306 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 16, 2017
Diffraction Modeling by a Soft–Hard Strip Using
Finite-Difference Time-Domain Method
Alper Uslu, Student Member, IEEE, Gokhan Apaydin, Senior Member, IEEE, and Levent Sevgi, Fellow, IEEE
Abstract—Diffraction by a strip with one face soft and the other
face hard boundary condition is modeled numerically using finite-difference time-domain method, and the results are compared to the method of moments, which was validated against physical the-ory of diffraction.
Index Terms—Diffraction, finite-difference time-domain (FDTD), hard boundary condition, high-frequency asymtotics, method of moments (MoM), scattering, soft boundary condition.
I. INTRODUCTION
I
NTERACTION of electromagnetic (EM) waves with objects generates scattered waves, formed primarily by reflected, re-fracted, and diffracted fields [1]–[21]. High-frequency asymp-totic (HFA) methods, such as geometric optics, physical optics, geometric theory of diffraction, uniform theory of diffraction, and physical theory of diffraction (PTD), have been used to ob-tain these fields for small wavelength compared to object size (see [2]–[7] for excellent tutorials and lists of related references). An HFA-based virtual tool has been developed for diffraction modeling in the canonical wedge problem [8]. Finite-difference time-domain (FDTD)-based diffraction modeling may be found for perfect electric conducting (PEC) and dielectric wedges in [9] and [10]. Another useful MATLAB-based wedge diffraction modeling virtual tool has been introduced in [11], where HFA and FDTD results may be observed comparatively. A two-step method-of-moments (MoM) approach has been introduced in [12] for the generation and discrimination of diffracted waves on the PEC wedges. Fringe waves that are a portion of diffracted fields have been revisited in a couple of publications [13]–[15]. The procedures of the discrimination of diffracted waves from double-edge structures using different numerical models have been given in [16] and [17]. Finally, modeling of backscattering and diffraction from wedge- and strip-type structures with dif-ferent boundary conditions (BC) may be found in [18]–[27]. The theory of diffraction at a PEC strip was developed using infiniteManuscript received January 14, 2016; accepted May 10, 2016. Date of publication May 30, 2016; date of current version February 27, 2017.
A. Uslu is with the Department of Electronics and Communications Engineer-ing, Dogus University, Istanbul 34722, Turkey (e-mail: alpuslu@gmail.com).
G. Apaydin is with the Department of Electrical-Electronics Engineering, Zirve University, Gaziantep 27260, Turkey (e-mail: gokhan.apaydin@zirve. edu.tr).
L. Sevgi is with the Department of Electrical and Electronics Engineering, Okan University, Istanbul 34959, Turkey (e-mail: levent.sevgi@okan.edu.tr).
Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LAWP.2016.2574361
number of multiple diffractions by Ufimtsev [22]. Buyukaksoy and Alkumru studied multiple diffractions at a soft–hard strip by using Wiener–Hopf technique [23]. Scattering by a strip with two different surface impedances has been investigated in [24]– [26]. Ufimtsev used the parabolic equation solution to obtain diffracted fields at a wedge with one face soft and the other face hard BCs in [27].
In this study, time-domain diffracted fields from a canonical two-dimensional (2-D) strip with one face soft (SBC) and the other face hard (HBC) using the FDTD method are obtained for the first time in the literature, and the results are compared against the existing MoM model [20]. The advantage of the proposed model over the existing MoM-based model is that broadband diffracted fields are obtained with a single-run via fast Fourier transform. In addition, visualization of fields in the time domain enables the understanding of diffraction phenomena for this structure. SBC behaves as PEC surface for the tangential components of electric field and as perfect magnetic conducting surface for the normal components of electric field; vice versa for HBC. Soft and hard surfaces are used to control the scattering in various EM applications such as antenna design. It is possible to obtain these surfaces artificially via corrugations that are oriented transversely for soft and longitudinally for hard [28]. Important aspects of soft/hard surface modeling were discussed in the special issue [29].
The letter is organized as follows: First, the problem is postu-lated in Section II. The novel FDTD-based procedure is pre-sented in Section III. Numerical tests and comparisons are given in Section IV. Finally, the conclusions are outlined in Section V.
II. PROBLEMPOSTULATION
The problem geometry is shown in Fig. 1. The cylindrical coordinatesρ, ϕ, z are used. Since the strip is assumed to be infinite alongz-direction, the problem may be reduced to 2-D and can be handled in polar coordinatesρ, ϕ. The width of the strip isL. The origin is at the midpoint of the strip, therefore the strip extends from edge to edge between(0, L/2) and (0, −L/2) on they-axis. The numbers 1 and 2 show top and bottom edges, respectively.
