Utilization of binary PSO algorithm and DDA method to investigate the plasmonic demultiplexer -based CPA filter

Optik 156 (2018) 968–974

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Optik journal homepage: www.elsevier.de/ijleo

Original research article

Utilization of binary PSO algorithm and DDA method to investigate the plasmonic demultiplexer -based CPA filter Mohamadreza Soltani Department of Electrical Engineering, Tiran Branch, Islamic Azad University, Tiran, Iran

a r t i c l e

i n f o

Article history: Received 17 August 2017 Accepted 14 December 2017 Keywords: Plasmonic nano particles Optimization algorithm Optical circuits

a b s t r a c t Here, we suggest the possibility of optical circuit design approach by employing the binary optimization of plasmonic nano disks. The proposed mechanism is based on combination of binary particle swarm optimization (BPSO) algorithm and discrete dipole approximation (DDA) method. BPSO, a group of birds including a matrix with binary entries responsible for controlling nano disks in the array, shows the presence with symbol of (‘1’) and the absence with (‘0’). The current research represents a nanoscale and compact 4 channels plasmonic Demultiplexer as optical circuit. It includes 8 coherent perfect absorption (CPA) –type filters. The operation principle is based on the absorbable formation of a conductive path in the dielectric layer of a plasmonic nano-disks waveguide. Since the CPA efficiency depends strongly on the number of plasmonic nano-disks and the nano disks location, an efficient binary optimization method based the BPSO algorithm is used to design an optimized array of the plasmonic nano-rod in order to achieve the maximum absorption coefficient in the ‘off’ state. © 2017 Elsevier GmbH. All rights reserved.

1. Introduction Surface plasmon polaritons have very good promising applications in the high density of optical component for integrated optical circuits and devices due to their removing of diffraction limit and light manipulation on subwavelength scales [1–3]. To realize photonic circuitry based on plasmonics, a variety of components are required: splitters [4–6], couplers [7,8], multiplexers and demultiplexers [9–12], switches [13], logic gates [14] etc., which have been studied recently. Plasmonic demultiplexers, which are key elements of plasmonic circuits for wavelength division multiplexing (WDM) system need to be developed, and a high resolution is necessary to achieve a narrow channel spacing in high capacity plasmonic networks. Plasmonic nanoparticles are particles whose electron density can couple with electromagnetic radiation of wavelengths that are far larger than the particle due to the nature of the dielectric-metal interface between the medium and the particles: unlike in a pure metal where there is a maximum limit on what size wavelength can be effectively coupled based on the material size [15]. What differentiates these particles from normal surface plasmons is that plasmonic nanoparticles also exhibit interesting scattering, absorbance, and coupling properties based on their geometries and relative positions [16]. These unique properties have made them a focus of research in many applications including solar cells, spectroscopy, signal enhancement for imaging, and cancer treatment. The optimization problems in the plasmonic nano-structure area

E-mail address: m soltani [email protected] https://doi.org/10.1016/j.ijleo.2017.12.060 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

M. Soltani / Optik 156 (2018) 968–974

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Fig. 1. Schematic view of the proposed of plasmonic nano Disk demultiplexer.

can be divided into two categories. In the first type the continuous optimization algorithm can be performed to engineer the geometrical metal nano-structures [17], whereas in the second type, the binary optimization algorithm can be used to control the presence (‘1’) or absence (‘0’) of the metal nano particles in the array [18]. In ref [19], binary TLBO algorithm was used and an optical switch based on the dimer plasmonic nano-rods has been proposed. In this paper BPSO algorithm is used to control the existence (‘1’) or non- existence (‘0’) of plasmonic nano particles to design an optical demultiplexer. In BPSO, a swarm consists of a matrix with binary entries control the presence (‘1’) or the absence (‘0’) of nano single nano-disk in the array and find the best array of nano-disks from all possible arrays. The selected array should be able to maximize the absorption coefficient in the “off” state and minimize the absorption coefficient in the “on” state. 2. Model description The proposed plasmonic demultiplexer has been shown schematically in the Fig. 1. Here, the feasibility of the plasmonic optical demultiplexer has been investigated by integrating the coherent perfect absorption (CPA) devices into integrated photonic waveguides. In this figure, the plasmonic demultiplexer has 8 CPA filters: two filters in the A frequency, two filters in the B frequency, two filters in the C frequency, and two filters in the D frequency. In this figure, the plasmonic demultiplexer has two arms and any arm consists of multi serials array of metallic nano disks. When the signals are applied to the waveguide and enter in the upper path, first and second CPA filters, suppose the A and B frequencies and then enters in the two branches in which upper and lower branches filter the C and D frequencies respectively. 3. Background of numerical method The wide range of applications and exploitation possibilities of optical plasmonic properties of metals demand numerical techniques which provide accurate modeling and analysis of problems involving metal-dielectric interfaces in visible and near-infrared bands. Since optical response of metals can be well-described in the classical framework based on Maxwell’s equations [20], differential formulations as the finite difference time domain (FDTD) [21] or the finite integration technique [22] have been commonly used to solve this kind of problems due to their easy implementation from differential equations. Mainly due to the high computational requirements of the differential techniques, alternative tools demanding a lower number of unknowns for a given problem such as surface integral equation (SIE) formulations solved by the well-known method of moments (MoM) [23] have increased their presence in the context of plasmonics [24–26]. Moreover, there are some numerical simulation methods to study the interaction between the light and metal nano particles such as FDTD [27], FEM (Finite Element Method) [28], DDA (Discrete-dipole Approximation) [29], Mie Theory [30] and Transition matrix (T-matrix) theory [30]. In this paper, DDA is used to study the optical properties of plasmonic nano particles. 4. Theory The plasmonic demultiplexer, depicted in Fig. 1, consists of eight arrays of metallic nano disk that are periodically arranged in the x y –plane of the quartz substrate. The devise is excited by a monochromatic incident plan wave E inc (r, t) = E 0 ejk(r ω t)

