5.1 Heterogeneous Beams—Volume Integral Equation Formulation

Oblique incidence volume integral equation formulation presented in this section is mainly based on [3]. Other relevant references for normal and oblique TM and TE incidence and volume formulation can be found in [6–10], and [11]. The cross section of the beam is shown in Fig. 5.1. Moreover, the incident field propagation vector components are and , in which , θ₀ is the angle between the incident field propagation direction and the beam axis (for normal incidence ) and ω is the angular frequency.

We start with the time harmonic {e ^jωt} Maxwell equations and introduce equivalent electric and magnetic currents to represent the polarization currents induced in a medium with relative permittivity ϵ_r(x,y) and permeability μ_r(x,y).

5.1

5.2

5.3

5.4

The z dependence of the oblique incident field is with k_z being the propagation constant in z direction and correspondingly, we assume that the z dependence of the total fields in the system is also . Thus, the total electric and magnetic fields can be described by

5.5

5.6

in which are the transverse electric and magnetic fields and are the z-components of the electric and magnetic fields. Suppose that the cylinder cross section is modeled by triangular cells of constant permittivity and permeability, but varying from cell to cell. The total scattering from the beam would be the superposition of the scattering from all these various homogeneous triangular cells. Let's define the operators and to obtain the differential equations for the magnetic field of each triangular cell:

5.7

Similarly, we obtain for the electric field

5.8

Substitution of (5.5) and (5.6) into (5.1)–(5.4) yields

5.9

5.10

5.11

5.12

Moreover, if we apply the operator on (5.9) and substitute (5.7), we obtain after some algebraic manipulations,

5.13

5.14

5.15

5.16

We can define the transverse and longitudinal equivalent polarization magnetic currents , J_mz and express them in terms of the z-directed electric and magnetic components of the fields.

5.17

Similarly, one can show that

5.18

and the transverse and longitudinal electric polarization currents and J_z are,

5.19

The equivalent polarization currents radiate in free space, but are unknown quantities since they depend on the total field components E_z and H_z. If we denote by the electric and magnetic fields due to the electric currents and the fields due to the magnetic currents by superposition we can rewrite the total fields,

5.20

5.21

We define and in which is the vector magnetic potential and is the vector electric potential. Thus, substitution of the vector potentials into Maxwell eqs. (5.1) and (5.2), and using some vector identities yields

5.22

5.23

in which Φ is the scalar electric potential

5.24

5.25

If we use Lorentz gauge , we obtain

5.26

and

5.27

Similarly, we obtain the Lorentz gauge for the magnetic currents with Φ_m being the magnetic scalar potential and

5.28

and

5.29

Finally, we get the scattered fields in terms of vector magnetic and electric potentials:

5.30

Similarly, for the magnetic field we obtain the expression

5.31

Substitution of (5.5) and (5.6) and the explicit expressions of the operators in terms of transverse and longitudinal components into (5.30), yields

5.32

From (5.32), we obtain the explicit form of the z-component

5.33

Similarly, for the magnetic scattered field we obtain:

5.34

and its associated z-component:

5.35

Given a z-directed line source with current , we can solve the differential equation for A_z in cylindrical coordinates (r, φ, z),

5.36

By symmetry considerations of the line source geometry, we can assume that . Thus,

5.37

and the solution [12] is,

5.38

Solution for J_x and J_y yields similar results for a_x(r) and a_y(r). H₀⁽²⁾(x) is the second kind Hankel function of zero order. If we repeat the differential equation solution for a magnetic line source current, we obtain the same results but for the electric vector potential components f_x(r), f_y(r)_, f_z(r). Moreover, displacement of the line source to r' location in (x,y) plane will change the solution to in which . If we extend the derivation to a 2D electric and magnetic current distribution and use superposition for the computation, we obtain the following result for the z-directed electric and magnetic fields,

5.39

and

5.40

in which * denotes convolution. The electric field integral equation (EFIE) and magnetic field integral equation (MFIE) are formulated by recognizing that the total field E_z(x, y) is equal to the summation of the incident field and the scattered field excited by the electric and magnetic polarization currents , such that

5.41

and

5.42

in which and with E₀ and H₀ being the amplitudes of the incident electric and magnetic fields, respectively. These two integral equations can be solved by method of moments (MoM) [3], using, for instance, point matching or linear basis functions B_n(x,y). Thus, the z-component of the electric and magnetic field can be expressed by

5.43

and

5.44

in which N is the number of basis functions and B_n(x,y) denotes a pyramid basis function centered at the n^th corner or node within the triangular-cell model of the cylinder cross section as shown in Fig. 5.2.

The basis function connects the triangular cells grouped around a given vertex and vanishes at every other node in the model. Within the p^th cell, there are three nodes i = 1,2,3 and the field can be described as

5.45

The coefficients a₁, a₂, and a₃ can be evaluated in terms of the field values at the cell vertices c₁, c₂, and c₃ with corresponding coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃) such that

5.46

Solution of (5.46) for (a₁, a₂, a₃) yields

5.47

with representing the area of the triangle cell. Accordingly, the basis function as shown in Fig. 5.2 can be derived from (5.45) and (5.47) for c₁ = 1, c₂ = 0, c₃ = 0.

