Aldo Auditore, Matteo Conforti,

Costantino De Angelis Dipartimento di Ingegneria dell’Informazione,

Università degli Studi di Brescia

Binary plasmonic waveguide arrays.

Modulational instability and gap solitons.

Alejandro B. Aceves Southern Methodist University, Dallas

Triantaphyllos R. Akylas Massachusetts Institute of Technology, Boston

2

Outline of the talk

Introduction

Linear Regime: tuning the coupling

coefficient

Nonlinear Regime: modulational instability

and dark solitons

Conclusions

3

Discrete optics in coupled waveguides offers a unique framework where to study discrete linear and nonlinear phenomena in both regular (uniform) and non regular (non-uniform) settings.

Introduction

C1 C1 C1

C2 C2 C2 C2

C1 C1 x

z

F. Lederer, G.I. Stegeman, D.N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics” , Physics Report 463, 1-126 (2008).

I.L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y.S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides” , Physics Report 518, 1-79 (2012).

4

CMT for directional couplers

),(),(),( zxzxzx ba ),(),(),( zxzxzx ab

The coupler

Waveguide a

Waveguide b

),( zx

z

),( zxa

z

),( zxb

z

M. Skorobogatiy et al. Phys. Rev. E 68 065601 (2003) ; D. Michaelis et al. Phys. Rev. E 68 065601 (2003)

5

CMT from reciprocity theorem ),(),(),( , ),(),(),( zxzxzxzxzxzx abba

11

11

EiH

HiE

a

PiEiJJEiH

HiE

aa

22222

22

,

PEiHEHE 11221

),( zx

z

6

CMT from reciprocity theorem

Field 1 is an ideal backward propagating mode of

the unperturbed structure (guide a):

PEiHEHE 11221

zizxhzxH

zizxezxE

aa

aa

exp),(),(

exp),(),(

*

1

*

1

Field 2 is a z-dependent superposition of the modes of waveguide a and b.

),()(),()(),,(

),()(),()(),,(

2

2

zxhBzxhAzxH

zxeBzxeAzxE

ba

ba

Field 2. Write the transverse field as:

Field 2.

From Maxwell equations it follows: ),()(),()(),,(

),()(),()(),,(

2

2

zxhBzxhAzxH

zxeBzxeAzxE

bzazz

bzb

aza

z

7

CMT from reciprocity theorem

PEiHEHE 11221

0

0

zCAdz

zdBi

zCBdz

zdAi

zL

bzazb

baa

za

eeeedzdxLI

C0

**

2

C

011 nnn EEC

dz

dEi

UNIFORM WAVEGUIDE ARRAY

8

Sign of the coupling coefficient For conventional couplers based on TIR, the structure is z

independent:

* xexexxdxC byaya

It is real and in single mode

operation can not change sign

Always positive

** xexexexexxdxC bzazb

bxaxa

TE

TM

9

Sign of the coupling coefficient For z-independent plasmonic couplers:

** xexexexexxdxC bzazb

bxaxa

Can be positive or negative Can be positive or negative

TM

10

Sign of the coupling coefficient Transverse magnetic (TM) surface plasmons in a single graphene layer

))0((2)3()1(

20

22

20

10

22

10

xEfi

z

r

r

r

r

satx

x

xxf

1

Auditore A. et al. Optics Letters 38, 631-633 (2013).

Gorbach A. V., Physical Review A 87, 013830 (2013).

10 r 20 r

Graphene Layer (tunable EM parameters)

11

Sign of the coupling coefficient Coupled TM surface plasmons in graphene double layer

Even Mode.

Out of phase superposition of

the isolated graphene

plasmons

Odd Mode.

In phase superposition of the

isolated graphene plasmons

10 r20 r

Graphene Layers

30 r

B. Wang, Appl. Phys. Lett. 100, 131111 (2012).

12

Sign of the coupling coefficient Coupled TM surface plasmons in graphene double layer

even

odd

Even Mode

Odd mode

02

evenoddC

13

Tuning of the coupling coefficient

Auditore A. et al., “Tuning of surface

plasmon polaritons beat length in graphene

directional couplers” Optics Letters in press.

