ON THE DIFFRACTION OF A PLANE ELECTROMAGNETIC WAVE BY AN ARRAY OF COAXIAL CYLINDERS

E. A. IVANOV, S. F. ILYUKEVICH

A rigorous solution is given for the problem of the diffraction of a linearly polarized plane electromagnetic wave by an array consisting of \(2N+1\) pairs of parallel coaxial cylinders forming a planar spatial array. An approximate solution of the problem, obtained from exact formulas, is presented for the case in which the wavelength of the excitation is much smaller than the distance between the centers of each pair of cylinders and certain other assumptions are satisfied. The approximate solution for the secondary field in the far zone is represented in the form of readily surveyable formulas suitable for numerical calculation and analysis.

The problem of the diffraction of a linearly polarized electromagnetic wave by an array composed of \(2N+1\) pairs of parallel circular cylinders of infinite length, situated in an unbounded homogeneous and isotropic space, is solved under the assumption that the physical properties of each pair of cylinders are identical (along the longitudinal axis all outer (inner) cylinders have equal radii \(\rho=a\) (\(\rho=b\)) of their cross sections and equal values of the constants \(\varepsilon,\mu,\sigma\)) and that the direction of propagation of the wave, determined by the unit vector \(\mathbf n\), forms an arbitrary angle \(\alpha\) with the line of centers of the cylinders in the plane of their normal cross section. The distance between the centers of each pair of cylinders is the same and equal to \(l\) (Fig.).

Normal cross section of the cylinder array. Coordinate systems

All pairs of cylinders are numbered: the central one is assigned the number \(s=0\), and all the others, situated symmetrically with respect to it, are assigned the numbers \(s=\pm 1,\ldots,\pm N\). The coordinate system \(Oxyz\) is introduced so that its origin coincides with the center of the cross section of the central pair of cylinders (\(s=0\)), the axis \(Oz\) is directed along the axes of the cylinders, and the axis \(Ox\)—along the line of centers. In addition, local coordinate systems \(O_sx_sy_sz_s\), \(s=0,\pm1,\ldots,\pm N\) (\(O_0x_0y_0z_0\) coincides with \(Oxyz\)), and cylindrical coordinates \(\rho_s,\varphi_s,z_s\), connected with the former by the formulas

\[ x_s=\rho_s\cos\varphi_s,\qquad y_s=\rho_s\sin\varphi_s,\qquad z_s=z_s,\qquad s=0,\pm1,\ldots,\pm N. \]

In what follows

By \(\Psi\) we denote the desired \(z\)-component of the vector \(\mathbf E\) of the secondary field, or the \(z\)-component of the vector \(\mathbf H\) of the secondary field, depending on the type of the diffracting plane wave \(\mathbf E,\mathbf H\). In the first case we shall be concerned with the diffraction of a transverse-magnetic wave (\(TM\)), when \(\mathbf E^0=\{0,0,E_z^0\}\), \(\mathbf H^0=\{H_\rho^0,H_\varphi^0,0\}\), \(E_z^0=Ae^{ik_1 n\mathbf R-i\omega t}\), and in the second, with the diffraction of a transverse-electric wave (\(TE\)), when \(\mathbf E^0=\{E_\rho^0,E_\varphi^0,0\}\), \(\mathbf H^0=\{0,0,H_z^0\}\), \(H_z^0=Ae^{ik_1 n\mathbf R-i\omega t}\) (in what follows the time factor \(\exp[-i\omega t]\) is omitted everywhere). The functions \(\Psi\) are assigned indices \(j=1,2,3\) and \(s=0,\pm1,\ldots,\pm N\), depending on the region in which it is considered. The problem of finding \(\Psi\) reduces to finding a single-valued solution of the two-dimensional Helmholtz equation

\[ \Delta\Psi_j+k_j^2\Psi_j=0,\qquad j=1,2,3, \tag{1} \]

\[ k_j^2= \begin{cases} k_1^2=\dfrac{\varepsilon_1\mu_1\omega^2}{c^2} & \text{in region (I)},\\[6pt] k_j^2=\dfrac{\varepsilon_j\mu_j\omega^2+4\pi\mu_j\sigma_j\omega i}{c^2},\quad j=2,3, & \text{in regions (II), (III),} \end{cases} \]

satisfying the boundary conditions

\[ \left. \begin{aligned} \Psi_0+\Psi_1&=\Psi_2^{(s)},\\ \alpha_1\,\frac{\partial(\Psi_0+\Psi_1)}{\partial\rho_s} &=\frac{\partial\Psi_2^{(s)}}{\partial\rho_s} \end{aligned} \right\}, \quad \rho_s=a,\quad s=0,\pm1,\ldots,\pm N, \tag{2} \]

\[ \left. \begin{aligned} \Psi_2^{(s)}&=\Psi_3^{(s)},\\ \alpha_2\,\frac{\partial\Psi_2^{(s)}}{\partial\rho_s} &=\frac{\partial\Psi_3^{(s)}}{\partial\rho_s} \end{aligned} \right\}, \quad \rho_s=b,\quad s=0,\pm1,\ldots,\pm N, \tag{3} \]

where \(\Psi_0=E_z^0\), \(\alpha_1=\dfrac{\mu_2}{\mu_1}\), \(\alpha_2=\dfrac{\mu_3}{\mu_2}\) in the problem for a wave of type \(TM\), and \(\Psi_0=H_z^0\),

\[ \alpha_1=\frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_1(4\pi\sigma_1-i\omega)},\qquad \alpha_2=\frac{\varepsilon_3(4\pi\sigma_3-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)} \]

in the problem for a wave of type \(TE\). The function \(\Psi_1\), defined in the unbounded region (I), is required, in addition, to satisfy the radiation condition at infinity.

