Metasurface Salisbury screen: achieving ultra ... - OSA Publishing

Vol. 25, No. 24 | 27 Nov 2017 | OPTICS EXPRESS 30241

Metasurface Salisbury screen: achieving ultrawideband microwave absorption ZIHENG ZHOU,1 KE CHEN,1,* JUNMING ZHAO,1 PING CHEN,1 TIAN JIANG,1 BO ZHU,1 YIJUN FENG,1,3 AND YUE LI2 1

School of Electronic Science and Engineering, Nanjing University, Nanjing, 210093, China Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China 3 [email protected] * [email protected] 2

Abstract: The metasurfaces have recently been demonstrated to provide full control of the phase responses of electromagnetic (EM) wave scattering over subwavelength scales, enabling a wide range of practical applications. Here, we propose a comprehensive scheme for the efficient and flexible design of metasurface Salisbury screen (MSS) capable of absorbing the impinging EM wave in an ultra-wide frequency band. We show that properly designed reflective metasurface can be used to substitute the metallic ground of conventional Salisbury screen for generating diverse resonances in a desirable way, thus providing large controllability over the absorption bandwidth. Based on this concept, we establish an equivalent circuit model to qualitatively analysis the resonances in MSS and design algorithms to optimize the overall performance of the MSS. Experiments have been carried out to demonstrate that the absorption bandwidth from 6 GHz to 30 GHz with an efficiency higher than 85% can be achieved by the proposal, which is apparently much larger than that of conventional Salisbury screen (7 GHz - 17 GHz). The proposed concept of MSS could offer opportunities for flexibly designing thin electromagnetic absorbers with simultaneously ultra-wide bandwidth, polarization insensitivity, and wide incident angle, exhibiting promising potentials for many applications such as in EM compatibility, stealth technique, etc. © 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (160.3918) Metamaterials; (050.6624) Subwavelength structures; (290.1350) Backscattering; (350.4010) Microwaves; (260.5740) Resonance.

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#307018 Journal © 2017

https://doi.org/10.1364/OE.25.030241 Received 12 Sep 2017; revised 15 Nov 2017; accepted 15 Nov 2017; published 17 Nov 2017


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1. Introduction As the specifically designed devices used for inhibiting the reflection and transmission of electromagnetic (EM) waves, EM absorbers [1–6] have attracted much attention due to their potential uses in many real-world applications, such as EM compatibility, wireless or solar energy harvesting, camouflage, and stealth technique, etc. EM absorbers composed of absorptive materials like carbon and ferrite are the most straightforward approach to achieve efficient and broadband absorption in microwave region, but they are usually bulky with large mass density, which may inherently limit their uses [7, 8]. In this regard, Salisbury screen employing an ultra-thin homogeneous resistive sheet mounted quarter wavelength above a metallic plate has emerged as a promising alternative approach, mainly due to its advantages of low-cost, light-weight and extremely simple structural design [9, 10]. However, the narrow operation bandwidth and discrete spectral absorption peaks still hinder its further development. Therefore, considerable efforts have been made to improve the bandwidth of conventional Salisbury screen (CSS) using the so-called Jaumann screen [11–13] or circuit analog (CA) absorbers [14, 15], of which wideband absorption are realized by multi-layered configuration or complex lossy frequency selective surface (FSS). Although the bandwidth is broadened, the total thickness of these EM absorbers is inevitably increased. Besides, Salisbury screen loaded with high-impedance surface (HIS) can have an additional narrow absorption band that closely depends on the dispersion of the HIS [16]. Metamaterials using metal/dielectric multilayered structures [17, 18] or loaded with chip resistors are efficient to absorb the incidence within thin-thickness [19, 20] but they usually require precise layer alignments and careful soldering, which is complex and time-consuming during the assembly procedure. For the above reasons, it is still required to simultaneously embrace highperformance, continuously ultra-wide band, wide-angle, light-weight, and easy-fabrication in one thin-thickness absorber design, and it is still necessary to further create practical deployable solutions. Recently, metasurfaces, i.e., the planar artificial structures composed of sub-wavelength engineered particles, have been proposed to fully manipulate EM waves in an unprecedented way. By spatially controlling the interfacial phase-discontinuities, the metasurfaces have enabled a variety of EM functionalities such as anomalous reflection/refraction [21, 22], flat lensing [23–25], propagating wave to surface wave coupling [26], polarization conversion [27, 28] and vortex beam generation [29, 30]. The resonant absorption at certain frequencies with relatively narrow bandwidth in CSS can be attributed to the non-dispersive reflection phase of constant π by the metallic ground plane employed in the CSS. Meanwhile, the metasurface is highly valued for its ability to generate arbitrary phase responses or even controllable dispersion properties [31, 32], which may provide more degrees of freedom in broadening and controlling the absorption bandwidth of the Salisbury screen. Here in this paper, we propose the metasurface Salisbury screen (MSS) concept that metasurface with designable reflection phases is employed to substitute the metallic ground in the CSS. The proposed MSS is quite different to the prior attempts that exclusively operate with materials for the screen and spacer, for example, by replacing the top resistive film with lossy FSS, which, more or less, sacrifice at least one of the advantages of thin-thickness [11– 14] or angular stability [15], etc. As a proof of concept, we have designed and fabricated a prototype of the metasurface containing four types of patterned reflective metasurface elements to work as the ground plane, and so as to form the MSS. Experiments have verified that ultra-wideband (6 GHz to 30 GHz, 133% relative bandwidth) absorption with an efficiency higher than 85% can be realized by the proposed MSS, showing a significantly improvement over that of the CSS (7 GHz to 17 GHz, 83% relative bandwidth). In addition, the superior performance can be well preserved as the oblique incident angle up to about 50° for both wave polarizations.


