IET Microwaves, Antennas & Propagation
Research Article
Nano-antenna synthesis for end-fire and
pencil-beam far-field radiation patterns using
vector spherical wave functions
ISSN 1751-8725
Received on 12th December 2019 Revised 1st April 2020
Accepted on 27th May 2020 E-First on 8th October 2020 doi: 10.1049/iet-map.2019.1105 www.ietdl.org
Seyed Sina Vaezi
1, Saeid Nikmehr
1, Ali Pourziad
11Electrical and Computer Engineering Department, Tabriz University, Tabriz, Iran E-mail: [email protected]
Abstract: Two sets of irregularly placed dielectric nano-spheres are synthesised as two nano-antennas. One of them has an end-fire and the other has a pencil-beam far-field radiation pattern. The electrical sizes (radiuses) and relative permittivities of all spheres are chosen to be 0.1 and 4, respectively. An incident plane-wave is illuminated on spheres. The aim is to find the optimal positions of spheres to scatter the incident wave similar to the end-fire and pencil beam radiation patterns. At first, the numbers of spheres of the nano-antennas are determined. Next, a random three-dimensional distribution of spheres is considered. Desired pencil-beam and end-fire patterns besides all scattered and incident waves are expanded to vector spherical wave functions. Therefore, the analysis is performed in the spectral domain of the spherical harmonics. The multiple scattering between spheres is considered. Coefficients of scattered waves are calculated by applying boundary conditions on the surface of all spheres. The aggregate difference of coefficients of total far-field scattering waves and coefficients of the desired far-field pattern is considered as a cost-function for an optimisation method. Particle swarm optimisation tool is used to optimise the positions of spheres to decrease the cost value. This process is done in several iterations. It is shown that by increasing the number of iterations, the value of cost function is decreased. So, this is an efficient method to design antennas including nano-antennas for any arbitrary radiation pattern and polarisation.
1 Introduction
Antennas are transducers of propagating electromagnetic (EM) waves into alternating currents and vice versa. This transformation is one of the main concepts which should be considered in antenna design procedures [1]. The propagating EM waves of the antennas have some radiation characteristics. These characteristics have different spectral values for different applications. As a method of the antenna design process, the current distributions and current paths on an antenna structure could be manipulated in a way that specific radiation characteristics could be achievable. This is usually done by shaping metallic parts of the antennas and defining current paths on it. Even the array antenna theory is based on the placement of current elements on theoretically determined positions. Some common and simple types of antennas are wire, dipole, microstrip, log-periodic, bowtie, traveling wave, and so on.
These antennas have well-specified characteristics. Besides metallic antennas, some antennas are constructed with dielectric materials like dielectric resonator antennas. These antennas convert resonating near-field waves into radiating far-field waves. This kind of conversion is also useful in design and synthesis procedures.
In antenna design, the similarity of characteristics of the designed antenna with desired features specifies the quality of antenna synthesis technique. One of the main features of the antennas is the far-field radiation pattern. Simple patterns are achievable by mentioned single element antennas. Some of the most useful patterns are wide-angle, end-fire (EF), broadside, and pencil-beam (PB) patterns. These patterns are widely used because of their various applications in energy harvesting, radar, military and medical cases, and so on. When more complex and specific patterns are required, some pattern synthesis methods should be implemented. First of all, the desired features should be modelled physically and mathematically. This modelling should be either exact or approximate. The next step is to configure the structure of the antenna and define the excitation of the antenna. There have been various analytical investigations in this field [1–11]. Besides analytical methods, numerical optimisation tools like genetic
algorithm and particle swarm optimisation (PSO) are widely used for synthesis procedure [12–15].
Nano-antenna (NA) is an extended concept of radio-frequency (RF) antennas into the optical frequency domain. Recent advances in sub-wavelength NAs have brought impressive improvements in energy harvesting, sensing, imaging, disease treatment, and deterrence. These antennas are mainly made of nanometer-sized particles that are called nano-particles. The interaction of these nano-particles with EM waves differ from their RF counterparts. In the optical domain, the assumption of the perfect electrical conductor (PEC) is no longer valid. This is because of collective oscillations of electrons on the surface of metallic particles. This phenomenon is called plasmonic resonance and causes metallic materials to behave as frequency-dependent dielectrics instead of PECs. Furthermore, some unique behaviors like plasmonic hybridisations occur in the optical domain. In this context, the transformation between the strongly localised near-field and radiating far-field waves is the main concern in design and analytical problems.
There are several ways to excite confined near-field waves of nano-antennas. Illumination of incident EM waves or placing quantum sources in the vicinity of them like quantum dots are some of these techniques. These excitation methods are the equivalents of voltage and current sources in the RF domain.
