Fu Liu, Do-Hoon Kwon, Sergei Tretyakov
IMAGE LICENSED BY INGRAM PUBLISHING
Reconfigurable reflectors have significant potential in future telecommunication systems, and approaches to the design and realization of full and tunable reflection control are now actively studied. Reflectarrays, being the classical approach to realize scanning reflectors, are based on the phased-array theory (the so-called generalized reflection law) and physical optics approximation of the reflection response. To overcome the limitations of the reflectarray technology, researchers actively study inhomogeneous metasurfaces, using the theory of diffraction gratings. To make these devices tunable and fully realize their potential, it is necessary to unify the two approaches and study reconfigurable reflectors from a unified point of view. Here, we offer a tutorial on reflectarrays and metasurface reflectors, explaining their common fundamental properties that stem from the diffraction theory. This tutorial is suitable for graduate and postgraduate students and hopefully will help to develop deeper understanding of both phased arrays and diffraction gratings.
During the past few years, many research groups have been studying the possible use of reconfigurable intelligent surfaces (RISs) in future wireless communication systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The main functionality of these RISs is to reflect incident waves (coming from a specified direction or directions) into desired directions. Basically, this is the same function as usually realized by reflectarray antennas. Most commonly, reflectarrays are used as flat or conformal equivalents of parabolic reflectors, while RISs are usually designed to reflect plane waves, but this is not a principal difference. Such RISs are equivalent to focusing reflectors with an infinite focal distance.
Realizations of anomalously reflecting metasurfaces are usually designed as phase gradient reflectors, which are made of reactive impedance boundaries with a linearly varying phase of the local reflection coefficient. However, recent research has shown that such realizations have a fundamentally limited efficiency, which degrades when the desired performance significantly deviates from that of uniform mirrors and retroreflectors [11], [12], [13], [14], [15]. This degradation is attributed to the excitation of parasitic propagating modes that scatter part of the incident power in unwanted directions. Actually, similar effects are known also for reflectarrays, which function well only if the reflected rays do not have to be tilted much. When the deflection angle is large (i.e., elements are close to the reflectarray edge), the specular reflection is not controlled and considered wasted, lowering the efficiency [16, Sec. 4.1.4]. The physical reason behind a deteriorating reflection efficiency with an increasing deflection angle is the wave impedance mismatch between the incident and deflected plane waves. Recognized first in perfect anomalously refracting metasurfaces [13], [15], [17], this wave impedance mismatch for efficiency reduction is analogous to the angle-dependent scan impedance mismatch in phased arrays. While it can be tolerated for conventional applications of reflectarrays, for the envisaged use of anomalous reflectors as RISs, this limitation can significantly compromise usability. Indeed, most usage scenarios of RISs assume that the reflected waves can be sent in any direction.
Recently, it was shown that advanced metasurfaces can control reflection theoretically perfectly, without any spurious scattering (except that caused by manufacturing imperfections and dissipation losses), e.g., [14], [15], [18], [19], [20], [21], and [22]. Different design approaches have been developed (we summarize and discuss them in the “Reflectarrays and Metamirrors” section). Interestingly, all of them are based on the theory of diffraction gratings and do not use the conventional design methods and topologies that have been developed for phased arrays and reflectarray antennas.
For the proper understanding and further development of devices for full and efficient control of wave reflection, it appears that it is necessary to analyze the basic principle of inhomogeneous reflectors, looking at both metasurfaces (realized with subwavelength structures) and reflectarrays (formed by repeating antenna elements at half-wavelength intervals) from a unified point of view. While these two techniques are different, they have fundamental similarities: both can be considered diffraction gratings. In this basic tutorial article, from the diffraction grating theory, we explain the fundamental principles behind any device that creates plane waves propagating in a certain direction. In the final section, we summarize and classify the currently known methods to design and realize anomalous reflectors and discuss current research challenges.
In the design of reflectarrays and reflecting metasurfaces, the main challenge is to ensure that the proper currents on the reflector are excited to create the desired reflected fields. However, first, one needs to know what current distribution is necessary to be realized. To this end, we discuss active arrays [23], assuming that we can fix any desirable current distribution over a planar surface. Our goal is to determine what current distribution we should set to create the desired propagating waves. Here, it is enough to consider sheets of electric surface currents. Then, we are able to properly determine the needed current profile over the reflector by using this simple model, although these sheets create waves on both sides (a ground plane or a complementary sheet of magnetic surface current can be introduced to realize one-side excitation).
For simplicity, we consider infinite arrays, and our desired reflected modes are plane waves. For infinite arrays, the most common design goal is to ensure that in the regions far enough from the array, where all the evanescent fields can be neglected, there is only one plane wave propagating in the desired direction, corresponding to a delta function array factor. For finite arrays, this goal is equivalent to the radiation pattern having only one main beam in the desired direction, without any grating lobes. In this sense, conclusions made for infinite arrays will hold also for finite arrays.
