4.1 Network model of angular regions in absence of sources

Let us consider an angular region φ₁ < φ < φ₂ without sources, the functions and satisfy the transmission line Eqs (17) and (19).

As already stated, taking into account the uniqueness theorem, two equations must relate the four quantities (q = 1, 2) in spectral domains (see Eq (2) for definition of q). In t-plane the two relations are algebraic and assumes the typical properties of transmission line in frequency domain, therefore the solution of Eqs (17) and (19) in terms of (q = 1, 2) in absence of internal sources yields the admittance parameters of the classical two-port network (see Fig 6b) without current generators constituted by a segment of transmission line (see also [3] p.181): (21) (22) where l_φ = φ₂ − φ₁

As in classical circuit theory for reciprocal and symmetrical circuits, in absence of sources, we can model the admittance matrix through a π two-port networks (see Fig 8) whose parameters are (23) Note that in Eq (23) we have reported the admittances in t-plane which are algebraic.

The network model reported in Fig 6b without current generators can be reinterpreted in w plane. We note that, by applying the inverse MF transform Eq (6) to (24) we obtain (25) where the convolution theorem is applied in the framework of the MF transform.

From Eq (25), we notice that the admittances are convolutional integral kernels in the w-plane. The inverse MF transforms of Eq (22) yield: (26) (27)

In Eqs (26) and (27) we have reported the admittances in w-plane. The model in w plane is still a two port network represented by Fig 6b without current generators, although the admittances are now convolutional operators, see Eq (25).

4.2 Sources concentrated along an azimuthal direction

Let us consider a homogeneous angular region with a source concentrated along the azimuthal direction φ_o, i.e. J_z(ρ, φ) = J_z(ρ)δ(φ − φ_o). With reference to Fig 4 the region S reduces to a line. With this source constraint Eq (19) can be rewritten as (28) since the source term in Eq (19) is null for φ ≠ φ_o.

The E_z-polarized plane wave Eq (29) is a particular case, where a point source is located at infinity toward the azimuthal direction φ_o. (29)

In this case, the direct computation of , i.e. , through spectral transformations is not possible since we cannot describe in closed form the source J_z(ρ, φ), therefore we need to resort to a different procedure, see Eqs (17), (18) of [34]. Since is a quantity due to the presence of the incident wave, we compute the current due to the incident wave through (17b) of [34] by using (18)s of [34] at . The jump of at φ = φ_o gives (30)

The procedure used in [34] is based on the computation of rotating waves for an incident plane wave without any structure/obstacles since is independent from the terminations (i.e the network parameters) of the angular region under consideration.

Since the representation of a plane wave with incident direction φ_o is modelled by a concentrated source in t, the network model of this kind of source is a current generator of intensity located at φ = φ_o.

Taking advantage of π network model of the previous subsection, the representation of an angular region φ₁ < φ < φ₂ illuminated by a plane wave with incident direction φ₁ < φ_o < φ₂ is constituted of two π networks, one for each free-of-source sub angular region P and Q (i.e φ₁ < φ < φ_o and φ_o < φ < φ₂), with a current generator located at φ = φ_o. This network representation in t plane is reported in Fig 9.

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Fig 9. The equivalent network in t plane for an angular region excited by a plane wave.

https://doi.org/10.1371/journal.pone.0182763.g009

4.3 Norton equivalent currents for a plane wave source

The estimation of the Norton equivalent currents for an angular region illuminated by a plane wave source is obtained through an auxiliary electromagnetic problem.

Fig 10 presents the problem of a PEC wedge with aperture angle 2π − (φ₁ + φ₂) in free space and illuminated by a plane wave at normal incidence with direction φ₁ < φ_o < φ₂ and electric field intensity E_o Eq (29). The free space angular region is subdivided into two homogenous regions P (φ₁ < φ < φ_o) and Q (φ_o < φ < φ₂). This problem allows to compute the Norton equivalent currents. According to the theory presented in the previous Sections, the problem can be modelled using a network in t-plane, see Fig 11, terminated by short-circuits. The two π networks are constituted of the following admittances: (31) (32)

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Fig 10. PEC wedge excited by an incident plane wave.

https://doi.org/10.1371/journal.pone.0182763.g010

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Fig 11. The equivalent network in t plane relevant to the problem described in Fig 10.

https://doi.org/10.1371/journal.pone.0182763.g011

By analyzing the network of Fig 11, we obtain the equivalent Norton currents (33)

Taking into account Eqs (21), (22) and (33) we obtain that the Norton model in t plane in presence of an incident plane source is (34)

The inverse MF transforms of Eq (33) yield the equivalent Norton currents in w-plane Eq (35) and by using the inverse mapping of η = −k cos(w) (see for instance [30]) we obtain the equivalent Norton currents in the η plane. (35) The expressions reported in Eq (35) of are compared to the ones obtained in the PEC wedge problem illuminated by a plane wave by using the method of rotating waves [34]. Eqs (21)–(23) and (31)–(33) define the complete Norton representation Eq (34) in t plane through algebraic equations, while via the inverse MF transform Eqs (25)–(27) and (35) define the Norton model in w plane Eq (36) through integral operators. (36) The Norton network model Eqs (34) and (36) of the angular region φ₁ < φ < φ₂ illuminated by an incident plane wave is Fig 6b.

5.1 Angular region terminated by a PEC surface and illuminated by a plane wave

Without loss of generality, let us suppose to a particular problem where the angular region 0 = φ₁ < φ < φ₂ = Φ_a is terminated by a PEC half-plane at Φ_a and illuminated by an E_z-polarized plane wave with incident angle φ_o, see Fig 12.

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Fig 12. Angular region delimited by a PEC half-plane.

https://doi.org/10.1371/journal.pone.0182763.g012

With reference to Fig 6b, in t plane taking into account Eq (34), we get the one port Norton representation (38) since .

In w plane Eq (38) becomes through Eqs (6), (7) and (25) (39) By using Eqs (26) and (35) the explicit form of Eq (39) is (40) that holds for Re[w] = 0 and PV stands for Principal Value due to the definition of MF transform. The network representation of Eqs (38) and (40) is reported in Fig 13 although the admittance are algebraic quantities in t plane and convolutional operators in w plane.

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Fig 13. Network models of the problem describe in Fig 12.

https://doi.org/10.1371/journal.pone.0182763.g013

We remark that Eq (40) holds independently from the media or sources located outside of the angular region under investigation, i.e. 0 < φ < Φ_a. Moreover, the validity of Eq (40) has been ascertained using a completely different procedure in η = −k cos w plane starting from the GWH formulation via Fredholm factorization [32, 35], see Section 7.1 for a short description.

Moreover, the numerical verification of Eq (40) has been performed for the auxiliary problem constituted by a PEC wedge with PEC faces located at φ = Φ_a and φ = −Φ_b for several values of Φ_a, Φ_b, φ_o. In particular we have considered the angular region defined by 0 = φ₁ < φ < φ₂ = Φ_a to test Eq (40). The comparison has been done with respect to the exact solutions of the PEC wedge problem in terms of and , see (73) and (74) of [8]. We compute the relative error of the approximated value of for Re[w] = 0, i.e. w = ju with , obtained via numerical integration of Eq (40) when substituting the exact expression of and taking into account specialized quadrature techniques to handle the PV integration as [42]. Fig 14 reports on top the absolute value of , on the bottom the relative error in log₁₀ scale obtained from the approximate expression of . The different curves refer to different sets of parameters Φ_a, Φ_b, φ_o.

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Fig 14. Validation of Eq (40).

