A semi-analytical method for analyzing inhomogeneous meta-surfaces under oblique incidence based on the polarizabilities of constituent particles and interaction constants is presented. The inhomogeneity is proposed in which a meta-surface is considered as a periodic array of repetitive super-cells. Our proposed method provides effective polarizabilities and therefore it gives a comprehensive view of meta-surface behavior. To validate our proposed method, several examples are illustrated, which include different cases of isotropic and bi-anisotropic arrays. In the first example, an array of different sizes of Gold spherical Nano-particles is considered. In the second example, a combined array of Gold (Plasmonic) and Si (dielectric) particles is analyzed and the third example includes an array of different sizes of mutually coupled Gold Nano-disks. The effective polarizabilities of all these arrays are computed under the illumination of oblique incident waves with different incident angles. The accuracy of this method is verified by comparing the computed reflection and transmission coefficients to the results produced by the HFSS software.
1. Introduction
Analyzing meta-surfaces is an important step in their design and synthesis, in different applications. Different analytical [1,2] and numerical [3] methods have been suggested for this purpose, in the literatures. In [1], a green function is employed to analyze a meta-surface made of Si spherical nano-particles; and, in [2], a green function is utilized to analyze a homogenized meta-surface under illumination of an electric dipole excitation. In [3], integral equation formulation and surface boundary condition are employed to study a meta-surface. Although the analytical methods are fast and provide an overall scheme of the meta-surface, they are complicated and only provide answers for some limited problems. On the other hand, while most problems can be solved using numerical methods, they are slower. In reality, however, the purpose of the analysis is to apply an effective method to predict the behavior of a meta-surface. In [2,4–6], meta-surfaces were modeled as homogenized surfaces; while different surface parameters such as polarizabilities, susceptibilities and surface impedances have been employed to model meta-surfaces in other instances. One well-known method for homogenizing meta-surfaces is that of generalized sheet transition conditions (GSTC) [6]. In this method, the surfaces’ impedance parameters are employed to model the meta-surface; and while the method is useful for solving a large variety of problems, the main drawback of this method is that impedance is not an intrinsic parameter of meta-surfaces, and it depends on both the arrays’ period and the specifications of the incident wave. Therefore, if the arrays’ period or the incident angle or polarization were to be changed, the equivalent parameters would need to be recalculated as well. To overcome this problem, another strategy was introduced in [5]. and in this method, a meta-surface was modeled using the two individual parameters of the single-particle’s polarizabilities as well as the interaction constants. Polarizability is an intrinsic property of a particle, and it does not change when changing the specifications of the incident wave, such as its polarization or incident angle. Analytical solutions have been proposed to calculate the polarizabilities of some limited cases [7–9]. In [4], these parameters were retrieved from the reflection and transmission coefficients. In [10], a numerical method was suggested for computing the polarizabilities of a particle in free space using its far-field scattering waves’ characteristics. This method was extended to substrated particles, in [11].
On the other hand, interaction constants model the arrays’ specifications and they also depend on the incident waves’ specifications. Therefore, if the mentioned parameters were to be changed, it would be sufficient to update just the interaction constants, and the first part of the analysis (the single-particle’s polarizabilities) could be used, unchanged. In [5], formulations were done for a homogenous array with a square arrangement, in free space and under normal incidence. In many realistic applications, meta-surfaces are illuminated by oblique incidences; therefore, these formulations were extended to the same problem under oblique incidence, in [12,13]. In [14], the theory of [5] is extended to a more general problem of analysis of asymmetrical meta-surfaces under normal incidence. Studying the polarizabilities and interaction constants of a meta-surface gives a comprehensive view of its specifications which is useful in designing and synthesizing of them. In [15], a generalized formulation to achieve Kerkers’ condition under oblique incidence are brought by studying their polarizabilities. Additionally, due to providing effective polarizabilities of constituent particles in an array, the theory of analyzing meta-surfaces under oblique incident introduced in [12] are employed in [16], to design a reflective optical surface; and in [17], this theory is used to find surface impedance modeling of an all-dielectric meta-surface.
On the other hand, inhomogeneous meta-surfaces have attracted much attention in recent years [18,19], due to their widespread applications in both wide-band and multi-band applications. Therefore, different methods for analyzing these types of meta-surfaces were suggested, in [20–23]. Based on the theory of [5], a semi-analytical method was introduced in [24] for analyzing inhomogeneous meta-surfaces under normal incidences. Employing inhomogeneous and multi-spectral meta-surfaces gives more degree of freedom in designing them in different applications; therefore, finding suitable answers for such problems is an important issue. However, analyzing inhomogeneous meta-surfaces under oblique incidence using their polarizabilities and interaction constants is an important problem needs to be addressed. In [23], a valuable retrieval method to study meta-surfaces under oblique incidences was introduced, which is applicable for studying a wide variety of problems. However, to calculate the effective polarizabilities of a meta-surface using that method, it is necessary to have a general knowledge of the polarizabilities, since the number of equations provided is less than the number of polarizabilities in that method [23]. Additionally, when using any retrieval method, if the period or the arrangement of the array are changed, the calculation should be repeated from the beginning. In this article, the method introduced in [24] is extended from normal incidences to the general form of inhomogeneous meta-surfaces under oblique incidences. Using this proposed method, all of the effective polarizabilities of a meta-surface can be computed separately, so it generates a comprehensive view of a meta-surface. Moreover, as mentioned in [24], inhomogeneity could create some effective polarizabilities, even if they do not exist in any of the constituent particles. Since in our method the interaction constants are considered separately, if the array specifications such as its period or particle arrangements are modified, it is sufficient to simply update these constants. The accuracy of our suggested method is explored using numerical examples, and the results are compared to the results of a high-frequency structure simulator (HFSS). We show that the two are in good agreement.
