The H-polarized wave scattering by FPFISG

It is well known, that the problem of the plane EM wave scattering by classical zero-thickness strip grating leads to external boundary problems for two-dimensional (2D) Helmholtz equation on some set of segments – cross-section of the finite grating [Reference Honl, Maue and Westpfahl1]. It is the external 2D mixed boundary problem for the case of IS grating. Making use of the SIE method immediately, this 2D mixed boundary problem can be transformed to 1D problem of the first kind SIE system solution [Reference Koshovy19, Reference Koshovy20]. The SIE system for both polarization of the plane EM wave can be written in the form:

(1)$$\int_S {} \phi \lpar {x}^{\prime}\rpar \cdot H_0^{\lpar 1\rpar } \lpar k\cdot \vert {x-{x}^{\prime}} \vert \rpar d{x}^{\prime} = F\lpar x\rpar .$$

Here the domain of integration S is a set of segments, $H_0^{\lpar 1\rpar } \lpar y\rpar$ is the first Hankel function of zero-order (the Green function of 2D space up to a constant [Reference Honl, Maue and Westpfahl1]) and k is the wave number. Unknown function φ(x) determines the surface current densities flowing on the strips. Right-hand part's function is essentially dependent on polarization and contains two sets of unknown constants for both EM wave polarizations. This is the principal difference between FPFISG under consideration and a pre-fractal grating of PEC strips [Reference Koshovy16, Reference Koshovy17]. So, to have a correct mathematical model of the plane EM wave scattering by FPFISG some transformations of the SIE system will be realized. Firstly, we ought to fill it with the real content of pre-fractal nature. For this purpose, let us consider some useful information about 1D SSF, as an ideal notion in pure mathematic.

The simplest class of SSF with variable HD

By definition, pre-fractal grating is a grating with the finite number of elements located according to a certain stage of 1D SSF creation. SSF can be clearly defined mathematically as a set whose Hausdorff dimension (HD) is strictly greater than its topological dimension [Reference Mandelbrot11, Reference Koshovy17]. There are a lot of 1-D fractals with variable HD in some segment of real line R. The most regnant representative of the 1-D SSF class is triadic SSF (or more shorter and convenient SSF2). Figure 1 presents four stages of the simplest 1D SSF2 creation process: initiator – a segment, creator (or the first stage of SSF2 creation) consists of two segments, which are defined by the parametrical equations: ± b + at (|t| < or = 1), and so on.

Fig. 1. Creation of the simplest 1D SSF2.

Thus, two geometrical measureless parameters β = kb, α = ka, (k is the wave number) present the simplest type of SSF. Its fractal (or Hausdorff) dimension (defined by the formula: HD = ln2/lnκ (κ = 1 + b/a = 1 + β/α > 2) is positive and zero topological dimension. The second stage of SSF2 creation consists of four segments, which are defined by the equations: ± b ± (b + at)/κ. It will be noted, that the HD is a real number and changes in the interval (0, 1) depending on the self-similarity coefficient κ. For the case of the classical perfect Smith-Cantor set, the coefficient κ = 3, so, well-known formula: HD = ln2/ln3 is arisen [Reference Mandelbrot11]. The mentioned two measureless parameters, α and β, represent input variables of the mathematical models of the SSF2 creation process. The output variables of the model should be considered as a set of 2^m (m is number of the SSF2 creation stage) functions, which strictly fix positions of the segments of a stage along the numerical axis. For example, the SSF2 creator is defined by the functions: ± β + α t. In such a way, using output functions of 1D SSF creation model, we transform the SIE systems and fill them with the real content of pre-fractal strips location [Reference Koshovy15–Reference Koshovy17].