Absolute value of on top and relative error in log₁₀ scale on bottom obtained from the approximate expression of given by Eq (40) while substituting the exact expression of and performing the numerical PV integration. Test cases are relative to different sets of parameters [Φ_a, Φ_b, φ_o], where φ = Φ_a and φ = −Φ_b are the azimuthal directions of the PEC faces of the wedge and φ = φ_o is the azimuthal direction of the incident E_z-polarized plane wave.

https://doi.org/10.1371/journal.pone.0182763.g014

In particular Fig 14b shows that the relative error is scattered and is approximately around 10⁻¹⁵ ÷ 10⁻⁹. The different performances are due to the numerical implementation of the PV integration in Eq (40).

6.1 Step 1

Since Eq (40) holds for Re[w] = 0 from the definition of inverse MF transform Eq (6), first, we want to extend the formulation for Re[w] < 0. We observe that the kernel Eq (41) in Eq (40) has kernel singularities located at w′ = w + 2nΦ_a () that constitute lines of singularities (for any complex w we have the singularity located at w′ = w + 2nΦ_a). Limiting the extension to the strip −2Φ_a < Re[w] < 0, only w′ = w gives contribution: for any w we have the pole w′. (41)

Let us consider the indented contour γ, which is the vertical line Re[w′] = 0 indented near w′ = w through a small semi-circumference on the right side of the imaginary axis. Taking into account the integral part of Eq (40), by deforming the contour integration line from M (Re[w′] = 0) to γ we obtain (42) The term out of the integral is related through the factor 1/2 to the residue due to the pole w′ = w of and it yields (43)

Substituting Eq (42) into Eq (40) with Eq (43) we obtain (44) that holds for −2Φ_a < Re[w] < 0. Similarly, by modifying the side of indentation of γ, we obtain: (45) that holds for 0 < Re[w] < 2Φ_a.

To extend Eqs (44) and (45) respectively in the half planes Re[w] < 0 and Re[w] > 0, we need to consider the entire set of kernel singularities w′ = w + 2nΦ_a (). In particular Eq (44) is generalized to the half planes Re[w] < 0 by adding on the right side of the equation the following term (46) where u(⋅) is the unit step function, due to the kernel singularities that cross and are captured by the integration line M (Re[w′] = 0) while moving w towards negative real values (note in Eq (46)).

We observe that the kernel singularities generate barriers in the analytic extension of the models in w plane, see Eqs (44) and (45), and we refer to this effect with the key term kernel singularity lines since in general w and w′ span their entire complex planes and not just real values.

To extend Eq (45) in the half planes Re[w] > 0 the additional term is (47)

As already done for Eq (40) in the previous Section, Eqs (44) and (45) have been numerically verified by considering the exact solutions of the PEC wedge problem in terms of and , see (73) and (74) of [8]. Both Eqs (44) and (45) are integral equations that relate to through the kernel Eq (41).

To test Eq (44) and to show the importance of Eq (46) we use the auxiliary problem constituted by a PEC wedge with PEC faces located at φ = Φ_a and φ = −Φ_b for several values of Φ_a, Φ_b, φ_o. To implement Eq (44) we refer to the angular region 0 = φ₁ < φ < φ₂ = Φ_a.

Fig 15 reports on the left the absolute value of for , −2π < w < 0, obtained from the exact expression (74) of [8] and the numerically estimated expression Eq (44) with-and-without Eq (46) while substituting the exact expression of (73) of [8] and performing the numerical integration. The right side of the figure reports the relative error in log₁₀ scale obtained from the approximate expression of given by Eq (44) with Eq (46).

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Fig 15. Validation of Eq (44) with the additional term Eq (46).

Left: absolute value of for , the exact value (solid thick gray line), the approximate value obtained through Eq (44) (gray dash-dot thin line) and the approximate value obtained by adding Eq (46) to Eq (44) (solid thin black line) while substituting the exact expression of and performing the numerical integration. Right: relative error in log₁₀ scale obtained from the approximate expression of given by Eq (44) with Eq (46). Test cases are relative to different sets of parameters: on top , on bottom . We recall that φ = Φ_a and φ = −Φ_b are the azimuthal directions of the PEC faces of the wedge and φ = φ_o is the azimuthal direction of the incident E_z-polarized plane wave.

https://doi.org/10.1371/journal.pone.0182763.g015

From Fig 15 we note that the approximate is valid for any real w < 0 and the relative errors are scattered and are approximately around 10⁻¹⁶ ÷ 10⁻¹¹. Eq (45) together with Eq (47) shows similar properties.

From here on, we define the line M as the vertical integration line Re[w] = 0, the line M₁ the vertical integration line Re[w] = −π/2 and the line the vertical integration line Re[w] = −Φ_a/2. We note that the line M₁ is a line that corresponds to the imaginary axis of η if k is real.

We observe that Eqs (44) and (45) are integral formulations with a singular non-compact kernels, that cannot be easily estimated using simple quadrature techniques: in particular we cannot numerically evaluate these equations while w belongs to M (due to the kernel singularity at w′ = w).

We state that Eqs (44) and (45) respectively with Eqs (46) and (47) are Norton representations Eq (39) with a convolutional operator admittance and a source generator in w plane that can be graphically represented as in Fig 13b.

From a mathematical point of view, Eqs (44) and (45) respectively with Eqs (46) and (47) are expressions of that constitute an analytic function, while Eqs (44) and (45) by themselves constitute sections of the analytic function , i.e. Eqs (44) and (45) are sectionally analytic functions [44].

6.2 Step 2

The procedure to obtain a well-behaved kernel K_new(w, w′) for Eqs (44) and (45) consists of a mathematical artifice. In the following we make reference to the implementation for Eq (44); the procedure to handle Eq (45) is straightforward.

The original kernel K_o(w, w′) Eq (41) is modified by adding a null term K₁(w, w′) Eq (48), that improves the mathematical behavior of the integral equation. (48) Because of the symmetry of the integrand in Eq (49), i.e. the integrand is an odd function of w′ (see Section 3), we obtain that (49) for −2Φ_a < Re[w] < 0.

By considering Eqs (41) and (49) the integral term Eq (50) in Eq (44) (50) results to be (51) for −2Φ_a < Re[w] < 0.

Eq (44) can now be rewritten using Eq (51) as (52) that holds for −2Φ_a < Re[w] < 0 and where the new kernel is (53)

The new kernel shows kernel singularity lines located at w′ = ±(w + 2nΦ_a) and w′ = ±(w + 2nπ) (). While extending Eq (52) for any Re[w] < 0 we need to take care of these singularity lines (except the one for n = 0 already taken into account in Step 1). Moreover, since we have added the null term (null in −2Φ_a < Re[w] < 0), we have also to consider its extension for any w, in particular such that Re[w] < 0, taking care of its kernel singularity lines that are the same of the new kernel.

We observe that: 1) the poles w′ = ±(w + 2nπ) yield no contributions for any Re[w] < 0 because they are canceled by the contributions that comes from the extension of Eq (49) for any Re[w] < 0; 2) the poles w′ = ±(w + 2nΦ_a) yield contributions that, together with the ones that comes from the extension of Eq (49) for any Re[w] < 0, give Eq (46) as already studied for the extension of Eq (44). Moreover, from a different point of view, the correction terms Eq (46) is the contribution of the poles w′ = ±(w + 2nΦ_a) reorganized using the symmetry of kernel, the symmetry of and the crossing direction of the line M performed by the kernel singularities. With the same properties we demonstrate that w′ = ±(w + 2nπ) do not give contributions.

