Diffraction of Electromagnetic Waves Radiated by an Electric Dipole on Two Disks with a Common Axis of Rotation

E. A. Ivanov

A rigorous solution is given of the problem of diffraction of electromagnetic waves, radiated by an elementary electric dipole, on two infinitely thin and perfectly conducting disks of different radii having a common axis of rotation, when the direction of the moment of the dipole \(\vec p\), located on the axis of the disks at an arbitrary point of it (including at the center of one of the disks), forms an arbitrary angle \(\alpha\) with the axis of the disks. The analogous problem for the case of a magnetic dipole was considered by us in sufficient detail in [1—4]. Taking into account that the method we use for solving the problem in the case of an electric dipole is based on the same ideas as in [1—4], we confine ourselves here to only a schematic exposition of the theory of the solution, with an indication of the final results.

1. If we introduce into consideration the electric Hertz vector \(\vec \Pi\) of the total electromagnetic field \(\vec E, \vec H\), assuming here that \(\vec \Pi=\vec \Pi^{(1)}+\vec \Pi^{(2)}\), where \(\vec \Pi^{(1)}=\vec p\,e^{ikR}/R\) is the Hertz vector of the primary field of the dipole \(\vec E^{(1)}, \vec H^{(1)}\), and \(\vec \Pi^{(2)}\) is the Hertz vector of the secondary field \(\vec E^{(2)}, \vec H^{(2)}\), caused by the presence of the disks in space (the time dependence is assumed to be given by the factor \(e^{-i\omega t}\), which we omit everywhere), then \(\vec E,\vec H\) are determined through \(\vec \Pi\) from the relations [5]

\[ \begin{gathered} \vec E=\operatorname{grad}\operatorname{div}\vec \Pi+k^2\vec \Pi,\\ \vec H=-ik\,\operatorname{rot}\vec \Pi \end{gathered} \qquad \left\},\quad k=\frac{2\pi}{\lambda}. \right. \tag{1} \]

Since \(\vec p=\vec p_{\mathrm{v}}+\vec p_{\mathrm{h}}\), where \(\vec p_{\mathrm{v}}=\vec p\cos\alpha\), \(\vec p_{\mathrm{h}}=\vec p\sin\alpha\), and \(\vec \Pi^{(1)}=\vec \Pi_{\mathrm{v}}^{(1)}+\vec \Pi_{\mathrm{h}}^{(1)}\), \(\vec \Pi_{\mathrm{v}}=\vec p_{\mathrm{v}}e^{ikR}/R\), \(\vec \Pi_{\mathrm{h}}=\vec p_{\mathrm{h}}e^{ikR}/R\), while \(\vec \Pi=\vec \Pi_{\mathrm{v}}+\vec \Pi_{\mathrm{h}}\), where \(\vec \Pi_{\mathrm{h}}=\vec \Pi_{\mathrm{h}}^{(1)}+\vec \Pi_{\mathrm{h}}^{(2)}\), \(\vec \Pi_{\mathrm{v}}=\vec \Pi_{\mathrm{v}}^{(1)}+\vec \Pi_{\mathrm{v}}^{(2)}\) (here and everywhere below the subscript “v” is assigned by us to the vertical component, and the subscript “h” to the horizontal component of the vectors under consideration), then, by virtue of the linearity of the problem we are solving with respect to \(\vec \Pi\) and \(\vec E,\vec H\), it, as in [4], can be split into two particular problems:

a) the problem of diffraction of the field of a vertical electric dipole with moment \(\vec p_{\mathrm{v}}=\vec p\cos\alpha\) on two disks, and

b) the problem of diffraction of the field of a horizontal electric dipole with moment \(\vec p_{\mathrm{h}}=\vec p\sin\alpha\) on two disks.

Then the solution of the general problem will be represented in the form of the sum of the corresponding solutions of problems a) and b).

