Validation and implementation of a near-field physical optics formulation

Transcription

1 Validation and implementation of a near-field physical optics formulation Stéphane R. Legault and Aloisius Louie Defence R&D Canada Ottawa TECHNICAL MEMORANDUM DRDC Ottawa TM August 24

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3 Validation and implementation of a near-field physical optics formulation Stéphane R. Legault Defence R&D Canada Ottawa Aloisius Louie Defence R&D Canada Ottawa Defence R&D Canada Ottawa Technical Memorandum DRDC Ottawa TM August 24

4 Her Majesty the Queen as represented by the Minister of National Defence, 24 Sa majesté la reine, représentée par le ministre de la Défense nationale, 24

5 Abstract Previously proposed modifications to physical optics for improving the accuracy of scattered field computations are validated within the context of actual scattering problems. Results suggest that the procedure can potentially extend the region of validity of physical optics from the far field to short ranges lying well within the near field. Furthermore, in the case of flat plates, the angular region over which accurate results are obtained remains approximately the same even at shorter ranges. Finally, computations using an actual ship model verify that the improved formulations captures variations in the scattered field which would otherwise go unnoticed using a standard formulation. Résumé Une approche, établie antérieurement, visant à améliorer la performance de la méthode de l optique physique pour le calcul de surface équivalente radar en champ proche est validée dans le cadre de problèmes de diffusion. Il est ainsi vérifié que cette méthode permet d étendre la région de validité de l optique physique profondément dans le champ proche à des distance qui sont de beacoup inférieures aux limites habituelles associées avec le champ lointain. En effet, les calculs de surfaces équivalentes radar en fonction de l angle d incidence pour une plaque carrée indiquent que l exactitude de la méthode modifiée en champ proche est comparable à celle typiquement obtenue en champ lointain avec la méthode standard. Finalement, des calculs dans le cas d un navire en champ proche illustrent que le comportement du champ diffus en fonction de la distance peut différer largement des valeurs obtenues en champ lointain. DRDC Ottawa TM i

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7 Executive summary The nature of large scale electromagnetic scattering problems those where the dimensions involved span thousands of wavelengths presents challenges both for the measurement as well as the numerical modeling of the radar cross section. The latter is usually defined in a region called the far field, lying at large distances from the scatterer, and its direct measurement for a naval vessel can be problematic since the observation point is typically situated relatively close to the scatterer. This in turns leads to difficulties with numerical modeling: the computational algorithms used, such as physical optics, are usually tailored to perform far-field computations and their accuracy is therefore suspect in scenarios duplicating actual measurements. This work provides numerical results which validate a previously proposed version of physical optics modified to perform computations in the near field. The approach has the advantage of not requiring significantly greater computational resources while remaining easy to implement, making the modification of existing codes a simple task. The modified physical optics technique is verified herein by examining simple flat plate scatterers. Computations demonstrate that the proposed physical optics formulation effectively extends the region of validity to much shorter ranges while maintaining the same level of accuracy obtained in the far field. The required modifications to RAPPORT, a physical optics code used by DRDC, are discussed and the modified code is validated. It is then used to compute the RCS of CFAV QUEST at various ranges, confirming that the modified code allows the observation of range-dependent behaviours that significantly differ from far-field values obtained with standard physical optics implementations. Future efforts will consist of adding further refinements to the numerical model. Chief among those is the combination of the near-field formulation with sea surface modeling using, initially, a lossy ground plane. This will improve the ability to to model multipath scattering effects in the near field. The resulting code will then be used to assess the performance of the near-field formulation when duplicating RCS measurements of actual naval vessels. Legault, Stéphane R. and Louie, Aloisius. 24. Validation and implementation of a near-field physical optics formulation. DRDC Ottawa TM Defence R&D Canada Ottawa. DRDC Ottawa TM iii

8 Sommaire L évaluation de la surface équivalente radar (SER) au moyen de méthodes expérimentales ou numériques s avère souvent très difficile dans le cas où les diffuseurs ont des dimensions de l ordre de milliers de longeurs d onde. En effet, puisque la SER est typiquement définie dans le champ lointain, région située à de grandes distances du diffuseur, il peut être difficile, voir même impossible, de la mesurer directement puisque le point d observation doit souvent se trouver à de courtes distances du diffuseur. Par exemple, le champ lointain pour un navire peut se situer au delà d un millier de kilomètres alors que les mesures de SER doivent s effectuer en champ proche à des distances inférieures à dix kilomètres. De plus, la reproduction de telles mesures à l aide de méthodes computationnelles pose également des difficultés puisque les algorithmes requis pour des structures aussi grandes font souvent appelle à des méthodes dites haute fréquence, comme l optique physique, optimisées pour les calculs en champ lointain. Ce document fourni des résults numériques servant à valider une approche, établie antérieurement, qui améliore la performance de la méthode de l optique physique en champs proche, c est-à-dire à de courtes distances du diffuseur. Cette version modifiée de l optique physique est évaluée en examinant sa performance dans le cas où de simples surfaces carrées agissent comme diffuseurs. Il est ainsi démontré que la nouvelle méthode permet d obtenir la même précision qu en champ lointain mais à des distance beaucoup plus courtes. Elle est d autant plus attrayante puisque qu elle ne nécéssite pas une charge supplémentaire de calculs beaucoup plus élevée tout en demeurant facile à implémenter. Les modification nécessaire au logiciel RAPPORT, qui utilise des algorithmes basés sur l optique physique, sont discutées et la version modifiée du logiciel est validée. Cette dernière est par la suite utilisée pour calculer la surface équivalente radar du navire de recherche QUEST et ceci démontre que cette méthodologie permet l observation de variations en fonction de la distance qui ne pourraient être reproduites avec des méthodes standards. Les travaux futurs viseront à poursuivre l amélioration des modèles numériques dans le but d offrir des techniques fiables pour estimer la surface équivalente radar de diffuseurs situées au sein d environements complexes. Une importante étape sera la combinaison de la formulation en champ proche avec un modèle de la surface de l océan initialement basé sur un plan conducteur avec perte. Ceci mènera à une amélioration de la modélisation des effets de trajet multiple sur le champ diffusé en champ proche dans le cas de navires. Legault, Stéphane R. and Louie, Aloisius. 24. Validation and implementation of a near-field physical optics formulation. DRDC Ottawa TM R & D pour la défense Canada Ottawa. iv DRDC Ottawa TM

9 Table of contents Abstract i Résumé i Executive summary Sommaire iii iv Table of contents v List of figures List of tables vii ix 1. Introduction Radiation integrals Geometry and parameters of interest Field expressions Numerical results Accuracy of modified physical optics formulation The 4λ plate The 1λ plate Near-field version of RAPPORT RAPPORT modifications The 1λ plate RCS of CFAV QUEST Conclusion References Annexes DRDC Ottawa TM v

10 A Electric field expressions A.1 Exact formulation A.2 Far-field approximation A.3 Expression with arbitrary expansion point A.4 Expressions with arbitrary expansion point refined phase and magnitude B Method of moments: Impedance matrix elements B.1 The A mn elements B.2 The B mn elements B.3 The C mn elements B.4 The D mn elements C Method of moments: Partial convergence study vi DRDC Ottawa TM

11 List of figures 1 The PEC flat plate geometry used showing the incident vector and the polarization directions Surface electric currents excited on a 4λ 4λ square PEC plate The monostatic co-polarized RCS of a 4λ 4λ square plate as a function of incidence angle θ i when observed at various ranges for a θ-polarized incident wave The monostatic cross-polarized RCS of a 4λ 4λ square plate as a function of incidence angle θ i observed at various ranges for a θ-polarized incident wave The bistatic co-polarized RCS of a 4λ 4λ square plate as a function of scattering angle θ s observed at ranges of 1 m and 5 m for a θ-polarized incident wave The bistatic cross-polarized RCS of a 4λ 4λ square plate as a function of scattering angle θ s at ranges of 1 and 5 m for a θ-polarized incident wave Method of moments and physical optics surface currents on a 4λ 4λ square plate The monostatic co-polarized RCS of a 4λ 4λ square plate as a function of incidence angle θ i observed at r = 1 m and r = 1 m The bistatic co-polarized RCS for a 4λ 4λ square plate observed at r = 1 m r = 1 m as a function of scattering angle θ s The monostatic co-polarized RCS of a 1λ 1λ square plate as a function of range when θ = φ = The monostatic co-polarized RCS of a 1λ 1λ square plate as a function of incidence angle for various ranges when illuminated by a θ-polarized wave (continued) The monostatic co-polarized RCS of a 1λ 1λ square plate illuminated by a θ-polarized wave as a function of incidence angle θ i for various ranges The far-field monostatic co-polarized RCS of a 1λ 1λ square plate as a function of range when θ = φ = The monostatic co-polarized RCS σ and σ of a 1λ square plate as a function of observation range r The monostatic co-polarized RCS of a 1λ square plate as a function of incidence angle CAD model of CFAV QUEST DRDC Ottawa TM vii