The advantages of Eq (52) with respect to Eq (44) is that the new kernel Eq (53) is regular at w′ = w, although singular at w′ = −w. This singularity (w′ = −w) does not appear along M₁ (i.e Re[w] = −π/2) and thus the new kernel is compact if the line of integration M is warped into M₁. Moreover with integration contour M₁ we obtain quick convergence in the numerical estimation of the integral representation (see Step 3) due to its new kernel Eq (53).

To validate Eq (52) with Eq (46) we have repeated the same tests of validation already done for Eq (44) in step 1, by obtaining similar results as the one reported in Fig 15 for Eq (44) that we omit.

Similarly we can rewrite Eq (45) as (54) that holds for Re[w] > 0 by adding Eq (47).

As already discussed in Step 1 the additional terms Eqs (46) and (47) take into account the complete set of kernel singularity lines w′ = ±(w + 2nΦ_a) ().

Again, from a mathematical point of view, Eqs (52) and (54) respectively with Eqs (46) and (47) are expressions of that constitute an analytic function, while Eqs (52) and (54) by themselves constitute sections of the analytic function [44].

6.3 Step 3

The convergence properties of the new kernel Eq (53) along M₁ (Re[w′] = −π/2) suggest to warp the integration contour M (Re[w′] = 0) of Eq (51) into M₁, since the kernel is compact on this line.

With reference to Eq (52), while warping the integration contour M of Eq (51), we need to avoid that the integration contour crosses the spectral observation point w, since Eq (52) has been derived from Eq (40) selecting the contour γ indented on the right to overcome the PV integration, as described in Step 1. If we warp the integration contour of Eq (52) by crossing point w, Eq (52) is not anymore valid. In fact, in this case, the correct formulation of the problem is derived from Eq (40) selecting the contour γ indented on the left side, thus yielding Eq (54). Therefore, if one warps the integration contour of Eq (52) crossing point w, Eq (52) becomes Eq (54) and viceversa.

Once correctly warped M into M₁ in Eq (52) (by keeping point w on the left side of the contour), we make w tend to w′. We note that the kernel K_new(w, w′) is not singular on M₁, i.e. while w, w′ ∈ M₁. This procedure is useful to obtain FIEs for practical problems where the unknowns are discretized inside and outside the integral sign along the same path, for example M₁.

The contour deformation from M into M₁ can capture two kinds of singularities that belong to the strip :

• the non-structural (source) singularities of
• the kernel singularity lines of K_new(w, w′)

For this reason the contour deformation yields further terms out of integration in the integral equations.

In general, while warping the contour M into M₁ in Eq (51), we obtain: (55) The term I₁ is (56) with w on the left of M₁.

and I_K− are the terms respectively due to non-structural (source) singularities and kernel singularities.

Concerning the non-structural singularities, and with reference to an angular region problem with plane wave illumination, we note that these singularities are located in the 2nd and 4th quadrant of proper and improper sheets of plane η = −k cos w, since they correspond to real values of w (see Laplace transform Eq (1) and the mathematical definition of a plane wave).

We recall that the w plane presents all the Riemann sheets of η in a unique plane w: the proper and improper sheets are related to the definition of and they are linked to w region through the mapping η = −k cos w, see Appendix I of [30].

In w plane shows all the potential GO singularities w_GO even if they do not correspond to effective physical GO fields. The number and the location of these singularities depend on the geometry, on the media and on the sources of the entire problem where the angular region of aperture angle Φ_a under consideration is one of the sub-domains. The poles are related to potential incident, reflected, transmitted, multiple reflected and transmitted waves of the entire problem. The study of these poles requires the complete formulation of the problem since it depends on the whole geometry where the angular region constitutes a sub-domain.

In particular, for example, while considering an angular region terminated by PEC faces −Φ_b < φ < Φ_a, the complete set of singularities w_GO of (from (73) of [8]) are (57) for .

In particular we note that w_o = −φ_o is the pole of the incident wave, w_ra = −2Φ_a + φ_o is the reflected wave from face A, w_rb = −2Φ_b − φ_o is the reflected wave from face B, w_rbra = −2Φ_b + w_ra is the reflected wave first reflected from A than from B, w_rarb = −2Φ_a + w_rb is the reflected wave first reflected from B than from A…

The contributions are obtained using the residue theorem (58) where w_s are the w_GO with −π/2 < Re[w_GO] < 0, i.e. between M and M₁. Note that Eq (58) is exacltly known and computed by substituting with (the GO component of the spectrum), since is regular in w = w_s.

Concerning the kernel singularity lines of K_new(w, w′), they are located at w′ = ±(w + 2nΦ_a) and w′ = ±(w + 2nπ) () and therefore they could cross the line M₁ while moving the observation spectral point w (as discussed previously we note that the case n = 0 is already taken into account in Step 1).

The contributions of I_K− are derived using the residue theorem and the kernel singularities are captured if they cross M₁. Considering the symmetry of the kernel and of we obtain (59) with Re[w] < 0 and where u(⋅) is the unit step function that limits the series to be finite sums.

Note in Eq (59) that, while integrating along M₁, the kernel singularities (w′ = ±(w + 2nΦ_a) and w′ = ±(w + 2nπ)) yield line of singularities that are shifted according to −π/2 and parallel to M₁. In general we state an important property: the lines of singularities are always parallel to the integration line in w plane even if it is not straight as M₁.

We note that these contributions are independent from the source, i.e. they are structural. Moreover, the number of captured kernel singularities increases with decreasing of the aperture angle Φ_a.

Eq (52) can now be rewritten for Re[w] < 0 as (60)

Note that each strip defined by unitstep functions of Eq (59) in Eq (60) defines a section of the analytic function [44], although is an analytic function in the whole w plane.

Eq (60) is a Norton equation in w plane as it can be rewritten as (61) where is the admittance operator , is a current controlled by the GO fields of the entire problem and is the Norton current Eq (35).

The network representation of Eq (61) is reported in Fig 16 and it constitutes a modified Norton representation due to the presence of the controlled generator that depends on the GO terms of the entire problem. We note that the use of controlled generator is due to the warping of integration line from M to M₁.

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Fig 16. Modified Norton equivalent network corresponding to Eq (61).

https://doi.org/10.1371/journal.pone.0182763.g016

By considering Eqs (54) and (47), we can repeat the whole procedure to derive a Norton model which is valid for Re[w] > 0 with integration line M₁.

The general form of the modified Norton equation for any w is: (62) where (63)

We note that the sign symbol in Eq (62) select one of the two expressions derived from Eqs (44) and (45). The delimitation is set at π/2 since the regularization process has moved the discontinuity from w = w′ to w = −w′ and the integration line in Eq (62) is M₁ instead of M. Note in Eq (63) that, while integrating along M₁, the kernel singularities (w′ = w ± 2nΦ_a and w′ = w ± 2nπ) yield lines of singularities that are shifted according to −π/2 and parallel to M₁, see the arguments of the unitstep functions.

The Eq (62) is a generalization of Eq (60) that can be interpreted as Norton model Eq (61) which is valid for any w, where is the admittance operator (64) is a current controlled by the GO fields of the entire problem and is the Norton current Eq (35), see Fig 16 for the network representation.

We recall that the aim of the proposed procedure is to obtain FIEs for practical problems where the unknowns are discretized inside and outside the integral sign along a same path and therefore we need to pay particular attention on the properties of Eq (62) along the integration path, for example w ∈ M₁.

In general, the FIEs in are obtained from coupling Norton models of contiguous geometrical subdomains and by eliminating via substitution the unknown currents, i.e .