1. To solve problem a), two local coordinate systems \(O_s x_s y_s z_s\), \(s=\pm 1\) (see the figure), and local coordinates \(\xi_s,\eta_s,\varphi_s\) of an oblate spheroid are introduced, related to Cartesian coordinates by the relations

\[ x_s=a_s\left[(\xi_s^2+1)(1-\eta_s^2)\right]^{1/2}\cos\varphi,\quad y_s=a_s\left[(\xi_s^2+1)(1-\eta_s^2)\right]^{1/2}\sin\varphi, \]

\[ z_s=a_s\xi_s\eta_s, \tag{2} \]

where \(s=\pm 1\); \(a_s\) is the radius of the \(s\)-th disk (the surface of the \(s\)-th disk is determined by the value \(\xi_s=0\)); \(0\le \xi_s<\infty\); \(-1\le \eta_s\le +1\); \(0\le \varphi_s\le 2\pi\). Then in these coordinates \(\vec E_{\mathrm v}=\{E_{\mathrm v,\xi_s},E_{\mathrm v,\eta_s},0\}\), \(\vec H_{\mathrm v}=\{0,0,H_{\mathrm v,\varphi}\}\), where

\[ E_{\mathrm v,\xi_s} = \frac{i}{c_s(\xi_s^2+\eta_s^2)^{1/2}} \times \frac{\partial}{\partial\eta_s} \left[(1-\eta_s^2)^{1/2}H_\varphi\right]; \tag{3} \]

\[ E_{\mathrm v,\eta_s} = \frac{-i}{c_s(\xi_s^2+\eta_s^2)^{1/2}} \times \frac{\partial}{\partial\xi_s} \left[(1+\xi_s^2)^{1/2}H_\varphi\right], \quad c_s=ka_s,\quad s=\pm1, \tag{4} \]

\[ H_{\mathrm v,\varphi}=H_{\mathrm v,\varphi}^{(1)}+H_{\mathrm v,\varphi}^{(2)}, \]

where, in the coordinates of the \(s\)-th disk, \(H_{\mathrm v,\varphi}^{(1)}\) is given by the following series in spheroidal wave functions [6–8] (the wave spheroidal functions are defined as in [9]):

\[ H_{\mathrm v,\varphi}^{(1)} = \frac{4k^2|\vec p_{\mathrm v}|}{a_s(\xi_{0s}^2+1)^{1/2}} \times \]

\[ \times \sum_{n=1}^{\infty} \varepsilon_{ns} \frac{\sigma_{1n}(-ic_s)S_{1n}(-ic_s,\eta_s)} {N_{1n}(-ic_s)} \times \]

\[ \times \begin{cases} R_{1n}^{(1)}(-ic_s,i\xi_s)\,R_{1n}^{(3)}(-ic_s,i\xi_{0s}), & \xi_{0s}>\xi_s,\\[4pt] R_{1n}^{(1)}(-ic_s,i\xi_{0s})\,R_{1n}^{(3)}(-ic_s,i\xi_s), & \xi_{0s}<\xi_s, \end{cases} \tag{5} \]

and \(H_{\mathrm v,\varphi}^{(2)}\) is found as the solution of the boundary-value problem for the Helmholtz equation
\[ \Delta_{\xi,\eta,\varphi}H_{\mathrm v,\varphi}^{(2)}+k^2H_{\mathrm v,\varphi}^{(2)}=0, \]
satisfying, on the surface of each disk, the boundary condition
\[ \frac{\partial}{\partial\xi_s} \left[(\xi_s^2+1)^{1/2} \left(H_{\mathrm v,\varphi}^{(1)}+H_{\mathrm v,\varphi}^{(2)}\right)\right]=0, \quad \xi_s=0,\quad s=\pm1, \]
and satisfying the radiation condition at infinity. This boundary-value problem is solved as in [6]. Its solution is given by the formulas:

\[ H_{\mathrm v,\varphi}^{(2)} = \sum_{s=\pm1} H_{\mathrm v,\varphi}^{(s)}, \quad H_{\mathrm v,\varphi}^{(s)} = \sum_{n=1}^{\infty} n R_{1n}^{(1)\prime}(-ic_s,i0)\,X_n^{(s)} \times \]

\[ \times R_{1n}^{(3)}(-ic_s,i\xi_s)S_{1n}(-ic_s,\eta_s), \tag{6} \]