2. Design principle of metasurface Salisbury screen The intrinsically simple geometry of the CSS makes it an attractive option for deployment where EM absorption is required. The geometry of CSS with a total thickness l is illustrated in Fig. 1(a), where a resistive film of 377 Ω/ is suspended above a metallic ground. The Salisbury resonance occurs when the summation of the phase accumulation as the wave passes through and reflects by the whole CSS reaches 2π. In other words, at resonance, the condition of 4π f n l / c − π = 2nπ should be satisfied, where n = 0, 1, 2… corresponds to its nth order resonance, respectively. However, this design strategy only works for discrete frequencies as schematically shown in Fig. 1(b), since the desired phase accumulation cannot be preserved in a wide band due to the intrinsic non-dispersive phase response of the metallic ground. Apparently, the constant reflection phase of π by the metallic ground limits the achievable attempts aiming to extend the bandwidth of CSS. Fortunately, metasurfaces have provided new opportunities for arbitrarily designing the spectral reflection phases. Figure 1(c) schematically illustrates the MSS based on a reflective metasurface comprising several kinds of metasurface elements with distinct phase responses. These metasurface elements are independently designed to have different Salisbury resonances, which, in turn, give rise to the reflection dips corresponding to each MSS element successively appearing along the frequency axis, as depicted in Fig. 1(d). The reflection coefficient of each curve can be carefully designed to achieve almost equal distribution of the resonant frequencies of the MSS elements across the whole frequency band of interest. As a result, while at specific frequencies the MSS can support single Salisbury resonance dictated by the corresponding metasurface element, at other frequencies, the reflection of the MSS can also be well restrained attributed to the multiple Salisbury resonances dictated by the adjacent resonant behaviors, thus realizing a relatively good EM absorber immune from frequency change within a broadband. Figure 1(e) schematically shows the proposed MSS, where MSS super cell containing 3 × 3 same elements are distributed in a finite plate. The 3 × 3 configuration is adopted to better comply with the periodic boundary hypothesis used in metsurface element simulation. As an example, we choose four types of MSS elements in the present scheme, and all of them are formed by placing a resistive sheet of surface impedance Rs = η0 = 377Ω/ above the metasurface element with an air spacing distance of d. The general equivalent transmissionline model of the MSS element is illustrated in Fig. 1(f). The air spacer is represented by a transmission line with the intrinsic impedance η0 and the propagation constant β0 = ω/c, while the resistive sheet is equivalent to a lumped resistor in parallel to the transmission line. The whole transmission line is terminated by a lossless load of ZL, which will vanish to zero in a CSS. However, this term can be arbitrarily designed by the metasurface, given as