For w ∈ M₁, we observe that no kernel singularity line is captured, when Φ_a is obtuse, i.e. I_K = 0 for w ∈ M₁ and Φ_a > π/2. In fact taking into account the negative poles w′ = −(w + 2nΦ_a) () we observe that these kernel singularities remain on the left of the integration line M₁ if −(w + 2nΦ_a) < −π/2 therefore Φ_a > π/(2n) (). On the contrary, with acute Φ_a we need to pay attention to the kernel singularities located in −π/2 < Re[w] < 0.

As final validation of the Norton model in the w plane, we test Eq (62) in the entire complex plane w by computing in log₁₀ scale the relative error of the approximated value of obtained via numerical integration of Eq (62) when substituting the exact expression of for the auxiliary entire problem constituted by a PEC wedge with PEC faces located at φ = Φ_a and φ = −Φ_b, see (73) of [8]. In particular, we present in Fig 17 the results for , where the extremely acute angle Φ_a magnifies the number of kernel singularities captured while moving the spectral observation point w, see Eq (63).

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Fig 17. Validation of Eq (62).

Absolute value in log₁₀ scale for −4π < Re[w] < 0, on the bottom the relative error in log₁₀ scale obtained from the approximate expression of given by Eq (62) while substituting the exact expression of (73) of [8] and performing the numerical integration. . The colors at the bottom delimits zone with relative errors lower than 10⁻¹⁴ (dark blue), lower than 10⁻¹² (blue) and lower than 10⁻¹¹ (light blue).

https://doi.org/10.1371/journal.pone.0182763.g017

Fig 17 reports on top the absolute value in log₁₀ scale for −4π < Re[w] < 0, where it is possible to notice the GO poles related to the incident wave w_i = −φ_o and the reflected wave from face A w_RA = −2Φ_a + φ_o. The replica of the spectra is also shown.

The bottom of Fig 17 shows that the relative error is scattered and is approximately around 10⁻¹⁶ ÷ 10⁻¹² except for the local error along the lines of the kernel singularities w′ = ±(w + 2nπ) () (here w′ = −π/2 due to the selection of the integration line M₁).

Similar results and errors are obtained considering any aperture angle Φ_a: we observe that the number of kernel singularities and non-structural singularities concentrate toward small value of w for small aperture angles.

We assert that Eq (62) is generally valid with the following limitations: 1) kernel singularities do not belong to M₁, i.e. kernel singular lines do not overlap M₁, 2) singularities are with order 1. Generalization is straightforward.

The complete Norton model in w plane represented by Eq (62) can be re-phrased in η plane by carefully applying the inverse mapping η = −k cos w (see [30] for example). We recall that the w plane presents all the Riemann sheets of the η plane in a unique plane. The pictorial network representation of Fig 16 is also valid in η plane.

6.4 Step 4

As already stated the solution of practical problems is obtained via the numerical evaluation of FIEs that are sampled and discretized along the same (integration) line. The FIEs are obtained from coupling Norton models of contiguous subdomains and by eliminating via substitution the unknown currents, i.e starting from Eq (62) we eliminate in w plane.

The presence of I_K Eq (63) in Eq (62) makes Eq (62) an integral-difference equation that is cumbersome to treat, since we need to consider samples of the unknowns along different lines parallel to M₁.

We propose two methods to remove the presence of the difference terms.

The first method is to use a modified w plane, i.e. , and to integrate along the line (). Starting from Eq (40) and following Step 1, 2, 3 in with integration line , we obtain a new Norton model without the presence of kernel singularity lines. In fact, in this case, the kernel singularities are located at and with . The resulting FIEs do not contain any difference term in plane also in the case of small aperture angles. This method has been extensively used in recent papers for angular regions problems without network representations, see [8, 30, 31, 45].

However, for spectral observation points/samples w ∈ M₁ only the first sum in I_K Eq (63) can be different from zero and in particular is non zero if Φ_a is acute (less than π/2). We define (65)

Starting from Eq (62) with Eq (65) while sampling and integrating on M₁, the second method rewrites the equation in η plane warping the contour from M₁ to the real axis of the plane η. It resorts to the inverse mapping of η = −k cos w (see for instance [30]), to the regularity properties of the unknowns and to the application of the Cauchy Theorem in η plane. In particular we anticipate that the difference terms in w are transformed into integrals in η.

In this paper we focus the attention on the second method for two reasons. The first is that we want to illustrate a general procedure to study complex problems constituted by different geometries and media by obtaining simple network representations and the presence of different media does not allow to define an unique w plane because of its dependence, by definition, on the propagation constant k (see [31] to deal with multiple w planes). The second is that the possible presence of finite layers introduce the need to handle modal representation of the field that in w plane exhibits infinite replica of the relevant structural poles. On the contrary, in the η plane, the handling of the structural poles is straightforward because of the one-to-one relationship between the structural poles and the modal expansion of the field, see Section 7 for a practical example.

Taking into account Eq (62) with Eq (63), we recall that for vanishing imaginary part of k, M₁ is the imaginary axis of η plane (Re[η] = 0) with opposite versus. We can warp M₁ of Eq (62) in η plane into the real axis Im[η] = 0, because no poles are in the 1st and 3rd quadrant of η plane, see Figs 13 and 14 of [30]. In fact, concerning the non-structural singularities, and with reference to an angular region problem with plane wave illumination, we note that these singularities are located in the 2nd and 4th quadrant of proper and improper sheets of η plane, since they correspond to real values of w. The number and the location of these singularities depend on the geometry/media of the problem inside and also outside the angular region under consideration. The study of the singularities requires the complete formulation of the problem where the angular region under consideration is one of the sub-domains.

Moreover the warping in η plane does not yield any error in the contribution of the kernel singularities in I_K Eq (63) since, by considering the w plane, also the lines of the kernel singularities are shifted via the deformation (we recall that in w plane the kernel singularity lines are parallel to the line of integration, see Step 3). We note that the number of kernel singularities to be considered depends on the spectral observation points.

If the observation points lyes on the integration line (for example the real axis of the η plane), the equation preserves the same contributions of captured kernel singularities for integration line M₁ Eq (65).

Conversely, when only the spectral observation point is moved from the integration line to other position in complex plane η or w, the kernel singularities’ contribution to be considered is dictated by the general term Eq (63).

Therefore, by using the inverse mapping of η = −k cos w, considering the opposite versus of M₁ and for real values of η, in the proper η plane Eq (62) becomes: (66) for Im[η] = 0 and where (67) is the free-space spectral admittance, is the free-space spectral propagation constant (with ξ(0) = k), Z_o = 1/Y_o is the free space impedance and where (68) (69) In Eq (66) is defined in Eqs (39) and (40), and in Eq (58).

The terms in the summation reported in Eq (66) comes from the mapping of the difference terms Eq (65) in the η plane, i.e. (70)

We recall that, in the derivation of Eq (66), the kernel singularities (that give the difference terms) are captured only for Φ_a ≤ π/(2n) with , see Eq (65). For such values, taking into account that sin(2nΦ_a) > 0 and that for real η, it yields that p_n(η) for real η are located in the upper-half proper sheet of η plane, i.e. Im[p_n(η)] > 0, see Eq (68).

From the Cauchy’s integral formula [26] we obtain an exact expression of V₁₊(p_n(η)) while p_n(η) is located in the upper-half proper η plane (71) where the term is constituted of the non-standard plus singularities due to effective GO terms of the entire problem under investigation (non-standard non-structural poles).

Note that Eq (71) must be interpreted in the proper sheet of η plane and it is obtained by closing the contour in the upper-half plane.

By selecting the real axis of η plane as integration line and for observation points out of the real axis η, p_n(η) can be located in the lower-half proper η plane. Due to Cauchy Theorem, when p_n(η) is located in the lower-half proper η plane, the rhs of Eq (71) is zero.

