## 1. Introduction

[2] Wave characterization and modeling for arrays with a large number of elements have been subject of various studies in the recent years. Assuming the structure as infinite when using a full-wave model, though simple and efficient it may not be satisfactory when array-truncation effects are relevant. Owing to the localization of the truncation-induced diffracted waves for large arrays, many physical insights of the wave processes can be extracted from a canonical problem such as a semi-infinite array of impressed electric line sources or dipoles [*Kildal*, 1984; *Carin and Felsen*, 1993; *Felsen and Carin*, 1994; *Capolino et al.*, 1998, 2000a, 2000b]. In the present paper, we deal with the exact analysis of a semi-infinite array of narrow perfectly conducting strips illuminated by a plane wave. The same structure was analyzed by *Carin and Felsen* [1993] with a hybrid (ray) (Floquet) (MoM) efficient formulation; however, truncation field effects were accounted for by using a Kirchhoff approximation; i.e., the current on the strips were assumed as those of the array assumed as infinite. Successively, truncation effects were numerically refined by *Neto et al.* [2000a, 2000b], *Çivi et al.* [2000], and *Craeye et al.* [2004] using a MoM with basis functions shaped as truncation-induced diffracted fields.

[3] Here, the problem of a semi-infinite array of narrow perfectly conducting strips is solved rigorously by using a discrete Wiener-Hopf technique which has the advantage of being both exact and analytically explicit. This electromagnetic problem reduces to the scalar case where the Dirichlet boundary condition is imposed on the strips. The Wiener-Hopf method is a well established technique. Electromagnetic or acoustic problems usually involve branch point singularities, and the general formalism is given by *Noble* [1958], *Kobayashi* [1990], and *Jones* [1964] where the method was applied in the classical spectral wave number along a cartesian axis. In other fields, like digital signal processing or control theory, the Wiener-Hopf method is combined with the Z transform of successions of samples but they do not deal with branch point singularities because the spectral kernels to be factorized are usually rational. There are a few papers where the Wiener-Hopf method was applied in the Z-transformed domain to solve diffraction problems with a discrete set of scatterers, thus involving branch points singularities [*Fel'd*, 1958; *Hills and Karp*, 1965; *Koughnett*, 1970; *Wasylkiwskyj*, 1973; *Linton and Martin*, 2004].

[4] In the work of *Fel'd* [1958] the problem was formulated for a semi-infinite array of cylinders (as in the work of *Hills and Karp* [1965]) by using the Z transform and the factorization method [*Fel'd*, 1958, equation (17)], but, as the authors themselves admit, their formulation leads to complicated integrals and the problem is instead solved by a variational principle.

[5] The discrete Wiener-Hopf technique was applied in the remarkable paper by *Hills and Karp* [1965] to treat a semi-infinite grating of small cylinders under the hypothesis of electrically large interelement spacing (*d* ≫ *λ*, with *d* the array period and *λ* the free space wavelength).

[6] In the work of *Koughnett* [1970] a beautiful analysis in terms of Z-transformed quantities was given for an array of dipole antennas, and a formal solution was provided as the sum of factorized terms that, however, were not explicitly evaluated. Only the case with mutual couplings set to zero after a certain distance was numerically solved. In this way the difficulty arising from treating branch point singularities was avoided. In general, truncation effects vanish after a certain number of array elements but there are important cases where truncation effects may extend over a large portion of the array; e.g., when the period is much smaller than the wavelength or when near “resonance” conditions occur.

[7] In the work of *Wasylkiwskyj* [1973] the Z domain Wiener-Hopf method was extensively explained but the factorization was not performed in the way shown in this paper, though various formulas and simplifications were there provided. Furthermore, the aim of that paper was about showing truncation effects in input parameters for arrays of minimum-scattering dipole antennas [*Wasylkiwskyj and Kahn*, 1970], and not about diffractive effects arising form the array truncation. No high-frequency concepts were discussed by *Wasylkiwskyj* [1973].

[8] In the work of *Nishimoto and Ikuno* [1999] a strip grating as in this paper was analyzed, in contrast to gratings of small cylinders as in the works of *Fel'd* [1958] and *Hills and Karp* [1965]. There, the problem was not solved with the Wiener-Hopf method but some interesting properties of the diffracted current were shown.

[9] In the work of *Linton and Martin* [2004] the Z domain Wiener-Hopf method was shown to solve semi-infinite arrays made of strips and of cylindrical scatterers, with Dirichlet boundary condition (soft case). There, diffraction effects were extensively discussed, but the analysis is different than that in our paper though there are similarities.

[10] In summary in the works of *Fel'd* [1958], *Koughnett* [1970], and *Wasylkiwskyj* [1973], no diffraction effects and high-frequency concepts are emphasized as in the works of *Hills and Karp* [1965], *Nishimoto and Ikuno* [1999], *Linton and Martin* [2004], as well as in the present paper.

[11] In our formulation many new aspects are introduced compared to *Hills and Karp* [1965] and the other cited papers: (1) Arbitrary basis functions can be chosen to shape the current on the strip elements also permitting to treat both TE and TM polarizations, with respect to the direction of the strips, though in this paper we limit our analysis to the TM case; (2) The discrete Wiener-Hopf is implemented using a *Z* transform of the sampled distribution of currents. Here, the topology (critical points) of the *z* spectral plane is shown explicitly and discussed in details for a general semi-infinite structure. A complete correspondence between the discrete *z* spectral representation and the standard continuous plane wave *k*_{x} spectral wave number representation (the semi-infinite array is periodic along *x*) is established; (3) Solutions for any interelement spacing *d* as well as for the two limit cases of large (*d* ≫ *λ*) and small (*d* ≪ *λ*) *d*, compared to the wavelength *λ*, are automatically obtained from our formulation; (4) We provide an approximate factorization that is useful to approximate the currents on the strips and to numerically perform the exact factorization; (5) Asymptotics is performed via path deformation and SDP (steepest descent path) evaluation directly in the *z* plane (showing the correspondence to the *k*_{x} plane); (6) Asymptotic results are performed in a totally uniform fashion for the current on the strips. The diffracted current is found asymptotically as a sum of Floquet waves (solution for the infinite array) plus a diffracted current arising from the truncation of the array; (7) The general spreading factor *n*^{−3/2} of the diffracted terms versus strip number *n* is obtained and discussed also showing its range of validity.

[12] Our solution is simple to use and offers a net physical insight into the behavior of the current in truncated periodic structures. We emphasize that our closed form solution can be directly applied to finite array of strips as long as the current diffracted at one edge of the array does not significantly couple with the other edge of the array.