Significant challenges still exist in estimating the radar characteristics of a complex electrically large target using numerical evaluation methods, despite constant development in computational capabilities and algorithms. Physical optics (PO), relying on assumptions attuned to large scale problems, has been widely adopted with efficiency and accuracy to estimate the radar cross sections (RCS) of electrically large targets. An important assumption in the simplification of the PO method is that the observation point should be far away from the source. As the standard PO code is usually optimized in far field conditions, it cannot be applied directly to the evaluation of near field scattering due to the finite distance between the source and observer.

In recent years, to overcome the shortcomings of PO in the near field zone, several approaches have been used to modify the standard PO for near field calculations. The physical theory of diffraction and the method of shooting and bouncing rays were modified with a spherical wave by Jeng for near field computation.^[1] A near field RCS calculation method with exact Green function was introduced by Neto^[2] and Chen et al.^[3] The accuracy of this algorithm is excellent, but the efficiency is undesirable for electrically large targets due to the requirement for numerical integration of the surface currents. An approach proposed by Legault overcomes this difficulty by means of locally expanded phase approximations combined with surface partitioning.^[4] Papkelis et al. derived an accurate and time-efficient analytical PO method for near field calculation of rectangular plates.^[5] The near field formulation of PO and equivalent edge currents was developed in Refs. [6] and [7]by introducing the concept of the distinct wave propagation vector. A PO near field refinement was introduced in Ref. [8] for dielectric and perfect electric conducting (PEC) targets. In Refs. [9] and [10], a near field fast PO algorithm for large convex objects was presented, using phase and amplitude compensation factors. In Ref. [11], the influence of the antenna and distance as well as spherical wave irradiation was taken into account in near field calculation. A computational method of near field RCS was proposed using the surface element as the calculating unit by Cheng.^[12,13]

In this work, we propose a modified physical optics (MPO) algorithm for the analysis of complex electrically large targets with PEC surface in the near field zone. The differences between the incident wave and antenna radiation pattern in either the near or far range are taken into consideration. A more accurate representation of the Green function in the near field is also employed in this algorithm. The proposed algorithm can be seen as a near field extension of the standard PO based on the principles presented in Refs. [3] and [5]. Each meshed facet is treated as an individual object. Every single facet has its own incident wave expressed with a spherical wave front and radiation gain described by the antenna radiation pattern. Furthermore, the far field assumption is replaced by a more accurate phase approximation of the Green function to overcome the shortcomings of the standard PO in the near field. The formulation of MPO is derived on the basis of the proposed principle and applied to discuss near field scattering characteristics. This algorithm preserves the simplicity of refined PO in Ref. [3] and is more accurate than the method in Ref. [5].

This paper is arranged as follows. In Section 2, the assumptions in the far field are introduced. The near field modifications in physical optics are derived in detail in Section 3. In Section 4, the accuracy of the presented algorithm and the near field characteristics of the targets are discussed via several numerical examples. Section 5 concludes the current work and looks forward to future work.

According to the electric field integral equation (EFIE), the scattered field E_s(r) for an electrically large PEC object of maximum dimension D with surface S and surface normal , is given by^[14] 1 where S_lit denotes the lit portion of the surface; Γ is the boundary of the lit regions and the shadow regions; E and H are the total electric field and the total magnetic field in dS, respectively; and ψ₀ is the free space Green function 2 In far field conditions, the observation point is far away from the source points, so the following approximations are taken.

1) The unit incident wave vector is directed towards the origin of the centroid coordinate and the unit scattering direction vector is parallel to the observation point r′.

2) The antenna electric field radiation intensity of every facet in the lit regions is equal and the scattered power contributed by each single facet is equal.

3) The Green function may be approximated as follows:

(i) |r| = r, |r′| = r′, and r ≫ r′, so the amplitude term can be simplified as |r−r′| ≈ r.

(ii) As is well known, the scattered field is more sensitive to change of phase, so higher accuracy is required in its approximation. By using a Taylor series, the phase term is expressed as 3 Discarding terms o(1/r) in the series, the Green function is written as 4 The corresponding gradient of the Green function is approximated as^[15] 5

Only in the far field, such as r → ∞ or r′ → 0 for finite r, can the phase approximation of the proposed simplification be qualified. According to the criterion presented in Ref. 16, a maximum phase error is given by ∇ φ = π/8 rad, and the acceptable minimum observation range r_ff is estimated using 6 where λ is the wave number. r_ff is thus proportional to the square of max(r′). In other words, the accuracy of the simplified Green function decreases with increasing r′ for a given observation point.

Based on the presented approximations, equation (1) can be simplified as 7

In this section, an MPO algorithm is described to compute the near field scattering. Corresponding to the three assumptions proposed in the far field, more accurate approximations are exploited and refined formulations qualified for near field calculations are derived.

In this section, the main performance of the modified algorithm is demonstrated via several representative simulations. The section is divided into two parts. In the first one, the accuracy of the modified physical optics algorithm is verified in both the angle and range. The second part focuses on the variations of the near field scattering characteristics with range, frequency, and azimuth angle. In all the following examples, the structure is illuminated by a vertically polarized wave without special declaration. All of the following calculations are performed on a PC with Quad 3.20 GHz CPUs and 3.94 GB memory.