12 16 The RCS of CFAV QUEST as a function of range for two azimuth angles and three frequencies Polar plot of the computed RCS of CFAV QUEST for far-field and near-field observation C.1 The monostatic co-polarized RCS of a 4λ 4λ square plate as a function of incidence angle for two different surface element densities viii DRDC Ottawa TM

13 List of tables 1 Definition of the quantities ˆQ, I 1 and I 2 appearing in (9) for each of the cases of interest. The Green functions g are defined in Table Definition of the Green functions used in Table DRDC Ottawa TM ix

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15 1. Introduction Despite constant advances in computational capabilities, estimating the radar cross section (RCS) of electrically large scatterers using electromagnetic (EM) numerical models still presents significant challenges. Indeed, geometries with dimensions on the order of thousands of wavelengths such as naval vessels cannot be modeled using accurate full-wave techniques like the method of moments (MoM) or the finite element method due to the prohibitive quantity of computer memory required. For example, the problem of scattering from an aircraft at X band is sufficient to push current capabilities to their limits: computing the RCS of the virtual VFY218 aircraft at 8 GHz using a variation of MoM required 48 Gb of memory and 7.4 hours of computation time on 128 processors [1] ( see also [2, chapter 9]). One must turn instead to high-frequency techniques, such as geometrical optics or physical optics (PO), which combine efficiency and reasonable accuracy by relying on assumptions attuned to large scale problems. One of the high-frequency codes used by DRDC to solve electrically large scattering problems is RAPPORT [3], which consists of a PO based formulation used in conjunction with ray tracing to account for multiple interactions. Though approximate in nature, it can provide good accuracy as long as proper attention is paid to underlying simplifying assumptions. In particular, one must be careful if computations must be performed at relatively short distances from the scatterer since PO codes are usually optimized for far-field RCS computations: one of the key simplifying assumptions in physical optics is tied to the range at which the scattered fields (or the RCS) are computed. This originates from the definition of RCS, σ = lim r 4πr 2 E s (r) E i 2, (1) with requires the observation range r to recede to infinity. Here E i and E s (r) denote the incident and scattered field, respectively. Although at first glance the requirement on r may appear awkward, it actually provides a range independent definition of RCS equally applicable to all finite scatterers. With respect to PO it also leads to a much simpler mathematical formulation, making the surface radiation integrals amenable to closed-form solutions without any sacrifices in accuracy when the observation range is infinite (or relatively large) [5, Annexes A and B]. This makes physical optics well suited for traditional far-field radar cross section computations but potentially ill suited for computations at finite ranges. In practice, particularly when RCS measurements are concerned, one must resort to large finite values of r since infinitely removed observation distance are not feasible. The acceptable minimum range r ff at which far-field behaviour can still be observed is estimated using r ff = 2D2 λ, (2) where D is the maximum dimension of the scatterer. The above is the traditional far-field condition; see the work of Kouyoumjian and Peters [4] for one the earliest references on this topic as well as [5] for further discussion. It is revealing to examine (2) in the context of a ship at X band: for a 135 m ship and a frequency of 1 GHz it yields an estimate of 1215 DRDC Ottawa TM

16 km for the inner reaches of the far field. Given that measurements are typically carried out at distances of less than 1 km and regardless of the approximate nature of (2) one can legitimately question the validity of using standard PO based formulations to reproduce actual measurements. In fact, measurements are actually conducted in the inner reaches of the near field. Its location is estimated [5] with D 3 r nf =.62 λ, (3) which yields, using the above values, a near-field limit of 7 km. These concerns with the applicability of physical optics at such short ranges stimulated an effort aimed at increasing accuracy in the near field. It was shown in [5] that by subdividing the surface of the scatterer and using locally expanded Green functions, the accuracy could be dramatically improved at short distances without incurring significant extra computational costs. This conceptually simple approach leads to expressions of the same nature as the ones briefly stated without supporting derivations by Gordon [6]. The objective of this work is to provide a validation of the formulation proposed in [5] by comparing numerical results from standard and the modified (near-field) physical optics with MoM results. The latter serve as a reference solutions throughout given their relatively high accuracy. The performance of the associated radiation integrals is also examined for the case of a flat square plate. Since much of this work examines the performance of the PO techniques as a function of range, comparisons are made using the modified radar cross section definition σ(r) = 4πr 2 E s (r) E i 2, (4) which recovers the standard expression (1) as r. The phase accuracy of the results is considered using the square root of the RCS σ(r) = 2 πr ê s E s (r) ê i E i, (5) a quantity related to (4) but which preserves phase information. The unit vector ê s above (and similarly for ê i ) corresponds to the polarization direction of E s (r); their scalar product therefore yields the magnitude and phase of the field E s (r). The latter part of this work also examines the implementation of the proposed scheme in RAPPORT. Details are provided on the modifications required and a validation is carried out by revisiting the flat plate geometry. Finally, the RCS of the naval research vessel CFAV QUEST is computed in order to gauge the impact of the near-field formulation compared to the standard PO formulation. 2 DRDC Ottawa TM

17 2. Radiation integrals It is insightful to first consider the relative accuracy of the various radiation integrals involved prior to studying in detail the performance of the near-field version of PO in subsequent sections. The integrals considered include both an exact representation as well as approximate ones distinguished by the degree of approximation used for the free-space Green function and its derivatives. The comparison is carried out by employing the same surface currents 1 J MM obtained using the method of moments for good accuracy as sources in the various radiation integrals. In this fashion, discrepancies observed in the computed RCS values obtained with the various formulations will be attributed to the radiation integrals themselves. This is an important preliminary step since forthcoming comparisons in latter sections examine the RCS values obtained from the method of moments, which relies on accurate surface currents J MM, and RCS values obtained from the method of physical optics, which relies on approximate surface currents J PO. 2.1 Geometry and parameters of interest The surface currents considered are those excited on a perfectly electrically conducting (PEC) 4λ 4λ square plate lying in the xy plane and illuminated by a θ-polarized plane wave with wavelength λ = 1 m. The reader is referred to Figure 1 which depicts the geometry of the problem. Note how the polarization of the incident field is a function of angle of incidence θ i. In accordance with the convention used for a spherical coordinate system, the angle θ such as the incidence angle θ i and the scattering (observation) angle θ s is defined with respect to the z 1 To alleviate the notation, the surface currents are denoted by J instead of J s. All the currents mentioned herein are surface currents with units of A/m. x ˆφ ˆki ˆθ y θ i z Figure 1: The PEC flat plate geometry used showing the incident vector ˆk i together with the polarization directions ˆθ and ˆφ. The triplet (ˆk i, ˆθ, ˆφ) is orthogonal. DRDC Ottawa TM