However, these techniques have some important considerations like multiple scattering and interaction between particles. Multiple scattering could be solved by applying boundary conditions.
In addition to near-field waves, far-field radiating waves should also be considered. To do this, the interaction of radiating EM waves with nano-particles must be analysed. Different analytical and numerical methods are used to investigate the interaction of nano-particles with incident light [16–22]. In some cases, EM waves are transformed from the spatial domain to a spectral domain. Fourier, Laplace, and Bessel transforms are some of these conversions. Another useful transformation is based on vector spherical wave functions (VSWF). This transformation is used when dealing with spherical boundaries and structures [23–31].
These vector functions are solutions of the Helmholtz equation.
Furthermore, they form an orthogonal set of base functions which
makes them suitable choices for the expansion of EM waves to VSWFs [23]. There have been some researches on VSWFs for EM applications including calculating resonances of spherical structures, scattering, expansion of specific EM waves like plane- waves or Gaussian beams and also dyadic Green's functions, antenna design, and so on [24–31]. Numerical optimisation methods are also widely used for antenna design purposes [12–15, 32–35]. These tools usually minimise or maximise some defined cost functions or fitness functions.
In this paper, two sets of irregularly positioned dielectric nano- spheres (nano-particle) are designed as two NAs. These antennas are excited by an incident plane wave and have EF and PB radiation patterns. These well-known patterns are the inputs of the synthesis problem and the outputs are the three-dimensional (3D) positions of similar homogenous particles of the two sets. Relative permittivities and electrical sizes of radiuses of all spheres are set to be 4 and 0.1, respectively. At first, the positions are selected randomly. The incident plane-wave induces electrical dipoles in each particle. These dipoles could be considered as elements of an array antenna shaping the radiation pattern based on their amplitudes and phases. The illumination of particles by plane EM wave results in the interaction between them. This interaction is described by multiple scattering. Boundary conditions are applied on the surfaces of all spheres. This will distinctly identify the scattering waves from all spheres. The aggregate of these individual scattering waves will be the total scattering. The total scattering pattern should be similar to the desired EF or PB radiation patterns. To implement the mentioned process, all EM waves including desired patterns, incident wave, and scattered waves are expanded to VSWF. This process will lead to a unique solution to the problem of scattering. The total scattering of any 3D arrangement of particles including nested ones could be analysed using VSWF expansion of incident and scattered waves. This problem could be solved by numerical iterative methods. However, it will be shown that for homogenous spheres which are very small compared to the wavelength, no iterative method is needed. As the scattering pattern is assumed to be arbitrarily defined, the PSO tool is applied. The variables of PSO are the positions of the particles.
The aim is to find the optimal values of these variables to have the desired scattering pattern. The variables are changed on each iteration based on pre-defined algorithms. On each iteration, as the positions are determined, the scattering is calculated. As all EM waves are expanded to VSWFs, the coefficients of the total scattering field are compared to the coefficients of the desired patterns. The difference between these two sets of coefficients is the cost function of PSO. This process is repeated and on each iteration, the positions are updated based on the calculated cost function. The iterations are repeated until the cost value is decreased to an acceptable value. Since the scope of this paper is to design nano-antennas, design frequency is chosen to be 600 THz and the radiuses of spheres are below 10 nm.
2 Theoretical basis for problem
The theoretical basis of the problem consists of several steps. First, the scattering from multiple nested particles is investigated. The method introduced to solve the scattering problem could be solved by iterative methods. Next, the specific condition of small spheres
is considered to simplify the calculation of scattering. Furthermore, a method is used to expand the arbitrary pattern into VSWFs.
Finally, an optimisation tool is implemented to synthesise the antenna for an arbitrary radiation pattern.
