II. THEORY OF PGSs

When there are constant phase changes (phase jumps) along with an interface between free space and a surface then an anomalous reflection would occur and can be described by the generalized law of reflection [Reference Yu1]. Based on this principle a surface with certain phase jumps (phase change or phase gradient) across its aperture and along the air–surface interface can manipulate the reflected energy (amplitude and phase). In 1D phase gradient reflective surface the phase change is either along x- or y-axis and can be denoted as Δϕ/Δx or Δϕ/Δy. In case of a surface that consists of a number of periodic unit cells, the relation between the phase change along the interface (Δϕ/Δx) and the reflected wave amplitude/phase can be derived starting from the generalized Snell's law of reflection [Reference Yu1] as in equation (1).

(1)$$\sin (\theta _r) - \sin (\theta _i) = \displaystyle{1 \over {n_ik_o}}\displaystyle{{\Delta \varphi} \over {\Delta x}} = \displaystyle{{\lambda _o} \over {2\pi}} \displaystyle{{\Delta \varphi} \over {\Delta x}},$$

where λ _o is the free space wavelength of the incident EM waves, θ _r and θ _i are the angles of reflection and incidence, respectively, k _o is the free space wave vector, n _i is free space refractive index. Usually, the PGS is realized using a number of periodic unit cells of same inter-element spacing (P) and a pre-defined number of unit cells (N) to cover the full 360° phase range. Taking θ _i = 0° and n _i = 1 then the direction of the reflected beam can be calculated using formula (2).

(2)$$\eqalign{\theta _r & = \arcsin \left( {\displaystyle{{\lambda _o} \over {2\pi}} \displaystyle{{\Delta \varphi} \over {\Delta x}}} \right) = \arcsin \left( {\displaystyle{{\lambda _o} \over {2\pi}} \displaystyle{{2\pi} \over {NP}}} \right). \cr & \Rightarrow \theta _r = \arcsin \left( {\displaystyle{{\lambda _o} \over {NP}}} \right)}$$

In this way, the incident EM wave at a certain frequency can be reflected/steered to a predefined angle based on the design parameters (i.e. N and P) of the PGS.

In this paper the unit cell used to realize the 1D and 2D phase gradient dielectric perforated reflective surfaces is all dielectric unit cell as shown in Fig. 1(a). The unit cell has four air-filled drilled holes and the diameter of those holes determines the reflection phase value as the effective permittivity of the dielectric substrate is altered after drilling the holes. With such super cell configuration, the phase change (phase jumps) along the interface is about 72° (2π/5 = 72°). All unit cells are simulated under the same boundary conditions using 3D high-frequency full-wave EM simulator, where the unit cell is surrounded by Master-Slave boundary conditions and terminated to a floquet port and backed by a solid perfect electrical conductor (PEC) ground. Then the floquet port is embedded to the top surface of the unit cell. The reflection phase characteristics of the five unit cells are then computed and presented in Fig. 1(b) and at 70 GHz there is a constant phase difference of about 70° between the reflected waves from those unit cells based on the diameter of the drilled air holes. Moreover, full wave simulations are performed to compute the backscattered E-field distribution of the individual unit cells (located at z = 0) composing the super unit cell under normal incidence and the results are plotted in Fig. 1(c). As can be seen, each unit cell provides a specific discrete phase shift as the radius of the unit cells increased gradually from 0.21 to 0.3 mm.

Fig. 1.

The layout and the reflection characteristics of the unit cell used to realize the 1D and 2D phase gradient dielectric surfaces. (a) Super unit cell consists of five sub-unit cells: A = 1.5 mm, R ₁ = 0.21 mm, R ₂ = 0.23 mm, R ₃ = 0.25 mm, R ₄ = 0.27 mm, R ₅ = 0.3 mm, h = 1.27 mm, ε _r = 10.2. (b) Reflection phase versus frequency curve of the unit cells. (c) E-field distribution of the super unit cell where dФ is the phase change.

