Fully analytical circuit-like approach for the TE scattering by narrow-slit printed gratings

Fully Analytical Circuit-like Approach for the TE Scattering by Narrow-Slit Printed Gratings Ra´ ul Rodr´ ıguez-Berral, Francisco Mesa Dpt. of Applied Physics 1 University of Seville, 41012 Seville, Spain Email: [email protected], [email protected] Francisco Medina Dept. of Electronics and Electromagnetism University of Seville, 41012 Seville, Spain Email: [email protected] Abstract— A fully analytical solution for the scattering of a uniform TE polarized plane wave by an infinite periodic printed array of narrow slits is provided in this contribution. The model is based on some simplifications made on the formulas arising from a rigorous full-wave formulation. The present approach gives the topology of the transverse equivalent network as well as closed-form expressions for all its components, and overall very accurate results. I. INTRODUCTION The study of the scattering by periodic gratings has been a topic of intense research since long [1]–[3]. The specific case of metallic gratings has received considerable attention, being a notably interesting contribution to the modeling of this phenomenon the proposal of equivalent transmission-line circuits that account in a simple manner for the behavior of the reflected and transmitted fields. In the very general case, only very complicated networks can be obtained [4]. However, in some cases, it is possible to find simple equivalent circuits that model very accurately the original structure [5], [6]. In this paper we present a transverse equivalent network to model the scattering of an obliquely incident TE-polarized plane wave impinging on a periodic narrow-slit grating printed on dielectric slabs. Closed-form expressions are obtained for all the circuit parameters. Surprisingly good accuracy is verified by comparison with method-of-moments results. II. FORMULATION Consider the situation shown in Fig.1, where a TE-polarized oblique incident plane wave impingues on a periodic grating. The tangential electric field at both sides of the screen can be written in terms of the following Floquet-series expansion: E(y) = (1 + R+ ∞ X n=1 Ene−jkny+E−nejkny)e−jkty (1) where the amplitude of the incident field has been normalized to unity, Ris the complex reflection coefficient, and kn=2πn p(2) kt=k0sin θinc .(3) w Fig. 1. A TE-polarized incident plane wave impingues obliquely on a periodic slit grating printed on a dielectric slab. The magnetic field at both sides of the screen is given by Hleft(y) = Y(1) 0(1 −R) − ∞ X n=1 hY(1) nEne−jkny+Y(1) −nE−nejknyie−jkty(4) Hright(y) = Y(in) 0(1 + R) + ∞ X n=1 hY(in) nEne−jkny+Y(in) −nE−nejknyie−jkty(5) where Y(εr) nis the wave admittance of the n-th spatial harmonic inside a dielectric medium with relative permittivity εr, given by Y(εr) n=η−1 0 β(εr) n k0 (6) with η0=pµ0/ε0being the free-space impedance and β(εr) ±n=qεrk2 0−(kt±kn)2(7) the wavenumber along z. In (4) we actually have free space, εr= 1, and therefore Y(εr) n≡Y(1) n. The Y(in) nadmittances in (5) represent the input wave admittances to the dielectric slab seen from the right hand side of the screen, and can therefore be written as Y(in) n=Y(εr) n Y(1) n+ jY(εr) ntan β(εr) nd Y(εr) n+ jY(1) ntan β(εr) nd.(8) 978-1-4673-1088-8/12/$31.00 ©2012 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 02,2020 at 15:37:51 UTC from IEEE Xplore. Restrictions apply. Now we make the assumption that the tangential electric field in the aperture (slit) can be expressed as Ea(y) = A f(y)(9) where f(y)is a normalized field profile. The Encoefficients can be given in terms of the aperture field using the following identity: Zp/2 −p/2 E(y) ej(kt±kn)ydy=Zw/2 −w/2 Ea(y) ej(kt±kn)ydy . (10) Thus, introducing (1) in (10) we obtain (1 + R)p=Af0w(11) Enp=Awfn(12) with f±n=1 wZw/2 −w/2 f(x) ej(kt±kn)ydy , (13) and therefore A=1 + R f0 p w(14) En= (1 + R)fn f0 .(15) Since the magnetic field is continuous through the slit, we can write Hleft (y) = Hright (y)for |y|<w/2.(16) Following a Galerking-like testing scheme, we can project (16) over the aperture field profile to obtain the following equation: Zw/2 −w/2 f(y)Hleft (y)−Hright (y)dy= 0 .(17) Substituting (4) and (5) into (17) and taking into account (15), we obtain R= Y(1) 0−Y(in) 0−f−2 0X n6=0 Y(1) n+Y(in) nf2 n Y(1) 0+Y(in) 0+f−2 0X n6=0 Y(1) n+Y(in) nf2 n (18) where the sum extends over all non-zero integers (positive and negative harmonics). Looking at (18) we can see that the effect of the slit grating can be accounted for by the following equivalent admittance: Yeq =f−2 0X n6=0 Y(1) n+Y(in) nf2 n.(19) For narrow slits (in practice for w/λ0/0.2), the following field profile is expected to give a good approximation of the aperture field: f(y) = 4 πs1−2y w2 .(20) ... ... Fig. 2. Equivalent network obtained for the scattering of a TE-polarized incident plane wave impinging obliquely on a narrow-slit periodic grating. This profile implies that the coefficients f±nare given by f±n= 2 J1 (kt±kn)w 2 (kt±kn)w 2 (21) where J1(·)is the Bessel function of the first kind and order 1. For higher-order harmonics that are well below cutoff, we can make the following approximations: kt±kn≈ ±kn(22) β(εr) n≈ −jkn(23) Y(εr) n≈1 jω ckn η0 (24) with cbeing the speed of light in vacuum. In (24) we observe that the admittances account for frequency-independent inductances given by Ln=η0/(ckn). Note that the Lninductances do not depend on the permittivity. The input admittances Y(in) n can as well be approximated by Y(in) n≈Y(εr) n≈1 jωLn (25) and, therefore, the equivalent admittance defined in (19) can be written as Yeq ≈ N X n=−N n6=0 AnY(1) n+Y(in) n+1 jωLho (26) where An= (fn/f0)2[f0and f±nare given by (21)], Nis the number of positive or negative harmonics (the larger of the two) that are propagative or almost propagative inside the dielectric, and 1 Lho =8c wη0J1(ktw/2) ktw/2−2∞ X n=N+1 J2 1(knw/2) knw/2.