2. Theory
Suppose we have an array of multiple particles under oblique incidence, with an angle of ($\varphi _0$, $\theta _0$) as shown in Fig. 1(a). This meta-surface can be considered as a periodic array of super-cells, as shown in Fig. 1(b). As different particles exist in the meta-surface, different electric and magnetic moments, $\vec p_i$ and $\vec m_i$ are induced on the meta-surface, where the index, $i$ is related to the $ith$ particle. These moments are related to the local fields, as follows [24]:
(1)$$\left[ {\begin{array}{c} \vec p_i\\ \vec m_i \end{array}} \right] = \left[ {\begin{array}{cc} {{{\overline{\overline {\hat \alpha } }}^{ee}_{i}}} & {{{\overline{\overline{\hat \alpha} } }^{em}_{i}}}\\ {{{\overline{\overline {\hat \alpha }} }^{me}_{i}}} & {{{\overline{\overline {\hat\alpha }} }^{mm}_{i}}} \end{array}} \right].\left[ {\begin{array}{c} {{\vec E^{loc}_{i}}}\\ {{\vec H^{loc}_{i}}} \end{array}} \right].$$
where $\vec E_i^{loc}$ and $\vec H_i^{loc}$ are the local electric and magnetic fields at the position of the $ith$ particle, respectively. These local fields are the superposition of the incident and interaction fields caused by the other particles at the position of each particle, as follow: (2)$${\vec E^{loc}_{i}} = {\vec E_{inc}} + \sum_{j = 1,i \ne j}^N {{{\vec E}_{{\mathop{\rm int}} ,ji}}} ,$$
(3)$${\vec H^{loc}_{i}} = {\vec H_{inc}} + \sum_{j = 1,i \ne j}^N {{{\vec H}_{{\mathop{\rm int}} ,ji}}} ,$$
To find the effective polarizabilities of the array, similar to [24], the first step is to compute the effective polarizabilities of the super-cell. For this purpose, it is necessary to calculate the inter super-cell interaction constants. Therefore, in the first step, all the particles inside the super-cell are replaced by the electric and magnetic dipoles (as shown in Fig. 1(c)). The electromagnetic fields of an electric dipole are calculated as follows [25]: (4)$${\vec E_p}(r) = \{ (jk + \frac{1}{r})[\frac{{3\hat r(\hat r.\vec p) - \vec p}}{r}] + \,{k^2}\hat r \times (p \times \hat r)\} \frac{{{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}},$$
(5)$${\vec H_p}(r) ={-} j\omega (jk + \frac{1}{r})(\vec p \times \hat r)\frac{{{e^{ - jkr}}}}{{4\pi r}},$$
Since the observation point is at the meta-surface plane, the normal vector, $\hat r$, is considered to be $\hat r = cos(\phi )\hat x+sin(\phi )\hat y$. Therefore, the electromagnetic fields of each electric dipole can be written in the following terms: (6)$$\begin{aligned}{E_x} =& [(j\frac{k}{r} + \frac{1}{{{r^2}}})((3{\cos ^2}\varphi - 1){p_x} + 3\sin \varphi \cos \varphi {p_y})\\ &+ {k^2}({\sin ^2}\varphi - 1)]\frac{{{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}},\,\,\, \end{aligned}$$
(7)$$\begin{aligned}{E_y} =& [(j\frac{k}{r} + \frac{1}{{{r^2}}})((3{\sin ^2}\varphi - 1){p_x} + 3\sin \varphi \cos \varphi {p_y})\\ &+ {k^2}({\cos ^2}\varphi - 1)]\frac{{{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}},\, \end{aligned}$$
(8)$${E_z} = ( - (j\frac{k}{r} + \frac{1}{{{r^2}}}) + {k^2})\frac{{{p_z}{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}}\,$$
(9)$${H_x} ={-} j\omega (jk + \frac{1}{r})\sin \varphi \frac{{{p_z}{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}}\,$$
(10)$${H_y} = j\omega (jk + \frac{1}{r})\cos \varphi \frac{{{p_z}{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}}\,$$
(11)$${H_z} = j\omega (jk + \frac{1}{r})({p_x}\sin \varphi - {p_y}\cos \varphi )\frac{{{e^{ - jkr}}}}{{4\pi {\varepsilon _0}r}}\,$$
The interaction fields of $jth$ particle on the $ith$ particle are related to the polarizabilities, as follows [24]: (12)$${\vec E_{{\mathop{\rm int}} ,ji}} = \overline{\overline \beta } \,_{ji}^{ee}.{\vec p_j} + \overline{\overline \beta } \,_{ji}^{em}.{\vec m_j},$$
(13)$${\vec H_{{\mathop{\rm int}} ,ji}} = \overline{\overline \beta } \,_{ji}^{me}.{\vec p_j} + \overline{\overline \beta } \,_{ji}^{mm}.{\vec m_j},$$
Under a normal incidence, the incident field reaches all the particles with the same amplitudes and phases. Conversely, under an oblique incidence, the particles are excited by the same amplitude, but there is a phase difference due to the different traveling paths of the incident wave in reaching the particles, as shown in Fig. 1(d). This phase difference is computed as $D_i=k.r_i$ where $D_i$ is the phase difference of the $ith$ particle, and $k=k_0sin(\phi _0)\sin ({\theta _0})\hat {x}+k_0cos(\phi _0)\sin ({\theta _0})\hat {y}-k_0\cos ({\theta _0})\hat {z}$ and $\vec r_i=r_isin(\phi )\hat {x}+r_icos(\phi )\hat {y}$ are the wave-number and the distance of the $ith$ particle from a considered reference on the meta-surface, respectively. Without causing any loss in generality, suppose that the reference point is in the middle of the super cell. As a consequence of the phase difference, each particle would be excited by the incident wave with the same amplitude, but with different phases. Therefore, the electric and magnetic moments, respectively, could be considered as follows: (14)$${{p_i}(r') = {p_i}{e^{ - j{D_i}}}},$$