5.48

Substitution of (5.43), (5.44) into (5.41) and (5.42) and evaluation at discrete points (x_m,y_m) with m = 1..N in the cylinder cross section yields the matrix equation

5.49

in which

5.50

5.51

5.52

5.53

Because the z-components of the fields are linear functions within this representation, transverse fields and transverse current densities are constant within a triangular cell. Let us denote

5.54

in which s_p denotes the area of the p^th triangle with a vertex at (x_n, y_n). H₀⁽²⁾ and H₁⁽²⁾ are the second kind Hankel functions of zero and first order, respectively. The convolutions in (5.54) are two-dimensional integrals. In general, all of the convolution integrals must be evaluated by numerical quadrature. Substitution of (5.48) into (5.50)–(5.53), evaluation of the vector operations, and substitution of (5.54) yields the following:

5.55

5.56

5.57

5.58

in which P_n is the total number of triangles with vertex at (x_n, y_n). Each entry in (5.55)–(5.58) represents a contribution from a source that is distributed over several adjacent triangular cells. The electric and magnetic fields vectors in (5.49) can be evaluated by matrix inversion.

An important parameter, which characterizes the scattering from a beam, is the IFR. The IFR is equal to the ratio of the maximum forward far-field scattering field to the hypothetical forward maximum radiated far field from a 2D aperture with a width equal to the optical projected width of the beam on the incident plane wave front. The aperture field distribution is equal to the incident field [12]. The IFR depends on polarization, such that IFR_e is related to TM incident wave and IFR_h is related to TE incident wave. Usually, the maximum forward scattering occurs in the propagation direction of the incident wave, as shown in Fig. 5.3, with the maximum forward-scattering direction (θ = π – θ₀, φ = φ₀).

Eq. (5.39) can be rewritten in the form

5.59

in which and . S denotes the cross-section area of the beam and the electric and magnetic current distributions have been computed numerically using the MoM, as explained above. Next, we need to evaluate the scattered field for far-field approximation and use the expressions of Hankel functions for large arguments [13]:

5.60

Evaluation of the phase term in (5.60) for , φ = φ₀ yields

5.61

Evaluation of (5.59) for far field approximation using (5.60), (5.61) and the representation of in Cartesian coordinates yields

5.62

Similar derivation enables to compute the maximum radiated electric field in the direction θ = π – θ₀, φ = φ₀ by an infinite aperture in z direction with width w positioned at the origin embedded in an infinite conductive plane and illuminated by the same TM polarization oblique incident field as the beam:

5.63

in which . The ratio of (5.62) and (5.63) results in the expression of IFR_e, such that

5.64

By a similar derivation eq. (5.40) can be rewritten for far field approximation for the maximum forward scattered magnetic field by,

5.65

In a similar fashion, we consider the maximum radiated TE polarized magnetic field in the direction θ = π − θ₀, φ = φ₀ by an infinite aperture with width w,

5.66

The ratio of (5.65) and (5.66) results in the expression of IFR_h, such that

5.67

An alternative figure of merit to describe the scattering from the scatterer is the radar cross section (RCS). For TM polarization the RCS is defined by

5.68

and for TE polarization,

5.69

Fig. 5.4 shows the bistatic scattering of a rectangular beam 1.38 × 6.2 in.² with ϵ_r = 4.6, tan δ = 0.014 illuminated normally by a plane wave on its narrow side at 5 GHz for TE_z and TM_z polarizations. The simulation was performed with HFSS commercial software from ANSYS. One can observe a higher scattering of the TE_z polarization.

Fig. 5.5 shows the IFR (amplitude and phase) as a function of frequency for two orthogonal polarizations and an incident plane wave on the beam narrow side. The beam is dielectric with mechanical and electrical properties as shown in Fig. 5.4. It is interesting to note that the IFR_h is higher and more oscillating compared to IFR_e.

5.2 Homogeneous Beams—Surface Integral Equation Formulation

A volume discretization is necessary to model properly a heterogeneous scatterer and as such is computational intensive. In case of homogeneous, layered geometries, and conductive cylinders, they can be treated with surface integral equation formulation and consequently reduce the required number of unknowns in the MoM numerical algorithm [14]. Therefore, this method is considered as a more efficient alternative to the volume integral equation discussed in the previous section. In the derivation, the equivalence principle [15] is used. Fig. 5.6 shows the beam geometry and the equivalent sources for the external and internal methods.

For the EFIE, we consider the tangential electric field at S⁺ and S^- such that

5.70

in which , and , are the magnetic and electric vector potentials in free space and in the homogeneous beam, respectively. Similarly, for the MFIE we consider the tangential magnetic field at S⁺ and S^- such that

5.71

The z dependence of the oblique incident field is and correspondingly, we assume that the z dependence of the total fields in the system is also , in which . The electric and magnetic fields can be described by (5.5), (5.6) and the incident fields for the TM and TE cases are and with E₀ and H₀ being the incident electric and magnetic field intensity, respectively. Let's define the operators and . Substitution of these operators in (5.70) and (5.71) results in

5.72

and

5.73

in which

5.74

where i = 0,d and . is the unit vector tangent to the cylinder contour, as shown in Fig. 5.7.

5.75

Eq. (5.72)a and (5.73)a should be evaluated with the observer an infinitesimal distance outside S, while (5.72)b and (5.73)b should be evaluated with the observer an infinitesimal distance inside S. Equations (5.72) and (5.73) can be solved with MoM with N pulse basis functions on the cylinder contour used to represent the unknowns J_z, J_t, J_mz, J_mt and delta functions as testing functions enforced at the center of each of the cells of the model. The result is a matrix 4 × 4 block structure:

5.76

where

5.77

in which , and φ_m is the polar angle defining the outward normal vector to the n^th strip in the model shown in Fig. 5.7.