18 axis. In the monostatic case θ s = θ i. The three unit vectors (ˆk i, ˆθ, ˆφ) shown in Figure 1 form an orthogonal triplet; in this particular instance ˆθ lies in the xz plane and ˆφ is parallel to the yz plane. The currents are computed using an electric field integral equation (EFIE) version of the method of moments; the implementation used relies on rooftop basis functions with pulse ( razor blade ) testing functions as described in [7, Section 1.1]. The information provided therein is quite complete although final expressions for the impedance matrix elements are not explicitly provided. They are included in Annex B for completeness. The plate is modeled using a discretization of ten surface elements per wavelength resulting in a grid of 4 4 surface elements with.1 m edge length. As discussed in Annex C, it turns out that this grid density is sufficient for our purposes. Sample currents for the case of normal incidence are provided in Figure 2 which depicts both the co-polarized currents J x and cross-polarized currents J y. For normal incidence, the incident θ-polarized electric field is collinear with ˆx; it is effectively x-polarized. Note that, as expected, there is no current flow perpendicular to the edges and the cross-polarized (y-directed) components are much lower than the co-polarized (x-directed) ones. As discussed at length in [5] and more briefly in the introduction, the relevant far-field and near-field range thresholds are function of the maximum dimension of the structure as well as the wavelength. For a plate with dimensions of 4 m, the corresponding maximum dimension is D = 4 2 m so that the far-field limit is r ff (D) = 2D2 λ = 64 m, (6) and the near-field limit r nf (D) =.62 D 3 λ = 8.34 m. (7) In rough terms, all the representation are expected to provide good accuracy at ranges on the order of r ff (D) = 64 m or more. However, the accuracy of the approximate representations especially the one based on the far-field approximation will diminish as the range decreases and the observation point moves deeper into the near field, closer to the scattering body. Following from the discussion in [5] regarding the use of Green functions with local expansion point, we are also interested in the far-field value associated with the dimensions of the surface elements. Since each surface element has.1 m edges, their corresponding maximum dimension the diagonal corresponds to d = 2/1. The far-field limit of each element is therefore r ff (d) = 2d2 λ =.4 m, (8) about three orders of magnitude smaller than r ff (D). However, as mentioned in [5], the accuracy of the formulation cannot be arbitrarily increased at extremely short ranges by increasing the surface element density. Indeed, the physical optics formulation assumes r max(1, k, r ) [5, Annex A] and this should constitute the fundamental range limit at which accurate results can be obtained, irrespective of the element size used. 4 DRDC Ottawa TM

19 ٠.٠١٥ Jx [A/m] ٠.٠١ ٠.٠٠٥ ٠ ٢ ١ ٠ y [λ] ١ ٢ ٢ ١ ٠ x [λ] ١ ٢ (a) Co-polarized currents J x.15 Jy [A/m].1.5 J x [A/m] 2 1 y [λ] x [λ] 1 2 (b) Cross-polarized currents J y Figure 2: (a) Co-polarized and (b) cross-polarized surface currents excited by a normally incident x-polarized plane wave on a 4λ 4λ square PEC plate when λ = 1 m. The currents were obtained using the method of moments. DRDC Ottawa TM

20 Table 1: Definition of the quantities ˆQ, I 1 and I 2 appearing in (9) for each of the cases of interest. The Green functions g are defined in Table 2. Condition ˆQ I 1 (r, r n) I 2 (r, r n) Exact ˆR ( 1 j kr 1 ) k 2 R 2 g (r, r n) ( 1 k 2 1 j3k R 3 ) R 2 g (r, r n) Far-field ˆr g ff (r, r n) g ff (r, r n) Near-field ˆρ n g nf (r, r n; r n) g nf (r, r n; r n) (phase) Near-field ˆρ n g nf2 (r, r n; r n) g nf2 (r, r n; r n) (mag and phase) 2.2 Field expressions As shown in more detail in Annex A, surface electric currents lying in the xy plane give rise to the radiated electric fields E x (r) = jkz [ ] J x (r n)i 1 (r, r n) (ˆx ˆQ)( ˆQ J(r n))i 2 (r, r n) S n (9a) n E y (r) = jkz [ ] J y (r n)i 1 (r, r n) (ŷ ˆQ)( ˆQ J(r n))i 2 (r, r n) S n (9b) n E z (r) = jkz [ ] (ẑ ˆQ)( ˆQ J(r n))i 2 (r, r n) S n, (9c) n which are discretized representations using Riemann sums together with constant currents over each element of the radiation integrals. The wavenumber is identified by k, the intrinsic impedance by Z, and S n is the surface area of the nth surface element. The quantities ˆQ, I 1 and I 2 are defined in Table 1. The unit vector ˆQ points from the source to the observation point. Depending on the representation used, it may (i) be exact so that ˆQ = ˆR, (ii) be globally approximated in the case of the far-field approximation where ˆQ = ˆr, or (iii) locally approximated about an expansion point r n such that ˆQ = ˆρ n where ˆρ n = ρ n ρ n (1) with ρ n = r r n. (11) The scalars I 1 and I 2 are used to select exact or approximate formulations as indicated in Table 1. The four Green functions identified therein are defined in Table 2. They correspond to (i) an exact Green function g, (ii) a far-field approximation g ff of the Green function, (iii) a Green function g nf with an arbitrary expansion point r n for the phase approximation and (iv) a Green function g nf2 with an arbitrary expansion point r n used in both magnitude and phase. The far-field approximation of the Green function corresponds to the one used in standard physical 6 DRDC Ottawa TM

21 Table 2: Definition of the Green functions used in Table 1. Condition Exact free-space Far-field approximation Near-field approximation (refined phase) Near-field approximation (refined magnitude and phase) Green function g (r, r ) = e jk ˆR R 4πR g ff (r, r ) = e jkˆr R 4πr g nf (r, r ; r n) = e jk ˆρ n R 4πr g nf2 (r, r ; r n) = e jk ˆρ n R 4π ρ n optics (PO) implementations. The main interest lies mainly with formulation (iii) using arbitrary expansion points for the phase based on the formulation discussed in [5] and, to a lesser extent, formulation (iv) using arbitrary expansion points for both magnitude and phase. All of the formulation should agree when the observation point is sufficiently far from the scatterer (at least 64 m, say). However, formulations (iii) and (iv) should clearly outperform the standard far-field approach (ii) at short ranges. 2.3 Numerical results By means of the surface currents obtained with the MoM, the RCS values are now computed using the field expressions (9). Monostatic and bistatic RCS values for both the co-polarized and cross-polarized cases are considered. The monostatic case is first examined. The results for monostatic RCS are shown in Figures 3 and 4 for co- and cross-polarization respectively. Both figures provide plots of σ in terms of the incidence angle θ i from at boresight to 9 at grazing. Examination of the figures shows that the relative performance of the different radiation integrals is similar for both the co- and cross-polarization cases. Excellent agreement is obtained between all radiation integrals when the observation range r = 1 m, as expected since r ff = 64 m. The agreement is still good when r = 5 m. Both of the approaches based on localized expansions still perform very well when r = 1 m, particularly the one with improved magnitude and phase, whereas the standard far-field approximation based radiation integral exhibits discrepancies over the entire angular range. All three approximate formulations fail when r = 2 m. The likeliest explanation is the failure to fulfill r max(1, k, r ), a condition that must be satisfied in order for the approximate representations to hold. The interested reader is referred to [5, Annex A] for more details. For the bistatic case, the illuminating wave normally impinges on the plate and the observation angle θ s varies from in the normal direction to 9 at grazing angle. Results are shown in DRDC Ottawa TM

22 Figure 5 for the co-polarized case and Figure 6 for the cross-polarized case. Due to the similarity with the relative behavior of the curves in the monostatic case, results are only shown for ranges of 5 and 1 meters. It can be appreciated that the standard far-field PO formulation breaks down as the range decreases. The best performer at r = 1 m is the formulation based on locally expanded magnitude and phase followed closely by the one with refined phase only. Observe how the x-directed co-polarized currents do not provide any contribution as the observation angle nears 9, in contradistinction with the y-directed cross-polarized currents. As expected, the radiation integrals based on the locally expanded Green functions outperform the standard far-field approximation for short observation ranges despite having similar computational complexities. The approximation based on the Green function with refined magnitude and phase g nf2 is the best performer; the one relying solely on a refined phase, g nf, fares slightly worse. However, the two formulation will have comparable accuracies if r r. This is typically the case for near-field observation of ships where scatterers have dimensions on the order of a hundred meters and the observation range is on the order of thousands of meters. The absolute accuracy of this approximation is now characterized in the context of a physical optics formulation by comparing it with method of moments solutions. Attention is restricted to the representation using a refined phase in what follows. 8 DRDC Ottawa TM

23 θ i [deg] σ θθ θ[dbsm] i [deg] σ θθ θ[dbsm] i [deg] σ θθ θ[dbsm] i [deg] σ θθ [dbsm] Method of moments Green function with far-field phase approximation Green function with locally expanded phase Green function with locally expanded magnitude and phase 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ i [deg] (a) r = 1 m θ i [deg] (b) r = 5 m 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ i [deg] (c) r = 1 m θ i [deg] (d) r = 2 m Figure 3: The monostatic co-polarized RCS of a 4λ 4λ square plate (λ = 1 m) as a function of incidence angle θ i observed at various ranges for a θ-polarized incident wave. Curves are provided for the method of moment solution and various approximate Green functions. DRDC Ottawa TM

