### I. INTRODUCTION

Retro-reflectors are devices or surfaces with attractive characteristics that reflect incident waves in an incoming direction as opposed to a specular direction. Since this characteristic serves to increase the target’s monostatic radar cross section (RCS), retro-reflectors are extensively used in fields dealing with millimeter waves and microwaves, such as the military and civilian industries [1–4]. The types of retro-reflectors include 3D structure types, such as the corner cube reflector and cat’s eye retroreflector, and 2D structure types, such as the Van Atta array and metasurfaces [5–7]. Among these, the 2D structure retrodirective metasurface (RDM) has been gaining attention as a next-generation retro-reflector due to its advantages, such as its thin planar structure, light weight, and easy fabrication [5–7]. However, the RDM has a limited retro-reflection angle since it is a passive retro-reflector. Therefore, it is difficult to use it in radar in real time or in a wireless power system with multiple targets [1–4].

Notably, some research on reconfigurable metasurface (RMS) using PIN diodes has recently been conducted [8–10]. PIN diodes exhibit several distinct advantages, including fast switching speed, low loss, and a simple structure. In this context, it is anticipated that the PIN diode-based N-bit reconfigurable retrodirective metasurface (N-bit RRDM), as an application of RMS, can overcome the above-mentioned challenges. The PIN diode-based N-bit RRDM is a retro-reflector that freely determines the angle of retro-reflection by electronically controlling the reflection phase of the unit cell and the phase gradient of the surface through the PIN diode, which is mounted on the unit cell. In such a case, however, the reflection phase of the unit cell is controlled by a quantized N-bit phase (360°/2

^{N}), which is based on the number (N) of mounted PIN diodes [8, 9].Therefore, the N-bit RRDM does not fully implement the phase gradient required for retro-reflection. These characteristics may sometimes cause problems such as beam gain reduction and spurious beam errors. In particular, spurious beams occurring in 1-bit RRDM relate to a completely different concept from that of the grating lobe.

This paper proposes a PIN diode-based N-bit RRDM whose characteristics are theoretically analyzed to propose the criteria for an N-bit RRDM design that eliminates the problems of beam gain reduction and spurious beam errors. The proposed metasurface is analyzed using generalized Snell’s law [11] and array factor theory [12]. Furthermore, the theoretical analysis is validated through full-wave simulation and experimentation using the 1-bit metasurface.

### II. THEORY AND ANALYSIS OF N-BIT RRDM

The PIN diode-based N-bit RRDM is a metasurface composed of N PIN diodes mounted on each of its M unit cells, as shown in Fig. 1. The N-bit RRDM controls the phase of the unit cell related to the N-bit by switching the PIN diodes on/off. As a result, the quantized phase is considered 360°/2d Φ I d e a l d x ) required for this operation can be calculated as Eq. (1) using generalized Snell’s law. Additionally, Φ

^{N}. If a plane incident wave with an incident angle*θ**retro-reflects at this N-bit RRDM, the reflective angle of*_{i}*θ**will be equal to −*_{r}*θ**. Effectively, the ideal reflection phase gradient (*_{i}*(*_{Ideal}*m*), which is the ideal reflection phase value of the*m*^{th}unit cell, can be calculated using Eq. (2), as noted below:Here,

*n*_{0}indicates the refractive index of free space,*λ*_{0}refers to the wavelength in free space, and*d**represents the distance between adjacent cells. Following this, the N-bit RRDM implements the reflection phase acquired from Eq. (2) using the quantized N-bit phase. Therefore, the reflection phase (Φ*_{x}*(*_{N–bit}*m*)) of the*m*^{th}unit cell in the N-bit RRDM can be expressed as Eq. (3):Here,

*R*(*X*) refers to an integer value obtained by rounding off*X*. The phase of the unit cell is quantized to only 0° and 180° when N is 1. Subsequently, as N increases, the quantized phase difference becomes smaller. At the same time, a continuous retrodirective metasurface (continuous RRDM), which controls the phase continuously, can implement a perfect reflection phase gradient, with the controlled reflection phase (Φ*(*_{Continuos}*m*)) of the*m*^{th}unit cell being equal to Φ*(*_{Ideal}*m*). Due to this difference between the N-bit and continuous RRDMs, they operate differently. Furthermore, the array factor (*AF*(*θ*)) of the N-bit and continuous RRDMs can be calculated by changing N from 1 to 4. Fig. 2 shows the normalized value of*AF*^{2}on using 1, 2, 3, and 4-bit RRDMs, with the retroreflection angles being 5° and 25°. The cell distance is 0.35*λ*_{0}and the number of cells is 20.Moreover, since the magnitude of

*AF*(*θ*)^{2}and the gain of the beam are proportional to each other, it is possible to assess the reduction in gain caused by the phase quantization by comparing the*AF*(*θ*)^{2}of the N-bit and continuous RRDMs. The ratio of the maximum peak value (*Peak*) of the N-bit and continuous RRDMs can be considered the quantization efficiency (η*), as shown in Eq. (4) [10]:*_{Quantization}In accordance with Fig. 2, the value of η

*converges to 1 as N increases, indicating an improvement in the gain reduction.*_{Quantization}Meanwhile, in the case of 1-bit (N = 1) RRDM, Fig. 2(a) shows a spurious beam error at the same level as the main lobe occurring in an undesired direction (

*θ*=−15.5°). This spurious beam was analyzed using array factor theory, according to which the*AF*(*θ*) of the N-bit RRDM (*AF**(*_{N–bit}*θ*)) can be denoted as Eqs. (5) and (6), noted below:Here,