(Z

L

− η0 ) / ( Z L + η0 ) = e jψ ( f ) ,

(1)

where ψ ( f ) is the reflection phase of the metasurface element. After some analytical manipulations, the reflection of MSS element can be derived as:

Γ(f ) =

1 − e j (2 β d −ψ ( f )) . 3e j (2 β d −ψ ( f )) + 1 0

0

(2)

Since the transmission wave is totally blocked by the metasurface ground, the absorption of the MSS element can be written as A ( f ) = 1 − Γ( f ) . The above equations highlight the 2

dependence of absorption on the reflection phase of the metasurface element, and give an analytical solution to the excitation of MSS resonance: 4π f n d / c −ψ ( f n ) = 2nπ , n = 0, 1, 2 …

(3)


Since the reflection phase ψ ( f n ) of the metasurface ground can be arbitrarily designed, the air spacing distance no longer obeys the λ/4 resonance condition, but closely depends on the ψ ( f n ) . For a given parameter d, the MSS resonance occurs at the frequencies where Eq. (3) is satisfied. In other words, the absorption peaks in the frequency spectrum can be flexibly altered by pre-designing the metasurface reflection phases. Furthermore, by manipulating the dispersive ψ ( f ) , one can simultaneously excite several high-order MSS resonances in a given bandwidth, thus decreasing the intervals between adjacent resonances.

Fig. 1. (a) Schematic of the CSS with total thickness l, and (b) its reflection curve. (c) Schematic of the MSS with metasurface ground comprising four different kinds of metasurface elements. Each color represents a kind of metasurface element. (d) The colored reflection curves correspond to the four colored elements in Fig. 1(c), respectively. Each curve represents the role that the element plays in the contribution to the overall reflection curve of the MSS, which is conceptually depicted by the dash line as a collective result of the reflection curves from these elements. The marks on the curves are the MSS resonant modes. For convenience, we denote each resonance frequency as fi,j where the subscript “i” represents the element type and “j” corresponds to the jth MSS resonance. The thickness of the metasurface is denoted by h and the separation between the resistive sheet and the metasurface is denoted by d. The total thickness of the CSS or MSS is l. (e) Schematic layout of the MSS based on the metasurface comprising optimized distribution of four metasurface elements with arbitrarily designable reflection phases. (f) The equivalent circuit model of the MSS-element with controllable ZL dictated by the metasurface pattern.

3. Simulation and optimization

The key point for achieving the ultra-wideband MSS is to choose suitable metasurface elements. Based on the aforementioned design principle, four types of metallic structures printed on a grounded dielectric substrate are designed to realize the MSS elements, as illustrated in Fig. 2. The air separation distance between the metasurface and the resistive sheet (not shown in figures) is set as d = 4 mm. Copper film with a thickness of 0.018 mm is used to form the top patterns and the ground plane, while the Taconic RF-35 is used as the dielectric layer with a thickness of h = 2.2 mm and a relative permittivity of 3.5 and loss tangent of 0.002. The total thickness of the MSS element is l = 6.2 mm. The metallic patterns have different shapes: square block with a side length of 2.9 mm [Fig. 2(a)], square block (with a side length of 7.2 mm) subtracted by its inscribed circle [Fig. 2(b)], circular patch with a radius of 3.1 mm [Fig. 2(c)], and a pair of orthogonally arranged double-head arrows with arm length of 6 mm, wire width of 1 mm. Here, we denote the four metasurface elements with different pattern shapes shown in Figs. 2(a) - 2(d) as the element “1”, “2”, “3” and “4”, respectively.


The simulated reflection phase responses of the metasurface elements and the reflection coefficients of the corresponding MSS elements are shown in the right panel of Fig. 2. According to Eq. (3), the MSS resonant frequencies are the intersections of the lines ϕ n ( f ) = 4 f π d / c – 2nπ , n = 0,1, 2 … (dashed line in Fig. 2) and the reflection phase curve of the metasurface elements. High-order MSS resonances are simultaneously excited within the investigated bandwidth (6 GHz - 30 GHz), and there are totally about dozen resonances in the MSS. For convenience, we denote each resonance frequency as f i , j where the subscript “i” represents the element type and “j” corresponds to the jth MSS resonance. The design aims to decrease the intervals between the adjacent resonances and increase the resonance types in the operation band. Therefore, when these elements are utilized to form the MSS, multiple MSS resonances should be simultaneously trigged under the illumination of incidence within the operation frequency band. As a result, most of the incident energy can be efficiently absorbed by the resistive film.

