Fundamental properties of the field at the interface between air and a periodic artificial material excited by a line source

An efficient algorithm based on a moment-method formulation is presented for the evaluation of the field produced by a line source at the interface between an air superstrate and a one-dimensional-periodic artificial-material slab. The formulation provides physical insight into the nature of the fields via path deformation in the complex wavenumber plane. From an asymptotic analysis in the complex wavenumber plane it is found that the space wave produced by a line source consists of an infinite number of space harmonics that decay algebraically as x/sup -3/2/. Guided modes may also exist and be excited, including leaky modes.

Abstract-An efficient algorithm based on a moment-method formulation is presented for the evaluation of the field produced by a line source at the interface between an air superstrate and a one-dimensional-periodic artificial-material slab. The formulation provides physical insight into the nature of the fields via path deformation in the complex wavenumber plane. From an asymptotic analysis in the complex wavenumber plane it is found that the space wave produced by a line source consists of an infinite number of space harmonics that decay algebraically as 3 2 . Guided modes may also exist and be excited, including leaky modes.

I. INTRODUCTION
P ERIODIC artificial surfaces and materials such as electromagnetic bandgap (EBG) structures [1], artificial magnetic conductors [2] and artificially soft surfaces [3] have been used recently to modify the radiation pattern and other characteristics of sources located near or within them. For example, artificial EBG materials have been used to suppress surface-wave propagation on dielectric substrates [2], [4], [5]. Artificial surfaces and materials have also been used to obtain highly directive antenna patterns in the microwave and millimeter-wave ranges [6], [7]. Artificially soft surfaces have found use in several fields, including applications that require the attenuation of the spatial field produced by a source along an interface [8].
In the present investigation an efficient numerical scheme for evaluating the field produced by a line source above an artificial material (or any other periodic structure) that is periodic in one-dimension (1-D) is first examined. For simplicity, a twodimensional (2-D) problem is considered (see Fig. 1), which is invariant along the dimension, with a 1-D-periodicity along (in the following, a material periodic in one dimension is called 1-D-periodic material.) An extension of the method to 2-D-periodic structures, periodic along and , is possible, but is not considered here. The focus is then placed on some fundamental properties pertaining to the nature of the field along the interface between air and the material. The results are directly applicable Manuscript  to determining the coupling between sources located in proximity of a periodic artificial structure, such as a wire medium of finite thickness. The periodic artificial material consists of a periodic (along ) structure made of layers of conducting strips or cylinders, with period . A finite number of layers may be stacked along to form an artificial material slab with a finite thickness as shown in Fig. 1(a), or there may be a single layer of elements, as for the corrugated structure of Fig. 1(b) or the strip grating of Fig. 1(c). An electric line source in the direction (parallel to the periodic elements) is either placed inside or outside the artificial material, at . (The method could be extended to treat the case of a dipole excitation near the 1-D periodic structure, but this is not considered here.) The problem is thus one of transverse electric (TE) (to ) polarization. Although transverse magnetic (TM) polarization could be treated in a similar fashion, the TE case has been selected here for two reasons: first, for consistency with the EBG material consisting of the wire medium in Fig. 1(a), since the TE polarization is most affected by the presence of the wires. Second, to numerically isolate and study the space wave fields on the structure, which is easier in the absence of guided modes. Indeed, as shown in [9], for TM polarization, periodic structures such as the corrugated structure have propagating modes for low frequencies since the structure behaves as an inductive surface, with mode suppression occurring when the depth of the teeth is approximately a quarter wavelength (the structure acts as an artificially soft surface at this point). For the TE case it is possible to obtain a modeless structure, so that the total field excited by the source is the same as the spatial (space-wave) field.
An efficient field evaluation is obtained in Sections II and III using the "array scanning method" [10]- [12]. To improve the computational efficiency of the method, a 2-D Ewald acceleration scheme [13], [14] is used to improve the convergence of the periodic free-space Green's function. The resulting field from the line source then has a representation in the form of an integral in (over the Brillouin zone). In Section IV an alternative representation of the field from the line source is obtained by "unfolding" the integration over the Brillouin zone onto the entire real axis in the plane. This allows for a convenient path deformation to enclose any singularities in the complex wavenumber plane, including pole and branch-point singularities. In Section V it is shown how the complex wavenumber plane for such problems has an infinite number of periodically spaced branch points, and also a periodic set of poles (assuming that a guided mode exists). (This was anticipated in [15] and demonstrated in [16] for a specific type In all cases, a denotes the periodicity along x. (a) The material is an infinite periodic array of metallic cylinders with period a that is truncated in the negative z direction after a finite number of layers. In the figure, the periodic supercell (i.e., unit cell with multiple conductors) n = 2 is highlighted. The source is located in the n = 0 supercell [arrows point to three of the conductors within the n = 2 supercell in Fig. 1(a)]. S denotes the surface of the conductors in the nth supercell. The volumetric region of the nth supercell is denoted by V . (b) A corrugated surface consisting of metallic teeth above a ground plane. The thickness of the teeth is h. (c) A metallic strip grating. Also shown is an expanded view of one of the strips, with a sketch of the current shape for a small strip width w. of structure. An infinite number of periodically spaced branch points has also been found in a similar problem [17] where the waves arising at the truncation of a periodic set of metallic strips have been rigorously analyzed.) The residue evaluations at the set of poles yields the modal amplitudes of the Floquet harmonics of the guided mode (if any) on the periodic structure, while the branch points determine the space-wave field radiated by the line source. In Section VI a structure consisting of a periodic arrangement of metallic strips is used as an example, since closed-form expressions for the integrand are available in the case of narrow strips. Asymptotic evaluations involving path deformations into steepest-descent paths are used to determine the field behavior on the interface with increasing distance from the source. In Section VII, it is shown that the general conclusions are valid for a line-source excitation of any artificial material structure comprising a periodic arrangement of conducting objects that are invariant in the direction, the structure being infinite and periodic in the direction with a finite extent in the direction. In Section VIII, results are presented for the structures of Fig. 1 to confirm the validity of the conclusions.