(15)$${{m_i}(r') = {m_i}{e^{ - j{D_i}}}},$$
By inserting Eqs. (14)–(15) into Eqs. (6)–(11), and combining the resulting relations with Eqs. (12)–(13), the following interaction constants can be achieved for each particle: (16)$$\beta _{xx,ji}^{ee,o} = \{ (j\frac{k}{{{r_{ji}}}} + \frac{1}{{{r_{ji}}^2}})[3{\cos ^2}{\varphi _{ji}} - 1] + {k^2}{\sin ^2}{\varphi _{ji}}\} \frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}}\,$$
(17)$$\beta _{yy,ji}^{ee,o} = \{ (j\frac{k}{{{r_{ji}}}} + \frac{1}{{{r_{ji}}^2}})[3{\sin ^2}{\varphi _{ji}} - 1] + {k^2}{\cos ^2}{\varphi _{ji}}\} \frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}}\,$$
(18)$$\beta _{zz,ji}^{ee,o} = \{ - (j\frac{k}{{{r_{ji}}}} + \frac{1}{{{r_{ji}}^2}}) + {k^2}\} \frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}}\,$$
(19)$$\begin{aligned}\beta _{xy,ji}^{ee,o} = \beta _{yx,ji}^{ee,o} &= \{ (j\frac{k}{{{r_{ji}}}} + \frac{1}{{{r_{ji}}^2}})3\sin {\varphi _{ji}}\cos {\varphi _{ji}}\\ &+ {k^2}\sin {\varphi _{ji}}\cos {\varphi _{ji}}\} \frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}},\end{aligned}$$
(20)$$\beta _{xz,ji}^{me,o} ={-} \beta _{zx,ji}^{me,o} ={-} j\omega (jk + \frac{1}{{{r_{ji}}}})\sin {\varphi _{ji}}\frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}}\,$$
(21)$$\beta _{yz,ji}^{me,o} ={-} \beta _{zy,ji}^{me,o} = j\omega (jk + \frac{1}{{{r_{ji}}}})\cos {\varphi _{ji}}\frac{{{e^{ - jk{r_{ji}}}}{e^{ - j{D_i}}}}}{{4\pi {\varepsilon _0}{r_{ji}}}}\,$$
where $r _{ji}$ and $\varphi _{ji}$ are the distance and phase between the $jth$ particle and the $ith$ particle, respectively. As it is expected, the different excitation fields result in different interaction constants; so, by considering the explained phase differences on each particle in the super cell, the inter super cell interaction constants of this problem can be related by those of a normal incidence [12]. By considering the explained phase difference on each particle in the super cell, the inter super cell interaction constants of this problem are related by those of normal incidence [24] as follows: (22)$$\overline{\overline \beta }_{ji}^o = \overline{\overline \beta }_{ji}^n.{{\overline {\overline{D}}}_i},$$
(23)$${\overline{\overline D}_i} = \left[ {\begin{array}{ccc} {{e^{ - j{D_i}}}} & 0 & 0\\ 0 & {{e^{ - j{D_i}}}} & 0\\ 0 & 0 & {{e^{ - j{D_i}}}} \end{array}} \right],$$
where ${\overline {\overline \beta } _{ji}^o}$, ${\overline {\overline \beta } _{ji}^n}$ and $\overline {\overline D}$ are the interaction constants of $jth$ particle on $ith$ particle under oblique incidence, those interaction constants under normal incidence and the delay matrix, respectively. The next step is the calculation of the effective polarizabilities of the super cell; and by substituting the incident and interaction fields into Eqs. (1) and (2), we have the following relations: (24)$$\begin{aligned}\left[{\begin{array}{c} {{{\vec p}_i}}\\ {{{\vec m}_i}} \end{array}} \right]&= \left[{\begin{array}{cc} {\overline{ \overline \alpha} _i^{ee}} & {\overline{ \overline \alpha} _i^{em}}\\ {\overline{ \overline \alpha} _i^{me}} & {\overline{ \overline \alpha} _i^{mm}} \end{array}}\right].\\ &\left[ {\begin{array}{c} {{e^{ - j{D_i}}}{{\vec E}_{inc}}+ \sum_{j = 1,i \ne j}^N {\overline{ \overline \beta} _{ji}^{ee,o}.{{\vec p}_j}} \, + \sum_{j = 1,i \ne j}^N {\overline{ \overline \beta} _{ji}^{em,o}.{{\vec m}_j}} }\\ {{e^{ - j{D_i}}}{{\vec H}_{inc}} + \sum_{j = 1,i \ne j}^N {\overline{ \overline \beta} _{ji}^{me,o}.{{\vec p}_j}} \, + \sum_{j = 1,i \ne j}^N {\overline{ \overline \beta} _{ji}^{mm,o}.{{\vec m}_j}} } \end{array}} \right].\ \end{aligned}$$
Note that, as mentioned previously, the incident wave reaches each particle with a phase delay. By writing Eq. (24) for all of the constituent particles of the super cell, the incident waves can be related to the electric and magnetic moments, as follows: (25)$$\left[ {\begin{array}{c} {{{\vec p}_1}}\\ {{{\vec m}_1}}\\ \vdots \\ {{{\vec p}_n}}\\ {{{\vec m}_n}} \end{array}} \right] = \,{\overline{\overline \alpha } _t}.\left[ {\begin{array}{c} {{{\vec E}_{inc}}}\\ {{{\vec H}_{inc}}}\\ \vdots \\ {{{\vec E}_{inc}}}\\ {{{\vec H}_{inc}}} \end{array}} \right],$$
