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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
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Outline of the talk
Introduction
Linear Regime: tuning the coupling
coefficient
Nonlinear Regime: modulational instability
and dark solitons
Conclusions
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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).
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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)
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CMT from reciprocity theorem ),(),(),( , ),(),(),( zxzxzxzxzxzx abba
11
11
EiH
HiE
a
PiEiJJEiH
HiE
aa
22222
22
,
PEiHEHE 11221
),( zx
z
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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
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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
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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
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Sign of the coupling coefficient For z-independent plasmonic couplers:
** xexexexexxdxC bzazb
bxaxa
Can be positive or negative Can be positive or negative
TM
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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)
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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).
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Sign of the coupling coefficient Coupled TM surface plasmons in graphene double layer
even
odd
Even Mode
Odd mode
02
evenoddC
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Tuning of the coupling coefficient
Auditore A. et al., “Tuning of surface
plasmon polaritons beat length in graphene
directional couplers” Optics Letters in press.
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Plasmonic arrays: diffraction management
C1 C1 C1
C2 C2 C2
x
z
m
C1>0
C2
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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
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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).
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xz
zxnzxn
kcck
zknkiBBzknkiAA
cos21
)](exp[)],(exp[
1
2
1
c1=-1+0.25
Gap solitons
Nonlinear regime: discrete dispersion relation
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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
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22
)](exp[),()],(exp[),(
xz
zxzx
kk
zkxkiBzxwzkxkiAzxu
ε =0.25
Nonlinear regime: continuous system
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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
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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
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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)
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Nonlinear regime
Dark solitons
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Nonlinear regime: dark solitons
Solitary wave solutions correspond to trajectories emanating from
unstable fixed points with μ0 = ± p/2
P1
P2
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Nonlinear regime: dark solitons
P1
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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).
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Nonlinear regime: modulational instability.
Plane wave solutions: dispersion relation =0
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Nonlinear regime: modulational instability
kx=1
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Nonlinear regime: modulational instability.
kx=1
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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
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Nonlinear regime: modulational instability. CW solutions.
=1 =0.
5 =0 I0=1
Continuous problem
Discrete problem
202
0000 , exp),(,exp),( BAInikzikBznBnikzikAznA xzxz
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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.
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Nonlinear regime: discrete solitons.
zinDznBzinCznA exp),(,exp),(
2
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Nonlinear regime: discrete solitons.
zinDznBzinCznA exp),(,exp),(
5
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Light propagation in waveguide arrays with
alternating positive/negative couplings and
binary Kerr nonlinearity:
o Linear regime;
o Nonlinear regime.
Conclusions