Convolution type operators for wave diffraction by conical structures

Authors


Abstract

[1] The analytical regularisation technique for rigorous solution of dual series equations in diffraction theory for structures, which consist of coaxial perfect (perfectly conducting, rigid or soft) conical sections, is proposed. The conical sections do not form biconical regions and the continuation of their generatrices has a single crossing point. The proposed approach is based on the mode matching technique as well as on establishing of the rule for correct transition to the infinite systems of linear algebraic equations and on receiving the solutions, which provide the fulfilment of all the necessary conditions for all the boundary value problems considered here: Neumann, Dirichlet, scalar electromagnetic problem. These systems are proved to be regularized by a pair of operators, which consist of the convolution type operator and the corresponding inverse one. The elements of the inverted operator are founded analytically using the factorization procedure for kernel functions for each boundary value problem. The peculiarities of the far field formation by one- and two-section cones in the case of their axially symmetric excitation by TM- electromagnetic wave are analyzed numerically.

1. Introduction

[2] The study of wave scattering by sections of perfect (perfectly conducting, rigid or soft) conical surfaces has a fundamental role in diffraction theory. Such problems are models for study of different physical phenomena. Analysis of the scattering and diffraction by the conical sections as well as radiation from them has been of great interest recently due to prediction and reduction of the radar cross section (RCS) of different targets, wide-band antennas as well as horn and phased antennas design. This problem serves as a simple model of duct structures such as jet engine intakes of aircrafts and defects' occurring on surfaces of complicated bodies.

[3] Some of the conical scatterers have been analyzed thus so far using a variety of different analytical asymptotical and numerical methods by Ufimtsev [1962], Northover [1965a, 1965b], Pridmore-Brown [1968], Senior and Uslenghi [1971], Bevensee [1973], Syed [1981], Eremin and Sveshnikov [1987], Goshin [1987], and Davis and Scharstein [1994]. It is known that fields of perfect conical scatterers can be completely described within discrete modal representation. So, the most adequate mathematical approach, which could be used for studying the corresponding diffraction processes, is the mode matching technique. Such an approach is widely used in field analysis of canonical structures, but its formal application is known to become substantially complicated for wave diffraction analysis in resonance and quasioptical frequency bands. At the same time, there are well known approaches, which allow the substantial improvement of the possibilities of the mode matching in the case of internal boundary diffraction problems for bifurcations of infinitely extended (rectangular or cylindrical) waveguides [Mittra and Lee, 1971; Shestopalov et al., 1984]. Accounting the importance of conical structures for solution of a wide range of problems in technical physics, and the complexity of rigorous analysis of the corresponding diffraction problems the analytical regularisation technique was developed by Kuryliak [2000a, 2000b] and Kuryliak and Nazarchuk [2000, 2006] for obtaining of the solutions. The proposed method includes four principal steps: (a) application of the mode matching technique and derivation of the rule for correct receiving of the infinite system of linear algebraic equations (ISLAE); (b) separation of the matrix operator of convolution type: A:{apn = (ξpzn)−1} from the static part of ISLAE, where ξp, zn are the functions which are linear depended from the indexes at p, n → ∞; (c) finding of the inverse operator A−1 in analytical form; (d) application of the pair of operators A and A−1 for the development of analytical regularisation technique, which includes the separation of the convolution type operator A from ISLAE, its analytical inversion by the operator A−1 and transition to the ISLAE of the second kind.

[4] The main reason for application of this technique is that operators of the convolution type consist of the principal part of the asymptotic of ISLAE for initial diffraction problem. It means that if the indexes of the matrix elements of ISLAE tend to infinity, the matrix elements of this system tend to the elements of the matrix operators of the convolution type for any geometrical parameters and frequency. The effectiveness of the proposed technique for solution of wave diffraction problem is stipulated by the analytical inversion of the principal part of its asymptotic. This leads to improvement of the convergence of the reduction procedure and allows obtaining the solution with prespecified accuracy for any parameters of the problem. In this paper the main principles of such an approach are demonstrated. This approach can be used to obtain the rigorous solutions for a wide range of problems, particularly, for the structure with coaxial sections with the same efficiency as it is made for waveguides. The time factor is assumed to be eiωt and suppressed throughout this paper.