II. THE ARRAY SCANNING METHOD
The array scanning method (ASM) (as the method was called in [11], though it had seen previous use, e.g., in [12]) is an analytic procedure that synthesizes the field from a single source in terms of a spectral wavenumber integration over a phased array of sources, as shown in Fig. 2. Therefore, a convenient numerical evaluation of the aperiodic (single source) excitation of an infinite periodic structure such as the EBG material slab in Fig. 1 can be obtained using the ASM. The first step is to note the following relation between an infinite periodic array of impressed linearly-phased line sources with currents directed along , and the corresponding single line source (1) where is an impressed wavenumber along . The single line source is thus synthesized from the periodic phased array of line sources spaced along the axis by integrating in the wavenumber variable over the Brillouin zone. The electric field at any point produced by the periodic array of phased line sources in free space (the field that is incident on the periodic structure from the phased array of sources) is denoted as (2) where (3) is the periodic Green function for the magnetic vector potential component produced by the phased array of line sources, in which (4) are the Floquet mode wavenumbers along and , respectively, with the homogeneous-space ambient wavenumber. There are an infinite number of branch points in the plane, located at The th branch point corresponds to the square root involved in that appears in . The top Riemann sheet of the plane for the branch point is defined as . The field produced by the periodic phased array of line sources near the EBG slab is denoted as . By the same weighted superposition used in (1), the electric field produced by the single source in that periodic environment is then given by (6) The calculation of , which involves the periodic moment method, is discussed in the next section.