In the next step, the IFR of the cylinder can be evaluated. For this task, we need to evaluate the forward-scattered far-field approximation using the expressions of Hankel functions for large arguments [13]. The maximum forward-scattering occurs in the propagation direction of the incident wave, (θ = π – θ₀, φ = φ₀). In the TM case based on (5.72), the z-directed scattered electric field is

5.78

Evaluation of the scattered field for far-field approximation, using the expressions of Hankel functions for large arguments in eq. (5.60) and the phase argument for (θ = π – θ₀ φ = φ₀) described in eq. (5.78) yields

5.79

Derivation of the maximum radiated electric field in the direction θ = π – θ₀, φ = π/2 by an infinite aperture with width w, as shown in eq. (5.63) and taking the ratio of (5.79) and (5.63) yields the IFR_e for the homogeneous dielectric cylinder,

5.80

By similar derivation for the TE case we obtain the forward scattered magnetic field:

5.81

and the IFR_h is

5.82

Fig. 5.8 shows the back-scattering RCS from a circular dielectric cylinder with radius a = 24 mm, for normal incidence TM polarized as a function of frequency. The cylinder is a lossy dielectric with ϵ_r = 4 and σ = 0 (solid line), 0.017 S/m (dot line), 0.05 (dash-dot line) S/m.

One can observe that increase in the cylinder conductivity reduces the back scattering.

Fig. 5.9 shows the RCS of a lossless elliptic cylinder (ϵ_r = 2) with major axis radius a = 38 mm and minor radius axis b = 19 mm illuminated by normal incidence TM polarized wave at angle φ_inc = 60 deg with respect to x-axis.

One can observe the forward and backward scattering as a function of frequency in the range 1–5 GHz. In the scattering pattern at 5 GHz, one can observe the forward scattered lobe with peak at φ = 60 deg and a minor lobe in the direction of the specular reflection at φ=120 deg. Fig. 5.10 shows the comparison of the numerical SIE solution (MoM with N = 20 contour divisions) and the exact solution of the TE bistatic RCS of three types of circular cylinders with parameters (ϵ_r = 1, μ_r = 10, k₀a = 0.7), (ϵ_r = 9, μ_r = 5, k₀a = 0.7), and (ϵ_r = 9, μ_r = 1, k₀a = 2).

The agreement of the numerical solution with the exact solution is good but slightly inferior to that of the TM polarization, as shown in Fig. 5.8.

5.3 Conductive Beams—Surface Integral Equation Formulation

In case of conductive beams, we can use the derivation of the surface integral equation with adaptation to the perfect electric conductor (PEC) boundary conditions on the beam surface [16]. Thus, for oblique incidence as shown in Fig. 5.3 and TM case, we can consider only the z-directed scattered electric field due to z-directed induced currents and the corresponding EFIE based on (5.72). In this case, no transverse electric and magnetic currents are induced; therefore,

5.83

Similarly, based on (5.71)a, one can derive the MFIE. We start with the boundary condition , which simplifies to

5.84

Numerical solution of the integral equations (5.83) and (5.84) encounters singularity problems, due to internal resonances. These internal cavity resonances are the result of the homogeneous solution of the integral equations and they corrupt the solution at these frequencies. The root of the problem is that the surface integral equation involves only data on the mathematical surface of the scatterer and cannot distinguish between “inside” and “outside” in order to produce the desired external solution. Since the interior resonances are different for the EFIE and MFIE, the effect of the interior resonances can be minimized if we combine these integral equations into the so-called combined field integral equation (CFIE) [5]:

5.85

In eq. (5.85), we linearly combine eqs. (5.83) and (5.84) using the scale factors α and (1 – α)η₀ to employ the same units, where α is a real value between 0 and 1 and η₀ is the characteristic impedance of the host medium. However, this CFIE remains valid if is purely imaginary, which would occur if k_z > k. There are no interior resonances frequencies with k_z>k, and consequently, the conventional EFIE and MFIE alone would be more efficient choice throughout the invisible region of the spectrum. Eq. (5.85) can be solved numerically using the point matching MoM. The incident field for the TM case is and . Using N pulse basis functions and N Dirac testing functions produces the matrix equation

5.86

in which j₁…j_N are the z-directed currents and e₁…e_N are the z-directed incident fields on the beam contour. Using the convention outlined in Fig. 5.7, the definition of and in eq. (5.75) and the approximations made for R_mm→0 in (5.77), the diagonal and off-diagonal entries in (5.86) are determined by

5.87

and

5.88

in which and the entries of the excitation column vector are

5.89

Matrix inversion of (5.87) results in the z-directed currents on the beam contour. The knowledge of the currents enable to compute the IFR_e based on (5.80), given that in this case only z-directed electric currents are excited,

5.90

where w is the optical projected width of the conductive beam on the incident wave plane and J_z is defined in Fig. 5.3 with the maximum forward-scattering direction (θ = π – θ₀, φ = φ₀).

