

+ =− γ

Mean field and tight binding approximations

T∂

coupling or discrete diffraction


Array of Array of Coupled Cavities Coupled Cavities -- LimitsLimits

evanescent coupling



22

02

( ) (( ) ) ( ) ( )u xi i u x uD u x uT x

x E∂ + + Δ + γ∂

=∂

∂+


T x∂∂

‘ordinary‘ continuous model for wide beams and normal incidence

“continuous” limitC → ∞

Cavity Solitons


Array of Array of Coupled Cavities Coupled Cavities -- LimitsLimits

evanescent coupling




+∂

+ + Δ +∂

+ =− γ


T∂

“anti-continuous” limitC=0C 0

arbitrary shapes



evanescent coupling




+∂

+ + Δ +∂

+ =− γ


T∂

increasing C“anti-continuous” limitC=0

“continuous” limitC → ∞C 0

arbitrary shapes Cavity SolitonsDiscrete Cavity Solitons


Discrete DiffractionDiscrete Diffraction

1D 1D waveguide arraywaveguide array

NM: Bloch waves → DR: z 2 ( )cos( )xC k dk = β + ω


Discrete DiffractionDiscrete Diffraction

1D 1D waveguide arraywaveguide array

NM: Bloch waves → DR: z 2 ( )cos( )xC k dk = β + ω

kz positive

zero curvature

πK=kxd

πβ

negative2C

−π π


T. Pertsch, U. Peschel, and F. Lederer, Phys. Rev. Lett., 88 (02) 093901Discrete DiffractionDiscrete Diffraction

z 2 ( )cos( )xC k dk = β + ω

anomalous diffraction anom. refraction and diffractioncanonical


OutlineOutline






Discrete Cavity Discrete Cavity Soliton Soliton –– Simplest SolutionsSimplest Solutions

1) focusing nonlinearity

→ bright solitons in continuous limit

8

u n|2

4

6

pow

er |u

0

2

0 5 10 15

peak

p

→ coupling (discrete diffraction) controls soliton formation

0 5 10 15coupling C Hopf instability


p g ( )

Discrete Cavity Discrete Cavity Soliton Soliton –– Simplest SolutionsSimplest Solutions

2) defocusing nonlinearity

→ dark solitons in continuous limit

4u n|2

2

3

4

pow

er |u

0

1

0 0.1 0.2 0

peak

li C0 0.1 0.2

→ bright solitons disappear for large coupling

coupling C


Effect of an Inclined Holding BeamEffect of an Inclined Holding Beam

2u∂ i

inclined holding beam

( ) 21 1 02 ( )n n n n n n nui C u u u i u u u ET + −

∂+ + − + + Δ + γ =

∂iqne


Effect of an Inclined Holding BeamEffect of an Inclined Holding Beam

2u∂ i

inclined holding beam

( ) 21 1 02 ( )n n n n n n nui C u u u i u u u ET + −

∂+ + − + + Δ + γ =

∂iqne

“anti-continuous” limitC = 0

no power transfer;

“continuous” limitC → ∞

translational symmetry;no power transfer;→ at rest - q < qcrit → at rest

- q > qcrit → motion

translational symmetry;→ motion


Reminder Reminder -- Discrete Diffraction Discrete Diffraction

z 2 ( )cos( )2 cos( )