24 Method of moments Green function with far-field phase approximation Green function with locally expanded phase Green function with locally expanded magnitude and phase σφθ [dbsm] 2 4 σφθ [dbsm] θ i [deg] (a) r = 1 m θ i [deg] (b) r = 5 m σφθ [dbsm] 2 4 σφθ [dbsm] θ i [deg] (c) r = 1 m θ i [deg] (d) r = 2 m Figure 4: The monostatic cross-polarized RCS of a 4λ 4λ square plate (λ = 1 m) as a function of incidence angle θ i observed at various ranges for a θ-polarized incident wave. Curves are provided for solutions obtained using the method of moments and various approximate Green functions. 1 DRDC Ottawa TM

25 θ s [deg] σ θθ θ[dbsm] s [deg] σ θθ θ[dbsm] s [deg] σ θθ θ[dbsm] s [deg] σ θθ [dbsm] Method of moments Green function with far-field phase approximation Green function with locally expanded phase Green function with locally expanded magnitude and phase 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ s [deg] (a) r = 5 m θ s [deg] (b) r = 1 m Figure 5: The bistatic co-polarized RCS of a 4λ 4λ square plate (λ = 1 m) as a function of scattering angle θ s observed at (a) r = 5 m and (b) r = 1 m for a θ-polarized incident wave. Curves are provided for solutions obtained using the method of moments and various approximate Green functions. Method of moments Green function with far-field phase approximation Green function with locally expanded phase Green function with locally expanded magnitude and phase 2 2 σφθ [dbsm] 4 6 σφθ [dbsm] θ s [deg] (a) r = 5 m θ s [deg] (b) r = 1 m Figure 6: The bistatic cross-polarized RCS of a 4λ 4λ square plate (λ = 1 m) as a function of scattering angle θ s observed at (a) r = 5 m and (b) r = 1 m for a θ-polarized incident wave. Curves are provided for solutions obtained using the method of moments and various approximate Green functions. DRDC Ottawa TM

26 3. Accuracy of modified physical optics formulation A comparison between the proposed PO formulation based on the Green function with refined phase g nf of the previous section (see also [5]) and accurate solutions obtained with the method of moments is now presented. Owing to the high accuracy of MoM results, this comparison essentially provides a measure of the absolute accuracy of the modified PO formulation. For reference, standard PO results are also supplied. The expressions used for radiating the MoM currents are the same as the ones provided in the previous section for the exact formulation. The ones used for the PO formulations correspond to the far-field and near-field (refined phase only) expressions based on (9) but with the approximate PO currents being used as sources; this effectively recovers the expressions provided in [5]. The distinction between the PO currents J PO and the high-accuracy MoM currents J MM is shown in Figure 7 for the case of a 4λ square plate illuminated at normal incidence. It is seen that in the case of co-polarized currents the PO currents essentially provides a zeroth order approximation of the exact currents. This comes as no surprise since the PO currents correspond to specular currents unperturbed by the edge currents which lead to ripples in the total surface currents. For the cross-polarized currents, which are significantly lower than the co-polarized ones, the corresponding PO currents are nil. This is a consequence of the inability of PO to account for depolarization due to diffraction (or edge) effects. Despite the seemingly coarse representation provided by the PO currents, they can nevertheless yield good accuracy for the co-polarized case, particularly in regions where the specular component of the scattered field is dominant. Besides examining below the 4λ flat square plate geometry used in the previous section, a 1λ plate is also examined in the spirit of considering a more electrically large structure. In the case of the latter, the relationship between surface element density and accuracy is considered for the first time. 3.1 The 4λ plate Consider first the case of monostatic scattering; results are shown in Figure 8 which compares MoM results with the standard PO formulation as well as the modified near-field formulation. As expected, see Figure 8(a), both PO techniques used agree very well at a range of r = 1 m over incidence angles of to 3. This is in concordance with the widely accepted rule of thumb of restricting observation angles within 3 of boresight for accurate backscattering results using physical optics with planar structures. Essentially, PO provides good agreement in the region where the specular component dominates but fares rather poorly where the diffraction currents dominate such at grazing angles of incidence. Much more interestingly, the refined PO formulation still displays similar agreement at the much shorter range of r = 1 m, well within the near field, where it outperforms the standard PO formulation which fails to recover the correct values at normal incidence and fails to account for the filling in of the far-field nulls. Following (7) the near-field boundary lies at 8.34 m: roughly speaking, the refined formulation essentially extends the region of validity of PO from the far-field boundary to the near-field boundary. This effectively verifies the results anticipated in [5]. The conclusions are the same 12 DRDC Ottawa TM

27 Jx [A/m].1.5 Jx [A/m] y [λ] x [λ] 2 1 y [λ] x [λ] 1 2 (a) MoM currents (b) PO currents Jy [A/m].1.5 Jy [A/m].1.5 J x [A/m] 2 1 y [λ] x [λ] 1 2 J x [A/m] 2 1 y [λ] x [λ] 1 2 (c) MoM currents (d) PO currents Figure 7: Co-polarized [cross-polarized] surface currents J s obtained with (a) [(c)] the method of moments and (b) [(d)] physical optics on a 4λ 4λ square plate observed at normal incidence when λ = 1 m. DRDC Ottawa TM

28 θ i [deg] σ θθ θ[dbsm] i [deg] σ θθ θ[dbsm] i [deg] σ θθ [dbsm] Method of moments Standard PO PO with locally approximated phase 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ i [deg] (a) r = 1 m θ i [deg] (b) r = 1 m Figure 8: The monostatic co-polarized RCS of a 4λ 4λ square plate (λ = 1 m) observed at (a) r = 1 m and (b) r = 1 m as a function of incidence angle θ i for a θ-polarized incident wave. Results obtained with the method of moments, standard PO, and PO with local phase approximations are compared. in the case of bistatic scattering which is illustrated in Figure 9. Again, both PO techniques agree well with the MoM values up to around 3 when r = 1 m. At r = 1 m the distinction between the modified PO and the standard PO techniques is more pronounced; the modified PO again provides superior agreement with the MoM values up to approximately DRDC Ottawa TM

29 θ s [deg] σ θθ θ[dbsm] s [deg] σ θθ θ[dbsm] s [deg] σ θθ [dbsm] Method of moments Standard PO PO with locally approximated phase 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ s [deg] (a) r = 1 m θ s [deg] (b) r = 1 m Figure 9: The bistatic co-polarized RCS for a 4λ 4λ square plate (λ = 1 m) observed at (a) r = 1 m and (b) r = 1 m as a function of scattering angle θ s for a θ-polarized incident wave. Results obtained with the method of moments, standard PO, and PO with local phase approximations are compared. 3.2 The 1λ plate Consider now a 1λ plate which may be considered as the smallest size possible for an electrically large scatterer. Once again, MoM and PO solutions are compared but in this case the MoM results were obtained using JUNCTION. Since the results for the 4λ square plate in the previous section led to similar conclusions for both the monostatic and bistatic cases, it is now sufficient to restrict attention to the case of monostatic scattering. In the case of a 1λ plate with λ = 1 m, the far-field and near-field range thresholds are r ff (D) = 2D2 λ = 4 m, (12) and r nf (D) =.62 D 3 λ = 33 m. (13) The RCS σ(r) as a function of range along a line normal to the plate (the z axis in Figure 1) is first examined. Since the accuracy of the refined PO formulation can be improved, in principle at least, by increasing the surface element density it is important to examine the accuracy for various surface element grid densities. To this end, computations are carried out with the modified PO formulation using 2 2, 5 5 and 1 1 surface element grids. Based on the maximum dimension of the surface elements, these three grid densities have respective far-field boundaries of 1, 16 and 4 m. The results provided in Figure 1 show that even the coarsest grid density provides good agreement with MoM down to a range of 2 m whereas the two DRDC Ottawa TM

30 r [m] σ r θθ [m] [dbsm] σ r θθ [m] [dbsm] σ r θθ [m] [dbsm] σ r θθ [m] [dbsm] σ θθ [dbsm] Method of moments Standard PO Near-field PO, 2 2 surface element grid Near-field PO, 5 5 surface element grid Near-field PO, 1 1 surface element grid σ θθ [dbsm] r [m] Figure 1: The monostatic co-polarized RCS of a 1λ 1λ square plate as a function of range when θ = φ =. The structure is illuminated by a θ-polarized wave and λ = 1 m. Represented are the method of moments solution, the standard PO solution and the near-field PO solution with 2 2, 5 5 and 1 1 surface element grids. 16 DRDC Ottawa TM

