Wave Scattering by a Planar Junction Between Anomalously Reflecting Metasurface and Impedance Sheets

Metasurface technology has become one of the promising structures for controlling and manipulating of the phase and amplitude characteristics of the electromagnetic waves in recent years. For this reason, the determination of the exact diffracted fields is also crucial especially in the edge or discontinuity points of these types of materials. In the paper, the discontinuity problem by the junction between the layers of an anomalously reflecting metasurface and an impedance half-screen is taken into consideration. These two planar layers have a common edge discontinuity where the diffracted waves occur. The scattered waves from the structure are obtained by the novel technique of the method of transition boundary, which especially offers the relation between the diffracted and scattered geometrical optical waves at the transition boundaries. The diffracted wave expressions are derived by the technique of the method of transition boundary and the uniform wave expressions are obtained by the method of uniform geometrical theory of diffraction. The total wave result of the method of transition boundary is also compared and verified by the well-known theory of physical optics. The behaviors of the total wave and its components are analyzed numerically.


I. INTRODUCTION
The main difference in the anomalous reflection is the waves do not obey Fermat's principle. It is observed in the metamaterials and metasurfaces, which are artificial constructions and are generally constructed with a desired permittivity and permeability [1]. As a result of the increase in the number of studies in the field of the metamaterials and metasurfaces technology, these artificial constructions are also used to excite the anomalously reflected waves on the scattered waves [2]- [5]. The anomalous effects of these structures are used in a wide range for different purposes with different techniques such as forming metasurfaces by using several resonators, controlling of terahertz (THz) wave by using coding metasurfaces, controlling of both anomalous reflection and phase difference over supercells, creating an anomalous reflection by using cylindrical nanorods and forming gradient metasurfaces for anomalous reflection in optical frequencies [6]- [10]. These are some of the purposes for the use of metamaterials and metasurfaces.
The scattering problem is a phenomenon in the field of electromagnetics. Especially, the determination of the exact diffracted waves by the edge or discontinuity points of the scatterers [11]- [16] is very crucial for the control of the scattering waves or constructing the type of meta structures having electromagnetic invisibility with having reflectionless property or very low-scattering [17]- [19]. The scattering and diffraction problems on metasurfaces have also been studied by several authors in recent years [20]- [24]. Umul investigated the effect of the discontinuities on the anomalous reflection characteristics of metasurfaces in several studies and investigated the metasurface structures with the different discontinuity problems [24], [27]- [33]. After his several scattering problem studies, he has realized that the diffracted waves have a direct relation with the scattered geometrical optical (GO) waves at the transition boundaries. For this reason, Umul has transformed these studies into a novel technique, which is named as the method of transition boundary (MTB) [34]. The technique is applied In this paper, we will be investigating the scattering properties of waves by a discontinuous junction point between an anomalously reflecting metasurface screen and an impedance half-screen by the novel technique of the MTB [34]. To the best of our knowledge, such a kind of scenario has not been investigated yet. In the literature, the existing studies do not include such a junction discontinuity problem with the inclusion of an anomalously reflecting metasurface. First of all, the initial fields will be determined. Then, the total and scattered GO waves will be obtained for the problem under consideration. After that, the diffracted wave expression will be derived by the superposition of the diffracted fields. The uniform diffracted wave will be expressed by using the uniform geometrical theory of diffraction (UTD) [44]. The behaviors of the total, total GO, and diffraction field expressions will be analyzed numerically.
A time factor of ( ) exp jt  is assumed and suppressed throughout the paper.  and t are the angular frequency and time, respectively.

II. DEFINITION OF THE PROBLEM
An anomalously reflecting metasurface half-plane and impedance half-plane which have a common planar edge discontinuity are located at

III. MTB SOLUTION OF THE PROBLEM
In order to determine the exact diffracted waves in our problem, we will use the novel technique of MTB that was introduced by Umul offering an easiness to determine exact diffracted waves [34]. The method is especially a competent technique for the determination of the diffracted waves by the discontinuities at the transition boundaries.
The diffracted wave has the general expression of at high-frequencies [45]. f is a function that will be determined. E  and 0  are the angles of observation and incidence, which are measured from the surface of the scatterer. E R is the distance between the edge and observation points. The general geometry that defines (2) is given in Fig. 2.
The scattered GO field has the expression of for the geometry in   [46], [47].
We will apply the technique of MTB [34] consisting of five main steps to find the total wave and its components for our metasurface junction problem by taking into consideration these steps.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