III. FIELD PRODUCED BY A LINE-SOURCE ABOVE A PERIODIC MATERIAL
The electric field in Fig. 1 is polarized along the direction, since there is no variation along the axis. For simplicity, we consider here only metallic scatterers, e.g., as those shown in Fig. 1. We denote by and the surface current in the direction on the metallic conductors and the electric field directed along at any point, respectively.
The current on the surface of the conductors (posts) within the supercell due to the phased array of line sources is found by solving the EFIE (7) for , where the periodic Green's function is accelerated using the 2-D Ewald method [2], [4]. Note that is a periodic function of with period . The electric field that is scattered by the periodic structure from the phased array of line sources is determined by integrating over the post currents as (8) with the integral performed over the post currents within the unit supercell by using the periodic Green's function . Note that is also a periodic function of with period . The scattered field in the th supercell from the single line source is then found from the field within the zeroth supercell in the phased-array problem as (9) where . The total field is obtained by adding the scattered field (9) to the incident field produced by the line source, It has been observed that the integrand in (9) has a branch point singular behavior at that may result in a numerical inefficiency in the numerical integration of (9). (There are an infinite number of branch points, as seen from (5), although only these two branch points are encountered for many practical situations, where .) To overcome this difficulty, the total electric field in (10) could alternatively be obtained by representing the incident electric field in terms of its spectral representation (11) with given in (2). The total electric field (10) is thus expressed as (12) where (13) While the integrand in (9) at possesses a singularity behavior of the type , the integrand in (12) instead contains weaker branch point singularities of the type . (Physically, this corresponds to the fact that the total spatial field along the interface decays faster than does the scattered field alone.) These features are established in Appendix A and Section VII. Because the integrand in (12) is less singular than that in (9), the integration requires fewer-integration points near the branch point singularities at .

IV. UNFOLDING THE INTEGRATION PATH
The integrand in (9) is a periodic function of with period . Indeed, is periodic because is excited by a periodic (in ) phased array of line sources. After inserting (8) into (9), and using the explicit form of the Green's function in (3), (8) is written as (14) Since the term is periodic in , applying the shift of variables for every term of the sum leads to (15) which eliminates the sum and expresses the scattered field as a continuous integration over the entire axis, physically corresponding to a continuous-spectrum plane wave expansion of the scattered field.

V. THE COMPLEX PLANE AND FIELD REPRESENTATION
In addition to the two branch-point singularities introduced by the term in (15), the periodic function introduces a periodic set of branch-point singularities. Furthermore, this function may also exhibit a periodic set of poles, each one representing modal propagation along . The branch point singularities at in (5) of the spectral function arise from the periodic Green's function in (7) and are shown in Fig. 3. This figure also shows a possible set of periodic pole singularities, representing a leaky mode on the structure (with a complex wavenumber). Complex poles are located symmetrically with respect to both the real and imaginary axes, though only one set of poles is shown here for simplicity (the set that is shown corresponds to a physical leaky mode in the fourth quadrant of the fundamental Brillouin zone). If the mode is a physical leaky mode radiating in the forward direction, then it is on the improper sheet with respect to its nearest branch point, and on the top sheet of all other branch points. This corresponds to a mode for which all of the space harmonics (Floquet waves) of the guided mode on the structure are proper (decaying vertically) except for the one that is a fast wave, i.e., that with wavenumber smaller than . If the mode is a physical leaky mode radiating into the backward region, then all of the poles are on the top sheet of all the branch points. (In this case the pole located in the fundamental Brillouin zone would have a negative phase constant.) As shown in Fig. 3, the original integration path on the real axis can be deformed around the spectral singular points to highlight the space-wave and modal contributions. When evaluating the total field, the path deformation leads to the representation (16) where the modal field arises from the reside evaluations at the periodic pole locations, with the residue at each location determining the amplitude of the corresponding Floquet mode contribution to the guided leaky mode. The space-wave field arises from the evaluation of the integral around each branch point.
In the case of , the vertical paths shown in the figure are the steepest descent paths. One can infer that the space wave arises from all of the branch points, and consists of an infinite number of space harmonics.
From an asymptotic evaluation of the spectral integral carried out in Appendix B, it is seen that each space harmonic that is part of the space-wave field has a spreading factor along the interface. The remaining spatial integral in (15) determines the weight of each decaying spatial harmonic.