Similar derivation can be applied for the TE case. The incident field for the TE case is and . In this case, we encounter only components of the excited electric current . The boundary conditions on the beam surface is . Substitution of (5.27) for and (5.35) for yields the EFIE; in this case, i.e.,

5.91

The MFIE is based on the boundary condition , which can be simplified using (5.35) to

5.92

To avoid the interior resonances excited in the EFIE and MFIE formulations in a similar fashion to the TM case, we linearly combine the two integral equations into the CFIE for the TE case:

5.93

In (5.93) the scale factors α and (1 – α)η₀ are used to employ the same units, where α is a real value between 0 and 1 and η₀ is the characteristic impedance of the host medium. Eq. (5.93) can be solved numerically using the MoM with N pulse basis functions and N Dirac delta-testing functions to obtain the matrix eq. (5.94):

5.94

in which j_t1…j_tN are the tangential currents on the beam contour and e₁…e_N are the tangential incident fields on the beam contour. Using the convention outlined in Fig. 5.7, the definition of and in eq. (5.75) and the approximations made for R_mm→0 in (5.77), the diagonal and off-diagonal entries in (5.94) are determined:

5.95

and

5.96

in which and the entries of the excitation column vector are

5.97

Matrix inversion of (5.94) results in the tangential currents on the beam contour on the beam contour. The knowledge of the currents enables to compute the IFR_h based on (5.82) with the maximum forward-scattering direction coinciding with axis (θ = π – θ₀, φ = φ₀). Given that in this case only transverse electric currents are excited,

5.98

where w is the optical projected width of the conductive beam on the incident wave plane and J_t is defined in Fig. 5.3.

Fig. 5.11 shows the comparison of the numerical solutions of the EFIE, CFIE, and the exact solutions for the TM current density on a circular PEC cylinder of radius 0.82λ₀ [5].

One can observe the distortion in the current distribution computed with EFIE due to the internal singularities compared to the exact solution. All that is in contrast to the numerical solution using CFIE, in which exists a nice agreement with the exact solution. Fig. 5.12 shows the comparison of the EFIE, CFIE, and exact solutions for the TM bistatic radar cross section scattering patterns from the circular PEC cylinder of radius a = 0.82λ₀ [5]. In this case, good agreement is obtained for both numerical solutions (EFIE and CFIE) and the exact solution. This observation implies that the internal singularities affect the current distribution on the scatterer and its near field but almost have no effect on the scattered far field. This can be attributed to the fact that the internal singularities affect the reactive scattered fields of the beam and have almost no effect on the real scattered fields, which determine the scattered far field.

The exact solution of the normalized scattered far field (r→∞) from a circular PEC cylinder is derived using cylindrical wave functions and is described in [15] by

5.99

with its corresponding IFR_e given by

5.100

where J_n and H_n⁽²⁾ are the Bessel and outgoing Hankel functions and θ₀ is the tilt angle between the incident wavefront and the cylinder axis. Similar derivation using the cylindrical wave functions for the TE case yields

5.101

with the IFR_h given by

5.102

In general, the IFR_e is larger in magnitude than IFR_h and has a positive phase angle compared to a negative phase angle for the HP. Both IFR's approach the value of –1.0 + j0.0 as the radius increases. IFR_h approaches this limit from below and IFR_e approaches the limit from above.

Another example that demonstrates the effect of the internal resonances on the induced current distribution on a pie-shaped PEC cylinder for TM polarization is shown in Fig. 5.13. In a similar fashion to the circular PEC cylinder computation of the current distribution using the CFIE formulation solves the problem almost entirely. In Fig. 5.13 the numerical solution is compared to the induced current computed using physical optics approximation, which is nonzero only on the illuminated surfaces.

5.4 Tuned Beams—Surface Integral Equation Formulation

The main objective in the design of space frame radomes made of homogeneous dielectric beams is to reduce their forward scattering. This objective can be accomplished by inserting periodic vertical and horizontal conductive strips in the dielectric beam with periodicity Δ_z in z direction, as shown in Fig. 5.14. Consequently, the strips can be arranged into unit cells and it is sufficient to study the behavior of the electromagnetic field within a single reference unit cell. The induced currents in the conductive strips offset the effect of the electric polarization currents induced in the homogeneous dielectric beam and reduce the forward scattering of the beam for a limited frequency bandwidth. The analysis of the scattering for these tuned beams is possible by formulating the EFIE of the problem by forcing the tangential electric field along the conductive strips of the beam to be equal to zero. Evaluation of the total electric field at the strip locations is done in two steps. In the first step using the coupled surface integral equation of the unloaded homogeneous beam as outlined in Chapter 5.2, the equivalent surface electric currents and surface magnetic currents on the beam contour S, as shown in Fig. 5.6, are evaluated using the MoM using eq. (5.72) for the TM case and eq. (5.73) for the TE case. In the second step, an EFIE is constructed in terms of the unknown induced currents on the conductive strips and enforcing the total electric field on the strips to vanish. The total electric field at the strip locations is equal to the field radiated by the surface equivalent sources on the beam contour S for the unloaded homogeneous beam and the scattered electric fields by the vertical and horizontal conductive strips. The scattered fields by the conductive strips are formulated in terms of a z-directed periodic array of vertical and horizontal current elements as shown in Fig. 5.14.

Without loss of generality, we assume that the horizontal wires are y-directed with center coordinates (x_r,y_r,z_r) and the vertical wires are z-directed with center coordinates (x_q,y_q,z_q). The length of the vertical strips in the unit cell is L_v and the length of the horizontal strips is L_h with the strips width being w. The current singularity near the strip edges [17] is taken in consideration for the vertical strips by assuming that the transverse current distribution is and for the horizontal strips. Accordingly, the z-periodic currents on the conductive strips of the beam can be represented by a Fourier series:

5.103

Here, with n = 0,1,…, denoting the Floquet harmonics [17] index, Δ_z being the unit cell length in z direction and . The center locations of the horizontal strips are (x_r,y_r,z_r) with r = 1,…,R and R being the total number of horizontal strips in the unit cell. The center locations of the vertical strips are (x_q,y_q,z_q) with q = 1,…,Q and Q being the total number of vertical strips in the unit cell. The electric field scattered by this current distribution can be described by

5.104

in which and where , , and v being the unit cell volume. The EFIE of the problem is set by forcing the tangential components of the total electric field to vanish along the conductive strips. The location along the strips is denoted by r_c.

