Electromagnetic characterization of tuneable graphene‐strips‐on‐substrate metasurface over entire THz range: Analytical regularization and natural‐mode resonance interplay
2021; Institution of Engineering and Technology; Volume: 15; Issue: 10 Linguagem: Inglês
10.1049/mia2.12158
ISSN1751-8733
AutoresFedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Yuriy Rapoport,
Tópico(s)Plasmonic and Surface Plasmon Research
ResumoIET Microwaves, Antennas & PropagationVolume 15, Issue 10 p. 1225-1239 ORIGINAL RESEARCH PAPEROpen Access Electromagnetic characterization of tuneable graphene-strips-on-substrate metasurface over entire THz range: Analytical regularization and natural-mode resonance interplay Fedir O. Yevtushenko, Corresponding Author fedir.yevtushenko@gmail.com orcid.org/0000-0003-3119-8315 Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorSergii V. Dukhopelnykov, Corresponding Author dukh.sergey@gmail.com orcid.org/0000-0002-0639-988X Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Department of Applied Mathematics, V. N. Karazin, Kharkiv National University, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorTatiana L. Zinenko, Corresponding Author tzinenko@yahoo.com Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorYuriy G. Rapoport, Corresponding Author yuriy.rapoport@gmail.com Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this author Fedir O. Yevtushenko, Corresponding Author fedir.yevtushenko@gmail.com orcid.org/0000-0003-3119-8315 Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorSergii V. Dukhopelnykov, Corresponding Author dukh.sergey@gmail.com orcid.org/0000-0002-0639-988X Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Department of Applied Mathematics, V. N. Karazin, Kharkiv National University, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorTatiana L. Zinenko, Corresponding Author tzinenko@yahoo.com Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, Kharkiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this authorYuriy G. Rapoport, Corresponding Author yuriy.rapoport@gmail.com Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Correspondence Fedir O. Yevtushenko, Sergii V. Dukhopelnykov, Tatiana L. Zinenko, Laboratory of Micro and Nano Optics, Institute of Radio-Physics and Electronics NASU, 12, Ac. Proskura st.,Kharkiv, 61085, Ukraine. Email: fedir.yevtushenko@gmail.com, dukh.sergey@gmail.com and tzinenko@yahoo.com Yuriy G. Rapoport, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, City of Kyiv, Kyiv, 01601, Ukraine. Email: yuriy.rapoport@gmail.comSearch for more papers by this author First published: 03 June 2021 https://doi.org/10.1049/mia2.12158Citations: 1 [Correction added on 18 June 2021, after first online publication, "DESIRED SAMPLING ERROR" was corrected to "DUAL SERIES EQUATION", on page 4, section 4 heading. Correction added on 29 July 2021, on page 6, equation 34, in the denominator, "m" was corrected to "±"] AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract Scattering and absorption of the H-polarized plane wave by the infinite grating of flat graphene strips are considered in the environment met most frequently—on or at the surface of a dielectric-slab substrate. The full-wave meshless code is based on the analytical semi-inversion using the Riemann–Hilbert problem solution. This leads to a Fredholm second-kind matrix equation for the Floquet harmonic amplitudes that guarantees code convergence and provides easy control of computational error, which can be reduced to machine precision. The matrix elements are combinations of elementary functions, and therefore, the code is accurate and quite economical. This enables computation of the reflectance, transmittance, and absorbance as a function of the frequency in the wide band from static case to 10 THz. Numerical results show that such a metasurface with micrometre-sized strips is a composite periodic open resonator. It is highly frequency-selective thanks to the interplay of three types of natural modes—low-Q slab, moderate-Q plasmon strip, and ultra-high-Q lattice—that do not exist in the absence of the substrate. Varying the chemical potential of graphene, one can manipulate the electromagnetic characteristics of the metasurface at a fixed frequency from almost total transmission to almost total reflection. 