*k*_{0}(=2*π*/*λ*_{0}) indicates a propagation constant in free space and ΔΦ*(*_{N–bit}*m*) denotes a difference in the reflection phase between the*m*^{th}and*m*–1^{th}cells when retro-reflection occurs in the N-bit RRDM. In addition,*Ψ**(*_{N–bit}*m*,*θ*) refers to a progressive phase—the relative phase between the*m*^{th}and*m*–1^{th}cells in the direction*θ*°.With regard to the N-bit RRDM, let us consider defining

*Ψ**(*_{N–bit}*m*,*θ**) and*_{re}*AF**(*_{N–bit}*θ**) as the progressive phase and the array factor in the*_{re}*θ**(=−*_{re}*θ**) direction, respectively. Furthermore, let us consider defining*_{i}*Ψ**(*_{N–bit}*m*,*θ**) and*_{s}*AF**(*_{N–bit}*m*,*θ**) as the progressive phase and the array factor in the*_{s}*θ**direction—in which the spurious beam error occurs— respectively. Considering these assumptions, according to Fig. 2, since the spurious beam error is of the same level as the main lobe (retro-reflection), it can be accomplished that*_{s}*AF**(*_{N–bit}*θ**) =*_{re}*AF**(*_{N–bit}*m*,*θ**). This can be expressed as Eqs. (7) and (8), since*_{s}*AF**(*_{N–bit}*θ*) represents the even function of*Ψ**(*_{N–bit}*m*,*θ*).Furthermore, since

*θ**is equal to −*_{re}*θ**, Eqs. (7) and (8) can be formulated as Eqs. (9) and (10), respectively, when retroreflection occurs.*_{i}Here,

*a*is an integer. Notably, Eq. (9) has a quantization term (ΔΦ*(*_{N–bit}*m*)), which is satisfied for all*m*only when N = 1. Therefore, Eq. (9) represents the spurious beam error information in the 1-bit RRDM. Meanwhile, since Eq. (10) does not contain the quantization term (ΔΦ*(*_{N–bit}*m*)), it can be satisfied, regardless of N. Effectively, this is similar to the conditions under which the grating lobe occurs.### III. RESULTS AND VERIFICATION

The average value of η

*, calculated by changing the*_{Quantization}*dx*and*θ**, were found to be 43.6% for 1-bit, 82.2% for 2-bit, 95.3% for 3-bit, and 98.8% for 4-bit. The number of spurious beam errors is presented in Fig. 3 in terms of the*_{re}*dx*and*θ**by considering the results obtained from Eqs. (9) and (10). For instance, in the 1-bit RRDM, no spurious beam error occurs if the cell distance is 0.35*_{re}*λ*_{0}and the retro-reflection angle is 25°, as shown in Fig. 3(a). Meanwhile, when the retroreflection angle is 30° and the cell distance is 0.75*λ*_{0}in the 2- bit RRDM, one spurious beam error occurs, as shown in Fig. 3(b), with the angle derived using Eq. (10) being −56.4°.
Fig. 4 presents schematic diagrams of the simulations and the measurement method employed to verify the analyzed results. Notably, this study adopted the ring patch structural unit cell used in [10] as the 1-bit RRDM’s unit cell. The measurement and simulations were conducted at 10.1 GHz for the 1-bit RRDM, arranged as 12 × 12 (126 mm × 126 mm). Each unit cell (10.5 mm × 10.5 mm) was mounted with one PIN diode [13], whose phase was controlled by a 1-bit by switching the PIN diode on/off. In Simulation 1, as depicted in Fig. 4(a), the bistatic RCS result is obtained by simulating a case in which the incident wave is a plane incident wave moving in the direction (25°, 0°). In Simulation 2, the incident wave is radiated from the Tx horn antenna fixed at (60 cm, 25°, 0°), resulting in scattered power patterns that indicate the relative power value of the wave scattered in the 1-bit RRDM. This phenomenon bears the same meaning as the RCS, which expresses scattered power in an area.

In both Simulation 3 and the measurement, the Tx horn antenna is fixed at (60 cm, 25°, 0°), while the Rx horn antenna is moved to the arc of (60 cm,

*θ*°, 0°), ultimately resulting in*S*_{21}. Subsequently, the bistatic RCS was calculated using*S*_{21}and the radar range equation. It is noted that the retro-reflected wave can be measured from*S*_{1}using only one Tx horn, while the other waves can be measured using the Tx and Rx horn antennas.The bistatic RCS and the scattered power pattern obtained from the three simulations and the measurement were normalized and plotted, as depicted in Fig. 5. For normalization, each simulation and measurement was performed again using the conducting plate (126 mm × 126 mm). Subsequently, the resulting RCS values in the specular direction were used as the normalization factor. As observed in Fig. 5, the results of the simulations and the measurement conducted for the different methods exhibit similar shapes. The main lobe is formed at around 25°, indicating a retro-reflection angle, with no spurious beams. Meanwhile, the beam around −25° is in the specular direction, exhibiting a lower level than the main lobe. Overall, the measured results indicate relatively good agreement with the simulated results, thus verifying the analysis and theory of RRDM presented in this study.

### IV. CONCLUSION

This paper proposed a PIN diode-based N-bit RRDM that can electronically control the retro-reflection angle. The array factor of the N-bit RRDM achieved a lower value than that the continuous RRDM due to phase quantization, resulting in gain reduction. However, undesired spurious beam errors still occurred. This phenomenon was theoretically analyzed and arranged using generalized Snell’s law and the array factor theory. The obtained results show that the gain reduction can be controlled by the selection of N. In particular, it was found that when N increased from 1 to 2, the quantization efficiency increased dramatically from 43.6% to 82.2%. Furthermore, it was established that the occurrence of a spurious beam can be controlled by the selection of

*d**and*_{x}*θ**. Moreover, the theoretical results were successfully verified through simulation and measurement.*_{re}