In § 1 a rigorous solution of problem (1)—(3) is given, obtained there by the same method as the solution of the problem of diffraction of a plane wave by two pairs of coaxial cylinders in [1]. In § 2 an approximate solution of the problem is given, carried through to formulas suitable for numerical calculation and analysis, for the case when \(k_1l\gg1\) and certain other assumptions are fulfilled, analogous to those given in [2] in the problem of diffraction of a plane wave by an array of \(2N+1\) metallic cylinders.

§ 1. Rigorous solution of the problem

We shall seek the function \(\Psi\) in each of the regions (I)—(III) in the form of series

\[ \Psi_1=\sum_{s=-N}^{N}\sum_{n=-\infty}^{\infty} x_n^{(s)}H_n^{(1)}(k_1\rho_s)e^{in\varphi_s}, \tag{4} \]

\[ \Psi_2^{(s)}=\sum_{n=-\infty}^{\infty} y_n^{(s)}J_n(k_2\rho_s)e^{in\varphi_s} +\sum_{n=-\infty}^{\infty} z_n^{(s)}H_n^{(1)}(k_2\rho_s)e^{in\varphi_s}, \tag{5} \]

\[ \Psi_3^{(s)}=\sum_{n=-\infty}^{\infty} u_n^{(s)}J_n(k_3\rho_s)e^{in\varphi_s}, \qquad s=0\pm1,\ldots,\pm N \tag{6} \]

in terms of the proper wave functions of a circular cylinder. Then, if we first write \(\Psi_0\) in the coordinates of the \(s\)-th pair of cylinders in the form of the series

\[ \Psi_0=Ae^{ik_1 n R_{0s}}\sum_{n=-\infty}^{\infty} i^n J_n(k_1\rho_s)e^{in(\varphi_s-\alpha)} \]

and then apply the addition theorem

\[ H_n^{(1)}(k\rho_j)e^{in\varphi_j} = \sum_{m=-\infty}^{\infty} J_m(k\rho_s)H_{n-m}^{(1)}(k\rho_{js}) e^{im\varphi_s-i(n-m)\varphi_{js}} \]

\[ (j\ne s;\; j,s=0\pm1,\ldots,\pm N,\; \rho_{js}>\rho_s,\; \rho_{js}=|j-s|l,\; \varphi_{js}=0,\ \text{if } j<s, \]
\[ \text{and } \varphi_{js}=\pi,\ \text{if } j>s), \]
from (2), (3) for the coefficients \(x_n^{(s)}\) we obtain the infinite system of linear equations

\[ x_n^{(s)}+\sum_{\substack{j=-N\\ j\ne s}}^{N}\sum_{m=-\infty}^{\infty} \alpha_{nm}^{(js)}x_m^{(j)}=f_n^{(s)}, \tag{7} \]

the solutions of which are related to the coefficients of the series (5), (6) by the relations

\[ y_n^{(s)}=\frac{\Delta_1\Delta_5}{\Delta_2\Delta_5+\Delta_3\Delta_4}\,x_n^{(s)},\qquad z_n^{(s)}=\frac{\Delta_1\Delta_4}{\Delta_2\Delta_5+\Delta_3\Delta_4}\,x_n^{(s)}, \]

\[ u_n^{(s)}=\frac{\Delta_1\Delta_6}{\Delta_2\Delta_5+\Delta_3\Delta_4}\,x_n^{(s)}. \tag{8} \]

Here it is assumed that

\[ \alpha_{nm}^{(js)}=\Delta_n' H_{m-n}^{(1)}(k_1|j-s|l)e^{-i(m-n)\varphi_{js}}, \tag{9} \]

\[ f_n^{(s)}=A\Delta_n e^{ik_1 n R_{0s}-i\left(\alpha-\frac{\pi}{2}\right)n}, \qquad \Delta_n'=-\frac{\Delta_2\Delta_5+\Delta_3\Delta_4}{\Delta_5\Delta_7+\Delta_4\Delta_8}. \tag{10} \]

\[ \Delta_1=k_1 \left| \begin{array}{cc} H_n^{(1)}(k_1a) & J_n(k_1a)\\ H_n^{(1)\prime}(k_1a) & J_n'(k_1a) \end{array} \right|, \qquad \Delta_2=k_1 \left| \begin{array}{cc} J_n'(k_1a) & J_n(k_1a)\\ \dfrac{k_2}{\alpha_1 k_1}J_n'(k_2a) & J_n(k_2a) \end{array} \right|, \]

\[ \Delta_3=k_1 \begin{vmatrix} J'_n(k_1a) & J_n(k_1a)\\[3pt] \dfrac{k_2}{\alpha_1 k_1}H_n^{(1)'}(k_2a) & H_n^{(1)}(k_2a) \end{vmatrix}, \quad \Delta_4=k_2 \begin{vmatrix} \dfrac{k_3}{\alpha_2 k_2}J'_n(k_3b) & J'_n(k_2b)\\[3pt] J_n(k_3b) & J_n(k_2b) \end{vmatrix}, \tag{11} \]