Correct mathematical model

To have a correct mathematical model, it is necessary to transform the system of SIE equation (1), which can simulate the plane H-polarized EM wave scattering by a FPFISG. Firstly, the logarithmic singularity of the diagonal kernel functions has to be separated in the simplest form and the SIE system takes the form:

(2)$$\matrix{\int_{-1}^1 {\,j_\kappa (t)} {\rm ln}\left| {\tau -t} \right|dt + \sum\nolimits_{m = 1}^M {} \int_{-1}^1 {\,j_m(t)R_{\kappa m}(\tau -t)} dt = f_\kappa (\tau ), \hfill \cr \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left| \tau \right| < 1,\;\;\;\;\;\kappa = 1,...,M. \hfill} $$

New unknown function $j_\kappa ^{} \lpar t\rpar$ is proportional to values of old unknown function φ(x) on the segment of number κ. Functions Rκm(u) represent now regular kernels:

$$R_{\kappa \kappa }\lpar u\rpar = C + \ln \displaystyle{\alpha \over {2i}} + \sum\limits_{m = 1}^\infty {\displaystyle{{{\lpar {-}1\rpar }^m} \over {{\lpar m!\rpar }^2}}} \left({\displaystyle{{\alpha u} \over 2}} \right)^{2m}\left[{C + \ln \displaystyle{{\alpha \vert u \vert } \over {2i}}-\sum\limits_{n = 1}^m {{1 \over n}} } \right]\comma \;$$

(3)$${R_{\kappa m}\lpar u\rpar = \pi \cdot H_ 0^{\lpar 1\rpar } \lpar \vert {x_\kappa^{} \lpar 0\rpar -x_m^{} \lpar 0\rpar + \alpha u} \vert \rpar /2i}\comma \;\kappa \ne m.$$

Here C≈0.5772… is the Euler constant; the functions $x_m^{} \lpar t\rpar = x_m^{} \lpar 0\rpar + \alpha t$ are ordered linear functions – some stage output functions of 1D SSF creation [Reference Koshovy15–Reference Koshovy17]. In Fig. 2 one can see FPFISG, which consists of 8 IS, located according to the third stage of the simplest 1D SSF2 creation (in Fig. 1 the third level of 8 segments).

Fig. 2. FPFISG consisting of 8 IS.

The right-hand part functions of the SIE system equation (2) are presented by the expressions:

$$f_\kappa \lpar \tau \rpar = d_{1\kappa }\sin \lpar \chi \alpha \tau \rpar + d_{2\kappa }\cos \lpar \chi \alpha \tau \rpar + d_{0\kappa }\exp \lpar iq_1\alpha \tau \rpar .$$

The known constants are defined by the expressions:

$$\chi = \sqrt {1-w^2} \comma \;\,d_{0\kappa } = 2\pi i\cdot \exp \lpar iq_1x_\kappa \lpar 0\rpar \rpar /k\lpar q_2 + w\rpar .$$

Where measureless impedance parameter w is defined by the ratio of a surface strips' impedance and the wave resistance of a medium around the strips. Constants d _1κ, d _2κ are unknown. Here, it is possible to transfer to PEC strip grating by putting w = 0 [Reference Koshovy21].

Secondly, the correct transition from the system of the first-kind SIE with logarithmic (weak) singularity to a system of SIE with Cauchy type (strong) singularity must be provided. The resulting first-kind SIE system with strong singularity after the transformation has the form:

(4)$${\int_{-1}^1 {} \displaystyle{{\,j_\kappa (t)} \over {\tau -t}}dt + \sum\nolimits_{m = 1}^M {} \int_{-1}^1 {} j_m(t)R_{\kappa m}^{\prime} (\tau -t)dt = f_\kappa^{\prime} (\tau )}$$

Partial derivatives of diagonal kernels are defined by the formulas:

$$\scale97%{ \eqalign{{R}^{\prime}_{\kappa \kappa }\lpar u\rpar = & \sum\limits_{m = 1}^\infty {\displaystyle{{{\lpar {-}1\rpar }^m} \over {{\lpar m!\rpar }^2}}} \left({\displaystyle{\alpha \over 2}} \right)^{2m}u^{2m-1}\left\{{2m\left[{C + \ln \displaystyle{{\alpha \vert u \vert } \over {2i}}-\sum\limits_{\kappa = 1}^m {{1 \over \kappa }} } \right] + 1} \right\}\comma \;\cr & {R}^{\prime}_{\kappa m}\lpar u\rpar { = \alpha \displaystyle{{\pi i} \over 2}H_ 1^{\lpar 1\rpar } \lpar \vert {\rho_{\kappa m} + \alpha \cdot u} \vert \rpar }.} }$$