This behavior also is in accordance with step functions defined in w plane Eq (65), in particular the step functions are successfully implemented in Eq (71) through contour integration in η plane.

Substituting Eq (71) into Eq (66) we obtain the Norton equation in η plane for real η value (72) where is the admittance operator (73)

We recall that is a current controlled by the GO fields of the entire problem Eq (58) and I_1N+(η) is the Norton current (74) with η_o = −k cos(φ_o) and .

The Norton model Eq (72) is obtained for observation points that lye on the integration line (here the real axis of η plane), for extension see next Step 5. Also in η plane, the FIEs are obtained from coupling Norton models Eq (72) of contiguous subdomains and by eliminating via substitution the unknown currents, i.e I₁₊(η).

Let us consider three characteristic problems to show the properties of Eq (72): PEC wedges with , Φ_b = 1.2π, φ_o = 0.1π. The three problems shows that the number of I_K1 terms Eq (65) to be considered in Eq (73) of Eq (72) is respectively 0, 1 and 2.

Fig 18 shows on the left side the approximate spectrum of I₁₊(η) for real η obtained through Eq (72) while substituting the exact spectrum of V₁₊(η) (taken from (73) of [8] with η = −k cos w) and performing the numerical integration. On the right the relative errors are computed with respect to the exact spectrum of (see (74) of [8]).

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Fig 18. Validation of Eq (72).

Left: absolute value of I₁₊(η). Right: relative error in log₁₀ scale obtained from the approximate expression of I₁₊(η) given by Eq (72) while substituting the exact expression of V₁₊(η) and performing the numerical integration. Test cases are relative to different sets of parameters: , Φ_b = 1.2π, φ_o = 0.1π. We recall that φ = Φ_a and φ = −Φ_b are the azimuthal directions of the PEC faces of the wedge and φ = φ_o is the azimuthal direction of the incident E_z-polarized plane wave.

https://doi.org/10.1371/journal.pone.0182763.g018

As expected since we have performed the integration and the testing along the same line, we obtain full convergence of Eq (72) along the real axis η.

The selection of the same line for sampling and integrating yields numerical estimation of the unknowns that are valid in selected strips of complex spectral planes: the approximate Norton representations and the approximate solutions of the corresponding FIEs are only analytic elements of analytic functions, i.e the representation are valid only in a section C_o of the complex plane η [44]. For example, Eq (72) is valid in a strip containing the real axis η. The validity strips are related to the selection of I_K terms Eq (63) according to Φ_a while sampling and integrating along a specified integration line. For example, while sampling and integrating on M₁ in w plane or on the real axis of η plane, we consider just I_K1 Eq (65), however, while reconstructing the representation of the solution in the entire complex plane (either w or η) we need to consider the entire expression I_K Eq (63) as reported in Eq (62) in w plane and in the related expression for η plane. We recall the kernel singularity lines due to I_K Eq (63) are lines parallel to the selected integration lines.

6.5 Step 5

The Norton representation obtained for real η Eq (72), see Fig 18, provides accurate spectra only in a region C_o near the real axis of η plane. In particular, Fig 19 shows that the validity of Eq (72) is limited for real values of w and in general for complex values of w such that η ∈ C_o with η = −k cos w. The size and the shape of C_o is defined by the kernel singularity lines I_K Eq (63) originated from the singularity of the kernel Eq (53) in Eq (62) in w plane and its transposition in η plane (see step 4).

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Fig 19. Validation of Eq (72) along the real axis w, by substituting η = −k cos w.

Left: absolute value of (exact = thick solid gray line, approximate = solid black line). Right: relative error in log₁₀ scale obtained from the approximate expression of I₁₊(η) for η = −k cos w given by Eq (72) while substituting the exact expression of V₁₊(η) and performing the numerical integration. Test cases are relative to different sets of parameters: , Φ_b = 1.2π, φ_o = 0.1π. We recall that −π < w < 0 is the portion of real axis of w plane in the proper sheet of η plane.

https://doi.org/10.1371/journal.pone.0182763.g019

However a complete and rigorous theory for the class of problems reported in Fig 3 requires the knowledge of the spectra in a region of complex plane η that is wider than C_o and that includes also the improper sheet of η plane.

In this framework, we state that both complex planes, w and η, are important to derive: 1) the formulations of the problem, 2) the subdomain (section) of complex planes where the numerical representations/solutions holds, 3) the physical properties of the solutions, 4) the field components.

For a general problem constituted by angular regions coupled with layered regions, we obtain the complete spectra of the problem through the knowledge of at least a starting approximated spectra in the fundamental strip S_o (−π < Re[w] < 0) through suitable recursive equations obtained by the GWHEs of the entire problem (see Section 7 for a practical case). We note that, in order to effectively apply recursive equations in problems restricted to angular regions the initial approximate spectra can be limited to −Φ_a < Re[w] < 0 (see applications in [30] and [31]), on the contrary when angular regions are coupled to layered regions we need −π < Re[w] < 0 due to the presence of layers.

Since initial approximate representations in the entire complex plane w and η are difficult to be obtained, we resort to extended spectra that are necessary to the computation of several field components: reflected and refracted plane waves, diffracted waves, surface waves, lateral waves, leaky waves, mode excitations, near field and so on…

Fortunately we experienced that, to complete the analysis of field in several practical cases, it is sufficient to know the spectra in limited regions even smaller than S_o.

Since the image of C_o in the w plane is only an initial strip S_s ⊂ S_o (see for practical examples Fig 19), the aim of step 5 is to resort to Eq (62) with Eq (63) that is valid in the entire complex plane w, to enlarge the image of C_o to a strip useful to estimate the fields components and possibly to the fundamental strip S_o. We recall the kernel singularity lines due to I_K Eq (63) are lines parallel to the selected integration lines.

Although the validity regions are clear in w plane see Eq (62) with Eq (63), the possibility to extend the approximate expression obtained in η plane in wider strip can be obtained first by dealing with kernel singularities in η plane, second by resorting to difference equations in w plane (obtained from the WH formulation) once the initial spectra is sufficient, i.e. in general problems −π < w < 0.

We note that limiting the required strip to S_o simplifies the general form of Norton representation Eq (62), in particular in Eq (63) for in −π < Re[w] < 0 the 2nd, the 4th, the 6th, the 7th and the 8th sums are zero due the unit step functions. We focus the attention of the singularities lines given by the 1st, the 3rd and the 5th sums in Eq (63) which are respectively , , and respectively related to the kernel singularities p_n(η) = −k cos(w + 2nΦ_a) Eq (68), and p_−n(η) = −k cos(w − 2nΦ_a) with . Notice that p_n(η) and are the same kernel singularity but their effect is shifted in w plane due to the step functions (see Eq (63)).

The initial strip S_s and the extended strips (while considering additional kernel singularities) strongly depend on Φ_a. These strips in η plane are cumbersome regions that usually span the proper and improper sheets of η plane (the sheets are related to the definition of and they are linked to w region through the mapping η = −k cos w, see Appendix I of [30]).

Moreover the location of kernel singularities p_n(η) can span from the upper half plane to the lower half plane or the improper sheet of the η plane.

According to the Cauchy’s integral theorem, when p_n(η) goes into the lower half plane, Eq (71) is not a valid representation of V₁₊(p_n(η)) and it must be replaced by (75) where the term is constituted of the standard plus singularities due to effective GO terms of the entire problem under investigation (standard non-structural poles) and b_r is the contour around the structural branch lines of the branch points k. We notice that in presence of dishomogeneous media we have multiple k to be handled, one for each material. Note that Eq (75) must be interpreted in the proper sheet of η plane and it is obtained by closing the contour in the lower-half plane.