A. INITIAL WAVE
MTB offers the term of the initial wave ( ) in u instead of the incident wave of i u . The usage of this term has a great advantage to define the problem by excluding a part of the structure that causes discontinuity and diffraction. In the half-plane or wedge type construction problems, their presence directly causes discontinuity. For this reason, the initial waves are directly equal to the incident wave in these types of problems. However, in our problem, the discontinuity arises from the junction point of two different planes, and it is enough to exclude one of the half-screens for canceling the discontinuity. Therefore, the initial wave is different from the incident wave in our problem.
In order to determine the initial wave in our problem, we exclude the first half-plane of the anomalously reflecting metasurface with the impedance of 1 Z from the space. In this case, there will only be a whole impedance half-screen having the impedance of 2 Z at the plane of 0 y = . The reflection coefficients of the first and second halfplane of 1 R and 2 R can be given as respectively. sin i  is equal to  is a ratio that can be expressed as greater than one in some cases and the numerical results when the resistivity of the half-plane is especially greater than the impedance of the free space.
r  is the angle of reflection that can be chosen as a different value from the incident angle of 0  for providing the anomalous reflection condition. As a result, the initial wave can be defined by by excluding the first half-plane of the anomalously reflecting metasurface in this case by applying the first step of the MTB.

B. TOTAL GO WAVE
In order to determine the diffracted wave, the scattered GO wave must be first obtained. Before we obtain the scattered GO wave, the total GO wave must be obtained by considering the whole structure with the inclusion of both half-planes and applying the second step of the MTB. Hence, the general expression of the total GO wave can be written as according to the geometry in respectively. The total GO wave reads where r u can be written as where s x is the coordinate of the reflection point which can be defined by According to (14), a phase shift accompanies the amplitude variation of the reflected wave [29].

C. SCATTERED GO WAVE
The scattered GO wave is found by according to the third step of MTB. Hence, the scattered GO field can be expressed as according to (15).

D. DIFFRACTED WAVE
There are two transition boundaries in our problem. One of them is 1 f , which arises from the anomalously reflecting metasurface and the other one is This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
where the term of split function K + can be expressed by the relation of ( ) ( ) for the nonuniform diffracted wave.

E. UNIFORM DIFFRACTED WAVE
In order to obtain the uniform field expressions, the uniform form of the asymptotic relation of can be used [29]. The uniform form of the relation in (32) can be given as for the uniform diffracted wave.

IV. PHYSICAL OPTICS SOLUTION
In this section, we will offer a physical optics (PO) solution for the problem. We will compare and verify the results of MTB solution of the problem with the well-known theory of PO solution. The general scattered field definition of the PO expression can be given by for an impedance half-plane problem [26], [49]. The PO definition of total scattered field of the problem can also be represented by 12 , As a result, the total field expression of the PO solution that can be obtained by the sum of the total scattered PO field in (40) and the incident field in (1) can be represented by 11 .

V. NUMERICAL RESULTS
In this section, we will analyze the behaviors of the total field and its components numerically. The distance of observation is taken as 3 where  is the wavelength. The angle of observation ( )  changes from 0° to 180° since the junction combination of two planar planes is located at The different values will be taken into consideration for  [48]. Until 150°, the diffracted, reflected GO and incident GO waves interfere with each other. After this value, which is the reflection boundary, only the incident GO and diffracted waves exist.
It can be seen from Fig. 4 that there are interferences between two surfaces of anomalously reflecting This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. metasurface and impedance half-plane. The reflection boundaries are also read in the degrees of 120° and 150° which are the degrees of reflection boundaries of impedance half-plane and anomalously reflecting metasurface, respectively. The behavior of the total GO and diffracted waves can be observed clearly. Fig. 5 depicts the variation of the total wave concerning the observation angle for different values of 1 sin and 2 sin . The angle of reflection is 30°. The main difference is observed in the intensity of the reflected GO wave and accompanies with  is smaller than the reflection boundary of 150°. After the reflection boundary, only the diffracted waves are the cause of the difference between the intensities.  Fig. 6 shows the variation of the total wave for different values of r  . A phase difference is seen before 150°. The difference between the reflection angles also affects the intensity of the total wave. The two total waves are in harmony after 150°.  In Fig. 7(a) and Fig. 7(b), the comparison of two solutions, obtained by the techniques of MTB in Section III and PO in Section IV, is given for different values of

VI. CONCLUSIONS
In this paper, we investigated the scattering problem of waves by a planar junction between anomalously reflecting metasurface and impedance half-planes. The effect of the phase shift and the amplitude variation of the anomalously reflecting metasurface is taken into consideration during the evaluation of the expressions. The expressions of the total, total GO and diffracted waves are obtained by using the novel technique of MTB and the behaviors of the waves are also studied numerically. The result of MTB is also compared and verified by the well-known theory of PO. The comparison shows that the total field result of MTB is also in agreement with the total field result of PO. In these types of problems, the surface waves can also be obtained [50], [51]. After the scattered wave result is obtained by the technique of MTB, the scattered wave expression of the MTB is first transformed into the complex integral as in the several diffraction problems applied in [34], then the surface waves can be obtained by the saddle point evaluation of the complex integral. This study can also be considered in a future work paper.