Fig. 1 shows the multi-dipole elements with single-, three-, and five-dipoles. The elements are symmetric with respect to the axis of the longest dipole at the center. The multi-dipole elements are designed on a substrate with a thickness of h = 3.175 mm, relative permittivity of ɛ _r = 2.2, and loss tangent of tanδ = 0.0009. The element spacing is fixed at p = 10 mm (0.5 λ ₀ at f ₀ = 15 GHz). The lengths of the side dipoles l ₁ and l ₂ are proportional to the length of the center dipole l ₀ with ratios r ₁ and r ₂, respectively (i.e., l ₁ = r ₁ × l ₀ and l ₂ = r ₂ × l ₀). The gap distance between the center dipole and the second dipoles is g ₁, and the gap distance between the second dipole and the third dipole is g ₂. The widths of the center dipole, the second dipole, and the third dipole are w ₀, w ₁, and w ₂, respectively.

The single-dipole element shown in Fig. 1(a) is a singleresonant structure. The number of resonance increases as the number of side dipoles increases. Therefore, the three- and five-dipole elements in Fig. 1(b) and (c) are dual- and tripleresonant structures, respectively (detailed analysis of the resonance mechanism of the multi-dipole element can be found in [5]). Fig. 2 shows reflection phase curves of the multi-dipole elements when l ₀ increases from 0.1 to 9.9 mm in 0.1 mm steps. The reflection phases were calculated under the assumption of infinite array of an identical element, which can be done by using plane wave of normal incidence on a single element in a waveguide composed of electric and magnetic walls [1]. The widths of the dipoles are w ₀ = w ₁ = w ₂ = 0.5 mm, and the gaps between the dipoles are g ₁ = g ₂ = 0.5 mm. The ratios between the dipole lengths are designed as follows: r ₁ = 0.65 and r ₂ = 0.5 for both of the three- and five-dipole elements. These dimensions of the elements are chosen based on various parametric studies so that the phase curves of the elements become linear and parallel within the maximally achievable frequency range.

From Fig. 2, it is observed that the achievable phase delay increases as l ₀ increases. In the case of the single-dipole element, the first resonance frequency (when the reflection phase is 0°) decreases as l ₀ increases. However, the reflection phase curves begin to converge before the reflection phase exceeds −360° (the second resonance), as indicated by the dashed red circle in Fig. 2(a). Such convergence of phase curves is not desirable for broadband reflectarray design because the desired reflection phase distribution cannot be obtained.

As shown in Fig. 2(b) and (c), it is possible to achieve continuously spreading reflection phase curves with frequency by adding additional resonant structures. The second dipole of the three-dipole element induces the second resonance, and the phase delay can be increased to be larger than 360°. By adjusting r ₁, the distance between the first and second resonant frequencies can be controlled so that highly nonlinear change of the reflection phase is avoided. However, the phase curves still converge before the reflection phase exceeds −720° (the third resonance), as indicated by the dashed red circle in Fig. 2(b). Since the phase curves begin to converge when f is larger than approximately 17 GHz, it is expected that the gain will begin to decrease as the frequency exceeds 17 GHz. Introducing the third dipole, the phase curves of the five-dipole element can continuously spread up to 20 GHz, as shown in Fig. 2(c).

Fig. 3 shows the reflection phases against l ₀ at various frequencies for the multi-dipole elements shown in Fig. 1. In the case of the single-dipole element, not only is the reflection phase range not wide enough to cover 360°, but also the phase curves are parallel to each other only in a narrow frequency range. In the case of the three-dipole element, the phase range is widened due to the increased number of resonances, and the phase curves are linear and parallel to each other in a wider frequency range when l ₀ is larger than 5.5 mm, which is achieved by adjusting the dimensions of the dipoles. However, the slope of the phase curves is still reduced at upper frequencies (f > 17 GHz), as expected from Fig. 2(b). In the case of the five-dipole element, the phase curves are almost linear and parallel to each other from 10 to 20 GHz.

In phased arrays, because the element spacing is close to 0.5λ ₀, the blindness angle is normally far from the bore-sight. In reflectarrays, however, the angles from a feeder to most elements are generally small. For instance, the maximum incident angle is only 26.5° for a reflectarray with F/D = 1 (where F is the distance from the feeder phase center to the center of the reflectarray and D is the diameter of the reflectarray) when the feeder is on the axis of the reflectarray. The angles to the dominant elements that are under relatively high levels of illuminated field from the feeder are close to 0°. Therefore, strong coupling between the surface wave and the incident wave (or forced resonance of the surface wave) does not occur.

However, when the element spacing of phased array is increased, the scan blindness can occur at an angle that is close to the bore-sight [17]. Similarly, the surface wave can be excited on a reflectarray even with the plane waves with small incident angles or, in extreme cases, with normal incidence as frequency increases. Because the surface wave mode that limits the bandwidth enhancement is the dominant mode with the lowest excitation frequency, in this work, f _sw is considered for the TM₀ mode only and can be derived using the relation between the grating lobe angle and the scan blindness angle, as follows.

