Enhancement of Radar Detection Accuracy Using H-Beam Wave Polarization in Random Media

This work addresses the range in which the accuracy of object identification is enhanced regardless of radar parameters. We compute the radar cross-section (RCS) of conducting objects in free space and random media. We use beam wave incidence and postulate its coherency with a finite width around an object located in the far field. Accordingly, we examine the impact of radar parameters on the RCS, where these parameters include the incident angle, target size and complexity, medium fluctuation intensity and the beam size of the incident waves. H-wave polarization is assumed for the waves’ incidence.


Introduction
Calculation of electromagnetic waves from objects has been considered by researchers over the years [1][2][3]. The scattering waves problem was analyzed through proposing a numerical method as a boundary value problem in the far field [4][5][6]. In order to formulate the scattering waves, this method assumes an operator that uses the current generated on the surface of the object. Assuming the approximation of scalar waves, we can use this method efficiently in free space as well as in a random medium for an arbitrary contour cylinder. Accordingly, we can call this a current generator method (CGM).
The impact of incident wave features on the RCS of a limited-size target is evident, especially in a disordered medium [7]. Specifically, one of these features is the nature of the incident waves when it is either plane or beam waves. The more coherent components of rays scattering from the illumination region are added up and compared to the case of an incident beam wave that has a bounded width surrounding the target. CGM is a normalized numerical method useful to present a solution for the scattering problem in a wide range of frequencies in the far field. Our results agree excellently with the case that postulates a cylinder having a circular shape in the free space [8]. Years ago, CGM precision was also confirmed through a comparison with the FDFD method, and good compatibility was shown, wherein the error rate was less than 5% for cylinders in disordered media, while the error margin was quite limited in the free space [9,10].
In this paper, we work on accuracy enhancement of the remote sensing of conducting targets in disordered media, such as planes in turbulence. Our study handles various radar parameters, including the illumination region curvature, object size and complexity, and medium fluctuation intensity. In [11], we have used a plane wave incidence where it is quite hard to maintain its infinite width around a large object's cross-section in the far field. Therefore, we should consider a more practical detection technique where we utilize a beam wave with a finite beam width that can surround a large object. In this regard, we use the CGM to calculate the laser radar cross-section (LRCS) [12] of a conducting target in turbulence. Illuminations from non-convex surfaces of the object are assumed, where the size of the object is about 2.5 times greater than the wavelength for wave propagation through free space. We use the spatial coherence length (SCL) of the waves surrounding the object to refer to the inhomogeneities and the fluctuation intensity of the disordered medium. We postulate a horizontal polarization (H-wave incidence) and solve the problem in a two-dimensional configuration. The time factor exp(-iwt) is considered, but is eliminated in the formulation part.

