### I. Introduction

### II. Bandwidth Enhancement of Single-Layer Microstrip Reflectarray Elements

### 1. Element Configuration

*h*= 3.175 mm, relative permittivity of

*ɛ*

*= 2.2, and loss tangent of tan*

_{r}*δ*= 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.

### 2. Bandwidth Enhancement

*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.

*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.

*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).

*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.

### III. Limit of the Bandwidth Enhancement

The reduction of the reflection magnitude at the frequency of the surface wave coupling

*f*_{sw}The distortion of the reflection phase at the frequencies close to

*f*_{sw}

*f*

*is almost independent from the element shape or the number of resonant structures but is dominated by array parameters, such as the element spacing and the thickness and permittivity of the substrate. Therefore, this surface wave excitation condition fundamentally limits the reflectarray bandwidth enhancement.*

_{sw}### 1. The Condition of Surface Wave Excitation

*λ*

_{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.

*f*

*is considered for the TM*

_{sw}_{0}mode only and can be derived using the relation between the grating lobe angle and the scan blindness angle, as follows.

*p*(=

*p*

*=*

_{x}*p*

*) on a dielectric substrate with thickness*

_{y}*h*and relative permittivity

*ɛ*

*, the grating lobe occurs at angle*

_{r}*θ*

*[18].*

_{gr}*θ*

*can then be calculated as follows:*

_{sb}*β*

*is the propagation constant of the TM*

_{sw}_{0}surface wave on the unloaded grounded dielectric substrate. For a rigorous solution,

*β*

*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*

_{sw}*β*

*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*

_{sw}*β*

*is found in [19].*

_{sw}*f*

*under a normally incident plane wave can be derived with*

_{sw}*θ*

*= 0, as follows:*

_{sb}*c*

_{0}is the speed of light in free space. Note that

*f*

*in Eq. (4) is dependent on the array parameters such as*

_{sw}*p*,

*h*, and

*ɛ*

*. In the following part of this section, it will be shown that*

_{r}*f*

*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.*

_{sw}### 2. Effects of the Surface Wave

*p*= 10 mm,

*h*= 3.175 mm, and

*ɛ*

*= 2.2),*

_{r}*f*

*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*

_{sw}*l*

_{0}. It is observed that the rapid reduction of the reflection magnitudes occurs near

*f*

*= 23.03 GHz. The differences between*

_{sw}*f*

*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*

_{sw}*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 TM

_{0}mode surface wave that is excited by the incident plane wave is propagating in the

*y*direction. 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.

*f*

*, 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*

_{sw}*f*

*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*

_{sw}*φ*

*, then the reflection phases near*

_{g}*f*

*converge to*

_{sw}*φ*

*–2*

_{g}*πn*(

*n*= 0, 1, 2, …) as shown in Fig. 6(a).

*f*

*is almost independent from*

_{sw}*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*

*, 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*

_{sw}*f*

*. 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.*

_{sw}*f*

*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*

_{sw}*h*and

*p*, respectively. The values of

*f*

*calculated using (4) are given in Table 1 for comparison. The maximum values of*

_{sw}*l*

_{0}are 9.9 mm when

*p*= 10 mm, and 14.9 mm when

*p*= 15 mm, respectively. Because

*f*

*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*

_{sw}*f*

*from the full wave simulation in Table 1. From Table 1, it is observed that the values of*

_{sw}*f*

*from full wave simulation for various cases are close to*

_{sw}*f*

*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.*

_{sw}### IV. Broadband Reflectarray Design

*D*= 300 mm (15

*λ*

_{0}at

*f*

_{0}= 15 GHz). The distance from the feeder phase center to the center of the reflectarrays is

*F*= 426 mm (21.3λ

_{0}at

*f*

_{0}= 15 GHz), and the feeder is located on the axis of the reflectarray surface. Taconic TLY-5 substrates (

*ɛ*

*= 2.2, tan*

_{r}*δ*= 0.0009) with

*h*= 3.175 mm are used as the substrates for the reflectarrays.

*φ*

*of the*

_{r}*i*

^{th}element at the position

*x*=

*x*

*is*

_{i}*R*

*is the distance from the phase center of the feeder to the*

_{i}*i*

^{th}element,

*R*

*is the distance from the phase center to the edge element, and*

_{max}*Δφ*is a phase offset that can be arbitrarily added to the reflection phase. Note that

*Δφ*is a function of frequency. Theoretically, the phase offset

*Δφ*can be changed with frequency without affecting the radiation characteristics of a reflectarray because

*Δφ*is not a function of

*x*

*.*

_{i}*f*

_{0}= 15 GHz (which is a conventional reflectarray design method). In this way, the bandwidth enhancement that only corresponds to the element change can be observed. For instance, it is expected that the single-dipole element reflectarray will have the narrowest gain bandwidth among the four reflectarrays because the linearity of the phase curves in Fig. 3 are rapidly reduced as frequency increases from

*f*

_{0}. In addition, it is expected that the gain at

*f*

_{0}will be lowest because the achievable phase range at

*f*

_{0}is less than 360°. The gains at

*f*

_{0}and upper frequencies will be enhanced by using the three-dipole element, as the phase range at

*f*

_{0}is larger than 360° and the linearity of the phase curves in Fig. 3 are well conserved until

*f*= 17 GHz. The bandwidth enhancement up to

*f*= 20 GHz will be achieved with the five-dipole element as the linearity of the phase curves are further improved, as shown in Fig. 3. The phase curves of the seven-dipole element are similar to those of the five-dipole element (not shown for brevity), and no further bandwidth enhancement will be achieved due to the surface wave excitation discussed in Section III.

*f*

_{0}= 15 GHz is

*G*

_{0}= 31.0 dBi, and the corresponding aperture efficiency is

*η*

*= 59.1%. In the case of the five-dipole element reflectarray,*

_{a}*G*

_{0}= 31.4 dBi and

*η*

*= 64.1%. In the case of the seven-dipole element reflectarray,*

_{a}*G*

_{0}= 31.0 dBi and

*η*

*= 59.1%. The detailed radiation characteristics of the three reflectarrays at 15 GHz are summarized in Table 2.*

_{a}*f*< 17 GHz), but the side lobe levels are increased and the gain is reduced as frequency increases in the case of the three-dipole element reflectarray, as expected. Also note that the gains of the seven-dipole element reflectarray are similar to those of the five-dipole element reflectarray for the entire frequency range of interest, and no further bandwidth enhancement is observed with the increased number of dipoles, as predicted.