VI. CANONICAL EXAMPLE: STRIP GRATING IN FREE SPACE
Some properties derived from the above discussion are illustrated for the simple case of a single-layer periodic structure consisting of an infinite periodic arrangement of narrow conducting strips located at and excited by an electric line source at . We assume a fixed current distribution on each strip, proportional to the basis function defined about the center of each strip. This is a good approximation when , with the free-space wavelength. This simple case allows for an analytic solution for the strip current in the 0th unit cell when the structure is illuminated by the phased array of line sources, as (17) with (18) where the Bessel function of zeroth order, , is the Fourier transform of the basis function. From this current representation it is immediate to see that the infinite periodic arrangement of branch points is as shown in Fig. 3. The expression for the scattered field in (15) is represented as (19) with (20) It can be shown (see also the general discussion in Section VII) that near the branch point at and the current function in (20) behaves as As detailed in Appendix B, the dominant term arising from the constant at the branch yields the field with , which exactly cancels the incident field at any observation point . At the higher-order branch points with the current in (20) behaves as where .
As shown in Appendix B, the higher-order asymptotic contribution arising from the term of (21) at the branch point, as well as the dominant contributions from the other branch points in each th region (see Fig. 3) provides a spatial wave that varies along the interface as (23) with propagation wavenumbers for defined in (5), and being coefficients that are defined in Appendix B. Hence, the space wave along the periodic artificial material interface decays algebraically as , and consists of an infinite number of space harmonics.

VII. ASYMPTOTIC BEHAVIOR OF THE SPATIAL WAVE AT THE INTERFACE
The properties observed above for the simple analytical canonical problem of the conducting strip grating are here generalized to structures as those in Fig. 1(a) and (b). For observation points sufficiently away from the source, and along the air interface of the periodic material, i.e., for , an asymptotic evaluation based on the steps reported in Appendix B is carried out for the general case involving the radiation integral in (15). To this end, the integral (15) is rewritten as (24) where is now the 2-D Fourier transform of the post current with transform variables , as defined in (29). In order to factorize the observer and the source terms in (24) it has been assumed for simplicity that the observation point is slightly above the periodic material, i.e., for all . Once the definition of is used in (7) it is possible to observe that for (note that (21) assumes that ). This follows from substituting (3) into (7), multiplying both sides by , and then taking the limit as approaches . The branch points at [see (5)] appear in the higher-order expansion terms of near , as already seen for the strip grating case. By using the same argument as in Section III, once the incident field is included in (24), an expansion of the total field becomes (assuming here that ) An asymptotic evaluation of the integral is carried out by deforming the original integration path along the real axis onto the infinite number of vertical path contributions shown in Fig. 3. In the case that the source and observation points have approximately the same z-coordinate, i.e., , and , the paths coincide with the steepest-descent paths passing through . Note that asymptotically, the dominant contribution at vanishes because . An asymptotic evaluation of the higher order term at and of the dominant contributions at , with , as shown in Appendix B leads to a general representation of the spatial field in terms of an infinite number of space harmonics with a spreading factor as in (23), where the propagation wavenumbers are given in (5). Thus, the general form of the spatial field excited by a line source over a general periodic structure is observed to be the same as (23) for the canonical strip grating structure. In summary, from this asymptotic evaluation it is clear that the spatial field contribution in and decays as and , respectively. This is evidently the first time that such a conclusion has been reached for a line source over a general periodic structure. This property is expected to be relevant for the estimation of the coupling between two sources near a 1-D periodic artificial material. The conclusion should remain valid as the period tends to zero, in which case the periodic artificial material slab approaches a homogeneous artificial slab (e.g., a metamaterial slab with a negative permittivity).