5.105

in which

5.106

where denotes points on the beam contour. The electric currents, and magnetic currents, are the Fourier series of the currents and evaluated in advance, as explained in section 5.2. The currents on the strips can be determined by solving (5.105) using the MoM with pulses as basis functions and delta functions as test functions. The matrix inversion used in the MoM is performed for each harmonics n. Only the zeroth order harmonics (n = 0) is propagating, but few harmonics are needed to evaluate the total field in the beam. The next essential step is evaluation of the scattered far field of the tuned beam. As an interim step, we need to compute the total electric and magnetic field in the beam medium given the beam contour surface currents , and the evaluated conductive strip currents using

5.107

Using (5.107), the induced equivalent volumetric electric and magnetic currents in the beam volume can be computed using (5.17) and (5.19) to represent the induced polarization currents in the beam. Next, the forward far field can be computed using the induced electric and magnetic polarization currents in the beam and the electric currents on the strips. For this step, only the n = 0 propagating harmonics is needed. Without loss of generality, in the special case of incident field propagating in φ = π/2 direction and maximum forward-scattered electric field also in φ = π/2 direction, we can use (5.107) to compute the far-field approximation and the IFR values using (5.64) and (5.67).

Fig. 5.15 shows the geometry of a rectangular dielectric cylinder (ϵ_r = 3) loaded periodically with two horizontally and two vertically oriented PEC strips in the unit cell [17] with parameters s^v = 0.6, s^h = 0.8, and p = 0.75. The widths of the strips are w^v = w^h = 0.01L.

Fig. 5.16 shows the simulated RCS of the loaded and unloaded cylinder as a function of frequency for TE and TM polarizations. One can observe that for normal incidence and TE/TM polarizations, the RCS of the loaded dielectric cylinder is reduced by approximately 30 dB in a narrow frequency bandwidth, with center frequency 2.7 GHz for TE polarization and 3.1 GHz for TM polarization.

In a different example, a rectangular dielectric beam with electrical parameters ϵ_r = 4.6, tan δ = 0.014, and dimensions 2 × 0.4 in.² is considered. The beam is tuned with a grid of eight vertical conductive strips located on its center line, spaced 0.277 in. apart and with a strip width of 0.062 in. Fig. 5.17 shows the IFR dependence of the tuned and untuned cylinder on frequency. The data presented is for the TM polarization and normal incidence. The computed and measured data are compared.

At the tuning frequency of 5.6 GHz, the magnitude of the IFR drops significantly in comparison to that of the untuned beam. In addition, the IFR phase goes through the 180 deg value close to the tuning frequency, a feature common to resonant devices. Fig. 5.18 shows the field distribution (amplitude and phase) of the untuned and tuned beams inside the cross section of the dielectric beam at 5.6 GHz.

As expected the total electric field goes to zero at the locations of the strips. Moreover, the phase of the electric field is relatively more uniform as compared to that of the untuned beam. The uniformity of the phase distribution is a good indicator of the transparency of the beam because it resembles the uniform phase front of the incident field. Fig. 5.19 shows a comparison between the computed scattering pattern of the untuned and tuned dielectric beams at the tuning frequency of 5.6 GHz.

One can observe a significant reduction of the scattering levels in the forward direction, as compared to that of the untuned beam.

A different approach to the dual polarization tuning problem (forward scattering reduction) is described in [19]. The concept is based on replacement of soft surfaces by hard surfaces in the dielectric beam for both TE_z and TM_z polarizations. The presence of hard surfaces on a dielectric or conductive beam is desirable to reduce the beam blockage. Generally, a hard surface supports waves with maximum value of the E-field, like normal electric field on a conductive surface, whereas the soft surface makes the amplitude of the tangential E-field zero at a conductive surface. This effect is not allowing propagation and increases scattering. In case of propagation along a conductive surface, if the E-field excited is normal to the surface it can propagate along the surface with a strong intensity as the boundary condition is hard with . On the other hand, if the E-field is transverse to the direction of propagation and tangential to the surface, the wave is effectively stopped from propagating along the conductive surface and this increases scattering. Thus, the electric conductor has a strongly polarization dependent boundary condition for waves propagating along it. Accordingly, we are interested to transform the beam surfaces to hard surfaces to reduce scattering. This desire for a dielectric beam for TE_z polarization can be obtained by changing the profile of the beam to an oblong shape and strip loading its outer surface with inter-element strip spacing of less than λ/2 to transform the strip periodic structure to a full conductive surface. However, in case we don't have the flexibility to change the cross section of the beam due to mechanical constraints and for instance, still need to reduce the forward scattering of a rectangular cross section dielectric beam, a different approach should be adopted. In this case, two scattering reduction mechanisms are considered, as shown in Fig. 5.20a.