2.1 Multiple scattering from multiple spheres
EM waves are solutions of the vector Helmholtz equation. VSWFs which are a set of orthogonal functions are also solutions to this equation. Thus, these vector functions could be used as basis functions of the expansion of EM waves [22, 36, 37]. Considering A to be a vector function satisfying Helmholtz equation, the expansion could be written as
A=
∑
p= 1
2
∑
n= 1
N
∑
m= − n n
amnpNmnpν kr (1) In the equation above, indexes m, n, and p are the order, degree, and mode of expansion, respectively. Index p has a value of 1 for transverse magnetic (TM) mode and 2 for transverse electric (TE) mode. The order of expansion of A is represented by N which is determined by accuracy and spatial domain of expansion. This coefficient tends to infinity for an ideal expansion. Coefficients amnp are the expansion coefficients of expansion of A. These coefficients could be easily calculated using the orthogonality property of basis functions. Nmnpν kr represents the VSWF of order n, degree m, and mode p. EM waves could be propagative or non- propagative waves. Superscript ν is 1 and 2 for propagative and non-propagative waves, respectively. The propagation constant is defined by k. The two modes (TE and TM) of VSWFs are defined in two sets of equations as follows:
Nmnν1kr =1
k∇ × Nmnν2
kr (2)
Nmnν2kr = 2 n n+ 1
1/2
∇ × rψmnν kr (3)
In the equations above, ψmnν kr is the scalar spherical wave function. This function satisfies the scalar Helmholtz equation and is defined as
ψmnν kr = jnkr Ymncosθ, ϕ , ν = 1
hnkr Ymncosθ, ϕ , ν = 1 (4) This function is defined in spherical coordinates (r, θ, ϕ). jnand hn
represent the spherical Bessel and Hankel functions of order n, respectively. Also, Ymn is the spherical harmonic of order n and degree m defined as
Ymncosθ, ϕ = 2n + 1 4π
n− m ! n+ m !
1/2
Pnmcosθ exp jmϕ (5)
Pnmcosθ is the associated Legendre polynomial with order n and degree m.
Equation (1) is the expansion formula that could be used to represent electric and magnetic field intensities in terms of spherical harmonics shown in (2) and (3). In addition to EM field intensities, electric and magnetic vector potentials could also be expanded to spherical functions using (1).
The first step in the calculation of scattering is to define the physical structure. This structure is composed of several spheres that have spatial 3D distribution. Fig. 1 shows a general case. All spheres are described by their different relative permittivities, 3D positions, electrical sizes of their radiuses, and also their host mediums. The host medium of a particle is defined as a sphere that surrounds that particle, although the permeability and chirality factors could also be included. Spheres might be nested as spheres 3 and 4 in Fig. 1. Some of them that include no other particle inside are homogenous like spheres 1, 2, and 4. Sphere 0 is the virtual Fig. 1 Arbitrary set of nested spheres
surrounding medium of the structure. All sizes including radiuses and distances are applied in electrical size format in formulations.
The electrical size is 2πa/λ, where a is the physical size and λ is the operating wavelength.
The structure is illuminated by an incident plane wave. The goal is to calculate total scattering from the structure. All spheres are excited by the incident wave and also the scattered waves from other ones. This is called the multiple scattering that describes the interactions between particles. Different kinds of propagating and non-propagating waves are excited inside and outside of all spheres. Outside of each sphere, there has been a scattered propagating wave which travels in the outwards direction from the centre of that sphere. Furthermore, another incoming wave is excited outside which travels towards the centre. The incoming wave includes both the incident field and the scattered waves of all other spheres if the sphere is external (the sphere is not inside another one). For the case of sphere 0, the former is the total scattering from the structure and the latter is the incident plane wave. If the outside scattering fields are calculated for all spheres, the problem is completely solved. The electric fields outside sphere i could be written as
Eexti r− ri = Escai r− ri + Eregi r− ri (6) In the equation above, the numbers of superscripts (i) denote the numbers of spheres. Eexti , Escai , and Eregi are the total external, scattering, and incoming electric fields outside the sphere i. r and ri are the position vectors of spheres 0 and i, respectively. The arguments of fields show that the fields are expanded in local coordinates of sphere i. Inside every particle, there exists a regular non-propagating wave that travels towards the centre. Also, there are some travelling scattered waves from spheres that are inside that. For homogenous particles, no internal scattered waves will exist. So, the electric fields inside sphere i are described as
Einti r− ri = Eregi r− ri +
∑
j∈ Ninti
Escaj r− rj (7)
Referring to (6) and (7) it could be seen that electric fields are described in local coordinates. This simplifies the process of applying boundary conditions. Therefore, VSWFs must be defined in spherical local or global coordinates systems. Local coordinates are defined for every sphere with the origin of the centre of that sphere. Also, the origin of the global coordinates system is the centre of the virtual sphere 0 in Fig. 1. As a result, all fields (incident, regular, or scattering) shown in (6) and (7) could be expanded to their local VSWFs using (1)
Eir− ri =
∑
p= 1
2
∑
n= 1
N
∑
m= − n n
amnpi Nmnpν k r− ri (8)
The incident field is the excitation of the structure. This field should be defined completely. Some of the simple and well-known definitions of the incident wave are plane waves or Gaussian beams. The VSWF expansion coefficients of these known incident waves in global coordinates are calculated and reported [25, 36].