2. Formulation of the Basic Problems

[5] Let us consider wave diffraction by finite conical scatterer Q:{r ∈ (0, c1); θ = γ; ϕ ∈ [0,2π)}, where (r, θ, ϕ) are spherical coordinates, c1 and γ are conical length and opening angle (aperture). Let us analyze the cases, when the cone Q is perfectly rigid or perfectly soft. The diffraction problem can be formulated as a value boundary problem for the Helmholtz equation

equation image

where Um = Um (r,θ) with m = equation image are unknown Fourier expansion coefficients of the scalar potential U = U (r,θ,ϕ); k is wave number (k = k1 + ik2, k1, k2 > 0);

equation image

[6] In addition, Um(r,θ) should satisfy some boundary conditions on Q:

equation image
equation image

Here Um(i) = Um(i) (r,θ) is the scalar potential of an incident wave.

[7] Function Um (r,θ) should also satisfy the Sommerfeld radiation condition at infinity:

equation image

and condition of local energy limitation in arbitrary finite volume, which can be reduced to the fulfilment of Meixner condition on edge of surface Q:

equation image

where κ1(2) are constants.

[8] We shall seek the solution of the formulated above problems using the variable separation method, and represent the diffracted field Um(r,θ) for each region

equation image

by the series of eigen functions of Helmholtz equation as follows

equation image
equation image
equation image

Here xmn(1), ymp(1,1), ymk(1,2) are unknown expansion coefficients; Pκ−1/2m (·) are associate Legendre functions; Kν(·) and Iν(·) are Macdonald and modified Bessel functions; ρ = sr, ρ1 = sc1, s = − ik, i is imaginary unit; zmn = n + m − 1/2; νmp and μmk are dependent on cone aperture angle γ (γπ /2) positive growing sequences of the simple roots of the transcendent equations look like

equation image

in the case of perfectly rigid cone (Neumann boundary conditions) and

equation image

in the case of perfectly soft cone (Dirichlet boundary conditions).

[9] The diffracted field will be considered as some perturbation brought by cone Q into the incident field. Taking this into account, we can express the total field as

equation image

[10] Next, we assume that the radiation source is located in the region D2 and represent the incident field as the series of spherical functions:

equation image

Here Amn(0) are known values, ρ0 = sl, l is radial coordinate of the source.

[11] Then, according to (9), expressions (7a), (7b) define the total field, and (7c) represents the diffracted one. Now the problem is to find the unknown expansion coefficients from the conditions of mode matching at the spherical surface r = c1. These conditions allow to write the series equations, which are formal because of the singularities on edge. Therefore, we restrict ourselves by the finite number of addends in sums of these series equations and consider the limit transition to series. For this purpose we write the initial series equations as

equation image
equation image

Here

equation image

[12] To satisfy the Meixner edge conditions, the solution of series equations (11) should be found in a class of sequences with algebraic behavior at infinity

equation image

[13] The next step is the reduction of series equations to ISLAE. The solution of this problem is the main step of rigorous analysis of diffraction phenomenon on conical surfaces. To solve the mentioned problem, let us consider some auxiliary problems.

3. Basic Series Equations, Which Allow the Closed Form Solution

[14] Let us form new series equations from (11), substituting of cylindrical functions and their derivatives by asymptotic expressions [Lebedev, 1972], keeping only principal terms for the relations as

equation image

where ν are indices of the modified Bessel and Macdonald functions in (11).

[15] As a result the study of the equations (11) is reduced to analysis of series equations as follows

equation image
equation image

Here we limit ourselves by j-th addend in the representation of incident field.

[16] Next we consider the series equations (15) for Neumann and Dirichlet problems separately.