Here $H_1^{\lpar 1\rpar } \lpar y\rpar$ is the first Hankel function of the first order and the parameter of two indexes is defined by the expression: $\rho _{km} = x_\kappa ^{} \lpar 0\rpar -x_m^{} \lpar 0\rpar .$

To have a correct transition from equations (2) to (4) must be fixation before differentiating for restoring of the original equation (restoring condition) and unique solution's condition [Reference Koshovy19, Reference Koshovy22]. The unique solution condition can be written in the form [Reference Gakhov23]:

$${\int_{{-}1}^1 {} F_\kappa \lpar \tau \rpar {\lpar 1-\tau ^ 2\rpar }^{{-}0.5}dt = 0}.$$

Here the function:

(5)F_\kappa \lpar \tau \rpar = {\,f}^{\prime}_\kappa \lpar \tau \rpar -\sum\nolimits_{m = 1}^M \int_{{-}1}^1 {} j_m \lpar t\rpar R_{\kappa m}^{\prime} \lpar \tau -t\rpar dt.$$

Under this condition, the equation has a unique solution of the type (inversion formula of the simplest Cauchy's type SIE):

(6)$${\,j_\kappa \lpar t\rpar = \displaystyle{{\sqrt { 1-t^ 2} } \over \pi }\int\limits_{- 1}^ 1 {\displaystyle{{F_\kappa \lpar \tau \rpar \cdot d\tau } \over {\lpar t-\tau \rpar \sqrt { 1-\tau ^ 2} }}} }.$$

The SIE system equation (4) together with restoring and unique solution's conditions will be a correct dynamic mathematical model of the Н-polarized EM wave scattering by FPFISG in the general case.

It should be noted that the inversion formula equation (6) is the base to obtain a system of the second kind regular IE (see, for instance, [Reference Koshovy19]). But just obtained general mathematical model will be sufficient for working up asymptotic electrically narrow strip model. This model of the plane Н-polarized EM wave scattering by electrically narrow FIS grating can be referred to as a weakly or sparsely filled grating since the strips are narrow and rather greatly spaced. It allows obtaining the solution in an explicit or, using different terminology, in a closed-form [Reference Gakhov23, Reference Muskhelishvili24].

Scattering by sparsely filled FPFISG

So, under assumptions that the strips are electrically narrow and rather greatly spaced, the asymptotic mathematical model of sparsely filled FPFISG will arise. Making use of the asymptotic expressions for the kernels and right-hand parts of the first-kind SIE system equation (4) we can arrive at an explicit solution in the main approximation:

(7)$${\,j_\kappa \lpar x\rpar = 2ai\alpha \cdot \lpar w-q_2\rpar \cdot {\rm exp\lpar }iq_1x_\kappa ^{} \lpar 0\rpar \rpar \cdot \sqrt { 1-x^ 2} \lpar 1 + {\rm O}\lpar \alpha \rpar \rpar }.$$

This expression allows the transfer to zero strip impedance w = 0 or to the case of the plane H-polarized EM wave scattering by PEC strip gratings [Reference Koshovy16, Reference Koshovy17]. This is expected and important result since thanks to it, we obtain the explicit solution, which is practically useful itself and convenient for the validation of any numerical method based on the full-wave mathematical model of the plane H-polarized EM wave scattering by FPFISG: either the first-kind SIE system equation (4) or corresponding second-kind IE system together with restoring and unique solution's conditions.