where $\overline {\overline {\alpha }}^o_t$ is specified as Eq. (26) in which $\overline {\overline {I}}$ and $\overline {\overline {Z}}$ are identity matrix and zero matrix respectively and other parameters are defined as follows, (26)$$\begin{aligned}{\overline{\overline\alpha }^o _t}\, &= \,{\left[ {\begin{array}{ccccccc} {\overline{\overline I} } & {\overline{\overline z} } & {\overline{\overline \delta } \,_{21}^{e,o}} & {\overline{\overline \delta } \,_{21}^{m,o}} & \cdots & {\overline{\overline \delta } \,_{n1}^{e,o}} & {\overline{\overline \delta } \,_{n1}^{m,o}}\\ {\overline{\overline z} } & I & {\overline{\overline \gamma } \,_{21}^{e,o}} & {\overline{\overline \gamma } \,_{21}^{m,o}} & \cdots & {\overline{\overline \gamma } \,_{n1}^{e,o}} & {\overline{\overline \gamma } \,_{n1}^{m,o}}\\ {\overline{\overline \delta } \,_{12}^{e,o}} & {\overline{\overline \delta } \,_{12}^{m,o}} & I & {\overline{\overline z} } & \cdots & {\overline{\overline \delta } \,_{n2}^{e,o}} & {\overline{\overline \delta } \,_{n2}^{m,o}}\\ {\overline{\overline \gamma } \,_{12}^{e,o}} & {\overline{\overline \gamma } \,_{12}^{m,o}} & {\overline{\overline z} } & I & \cdots & {\overline{\overline \gamma } \,_{n2}^{e,o}} & {\overline{\overline \gamma } \,_{n2}^{m,o}}\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ {\overline{\overline \delta } \,_{1N}^{e,o}} & {\overline{\overline \delta } \,_{1N}^{m,o}} & {\overline{\overline \delta } \,_{2N}^{e,o}} & {\overline{\overline \delta } \,_{2N}^{m,o}} & \cdots & I & {\overline{\overline z} }\\ {\overline{\overline \gamma } \,_{1N}^{e,o}} & {\overline{\overline \gamma } \,_{1N}^{m,o}} & {\overline{\overline \gamma } \,_{2N}^e} & {\overline{\overline \gamma } \,_{2N}^{m,o}} & \cdots & {\overline{\overline z} } & I \end{array}} \right]^{ - 1}}\\ &.\left[ {\begin{array}{@{}ccccccc@{}} {\overline{\overline \alpha } _1^{ee}} & {\overline{\overline \alpha } _1^{em}} & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline \alpha } _1^{me}} & {\overline{\overline \alpha } _1^{mm}} & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline \alpha } _2^{ee}} & {\overline{\overline \alpha } _2^{em}} & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline \alpha } _2^{me}} & {\overline{\overline \alpha } _2^{mm}} & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\alpha _N^{ee}} & {\alpha _N^{em}}\\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\alpha _N^{me}} & {\alpha _N^{mm}} \end{array}} \right].{\left[ {\begin{array}{@{}ccccccc@{}} {{{\overline{\overline D} }_1}} & {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline z} } & {{{\overline{\overline D} }_1}} & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline z} } & {\overline{\overline z} } & {{{\overline{\overline D} }_2}} & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & {{{\overline{\overline D} }_2}} & \cdots & {\overline{\overline z} } & {\overline{\overline z} }\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {{{\overline{\overline D} }_N}} & {\overline{\overline z} }\\ {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & {\overline{\overline z} } & \cdots & {\overline{\overline z} } & {{{\overline{\overline D} }_N}} \end{array}} \right]_{6N \times 6N}}\, \end{aligned}$$
(27)$$\overline{\overline \delta } \,_{ji}^{e,o} ={-} \overline{\overline \alpha } \,_j^{ee}.\overline{\overline \beta } \,_{ji}^{ee,o} - \overline{\overline \alpha } \,_j^{me}.\overline{\overline \beta } \,_{ji}^{em,o},$$
(28)$$\overline{\overline \delta } \,_{ji}^{m,o} ={-} \overline{\overline \alpha } \,_j^{em}.\overline{\overline \beta } \,_{ji}^{ee,o} - \overline{\overline \alpha } \,_j^{mm}.\overline{\overline \beta } \,_{ji}^{em,o},$$
(29)$$\overline{\overline \gamma } \,_{ji}^{e,o} ={-} \overline{\overline \alpha } \,_j^{me}.\overline{\overline \beta } \,_{ji}^{mm,o} - \overline{\overline \alpha } \,_j^{ee}.\overline{\overline \beta } \,_{ji}^{me,o},$$
(30)$$\overline{\overline \gamma } \,_{ji}^{m,o} ={-} \overline{\overline \alpha } \,_j^{mm}.\overline{\overline \beta } \,_{ji}^{mm,o} - \overline{\overline \alpha } \,_j^{em}.\overline{\overline \beta } \,_{ji}^{me,o},$$