[17] 1. Neumann problem. Here indices of associate Legendre functions in regions 0 ≤ θ < γ and γ < θ ≤ π are determined by the condition (8a). Using the orthogonality property of associate Legendre functions [Bateman and Erdelyi, 1953], we can write the reexpansion formulas as

equation image

Here γ+:θ∈[0,γ), γ:θ∈(γ,π]; ηmp = νmp for superscript sign and ηmk = μmk– for subscript sign;

equation image
equation image

[18] Let us substitute expression (16) into formulas (15) and exclude the unknowns ymp(1,1) and ymk(1,2). Limited by the finite number of addends we arrive at systems of equations as in

equation image
equation image

Here

equation image

[19] Equations (18) are the double alternative type of systems, which appear in diffraction problems for plane waveguides theory [Mittra and Lee, 1971]. The solutions of systems (18), which depends on N, K, P parameters, can be expressed in the form

equation image

where superscript (k) indicates that cofactor with index n = k is omitted from the product.

[20] Accounting that the asymptotic of the indices νmp, μmk at p, k → ∞ looks like

equation image

with δ = 3/4, in the case of P, K → ∞ (N = P + K) and at large k (k < N), the following asymptotic estimation for (20) is valid:

equation image

[21] It follows from (22) that the solution of systems (18) depends on reduction rules. Also under condition that

equation image

the exponent indicator in (22) tends to zero. So, we get a solution in the class of sequences where sought values have algebraic character of decay

equation image

[22] Condition (23) is similar to the known reduction rule for ISLAE in the case of wave diffraction by bifurcation in the plane waveguide [Mittra and Lee, 1971]. Accounting to the relations (19) and (17a), we find using (24) that solution of the series equations (15)xmn(1) = O (nm−1) for n → ∞. It is easy to prove in the same way [Kuryliak, 2000a, 2000b; Kuryliak and Nazarchuk, 2006] that the expansion coefficients y(1,1), y(1,2) of the series equations (15) also satisfy the asymptotic condition (13).

[23] 2. Transition to the convolution type equation. Next we apply the alternative technique to obtain the solution of the equations (18). For this purpose we derive the inverse operator in analytical form. Let us introduce the growing sequence

equation image

According to sequence (25) let form ISLAE from (18a), (18b), letting P,K → ∞. We write this ISLAE as

equation image

where A(m) is matrix convolution type operator:

equation image
equation image

[24] Taking into account linear dependences of the main parts of asymptotics (21) with respect to integer parameters p and k, we can state that the condition (23) is fulfilled in the system of equations (26). So this system has a solution in the class of sequences (24).

[25] For finding an inverse operator, let introduce the meromorphic kernel function

equation image

[26] The expression (29) is the even function with respect to variable ν, regular in the strip Π:{∣Reν∣ < ξm1 (γ)}(ξm1 > 1/2), whereas beyond the strip Π it has simple real zeros and poles in points ±zmn and ±ξmn (n = equation image ), respectively. The function (29) allows factorization in the form

equation image

where M±(ν,γ;m) are split functions, regular in semiplanes Reν > − ξm1 and Reν < ξm1 respectively; M(ν,γ;m) = M+(−ν,γ;m), M±(ν,γ;m) = O(ν−1/2) for ∣ν∣ → ∞ in regularity regions;

equation image
equation image

Here Γ(·) is gamma function, ψ(·) is logarithmic derivative of gamma function.

[27] By taking into account analytical properties of the split function (31), let us introduce an inverse operator

equation image

where k, q = equation image, prime sign denotes the derivative relative to ν argument of M(ν,γ;m) function in points ν = ξmq, ν = zmk. At q, k ≫ 1 asymptotic estimation of (33) looks like

equation image

[28] Next we verify the correlation

equation image

where I is the unit operator.