14

Plasmonic arrays: diffraction management

C1 C1 C1

C2 C2 C2

x

z

m

C1>0

C2

15

Nonlinear regime: introduction

Waveguides based on plasmonic confinement offer a unique setting where to exploit positive and negative coupling regimes

Binary arrays offer a framework where to exploit a two–band structure in the linear and nonlinear regime (R. Morandotti et al, Opt. Lett. 29, 2890 (2004))

N.K. Efremidis et al., Phys. Rev. A 81, 053817 (2010)

M. Conforti, C. De Angelis, T.R. Akylas, Phys. Rev. A 83, 043822 (2011)

M. Conforti, C. De Angelis, T. R. Akylas, A. B. Aceves, Phys. Rev. A 85, 063836 (2012).

A. Auditore, M. Conforti, C. De Angelis, A. B. Aceves, Opt. Comm. 297, 125 (2013).

C2

C1 C1

C2

C1 C1

C2

x

z

C1>0

C2

16

Coupled mode equations

0|| 21111 nnnnnnnn EEEcEc

dz

dEi

0||

0||

2

11

2

11

nnnnn

nnnnn

BBAcAdz

dBi

AABBcdz

dAi

1

,,,12

,,,2

2

1121

2111

c

ccccBEnm

ccccAEnm

mmmnm

mmmnm

Even

Odd

Binary Array

Nonlinear regime: introduction

K. Hizanidis, Y. Kominis, N. Efremidis,

“ Interlaced linear-nonlinear optical waveguide

arrays” , Optics Express 16, 18296 (2008).

17

xz

zxnzxn

kcck

zknkiBBzknkiAA

cos21

)](exp[)],(exp[

1

2

1

c1=-1+0.25

Gap solitons

Nonlinear regime: discrete dispersion relation

18

Derive from the discrete system a continuous one by first order expansion around kx=0

c1=-1+ε

Nonlinear regime: long wavelength limit

0||

0||

2

2

wwudx

du

dz

dwi

uuwdx

dw

dz

dui

19

22

)](exp[),()],(exp[),(

xz

zxzx

kk

zkxkiBzxwzkxkiAzxu

ε =0.25

Nonlinear regime: continuous system

20

A.B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989). C. M. de Sterke et al., Phys. Rev. E 54, 1969 (1996).

C. Conti and S. Trillo, Phys. Rev. E 64, 036617 (2001). H. Alatas et al., Phys. Rev. E 73, 066606 (2006).

Nonlinear regime: solitons

0||

0||

2

2

wwudx

du

dz

dwi

uuwdx

dw

dz

dui

21

One dimensional Hamiltonian system, integrable by quadrature

With: g1,2(ξ ) = f (ξ ) exp[iθ1,2(ξ )] , P=|g1|

2-|g2|2, η=f2

2 , μ=θ1 – θ2

s=γ

d=γ

Nonlinear regime: Hamiltonian structure

22

Nonlinear regime

Bright solitons

Bright solitary wave solutions correspond to homoclinic trajectories emanating from unstable fixed points (η0, μ0) = (0, ±arccos(cos(Q)/ε) (H=0)

23

Nonlinear regime

Dark solitons

24

Nonlinear regime: dark solitons

Solitary wave solutions correspond to trajectories emanating from

unstable fixed points with μ0 = ± p/2

P1

P2

25

Nonlinear regime: dark solitons

P1

26

Nonlinear regime: modulational instability.

0||

0||

2

2

wwudx

du

dz

dwi

uuwdx

dw

dz

dui

For =0 we have no band gap.

M. Conforti, C. De Angelis, T. R. Akylas, A. B. Aceves, Phys. Rev. A 85, 063836 (2012).

A. Auditore, M. Conforti, C. De Angelis, A. B. Aceves, Opt. Comm. 297, 125 (2013).

27

Nonlinear regime: modulational instability.

Plane wave solutions: dispersion relation =0

28

Nonlinear regime: modulational instability

kx=1

29

Nonlinear regime: modulational instability.

kx=1

30

Nonlinear regime: modulational instability. Stability of solitons in the discrete system: can discreteness

change the stability of these solitons?

We thus consider the discrete problem:

0||

0||

2

1

2

1

nnnnnn

nnnnnn

BBAAAdz

dBi

AABBBdz

dAi

=0

31

Nonlinear regime: modulational instability. CW solutions.

=1 =0.

5 =0 I0=1

Continuous problem

Discrete problem

202

0000 , exp),(,exp),( BAInikzikBznBnikzikAznA xzxz

32

Nonlinear regime: modulational instability.

=1 =0.

5 =0 I0=1

Stability of CW solutions. On the upper branch, solutions become stable for high enough intensities

For example for kx=0, =1, =0, =0 we easily obtain on the upper

branch the following expression for the instability gain:

cos88cos44

2

1Im 20

2

0 IIg

Which is always zero (i.e. CW solutions are stable) for I0 bigger than 4.

33

Nonlinear regime: discrete solitons.

zinDznBzinCznA exp),(,exp),(

2

34

Nonlinear regime: discrete solitons.

zinDznBzinCznA exp),(,exp),(

5

35

Light propagation in waveguide arrays with

alternating positive/negative couplings and

binary Kerr nonlinearity:

o Linear regime;

o Nonlinear regime.

Conclusions