1 INTRODUCTION Recent progress in nanotechnologies has drawn broad attention to graphene as a material that can provide new functionalities to devices and systems that use electromagnetic waves, especially in the terahertz, infrared, and visible-light ranges. This interest is explained by graphene's good conductivity, which can be tuned with the aid of DC electric biasing [1-4]. In addition, graphene can support a surface plasmon-guided wave with an electric field orthogonal to its surface in the terahertz and infrared ranges at frequencies two orders lower than those of the noble metals [5]. These properties make graphene very promising for designing novel tuneable antennas, filters, sensors, and absorbers, to mention just a few possible devices. Today, the focus of research into the applications of graphene in electronics and photonics has shifted from wide-area sheets to patterned configurations in the form of strips, disks, and other flat forms [6]. One of the most intensively used configurations is a grating of parallel graphene strips [7-12]. In principle, graphene strips can be fabricated without substrate (suspended in air), thus improving their chemical stability [13]._Patterned graphene configurations are typically located, however, on the surface of flat dielectric substrates. For instance, chemical vapour deposition (CVD) technology can produce a controlled number of high-quality graphene monolayers on a large area of high-refractive-index substrate. It was used in [10] to manufacture double-layer gratings of nanosize-width CVD-graphene strips on a substrate of polished float-zone silicon for infrared sensing. To pattern strips from the graphene sheet, 100-keV electron beam lithography and etching in oxygen plasma were applied. Still, the technologies mentioned above are expensive. To reduce the cost and time of research and development, scientists have used the preceding modelling of the electromagnetic properties of patterned graphene. Of crucial importance here is the availability of the surface conductivity of non-patterned zero-thickness graphene in analytical form, known as the Drude model or the more sophisticated Kubo formalism [5]. Still, several aspects are non-trivial for accurate modelling and must fully account for Graphene's extremely thinness (1–2 nm) and finite and frequency-dependent conductivity as well as the presence of sharp edges, dielectric substrates, and superstrates. Comparative reviews of techniques employed for such modelling can be found in [14, 15] together with discussions of their limitations. In particular, it should be emphasized that the Fourier expansion technique (also known as 'rigorous coupled-wave analysis') is divergent in the H-polarization case [14], while commercial codes require the introduction of a nanoscale thickness of graphene that entails unnecessarily fine meshing and prohibitively long computation time [15]. Two analytical-numerical approaches stand out in this area: the method of singular integral equations (SIEs) solved using the Nystrom discretizations [16, 17] and the method of analytical regularization (MAR) applied to either SIE or other equivalent equations, casting them to Fredholm second-kind matrix equations [18-21]. The aim of our work is twofold. First, we would like to adapt the MAR version, which is based on the analytical solution of the Riemann–Hilbert problem (RHP) in complex calculus, to the analysis of wave scattering and absorption by an infinite grating of flat graphene strips lying on a flat dielectric-slab substrate. This MAR-RHP technique was developed earlier to