After computing the above equations with a similar procedure to [24], the effective polarizabilities of the super cell are computed. The next step is to compute the array’s effective polarizabilities. For this purpose, it is assumed that the meta-surface is a sheet with an inhomogeneous electric current, and magnetic densities of $P = \mathop \sum \nolimits_{i = 1}^N {{p_i}}/s$ and $M = \mathop \sum \nolimits_{i = 1}^N {{m_i}}/ s$ in which $N$ and $s$ are the total numbers of particles in the super cell, and on the surface of the super cell, respectively. The transversal and normal components of electric and magnetic current densities are defined as follows: (31)$$\vec P = {\vec P_t} + {\hat a_z}{P_n}\,$$
(32)$$\vec M = {\vec M_t} + {\hat a_z}{M_n}\,$$
The problem is thereby simplified to a square array under the illumination of an oblique incidence, with a new period, ${a'}$, which is equal to the center-to-center distance of the adjacent super-cells. Note that these theory is valid until the meta-surface is dipolar and the array period satisfies the condition of $ka'<3$ where $k$ is the wave number [24]. To compute the effective polarizabilities of the array, the interaction constants from [12] are employed; while to find the transversal components of the reflection and transmission coefficients of the meta-surface under an oblique incidence, the following relations should be employed [23]: (33)$$\vec E_t^{ref} ={-} \frac{1}{2}[j\omega (\overline{\overline Z} .\vec{P_t} \mp {\hat a_z} \times \vec{M_t}) \mp j({k_t}\frac{{{P_n}}}{\varepsilon }\, \pm \overline{\overline Z} .({\hat a_z} \times {k_t}))\frac{{{M_n}}}{\mu }],$$
(34)$$\vec E_t^{trans} = \vec E_t^{inc} - \frac{1}{2}[j\omega (\overline{\overline Z} .\vec{P_t} \pm {{\hat a}_z} \times \vec{M_t})\pm j({k_t}\frac{{{P_n}}}{\varepsilon }\, \mp \overline{\overline Z} .({{\hat a}_z} \times {k_t}))\frac{{{M_n}}}{\mu }],$$
where $\vec E_t^{ref}$ and $\vec E_t^{trans}$ are the tangential components of the reflected and transmitted fields, respectively, and $\overline {\overline Z}$ is defined as follows: (35)$$\overline{\overline Z} = \left[ {\begin{array}{cc} {{Z_{\bot}}{{\cos }^2}({\varphi _o}) + {Z_{||}}{{\sin }^2}({\varphi _o})} & {({Z_{\bot}} - {Z_{||}})\sin ({\varphi _o})\cos ({\varphi _o})}\\ {({Z_{\bot}} - {Z_{||}})\sin ({\varphi _o})\cos ({\varphi _o})} & {{Z_{\bot}}{{\sin }^2}({\varphi _o}) + {Z_{||}}{{\cos }^2}({\varphi _o})} \end{array}} \right],$$
where $Z_{\bot }$ and $Z_{||}$ are equal to $\eta.{cos{\theta _0}}$ and $\eta /{cos{\theta _0}}$, respectively. From Eqs. (33) and (34), it is clear that the transversal components of reflected and transmitted waves are affected by normal polarizabilities in addition to the parallel polarizabilities. Using these equations, the reflection and transmission coefficients are considered as the following matrixes for both TE and TM incident waves. Then, the reflection and transmission coefficients for any other incident waves can be found easily using a linear combination of these calculated coefficients. (36)$${R^{TE}} = \left[ {\begin{array}{cc} {R_{xx}^{TE}} & {R_{xy}^{TE}}\\ {R_{yx}^{TE}} & {R_{yy}^{TE}} \end{array}} \right],$$
(37)$${T^{TE}} = \left[ {\begin{array}{cc} {T_{xx}^{TE}} & {T_{xy}^{TE}}\\ {T_{yx}^{TE}} & {T_{yy}^{TE}} \end{array}} \right],$$
(38)$${T^{TM}} = \left[ {\begin{array}{cc} {T_{xx}^{TM}} & {T_{xy}^{TM}}\\ {T_{yx}^{TM}} & {T_{yy}^{TM}} \end{array}} \right],$$
(39)$${R^{TM}} = \left[ {\begin{array}{cc} {R_{xx}^{TM}} & {R_{xy}^{TM}}\\ {R_{yx}^{TM}} & {R_{yy}^{TM}} \end{array}} \right],$$
The unknown terms in the above matrix are defined in the Appendix.3. Numerical results
In this section, the explained theory is confirmed by different examples. For this purpose, three different inhomogeneous arrays are brought. In these examples, it is tried to consider various inhomogeneous array consisting of even isotropic or bi-anisotropic particles. Different dielectric (Si) and plasmonic (gold) materials are employed in these particles. The permittivities of the mentioned materials are taken from [26]. In the first example, an isotropic array of two different sizes of Gold spherical Nano-particles is brought. In the next example, a complicated array including both Si and gold spherical nano-particles is considered. The last example contains a bi-anisotropic array of mutually coupled Gold disks with two different sizes. Mie-scattering theory [27] is employed to calculate the dominant polarizabilities of spherical particles and the polarizabilities of third examples is calculated using induced current method [28].