In the first mechanism, two metal plates are inserted in the dielectric beam to form an internal parallel plate waveguide and the phase propagation through the waveguide is equalized to the phase through air, such that

5.108

in which ϵ_r is the dielectric constant of the beam, k₀ is the propagation constant in free space, l is the beam length, and n is an integer. The second tuning mechanism is strip loading of the outer surface with inter-element strip spacing of less than λ/2, which transforms the outer surface to a conductive surface or a hard surface boundary for TE_z polarization. The hard surface boundary condition for TM_z polarization is obtained by coating the metal plates of the beam by a thin dielectric layer of thickness t, as shown in Fig. 5.20a. Inside the dielectric coating, the propagation constant will be , which is larger than the k₀ outside the coating. The component of k_x along the dielectric surface must be equal to k₀. Therefore, we will get a normal component of the propagation vector, k_y inside the dielectric coating equal to . This component of k_d will transform the boundary condition at the metal surface to the boundary condition of an artificial magnetic conductor at the surface of the dielectric provided that,

5.109

Fig. 5.20b shows the absolute value of as a function of frequency. One can observe that for the tuned beam with dimensions and parameters outlined in Fig. 5.20a, a minimum in w_eq, which is proportional to the forward scattering, is obtained at 11.5 GHz.

Comparison of the two tuning techniques described above reveals that they both can achieve dual polarization forward-scattering reduction in a limited frequency band, but the tuning with vertical and horizontal conductive strips is more flexible from the manufacturing point of view, without the need to use special dielectric materials given a predetermined beam geometry, as may be required based on eqs. (5.108) and (5.109).

5.5 Scattering from Infinite Cylinders—Differential Equation Formulation

Volume integral equation solved by MoM as outlined in previous sections are versatile, but involve fully populated matrices and require relatively large computational effort to treat scatterers of moderate electrical size. On the other hand, surface integral equation matrices are populated as the volume integral equation matrices, but are smaller in size. However, they can accommodate only homogeneous, multilayer, and conductive problems. An alternative option to compute numerically the scattering from a composite inhomogeneous beam is to solve its differential equation using the finite element method (FEM) [4] and [5]. Comparison of the FEM computational effort to the solution with volume integral equations reveals that it is less computationally intensive, since the matrices involved are sparse, but they are slightly larger in size. The differential equation methods often include an additional region of space outside the scatterer to ensure that the scattered fields represent outward propagating solutions. This additional region must be terminated with a radiation boundary condition as will be discussed later. Fig. 5.21 shows the cylinder geometry confined to the volume Γ made of a composite material with permittivity ϵ_r(x,y), permeability μ_r(x,y) and PEC conductive surfaces denoted by ∂Γ_c. The surface that confines the computational domain is ∂Γ.

In the following, we will consider the general case of oblique scattering as discussed for the integral equation solution and described in [5]. The z dependence of the oblique incident field is and, correspondingly, we assume that the z dependence of the total fields in the system is also . In a similar fashion to the volume integral equation, the electric and magnetic fields in the system can be described by eqs. (5.5) and (5.6). Moreover, we assume that the cylinder cross section is modeled by triangular cells of constant permittivity and permeability, but varying from cell to cell. Then, the total scattering from the beam would be the superposition of the scattering from all these various homogeneous triangular cells. Furthermore, like in the case of the volume integral equation derivation, we employ the total fields E_z and H_z as the primary unknowns to be determined and obtain the transverse magnetic and electric field components within a homogeneous triangular cell via equations (5.16) and (5.18), respectively.

The electromagnetic fields generated in the vicinity of the scatterer can be determined by the Helmholtz equations for the TM case:

5.110

and similarly, for the TE case

5.111

Eqs. (5.110) and (5.111) illustrate the “strong” form of Helmholtz equations with the unknown being in a second-order differential operator, which are very sensitive to numerical solution. To make these equations numerically more stable, they can be converted into the so-called “weak” form. The fields in the vicinity of the scatterer must satisfy Maxwell's equations:

5.112

Multiplying both sides of (5.112) with the testing function and integrating over the domain Γ, we obtain

5.113

Now, recall the two-dimensional divergence theorem

5.114

with being the normal to the surfaces and and the vector identities

5.115

Substitution of (5.114) and (5.115) into (5.113) enables us to rewrite it in the form,

5.116

The nabla operator can be rewritten in the form with . Substitution of the transverse fields from equations (5.16) and (5.18) into (5.116) and utilizing the vector identity yields, after some algebraic manipulations,

5.117

and

5.118

Eqs. (5.117) and (5.118) constitute coupled “weak” differential equations describing the field throughout region Γ and is at the basis of the FEM. For normal incidence k_z = 0, we obtain two uncoupled equations one for the TM case (5.117) and the other for the TE case (5.118). Since the tangential electric field must vanish on the conductive surface, E_z is a known function on and satisfies the Dirichlet boundary condition. Consequently, all testing functions used to discretize (5.118) will vanish on , and the boundary integral will contribute nothing to the matrix equation. Similarly, in the boundary integral of eq. (5.116), Newman boundary condition is enforced by ignoring the integral over . In order to represent the complete scattering problem, it is necessary to augment these equations with additional information about the fields on , such as radiation boundary conditions (RBC), which will be explained in the following.