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Fig. 2. Schematic of oblate (a < b = c) ellipsoids.

where r, t, ω, k = ω/c = 2/, c, and  are the position vector, the time, the angular frequency, the wave vector, the speed of light, and the wavelength of the incident light, respectively. To calculate the E-field of each dipole, time harmonic component -j␻t of the E-field is left out. Local field arises from the incident light with polar (␪) and azimuth (␸) angles at each particle are: E inc (rs ) = E 0 ej

k rs

(1)

Where k=

2 [sin(). cos(), sin(). sin(), cos()] 

(2)

For the incident field with P-polarization: E0 = [sin( −

   ). cos() , sin( − ). sin(), cos(  − )] 2 2 2

(3)

and for the incident field with S-polarization: E0 = [cos( +

  ), sin( + ), 0] 2 2

(4)

When the applied field is parallel to one of the principle axes, the polarizability, ˛, is given by [31]: ˛ = Vε0

εr − 1 1 + L1 (εr − 1)

(5)

where εr = εparticle /εmedium is the relative dielectric function of the particle with respect to the medium, V is the particle volume, and L1 is the shape factor. The exact equation for L1 is given by [19]:

L1 =

1+f2 1 [1 − tan−1 (f )] f f2

, f2 =

b2 −1 a2

(6)

where a,b and c are seminal excess of an ellipsoid. The shape of a nano oblate is shown in the Fig. 2.In this simulation approximate a nano disk with a oblate (a < b = c) ellipsoid. The dipole moment induced in a single particle by a local electric field is given by [32]: Ps = ε0 ˛sE (rs )

(7)

Here, Ps is the induced dipole moment,˛s is Polarizability of the particle centered atrs , ELoc is local electric field, and ␧0 is permittivity of free space. The local field arises from two sources, appearing as two terms. The first term is incident light, Einc (rs ) = E0 ej k .rs , and the field radiated from each of the other N-1 radiating dipoles in the array. Combining these terms leads to local field at each dipole as follow [33]: Eloc,i = Einc,i + Edip,i = E0 ejkri −

 i= / j

Ai,j Pi

(8)

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where Ps is the dipole moment of the s-th particle and As ,h with s = / h is an interaction matrix with 3N × 3N matrixes as elements described by following equation [34–36]: Ai

j

=

eikrij



2

k rij × (rij × Pj ) +

rij3

(1 − ikrij ) rij2

 



[rij2 Pj

− 3rij (rij .Pj )]

(9)

wherers,h = rs − rh , rsh = rsh  and As,h are 3 × 3 matrices representing the interaction of two particles of s and h. Once the 3N-coupled complex linear equations, given by Eq.(9), are solved and each dipole moment Pi determined, and the optical absorption can be directly calculated from the dipole array as follow [37]: Qabs =

4k

 2 E0  a2

N  

 

Im Pi a−1 i

∗

Pi∗ −

2 3 k |Pi |2 3

(10)

i=1

4.1. Binary particle swarm optimization algorithm The PSO algorithm is an optimization procedure inspired by a colony such as birds which can improve its behaviors [38]. Any element of this colony is called a particle and moves in an n-dimensional space, correcting its trajectory based on the previous actions of itself and its neighboring particles. For each particle, velocity and displacement are updated based on the following relations [39]:







vik+1 = w vik + c1 r1 pki − xi k + c2 r2 pkg − xi k xik+1 = xik + vk+1 i

, i = 1, ...n



(11) (12)

where k is the number of the current iteration, n is the number of particles, w is the inertia weight, c1 and c2 are acceleration parameters, and finally r1 and r2 are random parameters between 0 and 1. The best position for the i-th particle which has been stored so far is represented as [39]: Pki = [pk1 , pki .............., pkn ] All the Pki

T

(13)

