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2015
AND gate in a nonlinear subwavelength dielectric
waveguide array
G. Mendoza-Gonzáleza and Erwin A.
Martı́-Paname ˜ noa,ba Benem´ erita Universidad
Aut´ onoma de Puebl a, Facultad de Ciencias
Fı́sico-Matem´ aticas, Posgrado en Fı́sica
Aplicada, Av. San Claudio y 18 sur, Colonia San Manuel,
Puebla 72570, Mexico.b Benem´ erita Universidad
Aut ́ onoma de Puebla. Laboratorio Nacional de
Superc´ omputo del Sureste de M ́ exico, Blvd.
Valsequillo y Ave. Las Torres, Puebla 72570, Mexico.
http://www.lns.buap.mx
[email protected]
Abstract: In this work we present numerical results
regarding the possibility to control
the output position of the light in nonlinear arrays of
subwavelength dielectric parallel
waveguides. The main result is an AND logic gate due the
interaction of two input beams.
© 2015 Optical Society of AmericaOCIS codes: 050.6624,
190.0190, 190.3270, 200.4660.
1. IntroductionA large number of optical effects can be observed
in discrete systems that are not possible in homogeneous media.
The discretized nature of light propagation gives rise to
phenomena like: discrete diffraction, diffraction management,
temporal and spatial discrete solitons and self-phase modulation
of the radiation [1–3]. At subwavelength scales,
new optical phenomena have been recently discovered such as
auto-accelerated beams propagating along curved
paths, plasmonic subwavelength solitons in nonlinear
metamaterials, nanoplasmonics, etc. [4, 5]. In the pursuit
of
low-dimensional optical systems, one can find reports that focus
on the waveguide dimensions. Particularly interest-
ing results have been found in nonlinear interactions. Foster et
al. [6, 7] focused on the balance between core size and
power confinement with the aim of maximizing nonlinear
interactions in waveguides. They found that the optimal core
size is subwavelength and that structures with asymmetric cross
sections maximize the effective nonlinearity. [6, 7].
The numerical study is based in to resolve Maxwell’s equations
considering the constitutive relations of the medium
in the time domain, for that, we apply the finite difference
time domain (FDTD) method [8, 9]. In this report we show
the propagation and interaction of optical radiation along
one-dimensional arrays of nonlinear parallel dielectric sub-
wavelength waveguides (SWG). A homogeneous dielectric medium,
with a refractive index lower than the waveguides
linear index, surrounds the whole system. The result of the
interaction is the behavior of AND logic gate.
1.1. Theoretical model
The nonlinear media considered in this work is isotropic and non
magnetic. In this case, the mathematical model for
light propagation is based on the following Maxwell’s
equations:
∂ H
∂ t =− 1
µ 0∇×E, (1)
∂ D
∂ t = ∇×H, (2)
This model will provide a set of partial nonlinear differential
equations for the six electromagnetic field components.
The model is closed considering the boundary and initial
conditions. The vector of displacement electric D
is:
D = ε 0(1+ χ (1)+ χ (3)
|E|2)E = ε 0(ε r + χ (3)
|E|2)E, (3)
where χ (1) and χ (3) are the linear and
nonlinear electrical susceptibilities of the medium,
respectively. ε r is the linearrelative
dielectric constant and is related to the linear refractive index
by n0 =
√ ε r . Considering the local intensity
I
(power per unit of area) for a monochromatic propagating wave
[10], as I = n0
ε 0/µ 0 |E|2. The nonlinear electricalsusceptibility
in function of the Kerr coefficient n2 is:
χ (3) = 2n2on2
ε 0/µ 0. (4)
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2015
2. Numerical results
The incident field is a monochromatic Gaussian beam, linearly
polarized in the x−direction with wavelengthλ =
800nm and amplitude E i = 4.5×
108 V /m. The values for the nonlinear medium are: n0
= 1.455 and n2 ≈2×10−16
m2/W . r = 300nm and
D = 850nm (see fig.1 (a)).
Ei2
Output
Input optical signals
c m1
Ei1
x
y
z
r D
θ
(a)
Output
(ii)
input
0
20
40
x [ ]μm
0 10-10
z[m]
μ
0 10-10
x [ m ]μ
0
2
4
E
x 1 0
[ V / m ]
8
(b)
(i)
"AND01.txt"
"AND10.txt"
"AND11.txt"
P e a
k A m p
l i t u d e
[ V / m ]
x 1 0
8
4
3
2
1
0
AND01 AND10 AND11
0 8 16 24 32 40z [ ]μm
(f)
0
2
4
0 10-10
x [ m ]μ
InOut
E
x 1 0
[ V / m ]
8
0 10-10
x [ m ]μ
InOut
0
2
4
E
x 1 0
[ V / m ]
8
( )c
(d) (e)
Fig. 1. (a)Schematic representation of the SWGs array. Normal
incidence of both Ei1 and Ei2 fields.
(b) Beams profiles, input (blue-line) output (red-line). (c)
Evolution of the propagation of light along
the array. (d,e) Beams profiles with only one input beam. (f)
Summary of behavior of AND logic
gate.
Figure 1(b,c) shows the interaction of two input beams,
forming one output in the central SWG, this result is
behavior of the AND logic gate –the output will be ”high”
(1) if and only if all inputs are ”high” (1). If
any input(s)are ”low” (0), the output is in a ”low” state–.
Fig. 1(d,e) show the behavior with only one input beam, the
output beamnot is self-trapped in the central SWG, we observe
radiative losses, for this reason, we have not a good output
beam
and we can consider lack of the output beam. (f) show the
summary of the behavior for different propagation length.
In analogy with microelectronics, for values between
1.5×108V /m to 3.0×108V /m we could consider an
active-highsignal, and before this value is an active-low
signal.
References
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