The cases of the plane E-polarized EM and acoustic waves scattering by sparsely filled finite flat gratings of IS were examined in proceeding papers [Reference Koshovy19, Reference Koshovy and Nosich25]. In the case of acoustic waves scattering by single flat IS both the main term of the asymptotic and the first-order correction term were derived [Reference Koshovy22]. So, in the same way the first-order correction term to the main term of the asymptotical expression equation (7) can be derived as well. But just now the basic integral characteristic – scattered EM field in the far-field zone will be obtained and examined firstly.

Having solution of the plane H-polarized EM wave scattering problem in the explicit asymptotical form equation (7) and using this expression it is easy to obtain integral characteristics of sparsely filled FPFISG under consideration. The scattered EM field around the IS grating can be represented in the form:

(8)$$\upsilon \lpar x\comma \;z\rpar = {-}\displaystyle{i \over 4}\int_S {} \phi \lpar {x}^{\prime}\rpar \lsqb {ikw\cdot H_0^{\lpar 1\rpar } \lpar kR\rpar } \left. { + \displaystyle{\partial \over {\partial z}}H_0^{\lpar 1\rpar } \lpar kR\rpar } \right]d{x}^{\prime}.$$

Here the function φ(x) is defined on the set of segments S and satisfies equation (1), the radical (in the argument of the first Hankel function of zero order) $R = \sqrt {{\lpar x-{x}^{\prime}\rpar }^2 + z^2}$. The observation point is convenient to specify in the polar coordinate frame, i.e. x = r cos θ and z = r sin θ. Then for a sufficient big distance from the grating (r > >1) in the general approximation we have the asymptotical expression:

(9)$$\upsilon \approx{-}\sqrt {\displaystyle{{i\pi } \over 8}} \displaystyle{{e^{ikr}} \over {\sqrt {kr} }}ik\lpar w + \sin \theta \rpar \int\limits_S {\phi \lpar {x}^{\prime}\rpar } \exp \lpar {-}ik{x}^{\prime}\cos \theta \rpar d{x}^{\prime}. $$

Separating the expression that characterizes the far-field distribution in dependence on the polar angle we find the formula:

(10)$$A\lpar \theta \rpar = {-}\displaystyle{{\sqrt i } \over 2}ik\pi \lpar w + \sin \theta \rpar \cdot \hat{\phi }\lpar k\cos \theta \rpar . $$

Here the cap over the letter φ stands for the symmetric integral Fourier transformation (SIFT). Substitution w = 0 leads to well-known expression for the case of PEC strip gratings. It gives us to be sure that the formula equation (10) is correct. Let us consider SIFT of the function φ(x) in details. Firstly, we have the expressions:

(11)$$\eqalign{& \sqrt {2\pi } \hat{\phi }\lpar k\cos \phi \rpar = \sum\nolimits_{m = 1}^M {\int_{{-}1}^1 {\,j_m\lpar t\rpar \cdot \exp \lpar {-}ix_m\lpar t\rpar \cos \theta \rpar dt} } \cr & \approx \sum\nolimits_{m = 1}^M {\,j_m\cdot \exp \lpar {-}ix_m\lpar 0\rpar \cos \theta \rpar .} } $$

Using the asymptotic formula equation (7) and integrating, it is easy to find the approximations:

$$j_m = \int_{{-}1}^1 {} j_m\lpar t\rpar dt\approx \pi ai\alpha \cdot \lpar w-q_2\rpar \cdot {\rm exp\lpar }iq_1x_\kappa ^{} \lpar 0\rpar \rpar .$$

Thus, the expression equation (10), that characterizes the far-field distribution in dependence on the polar angle, is determined by the approximate formula:

(12)$$A\lpar \theta \rpar \approx \sqrt i \pi \alpha ^2\lpar w + \sin \theta \rpar \displaystyle{{w-q_2} \over 2}\sum\limits_{m = 1}^M {\exp \lpar ix_m\lpar 0\rpar \lpar q_1-\cos \theta \rpar \rpar } $$

One can see that this final formula is accurate but simple enough. Due to its simplicity, it is powerful aim to make a lot of computer simulations. Firstly, the explicit dependence on the strip impedance w can be separated and used in numerical experiments for principal different values of w: real, imaginary and complex in general.