Fig. 2. The configuration of (a) element “1”, (b) element “2”, (c) element “3” and (d) element “4”. Reflection phases and the reflection coefficients of the corresponding MSS element are shown on the right panel. The dashed lines represent ϕ n ( f ) = 4 f π d / c – 2nπ , n = 0,1. For clear view of the metasurface elements, the resistive sheets are not shown in the figures. The MSS resonance occurs when the dashed line intersects with the phase curve of metasurface element, and at these frequencies the incident energy are dissipated on the resistive sheet, leading to nulls of the reflection curves of the MSS elements.

Now we consider a 9 × 9 array containing the four types of MSS super cell. Since the energy absorption is contributed by the multiple MSS resonances, the proportion of each element type and their spatial distributions have a crucial influence on the overall absorption performances, as well as the backward scattering patterns of the MSS. Here, a two-step optimization method is used to obtain the best performance of the synthesized MSS within a pre-determined frequency band. In the first step, we optimized the proportion of each element type to obtain a continuously operation band with maximum absorption efficiency, while in the second step, the spatial distribution of the four kind elements is optimized to avoid strong side-lobes of the scattering pattern. According to the reflective array theory [33], the total scattered electric field of the MSS under normal incidence can be derived from the far-field superposition of every individual MSS element: 

Es ( f ) = k

E0 a

2

4π r

g (θ , ϕ )

M

N

Γ

m,n

( f )e

jk ( sin ( θ ) cos( ϕ ) ma + sin ( θ ) sin ( ϕ ) na )

(4)

m =1 n =1

where k is the wave number, a is the periodicity of the MSS element, r is the observational radius in far-field region, E is the amplitude of the incident electric field, Γ ( f ) represents 0

m,n

the frequency-dependent reflection coefficient of the MSS element positioned at (ma, na), and g(θ, φ) is the element factor as a function of elevation angle θ and azimuth angle ϕ , given by


g (θ , ϕ ) = (1 + cos (θ ) )

(

) (

sin k a2 sin (θ ) cos ( ϕ ) sin k a2 sin (θ ) sin ( ϕ ) k a2 sin (θ ) cos (ϕ ) k a2 sin (θ ) sin ( ϕ )

)

.

(5)

The above Eq. (4) suggests that the total electric field should be as small as possible for an EM wave absorber. For an ideal EM absorber, the incident energy can be totally dissipated in the absorber and the right part of the Eq. (4) will be reduced to zero. As the first step, we use the differential evolution algorithm [34] to optimize the backward scattering along the surface normal (θ = 0°) within the optimization band from 6 GHz to 30 GHz. At this case, the optimization procedure can be reduced to solely optimize the proportion of each MSS 2 element type to obtain the minimum total reflection of 4 x ⋅ Γ ( f ) , where Γi ( f )