analyze scattering from zero-thickness perfectly electrically conducting (PEC) on-substrate strip grating [22-24]. Its modification for graphene strips has not been done so far and is expected to outperform other techniques, including existing MAR-based techniques. The latter use SIEs in the spatial or Fourier-transform domain solved with the Galerkin method-of-moments with judiciously selected expansion functions (namely, weighted Chebyshev polynomials) [14, 21]. They are available only for graphene-strip gratings embedded in a layer of dielectric. In contrast, our MAR-RHP technique is equally applicable to embedded and on-surface configurations; it also has a great advantage in not needing numerical integrations to fill in the matrix equation. Second, with the aid of such a trusted and efficient instrument, we aim to systematically research the resonance effects in the scattering and absorption of the terahertz waves by on-substrate graphene-strip gratings. Here, we focus our research on the so-called lattice-mode resonances, which do not exist on suspended graphene-strip gratings. They have been frequently overlooked or neglected earlier but became exposed recently [14, 21]. The lattice modes are specific natural modes of the gratings as periodic open resonators; they were first reported in [25] and then largely forgotten. However, they have attracted increasingly greater attention in recent times [19, 26-33]. This is because, in theory, they may have (for infinite gratings) various high-Q factors—for instance, if the substrate becomes thinner. The other natural modes of our composite scatterer, namely the plasmon modes of the strips and the slab modes of the substrate, do not have such a property. As noted in [14, 21], accurate quantification of such fine resonances appears to be possible only with the aid of truly sophisticated codes based on either MAR or SIE-Nystrom techniques. In our work, we study only H-polarized plane wave scattering and absorption because in the case of E-polarization, plasmon-mode resonances do not exist. In the numerical experiments, we select the strip width, grating period, and substrate thickness in dozens of micrometres. This places the frequencies of the substrate, plasmon, and lattice modes in the terahertz range. 2 PROBLEM FORMULATION We consider an infinite flat grating of zero-thickness graphene strips, located in the plane y = 0 with period р, as shown in Figure 1. This plane is the upper surface of a homogeneous dielectric layer (substrate) of thickness h and relative dielectric permittivity ε . The graphene strips are assumed infinite along the z-axis and have the width d . The H-polarized plane wave is incident at the angle α with respect to the x-axis and depends on time as e − i ω t . FIGURE 1Open in figure viewerPowerPoint Infinite flat graphene-strip grating laying on a dielectric substrate and illuminated by a plane H-polarized wave (a) and cross-sectional geometry and notations used (b) In the case of the H-polarization, the field components are ( E x , E y , 0 ) and ( 0,0 , H z ) . It is convenient to choose H z as the 'basic' component; we denote it U ( x , y ) . Then the incident field is a plane wave, U i n ( x , y ) = e i k 0 ( cos α x − sin α y ) , y > 0 , (1)where k 0 = ω / c = ω ( ε 0 μ 0 ) 1 / 2 with c being light velocity. The entire field is decomposed into a sum, U t o t = U i n + U ( 1 ) in domain #1 and U t o t = U ( 2,3 ) in domains #2 and #3. Thus, we obtain the following boundary value problem for determining the function U = U ( j ) , j = 1,2,3 : (I) it must satisfy the 2-D Helmholtz equation everywhere outside the strips and the slab interfaces: ( ∇ 2 + k 0 2 ε ( j ) ) U ( j ) ( r → ) = 0 , y ≠ 0 , y ≠ − h , j = 1,2,3 , (2) where we imply ε ( 1 ) = ε ( 3 ) = 1 , ε ( 2 ) = ε . (II) resistive boundary conditions at the graphene strips on the upper interface, which is at r → ∈ M : { y = 0 ; | x + n p | < d / 2 ; n = 0 , ± 1 , ± 2 , ... } , namely, 