[29] Let consider an integral

equation image

Here CR is a circle of radius R. Integrand (36) tends to zero as O(t−5/2) at ∣t∣ → ∞. Thus we have Inp(1) → 0 for R → ∞ on a system of regular contours. Replacing integral (36) by the sum of residues in points corresponding to simple poles in points t = ξmk (k = equation image ) and in point t = zmp, when p = n, as well using expressions (27) and (33), we find that

equation image

where δnp is Kronecker symbol.

[30] Applying the operator (33) to equation (26) and summing the series due to its transformation into the integral along of the closed contour in the complex sheet, we obtain the exact solution as follows

equation image

[31] Next using correlations (20), (31) and (38), for N > K > P we write:

equation image

[32] Tending N, K, P → ∞, we get from (39) that

equation image

[33] Under condition (23) right part of the expression (40) tends to 1 for all n < N and N, P, K → ∞. So, solution of equations (18), under the condition (23), and solution of the equation (26) coincide.

[34] 3. Dirichlet problem. Here indices of associate Legendre functions in regions 0 ≤ θ < γ and γ < θ ≤ π are determined from the condition (8b). Thus we can write the reexpansion formulas (16) with

equation image
equation image

[35] Substituting the formula (16) with coefficients (41a), (41b) into the left parts of series equations (15), we come to ISLAE in the form (26). In this case at p, k → ∞ for indices νmp, μmk the asymptotes (21) with δ = 1/4 exist. So, the matrices of obtained ISLAE also form the convolution type operator. To find the inverse operators we should consider the kernel function

equation image

which is even meromorphic one with the strip of regularity Π and simple real zeros and poles beyond Π in points ±zmn, ±ξmp(n, p = equation image) respectively. Next we shall factorize function (42) by the product (30) as follows

equation image

Here χm =χ0(γ) − ψ(3/4 + m/2) − Sm(γ) − Sm(πγ), Sm(γ) is given by formula (32), when βm = m/2 − 1/4; M±(ν,γ;m) are the split functions, regular in overlap semiplanes; M (ν,γ;m) = M+ (−ν,γ;m), M± (ν,γ;m) = O(ν1/2) for ∣ν∣→∞ in regularity regions.

[36] Having formed the inverse operator on the basis of (33), we can find an exact form solution of equation (26) as in the form (38). Accounting behavior of inverse operator at large indices, namely:

equation image

we find the asymptotic estimations of this solutions as

equation image

[37] Distinction of solution asymptotics for Neumann (24) and Dirichlet (45) problems is stipulated by difference in behavior of functions (31) and (43) at ∣ν∣ → ∞. Basing on the estimation (45) and asymptotic properties of expressions (41a), we can prove that expansion coefficients in series equations (15) for the case of Dirichlet problem also satisfy the asymptotic conditions (13).

4. Analytical Regularisation Procedure for Solution of the Basic Diffraction Problems

[38] Now we shall seek the solution of the series equations (11) in the class of sequences, defined by correlations (13). Using reexpansion formula for associate Legendre functions (16) and acting similarly as described in section 3, we come to the next system of equations

equation image
equation image

Here p = 1, 2, …, P; k = 1, 2, …, K; N = K + P, N(K, P) → ∞; W[f, g]t = f(t)g′(t) − f′(t)g(t);

equation image

Right parts for the system of equations (46) looks like

equation image

where indices ηmkequation image and ηmkequation image correspond to equation (46a) and (46b), respectively.

[39] From systems (46a), (46b) we form a new ISLAE, coming to the boundary N, P, K → ∞ and positioning equations in accordance with the sequence (25). As a results we get ISLAE as

equation image

where X(m) = equation image, A11(m) is a matrix operator

equation image

F1(m) = equation image is a known vector, which elements are given by the formula (48) with N → ∞.