xk C k dC q

= β + ω

= β +

anomalous diffraction anom. refraction and diffractioncanonical


Stability of Stability of Discrete Plane WavesDiscrete Plane Waves

Discrete model

( ) 21 1 02 ( ) iqnn n n n n n nui C u u u i u u u E eT + −

∂+ + − + + Δ + γ =

∂T∂

cw plane wave solution

eiqnnu b=

O. Egorov and F. Lederer, and Y. S. Kivshar,


,Opt. Exp. 15 (07) 4149

Stability of discrete plane wavesStability of discrete plane waves

Discrete model

( ) 21 1 02 ( ) iqnn n n n n n nui C u u u i u u u E eT + −

∂+ + − + + Δ + γ =

∂

perturbed plane wave

T∂

bistability condition

( )( )exp eiqnnu b a T iQn+ λ += ( )s 32 co 1C q −⎡ ⎤γ Δ + < −⎣ ⎦

modulational instability → ℜ(λ) > 0

perturbation

( )( ) ( )( )2 2c( , ) 1 2 cos 1 2 cos 1 32 s

os cos

sii nn

Q C Q b C Q b

iC

q q

qQ

qλ = − ± − + Δ + γ − + Δ + γ −

−


2 s sii nniC qQ

Modulational Modulational Instability Instability of PWof PW

3′γ Δ < − - bistable case

2.4

2.8

|b| Q=π3π/4 q=0

2 MI

1.2

1.6

π/2π/4

0 0.25 0.5 0.75 1q / π0.8

π/40 4 8 12E

normal diffractionE0

MI domains vs. modulation plane wave responseO. Egorov and F. Lederer, and Y. S. Kivshar,


,Opt. Exp. 15 (07) 4149



2.4

2.8

|b| Q=π3π/4 q=0.5 π

2

1.2

1.6

π/2π/4

tiny MI

0 4 8 12E0 0.25 0.5 0.75 1q / π0.8

π/4

diffraction arrestedE0

MI domains vs. modulation plane wave response




2.4

2.8

|b| Q=π3π/4 q= π

2

1.2

1.6

π/2π/4

MI

1.6 1.8 2 2.2 2.4E0 0.25 0.5 0.75 1q / π0.8

π/4

“anomalous” diffractionE0

MI domains vs. modulation plane wave response


Soliton SolutionsSoliton Solutions


Bright SolitonsBright Solitons

q = 0

2.5

|b|2.5 bright

q

2

|b|

1 5

2|u|

MI

1

1.5

1

1.5

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

0 2 4 6 8 10 12E0

0.5

l diff ti

bright DCS

3′Δ < − 1C = 1γ =

normal diffractionDCS domains vs. inclination

DCS domains vs. holding beamO. Egorov and F. Lederer, and Y. S. Kivshar,


,Opt. Exp. 15 (07) 4149

Dark Solitons on a PatternDark Solitons on a Pattern

q = 0

2.5 brightpatterndark DCS

2.5

|b|

q

1 5

2|u| pattern

dark DCS

MI2

|b|

1

1.5

darkbright DCS1

1.5

0 2 4 6 8 10 12E0

0.5dark

l diff ti

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

3′Δ < − 1C = 1γ =

normal diffractionDCS domains vs. inclination DCS domains vs. holding beam


Dark Dark SolitonsSolitons

q = π/2

2.4

dark DCS2.5

|b|

q

1.6

2dark DCS

MI2

|b|

0.8

1.2dark

bright DCS1

1.5

1.5 2 2.5 3 3.5 4E0

0.4

diff ti t d

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

3′Δ < − 1C = 1γ =

diffraction arrestedDCS domains vs. inclination DCS domains vs. holding beam


Dark Dark SolitonsSolitons

q = π

22.5

|b| dark DCS

q

1 2

1.62

|b| dark DCS

MI

0.8

1.2

dark1

1.5

bright DCS

1.7 1.9 2.1 2.3E0

0.40 0.2 0.4 0.6 0.8 1q / π

0.5

“ l ” diff ti

bright DCS

3′Δ < − 1C = 1γ =

“anomalous” diffractionDCS domains vs. inclination DCS domains vs. holding beam


OutlineOutline






Mobility of Discrete SolitonsMobility of Discrete Solitons

motion trapping

conservative systems

continuous system – Galileian invariance discrete system – no Galileian invariance

crq q>h/E

crq q<h/E Peierls - Nabarro potential

0 1 2 3x0

moving soliton

0 1 2 3x0

resting soliton


moving solitonresting soliton

Mobility Mobility -- Continuous Model Continuous Model →→ Translational ModeTranslational Mode

Linear stability analysis:

( ) ( ) ( ) Tsu x U x u x eλ= + δ

( ) ( ) ( ) Tsv x V x v x eλ= + δ U dxU U

soliton translational mode

shifted soliton

x

+ =1) continuous case(translational symmetry)