1 i k 0 ∂ ∂ y [ U i n ( x , y ) + U ( 1 ) ( x , y ) + 1 ε U ( 2 ) ( x , y ) ] | y = 0 = − 2 Z [ U i n ( x , 0 ) + U ( 1 ) ( x , 0 ) − U ( 2 ) ( x , 0 ) ] , (3-a) ∂ ∂ y [ U i n ( x , y ) + U ( 1 ) ( x , y ) − 1 ε U ( 2 ) ( x , y ) ] | y = 0 = 0 , (3-b) and transparent boundary conditions at the slots, which are at r → ∈ S : { y = 0 ; − ∞ < x < + ∞ } \ M , U ( 1 ) ( x , 0 ) + U i n ( x , 0 ) = U ( 2 ) ( x , 0 ) , (4-a) ∂ ∂ y [ U i n ( x , y ) + U ( 1 ) ( x , y ) − 1 ε U ( 2 ) ( x , y ) ] | y = 0 = 0 , (4-b) and similar conditions at the entire lower interface, y = ‒ h, − ∞ < x < + ∞ , U ( 2 ) ( x , − h ) = U ( 3 ) ( x , − h ) , (5-a) ∂ ∂ y [ 1 ε U ( 2 ) ( x , y ) − U ( 3 ) ( x , y ) ] | y = − h = 0 , (5-b) (III) the radiation condition, which means that at y → ± ∞ , the scattered field must contain only the 'outgoing' waves, and (IV) the condition of local finiteness of power: the power stored in any finite space domain D tends to zero if D → 0 ; this condition determines the edge behaviour of the function U: it must tend to zero as a square root of the distance to the strip edges. Conditions (I)–(IV) provide the uniqueness of the solution: if the function U exists, then it is unique. Additionally, the periodicity of the domain M, together with the shape of (1), entails the quasi-periodicity property, U ( x + p , y ) = e − i k 0 p cos α U ( x , y ) , (6)which enables the reduction of the analysis to a single period of the scatterer. 3 DUAL SERIES EQUATION The quasi-periodicity property (6) allows expansion of the unknown field in terms of the Floquet series in each of the domains #1, #2, and #3. On introducing the dimensionless notations, ϕ = 2 π x / p , ψ = 2 π y / p , θ = π d / p , ξ = 2 π h / p , κ = p / λ , (7)we can write these expansions as follows: in the upper half-space, U ( 1 ) ( ϕ , ψ ) = ∑ n = − ∞ + ∞ a n e i ( γ n ψ + β n ϕ ) , ψ > 0 , (8)in the dielectric substrate (domain #2), U ( 2 ) ( ϕ , ψ ) = ∑ n = − ∞ + ∞ ( b n e i γ n s l ψ + c n e − i γ n s l ψ ) e i β n ϕ , 0 > ψ > − ξ , (9)and in the lower half-space (domain #3), U ( 3 ) ( ϕ , ψ ) = ∑ n = − ∞ + ∞ d n e i ( − γ n ψ + β n ϕ ) , ψ < − ξ , (10)where unknown coefficients a n , b n , c n , d n are the amplitudes of the Floquet harmonics, and other notations are γ n = ( κ 2 − β n 2 ) 1 / 2 , γ n s l = ( κ 2 ε − β n 2 ) 1 / 2 , β n = n − β 0 , γ 0 = κ sin α , β 0 = κ cos α , (11) The reflectance and transmittance are the power fractions taken from the slab with grating to the upper and lower half-space, respectively. They are expressed via the Floquet harmonic amplitudes as P r e f = γ 0 − 1 ∑ | n − κ cos β | < κ γ n | a n | 2 , P t r = γ 0 − 1 ∑ | n − κ cos β | < κ γ n | d n | 2 . (12) The power absorbed in the metasurface can be found directly, as in [14, 21], or using the power conservation law, P a b s = 1 − P r e f − P t r (13) Substituting (9)–(10) into the conditions of (5), we obtain { ∑ n = − ∞ ∞ ( b n e − i γ n s l ξ + c n e i γ n s l ξ ) e i β n ϕ = ∑ n = − ∞ ∞ d n e i γ n ξ e i β n ϕ 1 ε ∑ n = − ∞ ∞ ( i γ n s l b n e − i γ n s l ξ − i γ n s l c n e i γ n s l ξ ) e i β n ϕ = ∑ n = − ∞ ∞ − i γ n d n e i γ n ξ e i β n ϕ ∑ n = − ∞ ∞ − i γ n d n e i γ n h + i β n ϕ (14) Since these series coincide over the entire period, we replace them with termwise equations and express the unknowns b n and c n in terms of d n : b n = 1 2 d n e i γ n ξ ( 1 − ε γ n s l γ n ) e i γ n s l ξ , c n = 1 2 d n e i γ n ξ ( 1 + ε γ n s l γ n ) e − i γ n s l ξ (15) According to graphene conditions (3-a) and (3-b) at the strips, for | ϕ | < θ , 1 κ ( − κ sin α e i β 0 ϕ + ∑ n = − ∞ ∞ a n γ n e i β n ϕ + 1 ε ∑ n = − ∞ ∞ ( γ n s l b n − γ n s l c n ) e i β n ϕ ) = 2 Z ( ∑ n = − ∞ ∞ ( b n + c n ) e i β n ϕ − e i β 0 ϕ − ∑ n = − ∞ ∞ a n e i β n ϕ ) , (16) − κ sin α e i β 0 ϕ + ∑ n = − ∞ ∞ a n γ n e i β n ϕ = 1 ε ∑ n = − ∞ ∞ ( γ n s l b n − γ n s l c n ) e i β n ϕ (17)at the slots, r → ∈ S , conditions (4-a) and (4-b) yield, for θ < | ϕ | < π , e i β 0 