Let us suppose that the reflected field that we want to create in the far zone is a set of propagating plane waves. We assume that this desired set of plane waves varies along the planar radiating surface (the coordinate x) as a periodic function. This means that the tangential wavenumbers (along x) of all radiated harmonics are in rational relations. The very important special case is when only one obliquely propagating plane wave is launched in angle ${\theta}$ (referenced to the surface normal) with tangential wavenumber ${k}_{t} = {k}_{0}\,{\sin}\,{\theta}$, where ${k}_{0} = {2}{\pi} / {\lambda}$ is the free space wavenumber and ${\lambda}$ is the wavelength. Then, the x-dependence of this field ${e}^{{-}{jk}_{t}{x}}$ is a periodic function, with period ${D} = {\lambda}{/}\,{\sin}\,{\theta}$. The case of launching aperiodically distributed fields can be, in principle, treated as a limiting case of the infinite period. Later, we also discuss possibilities to launch a single plane wave with aperiodic current distributions.
It is clear that the required current distribution should be in a phase synchronism with all the desired free space modes. Assuming that the radiating current distribution is periodic, with period D, it can be expanded into spatial Fourier series with tangential wavenumbers: \[{k}_{tn} = \frac{{2}{\pi}{n}}{D}, \qquad {n} = {0},{\pm}{1},{\pm}{2},{\ldots}{.} \tag{1} \]
In general, we should select the period D so that the tangential wavenumbers of all desired plane waves will be found among this set of numbers (for some values of index n). Let us consider the special case of launching only one plane wave at a certain angle ${\theta}$. In this case, it is enough to properly set only one Fourier harmonic of the current. The tangential wavenumber of the desired plane wave is \[{k}_{t} = {k}_{0}\,{\sin}\,{\theta} = \frac{{2}{\pi}}{\lambda}\,{\sin}\,{\theta}. \tag{2} \]
We need to select D so that at least one harmonic of the current distribution is in phase with the desired radiated wave. Comparing (1) and (2), the condition reads ${n} / {D} = \,{\sin}\,{\theta} / {\lambda}$; that is, ${D} = {n}{\lambda} / \,{\sin}\,{\theta}$. Since we want to send the energy in only one direction, it is reasonable to choose n as small as possible (that is, D as small as possible) to minimize the number of diffraction maxima or “open channels” (the directions where the array can radiate). Here, ${n} = {0}$ is not a valid solution since in that case, ${k}_{t} = {0}$, so we select ${n} = {1}$, which gives \[{D} = \frac{\lambda}{{\sin}\,{\theta}}. \tag{3} \]
In this case, the period of the radiating current is equal to the period of variations of the fields in the plane wave that we want to create. Obviously, this is an expected result.
Very importantly, we note that the array period D is greater than or equal to ${\lambda}$ for any angle ${\theta}$. The limit ${D}\,{\rightarrow}\,{\lambda}$ corresponds to ${\theta}\,{\rightarrow}\,{\pi} / {2}$, that is, to the endfire array. For small angles (radiation directions close to the normal), the period is very large. As a specific numerical example, we consider arrays that create a single plane wave in the ${\theta} = {70}^{\circ}$ direction, for which ${D}\,{\approx}\,{1.0642}{\lambda}$.
We have noted that it is desirable to select as small a period D as possible. More specifically, for a given period D, all harmonics whose tangential wavenumber ${k}_{tn}$ satisfies the inequality ${\left\vert{k}_{tn}\right\vert} = {2}{\pi}{\left\vert{n}\right\vert} / {D}\,{<}\,{k}_{0} = {2}{\pi} / {\lambda}$ are propagating modes. This corresponds to \[{\left\vert{n}\right\vert}\,{<}\, \frac{D}{\lambda}. \tag{4} \]
In our example case of ${\theta} = {70}^{\circ}$, we have ${\left\vert{n}\right\vert}\,{<}\,{1.0642}$. This means that if we set any periodic current on the antenna (using active sources) with this period, the radiation in the far zone can go only to 0, 70°, and ${-}{70}^{\circ}$ directions (corresponding to ${n} = {0},{1}$, and ${-}{1}$, respectively). All higher-order harmonics are evanescent. Note, however, that for scanning in other directions, the situation can be very different since we may need a rather large D compared with ${\lambda}$.
We see that from the diffraction theory point of view, any periodic antenna array that radiates in any direction except the normal direction is a diffraction grating because its period D is larger than the wavelength, and it radiates into an ${n}\,{≠}\,{0}$ spatial harmonic.
We see that any periodic reflectarray (radiating current distribution) can reflect (radiate) in more than one direction because ${D}\,{>}\,{\lambda}$; see (4). The next task is to set such a current distribution so that the waves in all the unwanted directions have zero amplitude.