The number and the location of kernel singularities to be considered are strongly dependent on the spectral observation points and on the aperture angle Φ_a. We have recognized three characteristic ranges of aperture angle Φ_a by the properties of the spectral unknowns for real negative w: obtuse π/2 < Φ_a < π, acute π/4 < Φ_a < π/2, hyper-acute 0 < Φ_a < π/4.

For obtuse angle no I_K1 term Eq (65) needs to be considered in numerical estimation of the Norton model Eq (72) for real η, instead, while numerically reconstructing the current in −π < w < 0 we need to consider I_K Eq (63). For acute and hyper-acute angles we need to consider I_K1 terms in the Norton model and I_K in reconstructing the current in −π < w < 0. For hyper-acute angles some of the kernel singularities can move from the upper half η plane to the lower half for −π < w < 0 (we recall that for real η the captured kernel singularities p_n(η) lies on the upper half η plane).

The same properties holds for the numerical solution of the to be-defined FIEs in terms of voltage unknowns since they are obtained by coupling Norton models and eliminating the current unknowns.

As shown in Fig 19 for the three test cases, we have that the region −π < Re[w] < 0 is limited for test case 1 (Φ_a, = 0.7π) on the left by (i.e. ), for test case 2 (Φ_a, = 0.4π) it is not limited (i.e. −π < Re[w] < 0), for test case 3 () it is limited on the left by and on the right by (i.e. )

By considering all the above properties, starting form Eq (62) with Eq (63) defined in w plane, we extend the validity of Eq (72) in η plane to model the approximation in −π < w < 0 considering the kernel singularities p_n and via Eq (71) and (75) and applying unite step functions in w plane as reported in Eq (63).

Therefore an extended form of Eq (66) while η = −k cos w with −π < w < 0 is: (76) Note that Eq (76) is written in η although it contains unit step functions in w plane that are cumbersome to be mapped in η plane. The presence of unitstep functions limits the series to be finite sums and the numbers of terms strongly depends on Φ_a.

We recall that for kernel singularities p_n located in the upper half η plane we need to use Eq (71) in Eq (76), on the contrary for kernel singularities in the lower half η plane we need to use Eq (75).

In the last case the rigorous numerical representation of I₁₊(η) via Eq (76) requires the integration of V₁₊(η) along the real axis of η plane together with the branch lines b_r that requires some effort to be numerically implemented.

To obtain an integral representation that do not sample branch lines, we can approximate V₁₊(p_n(η)) with p_n(η) in the lower-half η plane with different strategies: 1) shifting the contour of Eq (71) below p_n(η) and neglecting the horizontal integration line, i.e. we approximate V₁₊(p_n(η)) with captured GO poles; 2) closing the contour in the lower-half plane and neglecting the integration along b_r lines, see Eq (75); 3) using approximation based on continuing V₁₊(p_n(η)) expression Eq (71) by fixing the pole p_n(η) that goes in the lower half η plane just above the real axis η plane, i.e. for example at −k/100. In particular in the following numerical example we will demonstrate the performance of the 3rd method that ensures continuity in the transition region and keeps errors limited.

6.5.1 Numerical tests.

In Table 1 we have reported all the properties of kernel singularities of the Norton model Eq (72) and their influence in its analytical extension Eq (76) for real value of w in −π < w < 0 for the three characteristic test cases reported in the previous Sections, i.e. PEC wedges with , Φ_b = 1.2π, φ_o = 0.1π. Moreover Table 1 shows the same properties for a fourth test case in a more general form when Φ_a is reduced to π/9. As expected the decreasing of Φ_a increases the number of captured kernel singularities, i.e. reduces the dimension of the section where Eq (72) is analytic by itself.

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Table 1. Properties of kernel singularities and kernel singularity lines (we omit to report the dependence in η or w of kernel singularities).

https://doi.org/10.1371/journal.pone.0182763.t001

Note that the general properties of captured structural singularities are derived from Eq (63). For example the number of captured kernel singularities p_n while the observation points and the integration points lies on the same line (e.g. w ∈ M₁, real axis η) are 0 if π/2 < Φ_a < π (obtuse angular region) and n if π/(2(n + 1)) < Φ_a < π/(2n) for (acute angular region). A more complex rule holds for structural singularities p_n, and p_−n when the observation point is moved in −π < w < 0 as shown in the test cases reported in Table 1.

To avoid more cumbersome properties the line of integration must be regular, monotonic and cross the real w axis in one point. This requirement allows to deduce the property of kernel singularities for −π < w < 0.

Let us examine in detail test case 3 of Figs 18 and 19, i.e. , that is able to show all the properties of the equations. From Table 1 we notice that Eq (72) contains p₁(η) and p₂(η) Eq (68) (third column).

From Eq (72), the validity segment (real w) for test case 3 is since Eq (72) does not contain any further structural poles p_n for integer n > 2 and n < 0 or , that become necessary for analytical extension in w plane as clear in Eq (62) with Eq (63) and clearly reported in Eq (76) (see also fourth and fifth column of Table 1).

The two kernel singularities p₁ and p₂, while moving the observation point along the real axis of η plane, stay in the upper-half η plane. While moving the observation point along the real axis of w plane for −π < w < 0 the pole p₂ starts in the lower η plane at w = 0 and then it goes to the upper η plane at , on the contrary the pole p₁ starts in the upper η plane at w = 0 and then it goes to the lower η plane at . This property is expected in test case 3 since , see Eq (63) (see sixth column in Table 1).

At the same time is activated by the unit step functions Eq (76) when p₁ goes into the lower half plane, doubling its effect (see sixth column in Table 1). This properties is general and the singularity line related to each , i.e. , corresponds to the real value for which p_n goes into the lower half plane together with .

In test case 3, we note that for observation point −π < w < 0 we need to consider the contributions of p_n with n = 3, 4 that are activated respectively for and the contribution of p_n with n = −1 that is activated respectively for as reported in Eq (76) (see also Table 1 for a summary).

Fig 20 shows the validation of the general expression Eq (76) for −π < w < 0 for the three considered characteristic test cases. The results can be compared with the ones reported in Fig 19 obtained through Eq (72).

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Fig 20. Validation of Eq (66) along the real axis w, by substituting η = −k cos w.

Left: absolute value of (exact = thick solid gray line, approximate = solid black line). Right: relative error in log₁₀ scale obtained from the approximate expression of I₁₊(η) for η = −k cos w given by Eq (66) while substituting the exact expression of V₁₊(η), performing the numerical integration and considering the full properties of V₁₊(p_n(η)) in w plane. Test cases are relative to different sets of parameters: , Φ_b = 1.2π, φ_o = 0.1π. We recall that −π < w < 0 is the portion of real axis of w plane in the proper sheet of η plane.

https://doi.org/10.1371/journal.pone.0182763.g020

In particular we recall that for Φ_a < π/4 at least one p_n changes location in the half η planes. As expected we have full convergence when the kernel singularities lie on upper half η plane (see test case 1 and 2).

Since we have implemented an approximated technique (see previous subsection) to handle kernel singularities that lie in the lower half η plane for spectral observation points w that go toward w = −π (see test case 3) we experience a loss of precision moving away from the region where all the kernel singularities are in the upper half η plane.

However the loss of precision in test case 3 is localized for a restricted portion of spectra and the relative error is of the order of 10⁻² ÷ 10⁻³ that is acceptable in asymptotic estimations as demostrated in Section 6.6.

6.6 An example of implementation of FIE: The PEC wedge

As an example of validation, let us consider a PEC wedge (−Φ_b < φ < Φ_a) in a homogenous medium and illuminated by an E_z polarized plane wave with direction 0 < φ_o < Φ_a, see Fig 21a. We can subdivide the homogeneous angular region into two angular regions: 1) −Φ_b < φ < 0 and 2) 0 < φ < Φ_a.