0λ =( ) ( )x su x U xδ = ∂( ) ( )x sv x V xδ = ∂

translational mode

( ) ( ) ( ) ( )s x s su x U x a U x U x a= + ∂ ≈ +( ) ( ) ( ) ( )s x s sv x V x a V x V x a= + ∂ ≈ +( ) ( )x s ( ) ( ) ( ) ( )s x s s

2) Discrete case(loss of translational symmetry)

0λ ≠( ) ( )x su x U xδ ≈ ∂

quasi-translational

(loss of translational symmetry)


( ) ( )x s( ) ( )x sv x V xδ ≈ ∂ mode

Motion of DCS Motion of DCS -- PPerturbation Theoryerturbation Theory

1) Resting DCS

“odd” “even”

q < q crit

O. Egorov, U. Peschel, and F. Lederer


PRE 72 (05) 066603

Motion of Motion of DCS DCS -- PPerturbation Theoryerturbation Theory

1) resting DCS

1.5

"even"


2) R0.5

1

e λ

δuq < q crit

“odd” “even” “odd”

2) Rresting DCS are forced to move

-0.5

0Re

δu"odd"

motion → transition between „odd“ and „even“

q > q crit0 5 10 15 20 25

C1

-1

i l f h “ iC„ „

DCSs eigenvalue of the “quasi-translational” mode



PRE 72 (05) 066603

Motion of Motion of DCS DCS -- PPerturbation Theoryerturbation Theory

1) resting DCS

crit

( ) ( )odd evenC Cq K

Cℜλ − ℜλ

≈ ⋅


0 02q

critq C

q crit2) motion of DCS

qq critical

9 12 15 18 C0

0.005 restingDCS

C

DCSs



C1CPRE 72 (05) 066603

Moving DCS Moving DCS


QuasiQuasi--Continuous ApproachContinuous Approach O. Egorov, U. Peschel, and F. Lederer PRE 72 (05) 066603

aim: - to establish an analytical modely

idea: - to derive a quasi-continuous model for the DCS envelope by keeping properties of discrete diffraction (phase difference coupling strength)diffraction (phase difference, coupling strength)

→ soliton can be narrow9

U

5

7

U

envelope

-10 -5 0 5

3

-10 -5 0 5 10

, 1, 2,..nx n h n= =

x1 2 3 4


O. Egorov, U. Peschel, and F. Lederer PRE 72 (05) 066603QuasiQuasi--Continuous ApproachContinuous Approach

Continuous envelope:

7

9U( ) iqn nu x n h e u= ≡

Continuous envelope:

3

5

7

2 31 1h h h∂ ∂ ∂

Taylor expansion:envelop

-10 -5 0 5 -10 -5 0 5 10

2 31 11 2 6 ...n n x xx xxxu u h u h u h u± ≈ ± ∂ + ∂ ± ∂



( )(0) 2 cos 1D C q= −-20

- detuning

(1) 2 sinD C h q=

-4

-2

0

2

2

- inclination, velocity

2(2) coshD C q=

( )-2

0

2

2

- 2-order diffraction coefficient

( )3(3) 3 sinhD C q=-2

0

q−π −π/2 0 π/2


( )(1) (2) (2 3

21

3) (03

)02u u u u

u u u ui i i i u u u ED D D D∂ ∂ ∂ ∂ ⎡ ⎤+ + + + + Δ + + γ =⎣ ⎦∂ ∂ ∂ ∂ ( )13 02u u u uT x x x γ⎣ ⎦∂ ∂ ∂ ∂



( )(0) 2 cos 1D C q= −-20

- detuning

(1) 2 sinD C h q=

-4

-2

0

2

2

- inclination, velocity

2(2) coshD C q=

( )-2

0

2

2


( )3(3) 3 sinhD C q=-2

0

q−π −π/2 0 π/2- 3-order diffraction coefficient

i f

( )(1) (2) (2 3

21

3) (03

)02u u u u

u u u ui i i i u u u ED D D D∂ ∂ ∂ ∂ ⎡ ⎤+ + + + + Δ + + γ =⎣ ⎦∂ ∂ ∂ ∂

moving frame

( )13 02u u u uT x x x γ⎣ ⎦∂ ∂ ∂ ∂


Moving Discrete Cavity SolitonsMoving Discrete Cavity Solitons O. Egorov and F. Lederer, and Y S Kivsharand Y. S. Kivshar,Opt. Exp. 15 (07) 4149