Let us denote the current distribution as ${J}{(}{x}{)}$, where x is the coordinate along the antenna array plane. In our example of exciting a single plane wave, the tangential fields of the desired plane wave mode vary as ${e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$. As discussed in the preceding, the current distribution on the antenna should be a periodic function with the same period, ${D} = {\lambda} / {\sin}\,{\theta}$. But what specific function with this period should we select? An appropriate current distribution can be found by using the theory of waveguide excitation, e.g., [24, Sec. 4.12]. The fields excited in a waveguide by a given distribution of external currents are found as an expansion over the waveguide eigenmodes. In our case, the waveguide is free space, and its modes are plane waves that can, in general, propagate at any angle ${\theta}$. The amplitudes of excited plane wave modes are proportional to the excitation integrals (integrals of the current distribution and complex conjugate of the desired radiated field distribution, [24, eq. 4.108]). It is enough to integrate over one period.
Continuing discussing the specific example of launching a single plane wave along the ${\theta} = {70}^{\circ}$ direction, where the period ${D}\,{\approx}\,{1.0642}{\lambda}$, in fact, allows three propagating plane wave modes, the current distribution ${J}{(}{x}{)}$ should be such that \[ \mathop{\int}\nolimits_{0}\nolimits^{D}{J}{(}{x}{)}{e}^{{jk}_{0}{x}\,{\sin}\,{\theta}}{dx}\,{\rightarrow}\,{\text{maximum}} \tag{5} \] \[ \mathop{\int}\nolimits_{0}\nolimits^{D}{J}{(}{x}{)}{e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}{dx} = {0}, \qquad \mathop{\int}\nolimits_{0}\nolimits^{D}{J}{(}{x}{)}{dx} = {0}. \tag{6} \]
These are the coupling integrals for all three allowed propagating modes with the selected period of the array. These conditions ensure that the current distribution effectively launches the desired wave and that it is orthogonal to all the other existing propagating harmonics. Importantly, we do not impose any restrictions on the amplitudes of evanescent fields in the vicinity of the array, as the goal is to create the desired field in the far zone. However, note that in reflectarray and metasurface realizations, evanescent modes are to be controlled to optimize far-zone fields, and we briefly discuss this in the “Reflectarrays and Metamirrors” section.
Clearly, there are many possible solutions of ${J}{(}{x}{)}$ satisfying (5) and (6). The simplest and most obvious one is the current with a constant amplitude and linear phase gradient. For example, selecting ${J}{(}{x}{)} = {Ae}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$ (A is the complex amplitude), we maximize the integral in (5) because the product of the two exponentials is just unity. The integrals in (6) are zero because of double/single variations over the period. This is why the current does not radiate into ${-}{70}^{\circ}$ and normal directions.
Let us consider another case with a small tilt angle, say, ${\theta} = {5}$. We use the same simple theory and first select the period such that the array radiates the plane wave along this direction. Now the appropriate period is ${D} = {\lambda} / \,{\sin}\,{\theta}\,{\approx}\,{11.47}{\lambda}$. The plane wave Fourier harmonics will propagate when ${\left\vert{n}\right\vert}\,{<}\,{D} / {\lambda} = {11.47}$; see (4). Thus, our desired direction corresponds to ${n} = {1}$, but there are 22 more directions (11 on the left, 10 on the right side from the normal, and the normal direction) where the waves can also propagate. The current distribution should be such that \[ \mathop{\int}\nolimits_{0}\nolimits^{D}{J}{(}{x}{)}{e}^{{jk}_{0}{x}\,{\sin}\,{\theta}}{dx}\,{\rightarrow}\,{\text{maximum}} \tag{7} \] \[ \mathop{\int}\nolimits_{0}\nolimits^{D}{J}{(}{x}{)}{e}^{{jnk}_{0}{x}\,{\sin}\,{\theta}}{dx} = {0}, \quad {n} = {0},{-}{1},{\pm}{2},{\pm}{3},{\ldots}{.} \tag{8} \]
The current distribution with the ideal linear phase gradient ${J}{(}{x}{)} = {Ae}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$ clearly satisfies all these relations. For approximate (e.g., step-wise) settings, finding solutions for (7) and (8) is a nontrivial task. On the other hand, for small angles, the required linear phase variation is smooth, and the coupling to other propagating harmonics is weak. Thus, reasonably smooth discretization of the linear phase current creates nearly perfect single-wave fields in the far zone.
Next, we discuss the principle of diffraction grating (metagrating) realizations [18], [25], [26], [27], [28]. Instead of a continuous current distribution, the diffraction grating realizations excite the desired plane wave, with just a few small radiating elements placed in each period. To understand how it works, we assume that the current distribution over the antenna plane is a set of a few delta function sources. That is, we consider an array of small radiators with an isotropic element pattern.