Other fields stated in (6) and (7) including scattering and regular waves have unknown coefficients that must be determined. To this aim, the boundary conditions on the surface of all spheres should be applied. These conditions are the continuity of tangential components of electric and magnetic fields on all surfaces. It is considered that all spheres are homogenous. So, applying boundary conditions on all spheres relates the coefficients of scattering waves from one sphere to the coefficients of scattering waves from other spheres and also coefficients of incident wave [22]
amni ,s=
∑
t= 1 2
a¯ni,st
∑
j∈ Nexti
2
∑
l= 1 Lj
k
∑
= − l lHmn,kl,tkt0ri− rj aklj,t
+
∑
l= 1 L0
k
∑
= − l lJmn,kl,tkt0ri− r0 fkl0,t
(9)
Coefficients fkl0,t are the coefficients of the expansion of the incident field. Superscript 0 on fkl0,t indicates the expansion of incident wave in global coordinates. amni ,s are the coefficients of the expansion of scattered fields from sphere i (expanded in the local coordinate system of sphere i). Indexes s and t have the values of 1 or 2 for TE and TM modes, respectively. ri is the position vector for the centre of sphere i. r0 is the origin of the global coordinates system and could be considered as zero vector. kt0 is the propagation constant of mode t in sphere 0. Hmn,kl,tkt0ri− rj and Jmn,kl,tkt0ri− rj are addition coefficients translating the VSWF of order l, degree k, and mode t expanded in local coordinates of sphere j, into VSWF of order m, degree n, and mode t expanded in local coordinates of sphere i. These coefficients are fully described in [38]. Lj is the maximum Mie order of sphere j. This order is determined by the size and the relative permittivity of the sphere.
In the same way, L0 is the maximum Mie order of sphere 0. Nexti is the number of spheres that are outside sphere i. a¯ni,st is the generalised Mie coefficient of order n for sphere i and is determined by the radius and the relative permittivity of the sphere.
For every order, this coefficient is a 2 × 2 matrix (dyad) relating the TE and TM modes of the right side of (9) (t index) to the modes of the left side (s index). Equation (9) is the interaction equation.
Several methods like the bi-conjugate gradient method could be implemented to solve the interaction equation [22]. More complicated cases of nested particles are also studied in [22]. As the coefficients of local expansions of scattered fields (amni ,s) from external spheres are translated to the centre of structure through addition theorem [38]
amn0 ,s=
∑
i∈ Nint0
∑
l= 1 Li
∑
k= − l l
Jmn,kl,skriakli,s (10)
Nint0 is the number of external spheres. The coefficients amn0 ,s are the coefficients of total scattered field for all m, n, and s.
2.2 Eliminating the iterative method
A set of homogenous particles with equal relative permittivities are chosen. The electrical sizes of the radiuses of the spheres are also chosen to be very small. This reduces the interaction between them. To investigate the effect of decreasing the radiuses on the interactions, the right-hand side of (9) is considered. The first summation could be considered as the interaction of particles.
Although this summation is not the exact value of interactions between particles, this could be considered as a measure of interactions. The second part is the effect of incident plane wave on the scattering coefficients. Therefore, the ratio of interaction part to incident field part could be considered as a measure of interactions between spheres. This ratio could be written as
∑2t= 1a¯ni,st ∑2j∈Nexti ∑lL= 1j ∑kl= −lHmn,kl,tkt0ri− rj aklj,t
∑t2= 1a¯ni,st ∑lL= 10 ∑lk= −lJmn,kl,tk0tri− r0 fkl0,t (11) If the value of this ratio is negligible (e.g. under 0.1), (9) could be simplified to
amni ,s=
∑
t= 1 2
a¯ni,st
∑
l= 1 L0
∑
k= − l l
Jmn,kl,tkt0ri− r0 fkl0,t (12)
As the addition coefficient Jmn,kl,t is a function of positions and the coefficients fkl0,t are determined by the definition of the incident wave, this equation could be solved easily and needs no iterative methods to be solved. In Section 3, it is shown that spheres with electrical sizes of 0.1 have almost negligible interactions if the distances between them have electrical sizes >0.38.