For simplicity, we assume that the boundary is circular with radius a, and is located in free space. These assumptions simplify the right-hand side (RHS) of (5.117) and (5.118) such that

5.119

and similarly,

5.120

in which . To comply with the radiation boundary conditions on , the total electric and magnetic fields are expressed as a summation of the scattered fields E_z^s, H_z^s and the incident fields E_z^inc, H_z^inc. The scattered electric field on a circular boundary of radius r = a can be expressed in terms of cylindrical harmonics [15] by

5.121

where

5.122

and H_n⁽²⁾(x) is the n^th order Hankel function of second type. Accordingly,

5.123

Similarly, the plane wave incident field can be expressed in terms of cylindrical harmonics [15]

5.124

where J_n(x) is the n^th order Bessel function and

5.125

By a similar procedure to that of the scattered field as described in (5.123), we obtain for the incident field

5.126

Using the Wronskian relationship [15]:

5.127

and addition of (5.123) with (5.126), yields

5.128

Similar derivation for H_z, results in

5.129

Next we derive from (5.121) and (5.122)

5.130

and similarly, we obtain

5.131

Substituting (5.128)–(5.131) into (5.119) yields the RBC for circular boundary with radius a:

5.132

in which and

5.133

Similarly, (5.120) can be rewritten in the form

5.134

Although the summations in (5.133)a and (5.133)b are divergent, the required calculations in (5.132) and (5.134) are well behaved on , since the integrals involved and the cylindrical harmonic content of the functions E_z(φ) and H_z(φ) ensures that the integrals are convergent and relatively easy to compute. To ensure the convergence, the integration should be performed term by term before the summation. Eqs. (5.132) and (5.134) should be substituted into (5.117) and (5.118) to complete the formulation. The resulting equations can be discretized following the FEM procedure using a piecewise-linear representation B_n(x,y) for E_z(φ) and H_z(φ) and piecewise-linear testing functions B_m(x,y), as shown in Fig. 5.2 and described in eq. (5.48). Each basis function has unity amplitude at one node and vanishes at all other nodes in the mesh. Three basis functions overlap each triangular cell to provide a continuous piecewise-linear representation. Along the circular boundary the basis and testing functions, B_m(φ) may be expressed as

5.135

where for convenience, the φ coordinates of the three nodes associated with the m^th basis function are denoted as φ_m-1, φ_m, and φ_m+1. The incident uniform plane wave field with amplitude e₀ and at an angle (φ₀,θ₀) on a circular boundary of radius r = a and in terms of cylindrical harmonics can be expressed in terms of

5.136

Similarly,

5.137

Substitution of (5.43), (5.44), (5.132), (5.134), (5.136), and (5.137) into (5.117) and (5.118) results in the N × N matrix equation with N being the number of nodes in the solution domain:

5.138

in which

5.139

5.140

5.141

5.142

where

5.143

which only contributes to the preceding expressions if node m is located on the circular boundary . In addition, the excitation vector is given by

5.144

5.145

Matrix inversion of (5.138) enables to evaluate the electric and magnetic field distributions (e₁, e₂,….e_N) and (h₁, h₂,…,h_N) in the beam. The equivalent induced electric and magnetic polarization currents can be computed using (5.17) and (5.19). Next, the scattered far field and IFR_e and IFR_h can be computed using (5.64) and (5.67).

Fig. 5.22 shows a comparison of the E_z field produced on the surface of a circular dielectric cylinder with and ϵ_r = 50 – j20 by a TE wave incident at an oblique angle of 30 deg computed analytically, numerically with volume integral equation (VIE) and with partial differential equation (PDE) or FEM [20]. The agreement between the analytical and numerical results is very good and also between the VIE and FEM solutions.

References

1. 1 Kay, AF. Electrical design of metal space frame radomes. IEEE Trans. Antennas Propagat.,13(2), 188–202, 1965.
2. 2 Kennedy, PD. An analysis of the electrical characteristics of structurally supported radomes. The Ohio State University, Columbus, 1958.
3. 3 Michielssen, E, Peterson, AF, and Mittra, R. Oblique scattering from inhomogeneous cylinders using a coupled integral formulation with triangular cells. IEEE Trans. Antennas Propagat., 39(4), 485–490, 1991.
4. 4 Wu, RB, and Chen, CH. Variational Reaction Formulation of Scattering Problem for Anisotropic Dielectric Cylinders. IEEE Trans. Antennas Propagat., 34(5), 640–645, 1986.
5. 5 Peterson, AF, Ray, SL, and Mittra, R. Computational methods for electromagnetics. New York: IEEE Press, 1998.
6. 6 Richmond, JH. TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape. IEEE Trans. Antennas Propagat., 14(4), 460–464, 1966.
7. 7 Richmond, JH. Scattering by a dielectric cylinder of arbitrary cross section shape. IEEE Trans. Antennas Propagat., 13(3), 334–341, 1965.
8. 8 Peterson, AF, and Klock, PW. An improved MFIE formulation for TE-wave scattering from Lossy, inhomogeneous dielectric cylinders. IEEE Trans. Antennas Propagat., 36(1), 45–49, 1988.
9. 9 Ricoy, MA, Kilberg, SM, and Volakis, JL. Simple integral equations for two-dimensional scattering with further reduction in unknowns. IEE Proceedings, 136(4), 298–304, 1989.
10. 10 Rojas, RG. Scattering by an inhomogeneous dielectric/ferrite cylinder of arbitrary cross-section shape-oblique incidence case,” IEEE Trans. Antennas Propagat., 36(2), 238–246, 1988.
11. 11 Gupta, IJ, Lai, AKY, and Burnside, WD. Scattering by dielectric straps with potential application as target support structure. IEEE Trans. Antennas Propagat., 37(9), 1164–1171, 1989.
12. 12 Rusch, WVT, Appel-Hansen, J, Klein, CA, and Mittra, R. Forward scattering from square cylinders in the resonance region with application to aperture blockage. IEEE Trans. Antennas Propagat., 24(2), 182–189, 1976.
13. 13 Abramowitz, M, and Stegun, IA. Handbook of mathematical functions. New York: Dover, 1964.
14. 14 Wu, TK, and Tsai, LL. Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders. IEEE Trans. Antennas Propagat., 25(4), 518–524, 1977.
15. 15 Harrington, RF. Time-harmonic electromagnetic fields. New York: McGraw-Hill, 1961.
16. 16 Harrington, RF. Field computation by moment methods. New York: IEEE Press, 1968.
17. 17 Michielssen, E, and Mittra, R. Electromagnetic scattering from arbitrary strip loaded cylinders. In IEEE Ant. Propagat. Symposium, Chicago, Illinois, 1992.
18. 18 Shavit, R, Smolski, AP, Michielssen, E, and Mittra, R. Scattering analysis of high performance large sandwich radomes. IEEE Trans. Antennas Propagat., 40(2), 126–133, 1992.
19. 19 Kildal, PS, Kishk, AA, and Tengs, A. Reduction of forward scattering from cylindrical objects using hard surfaces. IEEE Trans. Antennas Propagat., 44(11), 1509–1520, 1996.
20. 20 Peterson, AF. Application of volume discretization methods to oblique scattering from high-contrast penetrable cylinders. IEEE Trans. Mic. Theory Tech., 42(4), 686–689, 1994.