where \(X_n^{(s)}\) are found as solutions of the infinite system of linear equations

\[ X_n^{(s)}+\sum_{m=1,3,\ldots}\beta_{nm}^{(-s,s)}X_m^{(-s)}=F_n^{(s)},\quad n=1,3,\ldots,\quad s=\pm1, \tag{7} \]

where

\[ \beta_{nm}^{(-s,s)}=\frac{mR_{lm}^{(1)'}(-ic_{-s},i0)}{nR_{ln}^{(3)'}(-ic_s,i0)}Q_{lnlm}(c_s,c_{-s};l_0,\theta_{-s,s}); \tag{8} \]

\[ F_n^{(s)}=-\frac{4k^2\left|\vec p_{\mathrm{B}}\right|\sigma_{ln}(-ic_s)R_{ln}^{(3)}(-ic_s,i\xi_{0s})} {a_s(\xi_{0s}^{2}+1)^{1/2}\,nN_{ln}(-ic_s)R_{ln}^{(3)'}(-ic_s,i0)}; \tag{9} \]

\[ Q_{lnlm}=\frac{2i^{\,n-m}}{N_{ln}(-ic_s)} \sum_{l=0,1}^{\infty}{}'\sum_{t=0,1}^{\infty}{}' d_l^{ln}(-ic_s)d_t^{lm}(-ic_{-s})\times \]

\[ \times\sum_{\sigma=|l-t|}^{l+t+2} i^\sigma b_\sigma^{(l+1,1,t+1,1)}h_\sigma^{(1)}(kl_0) P_\sigma(\cos\theta_{-s,s}). \tag{10} \]

Here \(l_0\) is the distance between the centers of the disks, \(\xi_{0s}\) is the radial coordinate of the dipole in the coordinates of the \(s\)-th disk, \(\varepsilon_{ns}=(-s)^{n-1}\) if the dipole is located in the gap between the disks, \(\varepsilon_{ns}=1\) if the dipole is located above the disks, and \(\varepsilon_{ns}=(-1)^{n-1}\) if the dipole is located below the disks, \(\theta_{-1,+1}=0\), \(\theta_{+1,-1}=\pi\), \(d_r^{mn}(-ic)\) are the coefficients of the wave functions \(R_{mn}\) and \(S_{mn}\) [9], \(N_{ln}(-ic)\) determines the norm of the angular functions \(S_{ln}(-ic,\eta)\) [10], \(h_\sigma^{(1)}(x)\) is the spherical Bessel function, \(P_\sigma(x)\) is the Legendre polynomial,

\[ \sigma_{1n}(-ic)=\frac12\sum_{r=0,1}^{\infty}{}'\frac{(2+r)!}{r!}\,d_r^{ln}(-ic), \]

\(b_\sigma\) are expansion coefficients,

\[ P_{j_1}^{m_1}(\cos\theta)P_{j_2}^{m_2}(\cos\theta) = \sum_{\sigma=|j_1-j_2|}^{j_1+j_2} b_\sigma^{(j_1,m_1,j_2,m_2)} P_\sigma^{m_2-m_1}(\cos\theta) \quad [10]. \]

The prime on the summation sign means that, in sums containing \(d_r^{mn}\), summation is carried out over even \(r\) if \(n-m\) is even, and over odd \(r\) if \(n-m\) is odd. Some remarks concerning the use of (7)—(10) for numerical computation will be made by us later.