For electrically small spheres, only the first order of Mie coefficients (a¯1,ist) are excited. This reduces the complexity of the calculation. On the other hand, as the spheres are considered to be external and excited by a plane wave, the summation over l and k in (12) could be replaced by the expansion coefficients of the plane wave in global coordinates multiplied by a phase shift [22]. The phase shift is a function of the positions
∑
l= 1 L0
∑
k= − l l
Jmn,kl,tkt0ri− r0 fkl0,t= exp jk(xicos α
+yisin α sin β + zicos β ) fmn0 ,t
(13) In (13), the angles α and β are Euler angles which determine the direction of propagation of incident plane wave. The direction is chosen to be on the x–z plane. Also, xi, yi, and zi are the Cartesian components of the position vector of the centre of sphere i. So, (12) is reduced to
amni ,s=
∑
t= 1 2
a¯ni,stexp jk xicos α + yisin α sin β + zicos β fmn0 ,t n= 1, 2, 3, …, Li, m = − n, …, n
(14) Equations (12) and (14) could be simplified to
amni ,s=
∑
t= 1 2
a¯ni,stAmn,tri, α, β n= 1, 2, 3, …, Li, m = − n, …, n
(15)
where
Amn,tri, α, β = exp jk xicos α + yisin α sin β + zicos β
fmn0 ,t (16)
As mentioned before, only the first order of Mie coefficients (a¯ni,st) is excited. As a result, the scattering coefficients (amni ,s) are excited only for the order of n = 1. So, (15) could be rewritten as
ami1,s=
∑
t= 1 2
a¯1,istAm1,tri, α, β , m= 0, ± 1 (17) Thus, (12) could be simplified [39]
amn0 ,s=
∑
i∈ Nint0
∑
k= − 1 1
Jmn,k1,s−kriaki1,s
=
∑
i∈ Nint0
∑
k= − 1 1
Jmn,k1,s−kri
∑
t= 1 2
a¯1,istAk1,tri, α, β (18)
As all spheres are of the same size and relative permittivity, a¯1,ist
will be equal for all of them (a¯1,ist= a¯1,st). The coefficients amn0 ,s are the expansion coefficients of the total scattering from the structure.
The total scattering field of any distribution of small homogenous spheres could be calculated by replacing the coefficients amn0 ,s in (1). There is no need for iterative methods and this decreases the calculation time.
2.3 Expansion of arbitrary patterns to VSWFs
The scattering pattern of a set of homogenous small spheres will be the same as the desired far-field scattering pattern if the coefficients of expansion of total scattering fields achieved in (18) are the same as the coefficients of expansion of the desired pattern.
In other words, the total scattering coefficients amn0 ,s are unknown and must be calculated using the expansion of the desired pattern
P∞θ, φ =
∑
p= 1
2
∑
n= 1 Np
m
∑
= − n namnp0 Nmnp3 r, θ, φ (19)
In the equation above, Np is the order of expansion of the far-field pattern into VSWFs. The vector function P∞θ, φ is the desired far-field pattern that depends only on the azimuthal and polar angles of the spherical coordinates system. The unknown coefficients must be calculated using (19) and applied to (17) for the synthesis process. One way to calculate these coefficients might be applying orthogonality of the VSFWs. However, the VSWFs are functions of all three coordinates including radial coordinate. As a result, applying the orthogonality to (19) does not yield a unique solution for the coefficients. Instead of using VSWFs, the far-field form of these vector functions could be used. Far-field forms of propagating VSWFs could be easily found by replacing Hankel functions by their asymptotic forms. The achieved vector functions that are described by Umnp are orthogonal, too
Umn2θ, φ = lim
kr→ ∞Nmn32
kr = ∇ΩYmnθ, φ × r^ Umn1θ, φ = lim
kr→ ∞Nmn31kr =1
k∇ × Umn2θ, φ
∇Ω= θ^∂
∂θ+ φ^ ∂
∂φ
(20)
These vector harmonics are not a function of the radial coordinate.
Therefore, the expansion of the desired far-field radiation pattern to vector spherical harmonics Umnp would lead to unique solutions.
This expansion is written as
P∞θ, φ = 1 jk
∑
p= 1
2
∑
n= 1 Np
m
∑
= − n n− jnamnpUmnpθ, φ (21)
Evaluation of unknown coefficients amnp is quite simple using the orthogonality of vector harmonics
amnp= jk
n n+ 1 − jn⟨P∞θ, φ , Umnp⟩ (22) where
⟨P∞θ, φ , Umnp⟩ =
∫
02π∫
0πP∞θ, φ ⋅ Umnp∗ sin θ dθdφ (23)It is noticeable that the far-field pattern is related to near-field wave P r, θ, φ as
P∞θ, φ = lim
r→ ∞ r= const .
P r, θ, φ
(24)
The calculated coefficients in (22) could be used as expansion coefficients of VSWFs in (19) [36, 39]
amnp0 = 1
jk − jnamnp (25)
The reason for choosing coefficients of (22) as expansion coefficients of P is that any selection of P which has an asymptotic form of P∞ could be considered as a solution to the problem. In other words, solutions for the synthesis problem for an arbitrary
pattern are not unique. So, if the coefficients of VSWF expansion of near-field pattern of an antenna are described by (25), this antenna would have a far-field pattern similar to the desired pattern and is a solution to the problem.