Let us try to reach the goal for radiation in ${\theta} = {70}^{\circ}$ by using only two such small active radiators per period, with arbitrary selected positions ${x} = {a}$ and b. The current distribution function becomes ${J}{(}{x}{)} = {A}{\delta}{(}{x}{-}{a}{)} + {B}{\delta}{(}{x}{-}{b}{)}$, where A and B are the complex amplitudes of the two point sources (line sources in the 2D scenario). Substituting this current distribution into (6), we get only a trivial solution ${A} = {B} = {0}$.
This means that we need at least three elements per period. Let us assume \[{J}{(}{x}{)} = {A}{\delta}{(}{x}{)} + {B}{\delta}{\left({x}{-} \frac{D}{3}\right)} + {C}{\delta}{\left({x}{-} \frac{2D}{3}\right)} \tag{9} \] and substitute it for (6); we get \[{B} = {-}{A}{\left({1} + {e}^{j \frac{{2}{\pi}}{3}}\right)} = {Ae}^{{-}{j} \frac{{2}{\pi}}{3}}, \qquad {C} = {-}{(}{A} + {B}{)} = {Ae}^{{j} \frac{{2}{\pi}}{3}} \tag{10} \] to ensure that there is no radiation in the normal and ${-}{70}^{\circ}$ directions. We see that in this equal spacing configuration, the three line sources have the same amplitude and a linear phase drop (a total of ${2}{\pi}$ over one period). By substituting this solution into (5), we find that the result is 3A (nonzero), meaning that in the far zone, there is a perfect plane wave in the desired direction. The positions of the small radiators can be varied, adding degrees of freedom in design. Here, we note that a recent paper [29] studied the excitation of plane waves by arrays of line currents. In that paper, excitation conditions are also imposed on evanescent modes, resulting in the conclusion that only continuous current distributions with a linear phase gradient can launch a single plane wave.
The preceding results are verified with numerical simulations, as presented in Figure 1. When the three line sources (2D point sources in the figure) are assigned according to (9) and (10), with ${A} = {1}{\text{ mA}}$, indeed, there is a perfect plane wave generated in the desired direction ${\theta}$ that is larger than the critical angle 30° in Figure 1(b) and (c). The critical angles correspond to ${D} / {\lambda}\,{\in}\,{\Bbb{N}}$ (${\theta} = {30}^{\circ}$ gives ${D} / {\lambda} = {2}$), which opens up more propagating modes. However, when ${\theta}\,{≤}\,{30}^{\circ}$, the scattered field is not a single plane wave, as more channels are open, as demonstrated in Figure 1(d). The working angles ${\theta}\,{>}\,{30}^{\circ}$ correspond to ${\left\vert{n}\right\vert}\,{<}\,{D} / {\lambda}\,{<}\,{2}$, which means that three point sources are enough for generating perfect plane waves when only three modes ${\left\vert{n}\right\vert}\,{<}\,{2}$ are propagating modes.
Figure 1. The radiation from three discrete radiators, with the current distribution given by (9) and (10). Note that the supercell shown here has a lateral shift along the x-axis. (a) The simulation setup, where the currents are flowing in the out-of-screen direction. (b)–(d) The scattered electric field patterns for angles ${\theta} = {70}$, 40, and 25°, respectively. PML: perfectly matched layer.
For the case of many propagating modes, the design complicates. For the preceding example of radiation in the direction of ${\theta} = {5}^{\circ}$, we need to satisfy a system of 23 equations. However, we can use a simple approach based on (9) and (10). Similarly, for N discrete sources that are evenly distributed over D, we can assume that the current distribution follows \[{J}{(}{N},{x}{)} = \mathop{\sum}\limits_{{m} = {0}}\limits^{{N}{-}{1}}{A}{e}^{{-}{j}{2}{\pi} \frac{m}{N}}{\delta}{\left({x}{-} \frac{Dm}{N}\right)} \tag{11} \] i.e., with the same amplitude and a linear phase drop of ${2}{\pi}$ over D. Associated with a uniform linear array of isotropic radiators with a linear phase, the radiation pattern of the source current (11) corresponds to the array factor in array theory [30]. Then, the radiation will be perfect if we select enough discrete points N, ensuring that integration in (8) satisfies the orthogonality conditions for all n that correspond to unwanted propagating channels. In general, if the periodicity allows propagation of ${2}{m} + {1}$ plane waves, one needs ${m} + {2}$ discrete sources to excite only one of these plane waves. This is also true for equally spaced segments, and it can be concluded from Table 1 in the next section.
Table 1. The absolute value of integrals (8) for the current distribution (12), with port number n and segment number N.