\[ \Delta_5=k_2 \begin{vmatrix} H_n^{(1)'}(k_2b) & J'_n(k_3b)\\[3pt] \dfrac{k_3}{\alpha_2 k_2}H_n^{(1)}(k_2b) & J_n(k_3b) \end{vmatrix}, \quad \Delta_6=\frac{1}{J_n(k_3b)} \begin{vmatrix} \Delta_5 & -H_n^{(1)}(k_2b)\\ \Delta_4 & J_n(k_2b) \end{vmatrix}, \]

\[ \Delta_7=k_1 \begin{vmatrix} \dfrac{k_2}{\alpha_1 k_1}H_n^{(1)}(k_1a) & J_n(k_2a)\\[3pt] H_n^{(1)'}(k_1a) & J'_n(k_2a) \end{vmatrix}, \quad \Delta_8=k_1 \begin{vmatrix} H_n^{(1)}(k_1a) & H_n^{(1)}(k_2a)\\[3pt] H_n^{(1)'}(k_1a) & \dfrac{k_2}{\alpha_1 k_1}H_n^{(1)'}(k_2a) \end{vmatrix}. \]

Just as in [1], it can be shown that none of the known methods for solving infinite systems can be applied directly, with justification, to system (7), since its matrix elements possess the same property as the matrix elements of system (19) of [1] as \(|m|\to\infty\). However, after replacing the coefficients \(x_n^{(s)}, y_n^{(s)}, z_n^{(s)}, u_n^{(s)}\) by the new \(X_n^{(s)}, Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\) according to the formulas

\[ x_n^{(s)}=J_n(k_1a)X_n^{(s)},\quad y_n^{(s)}=H_n^{(1)}(k_2a)Y_n^{(s)}, \tag{12} \]

\[ z_n^{(s)}=J_n(k_2b)Z_n^{(s)},\quad u_n^{(s)}=H_n^{(1)}(k_3b)U_n^{(s)} \]

from (7) for \(X_n^{(s)}\) we obtain the new system

\[ X_n^{(s)}+\sum_{\substack{j=-N\\ j\ne s}}^N\sum_{m=-\infty}^{\infty} \sigma_{nm}^{(js)}X_m^{(j)}=F_n^{(s)}, \tag{13} \]

where

\[ \sigma_{nm}^{(js)}=\frac{J_m(k_1a)}{J_n(k_1a)}\alpha_{nm}^{(js)},\quad F_n^{(s)}=\frac{f_n^{(s)}}{J_n(k_1a)}, \tag{14} \]

\(n=0,\pm1,\ldots,\ s=0,\pm1,\ldots,\pm N\). \(Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\) are found through \(X_n^{(s)}\) from (8) and (12), whose coefficient matrix, as in [1], defines a completely continuous form in the Hilbert space \(l^2\) (under the condition that the outer cylinders of the grating do not touch), and whose right-hand sides \(F_n^{(s)}\in l^2\). As in [1], system (13) is quasiregular and solvable by the method of truncation. Its solutions satisfy the condition \(\{X_n^{(s)}\}\in l^2\) (and then also \(\{Y_n^{(s)}\}\in l^2,\ \{Z_n^{(s)}\}\in l^2,\ \{U_n^{(s)}\}\in l^2\)); from this there immediately follows the absolute and uniform convergence of the series (4)—(6), if (8) and (12) are taken into account.

Determining \(X_n^{(s)}, Y_n^{(s)}, Z_n^{(s)}\), and \(U_n^{(s)}\) completely solves problem (1)—(3). In principle, system (13) is suitable for obtaining numerical results for any \(N\). However, computing it for large values of \(N\) is, in general, practically hardly feasible because of the complicated structure of system (13). In some special cases, however, from (13)

one can find approximate solutions for \(X_n^{(s)}\) (and consequently also for \(Y_n^{(s)}, Z_n^{(s)}, U_n^{(s)}\)), which make it possible, under certain additional assumptions, to obtain for the potential function \(\Psi\) an expression having a “closed” form.

§ 2. APPROXIMATE SOLUTION OF THE PROBLEM

Let us formally apply to the system (13) the method of successive approximations, taking as the zeroth approximations the values \(X_{n,0}^{(s)}=0\) for all \(s\). Then for the \((\nu+1)\)-st approximation we obtain

\[ X_{n,\nu+1}^{(s)} = F_n^{(s)} + \sum_{t=1}^{\nu}(-1)^t \sum_{\substack{s',n'\\ s'\ne s}} \sum_{\substack{s'',n''\\ s''\ne s'}} \cdots \sum_{\substack{s^t,n^t\\ s^t\ne s^{t-1}}} \sigma_{nn'}^{(s',s)} \sigma_{n'n''}^{(s'',s')} \cdots \sigma_{n^{t-1}n^t}^{(s^t,s^{t-1})} F_{n^t}^{(s^t)}, \]

whence, assuming that \(X_n^{(s)}=\lim_{\nu\to\infty}X_{n,\nu+1}^{(s)}\),

\[ X_n^{(s)} = F_n^{(s)} + \sum_{t=1}^{\infty}(-1)^t \sum_{\substack{s',n'\\ s'\ne s}} \sum_{\substack{s'',n''\\ s''\ne s'}} \cdots \sum_{\substack{s^t,n^t\\ s^t\ne s^{t-1}}} \sigma_{nn'}^{(s',s)} \sigma_{n'n''}^{(s'',s')} \cdots \sigma_{n^{t-1}n^t}^{(s^t,s^{t-1})} F_{n^t}^{(s^t)} . \tag{15} \]