1. Problem b) can be solved according to the same scheme as in [4], used there for the case of a magnetic dipole. For this purpose we represent \(\vec\Pi_{\mathrm{r}}^{(2)}\) in the form of the sum \(\vec\Pi_{\mathrm{r}}^{(2)}=\vec\Pi_{\mathrm{r}}^{(-1)}+\vec\Pi_{\mathrm{r}}^{(+1)}\), where \(\vec\Pi_{\mathrm{r}}^{(s)}\), \(s=\pm1\), is the Hertz vector of the secondary field due only to one \(s\)-th disk, and we assume that \(\vec\Pi_{\mathrm{r}}=\{\Pi_{\mathrm{r}x},0,\Pi_{\mathrm{r}z}\}\), where \(\Pi_{\mathrm{r}x}=\Pi_{\mathrm{r}x}^{(1)}+\Pi_{\mathrm{r}x}^{(2)}\), and \(\Pi_{\mathrm{r}z}=\Pi_{\mathrm{r}z}^{(2)}\) (local Cartesian coordinate systems \(O_sx_sy_sz_s\) are introduced so that the vector \(\vec p_{\mathrm{r}}\) lies in the plane \(O_sx_sz_s\), \(s=\pm1\)). Since then \(\vec\Pi_{\mathrm{r}}^{(1)}=\vec p_{\mathrm{r}}e^{ikR}/R\), problem b) will consist in finding the potential functions \(\Pi_{\mathrm{r}x}^{(2)}\), \(\Pi_{\mathrm{r}z}^{(2)}\), through which the fields \(\vec E_{\mathrm{r}}\), \(\vec H_{\mathrm{r}}\) themselves are then also determined. As in [1—4], the problem is first solved in cylindrical coordinates \(\rho,\varphi,z\), in which we set \(\Pi_{\mathrm{r}x}=\Phi(\rho,z)=\Phi_1(\rho,z)+\Phi_2(\rho,z)\), where \(\Phi_1(\rho,z)=\Pi_{\mathrm{r}x}^{(1)}\), \(\Phi_2(\rho,z)=\Pi_{\mathrm{r}x}^{(2)}=\Phi_{-1}(\rho_{-1},z_{-1})+\Phi_{+1}(\rho_{+1},z_{+1})\), and \(\Pi_{\mathrm{r}z}^{(2)}=\Psi(\rho,z)\cos\varphi=\Psi_{-1}(\rho_{-1},z_{-1})\cos\varphi_{-1}+\Psi_{+1}(\rho_{+1},z_{+1})\cos\varphi_{+1}\) \((\varphi_{+1}=\varphi_{-1}=\varphi)\), and in which the surface of the disks is specified by the equations \(z_s=0\),

\(s=\pm 1\) and \(\rho_s<a_s\). Expanding (1) in cylindrical coordinates and then taking into account that \(\Pi_{r\rho}=\Phi\cos\varphi\), \(\Pi_{r\varphi}=-\Phi\sin\varphi\), \(\Pi_{rz}=\Psi\cos\varphi\), for the tangential components of the vector \(\vec E_r\) we obtain the expressions:

\[ E_{r,\varphi}=-\sin\varphi\left[\frac{1}{\rho_s}\left(\frac{\partial\Phi}{\partial\rho_s}+\frac{\partial\Psi}{\partial z_s}\right)+k^2\Phi\right], \]

\[ E_{r,\rho}=\cos\varphi\left[\frac{\partial}{\partial\rho_s}\left(\frac{\partial\Phi}{\partial\rho_s}+\frac{\partial\Psi}{\partial z_s}\right)+k^2\Phi\right], \]

from which separate boundary conditions for \(\Phi\) and \(\Psi\) on the surface of each disk are obtained:

\[ \left. \begin{array}{r} \Phi=-\dfrac{C_s}{k^2}\\[6pt] \dfrac{\partial\Psi}{\partial z_s}=C_s\rho_s \end{array} \right\}, \quad z_s=0,\ \rho_s<a_s,\ s=\pm1, \tag{11} \]

where \(C_s\) are arbitrary constants [1—4, 7, 11]. Passing then again to the coordinates of the oblate spheroid (2), for determining \(\Phi_2\) and \(\Psi\) we arrive at the boundary-value problems:

\[ \Delta_{\xi,\eta,\varphi}\Phi_2+k^2\Phi_2=0,\quad \Delta_{\xi,\eta,\varphi}\Psi+k^2\Psi=0, \]

\[ \Phi_1+\Phi_2=-\frac{C_s}{k^2},\quad \frac{\partial\Psi}{\partial\xi_s}=C_s a_s^2\eta_s\sqrt{1-\eta_s^2} \quad \text{for } \xi_s=0,\ s=\pm1, \]

where it is assumed that \(\Phi_2\) and \(\Psi\) satisfy the radiation conditions at infinity. If the dipole does not lie on the surface of one of the disks, then the constants \(C_s\), \(s=\pm1\), are found from the additional requirement (Meixner condition [1—4, 7, 11, 12]) that the radial component of the total current induced on the disks vanish at their edges:

\[ j_{\rho_s}=0,\quad \xi_s=0,\ \rho_s=a_s,\ s=\pm1. \tag{12} \]