In actual realizations of antenna arrays, for example, using patch antennas, the radiating currents are not point sources; rather, they are small radiating elements. To analyze this case, we consider the discretization of the current distribution into segments. In each segment, the current is uniform, while for different segments, currents have different phases, still following the linear phase profile in the step-wise fashion. In this case, we can write the current distribution as \[{J}{(}{N},{x}{)} = {Ae}^{{j}{\Phi}{(}{N},{x}{)}}, \quad {\text{with }}{\Phi}{(}{N},{x}{)} = {-}{2}{\pi} \frac{{\text{Floor}}{\left(\frac{xN}{D}\right)}}{N}{-}{\delta}{\phi} \tag{12} \] where A is the complex amplitude, N is the segment number in each period D, ${\Phi}{(}{N},{x}{)}$ is the phase function corresponding to the discretized phase profile, ${\delta}{\phi}$ is a phase shift factor, and the function Floor(x) gives the greatest integer smaller than or equal to the argument x.
With such a segmented current distribution, we can calculate the integral in (8), and the results are listed in Table 1 for ${A} = {1}$ and ${\delta}{\phi} = {0}$. The value will indicate whether there is a plane wave propagating to each port n. As one can see, when ${N} = {3}$, the integration is nonzero for ports ${n} = {-}{5},{-}{2},{1},{4}$, and 7, meaning that if these ports correspond to propagating modes (depending on angle ${\theta}$), there will be plane waves propagating to those ports. As the number of segments N increases, more ports are suppressed, corresponding to zero integrals, and only the port ${n} = {1}$ is excited. Indeed, as N increases, the segmented current distribution is closer to the analytical uniform one with a linear phase. We further note that a change of ${\delta}{\phi}$ in (12) does not change the integration results at all, meaning that the phase can be arbitrarily shifted.
Actually, the point sources and uniform current segments may be replaced by suitable array elements, say, with currents of square shape or cosine shape. We check whether the radiation remains perfect for the same element number per period. Instead of (9) or (12), we assume \[{J}{(}{x}{)} = {AF}{(}{x}{)} + {BF}{\left({x}{-} \frac{D}{3}\right)} + {CF}{\left({x}{-}{2} \frac{D}{3}\right)} \tag{13} \] where ${F}{(}{x}{)}$ is the normalized current distribution function at each array element and A, B, and C are the complex amplitudes of the currents at the array elements. The radiation pattern of one array element with the current distribution ${F}{(}{x}{)}$ is the element pattern. For example, we can assume a square shape or a sine shape \[{F}{(}{x}{)} = \begin{cases}\begin{array}{ll}{1}&{\text{if }}\,{0}\,{<}\,{x}\,{<}\,{w} \\ {0}&{\text{otherwise}}\end{array} \end{cases}, \quad {F}{(}{x}{)} = \begin{cases}\begin{array}{ll}{\sin}{(}{\pi}{x} / {w}{)}&{\text{if }}\,{0}\,{<}\,{x}\,{<}\,{w} \\ {0} &{\text{otherwise}}\end{array} \end{cases} \tag{14} \] where w is the width of the array element and is smaller than ${D}/{3}$. Then, following the same approach as before, i.e., solving (6) with the current in (13) and (14), we find that the condition for radiation only to the desired angle ${(}{n} = {1}{)}$ is when the complex amplitudes are ${B} = {Ae}^{{-}{j}{2}{\pi} / {3}}{ }{\text{ and }}{C} = {Ae}^{{j}{2}{\pi} / {3}}$, which are independent of w. We note that this solution is exactly the same as the solution (10) for point source realization in the “Realization With Small Radiating Elements (Linear Phase Gradient)” section, clearly indicating that the point sources and uniform current sections can be replaced by other suitable radiating elements. This agrees with the principle of pattern multiplication in arrays [30]. The limitation is that the element pattern of these elements should not have a null in the desired direction of radiation.
It is important to note that the radiating elements of periodic arrays do not have to be equally spaced, and the phase shifts between the elements do not have to follow the linear phase advance law. Indeed, instead of (9), let us assume \[{J}{(}{x}{)} = {A}{\delta}{(}{x}{)} + {B}{\delta}{(}{x}{-}{a}{)} + {C}{\delta}{(}{x}{-}{b}{)} \tag{15} \] where ${0}\,{<}\,{a}{\text{ and }}{b}\,{<}\,{D}$. By substituting it into (6) to eliminate the wave radiation in the normal and ${-}{\theta}$ directions (${n} = {0}$ and ${n} = {-}{1}$), we find that the solution of B and C, in terms of A, reads \[{B} = {A} \frac{\sin{\left(\frac{{\pi}{b}}{D}\right)}}{\sin{\left[\frac{{\pi}{(}{b}{-}{a}{)}}{D}\right]}}{e}^{{-}{j}{(}\pi{-}\pi{a} / {D}{)}},\,{C} = {A} \frac{\sin{\left(\frac{{\pi}{a}}{D}\right)}}{\sin{\left[\frac{{\pi}{(}{b}{-}{a}{)}}{D}\right]}}{e}^{\frac{{j}{\pi}{b}}{D}}. \tag{16} \]
It is clear that when the radiating elements are not equally spaced, the required phase distribution does not follow the linear phase gradient law. In addition, the amplitudes of the currents are also changed. For example, when we set ${a} = {D} / {6}$ and ${b} = {2}{D} / {3}$, as in Figure 2(a), the required complex amplitudes of the second and third sources read ${B} = {(}{-}{3} / {4}{-}{j}\sqrt{3} / {4}{)}{A}$ and ${C} = {(}{-}{1} / {4} + {j} \sqrt{3} / {4}{)}{A}$, and with this setting, we can still get the desired single-wave radiation, as illustrated in Figure 2(a).