The left/right part of the strip has soft/hard BC. The line source is assumed on(ρ0, ϕ0). The locations of 360 receivers
are placed on the dashed circle with radiusρ. According to the scenario shown in Fig. 1, reflected fields are available between the lineL1and the lineL2. The incident field exists everywhere
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USLU et al.: DIFFRACTION MODELING BY SOFT–HARD STRIP USING FDTD METHOD 307
Fig. 1. Strip with one face SBC and the other face HBC. The left side is soft (u = 0), and the right side is hard (du/dn = 0).
except the shadow region between the line SB1 and the line SB2. The top and bottom edges (numbered 1 and 2 in Fig. 1) are responsible for the diffracted fields that exist everywhere.
We consider here the total and diffracted waves induced on a strip with zero impedance on the left face (u = 0, SBC) and infinite impedance on the right face (∂u/∂n = 0, HBC). The total fieldu (ρ, ϕ) around the strip satisfies the wave equation
∂2 ∂ρ2 + 1 ρ ∂ ∂ρ+ 1 ρ2 ∂2 ∂ϕ2 + k 2 u = I0δ(ρ − ρ0)1 ρδ(ϕ − ϕ0) (1)
and the Sommerfeld’s radiation condition at infinity lim ρ→∞ kρ ∂u ∂ρ − iku = 0 (2)
under a line source illumination atu (ρ0, ϕ0). Here, k is the
wavenumber. Note that∂u/∂n = ∂u/∂x on the right face for the scenario is pictured in Fig. 1. The source position goes to infinity to consider plane wave. Cylindrical wave illumination andexp(−iωt) time dependence are assumed in this letter.
III. FDTD-BASEDDIFFRACTIONMODELING
FDTD is a time-domain numerical method that utilizes the discrete form of Maxwell’s partial differential equations. Its application area spans a wide range of EM problems includ-ing antennas and propagation modelinclud-ing, microwave circuit de-sign, metamaterials, and biomedical applications. FDTD has also been used in scattering and diffraction analysis [9]–[11]. The model shown in Fig. 1 is infinite alongz-direction, hence 2-D TMz polarization (withHx,Hy,Ez components) is used for diffraction modeling. The computation space is discretized in FDTD modeling, therefore only near fields can be simulated, but if necessary (e.g., when antenna radiation patterns or radar cross section of an aircraft is needed), far fields are extrapo-lated using the equivalence principle. Also, abrupt truncation of the computation domain causes full reflections; therefore, absorbing boundary blocks (to simulate the free space) must be added. Here, convolutional perfectly matched layer (CPML) termination is used [30]. Broadband frequency responses may
Fig. 2. FDTD modeling of the SHBC strip (left face: SBC, right face: HBC, strip thickness is one FDTD cell).
be obtained via single FDTD simulation by using pulsed EM signals.
In 2-D FDTD analysis, 1000× 1000 cells add up to a total of one million FDTD cells, and this corresponds to a computation space of 25λ × 25 λ, if the cell size is chosen as λ/40 (which is tested to be necessary and sufficient in most of diffraction modeling problems using FDTD). The object under investiga-tion is located in the middle of this space. Simulainvestiga-tion space is covered with 10-cell-thick CPML. Also, 5λ space is left for the air on both axes. Therefore, any 20λ × 20λ object can be in-vestigated easily in this FDTD space. This corresponds far into quasi-optical and optical scattering frequency regime, therefore a comparison with HFA is possible.
Due to the finite nature of FDTD grid, the strip is modeled as one cell wide as shown in Fig. 2. FDTD is also capable of modeling thin structures, but an original Yees algorithm for modeling the strip is preserved. Assuming the left and right faces of the strip ati = N th and i = (N + 1)st cells, respectively, the SBC and HBC will be satisfied usingu = 0 or Ez(N) = 0 and
∂u/∂n = 0 or Ez(N + 1) = Ez(N + 2), respectively, along all vertical cells on the strip.
Diffracted fields are extracted from the total fields by applying a three-step procedure as:
Step 1: The FDTD simulation is run with the strip, and the
time-domain values of the fields at the receivers are recorded. The total fields are obtained at the end of this step.
Step 2: The strip is replaced by a full plane (i.e., the strip is
extended to infinite vertically on both ends) and the FDTD simulation is rerun. The recorded time-domain data at the re-ceivers on the source side only contain incidence and reflected fields.