ϕ + ∑ n = − ∞ ∞ a n e i β n ϕ = ∑ n = − ∞ ∞ ( b n + c n ) e i β n ϕ (18) − κ sin α e i β 0 ϕ + ∑ n = − ∞ ∞ a n γ n e i β n ϕ = 1 ε ∑ n = − ∞ ∞ ( γ n s l b n − γ n s l c n ) e i β n ϕ (19) Thanks to (3-b) and (4-b), Equation (17) is satisfied over the entire period. Therefore, on substituting b n and c n from (15) and introducing new coefficients (n = 0,±1,…), x n = ( − δ n , 0 κ sin α + γ n a n ) ( Γ n ) − 1 + 2 δ n , 0 , (20) Γ n = [ 1 γ n − ε γ n s l ( γ n s l − γ n ε ) e 2 i γ n s l ξ + ( γ n s l + γ n ε ) ( γ n s l − γ n ε ) e 2 i γ n s l ξ − ( γ n s l + γ n ε ) ] − 1 , (21)we arrive at the expression that links d n and x n : d n = ( x n − 2 δ n , 0 ) Γ n ε e i γ n ξ [ i γ n s l sin ( γ n s l ξ ) + γ n ε cos ( γ n s l ξ ) ] − 1 . (22) Note that if | n | → ∞ , the weight function in (21) behaves as Γ n = i | n | 1 + ε [ 1 + O ( κ cos α | n | ) + O ( κ 2 n 2 ) + O ( e − | n | 2 π h / p ) ] (23) 4 REGULARIZATION OF DUAL SERIES EQUATION To achieve analytical regularization, we introduce the function Δ n ( κ , ε , h / p , α , Z ) = | n | + i ( 1 + ε ) Γ n + i ( 1 + ε ) κ Z (24)and using the expressions (16)–(22), the following is the dual series equation (DSE) for the unknown coefficients, x n : { ∑ n = − ∞ ∞ x n | n | e i n ϕ = ∑ n = − ∞ ∞ x n Δ n e i n ϕ − i ( 1 + ε ) 2 Г 0 , θ < | ϕ | ≤ π , ∑ n = − ∞ ∞ x n e i n ϕ = 0 , | ϕ | < θ , (25) It can be verified that if all Δ n = 0 , then (25) forms the RHP on an arc of the unit circle in the complex plane. This problem has an analytical solution expressed via the Plemelj–Sokhotski formulas, as explained, for instance, in [18, 22, 23]. Note that when building this solution, the edge condition (IV) is used explicitly. If this procedure is applied to the full DSE (25), it yields an infinite matrix equation, x m = ∑ n = − ∞ ∞ A m n x n + B m , m = 0 , ± 1 , ± 2 , … , (26) A m n = Δ n T m n ( θ ) , B m = − i ( 1 + ε ) 2 Г 0 T m 0 ( θ ) , (27)here, the functions T m n ( θ ) are expressed via the Legendre polynomials P m of the argument u = − cos θ (see [18, 23]), namely, T m n ( θ ) = ( − 1 ) m + n 2 ( m − n ) [ P m ( u ) P n − 1 ( u ) − P m − 1 ( u ) P n ( u ) ] , m ≠ n , (28) T 00 ( θ ) = − ln 1 2 ( 1 + cos θ ) , (29) T m m ( θ ) = 1 2 | m | [ 1 + ∑ s = 1 | m | t s ( u ) P s − 1 ( u ) ] , m ≠ 0 , (30)where t 0 = 1 , t 1 ( u ) = − u , and t s ( u ) = P s ( u ) − 2 u P s − 1 ( u ) + P s − 2 ( u ) . The large-index asymptotics of the Legendre polynomials enable one to see that the following infinite sums are bounded: ∑ m , n = − ∞ + ∞ | A mn | 2 < ∞ , ∑ m = − ∞ + ∞ | B m | 2 < ∞ (31) This is exactly what is needed to state that Equation (26) is a Fredholm second-kind matrix equation in the space of number sequences l 2 . Hence, the convergence of its numerical solution for progressively larger truncation numbers N is mathematically guaranteed. In [22], the inverted part of DSE was slightly different: namely, it involved the weight | n | + c o n s t instead of | n | in (25). This provided slightly faster convergence; however, it led to the Legendre functions of a complex-valued frequency-dependent index. In contrast, expressions (27) are combinations of elementary functions. In addition, they need no numerical integrations and hence can be easily computed with machine precision. This is an important advantage with respect to the other MAR-like techniques, such as MAR-Galerkin in the spatial or Fourier-transform domains [9, 12, 14, 19-21]. Inspection of (23), (24), and (27) shows that both Δ n and A m n contain the terms proportional to the normalized frequency, κ = p / λ , and the terms proportional to e − | n | 2 π h / p . This means that the regularization, which is the semi-inversion of DSE, is performed via the analytical inversion of the static limit of the part corresponding to the strip grating on the interface between two media, air and dielectric. As a result, both the existence of the finite substrate thickness and finite conductivity of strips must shift the 'threshold' value of the matrix truncation number, after which the error starts descending to larger values than in the case of suspended PEC strips: N t h ≈ κ [ 1 + h ε 1 / 2 / p + ( 1 + ε ) | Z | ] . 