i

i

i = 1

th

represents the reflection coefficient of the i MSS element, and xi corresponds to its proportion. The optimized result of −8.2 dB low reflection can be achieved when { x1 , x2 , x3 , x4 } equals {29.64%, 24.69%, 34.56%, 11.11%}. Although the backward reflectivity of the MSS can be reduced to an ultra-low level within a broad frequency range by previous optimization procedure, it is still necessary to avoid strong side lobes of the scattering pattern, which is important in many applications, for example, in stealth technique. In this regard, the optimization of the spatial distribution of the elements should be seriously considered. For a certain spatial MSS element arrangement, the scattering E-field pattern can be analytically calculated according to Eqs. (4) and (5). Here, we define the objective function as:  K  | Es ( f k , θ , ϕ ) |  F(layout) =  Max   (6) , k =1 θ ,ϕ  | Esm ( f k , 0, 0) |  where θ (0° ≤ θ < 90°) is the elevation angle and φ (0° ≤φ< 360°) is azimuth angle for the whole upper half-space of the MSS, f k is the sample frequencies ranging from 5 GHz to 30  GHz with an interval of 1 GHz, and Esm ( f k , 0, 0) represents the backward scattered E-field of the metallic plate. For an EM absorber, the numerical value of objective function should be as small as possible. As the second step, simulated annealing algorithm [35] is employed to achieve the optimized layout with extremely low side-lobes. As shown in Fig. 3, strong scatterings in other directions (θ ≠ 0°) can be significantly reduced after the optimization procedure, and the side-lobes are uniformly much smaller than the main lobe along the surface normal (θ = 0°). Before the optimization, the maximum side-lobe levels of the Eplane patterns at the four frequencies are −12.7 dB, −10.7 dB, −12.04 dB, and −12.9 dB, respectively. In comparison, these side-lobes are uniformly reduced below −27 dB after the optimization. The simulated 3D scattering patterns are also shown in the insets of Fig. 3. The final optimized layout is schematically illustrated in Fig. 1(e). Since this is a nonlinear problem and considering the calculation complexity, the optimal solutions may be not unique. Therefore, MSS with other optimized spatial layouts might have similar absorption performances. Though the EM functionalities of the diffusion metasurfaces and the absorbers are similar, which is aimed to diminish the backward scattering of targets, the underlying working mechanisms of them are quite different. In a diffusion metasurface, due to the strong destructive interference from the elements of the metasurface, the incident wave can be efficiently redistributed into numerous directions in the whole backward half-space, thus leading to significant reduction of the backward scattering [36–38]. But in an EM absorber, the incident energy is dissipated by the structural resonance or the lossy materials. Here, the metasurface ground plays a crucial role to provide phase accumulation to generate multiple and closely-neighboring resonant modes, ensuring the incoming electromagnetic energy can be transformed into the ohmic loss on the resistive sheet efficiently.


Fig. 3. Normalized electric-field scattering patterns in the E-plane with and without optimization procedure at (a) 8 GHz, (b) 14 GHz, (c) 22 GHz and (d) 30 GHz, respectively. The insets are the optimized 3D backward scattering patterns with calibration to the backward scattering of a same-sized metallic slab.

4. Experimental verification

To validate the design principle, a prototype of the metasurface is fabricated by the standard printed circuit board (PCB) technique to form the MSS, while the resistive film is printed by resistive ink made of graphite powders. A microwave foam slab with a thickness of 4 mm and a relative permittivity of about 1.05 and loss tangent of 0.005 is utilized to mimic the air spacer. The fabricated metasurface and resistive sheet are shown on Figs. 4 (a) and 4(b), respectively. The measurement is carried out in a standard microwave anechoic chamber and the reflection is calibrated to a same-sized copper slab. A pair of horn antenna serving as the emitter and the detector is linked to the two ports of a vector network analyzer (Agilent E8363A). As shown in Figs. 4(c) and 4(d), the measured reflection below 0.15 can be preserved from 6 GHz to 30 GHz, which roughly agree with those of full wave simulation and analytical results both in x- and y-polarized incidence cases. In the theoretical model, each MSS element is treated as an EM radiator, and the final spatial distributions of the MSS elements are the same to that of the fabrication sample. The scattered electric field of the MSS is calculated using Eqs. (4) and (5). Since the reflection of the MSS is mainly associated to the individual MSS resonance mode, the reflection curves show several dips within the operation band, namely at the frequency of 9.5 GHz, 13 GHz, 20 GHz and 25.5 GHz, respectively. To better illustrate the underlying mechanism, the dominating MSS resonant modes f i , j responsible for each ultra-low reflection dip are marked in the Figs. 4 (a) and 4(b). These closely distributed neighboring resonances have enabled a continuously ultrawideband low reflection. As a result, Fig. 4(e) shows that the relative bandwidth (with respect to the center frequency) defined by 85% energy absorption can reach about 133% by the MSS, which shows a great improvement over that from a CSS (83% relative bandwidth) with same total thickness. Since the transmission equals zero due to the MSS being grounded by the reflective metasurface, the absorption is calculated by A( f ) = 1 − R ( f ) , where A(f) is the frequency-dependent absorption, and R(f) is the sum of scattering energy of the whole


backward half-space. In addition, the measured far-field scattering pattern in E-plane [Fig. 4(f)] verifies that the side-lobes are much smaller than the main-lobe, as a result of the optimized distribution of the MSS elements. The robust angular dependent performance is an important criterion to evaluate the EM absorber in many practical applications. The specular reflections of the MSS under different oblique incident angles (from 5° to 55°, with an interval of 10°) for both transverse electric (TE) and transverse magnetic (TM) polarization are shown in Fig. 5. The absorption and bandwidth performances can be well preserved up to about 50° of the incident angle for all wave polarizations, which shows an advantage of angular stability over CSS [15]. In order to give a quick evaluation of absorption performance, we have calculated the physical ultimate thickness dmin for an ideal absorbing structure characterized by the same absorption properties [6, 42-43]: d min ≈