III. 1D AND 2D PGSs DESIGN

Using the super cell in Fig. 1(a), building blocks of 1 × 5 and 2 × 10 super cells and occupied an area of 7.5 × 7.5 and 15 × 15 mm² are designed as shown in Figs 2(a) and 2(b). The design parameters of these dielectric PGSs are P = 1.5 mm and N = 5. The direction of the reflected beam under normal incidence can be calculated using equation (2) and it is found that θ _r ≈ 34.7° at 70 GHz. A series of full-wave simulations are performed to compute the backscattered characteristics of the PGS in Figs 2(a) and 2(b) and the results are presented in Figs 2(c) and 2(d) with the backscattered pattern of their equivalent bare PEC surface in Figs 2(e) and 2(f) for comparison purposes. These results show that under normal incidence of EM-wave (θ _i = 0°), anomalous reflection is generated and the direction of the reflected beam is at about θ _r ≈ 32.2° (θ _r  ≠ θ _i) which is in good agreement with the predicted angles calculated by the generalized law of reflection (equation (2)) which is not the case for their equivalent bare PEC plates for which both the incident and reflected beams have equal angles (θ _r = θ _i) according to Snell's law of reflection. Another phase gradient reflective surface of different unit cell distribution is composed as shown in Fig. 3(a) and its computed backscattered RCS versus frequency under normal incidence is presented in Fig. 3(b). As can be seen, a clear reduction of more than 6 dBsm in the backscattered RCS is considered from about 69.1–78.5 GHz. The out-band and in-band 3D backscattered RCS field pattern (computed using CST Microwave Studio) of this surface is presented in Figs 4(a) and 4(b), respectively, which shows that anomalous reflection (at 70 GHz) is generated as two reflected beams in θ _r direction as confirmed by the 2D field distribution in Figs 4(c) and 4(d) which shows that the dielectric surface act as PEC surface at out band frequencies.

Fig. 2.

The layout of the 1D PGS: (a) 7.5 mm × 7.5 mm and (b) 15 mm × 15 mm. Polar RCS patterns under normal incidence of a far-field plane wave: (c) 1D PGS in (a), (d) 1D PGS in (b), (e) Bare PEC plate: 7.5 mm × 7.5 mm. (f) Bare PEC plate: 15 mm × 15 mm.

Fig. 3.

The layout of the 1D phase gradient dielectric surfaces with its RCS reduction characteristics. (a) Surface #1: 15 mm × 15 mm. (b) RCS reduction curves.

Fig. 4.

The backscattered field distribution in front of the surfaces. The surface is placed in xy-plane symmetric with respect to the origin point (0, 0, 0). (a) Surface #1 at 85 GHz (out-band). (b) Surface #1 at 70 GHz (in-band). (c) 15 mm × 15 mm PEC Plate at 70 GHz. (d) Surface #1 at 70 GHz.

In 2D PGS case there is a phase change (discontinuity) along both x- and y-axis, i.e. Δϕ/Δx and Δϕ/Δy. In this case the reflected beam cannot be described by only one angle as in the case of the 1D PGS and two angles named θ _r1 and θ _r2 are required, where the angle θ _r1 is the angle between the incident and reflected beam and θ _r2 is the angle between x-axis and the projection of the reflected beam in xy-plane as can be seen in Fig. 5. Under normal incident of EM wave, θ _r1 and θ _r2 can be calculated as in equations (3) and (4). As can be seen in equations (3) and (4) there is more freedom in controlling the reflected beam in both azimuth and elevation planes.

(3)$$\eqalign{\theta _{r1} = & \arcsin \left[ {\displaystyle{1 \over {k_o}}\sqrt {{(d\varphi /dx)}^2 + {(d\varphi /dy)}^2}} \right] \cr = & \arcsin \left[ {\displaystyle{{2\pi} \over {k_o}} \times \sqrt {{(1/N_xP_x)}^2 + {(1/N_yP_y)}^2}} \right],} $$

(4)$$\theta _{r2} = \arctan (N_xP_x/N_yP_y).$$

Fig. 5.