(27) The above series does not depend on frequency and therefore it has to be computed only once. Taking now into account the final form of Yeq in (26), we observe that the narrow-slit periodic grating can be modelled by the equivalent circuit depicted in Fig.2, where all the elements are known in closed form. 978-1-4673-1088-8/12/$31.00 ©2012 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 02,2020 at 15:37:51 UTC from IEEE Xplore. Restrictions apply. 0 0.2 0.4 0.6 0.8 1 30 35 40 45 50 55 60 Transmission Coefficient |S21| Frequency (GHz) d = 1.5mm d = 3mm d = 5mm Fig. 3. Full-wave MoM (solid lines) and analytical (symbols) results for a printed grating with p= 5 mm, w= 1 mm, and εr= 2.2.θinc = 0 and N= 2 in Eq.(26). III. RESULTS In this section we present some examples of application of the above discussed equivalent-circuit approach to characterize the scattering of a TE-polarized wave impinging on different narrow-slit periodic gratings. First, in Fig. 3 we analyze the case of a slit grating printed on a low-permittivity substrate (εr= 2.2) for various thicknesses of the substrate. In the figure we compare our fully analytical results (symbols) with those computed using a full-wave Method-of-Moment technique with multiple basis functions (solid lines). For the thinnest substrate case (d= 1.5mm) in Fig. 3, we can see that there is a peak of total transmission followed by a transmission null, and that this Fano-type resonance [3] is also very accurately captured by the analytical (circuit-model) results. For thicker slabs, there appear different Fano-type resonances, which are again in complete agreement with the fully analytical results. In the second example shown in Fig. 4 we consider the case of a grating printed on a electrically thick dielectric slab (the relative permittivity of the substrate is now 10.2). In this case the transmission spectra shown in Fig. 4 is much more complicated than in the low-permittivity cases studied in Fig. 3. However, despite the complexity of the transmission spectra (multiple Fano-type resonances, maxima of partial transmission, etc), the agreement between the full-wave MoM results and the analytical ones is surprisingly good. As a final example, we show in Fig. 5 the transmission spectra for the case of the thickest slab previously analyzed in Fig. 3, but now under oblique incidence and including frequencies in the grating-lobe regime (this regime starts at 60/(1+sin θinc)GHz for the considered structure). Again, we can see an excellent agreement between the full-wave MoM results and our analytical data. In summary, we have presented a fully analytical circuitlike approach to study the TE scattering of a plane wave impinging obliquely on a narrow-slit periodic grating printed on a dielectric slab. Our analytical results have proved to be very accurate when compared with full-wave numerical data, even for very complicated transmission spectra. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 Transmission Coefficient |S21| Frequency (GHz) full-wave MoM circuit model Fig. 4. Full-wave MoM and analytical results for a slit grating (p= 5 mm, w= 1 mm) printed on a electrically thick dielectric slab (h= 3 mm and εr= 10.2). θinc = 0 and N= 4 in Eq.(26). 0 0.2 0.4 0.6 0.8 1 20 25 30 35 40 45 50 55 60 Transmission Coefficient |S21| Frequency (GHz) θ inc 0o 20o 40o 60o Fig. 5. Full-wave MoM (solid lines) and analytical results (symbols) for a slit grating printed on a dielectric slab under oblique incidence (p= 5 mm, w= 1 mm, d= 5 mm, εr= 2.2and N= 2 in Eq.(26)). ACKNOWLEDGMENT The authors would like to thank the Spanish Ministry of Science and Innovation, European Union FEDER funds and the Spanish Junta de Andaluc´ ıa for financial support (projects TEC2010-16948, TIC-4595 and Consolider CSD2008-00066). REFERENCES [1] R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phylosophical Magazine, vol. 4, pp. 396- 402, 1902. [2] Lord Rayleigh, “Dynamical theory of the grating,” Proc. Roy. Soc. (London), vol. A79, p. 399-416, 1907. [3] U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am., vol. 31, p. 213-222, 1941. [4] M. Guglielmi and A. A. Oliner, “Multimode network description of a planar periodic metal-strip grating at a dielectric interface - Part I: Rigorous network formulations, - Part II: Small-aperture and smallobstacle solutions,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 3, pp. 535–541, March 1989. [5] R. Rodr´ ıguez-Berral, F. Medina, F. Mesa, “Circuit model for a periodic array of slits sandwiched between two dielectric slabs,” Appl. Phys. Lett., vol. 96, p. 161104, April 2010. [6] M. Beruete, M. Navarro-C´ ıa, S. A. Kuznetsov, M. Sorolla, “Circuit approach to the minimal configuration of terahertz anomalous extraordinary transmission,” Appl. Phys. Lett., vol. 98, p. 014106, 2011. 978-1-4673-1088-8/12/$31.00 ©2012 IEEE Authorized licensed use limited to: Universidad de Sevilla. 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