Step 3: The strip is removed, and the FDTD simulation is run
in free space without having any objects. The recorded time-domain data at the receivers include only incident fields. Once the three-step procedure is completed, the time data obtained from step 2 are subtracted from the time data obtained
308 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 16, 2017
Fig. 3. FDTD-generated (top) total and (bottom) scattered fields around the SHBC strip.
from step 1 within left or right reflection boundaries depending on the incidence angle. This will yield incident (I) + diffracted (D) fields (I+ D) in the simulation space. Then, the time data obtained from step 3 is subtracted from the previously obtained (I+ D) data within illuminated region.
Once this step is completed, only diffracted fields exist in simulation area. A sample plot for both total and scattered fields around the SHBC strip is shown in Fig. 3. Here, the source is a plane wave coming from 45◦ angle and hits the strip from the hard face. As observed in the top plot, there is a shadow region behind the strip where only edge-diffracted waves appear. The bottom plot shows that forward scattering and specular reflections are dominant. Edge diffractions are also observable in the figures. Note that incident fields are subtracted from the top plot and scattered-only fields around the strip are obtained.
IV. EXAMPLES ANDCOMPARISONS
MATLAB algorithms are developed for FDTD-based diffrac-tion modeling and are run for different sets of parameters. The results are compared to the existing MoM-based model [20], which was validated before against the PTD solution. Examples given in Figs. 4 and 5 present total and diffracted fields around SBC, HBC, and SHBC strips.
In Fig. 4, the total and diffracted fields around both HBC and SBC strips, computed with both FDTD and MoM models, are shown. The strip size is 20 m, and this corresponds to 2λ at 30 MHz. The line source is 8λ away from the origin. The receivers/observers are located along a circular path having a radius of 7λ. The agreement is very good, as observed. The dominant diffraction is seen along the incident shadow boundary (ISB) and reflection boundary (RB). Note that, although the strip is infinitely thin, one-cell-thick strip is used in these examples.
Fig. 4. (left) Total and (right) diffracted fields around the strip for (top) HBC, ϕ0 = 60◦ and (bottom) SBC,ϕ0 = 120◦(f = 30 MHz, L = 2λ, ρ0= 8λ,
ρ = 7λ; solid: MoM, dashed: FDTD).
Fig. 5. (left) Total and (right) diffracted fields around the SHBC strip: (top), ϕ0 = 60◦and (bottom)ϕ0= 120◦(f = 30 MHz, L = λ, ρ0 = 8λ, ρ = 7λ;
solid: MoM, dashed: FDTD).
This approach produces artificial reflections represented by the dashed lines in the figures. They disappear when infinitely thin strip model is used in FDTD simulations (as discussed and shown in [10], [16], and [17]). Alternatively, to overcome the problem, the thin-wire subcell model given in [31], which uses quasi-static field approximation for functional variation of the fields in the transverse direction radial to the strip, can be used. Fig. 5 belongs to the same comparisons, but for the SHBC strip. As observed in Figs. 4 and 5, the diffraction from hard surface is stronger than the one from soft surfaces. Using SHBC
USLU et al.: DIFFRACTION MODELING BY SOFT–HARD STRIP USING FDTD METHOD 309
strip does not change the magnitude of diffracted fields signif-icantly at the RB compared to HBC strip. On the other hand, the magnitude of the diffracted fields at soft side of the strip is almost the average of the diffracted fields obtained for merely SBC and HBC strip in ISBs.
In terms of computational complexity, MoM requires less computational time compared to FDTD on the same cell/segment size, e.g., for the strip configuration as given in Fig. 5. MoM simulation takes approximately 13 s. On the other hand, FDTD simulation takes 50 s. One of the key factors for this time difference is that MoM-based diffraction model re-quires two steps, i.e., incident fields are calculated directly; on the other hand, FDTD requires three steps. Another factor is the storage of field components at all time-steps brings signifi-cant computational load to FDTD simulations. Note that MoM solves a system of equations (i.e., requires a matrix inversion), while FDTD uses a few simple iterative equations. Although MoM simulations last shorter than FDTD simulations here in this study, in general, FDTD is less time-consuming especially when the size of the problem is high.
V. CONCLUSION
Diffraction by a canonical strip with one face soft BC and the other face hard BC is modeled numerically in time domain using the FDTD method. Results are compared against MoM. Very good agreement between the proposed model and the ex-isting MoM-based model confirms the validity of the proposed model. The advantage of the proposed model over the existing MoM-based model is that broadband diffracted fields can be obtained in a single run. Also, diffraction from SHBC strip can be analyzed in time step by step.
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