Numerical experiments

To investigate strips impedance influence on the scattered EM field in the far-field zone let us examine the factor:

(13)$$W\lpar w\rpar = \lpar w + \sin \theta \rpar \lpar w-q_2\rpar .$$

It consists of two multipliers, linear with respect to the measureless strip impedance parameter w. So, there are two roots for real values of the parameter w: w ₁ = −sinθ, w ₂ = q ₂ = sinθ ₀, where θ ₀ is a fixed angle of the plane H-polarized EM wave incidence upon the grating. The factor W is a function of two basic arguments: w and θ. It gives the possibility to examine the dependence of the integral characteristic equation (12) on the impedance parameter w and polar angle θ. In Fig. 3 one can see six graphs of the function |W(w)| for the case of real argument and fixed values of polar angle.

Fig. 3. Graphs of the function |W(w)| for real argument.

The solid lines correspond to three values of polar angle θ = π/3, π/6 and 0, while points – to three values of θ = π/2, π/4 and π/9. One can see the second zero w ₂ = sin (π/3) at the impedance parameter value $w = \sqrt 3 /2$, the first zero is impossible for presented strips impedance values except for w = 0 and θ = 0.

If the argument of the function |W(w)| is imaginary, then one can see six graphs in Fig. 4 for the same 6 values of polar angle θ.

Fig. 4. Graphs of the function |W(w)| for imaginary argument.

It is easy to see that for this case impedance influence is greater than the previous one. For these impedance parameter values, both zeroes are impossible, except for θ = 0 and r = Imw = 0. The same is true and for complex values of the impedance parameter. Results of computations for the case of complex impedance parameter values w = r(1 + i) one can see in Fig. 5.

Fig. 5. Graphs of the function |W(w)| for complex argument.

Comparisons of the function |W(w)| graphs show that the strip impedance influence is the greatest for complex values of the impedance parameter w.

It is interesting and useful to consider dependences of the function |W(θ)| on the polar angle for some fixed values of the impedance parameter w: real, imaginary and complex. In Fig. 6(a) one can see three graphs of the function |W(θ)| for the real impedance parameter w = 0.5 (small asymmetrical solid double loop), the pure imaginary value of the impedance parameter w = 0.5i (symmetrical points loop) and complex value of w = 0.5(1 + i).

Fig. 6. Graphs of the function |W(θ)| for fixed values of impedance parameter.

Figure 6(b) presents three graphs of the function |W(θ)| for the same real strips impedance parameter, pure imaginary w = i and complex value of the strip impedance parameter w = 0.5 + i (imaginary part is twice greater than real one). It is interesting to see big asymmetry (with respect to the polar ray) for real values of the strips impedance parameter, less polar ray asymmetry for complex values of the strips impedance parameter. The graphs of the function |W(θ)| for pure imaginary values of the strips impedance parameter are symmetrical. The angle of the plane wave incidence upon the grating θ ₀ = π/3 for all cases.

Thus, one can see that the asymptotical model of the sparsely filled FPFISG is powerful aim for examining the strip impedance influence on scattering integral characteristics. After examining of the factor W(w), let us investigate direction patterns of FPFISG in general. The purpose of the next computer simulation is to show that the final formula equation (12) is effective.

Direction patterns of FPFISG

The sum in the approximate final formula equation (12) can be simplified according to the central symmetry of all stages of 1D SSF2 creation (see Figs1 and 2). For example, in the case of the third stage of SSF2 construction, the sum takes the form:

$$2\sum\nolimits_{m = 1}^4 {\cos \lpar x_m\lpar 0\rpar \lpar q_1-\cos \theta \rpar \rpar }.$$

The geometrical parameters x _m(0) defined by the formulas: $x_{1\comma 2}\lpar 0\rpar = \beta -\beta _2\mp \beta _3$, $x_{3\comma 4}\lpar 0\rpar = \beta + \beta _2\mp \beta _3$; where the measureless parameters: β = kb = 2bπ/λ, β ₂ = β/κ, β ₃ = β ₂/κ. So, the sum dependents on the polar angle (the basic argument) and two parameters: κ = 1 + β/α > 2 and q ₁ = cos θ ₀. The first parameter is the self-similarity coefficient, which defines HD of SSF2 under consideration, but the second one dependent on a fixed angle of the plane wave incidence upon the grating.