Illustration of 2D phase gradient anomalous reflection angles.

The five-unit cells in Fig. 1(a) are used to form the 2D building blocks of 2D PGS of different unit cell distribution as shown in Fig. 6 with their unit cell distribution map. The layout of the designed 2D PGS is presented in Figs 7(a) and 7(b) where the main diagonal and distribution of the unit cells are different. The computed backscattered RCS under normal incidence for both surfaces is presented in Fig. 7(c) and both surfaces have a very close scattering characteristics and a clear reduction in the monostatic RCS is considered compared with its equivalent PEC plate. Moreover, the direction of the reflected beams is calculated using equations (3) and (4) where θ _r1 = 53.8° and θ _r2 = 45° and computed by full-wave simulations through CST microwave studio as shown in Fig. 7(d) and it is found to be θ _r1 = 51.3° and θ _r2 = 45°.

Fig. 6.

The configuration of the 2D super unit cell of size 7.5 mm × 7.5 mm in which the main diagonal has unit cells have R = 0.3 mm, and R = 0.21 mm for (a) and (c), respectively. (b) and (d) shows the unit cell distribution map.

Fig. 7.

(a) Surface #2: 15 mm × 15 mm. (b) Surface #3: 15 mm × 15 mm. (c) Monostatic RCS characteristics under normal incidence. (d) The backscattered field distribution in front of the surfaces at 70 GHz.

IV. FABRICATION AND EXPERIMENTAL RESULTS

To further verify the presented designs, the 2D surface in Fig. 7(b) is fabricated using standard printed circuit board technology. The size of the fabricated square sample is 30 × 30 mm² and consists of 20 × 20 unit cells as shown in Fig. 8(a) with its equivalent bare PEC in Fig. 8(b). All fabricated surfaces are etched on 1.27 mm RO6010 high-frequency laminate. The geometrical parameters of the fabricated surface are the same as those used in the numerical simulations in Section III. The fabricated samples are mounted inside a dielectric holder (ε _r ≈ 1) for measurement in an anechoic chamber. A single pyramidal horn antenna is used in the measurement which is fabricated using 1 mm thick copper sheet and connected to the measurement facilities via waveguide transition part as shown in Fig. 8. For reflectivity measurements [Reference Yang, Jiang, Liu and Weng11] to be performed an Agilent N5245A network analyzer (calibrated using Agilent V11644A cal-kit) is used. The horn antenna is then connected directly to an Agilent millimeter wave VNA extender via 50 mm long WR-12 → WR-15 (and WR-12 → WR-10) waveguide transition parts. Microwave pyramidal absorbers are used to reduce the reflections from the unwanted surroundings. The horn antenna is placed as high as the fabricated sample in the experiment with enough distance (R) between them to avoid the near field effect. The distance R is calculated using a very well-known far-field region formula R > 2D ²/λ in [Reference Balanis12] where D is the surface aperture size and lambda is the free space wavelength. The results of the reflectivity measurements are presented in Fig. 8(c) and show a clear reduction in the backscattered (reflected) power in the boresight direction. This reduction is because the incident power is redirected away from the source (horn antenna), in other words, away from the boresight direction and not reflected back to the direction where it has come from. The measured results in Fig. 8(d) are a little deviated from the simulated results, this deviation can be attributed to number of factors such as: (i) the misalignments between the PGS under test and the horn antenna used in the measurements, (ii) in simulation the PGS is excited with a far-field plane wave which is not the case in the measurements where horn antenna is used as EM-wave radiator, and (iii) the manufacturing error of the PGS.

Fig. 8.

Photograph of the fabricated (a) surface #3 and (b) the equivalent bare PEC plate. (c) Horn antenna and waveguide extension used in the measurements. (d) Measured reflectance versus frequency from the dielectric surface and bare PEC plate under normal incidence.