31 higher grid densities provide good agreement down to 5 m or so. The regular PO solution diverges from the MoM solution at a range of 2 m, an order of magnitude larger than for the 2 2 surface element grid, roughly two orders of magnitude larger than for the finer grids. In order to evaluate if this still applies off-boresight, the monostatic RCS is now examined as a function of azimuth. Indeed, previous results see Figures 8 and 9 clearly indicate the more stringent cases correspond to angles of incidence going towards grazing. Figure 11 shows the behaviour of the RCS at four different ranges. Again, all the PO results agree as r and compare well with the MoM solution up to angles of approximately 3. The situation is more interesting when r = 33 m [Figure 11(b)]. The improvement that the modified PO formulation provides over the standard PO formulation is evident, and the two higher grid densities provide good agreement with the exact solution up to approximately 3. The limits of the techniques are pushed in Figures 11(c) and 11(d) where similar curves are provided when r = 2 m and r = 1 m. For the r = 2 m case, both of the higher grid densities provide good agreement with the MoM solution up to roughly 3. Even for the very stringent r = 1 m case the observation range r is now on the order of r the agreement remains good up to approximately 15. This reduction in the angular region of validity comes as no surprise since, at such a short range, the assumptions made when deriving the PO formulation [5] are being violated due to the relatively small value of r. Nevertheless, the modified PO formulation is still performing relatively well. DRDC Ottawa TM

32 θ i [deg] θ σ θθ i [deg] θ[dbsm] σ θθ i [deg] θ[dbsm] σ θθ i [deg] θ[dbsm] σ θθ i [deg] [dbsm] σ θθ [dbsm] Method of moments Standard PO Near-field PO, 2 2 surface element grid Near-field PO, 5 5 surface element grid Near-field PO, 1 1 surface element grid 5 4 σ θθ [dbsm] θ i [deg] (a) r 5 4 σ θθ [dbsm] θ i [deg] (b) r = 33 m Figure 11: The monostatic co-polarized RCS of a 1λ 1λ square plate (λ = 1 m) illuminated by a θ-polarized wave as a function of incidence angle θ i. Observation is made at (a) r, (b) r = 33 m, (c) r = 2 m and (d) r = 1 m. Shown are the method of moments solution, the standard PO approximation, and the near-field PO solution with 2 2, 5 5 and 1 1 surface element grids. 18 DRDC Ottawa TM

33 θ i [deg] θ σ θθ i [deg] θ[dbsm] σ θθ i [deg] θ[dbsm] σ θθ i [deg] θ[dbsm] σ θθ i [deg] [dbsm] σ θθ [dbsm] Method of moments Standard PO Near-field PO, 2 2 surface element grid Near-field PO, 5 5 surface element grid Near-field PO, 1 1 surface element grid 5 4 σ θθ [dbsm] θ i [deg] (c) r = 2 m 5 4 σ θθ [dbsm] θ i [deg] (d) r = 1 m Figure 11: (continued) The monostatic co-polarized RCS of a 1λ 1λ square plate (λ = 1 m) illuminated by a θ-polarized wave as a function of incidence angle θ i. Observation is made at (a) r, (b) r = 33 m, (c) r = 2 m and (d) r = 1 m. Shown are the method of moments solution, the standard PO approximation, and the near-field PO solution with 2 2, 5 5 and 1 1 surface element grids. DRDC Ottawa TM

34 4. Near-field version of RAPPORT RAPPORT is a high-frequency RCS prediction tool which employs a combination of physical optics together with ray tracing, the latter being used account for multiple interactions on the scatterer. The code was developed in 1993 by M.G.E. Brand [3] from the TNO Physics and Electronics Laboratory. In its original version, the reliance of RAPPORT on a standard PO formulation makes it unsuitable to perform RCS computations in the near field. This section provides an overview of the modifications required to enable RAPPORT to perform RCS computations in the near field based on the discussion in [5]. The 1λ plate geometry is revisited in order to provide a validation of the formulation and allows direct comparisons with results in the preceding section. Finally, RCS computations are performed for CFAV QUEST at X band in order to convincingly demonstrate the impact of employing the modified formulation in the near field. 4.1 RAPPORT modifications As mentioned above, the PO formulation must be modified in order to remove the shortcomings of the standard expressions based on far-field approximations. RAPPORT proceeds by subdividing the surface of the scatterer into a collection of triangular surface elements. The PO currents are obtained for each element multiple interactions are taken into account using a ray tracing algorithm and their contribution summed to obtain the total scattered field. The surface integral required is simplified by relying on the representation proposed by Gordon [8] which neatly expresses the triangular surface integral as the sum of the contributions of three line integrals over the edges. Consequently, in the original version of RAPPORT, the electric field E (n) (r) due to the nth surface element is proportional to E (n) ieik(ˆk s r) κ ˆn (r) kr κ ˆn 2 3 m=1 ( ) k v m sinc 2 κ v m e i k 2 κ (v m+v m+1 ). (14) where k is the wavenumber, κ = ˆk i ˆk s, ˆk s is the direction of scattering (observation), ˆk i is the direction of incidence, ˆn is the unit normal of the element, v m is the mth vertex of the triangular element and v m = v m+1 v m. It is understood that v 4 = v 1. As discussed in [5], the above standard PO formulation assumes an infinitely removed observation point and it is therefore ill-suited to near-field computations. Following the procedure describes in [5], expression (14) is easily adapted to near-field computations. To this end, introduce the unit vector (equivalent to the vector ˆρ n defined in (1)) ˆk n = r r n r r n, (15) which points from the center r n of the nth element to the observation point r. Letting ˆk s ˆk n in (14), then κ = ˆk i ˆk s κ n = ˆk i ˆk n. (16) 2 DRDC Ottawa TM

35 Making the appropriate substitutions in (14), the field contributed by the nth element becomes E (n) ieik(ˆk n r) κ n ˆn (r) kr κ n ˆn 2 3 m=1 ( ) k v m sinc 2 κ n v m e i k 2 κ n (v m +v m+1 ). (17) It must be emphasized that the above relies solely on a refined expression for the phase, the 1/r dependence of the magnitude an approximation assuming far-field observation is sufficiently accurate for the cases of interests. 2 The new implementation is now verified by resorting to the test case of a 1λ square plate discussed in the previous section. 4.2 The 1λ plate The 1λ plate is reconsidered since it enables comparisons to be easily drawn with results presented earlier in this work. The surface element grid used by RAPPORT consists of elements. For a 1λ plate, this gives a maximum element size of d =.44λ =.44 m if λ = 1 m. The corresponding far-field boundary is R ff = 2 d2 2 =.39 m. (18) However, the formulation should fail well before this range, as previously discussed on page 7. The case of monostatic scattering as a function of incidence angle θ i is reexamined. As a formality, the RCS in the far field for the standard PO formulation, the modified RAPPORT formulation as well as the method of moments is first compared. Results are shown in Figure 12 where both formulation are shown to agree well with each other. The results shown in Figure 11(a) are effectively duplicated; good agreement is obtained with the method of moments up to around 3 off boresight. Consider next the monostatic RCS as a function of range in the case of normal incidence. The results are illustrated in Figure 13 where both the RCS σ(r) as well as σ [see (5) in Section 1] are provided. Very good agreement is obtained down to ranges below 1 m, and this holds if the real and imaginary parts are examined separately. Interestingly, this remains true below 1 m, despite the fact that r is now comparable to r. The behaviour of the modified RAPPORT code is next examined as a function of range and angle in Figure 14. The accuracy obtained in the figure is essentially the same as the one obtained in Figure 11. Indeed, the refined PO formulation provides good accuracy up to angles of approximately 3 for ranges down to r = 2 m. The region of high accuracy is significantly reduced when r = 1 m and this duplicates the results shown in 11(d). The modified version of the RAPPORT code successfully duplicated the near-field computations of the previous sections, providing a good degree of confidence in the modifications made. Consider now the case of an actual ship, CFAV QUEST, in order to gauge the impact of the near-field formulation at short ranges compared to standard PO computations. 2 It could however be easily modified by letting 1/r 1/ r r n. DRDC Ottawa TM