We shall now assume that \(k_1 l\gg 1\) and is such that

\[ H_n^{(1)}(k_1 l)=(-i)^n H_0(k_1 l),\qquad H_0(k_1 l)= \sqrt{\frac{2}{\pi k_1 l}}\, e^{\,i\left(k_1l-\frac{\pi}{4}\right)}, \tag{16} \]

then

\[ \sigma_{n^\mu-1\,n^\mu}^{(s^\mu,s^{\mu-1})} = H_0\!\left(k_1|s^\mu-s^{\mu-1}|l\right) \alpha_{n^{\mu-1}}\beta_{n^\mu} \varepsilon_{s^\mu s^{\mu-1}}, \qquad \text{where }\alpha_n=\frac{i^n\Delta_n}{J_n(k_1a)}, \]

\[ \beta_n=(-i)^nJ_n(k_1a),\qquad \varepsilon_{s^\mu s^{\mu-1}} = e^{-i(n^\mu-n^{\mu-1})\varphi_{s^\mu s^{\mu-1}}} \]

\[ \left(\varepsilon_{s^\mu s^{\mu-1}}=1,\ \text{if }s^\mu<s^{\mu-1} \quad\text{and}\quad \varepsilon_{s^\mu s^{\mu-1}}=(-1)^{\,n^\mu-n^{\mu-1}},\ \text{if }s^\mu>s^{\mu-1}\right), \quad s^\mu\ne s^{\mu-1}; \quad s^\mu,s^{\mu-1}=0,\pm1,\ldots,\pm N. \]

Since

\[ \mathbf{R}_{0s^t} = \sum_{\mu=1}^{t}\mathbf{R}_{s^\mu s^{\mu-1}}+\mathbf{R}_{0s}, \qquad \mathbf{n}\mathbf{R}_{0s}=sl\cos\alpha, \]

and putting

\[ F_n^{(s)}=A\gamma_n^{(s)}e^{ik_1\mathbf{n}\mathbf{R}_{0s}}, \qquad \text{where } \gamma_n^{(s)} = \frac{\Delta_n e^{\,i\left(\frac{\pi}{2}-\alpha\right)n}}{J_n(k_1a)}, \]

from (15) we obtain that

\[ X_n^{(s)} = F_n^{(s)} + A\sum_{t=1}^{\infty}(-1)^t \sum_{n',n'',\ldots,n^t} \left\{ \prod_{\mu=1}^{t} \sum_{\substack{s^\mu=-N\\ s^\mu\ne s^{\mu-1}}}^{N} \varepsilon_{s^\mu s^{\mu-1}} \right. \]

\[ \left. {}\times H_0\!\left(k_1|s^\mu-s^{\mu-1}|l\right) \alpha_{n^{\mu-1}}\beta_{n^\mu} e^{\,ik_1\mathbf{n}\mathbf{R}_{s^\mu s^{\mu-1}}} \right\} \gamma_{n^t}e^{ik_1s^t l\cos\alpha}, \tag{17} \]

where

\[ \sum_{n',n'',\ldots,n^t} = \sum_{n'}\sum_{n''}\cdots\sum_{n^t}. \]

Let us split each of the sums \(S_\mu=\)

\[ = \sum_{\substack{s^\mu=-N\\ s^\mu\ne s^{\mu-1}}}^{N} \varepsilon_{s^\mu s^{\mu-1}} H_0\bigl(k_1|s^\mu-s^{\mu-1}|l\bigr) e^{ik_1 nR_{s^\mu s^{\mu-1}}}\, s^\mu s^{\mu-1}, \]
where
\[ nR_{s^\mu s^{\mu-1}} = \tau_{s^\mu s^{\mu-1}}\times |s^\mu-s^{\mu-1}|\,l\cos\alpha, \]
\(\tau_{s^\mu s^{\mu-1}}=1\), if \(s^\mu>s^{\mu-1}\), and \(\tau_{s^\mu s^{\mu-1}}=-1\), if \(s^\mu<s^{\mu-1}\), into two: \(S_\mu=\sum' + \sum''\), in the first of which the summation is carried out only for \(s^\mu<s^{\mu-1}\), and in the second—for \(s^\mu>s^{\mu-1}\), and denote the difference \(|s^\mu-s^{\mu-1}|\) by \(q^{\mu-1}\). Then
\[ \dot S_\mu = \sum_{q^{\mu-1}=1}^{N_{+s^{\mu-1}}} H_0\bigl(k_1 q^{\mu-1}l\bigr) e^{-ik_1 q^{\mu-1}l\cos\alpha} + \]
\[ +(-1)^{n^\mu-n^{\mu-1}} \sum_{q^{\mu-1}=1}^{N_{-s^{\mu-1}}} H_0\bigl(k_1 q^{\mu-1}l\bigr) e^{ik_1 q^{\mu-1}l\cos\alpha}, \]
where \(N_{+s^{\mu-1}}+N_{-s^{\mu-1}}=2N\). In each of the sums \(S_\mu\), the value of the difference \(q^{\mu-1}\) is determined relative to the number \(s^{\mu-1}\), fixed for the given sum, of the \(s^{\mu-1}\)-th pair of cylinders in the grating. It is obvious that each partial secondary wave of any scattering order, assigned to the \(s^{\mu-1}\)-th pair of cylinders, arises due to the excitation of this pair by secondary waves of lower scattering order, consisting of waves: (a) scattered by all pairs of cylinders located “to the left” of the \(s^{\mu-1}\)-th pair, and (b) waves scattered by all pairs of cylinders located “to the right” of the \(s^{\mu-1}\)-th pair. Hence it is clear that if the \(s^{\mu-1}\)-th pair is not central in the grating \((s^{\mu-1}\ne0)\), then the number of pairs of cylinders located “to the left” and “to the right” of the \(s^{\mu-1}\)-th pair is not the same. Therefore, in general, in the sums \(S_\mu\),
\[ N_{+s^{\mu-1}}\ne N_{-s^{\mu-1}}. \]