(p best) are evaluated by a fitness function. The best particle among all p best is represented as pkg (Gbest best value),

which in a minimization problem pkg is the smallest member of the Pki vector while for a maximization problem it is the largest member ofPki . PSO was designed for continuous problems, but cannot deal with discrete problems. A new version of PSO, called Binary PSO (BPSO), was introduced by Kennedy and Eberhart in 1997 and applied to discrete binary variables. After that, many optimization problems in various areas were solved by this method. The position in BPSO is represented by a binary vector and the velocity is still a floating-point vector; however, velocity is used to determine whether the probability changes from 0 to 1 or from 1 to 0 when the positions of particles are being updated. The equation for updating the positions is then replaced with: 1



sigmoid(vkid ) =

 xkid

=

1+e 1,

−vk



if rand < sigmid(vkid )

0

(14)

id

(15)

otherwise

Since the CPA depends strongly on the number of plasmonic nano-disk and the nano-disk location, BPSO algorithm has been used to control the presence (‘1’) or the absence (‘0’) of nano disk in the array and find the best array of plasmonic nano-disk from all possible arrays. In order to increase the CPA efficiency, the selected array should be able to maximize the ¨ state and minimize the absorption coefficient in on¨ ¨ state. absorption coefficient in off¨ 5. Simulation results The DDA method and BPSO algorithm have been used to engineering the plasmonic nano disk layouts to maximize the absorption coefficient in the “off” state. Consider a quartz substrate with 25 plasmonic nano-disks (5 × 5). There is a 25 nm gap between metallic nano-disk. In this simulation we assumed that nano disk have 10 nm diameters, 30 nm height (b = c). Since the CPA depends strongly on the number of plasmonic nano disks and the nano disks location, BPSO has been used to control the presence (‘1’) or the absence (‘0’) of nano disks in the array and find the best array of plasmonic nano disks from all possible arrays. The goal is to maximize the Qabs by optimizing 25 binary nano disks for 2D arrays (5*5). As seen, BPSO is an algorithm that minimizes a profit function. To use BPSO algorithm to optimize the absorption coefficient, the BPSO method should minimized the following function: CostFunction = − Qabs(i)

(16)

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Fig. 3. Absorption spectra of the optimized array (5*5) (filter in A, B, C and D frequency) in which the optimization is perform around: a) ␭A = 420 nm b) ␭B = 620 nm c) ␭C = 520 nm d) ␭D = 320 nm.

30

20

10

0 200

300

400

500

600

Fig. 4. Absorption spectra of four CPA filters have been compared in the one figure.

Where i is the specific wavelength related to A,B,C, and D frequency in which the optimization is carried out and Qabs is the absorption coefficient. At first, optimization algorithm is applied to optimize the absorption coefficient in the A frequency. Fig. 3a shows the absorption spectra of the optimized array (5*5) in which the optimization is perform around ␭A = 420 nm. In this figure, the gold and gray disks mean the presence and the absence of the nano disk in the array, respectively. Again BPSO algorithm is applied to design an array of plasmonic nano disks to have high absorption coefficient in the ␭B = 620 nm, ␭C = 520 nm, and ␭D = 320 nm. Fig. 3b–d demonstrates the absorption spectra of the optimized array (5*5) in which the optimization is perform around ␭B = 620 nm, ␭C = 520 nm, and ␭D = 320 nm. Finally the absorption spectra of four CPA filters have been compared in the Fig. 4. As it is obvious, there are four pick wavelengths for each of the CPA filters corresponding to the resonance wavelengths from Eq. (10). The main feature of interest in view of the current work is the resonant behavior of optimized metallic nano-disk under the excitation of an external EM radiation, which leads to a very strong amplification of the EM fields inside and in the near field range outside the particles. Correspondingly, such systems exhibit

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strong resonance peaks in the absorption of light, whose characteristics (position and line width) depend on both intrinsic geometrical factors (single-particle size and shape) and extrinsic parameters (dielectric constant of the host, proximity of a surface or other polarizable entities). Such features, not observed in the bulk counterparts, are a characteristic effect of the collective oscillations of the electrons gas confined inside the metallic structures, and are given the name of localized surface plasmon resonances. In Fig. 4, BPSO was used to control the presence or the absence of plasmonic nano particles to have strong resonance peaks in the absorption of light. Depending on the position and number of the plasmonic nano disk, local field of each nano particle minimized or maximized. Therefore according to Eq. (10) the absorption can be suppressed for one wavelength and maximized for the other wavelength. The splitting and broadening of the single-particle resonances are determined by the strength of the interactions between adjacent particles, and they can be tuned by varying the separation or the number of the particles forming the chain. 6. Conclusion This paper, has reported the numerical analysis of a plasmonic demultiplexer in a new configuration. Its performance is based on integrating the coherent perfect absorption (CPA) devices into integrated photonic waveguides with resonators in specific wavelength. The simple and compact presented structure is promised for integrated all optical circuits. Its structural parameters can be adjusted to the desired and demanded channels. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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