Figure 2. The radiation from ${N} = {3}$ discrete current sources that are arbitrarily positioned in one period (with a lateral shift along x). Each subfigure shows the simulation setup and radiated electric field pattern for ${\theta} = {40}^{\circ}$. The assumed values are (a) ${a} = {D} / {6}$ and ${b} = {2}{D} / {3}$, (b) ${b} = {2}{a} = {D} / {4}$, and (c) ${b} = {2}{a} = {D} / {10}$. The complex amplitudes B and C are obtained from (16), with ${A} = {1}{\text{ mA}}$.
If we set ${b} = {2}{a}$ in (16), we find the required complex amplitudes as ${B} = {2}{A}{\cos}{(}{{\pi}{a}} / {D}{)}{e}^{{-}{j}{(}{\pi}{-}\pi{a} / {D}{)}}$ and ${C} = {Ae}^{{j}2{{\pi}{a}} / {D}}$. If we further make a much smaller than D so that the three sources are clustered, the phase quickly varies among the three sources; i.e., the first and third sources have small phases of zero and ${2}{\pi}{a} / {D}$ ${(}{a} \ll {D}{)}$, while the middle source has a phase that is close to ${-}{\pi}$. Moreover, the amplitude of the middle source is close to two, while the amplitudes of the other two are one. As examples, in Figure 2(b) and (c), we perform two sets of simulations, with ${b} = {2}{a} = {D} / {4}$ and ${b} = {2}{a} = {D} / {10}$, respectively. The results show that these clustered sources can still generate the desired radiation without any unwanted scatterings. Anomalous reflectors based on periodic arrays of a few small scatterers in each period are often called metagratings [18].
We note that it is straightforward to analyze similarly the case of small tilt angles with more discrete point sources. Obviously, the realization of such fast variations of the reflection phase in reflectarrays is challenging. Thus, for the case when many plane waves can propagate, it is preferred to use equal spacing and a linear phase gradient.
In fact, these conclusions hold true not only for small point-like radiating elements but also for arrays of radiating segments. For example, if we assume the current distribution function as \[{J}{(}{x}{)} = {AF}{(}{x}{)} + {BF}{(}{x}{-}{a}{)} + {CF}{(}{x}{-}{b}{)} \tag{17} \] similar to (15) for the point sources, the required amplitudes are also found to be given by (16) and independent of the segment width w. For infinite arrays, the performance should not depend on the current distribution function and element pattern if the amplitudes A, B, and C are set properly and the element pattern has no null in the desired radiation direction.
The assumption of a real-valued function ${F}{(}{x}{)}$ corresponds to the case of small resonant antennas as array elements. This is the case when the current distribution over a unit element is a standing wave, and the current phase is uniform over the unit cell. Also, arrays of slots in metal sheets can be modeled in this way. However, it is possible to assume that the current distribution is a complex-valued function, and the conclusions will not change.
It is obvious that these models assume that the current distribution over each array element is fixed and independent from the excitation of this element and all the other elements in the array. In the case of reflectarrays and metasurfaces, this is equivalent to the assumption that the distribution of near fields in the vicinity of the array does not depend on the incidence angle.
To give a clear overview of the different models of the diffraction grating approach, in Figure 3, we summarize the different current distribution functions discussed in the preceding. As we can see in Figure 3(a) and (b), the equally spaced element realizations all have the linear phase profile, while for the nonequally spaced element design, the phase is not a linear ${2}{\pi}$ phase dropping over the period, and the amplitude is not a constant in Figure 3(c).
Figure 3. The required current distribution functions (amplitude and phase) for the diffraction grating approach. (a) The ideal current ${J}{(}{x}{)} = {Ae}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$ in the “The Optimal Current Distribution” section, the required currents for three radiators is given in (9) and (10) for Figure 1 [or, equivalently, (11), with ${N} = {3}$] in the “Realization With Small Radiating Elements (Linear Phase Gradient)” section, and that for five segments with the current in (12), with ${N} = {5}$ and ${\delta}{\phi} = {0}$ in the “Realization With Radiating Segments (Linear Phase Gradient)” section. (b) Currents with the form of (13) and (14), with patches showing the square shape and sine shape ${F}{(}{x}{)}$. (c) The nonequally spaced element realizations in the “General Periodic Current Distributions (Metagratings)” section, using (15)–(17). Cases 1 and 2 correspond to the ones in Figure 2(a) and (b). Case 3 is the “patch” realization in (17), with the sine shape current in (14) and ${a} = {2}{D} / {5}$, ${b} = {3}{D} / {4}$, and ${w} = {D} / {10}$. Here, ${A} = {1}$ is used in all figures.