This dependence of the upper limits of summation in \(S_\mu\) on \(s\) does not make it possible to simplify (17). However, expression (17) becomes suitable for obtaining a closed formula for (4) if assumptions similar to those given in [2] are made. Namely, we shall assume that the number of pairs of cylinders in the grating is unlimited: \(s=0,\pm1,\pm2,\ldots\). Then \(q^{\mu-1}=1,2,3,\ldots\), and the upper limits of summation in \(\dot S_\mu\) will no longer depend on \(s\): \(N_{\pm s^{\mu-1}}=\infty\) (in this case each pair of cylinders in the grating may be considered central). Let us further assume, as in [2], that at the observation point the effect is produced only by \(M\) pairs of cylinders of the grating (\(M\) large), each of which is excited by the action on it of waves scattered by \(N\) pairs of symmetric cylinder pairs adjacent to it (\(N\) is not necessarily equal to \((M-1)/2\)) (in each case we neglect the “edge effect” [2], i.e., the result of the action of all the remaining cylinders of the grating located “to the left,” “to the right” of the \(2N+1\) taken into account). As noted in [2], the theoretical results obtained there under this assumption agree well with experiment. Then \(N_{\pm s^{\mu-1}}=N\), and
\[ S_\mu=l_-+(-1)^{n^\mu-n^{\mu-1}}l_+, \]
where
\[ l_-=\sum_{q=1}^{N} H_0(k_1 ql)e^{ik_1 lq\cos\alpha},\qquad l_+=\sum_{q=1}^{N} H_0(k_1 ql)e^{ik_1 lq\cos\alpha}. \]
Substituting this expression for \(S_\mu\) into (17), and the latter into (4), we obtain that

\[ \Psi_1=\Phi_1+\Phi_2, \tag{18} \]

where

\[ \Phi_1=\sum_{s=-N}^{N}\sum_{n=-\infty}^{\infty} J_n(k_1a)F_n^{(s)}H_n^{(1)}(k_1\rho_s)e^{in\varphi_s}, \tag{19} \]

\[ \Phi_2=A\sum_{s=-N}^{N}\sum_{n=-\infty}^{\infty}\sum_{t=1}^{\infty}(-1)^t \sum_{n',n'',\ldots,n^t} \left\{ \prod_{\mu=1}^{t}\left(l_-+(-1)^{n^\mu-n^{\mu-1}}l_+\right) \times \right. \]

\[ \left. \times \alpha_{n^{\mu-1}}\beta_{n^\mu} \right\} \gamma_{n^t}J_n(k_1a)H_n^{(1)}(k_1\rho_s) e^{isk_1l\cos\alpha+in\varphi_s}. \tag{20} \]

The functions \(\Phi_1\) and \(\Phi_2\) have a quite definite physical meaning: the first of them gives the values of the secondary field at the observation point without taking into account there the result of mutual successive diffractions of the scattered waves on the grating, while the second, on the contrary, determines only the result of all successive diffractions on the cylinders of the grating. If the width of the grating is small in comparison with the distance to the observation point, when \(k_1\rho_s\gg 1\) and

\[ H_n^{(1)}(k_1\rho_s)=(-i)^n\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)-isk_1l\cos\varphi}, \]

\(\rho_s\simeq \rho-sl\cos\varphi,\ \varphi_s\simeq\varphi\), then, on the basis of (10), (14), we obtain

\[ \Phi_1 = A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2} \sum_{n=-\infty}^{\infty}\Delta_n e^{in(\varphi-\alpha)}, \tag{21} \]

where \(\delta=k_1l(\cos\alpha-\cos\varphi)\). In this form the function \(\Phi_1\) is the usual function for Fraunhofer diffraction gratings [3], giving the amplitude distribution as a function of the angle of observation. Here the summation over \(n\) describes the result of the action of one pair of cylinders, while the factor

\[ \frac{\sin(2N+1)\delta/2}{\sin\delta/2} \]

expresses the result of the interaction of all the cylinders of the grating. At the same observation point, located in the far zone, the function \(\Phi_2\) takes the form

\[ \Phi_2 = A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2} \times \]

\[ \times \sum_{n=-\infty}^{\infty}(-i)^nJ_n(k_1a)\alpha_n e^{in\varphi} \sum_{t=1}^{\infty}(-1)^t \times \]

\[ \times \sum_{n',n'',\ldots,n^t} \left\{ \prod_{\mu=1}^{t-1} \left(l_-+(-1)^{n^\mu-n^{\mu-1}}l_+\right) \alpha_{n^\mu}\beta_{n^\mu} \right\} \times \]

\[ \times \left(l_-+(-1)^{n^t-n^{t-1}}l_+\right)\beta_{n^t}\gamma_{n^t}. \tag{22} \]