If, however, the dipole is located on the surface of one of the disks, then, along with (12), one must use the reciprocity theorem for electric dipoles [13]

\[ \vec p_1\vec E_2=\vec p_2\vec E_1. \tag{13} \]

In the coordinates of the oblate spheroid [9]

\[ \Phi_1=\left|\vec p_r\right|\,2ik \sum_{n=0}^{\infty} \frac{S_{0n}(-ic_s,\eta_{0s})}{N_{0n}(-ic_s)} S_{0n}(-ic_s,\eta_s)\times \]

\[ \times \begin{cases} R_{0n}^{(1)}(-ic_s,i\xi_s)R_{0n}^{(3)}(-ic_s,i\xi_{0s}), & \xi_{0s}>\xi_s,\\ R_{0n}^{(1)}(-ic_s,i\xi_{0s})R_{0n}^{(3)}(-ic_s,i\xi_s), & \xi_s<\xi_{0s}, \end{cases} \]

and \(\Phi_2\) and \(\Psi\) are determined by us in the form of expansions:

\[ \Phi_2=\sum_{s=\pm1}\Phi_s,\quad \Phi_s=\sum_{n=0}^{\infty} \varepsilon_n R_{0n}^{(1)}(-ic_s,i0)Y_n^{(s)} R_{0n}^{(3)}(-ic_s,i\xi_s)S_{0n}(-ic_s,\eta_s), \tag{14} \]

\[ \Psi=\sum_{s=\pm 1}\Psi_s,\quad \Psi_s=\sum_{n=0}^{\infty} R_{1n}^{(1)'}(-ic_s,i0)\, Z_n^{(s)} R_{1n}^{(3)}(-ic_s,i\xi_s)\, S_{1n}(-ic_s,\eta_s), \tag{15} \]

where the coefficients \(Y_n^{(s)}\) and \(Z_n^{(s)}\) are solutions of infinite systems of linear equations:

\[ Y_n^{(s)}+\sum_{m=0,2,\ldots}\mu_{nm}^{(-s,s)}Y_m^{(-s)}=f_n^{(s)},\quad n=0,2,\ldots,\ s=\pm 1, \tag{16} \]

\[ Z_n^{(s)}+\sum_{m=0,2,\ldots}\tau_{nm}^{(-s,s)}Z_m^{(-s)}=\varphi_n^{(s)},\quad n=0,2,\ldots,\ s=\pm 1, \tag{17} \]

in which

\[ \mu_{nm}^{(-s,s)}= \frac{\varepsilon_m R_{0m}^{(1)}(-ic_{-s},i0)} {\varepsilon_n R_{0n}^{(3)}(-ic_s,i0)} \,Q_{0n0m}(c_s,c_{-s};l_0,\theta_{-s,s}), \tag{18} \]

\[ \tau_{nm}^{(-s,s)}= \frac{R_{1m}^{(1)'}(-ic_{-s},i0)} {R_{1n}^{(3)'}(-ic_s,i0)} \,Q_{1n1m}(c_s,c_{-s};l_0,\theta_{-s,s}), \tag{19} \]

\[ f_n^{(s)}=-2ik\left[\vec p_r\right]\, \frac{S_{0n}(-ic_s,\eta_{0s})R_{0n}^{(3)}(-ic_s,i\xi_{0s})} {\varepsilon_n N_{0n}(-ic_s)R_{0n}^{(3)}(-ic_s,i0)} + \frac{2d_0^{0n}(-ic_s)C_s} {\varepsilon_n k^2 N_{0n}(-ic_s)R_{0n}^{(1)}(-ic_s,i0)R_{0n}^{(3)}(-ic_s,i0)}, \tag{20} \]

\[ \varphi_n^{(s)}=-\frac{4}{5}\, \frac{a_s^2 d_1^{1n}(-ic_s)C_s} {N_{1n}(-ic_s)R_{1n}^{(1)'}(-ic_s,i0)R_{1n}^{(3)'}(-ic_s,i0)}, \quad \varepsilon_n= \begin{cases} 1,\ n=0,\\ n^2,\ n\ge 1. \end{cases} \tag{21} \]