5 CONVERGENCE AND VALIDATION To visualize the rate of convergence of the numerical solution, we compute the relative error, in the l 2 -norm, of the solution found with varying truncation order N compared with N = 400 and defined as follows: e x ( N ) = ( ∑ n = − 400 400 | x n N − x n 400 | 2 ) 1 / 2 ( ∑ n = − 400 400 | x n 400 | 2 ) − 1 / 2 (32) The results in Figure 2a correspond to the normal and inclined incidence, α = 90 o and 45°, on the grating with p = 70 μ m , d = 14 μ m, h = 10 μ m placed on the substrate with relative dielectric permittivity values ε = 2.25 , 5 , and 12 ; the frequency is 5 THz, which means κ = 1.16 . The graphene parameters are T = 300 K , μ c = 0.39 eV , and τ = 1 ps , which results in the relative surface impedance Z = 0.06 − i 1.81 . FIGURE 2Open in figure viewerPowerPoint (a) The error in the computation of the Floquet harmonic amplitudes using (26-26)–(30-30) versus the matrix truncation order for the grating with parameters indicated in the inset. (b) The error in the computation of the reflectance. The frequency is f = 5 THz , and the graphene impedance is Z = 0.06 − i 1.81 This value can be considered the near-field error. As shown in Figure 2a, it begins nearly exponential decay as soon as N becomes larger than N t h ≈ 20 − 30 , as explained in the previous section. In the analysis of plane wave scattering from gratings, normally the phenomena of reflection, transmission, and absorption in terms of power fractions are of primary interest. Therefore, we define and compute the far-field error as a function of N, e P ( N ) = | P N − P 400 | / P 400 , (33)where P is transmittance, reflectance, or absorbance; see (12). As seen in Figure 2b, with an increase in N over N t h , the error (33) starts decreasing similarly to near-field error; however, the value of that error is one to two orders smaller than for (32). As shown, the rate of convergence is the highest in the case of absence of the dielectric layer and normal incidence, while thinner and optically denser slabs entail larger values of N to achieve the same accuracy. In contrast, the filling factor, d / p , does not change N t h or the rate of convergence. Finally, as proof of validation, in Figure 3 we present a comparison of our results with those in fig. 3a of [34], computed by a conventional MoM code and a MAR-Galerkin with only one weighted Chebyshev polynomial approximating the strip current. Here, the absolute value of the zeroth-order Floquet harmonic is shown versus the filling factor, d/p, at the normalized frequency κ = 0.5 for resistive-strip grating with Z = 100 Ohm on dielectric substrate with ε = 2 and 4 and h = p / 5 illuminated by the H-polarized plane wave incident at α = 60 o . The PEC-strip case (Z = 0) is also shown for comparison. The corresponding curves visually overlap; the small discrepancy from MAR-Galerkin is explained by the low order of the latter and vanishes if d / p ≤ 0.3 . FIGURE 3Open in figure viewerPowerPoint Comparison of the results of figure. 3a of [34] and MAR-RHP using (26-26)–(30-30). Reflectance of resistive-strip grating with Z = 100 Ohm on dielectric substrate versus the ratio d/p at κ = 0.5 , h / p = 0.2 , and ε = 2 and 4. The plots for the PEC strip array in the free space and on the same substrate are also shown 6 RAYLEIGH ANOMALIES AND NATURAL MODES 6.1 Rayleigh anomalies Rayleigh anomalies (RAs) are associated with the branch points of the field U as a function of the frequency at γ m = 0 ; the existence of these branch points is the consequence of our assumption that the grating is infinite and the use of the Floquet series (8) and (10). Note that the frequencies at which γ m s l = 0 are not the branch poi
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