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Fig 21. The Concave PEC wedge.

(a) Geometry of the problem subdivided into two angular regions: 1) −Φ_b < φ < 0 and 2) 0 < φ < Φ_a; (b) Network model using coupling of modified Norton equivalent circuits.

https://doi.org/10.1371/journal.pone.0182763.g021

For region 2, the previous Sections illustrate the procedure to obtains a Norton model in w and η planes, starting form Eq (40) with Fig 13b and obtaining Eq (62) with Fig 16 and Eq (66).

Following the same procedure for region 1, by noticing that no concentrated source is present for −Φ_b < φ < 0 and that the current has opposite versus in the one port Norton model, we obtain the same Norton equations with the following differences: 1) the sign of the current , 2) absence of , 3) Φ_a must be replaced by Φ_b, 4) the structural pole contributions depend on the amplitude of Φ_b instead of Φ_a.

The complete network model is presented in Fig 21b where a concave wedge is reported. We have selected a concave wedge with small aperture angle Φ_a + Φ_b to highlight all the properties of the Norton model for angular regions, in particular for small angular regions.

By summing the equations obtained for region 1 and 2 we eliminate obtaining an equation in , in particular in (integral -difference equation) and V₁₊(η) (Fredholm integral equation).

To validate the FIE for real w values, we have compared the exact spectrum , with one obtained numerically through the FIE by substituting inside the integral the exact expression of V₁₊(η) (for the exact spectrum see (73) of [8]).

Fredholm theory guarantees the convergence via numerical procedure of integral equations of second kind [43]. Since the kernel of the FIE presents well suited behavior, we use a simple sample and hold quadrature scheme to obtain accurate and stable numerical solutions: we apply uniform sampling f(hi) with and modified left-rectangle numerical integration formula where A and h are respectively the truncation parameter and the step parameter for the integrals in u. This rule has been successfully applied in wedge problems [30, 31, 45]. The total number of samples is N = 2A/h + 1. We observe that as A → +∞ and h → 0, the numerical solution of the FIE converges to the exact solution [43]; consequently h has to be chosen as small as possible and A has to be chosen as large as possible. In practical numerical implementations A and h are finite and they are selected according to the kernel behavior (spectral bandwidth and shape) as demonstrated in the numerical validations and examples: for instance see next Section 7 which reports effective values of A and h. To improve the accuracy of the numerical procedure, in general, we use the expedient described in [3, 4, 32] based on contour warping.

Once obtained the numerical solution in terms of approximated spectrum of V₁₊(η), we can obtain the approximate spectrum of I₁₊(η) through the direct application of the Norton equation in η plane either of the angular region 1 or of the angular region 2. The validity of the approximated expressions are ensured for −π < w < 0 and in sections of the proper sheet of η plane according to the singularity lines related to the selected integration line.

Fig 22 shows on the left the plots of the exact spectrum and of the approximate spectrum obtained through the FIE for the three characteristic test cases already used int the previous Sections: PEC wedges with , Φ_b = 1.2π, φ_o = 0.1π. On the right the same figure the relative error is plotted in log₁₀ scale. We note that the spectrum is valid for at least any real −π < w < 0 and the relative errors are scattered and are approximately around 10⁻¹⁶ ÷ 10⁻¹¹ except for test 3 near −π where we have applied the approximation discussed in the previous Section for the kernel singularities that go into the lower half η plane.

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Fig 22. Validation of FIE along the real axis w, by substituting η = −k cos w.

Left: absolute value of (exact = thick solid gray line, approximate = solid black line). Right: relative error in log₁₀ scale obtained from the approximate expression of V_1d(η) for η = −k cos w given by while substituting the exact expression of V₁₊(η) and performing the numerical integration. Test cases are relative to different sets of parameters: , Φ_b = 1.2π, φ_o = 0.1π. We recall that −π < w < 0 is the portion of real axis of w plane in the proper sheet of η plane.

https://doi.org/10.1371/journal.pone.0182763.g022

To analytically extend the solution in the entire complex plane w we need to resort to difference equations obtained from the Wiener-Hopf formulation of the problem by eliminating the minus unknowns, as reported in [30].

Once the spectra are known, the procedure to obtain the total field from the spectra is reported in several papers, see for example in the context of impenetrable wedge structure [30]: first we apply the SDP (Steepest Descent Path) method to get GO contributions and GTD diffraction coefficients, then the UTD theory is applied to obtain the total field around the edge of the wedge.

To test the full procedure in particular for critical small aperture angles, we have numerically solved the FIE and computed the total field after the estimation of the GO and UTD component.

Fig 23 reports on the left the total field obtained from the spectra in terms of GO and UTD components for a significant test case with parameters . Note that region 2 is hyper-acute and region 1 is acute to highlight all the properties in terms of kernel singularities. On the right of Fig 23 the relative error with respect to the exact solution is reported in log₁₀ scale. For the numerical solution we have selected a simple sample and hold quadrature with coarse parameters A = 20, h = 0.025. The loss of precision is due to the approximate handling of kernel singularities that go into the lower half η plane for hyper-acute aperture angles. For non-hyper acute aperture angles the accuracy is perfect and the relative errors approximately goes to machine precision.

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Fig 23. Validation through the total field for the wedge problem.

Left: Total field obtained from the spectra in terms of GO and UTD components for the test case . Right: Relative error on the computation of the total field in log₁₀ scale of the numerical solution obtained with coarse numerical parameters A = 20, h = 0.025.

https://doi.org/10.1371/journal.pone.0182763.g023

The validation is asserted by the complete overlap of the total field obtained with exact spectra and the approximate spectra.

Below, we list the mathematical and physical properties of the significant test case.

From a mathematical point of view, region 2 has an aperture angle , thus it is hyper-acute. According to Table 1 we have to consider p₁ and p₂ in Φ_a as structural singularities while integrating along the real axis η and we have to consider for the analytical extension in −π < w < 0. Moreover, we need to pay attention to the properties of . On the contrary, according to Table 1, region 1 is acute since Φ_b = 0.4π and therefore we have to consider p₁ in Φ_b as structural singularity while integrating along the real axis η and for the analytical extension in −π < w < 0.

From a physical point of view, we note that the GO contributions are [in the following we report the outgoing angle from the edge of the wedge for each wave]: the incident wave ϕ_I = −π + ϕ_o = −0.9π, the wave due to the first reflection from face A , the wave due to the first reflection from face B ϕ_RB = π − 2Φ_b − ϕ_o = 0.1π with azimuthal support −Φ_b < ϕ < ϕ_RB and the wave due to the first reflection from face A and the second from face B with azimuthal support −Φ_b < ϕ < ϕ_RBRA. UTD estimation compensate the two shadow regions.

From a mathematical-physical point of view the GO contribution are related to the spectral poles Eq (57): [w_I, w_RA, w_RB, w_RBRA] = [−ϕ_o, −2Φ_a + ϕ_o, −2Φ_b − ϕ_o, −2Φ_b − 2Φ_a + ϕ_o]. Note that only the poles w_I, w_RA are captured in −π/2 < w, 0 and gives contributions in the FIE.

7.1 Network model

Region 1 is an angular region that can be modelled via Norton Eq (72) in η plane or Eq (62) in w plane by considering that the GO poles are derived by a new configuration where the face b of the angular region is substituted by a PEC plane.