Formulation
The problem's geometry is shown in Figure 1. We consider a cylinder of the mean size a surrounded by a sphere of a random medium having a radius L in which a L. In this problem, we postulate that the incident field is a beam wave generated and transversely parallel to the y-axis, which is the axis of the cylinder, using a line source f (r ). We can assign the wave scattering as u s (r), the wave incidence as u in (r), and the total wave as u(r) = u in (r) + u s (r). In this formulation, we can consider the Neumann condition in (1) based on the assumption of having incident waves with a horizontal polarization (H-wave incidence). Accordingly, we can obtain the wave u(r) on the contour S of the object. That is, Here, we use ∂/∂n to indicate the positive normal derivative on S at r. In (1), u(r) is representing E x .  Drawing on the condition in (2), we can solve the scattering problem using a twodimensional configuration. Hence, r can be expressed as r = (x, z).
Here, B(r, r) represents the local intensity and l(r) represents the local scale-size of the medium fluctuations, and k = ω √ ε 0 µ 0 is the free space wavenumber. When B(r, r) 1, the medium has very low fluctuation intensity while kl(r) 1 indicates that the medium particles have no sharp edges. Thus, we can presume that both the scalar waves and also the forward scattering approximation are possibly applied [13]. In this case, wave propagation will not undergo severe transitions. Therefore, re-incident wave contributions are hence trivial [4]. Consequently, we can ignore the depolarization of propagating waves. Practical conditions in (2) are applicable in turbulence.
We postulate a cylindrical conducting object, and its surface can be formulated by Here, δ and φ are the concavity and the rotation indexes, respectively. Aircraft may have a contour of a partially convex surface expressed by (3). Drawing on the current generator Y H , the scattered wave can be represented by where G(r | r ) is the coherent Green's function in the disordered medium defined by [4]: where the angular brackets indicate the ensemble average, G 0 is the free space Green's function, while the coherence attenuation index α(L) is expressed as where l 0 and B 0 are constant within the medium, as depicted in Figure 1. Surface currents can be obtained though applying the Y H operator on the incident waves covering S, and this relies on the object contour complexity. Y H is formulated using wave functions fulfilling the radiation condition, as well as the Helmholtz equation. Accordingly, we can get the surface current by We can refer to the "reaction" represented in (8), as shown in [14]. In (8), Φ M are the basis (modal) functions that comprise a set of wave functions which are valid with the radiation condition and the Helmholtz equation for propagation in free space M denotes the complex conjugate, and Φ T M represents the transposed vectors of Φ M . Here, M = 2N + 1 constitutes the total mode number.
, A H is a Hermitian matrix which is positive and definite and can be expressed as The element m, n is the inner product of φ n and φ m : Y H approaches the accurate operator in terms of the mean when M → ∞. For Hpolarization, we can obtain the average intensity of the monostatic wave as where W is the beamwidth. We can solve (11) approximately assuming M 22 as [11].
in which, where µ and γ are medium parameters and defined as It should be noted that α(L) > 2 in Equation (6), and this is required to attain the validity of Equation (12). In doing this, we assume B 0 to be a constant in the range of B 0 = 5 × 10 −ν where ν = 5, 6. As shown in Figure 1, L is the medium size. Here, n is an index which should be a positive value, and it indicates the thickness of the turning boundary between the free space and the disordered medium; n = 8 3 is assumed in Section 3. Accordingly, (11) Uisng (22), we can get the LRCS σ b expression as

Numerical Results
In a random medium, we can express the degree of the spatial coherence as [5]: where r 0 = (0, 0), r 1 = (ρ, 0), r 2 = (−ρ, 0), r t = (0, z). Here, we postulate B(r, r) = B 0 and kB 0 L = 3π; thus, α defined in Equation (6) is 56.28 π 2 and 81.77 π 2 for kl = 53 π and 77 π, respectively, and this implies that waves are quite incoherent within the medium. As a result, we can define SCL to equal 2kρ where | Γ | 0.37. The relationship between kl and SCL is shown in Figure 2, and accordingly, SCL has sizes equal to 5 and 6 owing to the presumed kl values. SCL is used to illustrate the medium impcat on the LRCS. Integrations expressed in (11) are computed by using the trapezoidal rule.

RCS Using E-Plane Wave Incidence
In Figure 3 and with incident plane waves, we show results in free space and three different SCLs assuming E-polarization. Our results agree perfectly with those shown in [8,15] where [8] obtained results for free space only. These results validate our forthcoming results for a beam wave incidence.

LRCS in Free Space
In Figure 4, LRCS results for an object in free space are presented. LRCS suffers from relatively high oscillations within ka < 8, and this is owing to the creeping waves effect in the resonance range. As ka magnifies, the strength of these oscillations reduces. With a more complex contour in terms of δ, the difference between illumination region scattering is more obvious due to the impact of the inflection point contributions. However, the performance of LRCS is almost indifferent for the incident angle (θ) at the high frequency band where ka > 8.

