## 2 Basic Equations for EM Wave Scattering by a Linear Array of Spheres  
In the present paper we consider a free-standing one-dimensional infinite array of dielectric spheres in air Figure 1. In what follows we refer all length quantities in terms of the period \(h\) of the array. We formulate the scattering theory by a periodic array of dielectric spheres in the form similar to the approach developed for a periodic array of dielectric cylinders [37; 61]  
\[\widehat{S}^{-1}\Psi=\widehat{L}\Psi=\Psi_{inc}. \tag{1}\]  
where the matrix \(\widehat{L}\) accounts for both the scattering matrix of the isolated sphere as well as the mutual scattering events between the spheres, \(\Psi_{inc}\) is given by the incident wave, and the column \(\Psi\) consists of amplitudes \(a_{l}^{m}\) of the multipole expansion of the scattering function. The exact expression of the matrix \(\widehat{L}\) was derived by [PERSON] et al. [24] for EM guided waves on a periodic array of dielectric spheres. For the reader's convenience we present the equations and notations from the above reference. We seek the solutions of the Maxwell equations, which obey the [PERSON] theorem:  
Figure 1: A periodic infinite array of dielectric spheres illuminated by a plane wave (blue arrow). The wave can be transmitted and reflected to discrete diffraction continua enumerated by integers \(m\) and \(n\) in accordance with Equations (17) and (21) shown by red arrows. \[\mathbf{E}(\mathbf{r}+\mathbf{R}_{j})=e^{j\beta j}\mathbf{E}(\mathbf{r}),\mathbf{H}( \mathbf{r}+\mathbf{R}_{j})=e^{j\beta j}\mathbf{H}(\mathbf{r})\]  
with the Bloch wave vector \(\beta\) directed along the array aligned with the \(z\)-axis (see Figure 1). Here \(\mathbf{R}_{j}=j\mathbf{e}_{z}\) is the position of the center of the j-th sphere and \(\mathbf{e}_{z}\) is the unit vector along the array. Scattered electromagnetic fields are expanded in a series over vector spherical harmonics \(\mathbf{M}_{n}^{m}\) and \(\mathbf{N}_{n}^{m}\)[1; 24]. \[\mathbf{E}(\mathbf{r})=\sum_{j}e^{i\beta j}\sum_{lm}[a_{l}^{m} \mathbf{M}_{l}^{m}(\mathbf{r}-\mathbf{R}_{j})+b_{l}^{m}\mathbf{N}_{l}^{m}( \mathbf{r}-\mathbf{R}_{j})],\] \[\mathbf{H}(\mathbf{r})=-i\sum_{j}e^{i\beta j}\sum_{lm}[a_{l}^{m} \mathbf{N}_{l}^{m}(\mathbf{r}-\mathbf{R}_{j})+b_{l}^{m}\mathbf{M}_{l}^{m}( \mathbf{r}-\mathbf{R}_{j})]. \tag{2}\]  
In series (2) the first/second terms presents TE/TM spherical vector EM fields. In absence of an incident wave [PERSON] et al. [24] derived the homogeneous matrix equation for the amplitudes \(a_{l}^{m},b_{l}^{m}\). \[Z_{TE,l}^{-1}a_{l}^{m}-\sum_{\
u}(a_{\
u}^{m}\mathcal{A}_{\
u l} ^{mm}+b_{\
u}^{m}\mathcal{B}_{\
u l}^{mm})=0,\] \[Z_{TM,l}^{-1}b_{l}^{m}-\sum_{\
u}(a_{\
u}^{m}\mathcal{B}_{\
u l} ^{mm}+b_{\
u}^{m}\mathcal{A}_{\
u l}^{mm})=0, \tag{3}\]  
where summation over \(\
u\) begins with \(max(1,m)\), and the so-called Lorenz-Mie coefficients are given by  
\[Z_{TE,l}=\frac{\hat{h}_{l}(kR)[r_{l}(k_{0}r)]^{\prime}_{r=-\hat{ h}_{l}(k_{0}R)[r_{l}(k_{0})]^{\prime}_{r=R}}}{\hat{h}_{l}(k_{0}R)[r_{l}(k_{0})] ^{\prime}_{r=-\hat{h}_{l}(kR)[r_{l}(k_{0})]^{\prime}_{r=R}}},\] \[Z_{TM,l}=\frac{\epsilon\hat{h}_{l}(kR)[r_{l}(k_{0}r)]^{\prime}_{ r=-\hat{h}_{l}(k_{0}R)[r_{l}(k_{0})]^{\prime}_{r=R}}}{\hat{h}_{l}(k_{0}R)[r_{l}(k _{0})]^{\prime}_{r=-R}}, \tag{4}\]  
where \(k=\sqrt{c}k_{0}\) and \(\epsilon\) is the dielectric constant of the spheres  
\[\widehat{S}^{-1}\Psi=\widehat{L}\Psi=\Psi_{inc}. \tag{1}\]0  
\[\widehat{S}^{-1}\Psi=\widehat{L}\Psi=\Psi_{inc}. \tag{1}\]1  
The coefficients  
\[\widehat{S}^{-1}\Psi=\widehat{L}\Psi=\Psi_{inc}. \tag{1}\]2  
\[\mathcal{G}(l,m;\
u,\mu;p)=\frac{(-1)^{m+\mu}}{\sqrt{4\pi}}\sqrt{(2l+1)(2\
u+ 1)(2p+1)}\left( \begin{array}{ccc}l&\
u&p\\ m&\mu&-m-\mu\end{array} \right)\left( \begin{array}{ccc}l&\
u&p\\ 0&0&0\end{array}