VIII. NUMERICAL EXAMPLES
A first example is shown in Figs. 4-6, where an electric line source is placed over an artificial material EBG slab consisting of three layers of periodic conducting cylinders with normalized radius in free space. The axes of the cylinders in the first row are located at . The source is located at . In the MoM calculations each cylinder has been discretized using 16 sub-domain pulse basis functions. In Fig. 4, the operating frequency corresponds to and is thus in the 0th band gap of the infinite EBG material [18]. The total field (normalized by multiplying with the period ) is plotted versus the distance from the line source parallel to the EBG interface, at points and , with denoting the supercell index. The total field is obtained by adding the scattered field in (9) to the incident field. In Fig. 4, it is seen that the total field is dominated by the space wave, and exhibits the expected algebraic decay of the space wave at both observer locations. (The curves are normalized to the exact curves for large .) This indicates the absence of guided modes for this structure at this particular frequency.
In Fig. 5, the field is evaluated along the interface for various frequencies, and the decay is compared with the algebraic decay normalized to the exact fields for large . These numerical experiments indicate that the field at large distance behaves like (26) Fig. 4. Spatial decay of the total field produced by a line source over the periodic EBG material of Fig. 1(a) made of 3 layers of periodic conducting cylinders. The field is evaluated at points r and r where n denotes the supercell index. The fields match well with a simple 1=n factor (normalized to the exact fields for large n).  Fig. 4, the field is evaluated at points r , where n denotes the supercell index, for various frequencies. The fields match well with a simple 1=n factor (which is normalized to the exact fields for large n) for the two lower frequencies. Fig. 6. For the same geometry of Fig. 5, the normalized weighting coefficient aw(r; r ) (in decibels relative to 1 V) of the space wave is plotted versus normalized frequency.
where denotes the observation point within the supercell. The weights are reported for various frequencies in Fig. 6. Although the weight expression (27) could be derived from (23), here it has been determined by simply matching the exact field with the fitting curve for large .
Note that is at the edge of the passband [18] where the material approximately behave like an artificial material with a zero permittivity [19].
At higher frequencies, such as , a leaky mode is propagating along the interface as can be seen from the interference between the space wave and the leaky wave in Fig. 5 (the interference subsides for larger distances, due to the exponential decay of the leaky mode). From a numerical search in the complex plane, it has been found that the wavenumber of the leaky wave pole (corresponding to the pole location in the zeroth Brillouin zone) is approximately The above phase and attenuation constants correlate well with the "subtracted field" on the interface that is obtained after the asymptotic spatial field is subtracted from the total field (the results are omitted here). The subtracted field exhibits an exponential decay, as expected. As before, the spatial field decays as . As a second example, in Fig. 7 the total field is evaluated along the interface of the corrugated structure shown in Fig. 1(b) with cm, cm, for various frequencies. The source is located at cm and the field is observed along the interface at locations (in cm) . The MoM calculations are performed discretizing the unit-cell tooth into 10 subdomain pulse basis functions and using image theory to account for the ground plane. For this geometry the space wave once again exhibits the expected algebraic decay for all the frequencies examined. Note that at GHz the teeth height is a quarter-wavelength in free space, which is the condition to realize an artificially soft surface [3], [8]. The frequency GHz is the cutoff frequency for the TE polarization analyzed here to propagate into the teeth region as a waveguide mode. At GHz the teeth are such that where is the wavelength of the fundamental TE polarized waveguide mode in the parallel plate waveguide with plate separation cm. These numerical experiments indicate that the field at large distance behaves like (26) with the weight reported for various frequencies in Fig. 8. Expression (26) coincides with (23) when the spatial harmonics are summed. It is worth noting that for GHz, the total radiated field still exhibits a spatial decay, in contrast to a decay expected at the interface between air and a PMC, due to the presence of the conducting teeth.
The above results also verify that for TE polarization, the corrugated structure does not support surface-wave (bound) guided modes. This can be explained by the fact that the interface acts as a capacitive reactance for frequencies below 40.36 GHz, so that modal propagation of surface-wave modes is prohibited. Also, above 37.5 GHz the periodicity is greater that a half wavelength, so that any guided mode would be a leaky mode. Hence, propagation of surface-wave modes is prohibited in all frequency regions. Fig. 7. Spatial decay of the total field produced by a line source over the corrugated surface shown in Fig. 1(b). The field is evaluated at points r where n denotes the supercell index. The field matches well with a simple 1=n factor (normalized to the exact fields for large n). Fig. 8. Weighting coefficient w(r; r ) (in dB relative to 1 V/m) for the space wave representation in (26) versus frequency. Note that the amplitude of the space wave is maximum at approximately the frequency where the depth of the teeth is a quarter of wavelength for the fundamental TE mode of the corresponding parallel plate waveguide defined by the teeth.