36 φ [deg] φ σ [deg] θθ [dbsm] φ σ [deg] θθ [dbsm] σ θθ [dbsm] Method of moments Standard PO RAPPORT with near-field PO 6 45 σ θθ [dbsm] φ [deg] Figure 12: The far-field monostatic co-polarized RCS of a 1λ 1λ square plate as a function of range when θ = φ =. The structure is illuminated by a θ-polarized wave and λ = 1 m. Represented are the method of moments solution, the standard PO solution and the near-field PO solution computed with RAPPORT using a surface element grid. 22 DRDC Ottawa TM

37 6 5 σ θθ [dbsm] σθθ [m] r [m] (a) 1 σθθ [m] 1 2 σ θθ [dbsm] r [m] (b) Figure 13: The monostatic co-polarized RCS (a) σ(r) and (b) σ(r) of a 1λ 1λ square plate (λ = 1 m) as a function of observation distance r when normally illuminated by a θ-polarized wave. The blue lines indicate results obtained with RAPPORT whereas method of moments solutions are indicated by diamonds. In (b) the solid line corresponds to Re σ whereas the dashed line corresponds to Im σ. DRDC Ottawa TM

38 θ i [deg] θ σ θθ i [deg] θ[dbsm] σ θθ i [deg] [dbsm] σ θθ [dbsm] Method of moments Standard PO RAPPORT with near-field PO 6 4 σ θθ [dbsm] θ i [deg] (a) r = 33 m. 6 4 σ θθ [dbsm] θ i [deg] (b) r = 2 m. 6 4 σ θθ [dbsm] θ i [deg] (c) r = 1 m. Figure 14: The monostatic co-polarized RCS of a 1λ 1λ square plate (λ = 1 m) as a function of incidence angle θ i when illuminated by a θ-polarized wave. Observation is made at (a) r = 33 m, (b) r = 2 m and (c) r = 1 m. Results are shown for the method of moments, RAPPORT with near-field PO formulation using a surface element grid and standard PO. 24 DRDC Ottawa TM

39 Figure 15: CAD model of CFAV QUEST. 4.3 RCS of CFAV QUEST In the spirit of providing sample results in the context of an actual application, RCS computations are now provided for the research vessel CFAV QUEST at microwave frequencies. The CAD model used for the computations is shown in Figure 15. The vessel is 75 m in length which has a corresponding far-field range of r ff = λ, (19) which translate to a range of 45 km at 12 GHz. The behaviour of σ(r) as a function of range as well as a function of observation angle is examined in what follows. It must be emphasized that the results were obtained without the presence of a lossy ground plane to mimic the ocean surface. This has no bearing on the conclusions herein, however a lossy ground plane should be included when modeling a ship in the ocean since this would account, at least partially, for multipath contributions and the associated interference (scintillation) as a function of range. This is worth examining carefully, rough ocean surfaces can increase the RCS of a ship by as much as 12 db [9]. First consider the RCS of QUEST when viewed from φ = 18 (aft) as well as φ = 33 with the observation point situated at a constant height of 3 m above sea level. 3 This implies that 3 The observation angle of 33 lies 3 to starboard of the fore direction, a region where no strong specular contributions are observed and where PO should still yield reasonably good accuracy. DRDC Ottawa TM

40 the incidence angle θ i, although always very close to grazing, is a function of range r and given by θ i = arccos 3 r. (2) The results are shown in Figure 16 for the frequencies of 8, 12 and 16 GHz for ranges of 1 to 1 km corresponding to θ i = 88.3 and θ i 9, respectively. Note how the far-field (steady state) value of the RCS is recovered at much larger ranges in the case of aft illumination, despite the fact that for a beam of 16.4 m the corresponding far-field limit are 14, 22 and 29 km respectively at 8, 12 and 16 GHz. This apparent failure of the far-field estimate may be partly due to the complex nature of the target (compared to a simple flat plate, say) and the presence of strong specular components when φ = 18. In comparison, the results for φ = 33 recover the far-field values at much shorter ranges. More interesting perhaps, and this applies to both cases, is the behaviour of the RCS values as a function of frequency. Indeed, from the RCS of a specular reflector of area A at boresight see [5, Annex C] for details given by σ = 4π A2 λ 2, (21) one anticipates the RCS to be proportional to the square of the frequency. However, this behaviour is clearly not observed in Figure 16 where, even at steady state, the RCS value at 16 GHz does not clearly dominate the ones at 8 and 12 GHz. Moreover, the relative behaviour of σ at different frequencies is seen to be quite complex in the near-field of the target. Finally, also note the slight numerical instability at ranges in the neighborhood of 1 km which is intriguingly predominant at the frequency displaying the lowest RCS. This is more than likely an indication that the implementation of the various phase terms needs to be refined to minimize numerical errors. In both instances, one must take notice of the large variations in σ(r) at ranges below 1 km, the ranges at which RCS measurements are typically made. Finally, the azimuthal behaviour of σ for QUEST as a function of angle is examined. Figure 17 shows the RCS of QUEST at 12 GHz for the cases of far-field observation and near-field observation at 1 km. As anticipated, the near-field values can differ quite strongly from the far-field ones. Importantly, it can be seen that they can be much higher ( φ = 6, φ = 3 ) or much lower (φ = 12, φ = 24 ) than the corresponding far-field values. Such variations are probably due to the nature of the dominant scattering centers at the aspect angle of interest and their behaviour as the observation distance changes. Furthermore, at a given range, the distinction between the near-field and far-field formulation will be heightened as the dimensions of the scatterer increase. 26 DRDC Ottawa TM

41 r [km] σ(r) r [km] [dbsm] σ(r) r [km] Near-field RAPPORT, 8 GHz [dbsm] σ(r) r [km] Far-field RAPPORT, 8 GHz [dbsm] σ(r) r [km] Near-field RAPPORT, 12 GHz [dbsm] σ(r) r [km] Far-field RAPPORT, 12 GHz [dbsm] σ(r) Near-field RAPPORT, 16 GHz [dbsm] Far-field RAPPORT, 16 GHz 8 7 σ(r) [dbsm] r [km] (a) φ = 18 (aft) 5 4 σ(r) [dbsm] r [km] (b) φ = 33 Figure 16: The RCS σ(r) of CFAV QUEST as a function of range when (a) φ = 18 (viewed aft) and (b) φ = 33 when the observation point is situated 3 m above ground. Results are shown for frequencies of 8 GHz, 12 GHz and 16 GHz. The horizontal dashed lines provide the far-field values. DRDC Ottawa TM

42 Far-field RAPPORT Near-field RAPPORT, 1 km range RADIAL SCALE σ [dbsm] φ= Figure 17: The computed RCS σ(r) of QUEST for far-field observation (red) and observation at a range of 1 km (blue) as a function of azimuth φ when θ = 89.6 and f = 12 GHz. 28 DRDC Ottawa TM

43 5. Conclusion The refined physical optics formulation previously described in [5] was validated using flat 4λ and 1λ square plate geometries. The enclosed results suggest that the proposed formulation extends the regions of validity of PO from the far-field boundary to, approximately, the nearfield boundary. They also demonstrate that the angular region of validity for PO, accepted as extending roughly 3 from boresight in the far field, remains the same in the near field using the new formulation. These findings apply both for the monostatic and bistatic cases. Implementation notes for RAPPORT were also provided; it was shown that the existing code could be easily modified to support near-field computations. The modified version of RAPPORT was used to successfully duplicate previous results obtained using the flat plates, thereby providing a measure of validation. Sample RCS results were also provided for CFAV QUEST which clearly demonstrate how the RCS at least as defined herein varies in the near-field of the ship as a function of range. These results also showed that the RCS does not necessarily scale proportionally with frequency for complex geometries. A polar RCS plot was also used to show that the near-field RCS may be lower or higher than the far-field estimate at short ranges. These variations are probably related to the nature of the scattering centers involved and may merit further study. Further work will be aimed at examining the differences between near-field RCS estimates and far-field estimates for actual geometries of interest. Modifications must also be made to the code to enable it to perform computations in the near field of a scatterer in the presence of a lossy ground plane mimicking a smooth ocean surface. Finally, the numerical implementation should be examined more carefully to isolate the cause of numerical instability observed at medium to long ranges. DRDC Ottawa TM