If we introduce here the notation

\[ Q_n=(-i)^nJ_n(k_1a)\alpha_n e^{in\varphi},\qquad Q_+=\sum_{n=-\infty}^{\infty}Q_n,\qquad Q_-=\sum_{n=-\infty}^{\infty}(-1)^nQ_n, \]

\[ P_+ = \sum_{n=-\infty}^{\infty}(-\alpha_n\beta_n),\qquad P_- = \sum_{n=-\infty}^{\infty}(-1)^n(-\alpha_n\beta_n), \tag{23} \]

\[ S_+ = \sum_{n=-\infty}^{\infty}(-\beta_n\gamma_n),\qquad S_- = \sum_{n=-\infty}^{\infty}(-1)^n(-\beta_n\gamma_n). \]

and then expand the sum in \(t\), open the brackets \(\bigl(l_-+(-1)^{n^\nu-n^{\nu-1}}l_+\bigr)\), sum over all summation indices \(n,n',\ldots,n^t,\ldots\), and group the terms in a definite way; as a result we obtain that

\[ \begin{aligned} \Phi_2 &= A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{\,i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2} \times \\ &\quad \times \Biggl\{ -\sum_{n=-\infty}^{\infty} Q_n \sum_{n'}\bigl[l_-+(-1)^{n'-n}l_+\bigr]\beta_{n'}\gamma_{n'} + \\ &\quad +\sum_{n=-\infty}^{\infty} Q_n \sum_{n',n''} \bigl[l_-+(-1)^{n'-n}l_+\bigr]\alpha_{n'}\beta_{n'} \bigl[l_-+(-1)^{n''-n'}l_+\bigr]\beta_{n''}\gamma_{n''} - \\ &\quad -\sum_{n=-\infty}^{\infty} Q_n \sum_{n',n'',n'''} \bigl[l_-+(-1)^{n'-n}l_+\bigr]\alpha_{n'}\beta_{n'} \bigl[l_-+(-1)^{n''-n'}l_+\bigr] \times \\ &\quad\quad \times \alpha_{n''}\beta_{n''} \bigl[l_-+(-1)^{n'''-n''}l_+\bigr]\beta_{n'''}\gamma_{n'''}\ldots \Biggr\} = \\ &= A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{\,i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2} [\Sigma_1+\Sigma_2+\Sigma_3+\Sigma_4], \tag{24} \end{aligned} \]

where

\[ \begin{aligned} \Sigma_1 &= Q_+S_+\Bigl\{ l_-+P_+l_-^2+(l_-^3P_+^2+P_-^2l_-^2l_+) +(P_+^3l_-^4+P_-^2P_+l_-^3l_+ \\ &\quad +P_-^2P_+l_-^3l_+ + P_-^2P_+l_-^2l_+^2) +(P_+^4l_-^5+P_-^2P_+^2l_-^4l_+ +P_-^2P_+^2l_-^4l_+ \\ &\quad +l_-^3l_+^2P_-^2P_+^2+P_-^2P_+^2l_-^4l_+ +P_-^4l_-^3l_+^3+P_-^2P_+^2l_-^3l_+^2 +P_-^2P_+^2l_-^2l_+^3)-\ldots \Bigr\}; \end{aligned} \]

\[ \begin{aligned} \Sigma_2 &= Q_-S_-\Bigl\{ l_+ + P_+l_+^2 +(l_+^3P_+^2+P_-^2l_+^2l_-) +(P_+^3l_+^4+P_-^2P_+l_+^3l_- \\ &\quad +P_-^2P_+l_+^3l_-+P_-^2P_+l_+^2l_-^2) +(P_+^4l_+^5+P_-^2P_+^2l_+^4l_- \\ &\quad +P_-^2P_+^2l_+^4l_-+l_+^3l_-^2P_-^2P_+^2 +P_-^2P_+^2l_+^4l_- \\ &\quad +P_-^4l_+^3l_-^2+P_-^2P_+^2l_+^3l_-^2 +P_-^2P_+^2l_+^2l_-^3)-\ldots \Bigr\}; \end{aligned} \]

\[ \begin{aligned} \Sigma_3 &= Q_+S_-P_-\Bigl\{ l_-l_+ +(P_+l_-^2l_+ + P_+l_-l_+^2) +(P_+^2l_-^3l_+ \\ &\quad +P_-^2l_-^2l_+^2+P_+^2l_-^2l_+^2+P_+^2l_-l_+^3) +(P_+^3l_-^4l_+ \\ &\quad +l_+^2l_-^3P_+P_-^2+P_-^2P_+l_-^3l_+ +P_+^3l_-^3l_+^2 \\ &\quad +P_-^2P_+l_-^2l_+^3+P_+P_-^2l_-^2l_+^3 +P_+^3l_-^2l_+^3+P_+^3l_-^4l_+) +\ldots \Bigr\}; \end{aligned} \]

\[ \Sigma_4 = Q_- S_+ P_- \{ l_- l_+ + (P_+ l_-^2 l_+ + P_+ l_- l_+^2) + (P_+^2 l_-^3 l_+ + \]
\[ + P_-^2 l_-^2 l_+^2 + P_+^2 l_-^2 l_+^2 + P_+^2 l_- l_+^3) + (P_+^3 l_-^4 l_+ + \]
\[ + P_+ P_-^2 l_-^3 l_+^2 + P_+ P_-^2 l_-^3 l_+^2 + P_+^3 l_-^3 l_+^2 + P_-^2 P_+ l_-^2 l_+^3 + \]
\[ + P_+ l_-^2 P_-^2 l_+^3 + P_+^3 l_-^2 l_+^3 + P_+^3 l_- l_+^4) + \ldots \}. \]