The systems (16), (17), and (7) are obtained from the corresponding boundary conditions on the surface of each disk by means of the addition theorem

\[ R_{mn}^{(3)}(-ic_{-s},i\xi_{-s})S_{mn}(-ic_{-s},\eta_{-s}) = \sum_{p=m}^{\infty} Q_{mpmn}(c_s,c_{-s};l_0,\theta_{-s,s})\times \]

\[ {}\times R_{mp}^{(1)}(-ic_s,i\xi_s)S_{mp}(-ic_s,\eta_s), \tag{22} \]

where

\[ Q_{mpmn}= \frac{2i^{p-n}}{N_{mp}(-ic_s)} \sum_{l=0,1}^{\infty}{}' \sum_{t=0,1}^{\infty}{}' d_l^{mp}(-ic_s)d_t^{mn}(-ic_s)\times \]

\[ {}\times \sum_{\sigma=|l-t|}^{l+t+2m} i^\sigma b_{\sigma}^{(l+m,m,t+m,m)} h_{\sigma}^{(1)}(kl_0)\,P_\sigma(\cos\theta_{-s,s}), \tag{23} \]

and the orthogonality properties of the angular functions \(S_{mn}(-ic,\eta)\). Systems analogous to (7), (16), and (17) were studied by us in [2—4]. As shown in [2—4], they prove to be quasiregular and solvable by the truncation method under the condition that \(2l_0>a_{+1}+a_{-1}\). However, in [4] it was established that numerical computation by formulas of the type (7)—(10), (16)—(21) can be carried out only under the condition that \(l_0>a_{+1}+a_{-1}\) (only under this assumption was it possible there to prove the absolute and uniform convergence of double series analogous to (8), (18), (19).

1. As is seen from (20) and (21), \(Y_n^{(s)}\) and \(Z_n^{(s)}\), after being found from (16), (17), will depend linearly on \(C_s,\ s=\pm 1\). Condition (12), which takes the form

\[ \sum_{n=0,2,\ldots} \varepsilon_n X_n^{(s)} S_{0n}(-ic_s,0) = \sum_{n=0,2,\ldots} Z_n^{(s)} S'_{1n}(-ic_s,0), \tag{24} \]

makes it possible to set up two equations for \(C_s,\ s=\pm 1\), if the dipole does not lie on the surface of one of the disks, and one equation, which is obtained from (24) for the index \(-s\), if the dipole lies on the surface of the \(s\)-th disk \((s=+1\) or \(s=-1)\). In this case the second equation for \(C_s,\ s=\pm 1\), is obtained from (13), which takes the form

\[ \left. \Phi_2^{n/l} \right|_{\substack{\xi_s=0,\ \eta_s=1\\ s=+1\ \text{or}\ s=-1}} = \left. \Phi_2 \right|_{\substack{\xi_s=\infty,\ \eta_s=1\\ s=\pm 1}} . \tag{25} \]

Here \(\Phi_2^{n/l}\) is an auxiliary potential function, similar to \(\Phi_2\), in the analogous problem of diffraction of a plane electromagnetic wave

\[ H_y=E_x=Ae^{-ikz_s},\quad A=-k^2|\vec p_r| e^{ikz_{0s}/z_{0s}},\quad H_x=H_z=E_y=E_z=0, \tag{26} \]

on two disks [2–4]. Determination of \(C_s\), and then of \(Y_n^{(s)}, Z_n^{(s)}\) and the functions \(\Phi_2,\Psi\), completely solves the problem of finding \(\vec E_r,\vec H_r\).

As was already noted above, the solution of the general problem is now found in the form of the sum of the solutions found for problems a) and b).

References

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Received by the editors
November 1, 1964

Institute of Mathematics, Academy of Sciences of the BSSR