44 References 1. Velaparambil, Sanjay, Chew, Weng Cho, and Song, Jiming (23). 1 Million Unknowns: Is it That Big?. IEEE Antennas and Progation Magazine, 45(2), Stone, W. Ross, (Ed.) (22). The Review of Radio Science , Somerset, NJ: Wiley-Interscience. 3. Brand, M. G. E. (1996). Radar signature Analysis and Prediction by Physical Optics and Ray Tracing. The RAPPORT code for RCS prediction. TNO Physics and Electronics Laboratory. (TNO report FEL-95-A97). The Hague, The Netherlands. 4. Kouyoumjian, R. G. and L. Peters, Jr. (1965). Range Requirements in Radar Cross-Section Measurements. Proceedings of the IEEE, 53, Legault, Stéphane R. (23). A Refined Physical Optics Formulation for Near-Field Computations. (DRDC Ottawa TR 23-83). Defence R&D Canada Ottawa. 6. Gordon, William B. (1996). Near Field Calculations with Far Field Formulas. In 1996 IEEE Antennas and Propagation Symposium Digest, Vol. 2, pp Peterson, Andrew F., Ray, Scott L., and Mittra, Raj (1998). Computational Methods for Electromagnetics, New York: IEEE Press. 8. Gordon, William B. (1975). Far-Field Approximation to the Kirchoff-Helmholtz Representations of Scattered Fields. IEEE Transactions on Antennas and Propagation, 23(5), Shtager, Evgeny A. (1999). An Estimation of Sea Surface Influence on Radar Reflectivity of Ships. IEEE Transactions on Antennas and Propagation, 47(1), Harrington, Roger F. (1961). Time-Harmonic Electromagnetic Fields, New York: McGraw- Hill. 3 DRDC Ottawa TM

45 Annex A Electric field expressions An approach to obtain radiated fields from electric currents is to use a magnetic vector potential formulation. It can be shown [1] that for surface electric currents, the scattered electric field may be written as E(r) = jkz (1 + 1k ) 2 S J(r ) e jkr 4πR ds (A.1) where the integral is the vector magnetic potential A(r). The above is first developed without resorting to any approximations. A.1 Exact formulation Expanding out the differential operators in (A.1), one obtains { E(r) = jkz where S ds J(r )g (r, r ) 1 k 2 1 k 2 S S ds R(R J(r )) ( jk ds J(r ) R + 1 R 2 ( k 2 R 2 j3k R 3 3 R 4 g (r, r ) = e jkr 4πR = e jk r r 4π r r ) g (r, r ) ) } g (r, r ), (A.2) (A.3) is the free-space Green function. The distance from source to observation is R = r r. Defining I 1 (r, r ) = ( 1 j kr 1 ) k 2 R 2 I 2 (r, r ) = 1 k 2 ( k 2 j3k R 3 R 2 g (r, r ), ) g (r, r ), (A.4a) (A.4b) then (A.2) is more compactly written as { } E(r) = jkz ds J(r )I 1 (r, r ) ˆR( ˆR J(r ))I 2 (r, r ). (A.5) S Since the MoM surface grid used to solve the problem has a resolution of λ/1, that is 4 4 surface elements, the integration procedure can be approximated with a Riemann sum by assuming constant currents over each element S n. Using the center r n of each element as sampling point, expression (A.5) then becomes E(r) = jkz { } J(r n)i 1 (r, r n) ˆR( ˆR J(r n))i 2 (r, r n) S n, (A.6) n DRDC Ottawa TM

46 where R is a function of both r and r n. Letting (x 1, x 2, x 3 ) = (x, y, z), a general expression for the vector components of the electric field is E xi (r) = jkz { } J xi (r n)i 1 (r, r n) (ˆx i ˆR)( ˆR J(r n))i 2 (r, r n) S n, (A.7) n where i = 1, 2, 3. The x 3 or z component simplifies for a plate in the xy plate since J z vanishes. Then E x3 (r) = jkz (ˆx 3 ˆR)( ˆR J(r n))i 2 (r, r n) S n (A.8) n with (A.7) applying for the x (or x 1 ) and y (or x 2 ) components. This simplification also applies to all the subsequent cases but will not be repeated. A.2 Far-field approximation Assuming r 1, one can then approximate the free-space Green function as [5] g (r, r ) = e jk ˆR R 4πR e jkˆr R 4πr = g ff (r, r ), (A.9) where ˆr R = r ˆr r. For large r, the expression for the electric field [5, Annex A] can then be simplified to E(r) = jkz (1 + 1k ) 2 J(r ) e jkr S 4πR ds jkz (Ī ˆrˆr) J(r )g ff (r, r )ds (A.1) and the components of the electric field may be written as S E xi (r) = jkz n { Jxi (r n) (ˆx i ˆr)(ˆr J(r n)) } g ff (r, r n) S n, (A.11) where i = 1, 2, 3. A.3 Expression with arbitrary expansion point The expressions for the fields when using a Green function with an arbitrary expansion point are now provided. Following the approach described in [5, Section 5], define ρ n = r r n r r n (A.12) and the Green function becomes g ff (r, r ) = e jkˆr R 4πr ˆr ˆρ n e jk ˆρ n R 4πr = g nf (r, r ; r n). (A.13) 32 DRDC Ottawa TM

47 The associated field expression is now [5, Section 5] E(r) jkz (Ī ˆrˆr) J(r )g ff (r, r )ds S ) ˆr ˆρ n jkz (Ī ˆρ n ˆρ n J(r )g nf (r, r ; r n)ds. (A.14) The field components may now be written as S E xi (r) = jkz n { Jxi (r n) (ˆx i ˆρ n )(ˆρ n J(r n)) } g nf (r, r n; r n) S n i = 1, 2, 3 (A.15) A.4 Expressions with arbitrary expansion point refined phase and magnitude This is essentially the same as above save that a more refined amplitude approximation is used for the Green function; the amplitude ρ n is now used instead of r (see [5, Section 5.2]). Explicitly, g nf2 (r, r ; r n) = e jk ˆρ n R (A.16) 4πρ n and the field expressions are E xi (r) = jkz n { Jxi (r n) (ˆx i ˆρ n )(ˆρ n J(r n)) } g nf2 (r, r n; r n) S n, (A.17) with i = 1, 2, 3. DRDC Ottawa TM

48 Annex B Method of moments: Impedance matrix elements The method of moments code used for the 4λ plate is based on the formulation presented in [7, Chapter 1]. Since final expressions for the impedance matrix elements A mn, B mn, C mn and D mn are not provided therein, they are provided together with the expressions required for the singular integrands. To simplify the presentation, we define which are used in the integration limits and x = x m x n, y = y m y n, (B.1) R = x 2 + y 2, (B.2) where x and y are dummy surface integration variables. B.1 The A mn elements The matrix elements A mn may be written as A mn = A 1 mn + A 2 mn, (B.3) where and A 1 mn = j 3 ak A 2 mn = j3ak y+b/2 y b/2 y+b/2 y b/2 { x a/2 2 x 3a/2 { x a/2 x 3a/2 x+a/2 x a/2 + x+3a/2 x+a/2 ( x ) 2a + x2 2a x+3a/2 x+a/2 } x+a/2 x a/2 e jkr R dxdy (B.4) ( ) 3 4 x2 a 2 ( x ) } 2a + x2 e jkr 2a 2 dxdy. (B.5) R The above integrals must be evaluated numerically using (say) Gaussian quadrature. The process is straightforward unless a singularity of the integrand whenever both x and y go to zero for the above integrands is captured within the integration limits. In this case the integration per se remains well defined since the singularity of the integrand is integrable. However, this may present difficulties for most quadrature techniques and it may be required to properly extract the singular portion of the integrand. The singularity in an integrand can be subtracted to produce the combination of a non-singular portion which can be evaluated numerically and a singular portion which can be evaluated analytically in closed form. The procedure is straightforward for all of the above integrals. Suppose the limits are defined by x, x 1, y and y 1 for convenience. We also define the rectangular surface area S = {(x, y) (x, y) [x, x 1 ] [y, y 1 ]} enclosed by these limits. As mentioned 34 DRDC Ottawa TM