It is not difficult to see that in \(\Sigma_1\) each following parenthesis is obtained from the preceding one by applying to it the transformation matrix

\[ A_1 = \begin{pmatrix} P_+ l_- & l_- P_+ \\[4pt] \dfrac{P_-^2 l_+}{P_+} & l_+ P_+ \end{pmatrix} \tag{25} \]

according to the following scheme: the second parenthesis is obtained from the first by adding the results of multiplying \(P_+ l_-^2\) by the elements of the first column of the matrix (25), and \(P_-^2 l_-^2 l_+\) by the elements of its second column, with the indicated order of the products preserved in the notation. Then the third parenthesis is obtained from the second by adding the results of successively multiplying the first term of the second parenthesis by the elements of the first column of (25), the second term by the elements of the second column, the third term again by the elements of the first column, and the fourth term by the elements of the second column, with the indicated order of the products preserved in the notation, and so on. The first parenthesis in \(\Sigma_1\) is obtained by applying the matrix (25) to the term \(P_+ l_-^2\), written in the form of the sum \(P_+ l_-^2 + 0\). As a result we find that

\[ \Sigma_1 = Q_+ S_+ \{ l_- + P_+ l_-^2 + A_1 P_+ l_-^2 + A_1^2 P_+ l_-^2 + \ldots \} = \]

\[ = Q_+ S_+ \{ l_- + (E - A_1)^{-1} P_+ l_-^2 \} \]

provided that \(\|A_1\| < 1\), where \(\|A_1\|\) is some norm of the matrix \(A_1\). Since the determinant \(\Delta\) of the matrix \((E - A_1)\) is equal to

\[ \Delta = |E - A_1| = 1 - P_+(l_- + l_+) + l_- l_+(P_+^2 - P_-^2), \]

and

\[ (E - A_1)^{-1} = \begin{pmatrix} \dfrac{1 - l_+ P_+}{\Delta} & \dfrac{l_- P_+}{\Delta} \\[6pt] \dfrac{l_+ P_-^2}{P_+ \Delta} & \dfrac{1 - l_- P_+}{\Delta} \end{pmatrix}, \]

then

\[ (E - A_1)^{-1} P_+ l_-^2 = \frac{P_+ l_-^2 + l_-^2 l_+(P_-^2 - P_+^2)} {1 - P_+(l_- + l_+) + l_- l_+(P_+^2 - P_-^2)} \]

and

\[ \Sigma_1 = Q_+ S_+ \frac{l_- - P_+ l_- l_+} {1 - P_+(l_+ + l_-) + l_- l_+(P_+^2 - P_-^2)} \tag{26} \]

(the matrix \((E - A_1)^{-1}\) is applied to \(P_+ l_-^2\) according to the previous scheme: \(P_+ l_-^2\) is represented in the form of the sum \(P_+ l_-^2 + 0\), followed by multiplication of the first term by the elements of the first column of the matrix \((E - A_1)^{-1}\),

and of the second term (zero)—to the element of the second column. In a similar way, by applying the transformation matrix

\[ A_2=\begin{pmatrix} P_+l_+ & P_+l_+\\[2mm] \dfrac{P_-^2l_-}{P_+} & P_+l_- \end{pmatrix} \tag{27} \]

the sum \(\Sigma_2\) is found, if \(\|A_2\|<1\), where \(\|A_2\|\) is any norm of the matrix (27):

\[ \Sigma_2=Q_-S_-\,\frac{l_+-l_-l_+P_+}{1-P_+(l_-+l_+)+l_-l_+(P_+^2-P_-^2)} . \tag{28} \]

The sums \(\Sigma_3\) and \(\Sigma_4\) are found with the aid of the matrix (25). For them

\[ \Sigma_3=\frac{l_-l_+Q_+S_-P_-}{1-P_+(l_-+l_+)+l_-l_+(P_+^2-P_-^2)}; \tag{29} \]

\[ \Sigma_4=\frac{l_-l_+Q_-S_+P_-}{1-P_+(l_-+l_+)+l_-l_+(P_+^2-P_-^2)} . \tag{30} \]

As a result, on the basis of (26), (28)—(30), from (24) we find that

\[ \Phi_2=A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{\,i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2}\times \]

\[ \times \frac{ Q_+l_-\,[S_+-l_+(S_+P_+-S_-P_-)] +Q_-l_+\,[S_--l_-(S_-P_+-S_+P_-)] }{ 1-P_+(l_-+l_+)+l_-l_+(P_+^2-P_-^2) }. \tag{31} \]

It is known that a necessary and sufficient condition for the inequality \(\|A\|<1\) to hold for some norm of the matrix \(\|A_1\|\) is the inequality \(|\lambda_i|<1\), which must hold for all eigenvalues \(\lambda\) of the matrix \(A_1\). From the characteristic equation

\[ \lambda^2-\lambda(l_-+l_+)P_+ + l_-l_+(P_+^2-P_-^2)=0 \]

for the eigenvalues of the matrix \(A_1\), and from the expressions for \(l_\pm\), it follows that the inequality \(|\lambda_i|<1\) can always be ensured by an appropriate choice of the parameter \(k_1l\), since \(l_\pm\to 0\) when \(k_1l\to\infty\).