In the preceding theory, we started from stating that current distributions for launching a single plane wave in a desired direction should be periodic functions, with the period related to the period of the radiated plane wave. One reason for that assumption is that the excitation current and desired field must be in a phase synchronism. The other reason is that by fixing the period as defined by the desired reflection angle or angles, one makes sure that the reflector can create a plane wave exactly in the desired direction. The design goal is then to properly distribute the reflected power among the allowed “open channels,” minimizing scattering into all directions except the desired one. From the practical point of view, the design of periodic structures reduces to the design of only one period, which is a great simplification.
However, the current distribution does not have to be a periodic function. Conditions (5) and (6) as well as (7) and (8) can be considered conditions on the spatial Fourier transforms of the current distributions if we assume that ${J}{(}{x}{)}$ is not necessarily periodic and extend the integration over the whole x-axis. Actually, Floquet periodic functions of the form \[{J}{(}{x}{)} = \mathop{\sum}\limits_{{m} = {-}{\infty}}\limits^{\infty}{A}_{m}{e}^{{-}{j}{\left({k}_{0}\,{\sin}\,{\theta} + \frac{{2}{\pi}{m}}{d}\right)}{x}} \tag{18} \] can satisfy the required conditions for launching only one plane wave in the ${\theta}$ direction. Formula (18) corresponds to an ideal current profile ${e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$ that is modulated by an arbitrary periodic function ${F}{(}{x}{)}$ with period ${d}{:}{J}{(}{x}{)} = {F}{(}{x}{)}{e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$. Expanding ${F}{(}{x}{)}$ into a Fourier series, we arrive at (18). Really, the ${m} = {0}$ term is the ideal current profile ${e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}$ (the uniform amplitude and linear phase gradient) that launches a single plane wave in the ${\theta}$ direction. Indeed, this function is obviously orthogonal to all other plane waves as eigenmodes of free space because \[ \mathop{\int}\nolimits_{{-}\infty}\nolimits^{\infty}{e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}}{e}^{{jk}_{0}{x}\,{\sin}\,{\theta'}}{dx} = {0}\,\,{\text{for all }}{\theta'}\,{≠}\,{\theta}{.} \tag{19} \]
Thus, if we ensure that all other members of the series (18) have this property, currents of the form (18) will excite only one propagating plane wave. To ensure this property, we demand that ${\left\vert{k}_{0}\,{\sin}\,{\theta} + {{2}{\pi}{m}} / {d}\right\vert}\,{>}\,{k}_{0}$ for all ${m}\,{≠}\,{0}$. In this case, the tangential components of the wave vectors of all harmonics with ${m}\,{≠}\,{0}$ are larger than ${k}_{0}\,{\sin}\,{\theta'}$ for all ${\theta'}$, and the orthogonality condition (19) is satisfied. It is convenient to rewrite this condition as \[{\left\vert{\sin}\,{\theta} + \frac{\lambda}{d}{m}\right\vert}\,{>}\,{1}. \tag{20} \]
Then, the condition for the required repeating period d of the current elements can be analyzed. For ${\theta}\,{\rightarrow}\,{0}$, the condition is satisfied if ${d}\,{<}\,{\lambda}$ (the most “dangerous” term is, obviously, the term ${m} = {-}{1}$). For ${\theta}\,{\rightarrow}\,{\pi} / {2}$, the condition is true if ${d}\,{<}\,{\lambda} / {2}$. That is why for scanning phased arrays, the period d is conventionally chosen to be equal to ${\lambda} / {2}$: this choice ensures that for any scan angle, no parasitic diffraction lobes will be created for infinite scanning phased arrays.
The case when the current distribution is not a periodic function is equivalent to the limit case when the period is infinity. As we see, when the period approaches infinity, although there are infinitely many allowed plane wave propagation directions, it is possible to excite only one propagating plane wave by properly phasing the currents at the phased-array elements.