For the first example, an array of differently sized gold spherical nano-particles is considered, as shown in Fig. 2. The dimensions of smaller and larger spheres are $R_1 = 40 nm$ and $R_2 = 45 nm$, respectively. The center to center distance between any of the adjacent particles is $a = 150 nm$. The dominant polarizability of a single gold spherical nano-particle is $\alpha _{ee}$. The effective polarizabilities of the array of Fig. 2 under the illumination of oblique incident wave with $\theta _0 = 30^o$ are plotted in Fig. 3. However, there is no magnetic polarizability in none of the constituent particles, this polarizability is created due to inhomogeneity. Note that this component is not dominant but here, it is brought to show how inhomogeneity leads to stimulating new polarizabilities. To prove the accuracy of our proposed method, the calculated reflection and transmission coefficients of this meta-surface under the illumination of plane wave incidence with $\theta _0 = 30^o$ is calculated using our proposed method and compared to the results of HFSS software for both TE and TM incidence in Fig. 4. As shown in this Figure, these results are in good agreement with each other.
In the next example, a more complicated instance is considered, involving a combined array of Si and gold spherical nano-particles, as shown in Fig. 5. This example has more entropy in comparison to the previous example since each of these materials has different single-particle dominant polarizabilities, and different resonance frequencies. Here, it is assumed that both of the spheres have a radius of $R = 40 nm$, and the center-to-center distance between any adjacent particle is $a = 150 nm$. The dominant polarizabilities of the Si spherical nano-particles with the mentioned dimensions are the electric polarizability, $\alpha _{ee}$, and magnetic polarizability, $\alpha _{mm}$. The dominant effective polarizabilities of the array of Fig. 5 under the illumination of a plane wave with the incident angle of $\theta = 45^o$ are shown in Fig. 6. As it is clear from this figure, there are two different resonances for the effective electric polarizabilities, since $\alpha _{ee}$ is the dominant polarizability of both of the particles; while the magnetic polarizabilities have only one resonance frequency for the Si particles. However, the terms of $\alpha _{ee}^{xy/yx}$ are not dominant, but they are brought to show inhomogeneity creates polarizabilities which do not exist in none of the constituent particles. From the results of Fig. 6, there are three different resonance frequencies in this structure. One resonance frequency belongs to gold spherical nano-particle which occurs at 550 THz. Si spherical nano-particles have two different resonance frequencies at 700 THz and 770 THz which belong to magnetic polarizability and electric polarizability, respectively. Therefore it is expected the reflection and transmission coefficients have three different resonance frequencies. For this array, the reflection and transmission coefficients achieved using the proposed method are compared to the HFSS results, as shown in Fig. 7, under the illumination of an oblique plane wave with an incidence angle of $\theta _0 = 45^o$. The results of both methods show good agreement with each other. However, the error is increased in the vicinity of 700 THz under illumination of TM incidence. This frequency belongs to magnetic polarizability of the Si spherical Nano-particles. As it is expected, three mentioned resonance frequencies exist in the reflection and transmission coefficients.
In the last example, a bi-anisotropic array is brought. This array is composed of two different sizes of mutually coupled gold nano-disks which are separated by a glass spacer. This array is shown in Fig. 8. In comparison to previous examples which are composed of isotropic particles, this example has more complications. The dimension of the particles are as: Total cylinders hight, $h = 70 nm$, the Glass hight, $h_g = 20 nm$, the radius of larger cylinder, $R_l = 45 nm$ and the radius of smaller cylinder, $R_s = 35 nm$. The center to center distance between any of the adjacent particles is $a = 175 nm$.
The dominant polarizabilities for the array of Fig. 8 are $\alpha _{ee}^{xx/yy/zz}$, $\alpha _{em}^{xy/yx}$, $\alpha _{me}^{yx/xy}$ and $\alpha _{mm}^{yy/xx}$ [29]. Induced current method [28] is employed to calculate the polarizabilities of the single particles. Therefore, it is expected that the error for this example increases in comparison to pervious examples. These dominant effective polarizabilities for this array under the illumination of plane wave incident with the incident angle of $\theta = 60^o$ are plotted in Fig. 9. The repetitive components are not plotted in this figure due to reciprocity [5]. The reflection and transmission coefficients are computed using the proposed method and compared with HFSS results in Fig. 10. The results are in suitable agreement with each other; therefore, the accuracy of our method is confirmed in these examples.
4. Conclusion
The theory of analyzing inhomogeneous meta-surfaces has been extended, here, to the general form of oblique incidences. Using our proposed method, the effective polarizabilities of such arrays can be found, which is useful in the design and synthesis of these types of meta-surfaces, for different applications. To confirm the proposed formulation, different examples which covered different cases of isotropic and bi-anisotropic particles were illustrated. It is tried to employ both dielectric (Si) and plasmonic (gold) materials in the constituent particles. Moreover, the arrays are considered under the illumination of different incident angles. The results of our proposed method were also confirmed by comparison to the results of the HFSS software.