Formula (31), having a “closed” form, is quite suitable for obtaining numerical results and studies concerning the secondary field in the far zone, for which now

\[ \Psi_1=A\sqrt{\frac{2}{\pi k_1\rho}}\, e^{\,i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2} \left[ \sum_{n=-\infty}^{\infty}\Delta_n e^{-in(\alpha-\varphi)} + \right. \]

\[ \left. + \frac{ Q_+l_-\,[S_+-l_+(S_+P_+-S_-P_-)] +Q_-l_+\,[S_--l_-(S_-P_+-S_+P_-)] }{ 1-P_+(l_-+l_+)+l_-l_+(P_+^2-P_-^2) } \right]. \tag{32} \]

If it is assumed that \(|k_i a|\ll 1\), \(|k_i b|\ll 1\), and in all series in \(n\) only the first terms are retained, then from (32) one can obtain well-observable formulas for \(\Psi\). Thus, in particular, retaining everywhere only the terms with summation index \(n=0\), from (32) we obtain that

\[ \Psi_1=A\sqrt{\frac{2}{\pi k_1\rho}}\,e^{i\left(k_1\rho-\frac{\pi}{4}\right)} \frac{\sin(2N+1)\delta/2}{\sin\delta/2}\, \frac{\Delta_0}{1+\Delta_0L}, \tag{63} \]

where \(L\) denotes the sum

\[ l_-+l_+=2\sum_{q=1}^{N}H_0(k_1ql)\cos(k_1ql\cos\alpha), \]

and \(\Delta_0\) is determined in the same way as \(-W_0\) in formula (35) of [1] for a wave of type \(TM\), i.e.

\[ \Delta_0=i\frac{\pi}{2} \left[ (k_2a)^2\left(\frac{\mu_1}{\mu_2}-\frac{k_1^2}{k_2^2}\right) - \left(\frac{\mu_1}{\mu_2}\right)(k_2b)^2 \left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right) \right]\times \]

\[ \times \left\{ 2- \left[ \frac{\mu_1}{\mu_2}(k_2a)^2 -(k_2b)^2 \left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right) \left(\frac{\mu_1}{\mu_2}\right) \right]\ln\frac{2}{\gamma k_1a} \right. \]

\[ \left. -\frac{\mu_2}{\mu_3}(k_3b)^2\ln\frac{2}{\gamma k_2b} -(k_2b)^2 \left(1-\frac{\mu_2}{\mu_3}\frac{k_3^2}{k_2^2}\right) \ln\frac{2}{\gamma k_2a} \right\}^{-1}, \]

or, as in formula (40) of [1], for a wave of type \(TE\), i.e.

\[ \Delta_0=i\frac{\pi}{2} \left\{ (k_2a) \left[ \frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)} -\frac{k_1^2}{k_2^2} \right] - \frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)} \times \right. \]

\[ \left. \times (k_2b)^2 \left[ 1- \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)} \frac{k_3^2}{k_2^2} \right] \right\} \times \]

\[ \times \left\{ 2- \left[ (k_2a)^2 -(k_2b)^2 \left( 1- \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)} \frac{k_3^2}{k_2^2} \right) \right] \right\} \times \]

\[ \times \frac{\varepsilon_1(4\pi\sigma_1-i\omega)} {\varepsilon_2(4\pi\sigma_2-i\omega)} \ln\frac{2}{\gamma k_1a} - (k_2b)^2 \left[ 1- \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)} \times \right. \]

\[ \left. \times \frac{k_3^2}{k_2^2} \right] \ln\frac{2}{\gamma k_2a} - (k_3b)^2 \frac{\varepsilon_2(4\pi\sigma_2-i\omega)} {\varepsilon_3(4\pi\sigma_3-i\omega)} \ln\frac{2}{\gamma k_2b} \}^{-1}. \]

Conclusion

For small values of \(N\), numerical results for \(\Psi\) can be obtained from the exact formulas by solving the system (13). For large values of \(N\), from the exact formulas one can obtain an approximate solution of the problem suitable for numerical computation and analysis under certain assumptions: \(k_1l\gg 1\) and with the condition that the “edge effect” be neglected.

References

1. Ivanov E. A., Il’yukevich S. F. Differential Equations, No. 1, 1965.
2. Twersky V. Journal of Appl. Phys., 23, No. 10, 1952. “On a Multiple Scattering Theory of the Finite Grating and the Wood Anomalies.”
3. Gorelik G. S. Oscillations and Waves. Fizmatgiz, Moscow, 1959.
4. Faddeeva V. N. Computational Methods of Linear Algebra. GITTL, Moscow, 1950.

Received by the editors
September 18, 1964

Institute of Mathematics, Academy of Sciences of the BSSR