[40] By direct verification we can find that in the case of real positive values of indices of the modified Bessel and Macdonald functions we can validate the following asymptotic estimations

equation image
equation image

[41] According to formulas (51), the principal part of the operator (50) asymptotic for q, n → ∞ coincides with the static part, which forms the convolution type operator (27). Separating it from (50) and using the inverted operator (33), we come from the system (49) to the second kind ISLAE as

equation image

[42] Taking into account (51), (34) and (44) we can prove [Kuryliak, 2000b; Kuryliak and Nazarchuk, 2006] that ISLAE (52) allows a unique solution in the class of sequences

equation image

with 0 ≤ σ < 1/2 and 0 ≤ σ < 3/2 for Neumann and Dirichlet problems respectively. Taking into account the correlations (51) and (53), the ISLAE (52) can be solved with pre-specified accuracy for any geometrical parameters and frequency using the reduction procedure. Unknowns in the system (52) bound with expansion coefficients in (11) through the relation (47). Accounting asymptotic behavior of the expression' (17a) and (41a) at n → ∞, we get that equation image for 0 ≤ σm < m + 1, which provides the fulfilment of the Meixner edge conditions at rc1 + 0, θγ.

5. Electromagnetic Axially Symmetric Wave Diffraction Problems

[43] We formulate the boundary problem for scalar function U = U(r, θ), which satisfies the Helmholtz equation (1) for m = 0. In this case the field components can be expressed from the following relations:

equation image

Here Z = equation image is the medium impedance; μ and ɛ are the permeability and permittivity respectively.

[44] Thus, for solution of the wave diffraction problem by finite perfectly conducting cone Q, we should find the function U(r,θ), which satisfies the following boundary conditions on Q:

[45] 1. For TM-waves (E-polarization)

equation image

[46] 2. For TE-waves (H-polarization)

equation image

The function U(r,θ) should also satisfy the Sommerfeld radiation condition (5a) and Meixner edge condition on edge of cone Q.

[47] For solution of these problems we represent the functions U(r,θ) and U(i) (r,θ) in the series (7) and (10) respectively with m = 0. Then the problem is to find unknown expansion coefficients from the conditions of tangential field components continuity at the spherical surface r = c1. Due to the relations (54), these conditions allow for writing the next series equations:

equation image
equation image

Here zn = n + 1/2, bn(1) = An(0)equation image (ρ0) equation image (ρ1)/ equation image; Pν−1/21 (± cosθ) = ±equation image [Lebedev, 1972].

[48] The equations for determination of the indices for Legendre functions in conical regions look like

equation image
equation image

As follows from (57b), the solution of TE-wave diffraction problems on finite perfectly conducting cone Q are reduced to the solution of the Dirichlet problem (1), (4) with m = 1 and can be solved by the analytical regularisation technique as in section 4.

[49] In the case of TM-wave diffraction by cone Q we apply for solution the reexpansion formula (16) with

equation image
equation image

where ηp = νp for superscript sign and ηk = μk - for subscript sign.

[50] The procedure of transition from the series equations (56) to the second kind ISLAE is as

equation image

Here X = {xn}n = 1, xn = q(zn, γ)xn(1);

equation image

where {ξn}n = 1 is growing the sequence defined as

equation image

the pair of regularisation operators A, A−1 looks as follows

equation image
equation image

where ±zn, ±ξp are simple real zeros and poles of the kernel function

equation image

which allows the factorization

equation image
equation image

Here the split functions M±(ν,γ) are regular in the half planes Reν > −1/2 and Reν < 1/2 respectively and M±(ν,γ) = O(ν−1/2}) for ∣ν∣ → ∞. Accounting to this we find the solution of equation (59)xn = O(n−1/2) and solution of the series equation (56)xn(1) = O(n−2) for n → ∞, which provides the fulfillment of Meixner edge conditions.

6. Generalized Theory

[51] The proposed approach allows creation of the mathematically rigorous theory for solution of scalar diffraction problems in the case of systems of coaxial conical surfaces without biconical regions. The most important for different technical application is the fragments of conical surfaces with two ribs or numbers of conical sections. Below we provide the analysis of these problems.