With a planar array with element spacing p (= p _x = p _y ) on a dielectric substrate with thickness h and relative permittivity ɛ _r , the grating lobe occurs at angle θ _gr [18].

The angle of scan blindness θ _sb can then be calculated as follows:

where β _sw is the propagation constant of the TM₀ surface wave on the unloaded grounded dielectric substrate. For a rigorous solution, β _sw must be the propagation constant of the surface wave of the dielectric substrate with the printed elements. However, practically, this loading effect of the printed elements is small enough to be ignored and therefore β _sw can be approximated by using the propagation constant of the unloaded dielectric substrate. The error of the blindness angle due to such approximation does not exceed a few tenths of a degree [14]. An approximate closed form of β _sw is found in [19].

Finally, f _sw under a normally incident plane wave can be derived with θ _sb = 0, as follows:

where c ₀ is the speed of light in free space. Note that f _sw in Eq. (4) is dependent on the array parameters such as p, h, and ɛ _r . In the following part of this section, it will be shown that f _sw is almost independent from the shape of the element or the number of resonant structures but dominated by the array parameters by comparing the full wave simulation results with the analytic solutions.

In the case of the multi-dipole elements in Section II (p = 10 mm, h = 3.175 mm, and ɛ _r = 2.2), f _sw is calculated as 23.03 GHz from (4). In Fig. 4, the magnitudes of the reflection coefficients of the five-dipole elements under plane wave of normal incidence are shown as black lines for different values of l ₀. It is observed that the rapid reduction of the reflection magnitudes occurs near f _sw = 23.03 GHz. The differences between f _sw and actual frequencies of the surface wave excitation are due to the loading effect of the printed dipoles which are not considered in (4). When l ₀ = 2.75 mm, the largest reduction of the magnitude occurs at 22.9 GHz. Fig. 5 shows electric field distribution excited by y-polarized plane wave of normal incidence on three identical five-dipole elements with l ₀ = 2.75 mm at 22.9 GHz. The elements are placed inside a waveguide composed of electric and magnetic walls to assume an infinite array of the elements. Note that three identical elements (instead of one element) are shown in Fig. 5 only for better visualization of the surface wave. It is observed in Fig. 5 that the TM₀ mode surface wave that is excited by the incident plane wave is propagating in the ydirection. Thus, most of the energy of the incident wave is excited as the surface wave, and the surface wave propagates until it is fully dissipated as dielectric and conductor losses. Such reduction of the reflected power significantly decreases the efficiency of the reflectarray.

In addition, the efficiency is degraded due to the convergence of the reflection phases near f _sw , as in Fig. 6(a) that shows the reflection phases of the five-dipole element within the frequency range from 20 GHz to 25 GHz. The convergence of the phases occurs for the following reason. After the partial energy of the incident wave is coupled to the surface wave, the rest of the energy that is not coupled experiences the grounded dielectric substrate. Thus, the reflection phase at f _sw becomes the same as the phase when a plane wave is normally incident on the grounded dielectric substrate (with no printed element pattern on the dielectric layer). As a result, assuming that the reflection phase of the grounded dielectric substrate is φ _g , then the reflection phases near f _sw converge to φ _g –2πn (n = 0, 1, 2, …) as shown in Fig. 6(a).

Because f _sw is almost independent from l ₀, even if the number of dipoles is further increased, the bandwidth enhancement is still limited due to the surface wave. In Fig. 4, the reflection magnitudes of the seven-dipole element are shown as red lines. The seven-dipole element is designed by adding two shorter dipoles (with the length l ₃ (= r ₃ × l ₀, r ₃ = 0.3), the width w ₃ = 0.5 mm, and the gap distance g ₃ = 0.5 mm) to the five-dipole element. The reduction of the reflection magnitudes is also observed near f _sw , as in the case of the five-dipole element. The reflection phases of the sevendipole element are shown in Fig. 6(b), where similar distortion (convergence) of the phase curves is also observed near f _sw . Thus, it is obvious that the bandwidth enhancement of the reflectarray is limited due to the surface wave excitation, even with the increased number of dipoles.

It should be noted that similar phenomena are observed with the single- and three-dipole elements. In Table 1, the values of f _sw extracted from full wave simulation are listed for the multi-dipole elements with different numbers of dipoles. They are designed with two different values of h and p, respectively. The values of f _sw calculated using (4) are given in Table 1 for comparison. The maximum values of l ₀ are 9.9 mm when p = 10 mm, and 14.9 mm when p = 15 mm, respectively. Because f _sw shifts slightly as the shape or size of the printed element changes, as in Fig. 4, frequency ranges where the reflection magnitude reduction is larger than 3 dB are given for the values of f _sw from the full wave simulation in Table 1. From Table 1, it is observed that the values of f _sw from full wave simulation for various cases are close to f _sw from (4) and not dominated by the element types. Therefore, it can be concluded that the bandwidth enhancement of a reflectarray with single-layer multi-resonant type elements is limited by the excitation of the surface wave, no matter how much the number of resonant structures is increased.