2.5

|b| dark DCS2

|b| dark DCS

MI

1

1.5

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

bright DCS

domains of resting DCSs 3′Δ < − 1C = 1γ =



2.5

|b|1.5 continuous

Moving DCSs

2

|b|

1

eloc

ity

continuous

dark DCSMI

1

1.5

0.5

ve

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

0 0.2 0.4 0.6q / π

0

g

moving DCSs in the quasi-continuous model 3′Δ < − 1C = 1γ =

q



1.5 continuous2.5

|b|Moving DCSs

1

eloc

ity

continuousdiscrete

2

|b|dark DCS

MI

0.5

ve

1

1.5

bright DCS

0 0.2 0.4 0.6q / π

00 0.2 0.4 0.6 0.8 1q / π

0.5

g

q

3′Δ < − 1C = 1γ =moving DCSs in the discrete model



1.5 continuous2.5

|b|Moving DCSs

1

eloc

ity

continuousdiscrete

2

|b|dark DCS

MI

0.5

ve

1

1.5

bright DCS

0 0.2 0.4 0.6q / π

00 0.2 0.4 0.6 0.8 1q / π

0.5

bi t bilitresting and moving DCSs

g

qbistabilityresting and moving DCSs




2.5

|b|Moving DCSs

2

|b|dark DCS

MI

1

1.5

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

Resting and Moving DCSscollision between resting and moving

di t C it S lit

g

Resting and Moving DCSs discrete Cavity Solitons




2.5

|b|Moving DCSs

soliton ménage à troissoliton ménage à trois

2

|b|dark DCS

MI

1

1.5

bright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

resting and moving DCSsCollision between resting and moving

Di t C it S lit

g

resting and moving DCSs Discrete Cavity Solitons



OutlineOutline






Subdiffractive Subdiffractive SolitonsSolitons O. Egorov and F. Lederer, and K. StaliunasOpt. Lett. (to appear August 1)

( )2 3

2(1) (3)1 02 3

(2) (0)u uu u

u u u ui iD iD i u u u ET x x x

D D∂ ∂ ∂ ∂ ⎡ ⎤+ + + + + Δ + + γ =⎣ ⎦∂ ∂ ∂ ∂2(2) coshD C q=

2.5

|b|Moving DCSs

2.5

|u| bright

q = π/2

2

|b|

1.5

2| | bright

dark DCSMI

1

1.5

1PWbright DCS

0 0.2 0.4 0.6 0.8 1q / π0.5

diff ti t d1.75 2 2.25E0

0.5 dark g

3′Δ < − 1C = 1γ =diffraction arrested

moving DCSs in the discrete model


O. Egorov and F. Lederer, and K. StaliunasOpt. Lett. (to appear August 1)Subdiffractive Subdiffractive SolitonsSolitons

fundamental higher-order

1.5

2|u|

2.5

|u| bright

fundamental higher-order

0.5

1

1.5

2| | bright

0 9

1.5

2.1|u|

1PW

10 20 30waveguides

0.3

0.9

10 20 30waveguides

1.75 2 2.25E0

0.5 dark

3′Δ < − 1C = 1γ =


O. Egorov and F. Lederer, and K. StaliunasOpt. Lett. (to appear August 1)Subdiffractive Subdiffractive SolitonsSolitons

2.5

|u| bright

1.5

2| | bright

1PW

1.75 2 2.25E0

0.5 dark

3′Δ < − 1C = 1γ =


ConclusionsConclusions

- MI and discrete cavity soliton formation depend strongly on the holding beam inclination

- beyond a critical inclination resting solitons start to move

- a quasi-continuous approach permits to identify moving DCSsDCSs

- subdiffractive DCS exist if second order diffraction dissapearsdissapears


falk lederer and oleg egorov - uni-dortmund.detdohnal/iciam_talks/lederer.pdf · falk lederer and...

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