With an alternative procedure we note that, Eq (72) and all the other equations reported in the previous Sections for the angular region can be obtained directly from the GWHE Eq (77) according to the following steps:

1. transformation of Eq (77) into a classical WH equation in α plane via the mapping Eq (67)
2. elimination of minus unknown I₂₊(−m) in α plane via Cauchy integration that yields the FIE
3. application of mapping Eq (67) and η = −k cos w
4. contour warping from the real axis of η plane to imaginary axis of η plane and then to M₁ and M

This procedure has been effectively applied without reference to network modelling to several wedge problems in plane, see [30, 31].

The main drawback to derive the FIE in η plane from the GWHE Eq (77) is that the above procedure do not highlights the validity regions of the numerically approximate formulations in particular for small aperture angles that yield to sectionally analytic representations [44]. In particular the kernel singularities and their relevant regions are cumbersome to treat directly in η plane.

Let us consider region 2.

We apply Cauchy integration to Eq (83) along the contour γ_1η which is the smile integration line defined in [32] and [3]. We obtain for (86) since I₁₋(η) is regular in and thus .

By applying Cauchy integration to V₁₊(η) and I₁₊(η) along the contour γ_2η which is the frown integration line defined in [32] and [3] we obtain (87) where and are known functions that depends on the Geometrical Optics poles related to the entire problem and located in the upper-half η plane.

Combining Eqs (86) and (87) and taking into account that (88) we obtain the integral equation: (89) where is the operator (90)

In particular for the problem under examination I_D(η) is (91) where (ν) is related to the GO components with incident angles 0 < φ_ν < min[Φ_a, π/2] (incident wave, reflected wave from face a and multiple reflected waves from face a when Φ_a is acute) and (92) with η_ν = −k cos(φ_ν) that corresponds to the pole generated by the GO waves ν, with that is the reflection coefficient from the PEC plane and .

Eq (89) is the Norton representation of region 2 with generalized admittance and (short-circuit) Norton current I_D(η). The Norton generator is controlled by the GO terms of the entire problem.

Fig 25 reports the network model of the problem represented through the Norton Eqs (72) and (89) in η plane.

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Fig 25. Network model of the test problem reported in Fig 24.

https://doi.org/10.1371/journal.pone.0182763.g025

7.2 From network model to the solution

From circuital analysis of the network represented in Fig 25 we obtain (93) where I_tot(η) is the contribution of the currents due to the generators I_1N+ Eq (74), I_GO Eq (58) and I_D Eq (91), and are the operator admittances respectively for the layer region 2 (see Eq (90)) and for the angular region 1 (see Eq (73)).

Eq (93) is a Fredholm integral equation with unknown V₁₊(η), see Appendix. In normal form the explicit expression of Eq (93) is (94) where (95) and V_tot(η) = Z_e(η)I_tot(η).

Since Eq (94) is a FIE (see Appendix), it has good convergence properties. The presence of infinite integration interval can be overcome through suitable mappings to reduce the infinite integration interval to a finite one with optimal functional properties of the integrand [47] or truncating the integration interval due to the convergence behaviour of the spectral unknowns [43].

Since the kernels of Eq (94) present well suited behavior, we use a simple sample and hold quadrature scheme with truncation parameter A and step parameter h to obtain accurate and stable numerical solutions as already discussed and done for the wedge problem, see previous Section 6.6 and reference therein.

The coefficient of the excited TE_x modes c[−η_i] with inside the planar guide −d < y < 0, x < 0 are computed through the residue of I₁₋(η) in −η_i which are the propagation constants of the excited regressive TE_x modes where (96)

We note from Eq (83) that (97) with Y(η) defined in Eq (81).

Since we can estimate a priori that the spectrum of I₁₊(η) does not contain −η_i poles, we observe that residues of I₁₋(η) can be computed from approximated spectra of V₁₊(η).

In particular, from Eq (97) we compute (98) with (99)

The numerical solution of the integral equations provide an analytical element of the spectra V₁₊(η), I₁₊(η), I₁₋(η) that is in the proper sheet of the η-plane or in a strip of w plane. To obtain the diffraction coefficients, in general we need to know the spectra also in the improper sheet of the η plane or in a region of w plane that corresponds to multiple sheets of η plane, in particular the real axis of w plane. Analytical continuation of the approximate spectra is possible by resorting to the original GWHEs of the Problems (77) and (83) reformulated in the w-plane that yields recurrence relations/difference equations in w plane capable of extend the known approximate spectra to the whole complex plane w [4]. The recurrence equations for plus unknowns are obtained via the elimination of minus unknowns exploiting the fact that plus unknowns are even functions of w.

Analytical extensions of and are obtained via: (100) (101) with (102) (103) where and where has been used to get more compact expressions.

As stated in the previous Sections for complex problem we need a starting spectra in −π < w < 0 to compute the far field components.

Once the complete spectrum is obtained, diffraction coefficients are then calculated through SDP method using the extended spectra in w, see for example [8, 30, 31].

Fig 26 reports on the left the total field obtained from the approximate spectra in terms of GO and UTD components for the test case of Fig 24 with [Φ_a, φ_o, d] = [0.8π, 0.55π, 0.55λ] where λ is the wavelength. The numerical solution is obtained with integration parameters A = 40, h = 0.2 according to the kernel behavior. For the explicit expression of the total field we make reference to Section IV of [31] where the GO and UTD components are estimated from the spectra and .

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Fig 26. Total field.

Left: Total field obtained from the approximate spectra in terms of GO and UTD components for the test case of Fig 24 with parameters [Φ_a, φ_o, d] = [0.8π, 0.55π, 0.55λ]. The numerical solution is obtained with integration parameters A = 40, h = 0.2 according to the kernel behavior. Right: Comparison of Total Field with FEM solution as described in the main text.

https://doi.org/10.1371/journal.pone.0182763.g026

In this problem we obtain a GO contributions of incident wave, reflected wave from face A with outgoing direction Φ_RA = −π + 2Φ_a − ϕ_o = 0.05π with support ϕ_RA < ϕ < Φ_a and reflected wave from the PEC plate with outgoing direction Φ_RD = π − ϕ_o = 0.45π with support 0 < φ < ϕ_RD.

On the right side of Fig 26 we compare our solution with the one obtained by in-house FEM [48] with the following setup: region truncated at 9λ with perfectly matched layer and discretization via triangles with max side length of λ/8. We note that the FEM solution is less accurate in particular for forward far field directions (small φ) due to the uncorrect modelling of the termination of the waveguide. In fact this uncorrect modelling generates small spurious reflected waves that radiate again. We also note that perfectly matched layer should entirely surround an object in classical scheme.

For the same problem we have estimated the field inside the waveguide by computing c[−η_i] Eq (99). Table 2 reports some of the values of η_i and the corresponding c[−η_i] normalized to Z_o. Note that with the problem’s parameters we have just one propagating mode.

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Table 2. Properties of TE_x modes.

https://doi.org/10.1371/journal.pone.0182763.t002

Through modal expansion, we can build the H_x(x, y) field: (104)

Fig 27 shows the plot of H_x(x, y) at x = [−λ/10, −λ/5, −10λ] with N_i = 100, although N_i = 20 is sufficient for convergence especially leaving the edge profile (the origin of the cartesian coordinates in Fig 24). Details on the implementation of this practical problem with arbitrary aperture angle Φ_a and multiple media are going to be reported in [46].

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Fig 27. Modal field.

Modal field Z_oH_x(x, y) and E_z(x, y) at x = [−λ/10, −λ/5, −10λ] with N_i = 100 for the test case of Fig 24 with parameters [Φ_a, φ_o, d] = [0.8π, 0.55π, 0.55λ].

https://doi.org/10.1371/journal.pone.0182763.g027