[52] 1. Wave diffraction analysis of two-section cone. Let us consider axially symmetric electromagnetic wave diffraction by two-sections' cone Q = Q1Q2, where Q1:{r ∈ (0,c1); θ = γ1; ϕ ∈ [0,2π)} is perfectly conducting finite cone and Q2 :{r ∈ (c2,∞); θ = γ3; ϕ ∈ [0,2π)} is semi-infinite truncated cone, c2 > c1 (see Figure 1a).

Figure 1.

Geometry scheme; (a) two-section cone; (b) (N + 1)-section cone.

[53] Let cone Q is excited by the axial symmetric TM-wave. This problem is reduced to four series equations, which take into account modification of field potential representation for the regions bounded by the conical surfaces and slit. The potential representation for the region D2:{r ∈ (c1,c2); θ ∈ [0,π]} looks like

equation image

[54] For the conical regions D1 and D3, created by the surfaces Q1, Q2, the sought field is given by series of eigen functions of Helmholtz equations for supplemental conical regions with aperture angles γ1, πγ1 and γ3, πγ3. Based on the results of section 5, we reduce this problem to the following second kind ISLAE:

equation image

Here unknown vectors equation image are bound with expansion coefficients of the diffracted scalar field potential (66) through expressions

equation image

where q(zp,γ) is defined in (58a), zp = p + 1/2; Aij are matrix operators, with the following elements:

equation image
equation image

Note that {ξk} in (69) is a growing sequence (61) of positive roots of equations (57a) at γ = γ1, and {ηk} is similar sequence of roots of (57a) at γ = γ3;

equation image
equation image

{Ai, Ai−1}i=12 are two pairs of regularisation operators, which are determined in (62), (63) for γ = γ1(3).

[55] In the case when radiated source is placed on the symmetry axis in region c1< r < c2 with potential

equation image

the expressions for known vectors F1 = {fk(1)}k=1, F2 = {fk(2)}k=1 in (67), are written as

equation image
equation image

[56] The diagonal matrix operators A11 and A22 in (67) correspond to diffraction problems on finite cone with aperture γ1 and on truncated semi-infinite cone with aperture γ3 respectively. Non-diagonal operators A12 and A21 account mutual influence of finite and semi-finite cones with different apertures on formation of diffraction field. Difference in aperture angles of conical elements of scattering structures, in addition to difference in indices {ξn}, {ηn} in expressions for matrix elements (69), is reflected by appearance of typical cofactors in the form of relations of Legendre functions (71). In the particular case when γ1 = γ3 = γ the cofactors (71) equal to 1 and equations (67) corresponds to the problem of wave diffraction by the semi-infinite perfectly conducting cone with a slit.

[57] 2. Wave diffraction analysis of (N + 1)-section cone. Let us consider the perfectly rigid (soft) conical structure

equation image

where N > 0; Q2k − 1:{r ∈ (c2k − 2, c2k − 1);θ = γ2k − 1; ϕ ∈ [0,2π)} with c0 ≡ 0, c2N+1 ≡ ∞; cn−1 < cn are conical sections (see Figure 1b). As it is proved in Kuryliak and Nazarchuk [2006], this problem can be reduced to the sets of linear algebraic equations

equation image

Here k = equation image are unknown vectors related to expansion coefficients for field representation in regions D2k:{c2k − 1 < r < c2k; θ ∈ [0,π]} with k = equation image, which looks like

equation image

where ρk = sck, equation imagemp(2i−1) = q(zmp,γ2i−1)xmp(1,2i), equation imagemp(2i) = q(zmp,γ2i+1)xmp(2,2i), i = equation image; Akl(m) are matrices with structures of elements as