λmax

λmin

ln R (λ ) dλ 2π 2

,

(7)

where R(λ) is the wavelength response of the reflection coefficient, λmin and λmax are the minimum and maximum wavelengths corresponding to the operation waveband. Here, λmin = 10 mm , λmax = 50 mm . After some calculations, the ultimate thickness for such a design is about 4.4 mm, approximate 73% of the thickness of the fabricated MSS. For comparison, we have investigated the ratio of ultimate thickness dmin to the real thickness d for some representative previous works, as shown in Table 1. Our performance is comparable to most experimental results in Table 1. Since the prototype MSS is just a proof-of-concept design, we believe its performance could be further improved and the profile could be reduced with more careful designed metasurface elements.

Fig. 4. Photographs of the fabricated (a) metasurface and (b) resistive sheet. Measured, simulated and theoretically optimized backward reflection for the normal incidence with (c) xpolarization, (d) y-polarization. (e) Measured frequency-dependent absorption of the MSS and CSS under the illumination of normal incidence. (f) The measured E-plane scattering pattern of the MSS under the illumination of normal incidence, with calibration to the reflection of metallic slab at normal incidence.


Fig. 5. Measured specular reflection for TE-polarized oblique incidence with electric field along (a) x- (b) y-direction, and TM-polarized oblique incidence with magnetic field along (c) x- (d) y-direction. Table 1. Comparison with other electromagnetic absorbers

Absorber dmin/d

[39] 64% (exp.)

[40] 65% (exp.)

[41] 70% (exp.)

[42] 81% (exp.)

[43] 95% (sim.)

Our work 73% (exp.)

5. Conclusion

In summary, we have proposed the concept of metasurface Salisbury screen for ultrawideband EM wave absorption, where the metasurface with flexibly controllable reflection phase features is employed to substitute the metallic ground of the Salisbury screen and thus to generate multiple Salisbury resonances in a continuously ultra-wide frequency band. In particular, four different MSS elements are carefully designed to form the MSS absorber via certain optimization algorithm. Experimental results demonstrate that the proposed MSS can realize an 85% absorption band from 6 GHz to 30 GHz (133% relative bandwidth) under the normal incidence showing a significant advantage over the CSS of same thickness. Furthermore, good angular performance is verified in the experiment. The proposed MSS can be applied wherever continuously ultra-wideband absorption, angular stability, polarizationinsensitive, and light-weight is simultaneously required. Since the design principle can be readily scaled to other frequency bands, the concept of MSS is quite promising for a wide range of applications such as stealth technique, EM compatibility, etc.


Appendix: Derivation of Eq. (4) and (5)

Fig. 6. An array composed of N × N MSS elements with periodicity a is shined bythe plane wave with an incident angle α. (a) The schematic of an array constituted by MSS elements  exposed to the incoming plane wave with wave vector along ei . The far-field scattering at the 

direction ei is analyzed. The near-field reflection of the MSS element under the incidence with TE polarization and TM polarization are shown in (b) and (c), respectively.

Here, we schematically demonstrate the MSS comprising N×N elements exposed to the incidence of plane waves in Fig. 6(a). The derivation of the enclosed formed of scattered fields is started by the assumption that, the edge effects and the mutual coupling between the elements are assumed to be ignored, and the near-field scattered wave of the MSS element obeys the reflection law for oblique incidence with TE polarization [Fig. 6(b)] and TM polarization [Fig. 6(c)]. For TE polarized incidence, the incident electric field and magnetic field can be written as follows:  Ei = E0 e jk (sin(α ) y + cos(α ) z ) xˆ, (8)  E0 jk (sin(α ) y + cos(α ) z ) Hi = η e ( sin(α ) zˆ − cos(α ) yˆ ) , 0

and the near-field reflected fields of the MSS element positioned at (pa, qa) are:  Er = Γ p,q E0 e jk (sin (α ) y + cos(α ) z ) xˆ, 