LRCS in a Random Medium
In Figure 5 and compared to [16], extended numerical results for LRCS of a convex portion of the conducting object are presented. We can observe that LRCS oscillates gradually while lessening with ka. The impact of the medium fluctuations together with the creeping waves produced on the surface of the target contour [17] result in such LRCS behavior. The impact of the creeping waves on the behavior of LRCS is more evident with H-wave polarization than for E-wave incidence.  At ka ≤ 3, scattered waves undergo the resonance region effect, where the incident surface is a typical convexity illuminated and covered by a beam wave when 2 kW > ka. Therefore, the beam illumination typically behaves as being nearly a plane incidence in the far field surrounding the target cylinder. In the meantime, these wide LRCS fluctuations are due to the influence of the generated waves that creep around the object in addition to the direct incident waves.
When ka increases, the effect of kW on the scattered waves becomes more obvious. In other words, the amount of scattered rays add up more components to the scattered waves based on the size of kW. When kW is large enough, more coherent components are added, and hence, the LRCS is greater. Generally, and irrespective of the object complexity δ, the wave coherence length SCL and the beamwidth 2 kW, LRCS behaves similarly. This is because the restriction of the illumination area and the amount of stationary points do not change much, and accordingly their waves contributions are quite close. When ka > 2 kW, it is noticed that LRCS minimizes with a bigger target size, and hence, the beam wave power is barely capable of radar detection; this is as a result of the surface current shortage.
Next, we extend our numerical results in [18] for LRCS in Figure 6, assuming a concave illumination region of the conducting target. We observe that LRCS behaves like a mountain reaching a peak at a similar ka value in the Mie region. LRCS enlarges to a maximum inflection point, and then it reduces in turn progressively with ka irrespective of the SCL size that implements the effects of the medium's fluctuations strength. Such LRCS performance versus ka is due to the inflection points contributions together with the stationary points as well [19,20]. Some similarity can be noticed when we compare this performance with the convex illumination region shown in Figure 5 and our work published in [21]. Due to the excrescent amount of contributions from the expanded illumination area, the LRCS maximum value is greater for the concave illumination region. Additionally, we can observe that LRCS suffers from substantial oscillations produced as a result of the impact of the randomness intensity, as well as the creeping waves, as pointed out earlier. As shown in Figures 5 and 6, when SCL enlarges, the scattering rays contribute more collectively to the coherent part of the waves, and as a result the LRCS augments. It should be noted that such LRCS elevation also occurs when kW is wider, as the characteristics of the beam wave incidence around the cylinder in the far field is similar to the the type of plane wave incidence, particularly when kw → ∞. The effects of kW and SCL are more clear when we consider the size of SCL to be quite wide around the object, as shown in Figure 7, where SCL = 30. In this regard, the behavior of LRCS has a similar performance as the plane wave propagation in [11].
Additionally, surface curvature of the object is one parameter that has a seeming impact on the scattered waves' intensity. Specifically, the influence of the concavity index δ is obviously noted when it is bigger, leading to the LRCS being magnified, particularly the amount of inflection point contributions from the concave portion, and this is owing to the illumination surface expansion.
From all results shown and discussed above, we can understand that radar parameters, including incident angle, medium fluctuations intensity, object size and complexity, in addition to beam width, obviously affect the performance of LRCS. To improve the accuracy of the LRCS of a conducting target in a disordered medium, it is desired to have a close performance of LRCS regardless of these parameters. In doing this, we should have ka → ∞, and this implies that a λ. According to [22], if we consider ka = 16 for an aircraft such as the F/A-18 in which its a = 8 m, the frequency f approximately equals 95 MHz, and this falls in the V HF. However, RCS will undergo fluctuations, so f should be higher than that prescribed range.

Conclusions
This work presents an investigation on the radar parameters that affect waves scattering from conducting targets with finite dimensions embedded in random media. Accordingly, we define the frequency range where the remote sensing of conducting targets in a random medium is precious. In doing this, we used the method of the current generator to compute the LRCS of the object and analyzed its performance to maximize the accuracy of radar detection. We considered a beam wave incidence having finite width around the object with H-wave polarization in the far field. Different incident angles, coherence length and object configurations were assumed. This paper focuses on the radar application of radar detection of airplanes flying through turbulence.
The enhancement of radar sensing accuracy requires that the target should be detected using waves where their wavelength is quite small, at about a λ, and as a result, LRCS would be indifferent to the impact of radar parameters, including the incident angle, beam wave size, and wave fluctuation intensity. In addition, the generated waves that creep around the object influence the LRCS performance marginally when using such high frequency range. This is applicable with LRCS of targets in disordered media and also free space propagation.