equation image

Here ɛ1 = 1, ɛ2 = ɛ3 = 3, ɛ4 = ɛ5 = 5…, Δqp(m;1) = ξmq2zmp2, Δqp(m;3) = ηmq2zmp2, Δqp(m;5) = λmq2zmp2, where {ξmq}q=1, {ηmq}q=1, {λmq} q=1… are sequences (25) of positive roots of equations (8) at γ = {γ1,γ3,γ5…}, respectively; dqn(m;kl) is expressed through Macdonald and modified Bessel functions and as well as through their derivatives with respect to argument. It is important to note that diagonal matrices Akk (m) in the formula (76) have the same structure as the matrix in ISLAE (49). On this base we can directly prove the following statement. The ISLAE for scalar diffraction problems by coaxial sections of perfectly conical surfaces with different apertures (conical elements do not form biconical regions and continuation of their generatrices has a single crossing point) allow regularisation by outlining static parts in diagonal matrixes, which form convolution type operators, and with the use of corresponding inverted operators.

[58] Finally this problem is reduced by our technique to the second-kind sets as follows

equation image

Here l = equation image; {Al, m), A−1l, m)}l=12N are the pairs of regularisation operators, which are determined in (27) and (33) for different values of cone sections aperture parameters γ = γ2k−1 with k = equation image.

7. Numerical Results and Discussion

[59] Let us consider the representative numerical examples of the radiation power and far field pattern for various physical parameters to discuss the scattering characteristics of different conical structures: finite cone, semi-infinite truncated cone and two-section cone.

[60] Numerical results shown here correspond to the case of the axially symmetric excitation of the conical scatterers by TM-wave produced by the radial electric dipole, which is located on the conical axis. For numerical analysis we have formed two sets (or set in the case of finite and semi-infinite truncated cones) of N × N matrix equations from ISLAE (67) and solved them numerically for obtaining all the physical quantities. The incident field calculation is based on the formulas (54) and (72) with An(0) = A0zn, where A0 = 2P0Z/l, P0 is the dipole moment. For convenience we assume that P0k = 1/(4π)[A] and Z = 1[Om]. The far field pattern is defined by formula

equation image

Since the conical structures are symmetric along the z axis, we have plotted the results only for the range 0° < θ < 180°.

[61] Let us analyze the scattering mechanism for the far field pattern formation of finite conical horn Q1, where γ1 < π/2. As a rule, the study of field radiation from finite conical horn is based on radiation analysis of semi-infinite cones. The corresponding characteristics are simple to find, using the known analytical solutions of diffraction problems for semi-infinite cones [Felsen and Marcuvitz, 1973; Bowman et al., 1987]. Therefore the main problem is accounting the finite dimensions of horns. It is particularly important to answer the question: under which conditions the far field of the finite cone can be approximated by the field of the semi-infinite cone in the radiation zone? In order to answer this question we shall study peculiarities of formation of the far field distribution for finite and semi-infinite cones. Plots in Figure 2 illustrate the far fields distribution for finite and semi-infinite cones with the rise in the generatrix length of the finite cone. Closing in the forms of these fields, as it is shown by studies, are defined by the number of the excited modes in semi-infinite cone and in conical horn. For example, the far field pattern of semi-infinite cone, that corresponds to curves 3 of Figures 2a and 2b, is mainly contributed by three modes (second – fourth). The second peak is formed because of the third mode and a final correlation between amplitudes of the first and the second peaks is established because of the fourth mode. In conical horn of finite length, which far field pattern corresponds to curve 1 in Figure 2a, the third and fourth modes of semi-infinite cone are not completely formed yet. They appear with the rise in the length of finite conical horn (see Figure 2b). The second peak is formed only at kc1 > 28 and further rise in horn length weakly influences the shape of far field pattern in lighted sector.

Figure 2.

Far field radiated from conical horn at γ1 = 50°, kl = 12.5 : 1 – total field, 2 – diffracted field, 3 – field of semi-infinite cone; (a) kc1 = 18; (b) kc1 = 30.