E jk sin(α ) y + cos(α ) z ) H r = Γ p,q η0 e ( ( sin(α ) zˆ + cos(α ) yˆ ) , 0

(9)

(10) (11)

where Γ p , q represents the specular reflection coefficient of the MSS element when it is illuminated by a TE-polarized plane wave at an oblique angle of α. Thus, the induced surface   electrical current J e and magnetic current J m can be calculated by:   E J e = zˆ × H r |s = −Γ p,q η 0 e jk sin(α ) y cos (α ) xˆ, 0

(12)






J m = − zˆ × Er |s = −Γ p, q E0 e jk sin(α ) y yˆ ,

(13)

and the corresponding vector potentials can be calculated as:     Ae = 4π1 r  e− jk ( r −es ⋅r ′) J e ds' = −

 p,q

    Am = 4π1 r  e− jk ( r −es ⋅r ′) J m ds' = −

Γ p ,q E0 cos (α ) e− jkr Ι(θ ,ϕ ,α , 4π rη0

 p,q

Γ p ,q E0 e− jkr 4π r

p,q)xˆ,

(14)

Ι(θ ,ϕ ,α , p, q)yˆ ,

(15)

where ( q +1) a

Ι(θ ,ϕ ,α , p,q)= qa



( p +1) a pa

e jk (sin(θ )cos(ϕ ) x′+ sin(θ )sin(ϕ ) y′+ sin(α ) y′) dx′dy′.

(16)

Finally, the far-filed scattered electric field can be derived as:

(

) )

       Es = jkes × Am − jωμ0 Ae − es ⋅ Ae es

(

= jk ( cos (ϕ ) ( cos (θ ) cos (α ) + 1)θˆ − sin (ϕ ) ( cos (θ ) + cos (α ) ) ϕˆ ) ⋅  p , q

Γ p ,q E0 e − jkr 4π r

(17) Ι(θ ,ϕ ,α , p, q).

Through similar derivation, the scattered electric field for TM incident case can be obtained as well, given by: 

Es

= − jk ( sin(ϕ )(cos(θ ) + cos(α ))θˆ + cos(ϕ )(cos(θ ) cos(α ) + 1)ϕˆ ) ⋅



Γ p ,q E0 e − jkr p,q

4π r

. (18)

Ι (θ ,ϕ , α , p, q )

At normal incident case ( α = 0 ), from the above equations I( θ ,ϕ , α , p,q ) reduces to

Ι(θ ,ϕ , 0, p,q) = e jk (sin(θ )cos(ϕ )( p +1/2) a +sin(θ )sin(ϕ )( q +1/2) a ) f (θ ,ϕ ),

(19)

where f(θ, φ) is the element factor which can be written as:

f (θ ,ϕ ) =

sin ( k a2 sin(θ ) cos(ϕ ) ) sin ( k a2 sin(θ )sin(ϕ ) ) . k a2 sin(θ ) cos(ϕ ) k a2 sin(θ )sin(ϕ )

(20)

Hence, the far-field scattered field under the normal incidence can be calculated as:  E a 2 (1+ cos(θ )) Es = k 0 4π r f (θ , ϕ )  p , q Γ p , q e jk (sin(θ ) cos(ϕ )( p +1/ 2) a + sin(θ ) sin(ϕ )( q +1/ 2) a )

=k



a a E0 a2 (1+ cos(θ )) sin ( k 2 sin(θ ) cos(ϕ ) ) sin ( k 2 sin(θ ) sin(ϕ ) )  4π r a a

k 2 sin(θ ) cos(ϕ )

p,q

k 2 sin(θ ) sin(ϕ )

. (21)

Γ p , q e jk (sin(θ ) cos(ϕ ) pa + sin(θ ) sin(ϕ ) qa )

Finally, we can obtain the Eq. (4) and Eq. (5). Funding

National Natural Science Foundation of China (NSFC) (61571218, 61571216, 61301017, 61371034). Acknowledgments

This work is partially supported by PAPD of Jiangsu Higher Education Institutions, and Jiangsu Key Laboratory of Advanced Techniques for Manipulating Electromagnetic Waves.