Let us perform some computer simulations of direction patterns on the base of the approximate formula equation (12). Figure 7 presents dependences of abs(A) = |A(φ)|/πα² on the polar angle calculated for the FPFISG corresponding to the third stage of the SSF2 creation with HD = ln2/lnκ. These two gratings consist of 8 strips. Initial geometrical measureless parameters of the first one are: α = π/8, β = 31π/8, κ = 1 + β/α = 32 (Fig. 7(a)) and for the second – α = π/2, β = 15π/2, κ = 16 (Fig. 7(b)).

Fig. 7. Direction patterns of FPFISG for HD = 0.2 and 0.25.

The cross-sectional sizes of these two gratings are different. For the first one β + α = 4π, so, the cross-sectional size is equal to 4 λ, for the second – 8 λ, here λ is the plane EM wave length. The solid line corresponds to the value w = 0.5 + 0.5i; while points – to the value w = 0 (grating of PEC strips). The angle of the plane EM wave incidence θ ₀ = π/2. One can see symmetry (with respect to the polar ray) of direction patterns in the case w = 0 (PEC strips) and asymmetry of direction patterns of non-PEC strip grating.

Figure 8 shows comparison direction patterns for the FPFISG with the same number of strips; initial geometrical measureless parameters are α = π/2, β = 15π/2, κ = 16.

Fig. 8. Direction patterns of FPFISG for θ ₀ = π/3 and θ ₀ = π/6.

The cross-sectional size of the grating is equal to 8 λ (β + α = 8π). The angles of plane wave incidence upon the grating are oblique and take the values: θ₀ = π/3 (Fig. 8(a)) and θ ₀ = π/6 (Fig. 8(b)). One can see, that direction patterns of FPFISG essentially dependent on angles of the plane wave incidence upon the grating and values of the strip impedance measureless parameter.

Figure 9 presents dependences of abs(A) = |A(φ)| /πα² on the polar angle calculated for the same FPFISG with different values of the parameter: w = 0.1 + 0.1i (Fig. 9(a)) and w = 0.1 + 0.5i (Fig. 9(b)). The angles of the plane wave incidence upon the grating take two values: θ ₀ = π/4 and θ ₀ = π/9.

Fig. 9. Comparing of direction patterns of FPFISG.

It is interesting to examine direction patterns of FPFISG, when the angle of the plane wave incidence upon the grating is small (sliding angle). To make this examination the simplest FPFISG corresponding to the SSF2 creator will be taken [Reference Koshovy16–Reference Koshovy18]. Figure 10 presents dependences of 100⋅|A(φ)| for two gratings with geometrical basic measureless parameters defined by the formulas: α = π/32 < 0.1, β = 31π/32 ≈ 3, κ = 1 + β/α = 32, HD = 1/5 (Fig. 10(a)) and α = π/32, β = 15π/32 ≈ 1.5, κ = 1 + β/α = 16, HD = 1/4 (Fig. 10(b)).

Fig. 10. Comparing of direction patterns.

The plane wave incidence angle θ ₀ = π/90 (sliding angle). The cross-sectional sizes of these gratings (consisting of 2 strips) are different as well. For the first one β + α = π, so, the cross-sectional size is equal to λ, for the second – λ/2. The solid line corresponds to a sufficiently small value of the impedance parameter w = 0.1 + 0.1i; while points – to the value w = 0 (grating of PEC strips). Comparing line graphs and point graphs one can see, that direction patterns of impedance (w = 0.1 + 0.1i) strip gratings are greater in 3–4 times than direction patterns of PEC (w = 0) strip gratings.