Problems

1. P5.1 A rectangular PEC beam with cross section 1 × 4 in.² is illuminated by a TM_z plane wave with amplitude 1V/m at 60 deg azimuth angle, as shown in the sketch. The operational frequency is 10 GHz.

   

   1. a. Write the problem characteristic EFIE with the unknown current distribution on the beam.
   2. b. Formulate the numerical solution using MoM (point matching). Compute and plot the current distribution on the beam for subdivisions Δ = λ/10, λ/30.
   3. c. Compute and plot the beam RCS σ_TM(φ) for subdivisions Δ = λ/10, λ/30.
   4. d. Repeat items (a) – (c) for TE_z incident plane wave and formulation through a MFIE.
   5. e. Repeat items (a) – (c) for TM_z incident plane wave using Galerkin MoM with linear basis functions as shown in the sketch:

      

   Compare solution to that obtained using MoM point matching. Discuss the results and differences.

2. P5.2 A rectangular PEC beam infinite in z-direction is covered on all its sides by a dielectric layer with thickness 1 cm and dielectric constant ϵ_r = 4. The cross section of the PEC beam is 1 × 4 in.² and it is illuminated by a plane wave TM_z polarized propagating along y-axis at 10 GHz and with amplitude 1 V/m as shown in the sketch

   

   1. a. Write the EFIE integral equation characterizing the problem.
   2. b. Formulate the numerical solution using point matching MoM volume version. Compute and plot the current distribution on the conductive beam for segmentation Δ_x × Δ_y =(λ/10)², (λ/30)².
   3. c. Compute and plot the RCS σ_TM(φ) for subdivisions Δ_x × Δ_y = (λ/10)², (λ/30)².
   4. d. Find the optimal dielectric layer thickness for minimum RCS σ_TM(π/2).

3. P5.3 A rectangular PEC beam with cross section 3 × 8 in.² is illuminated by a TM_z plane wave with amplitude 1V/m at 90 deg azimuth angle, as shown P5.1. The operational frequency is 10 GHz.

   1. a. Formulate the CFIE integral equation with the z-directed unknown current distribution on the beam. Assume coupling coefficient α = 0.2.
   2. b. Formulate the numerical solution using MoM (point matching). Compute and plot the current distribution on the beam for subdivisions Δ = λ/30.
   3. c. Compute and plot the scattering pattern from the beam.
   4. d. Compute and plot the IFR_e as a function of w/λ in the range [0.3–4.0] with w being the narrow width of the beam.

4. P5.4 Repeat problem P5.3 for TE_z polarization.

   1. a. Prove that the CFIE integral equation characterizing the problem is
      
   2. b. Formulate the numerical solution of the MFIE and CFIE (with α = 0.2) equations using point-matching MoM and use the conjugate gradient iterative method for matrix inversion.
   3. c. Compute and plot the current distribution induced on the beam for Δ= λ/20 segmentation.
   4. d. Compute and plot the RCS σ_TE(φ) for both formulations MFIE and CFIE. Discuss differences.

5. P5.5 Consider a dielectric rectangular beam with dimensions 2 × 0.4 in.² and dielectric constant ϵ_r = 4.6, as shown in the sketch:

   

   The beam is illuminated by a plane wave with amplitude 1V/m, at 5.6 GHz, TM polarized and propagating in the direction φ^inc = 90 deg.

   1. a. Formulate the EFIE integral eq. of the problem.
   2. b. Formulate the numerical solution of the EFIE equation using point-matching MoM (volume formulation). Compute and plot the RCS σ_TM(φ) for cross section segmentation Δ_x × Δ_y = (λ/20)².

      On the beam axis, (y = 0) 8 PEC strips were inserted with width 0.062 in. and spacing 0.277 in. between centers, as shown in the sketch:

      

   3. c. Formulate the new EFIE equation characterizing the problem.
   4. d. Formulate the numerical solution using point-matching MoM (volume formulation). Compute and plot the RCS σ_TM(φ) for cross section segmentation Δ_x × Δ_y = (λ/20)². Assume on the metal strips one basis function. Compare and discuss the results to (b) results.