[62] Next we shall study the perturbation introduced by the vertex cut into the field of semi-infinite cone. Figure 3 shows how the dipole position influences the distribution of the total radiation field in the case when truncated cone Q2 is very close to the plane screen with a circular hole (γ3π/2). As can be seen from the Figure 3, when the dipole is situated close to the center of circular hole, the cone practically do not perturb the field, which has a shape of dipole far field distribution in free space (see curve 1). Passing of the dipole along the cone symmetry axis leads to the appearance of field oscillations. Their amplitudes can exceed the maximum value of the incident field more than twice and there are observation angles in the lighted sector, for which the field value is closed to zero (see curve 3). This effect can be used, for example, for electromagnetic protection of the electronic devices.

Figure 3.

Influence of source coordinate on shape of the far total field pattern for truncated semi-infinite cone at kc2 = 24, γ3 = 89° : 1 − kl = 0.01, 2 − kl = 5, 3 − kl = 10.

[63] One of the most important practical problems is regulation of the electromagnetic energy penetration into supplement non-regular waveguide regions by optimal choice of their parameters. Let consider the truncated semi-infinite cone with the fixed aperture angle, which we shall illuminate by the field radiated from the finite conical horn (see Figure 1a). Curve 1 in the Figure 4 shows the dependence of the normalized power W radiated from the conical horn into free space as a function of the parameter γ1. We normalize power radiated from horn to the power radiated by the dipole into the free space. Curves 2 and 3 in this figure show the same dependences for radiation power which penetrates into the supplement conical regions {r > c2, 0 ≤ θ < γ3} and {r > c2, γ3 < θ ≤ π} respectively. We can observe from Figure 4 the deep oscillations of curves 1 and 2, which are very close in wide range of changes of horn aperture angle. It means that practically all normalized power radiated from the conical horn in the case of γ1 < 47° penetrates into the hole, formed by the vertex cut of semi-infinite cone. There are exceptions in extreme points, where there is growth of losses due to radiation into the lateral slit. From this figure we also observe, that the quantity of radiated power depends on the conical horn aperture angle essentially.

Figure 4.

Penetration of the normalized radiation power into the free space (1) and into the semi-infinite truncated conical regions {r > c2; 0 ≤ θ < γ3} (2) and {r > c2; γ3 < θ ≤ π} (3) as a function of the aperture angle γ1 for the finite cone at kc1 = 17.9; kc2 = 18.9; kl = 17.95; γ3 = 40°.

[64] To design the scatterers it is important to know the far field pattern formed by conical scatterers for limiting case, when their geometrical parameters ensure the extreme values of radiated power. Examples of far field distribution for such situations in the case of two-section cone are shown in Figure 5. Curves 1 and 2 in this figure illustrate the far field, which provides the second maximum and minimum on the curve 2 in Figure 4 respectively. Comparing these curves, we can see that slight change in the aperture angle of finite conical horn (Δγ1 = 4°) substantially influences the far field distribution. It is important to note that both curves in Figure 5 have the similar lobe structure. In contrast to it, the far field has strongly suppressed lateral lobes corresponding to curve 2 in Figure 5. Despite this, the main contributions into formation of each curve are brought by modes from the third to the fifth.

Figure 5.

Far total field pattern for two section conical structure at kc1 = 17.9; kc2 = 18.9; kl = 17.95; γ3 = 40°: 1 − γ1 = 33°, 2 − γ1 = 37°.

8. Conclusion

[65] The analytical regularisation technique for rigorous solution of dual series equations in diffraction theory for structures, which consist of a numbers of coaxial perfect conical sections, is proposed. The technique exploit the mode matching and is based on establishing of the rule for correct transition to the infinite systems of linear algebraic equations as well as on receiving the solutions, which provide the fulfilment of all the necessary conditions for all the boundary value problems considered here: Neumann, Dirichlet and scalar electromagnetic problem. These systems are proved to be regularizing by a pair of operators, which consist of the convolution type operator and the corresponding inverse one. The elements of the inverted operator are found analytically using the factorization procedure for each the kernel function. The peculiarities of far field formation by one- and two-section cones under the condition of their axially symmetric excitation by TM- electromagnetic wave are analyzed numerically.

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