2.4 Performing optimisation to nano-antenna synthesis In the previous section, it is shown that any desired far-field radiation pattern could be expanded to VSWFs. The coefficients of (25) are the inputs of the problem. An optimisation tool (PSO) is used to synthesise a set of spheres. This set has a scattering pattern with expansion coefficients described by (18). These coefficients should be approximately equal to the coefficients of (25) with acceptable accuracy. To implement the optimisation method, a cost function is defined. This cost function is the sum of differences of expansion coefficients of the scattered wave calculated using (18) and expansion coefficients of the desired pattern calculated using (25). Also, the variables of PSO are 3D positions. The variables are changed on each iteration based on algorithms of PSO. For every arrangement, the scattering coefficients are calculated using (18).
These coefficients are compared to the coefficients of the desired radiation pattern calculated using (25). The difference between the coefficients is the cost value of optimisation. This value should be minimised by optimising the positions of the spheres.
3 Results and discussion
The investigations are performed using the programming tool of MATLAB software and frequency-domain solver of CST studio software on a computer with 8 GB of RAM and 2.4 GHz core i5 CPU. MATLAB is used to numerically analyse the results of theory and CST Studio is used to numerically validate the results of the theory.
3.1 Interactions of spheres
As mentioned in Section 2.2, the interactions between spheres could be neglected if the value of (11) is ignorable. This section explores the conditions which lower this value. As a starting point, the first summation on the right-hand side of (9) is analysed as the interactions. To do this, two spheres are considered. The sum of coefficients of expansion of interaction waves in (9) around sphere
1 which are coming from sphere 2 is calculated for different radiuses. The results are represented in Fig. 2.
The values of n1 and n2 in Fig. 2 are the Mie orders of spheres 1 and 2, respectively. As the electrical size is increased, the Mie order will increase, too. Mie orders of 3, 4, 5, 6, 7, and 8 are equivalent to electrical sizes of 0.3, 0.6, 1.3, 1.9, 2, and 3. The horizontal axis is the electrical distance between the two particles.
It is shown that for every selection of radiuses (from three cases of Fig. 2) as the distance is increased, the sum of coefficients is decreased. Also, for a fixed value of electrical distance, as the sizes (orders) are decreased, the interactions will decrease, too. For better illustration, the value of (11) is calculated for three cases of Fig. 2. Also, some other selections of sizes including the case for two equal spheres with electrical sizes of 0.1 are investigated. As stated in Section 2.2, if this ratio is negligible, the first summation of (9) could be eliminated and the validity of (12) is confirmed.
The results are shown in logarithmic scale in Fig. 3. The horizontal axis represents a scale factor. For any selection, the distance between spheres is the sum of electrical sizes multiplied by the scale factor. In other words, the electrical distance is f × S1 + S2 . S1 and S2 are the sizes of spheres and f is the scale factor (the horizontal axis of Fig. 3). It is shown that for sizes of 0.1, as the scale factor reaches ∼1.9, the value of (11) is decreased to 0.1. As the scale factor is further increased, this value reaches the values
<0.01. The scale factor of 1.9 is identical to the electrical distance of 0.38 for two spheres with electrical size of 0.1. Therefore, if the electrical distances are >0.38, the first summation in (9) could be ignored. This could be used as a criterion for the synthesis procedure in the next sections. For a synthesised set of spheres with the assumption of negligible interactions, the calculations would be true if all the electrical distances are >0.38. Another point to be considered is that, as it is evident from Fig. 3, some ripples occur as the distance is increased. It is observable that the calculated electrical distances are approximately equal to π for all cases. This is equal to the physical distance of half-wavelength.
3.1.1 Analysis of scattering of a single sphere: The first step is to investigate the scattering pattern of a single sphere. This is the element pattern of a set of spheres that could be considered as elements of an array antenna. It is expected that a single sphere with a radius that is very small compared to the operating wavelength will act as an electric dipole element when illuminated by an incident wave. The direction of the induced electric dipole will be the same as the direction of the electric field of the incident wave in the vicinity of the sphere.