Appendix – Calculation of the reflection and transmission coefficients
In this section, the known terms in Eqs. (36)–(39) are brought. For the problem shown in Fig. 1(a), the electric and magnetic fields of an oblique TE incident wave are defined as:
(A1)$${\vec E_{inc}} = {E_0}(\sin ({\varphi _0}){\hat a_x} + \cos ({\varphi _0}){\hat a_y}),$$
(A2)$$\begin{aligned}{\vec H_{inc}} &= \frac{{{E_0}}}{\eta }(\cos ({\varphi _0})\cos ({\theta _0}){\hat a_x} - \sin ({\varphi _0})\cos ({\theta _0}){\hat a_y}\\ &- (\sin ({\varphi _0}) + \cos ({\varphi _0}))\sin ({\theta _0}){\hat a_z}), \end{aligned}$$
Similar relations as Eqs. (
A1) and (
A2) can be considered for electric and magnetic components of TM wave, respectively. By putting these fields at
$\phi = 0$ and
$\phi = 90$ planes in Eq. (
1), the electric and magnetic moments are brought. Then, using Eqs. (
33) and (
34), the reflection and transmission coefficients are achieved as follows,
(A3)$$\begin{aligned}R_{xx}^{TE} &= \frac{{ - j\omega }}{2}[\frac{\eta }{{\cos ({\theta _0})}}\alpha _{ee}^{xx} \mp \alpha _{em}^{xy}\,\, \mp \tan ({\theta _0})\alpha _{em}^{xz} \pm \alpha _{me}^{yx} - \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{yy}\\ & - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{yz} \mp \tan ({\theta _0})\alpha _{me}^{zx} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{zy} + \frac{{\sin ({\theta _0})\cos ({\theta _0})}}{\eta }\alpha _{mm}^{zz}, \end{aligned}$$
(A4)$$\begin{aligned}R_{yy}^{TE} &= \frac{{ - j\omega }}{2}[\frac{\eta }{{\cos ({\theta _0})}}\alpha _{ee}^{yy} \pm \alpha _{em}^{yx}\,\, \mp \tan ({\theta _0})\alpha _{em}^{yz} \mp \alpha _{me}^{xy} - \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{xx}\\ & + \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{xz} \mp \tan ({\theta _0})\alpha _{me}^{zy} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{zx} + \frac{{\sin ({\theta _0})\cos ({\theta _0})}}{\eta }\alpha _{mm}^{zz}, \end{aligned}$$
(A5)$$\begin{aligned}R_{xy}^{TE} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{yx} \mp \alpha _{em}^{yy}\,\,{\cos ^2}({\theta _0}) \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{yz} \mp \alpha _{me}^{xx} - \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{xy}\\ & - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{xz} + \eta \sin ({\theta _0})\alpha _{ee}^{zx} \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{zy} \pm \,{\sin ^2}({\theta _0})\alpha _{em}^{zz}, \end{aligned}$$
(A6)$$\begin{aligned}R_{yx}^{TE} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{xy} \pm \alpha _{em}^{xx}\,\,{\cos ^2}({\theta _0}) \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{xz} \pm \alpha _{me}^{yy} + \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{yx}\\ & - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{yz} + \eta \sin ({\theta _0})\alpha _{ee}^{zy} \pm \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{zx} \mp \,{\sin ^2}({\theta _0})\alpha _{em}^{zz}, \end{aligned}$$
(A7)$$\begin{aligned}T_{xx}^{TE} &= 1 - \frac{{j\omega }}{2}[\frac{\eta }{{\cos ({\theta _0})}}\alpha _{ee}^{xx} \mp \alpha _{em}^{xy} \pm \tan ({\theta _0})\alpha _{em}^{xz} \mp \alpha _{me}^{yx} + \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{yy} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{yz}\\ &\pm \tan ({\theta _0})\alpha _{me}^{zx} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{zy}\, + \frac{{\sin ({\theta _0})\cos ({\theta _0})}}{\eta }\alpha _{mm}^{zz}, \end{aligned}$$
(A8)$$\begin{aligned}T_{yy}^{TE} &= 1 - \frac{{j\omega }}{2}[\frac{\eta }{{\cos ({\theta _0})}}\alpha _{ee}^{yy} \pm \alpha _{em}^{yx} \mp \tan ({\theta _0})\alpha _{em}^{yz} \pm \alpha _{me}^{xy} + \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{xx} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{xz}\\ &\mp \tan ({\theta _0})\alpha _{me}^{zy} - \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{zx}\, + \frac{{\sin ({\theta _0})\cos ({\theta _0})}}{\eta }\alpha _{mm}^{zz}, \end{aligned}$$
(A9)$$\begin{aligned}T_{xy}^{TE} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{yx} \mp \alpha _{em}^{yy}\,\,{\cos ^2}({\theta _0}) \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{yz} \pm \alpha _{me}^{xx} + \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{xy}\\ & + \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{xz} - \eta \sin ({\theta _0})\alpha _{ee}^{zx} \pm \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{zy} \mp \,{\sin ^2}({\theta _0})\alpha _{em}^{zz}, \end{aligned}$$
