Shielding Effectiveness of a Metamaterial Slab

Finally, in the homogenization process, a finitethickness h eff has to be associated to the homogeneousslab equivalent to the periodic structure, which takes intoaccount the fringing fields at the top and bottom layersof the structure. In particular, an equivalent thicknessequal to Nd will be adopted, in order to best match thereflection performance of the actual periodic structureand that of its homogenized model, as it will be shown inthe next sections.III. SHIELDING EFFECTIVENESSLet us thus consider a uniform plane waveimpinging on a homogeneous planar screen whichseparates two half-spaces filled with vacuum and whichis characterized by its relative constitutive parameters ε r ,µ r , its conductivity σ, and its thickness h. As is wellknown, with the considered type of electromagneticsource, its electric and magnetic shieldingeffectivenesses coincide and are equal toESE = 20logSE = 20logEdB lin twhere E i and E t indicate the electric fields of the incidentand transmitted plane waves, respectively. Thecalculation of the shielding effectiveness can easily beperformed by means of the usual transmission-lineanalogy, thus obtaining [9]whereRdBdBi(6)SEdB = RdB + AdB + MdB(7)= 20log R=20logA = 20log A=20log e( Z + Z )0 S4ZZjkxSh0 S⎛Z0 S − j2kxShMdB= 20log M = 20log 1−⎜ ⎟ eZ0 + ZS⎝2− ZThe impedances Z 0 and Z S depend on the TE or TMpolarization of the incident plane wave and on the angleof incidence θ i throughZZη η µ= , Z =TE 0 TE 0 r0 i ScosθkxSη kTM i TM 0 xS0= η0cos θ , ZS=ε rc⎞⎠2(8)(9)2 iwhere kxS = k 0 µ rεrc − sin θ and εrc = εr − jσ / ω(being k0 = ω µ 0ε0 and ω = 2π f ).When a planar metal screen is considered, theshielding effectiveness expression (7) can be furthersimplified (the propagation constant k xS is very large andpractically unaffected by the angle of incidence), so thatSElincησ32πf⎛⎜⎝πησf⎞c ⎟⎠0 0 exp h(10)When a lossless wire-medium screen is considered,characterized by an effective permittivity as in (1) andoperating below the plasma frequency, the followingexpression for the shielding effectiveness is obtainedinstead:SElin⎡ ⎤c 1 ⎢ 2π⎥ exp⎢N⎥ (11)4dfdd2πln ⎢ ln ⎥4r⎢0 ⎣ 4r⎥0 ⎦The explicit calculation of the shielding effectiveness of alossy wire-medium screen is just a bit more involved,since the imaginary part of the effective permittivity (asgiven in (4)) has to be taken into account.IV.NUMERICAL RESULTSA. Validation of the Homogenized ModelIn order to validate the adopted analysis technique,based on the homogenization of the metamaterial, fullwavesimulations have been performed of actual periodicwire-medium layers in free space with a commercialsoftware based on the finite integration technique (CSTMicrowave Studio TM ) [10].In Fig. 2 a representative example is shown, for astructure constituted by N = 5 layers of perfectlyconducting wires with parameters d = 100 mm, r 0 = 0.1mm, ε rh = 1 (wires in vacuum). These values have beenchosen in order to obtain values of the shieldingeffectiveness not too large (lower than 80 dB in theconsidered frequency range), so avoiding inaccuracies due tonumerical noise.The homogenized slab has f p = 486 MHz, calculatedwith (2). The choice of the equivalent slab thickness is amore delicate matter. In fact, on one hand, the absence ofa dielectric host slab containing the wires does not allowone to identify ‘natural’ interfaces between the