4.1. Prediction accuracy of modified physical optics

Let us begin with the accuracy analysis of the modified physical optics in far and near field scattering predictions. The results for monostatic RCS of a PEC cylinder with 20 cm diameter and 30 cm length are shown in Fig. 1 for the far field and the near field respectively. Figures 1(a) and 1(b) provide plots of RCS as a function of pitch angle from 0° to 180°. The frequency is 10 GHz.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) The far field monostatic co-polarized RCS of a PEC cylinder with 20 cm diameter and 30 cm length as a function of pitch angle. In (a), the observation is made at r ≫ rff. The RCS by MLFMM solution and the MPO solution are presented. In (b), the observation is made at r = 0.3 × rff. The RCS by measured data[17] and the MPO solution are presented.

In Fig. 1(a), the far field RCS by the MPO fits well with the results calculated by the MLFMM in the commercial software FEKO. Figure 1(b) shows the RCS by MPO and measured data^[17] at 30% of r_ff and an agreement is obtained. It is seen that MPO performs well both in the near and far scattering field calculations. The major error in the calculated results is most probably due to the absence of consideration of edge scattering and creeping waves.

The results of the monostatic RCS for a 10 m × 10 m square plate varying with range are illustrated in Fig. 2. The frequency is 300 MHz and the observation range varies from 1 m to 10 km in the condition of θ = 90° and φ = 0°. In Fig. 2, excellent agreement is obtained between the RCS by MPO and the data in Ref. [18] calculated by the method of moment.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) The monostatic co-polarized RCS of a 10λ × 10λ square plate (λ = 1 m) as a function of observation distance when θ = 90° and φ = 0°. The RCS by method of moments solution in Ref. [16] and the MPO solution are presented.

4.2. Near field scattering characteristics of a generic missile

In the following examples, a missile-like target with PEC surface is studied. The size of this target with maximum extent in the rectangular coordinates (x, y, z) are 5.58 m, 2.49 m, and 1.06 m.

Firstly, the computational accuracy and efficiency of the proposed algorithm are studied compared with the very popular MLFMM in FEKO. The results of the monostatic RCS for the generic missile as a function of range are provided in Fig. 3. The frequency is set at 3 GHz for ranges of 10–1000 m. The pitch angle is set at θ = 90° and φ = 90°. The computational costs (including RAM and CPU time) for the two solutions are listed in Table 1.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) The monostatic co-polarized RCS of a missile-like target as a function of observation distance when θ = 90° and φ = 90°. The RCS by MLFMM solution in FEKO and the MPO solution are presented.

Table 1.

Table 1.

Table 1. Comparison of the calculation time and memory. . Solution MLFMM MPO Time/s 2223.80 12.91 Memory/MB 5538.21 3.28

Table 1. Comparison of the calculation time and memory. .

As sketched in Fig. 3, the results obtained by MLFMM and MPO have good agreement in most of the distances. The deviation of the plots is most probably due to neglecting edge diffraction and the creeping wave. Besides, it is easy to find from Table 1 that, compared with MLFMM, the MPO solution has a great saving in computing resources.

Next, the RCS of the missile-like target is calculated when viewed from the top (φ = 0°) and the broadside (φ = 90°) of the missile. The computed results are shown in Fig. 4 with the frequency set at 1 GHz, 2 GHz, and 3 GHz for ranges of 1–10000 m. The pitch angle is set at θ = 90°.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) The monostatic co-polarized RCS of missile as a function of range when (a) viewed from the top (φ = 0°) and (b) viewed from the broadside (φ = 90°). Results are shown for frequencies of 1 GHz, 2 GHz, and 3 GHz. The horizontal dashed lines provide the far field values.

It is seen that the shake of RCS in near range becomes more violent as the frequency increases. This irregular concussion implies difficulties in near field detection and track. Comparing Figs. 4(a) and 4(b), the near field RCS for φ = 0° recovers the far field values at much shorter ranges and the shake of the results is more violent for φ = 90°. The complex structure and strong specular components of the missile when viewed from the broadside may be responsible for this obvious distinction.

Finally, the results for the monostatic co-polarized RCS of the missile as a function of azimuth angle are calculated. Figure 5 shows the RCS of the missile at 3 GHz in the conditions of far field observation and near field observation at r = 0.3× r_ff. As expected, the two plots presenting far field RCS and near field RCS are different from each other. The variations of the scattered power contributed by single facets and the transference of the dominant scattering centers as the observation distance changes may be the reason for the difference.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) The monostatic co-polarized RCS of missile for far field observation (red) and observation at a range of r = 0.3× rff (blue) as a function of azimuth φ when θ = 90° and f = 3 GHz.

An MPO algorithm for scattered field calculation of complex electrically large targets with PEC surfaces for source/observation in the near field and the far field is presented. By setting up local reference coordinates for each facet, the distinct wave propagation vector, the radiation gain based on the modulation of the antenna radiation pattern, and the refined Green function are exploited to adjust the standard PO in the near field scattering calculation. The MPO algorithm retains the accuracy and computational complexities under far field assumptions and performs well both in far field and near field.

Based on the methods proposed, the calculation of near field scattering can be further developed. Future extensions could involve modified equivalent edge currents for edge diffractions and modified iterative physical optics for multiple bounces in cavities.