49 above, for any of the above integrals each of which covers a different surface area due to the differing limits the integrand has an integrable singularity if S. This difficulty is sidestepped by subtracting the singular portion of the integrand as (x, y) (, ) to obtain a well behaved expression and adding the required compensating term which, in many cases, can be evaluated in closed form. Explicitly, we have S R dxdy = e jkr S e jkr 1 dxdy + R S 1 R dxdy = S e jkr 1 dxdy + I. R The quantity I represents the closed form evaluation of the 1/R integral; it may be written as 1 I = R dxdy = y 1 ln x 1 + R 11 + x 1 ln y 1 + R 11 y ln x 1 + R 1 x ln y 1 + R 1 x + R 1 y + R 1 x + R y + R where S R pq = x 2 p + y 2 q. (B.6) (B.7) (B.8) When filling the elements of the impedance matrix, the limits of each integral in (B.4) must be checked to verify if the singularity at (x, y) = (, ) is captured. If so, the corresponding integral in (B.4) must be replaced with (B.6) as required. If no singularity is captured, then expression (B.4) can be directly numerically integrated. This procedure must also be carried for the integrals found in (B.5). The corresponding representations required for singular integrands are ( x ) 2a + x2 e jkr {( 9 2a 2 R dxdy = 8 + 3x ) 2a + x2 2a 2 e jkr 9 } 1 8 R dxdy I, S S ( ) 3 4 x2 e jkr a 2 R dxdy = S S (B.9) {( ) 3 4 x2 a 2 e jkr 3 } 1 4 R dxdy I. (B.1) For instance, if the second integral in (B.5) had a singularity within its surface of integration, the first and last integrals could be evaluated directly using the expressions in (B.5) while the second one would have to be replaced with expression (B.1). B.2 The B mn elements More straightforward than the A mn terms, the matrix elements B mn may be written as B mn = j 3 bk [ y { x x+a } e jkr y b x a x R dxdy y+b { x y x a x+a x } e jkr Singularities can be dealt with by replacing the relevant integral with (B.6). R ] dxdy. (B.11) DRDC Ottawa TM

50 B.3 The C mn elements From symmetry, the matrix elements C mn follow directly from B mn since C mn = b a B mn. (B.12) B.4 The D mn elements The matrix elements D mn are closely related to the A mn elements; one can obtain one from the other by letting x y. Hence, we obtain D mn = D 1 mn + D 2 mn, (B.13) where and D 1 mn = j 3 bk D 2 mn = j3bk x+a/2 x a/2 x+a/2 x a/2 { y b/2 2 y 3b/2 { y b/2 y 3b/2 y+b/2 y b/2 + y+3b/2 y+b/2 ( y ) 2b + y2 2b y+3b/2 y+b/2 } y+b/2 y b/2 e jkr R dydx (B.14) ( ) 3 4 y2 b 2 ( y ) } 2b + y2 e jkr 2b 2 dydx. (B.15) R These have the same form as the integrals required for the impedance matrix elements A mn ; expressions (B.6), (B.9) and (B.1) therefore suffice to deal with all the singular integrands encountered in this formulation. 36 DRDC Ottawa TM

51 Annex C Method of moments: Partial convergence study The widely accepted rule of thumb for minimum discretization levels in MoM is a surface element density of ten elements per wavelength for smooth scatterers. This may however be unacceptable in the case of geometries presenting sharp edges or discontinuities due to the singular behaviour of surface currents in the neighborhood of such features. The surface currents near the edges of a plate depicted in Figure 2 provide a good example. Their quickly varying nature is difficult to approximate and, unless specially designed basis functions are used, refining surface element meshes may lead to appreciable differences locally at least in surface current estimates. However, as discussed below, a density of ten surface elements per wavelength is sufficient given the nature of the comparisons made in this work. To verify the adequacy of the proposed ten elements per wavelength mesh, a short comparative study was performed using ten and twelve elements per wavelength. Uniform meshes are used for all cases considered here. The resulting RCS was computed using the exact integral representation (A.7) at ranges of 2, 1, 5 and 1 m. The corresponding curves are shown in Figure C.1. It is readily observed that the larger discrepancies occur in the regions where the diffraction currents are more important contributors: those for angles of incidence near grazing, especially at shorter observation ranges. This is expected following the above discussion since refining the mesh has more impact on the rapidly varying edges currents but less so on the slowly varying specular currents. The upshot is that refining the mesh has little or no effect for angles of incidence lesser than 3, the region where specular contributions dominate and where PO provides the best accuracy. Even in the most extreme case of grazing incidence with an observation range of 2 m [Figure 1(d)], both curves are very similar qualitatively. This suggests that the results obtained with the ten elements per wavelength mesh are sufficient for the purposes of this study; any benefits reaped from higher mesh densities would essentially occur in areas where PO itself is highly inaccurate. DRDC Ottawa TM

52 θ i [deg] σ θθ θ[dbsm] i [deg] σ θθ [dbsm] 1 elements per λ 12 elements per λ 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ i [deg] (a) r = 1 m θ i [deg] (b) r = 5 m 4 4 σθθ [dbsm] 2 σθθ [dbsm] θ i [deg] (c) r = 1 m θ i [deg] (d) r = 2 m Figure C.1: The monostatic co-polarized RCS of a 4λ 4λ square plate (λ = 1 m) as a function of incidence angle θ i observed at various ranges for a θ-polarized incident wave. Curves are provided for method of moments solutions using 1 and 12 elements per wavelength. 38 DRDC Ottawa TM

53 UNCLASSIFIED SECURITY CLASSIFICATION OF FORM (highest classification of Title, Abstract, Keywords) DOCUMENT CONTROL DATA (Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified) 1. ORIGINATOR (the name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g. Establishment sponsoring a contractor s report, or tasking agency, are entered in section 8.) Defence R&D Canada Ottawa Ottawa, Ontario, K1A K2 2. SECURITY CLASSIFICATION (overall security classification of the document, including special warning terms if applicable) UNCLASSIFIED 3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S,C or U) in parentheses after the title.) Validation and implementation of a near-field physical optics formulation (U) 4. AUTHORS (Last name, first name, middle initial) Legault, Stéphane R. Louie, Aloisius 5. DATE OF PUBLICATION (month and year of publication of document) August 24 6a. NO. OF PAGES (total containing information. Include Annexes, Appendices, etc.) 5 6b. NO. OF REFS (total cited in document) 7. DESCRIPTIVE NOTES (the category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report, e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.) 1 Technical Memorandum 8. SPONSORING ACTIVITY (the name of the department project office or laboratory sponsoring the research and development. Include the address.) Defence R&D Canada Ottawa Ottawa, Ontario, K1A Z4 9a. PROJECT OR GRANT NO. (if appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant) 9b. CONTRACT NO. (if appropriate, the applicable number under which the document was written) Project 11au13 1a. ORIGINATOR S DOCUMENT NUMBER (the official document number by which the document is identified by the originating activity. This number must be unique to this document.) 1b. OTHER DOCUMENT NOS. (Any other numbers which may be assigned this document either by the originator or by the sponsor) DRDC Ottawa TM DOCUMENT AVAILABILITY (any limitations on further dissemination of the document, other than those imposed by security classification) ( x ) Unlimited distribution ( ) Distribution limited to defence departments and defence contractors; further distribution only as approved ( ) Distribution limited to defence departments and Canadian defence contractors; further distribution only as approved ( ) Distribution limited to government departments and agencies; further distribution only as approved ( ) Distribution limited to defence departments; further distribution only as approved ( ) Other (please specify): Distribution limited to the Canadian Department of National Defence 12. DOCUMENT ANNOUNCEMENT (any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution (beyond the audience specified in 11) is possible, a wider announcement audience may be selected.) Unlimited UNCLASSIFIED SECURITY CLASSIFICATION OF FORM DCD3 2/6/87

54 UNCLASSIFIED SECURITY CLASSIFICATION OF FORM 13. ABSTRACT ( a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual). Previously proposed modifications to physical optics for improving the accuracy of scattered field computations are validated within the context of actual scattering problems. Results suggest that the procedure can potentially extend the region of validity of physical optics from the far field to short ranges lying well within the near field. Furthermore, in the case of flat plates, the angular region over which accurate results are obtained remains approximately the same even at shorter ranges. Finally, computations using an actual ship model verify that the improved formulations captures variations in the scattered field which would otherwise go unnoticed using a standard formulation. 14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus-identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.) radar cross section high-frequency methods physical optics near-field observations UNCLASSIFIED SECURITY CLASSIFICATION OF FORM

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