A sphere with a relative permittivity of 4 is assumed. The radius (electrical size) is considered to be 0.1 and 1. The incident wave is a plane wave with an electric field along the x-axis and propagating along the z-axis. Fig. 4 shows the scattering patterns for the two mentioned electrical sizes. These results are evaluated with MATLAB. The results achieved by CST software are shown in Fig. 5 and confirms the validity of Fig. 4. For the case of the electrical size of 0.1, it is shown that the scattering electric field pattern is completely similar to the pattern of a Hertzian electric dipole. As expected, the dipole moment is along the direction of the electric field of illuminated plane wave (x-axis). Also, CPU Fig. 2 Sum of coefficients of expansion of interaction waves
Fig. 3 Value of (11) versus electrical distance
Fig. 4 Normalised electric far-field radiation calculated with MATLAB for one sphere of electric size
(a) 1, (b) 0.1
times are calculated and depicted in Table 1. This table shows the efficiency of the theory which is coded with MATLAB software.
Forming pattern based on array antenna theory: Antenna radiation pattern is the multiplication of element factor and array factor based on the array antenna theory. As shown in Figs. 4 and 5, the far-field radiation pattern of a sphere with an electrical size of 0.1 could be considered as the element factor of the array antenna pattern. So, arbitrary far-field radiation patterns could be generated by properly arranging arrays of small particles. The amplitudes and phases of electric dipoles that are induced on particles with the illumination of plane wave are the main parameters of antenna array theory which must be considered.
If the incident plane wave propagating along the z-axis is not evanescent, the electric dipole moments induced on very small spheres would have approximately the same amplitudes. However, the phases of elements would be different. The phases of induced dipoles that are placed on the x–y plane will be equal because this plane is an equi-phase plane for incident plane-wave propagating along the z-axis. Although the phase difference between spheres with separation of d from each other along the z-axis will be βd (β is the propagation constant), an array along the x–y plane will have an element phase shift of zero and an array along the z-axis with half-wavelength spacing between elements will have an element phase shift of π.
Synthesis of NAs for EF and PB radiation patterns: The next step is to find two optimum sets of small spheres that generate EF and PB far-field radiation patterns. The radiation EF and PB patterns are well-known patterns and there is no need to define them here. The only parameters which are important to be determined are the beam-widths of the patterns. The beam-width for the EF pattern is chosen to be 40° in both directions and also the beam-width for the PB pattern is chosen to be 40° in one direction. The initial normalised EF and PB patterns are shown in Fig. 6. The aim is to synthesise antennas that generate approximately these ideal patterns. These radiation patterns are the
inputs of the synthesis problem. As the initial patterns are defined, the next step is to expand these patterns to the VSWF series using (21). This process models the inputs of the problem mathematically. The achieved coefficients should be equal to the scattering coefficients of mentioned two sets of spheres calculated using (18). The flowchart of the process of synthesising an antenna that would generate the desired radiation patterns (EF or PB) is shown in Fig. 7. The goal is to generate the initial patterns (either EF or PB) with a certain accuracy. This accuracy is related to the accuracy of the expansion of the initial pattern to the VSWF series and also the optimisation process. The order of expansion of initial patterns to VSWFs and also search area of positions in the optimisation tool could be calculated as the accuracy is defined.
Considering (21), the accuracy of the expansion process is related to the order of expansion Np. As the required accuracy of expansion is increased, the order of expansion will also be increased. This increment will result in a larger radius of sphere 0 in Fig. 1. Therefore, the spheres could be positioned in larger areas and as a result, the search area of positions will be increased.
Another parameter that should be known before the synthesis procedure is the number of spheres that generates the best matches for the antenna pattern.
To find the optimum number of spheres, the optimisation method is performed with 15 iterations for a varying number of spheres from 2 to 10 and the cost values are depicted in Fig. 8. This shows the convergence behaviours of different numbers. A decision-making procedure could be applied to Fig. 8 to select the count of spheres that leads to the best convergence criteria. The cost values or the slope of the decrement of cost values are the important parameters in the decision procedure. It could be easily understood that eight spheres will lead to better convergence for Fig. 5 Normalised electric far-field radiation pattern calculated with CST
for one sphere of electric size (a) 1, (b) 0
Table 1 CPU time investigated for far-field calculations Size of element
factor CPU time (s)
MATLAB CPU time (s) CST
Studio
0.1 0.9219 9180
1 0.2344 544
Fig. 6 Desired 3D normalised (a) EF patterns, (b) PB patterns
Fig. 7 Flowchart of method of antenna design for arbitrary pattern and polarisation
the case of EF pattern. Also, for the case of PB pattern, eight or nine spheres could be selected. For less complexity of calculations, eight spheres are chosen for this case. As the initial parameters are known, the numerical optimisation tool, PSO, is initiated. To configure the optimisation method, the population of PSO is 200.
