Radio Absorbing Composite Based on Square Resistive Patches

Abstract

Solutions to the diffraction problems are used to calculate the effective permittivity of a composite material consisting of alternating planar arrays of square resistive patches shifted relative to each other in a dielectric layer. The relaxation character of the dispersion is demonstrated. The dispersion characteristic can be varied in wide ranges using variations in the parameters of the structure. Relative shifts of the arrays substantially affect the dispersion of permittivity of the structure. A ratio of the wavelength to the period of the structure that provides adequacy of a homogenization procedure is estimated.

APPENDIX

Below, we present the final formulas for calculation of the RC for a plane electromagnetic wave that is normally incident on a structure consisting of square resistive patches in a dielectric layer. The wavelength is no less than the period of the structure. We employ consecutive numbering of elementary squares (ESs) of all arrays and consecutive numbering of all modes of the equivalent waveguide. Note that the TEM mode has number n = 1.

The following notation is used.

(i) x_p, y_p, and z_p are coordinates of the p-th ES (p = 1, 2, …, P).

(ii) Γ_n and W_n (n = 1, 2, …, M ⋅ N) are the propagation constants and impedances for the modes of the Floquet channel at z > z₀ (Fig. 2).

(iii) $T_{n}^{ \pm }$ and $R_{n}^{ \pm }$ are the transmission and reflection coefficients of forward (+) and backward (–) waves at the boundary z = z₀ (Fig. 2).

(iv) χ_nx and χ_ny are transverse wave numbers of the n-th mode.

(v) $A_{n}^{{(y)}}$ and $A_{n}^{{(x)}}$ are the normalized factors of electric-field components E_ny and E_nx of the n-th mode [13].

(vi) S_np = sin(χ_nxx_p)sin(χ_nyy_p).

(vii) C_np = cos(χ_nxx_p)cos(χ_nyy_p).

(viii) Ψ_np = $\int_{{{\sigma }_{p}}}^{} {\cos }$(χ_nxx)cos(χ_nyy)dxdy.

(ix) ${{\tilde {\Psi }}_{{np}}}$ = $\int_{{{\sigma }_{p}}}^{} {\sin }$(χ_nxx)cos(χ_nyy)dxdy,

where σ_p is the area of integration over surface of the p-th square.

The last integrals are easily calculated but we omit cumbersome calculated results.

(x) C_n = –W_n/(2|W_n|),

$\beta _{{yy}}^{{p(n)}}$ = C_n|$A_{n}^{{(y)}}$|²Ψ_np, $\beta _{{yx}}^{{p(n)}}$ = C_n($A_{n}^{{(y)}}$)*$A_{n}^{{(x)}}$Ψ_np,

$\beta _{{xx}}^{{p(n)}}$ = C_n|$A_{n}^{{(x)}}$|²${{\tilde {\Psi }}_{{np}}}$, $\beta _{{xy}}^{{p(n)}}$ = C_n($A_{n}^{{(x)}}$)*$A_{n}^{{(y)}}{{\tilde {\Psi }}_{{np}}}$.

(xi) $E_{{nmp}}^{ \pm }$ = exp(iΓ_n(z_m ± z_p)).

(xii) α_n is the reflection coefficient of the n-th mode with respect to electric field in the plane z = z_{k+ 1}.

(xiii) ${{\tilde {S}}_{m}}$ = $T_{1}^{ + }$(exp(iΓ₁z_m) + α₁exp(–iΓ₁z_m))/(1 – α₁$R_{1}^{ - }$).

(xiv) F_nmp = $R_{n}^{ - }$[$E_{{nmp}}^{ + }$ + α_n($E_{{nmp}}^{ - }$ + 1/$E_{{nmp}}^{ - }$)] + α_n/$E_{{nmp}}^{ + }$.

(xv) S_nmp = exp(iΓ_n|z_p – z_m|) + F_nmp/(1 – α_n$R_{n}^{ - }$).

(xvi) $A_{{mp}}^{{(x)}}$ = $\sum\nolimits_n^{} {{{S}_{{nmp}}}\beta _{{xx}}^{{p(n)}}}$S_nm – δ_mpρ, where δ_mp is the Kronecker delta.

(xvii) $A_{{mp}}^{{(y)}}$ = $\sum\nolimits_n^{} {{{S}_{{nmp}}}\beta _{{xy}}^{{p(n)}}}$C_nm.

(xviii) $B_{{mp}}^{{(x)}}$ = $\sum\nolimits_n^{} {{{S}_{{nmp}}}\beta _{{yx}}^{{p(n)}}}$S_nm.

(xix) $B_{{mp}}^{{(y)}}$ = $\sum\nolimits_n^{} {{{S}_{{nmp}}}\beta _{{yy}}^{{p(n)}}}$C_nm – δ_mpρ.

The SLAE for surface current densities {I_px}, {I_py} consists of two subsystems

(xx)

$$\left\{ \begin{gathered} \sum\limits_p^{} {(A_{{mp}}^{{(x)}}{{I}_{{px}}} + B_{{mp}}^{{(x)}}{{I}_{{py}}})} = 0, \hfill \\ \sum\limits_p^{} {(A_{{mp}}^{{(y)}}{{I}_{{px}}} + B_{{mp}}^{{(y)}}{{I}_{{py}}})} = - {{{\tilde {S}}}_{m}}, \hfill \\ \end{gathered} \right.$$

(xxi)

G = $\frac{{{{\alpha }_{1}}T_{1}^{ + }\, + \,\sum\nolimits_p^{} {({\text{exp}}(i{{\Gamma }_{1}}{{z}_{p}})\, + \,{{\alpha }_{1}}{\text{exp}}( - i{{\Gamma }_{1}}{{z}_{p}}))({{I}_{{px}}}\beta _{{xy}}^{{p(1)}}\, + \,{{I}_{{py}}}\beta _{{yy}}^{{p(1)}})} }}{{1 - {{\alpha }_{1}}R_{1}^{ - }}}$

is the quantity that is used to represent the RC.

(xxii) R = $R_{1}^{ + }$ + $T_{1}^{ - }$G is the RC with respect to electric field in the plane z = z₀