IX. CONCLUSION
An efficient algorithm for the evaluation of the field produced by a line source near a periodic slab of artificial material has been derived using an "array-scanning method", which relates the field of a single line source to that produced by a periodic phased array of line sources. The complex wavenumber plane provides insight into the nature of the field produced. The main results from this study are as follows. a) An efficient algorithm for the numerical evaluation of the field produced by a line source on top of the artificial material has been obtained. b) The nature of the complex wavenumber plane and the periodic arrangement of the branch point singularities has been examined. c) Based on the nature of the complex wavenumber plane, it was concluded that the space-wave field on the interface consists of an infinite number of space harmonics, each decaying algebraically as . d) Guided modes (including leaky modes) may also be excited, if these are supported by the structure. The amplitudes of the Floquet waves that make up the guided mode are determined by the residues at the periodic pole locations in the complex wavenumber plane. e) For a physical leaky mode, each pole in the periodic set is located on the lower sheet of the nearest pair of branch points, and on the top sheets of all others, when the mode radiates in the forward direction. When the mode radiates in the backward direction, the poles are located on the top sheet of all branch points. For a surface-wave mode, all of the poles are located on the top sheet of all branch points. (For the polarization and frequencies considered here, there were no surface-wave modes, however.) The decay of the spatial wave has been demonstrated by numerical results, and also by an asymptotic analysis of a canonical problem consisting of a periodic conducting strip grating. This work lays the foundation for further studies involving other surfaces, including artificial magnetic conductors and other EBG materials. A formulation for 2-D periodic structures excited by a dipole source is also possible.

APPENDIX A DETERMINATION OF THE SINGULARITY ORDER
We determine here the order of the singularity of the integrand in (9) at where is defined in (5). We first note that since for , because of (10) has the same singularities as for . Thus, has a periodic set of singularities of the type for . It remains to demonstrate that has also the same type of singularities for other observation points . To show this, for simplicity assume that the observation point is slightly above the periodic material interface, i.e., for all . Then, substituting the explicit spectral form of the periodic Green's function (3) in (8)  , for a different does not change and from (28) we infer that the singularity is still .

APPENDIX B ASYMPTOTIC EVALUATIONS
In this Appendix, we provide the asymptotic evaluation of two important spectral integrals in order to determine the spatial behavior of the fields. Consider the form (31) where and has branch points at , with as given in (5) and shown in Fig. 3. The th term denotes the integration along the th steepest-descent path in Fig. 3. After the path deformation depicted in Fig. 3, the integral is represented as the sum of all the integration paths around the branch points (corresponding to the terms ). The asymptotic evaluation is first performed for the contribution of the path at the branch , with , that renders . At this branch point, and the integral in (31) is written as (32) The term with vanishes, for it does not possess a branch point and the two parts of the corresponding vertical steepest descent path cancel. Asymptotically the integral is further reduced using as The root assumes opposite values on the integration path in the top and bottom Riemann sheets. Next, the change of variables , with is applied and is rewritten as where has been used for the top Riemann sheet. Therefore, is evaluated asymptotically as The root inside the integral assumes opposite values on the integration path in the top and bottom Riemann sheets. Next, the change of variable , with is applied and is rewritten as (38) where has been used on the top Riemann sheet. The remaining integral in (38) is evaluated exactly leading to (39) It follows, therefore, that for the constants in (23) are