The initial positions are defined randomly. These positions must be inside sphere 0 which is defined by the accuracy of expansion of the desired pattern. The scattering coefficients of this set are calculated using (18). The results are compared to the coefficients
of expansion of desired EF or PB pattern calculated using (21). If the difference (the cost value defined in Section 2.4) is acceptable, the synthesis process is terminated and the optimum positions are achieved. If the difference is not acceptable, the positions are reformed using the PSO method. This process is repeated until the cost value is decreased to an acceptable error.
The procedure is performed with 10 and 50 iterations of the PSO and the placements of eight spheres are achieved for each iteration. The 3D normalised radiation pattern of 10 and 50 iterations for both EF and PB patterns achieved by programming in MATLAB software are shown in Figs. 9 and 10. Achieved 3D electrical positions are shown in Tables 2 and 3.
The minimum values of mutual electrical distances for synthesised eight spheres of EF and PB patterns are 1.93 and 0.61, respectively. These values are equal to 0.31 and 0.1 of the operating wavelength. However, these distances are equal to scale factors of 9.65 and 3.05 for the case of two spheres with sizes of 0.1. Referring to Section 3.1, the interactions are ignorable because the minimum electrical distances are higher than the threshold value of 0.38 discussed in Section 3.1. To confirm the results of the theory which are implemented by the programming tool of MATLAB software, the positions in Tables 2 and 3 are used to simulate the scattering problem using CST Studio software.
Simulation results for 3D far-field patterns are plotted in Fig. 11.
It is demonstrated that the results are in good agreement with the results of the theory depicted in Figs. 9 and 10. Another important point that should be considered in an optimisation Fig. 8 Cost values for different sphere numbers and 15 iterations for EF
and PB patterns
Fig. 9 3D normalised EF pattern for a set of eight spheres achieved by (a) 10 iterations, (b) 50 iterations using MATLAB
Fig. 10 3D normalised PB pattern for a set of eight spheres achieved by (a) 10 iterations, (b) 50 iterations using MATLAB
Table 2 Positions of spheres achieved after 50 iterations of PSO in electrical size for EF pattern synthesis
Sphere number x-position y-position z-position
1 −1.03 0.78 −0.30
2 −0.95 −1.14 −0.15
3 −1.37 −4.77 0.22
4 1.52 −0.09 −0.25
5 −4.83 −2.94 −0.34
6 −0.15 4.44 0.48
7 0.35 0.05 −2.52
8 −5.00 0.76 −0.36
Table 3 Positions of spheres achieved after 50 iterations of PSO in electrical size for PB pattern synthesis
Sphere number x-position y-position z-position
1 0.12 0.14 1.74
2 −4.89 0.63 1.60
3 −0.07 4.28 −0.85
4 −0.65 5.00 2.35
5 −0.55 −4.35 −1.26
6 −2.46 −0.02 −1.54
7 0.52 4.21 −0.95
8 5.00 0.53 1.34
Fig. 11 3D normalised pattern for a set of eight spheres calculated with CST Studio for
(a) EF patterns, (b) PB patterns
process is the convergence study. The cost values of the optimisation process for two cases of EF and PB patterns are shown in Fig. 12.
For ten iterations, the cost values are 0.86 and 1.23 for EF and PB patterns, respectively. By increasing iterations to 50, these values are reduced to 0.54 and 0.63, respectively. It is clear from Fig. 12 that increasing iterations of the optimisation method results in lower cost values. Therefore, even more accuracies (e.g. 10 or 1%) are achievable. This shows the efficiency of the introduced method.
Panel b of Fig. 8 with cost values of 0.54, panel b of Fig. 9 with cost values of 0.63, and simulation results of Fig. 10 are the radiation patterns of two sets of eight spheres which could be used as NAs with EF and PB far-field radiation patterns. Two- dimensional (2D) results for 10 and 50 iterations, initial pattern, and also simulation results for two planes φ = 0 and φ = π/2 are shown in Figs. 13–16. Considering the half-power beam-width (HPBW) circle, the beam-width of the achieved sets of spheres for both cases of EF and PB patterns are shown in Table 4 in degrees.
Results are in good agreement with the desired features of EF and PB patterns. It is also noticeable that the side-lobe level will also be decreased by increasing the number of iterations.
4 Conclusion
A novel method is introduced and used to synthesise two sets of eight spheres with optimum placement as nano-antennas. These antennas have EF and PB radiation patterns. VSWF expansion and PSO tool are the two main methods used to this aim. It is demonstrated that this method is efficient to synthesise nano- antennas with the desired radiation pattern (EF and PB patterns in this paper). Higher accuracies are also achievable with higher iterations of the PSO tool.
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