(A10)$$\begin{aligned}T_{yx}^{TE} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{xy} \pm \alpha _{em}^{xx}\,\,{\cos ^2}({\theta _0}) \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{xz} \mp \alpha _{me}^{yy} + \frac{{\cos ({\theta _0})}}{\eta }\alpha _{mm}^{yx}\\ & + \frac{{\sin ({\theta _0})}}{\eta }\alpha _{mm}^{yz} - \eta \sin ({\theta _0})\alpha _{ee}^{zy} \mp \cos ({\theta _0})\sin ({\theta _0})\alpha _{em}^{zx} \pm \,{\sin ^2}({\theta _0})\alpha _{em}^{zz}, \end{aligned}$$
(A11)$$\begin{aligned}R_{xx}^{TM} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{yy}\, \pm \alpha _{em}^{yx} + \eta \sin ({\theta _0})\alpha _{ee}^{yz} - \frac{1}{{\eta \cos ({\theta _0})}}\alpha _{mm}^{xx} \mp \alpha _{me}^{xy}\\ & \pm \tan ({\theta _0})\alpha _{me}^{xz} \pm \tan ({\theta _0})\alpha _{em}^{zx} - \eta \sin ({\theta _0})\alpha _{ee}^{zy} + \eta \sin ({\theta _0})\tan ({\theta _0})\alpha _{ee}^{zz}, \end{aligned}$$
(A12)$$\begin{aligned}R_{yy}^{TM} &= \frac{{ - j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{xx}\, \mp \alpha _{em}^{xy} - \eta \sin ({\theta _0})\alpha _{ee}^{xz} - \frac{1}{{\eta \cos ({\theta _0})}}\alpha _{mm}^{yy} \pm \alpha _{me}^{yx}\\ & \mp \tan ({\theta _0})\alpha _{me}^{yz} \mp \tan ({\theta _0})\alpha _{em}^{zy} + \eta \sin ({\theta _0})\alpha _{ee}^{zx} - \eta \sin ({\theta _0})\tan ({\theta _0})\alpha _{ee}^{zz}, \end{aligned}$$
(A13)$$\begin{aligned}R_{xx}^{TM} &= 1 - \frac{{j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{yy}\, \pm \alpha _{em}^{yx} + \eta \sin ({\theta _0})\alpha _{ee}^{yz} - \frac{1}{{\eta \cos ({\theta _0})}}\alpha _{mm}^{xx} \pm \alpha _{me}^{xy}\\ & \mp \tan ({\theta _0})\alpha _{me}^{xz} \mp \tan ({\theta _0})\alpha _{em}^{zx} + \eta \sin ({\theta _0})\alpha _{ee}^{zy} + \eta \sin ({\theta _0})\tan ({\theta _0})\alpha _{ee}^{zz}, \end{aligned}$$
(A14)$$\begin{aligned}T_{yy}^{TM} &= 1 - \frac{{j\omega }}{2}[\eta \cos ({\theta _0})\alpha _{ee}^{xx} \mp \alpha _{em}^{xy} - \eta \sin ({\theta _0})\alpha _{ee}^{xz} \mp \alpha _{me}^{yx} + \frac{1}{{\eta \cos ({\theta _0})}}\alpha _{mm}^{yy}\\ & \pm \tan ({\theta _0})\alpha _{me}^{yz} \pm \tan ({\theta _0})\alpha _{em}^{zy} - \eta \sin ({\theta _0})\alpha _{ee}^{zx} + \eta \sin ({\theta _0})\tan ({\theta _0})\alpha _{ee}^{zz}, \end{aligned}$$
(A15)$$\begin{aligned}R_{xy}^{TM} &={-} \frac{{j\omega }}{2}[\eta \alpha _{ee}^{xy}\, \pm \frac{1}{{\cos ({\theta _0})}}\alpha _{em}^{xx} + \eta \tan ({\theta _0})\alpha _{ee}^{xz} \pm \cos ({\theta _0})\alpha _{me}^{yy} + \frac{1}{\eta }\alpha _{mm}^{yx}\\ & \mp \sin ({\theta _0})\alpha _{me}^{yz} \pm \sin ({\theta _0})\alpha _{me}^{zy} - \frac{{\tan ({\theta _0})}}{\eta }\alpha _{mm}^{zx} \mp \tan ({\theta _0})\sin ({\theta _0})\alpha _{me}^{zz}, \end{aligned}$$
(A16)$$\begin{aligned}R_{yx}^{TM} &={-} \frac{{j\omega }}{2}[\eta \alpha _{ee}^{xy}\, \mp \frac{1}{{\cos ({\theta _0})}}\alpha _{em}^{yy} - \eta \tan ({\theta _0})\alpha _{ee}^{yz} \mp \cos ({\theta _0})\alpha _{me}^{xx} + \frac{1}{\eta }\alpha _{mm}^{yx}\\ & \pm \sin ({\theta _0})\alpha _{me}^{xz} \mp \sin ({\theta _0})\alpha _{me}^{zx} + \frac{{\tan ({\theta _0})}}{\eta }\alpha _{mm}^{zy} \pm \tan ({\theta _0})\sin ({\theta _0})\alpha _{me}^{zz}, \end{aligned}$$
(A17)$$\begin{aligned}T_{xy}^{TM}&={-} \frac{{j\omega }}{2}[\eta \alpha _{ee}^{xy}\, \pm \frac{1}{{\cos ({\theta _0})}}\alpha _{em}^{xx} + \eta \tan ({\theta _0})\alpha _{ee}^{xz} \mp \cos ({\theta _0})\alpha _{me}^{yy} - \frac{1}{\eta }\alpha _{mm}^{yx}\\ & \pm \sin ({\theta _0})\alpha _{me}^{yz} \pm \sin ({\theta _0})\alpha _{me}^{zy} - \frac{{\tan ({\theta _0})}}{\eta }\alpha _{mm}^{zx} \mp \tan ({\theta _0})\sin ({\theta _0})\alpha _{me}^{zz}, \end{aligned}$$
(A18)$$\begin{aligned}T_{yx}^{TM} &={-} \frac{{j\omega }}{2}[\eta \alpha _{ee}^{xy}\, \mp \frac{1}{{\cos ({\theta _0})}}\alpha _{em}^{yy} - \eta \tan ({\theta _0})\alpha _{ee}^{yz} + \cos ({\theta _0})\alpha _{me}^{xx} - \frac{1}{\eta }\alpha _{mm}^{yx}\\ & \mp \sin ({\theta _0})\alpha _{me}^{xz} \mp \sin ({\theta _0})\alpha _{me}^{zx} + \frac{{\tan ({\theta _0})}}{\eta }\alpha _{mm}^{zy} \pm \tan ({\theta _0})\sin ({\theta _0})\alpha _{me}^{zz}, \end{aligned}$$
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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