If the radiation direction is fixed to angle ${\theta}$ and no scan is required, by choosing ${m} = {-}{1}$ in (20), the period d of the general periodic structure can be set for that specific angle as \[{d}\,{<}\,{\lambda} / {(}{1} + {\sin}\,{\theta}{)}. \tag{21} \]
This is recognized as the condition for avoiding grating lobes in arrays [30]. For instance, for our example of ${\theta} = {70}^{\circ}$, we need ${d}\,{<}\,{0.5155}{\lambda}$. This is also verified from the results of Figure 1:
The periodic “modulation function” ${F}{(}{x}{)}$ can be a complex-valued function. An important special case is the form ${f}{(}{x'}{)}{e}^{{jk}_{0}{x'}\,{\sin}\,{\theta}}$, defined in the region ${-}{d} / {2}\,{<}\,{x'}\,{<}\,{d} / {2}$. Periodically repeating this modulation, we have the current in the Mth period (where ${x} = {Md} + {x'}{)}$ as \begin{align*}{J}{(}{x}{)} & = {F}{(}{x}{)}{e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}} = {f}{(}{x'}{)}{e}^{{jk}_{0}{(}{x}{-}{Md}{)}\,{\sin}\,{\theta}}{e}^{{-}{jk}_{0}{x}\,{\sin}\,{\theta}} \\ & = {f}{(}{x'}{)}{e}^{{-}{jMdk}_{0}\,{\sin}\,{\theta}}{.} \tag{22} \end{align*}
This is the phased-array design approach. The structure is formed by an array of unit cells with size d (the array elements). The unit cells have the same current distribution function ${f}{(}{x'}{)}$, but the phase of their current linearly varies from cell to cell. The continuous linear phase profile is replaced by a discrete one. The most common assumption is the real-valued function ${f}{(}{x'}{)}$, which corresponds to arrays of, e.g., resonant dipoles, resonant patches, and horn antennas.
We see that the phased-array approach is quite different from the conventional approach to the design of diffraction gratings and metasurfaces for reflection control. Here, we start from fixing the unit cell size d, usually at ${d} = {\lambda} / {2}$. Assuming the same current distribution function over each unit cell (usually the resonant mode of an antenna array element), we adjust the phases of the unit cells to create constructive interference in the desired direction.
Figure 4 illustrates and compares these two approaches. Figure 4(a) and (b) correspond to the diffraction grating approach for designing anomalous reflectors. In this case, we select a proper period ${D} = {\lambda} / {\sin}\,{\theta}$ and then find a suitable current distribution (the solution is not unique). For the case of the discretized point radiators in Figure 4(b), for launching the wave to 70°, where we have three propagating Floquet modes, we need three discrete sources in each period.
Figure 4. The (a) metasurface design approach (effectively continuous periodic current distribution), (b) metagrating design approach (a few small scatterers in each period), and (c) phased-array design approach (in general, aperiodic current distribution).
In Figure 4(c), which describes the phased-array approach, we have an array of identical “patches” spaced by ${\lambda} / {2}$. We assume that all the elements have the same amplitude pattern (a cosine shape is shown) and that the phase is uniform over each unit cell. Then, we feed this array with a linearly varying phase source so that the phases vary from unit to unit as \[{\Phi}{(}{n}{)} = {k}_{0}{x}{(}{n}{)}\,{\sin}\,{\theta} = \frac{{2}{\pi}}{\lambda}{n} \frac{\lambda}{2}\,{\sin}\,{\theta} = {\pi}{n}\,{\sin}\,{\theta} \tag{23} \] with n being the unit index, as shown in the figure. Importantly, we note that this current distribution is, in general, not a periodic function. The periodicity condition reads \[{n}{\pi}\,{\sin}\,{\theta} = {2}{\pi}{m} \tag{24} \] where m is the number of unit cells that will have the same phase as cell number zero. Obviously, this condition can be satisfied only if ${\sin}\,{\theta}$ is a rational number.
Figure 5 gives an example of the active phased array for generating radiation toward ${\theta} = {70}^{\circ}$. Due to the nonperiodicity of the current distribution, we consider a finite phased array made of ${N} = {40}$ “patches” with width ${w} = {\lambda} / {4}$ and spacing ${d} = {\lambda} / {2}$. The current of this finite phased array is set to follow (22), with ${f}{(}{x'}{)}$ being a cosine shape [the inset of Figure 5(a)]. It indeed radiates in the desired direction, as shown by the electric field pattern in Figure 5(b) and the normalized far field (the black solid curve) in Figure 5(c). Due to the finiteness of the array, there are side lobes but no other main beams. When increasing the radiating aperture size by adding more “patches,” the pattern tends to a delta function. This is illustrated by the red dashed curve in Figure 5(c) for ${N} = {100}$.
Figure 5. The radiation from a finite phased array consisting of ${N} = {40}$ “patches,” with ${w} = {\lambda} / {4}$ and ${d} = {\lambda} / {2}$. (a) The simulation setup. The inset shows the current distribution. (b) The simulated electric field pattern for the ${N} = {40}$ phased array. (c) The normalized far field (electric field). The black solid (red dashed) line corresponds to the phased array consisting of ${N} = {40}$ ${(}{N} = {100}{)}$ “patches.”
If we choose the element spacing in the phased array equal to ${D} / {3} = {(}{1} / {3}{)}{\lambda}{/}\,{\sin}\,{\theta}$, the phase array becomes periodic, and it is equivalent to the diffraction grating picture for ${\theta}\,{>}\,{30}^{\circ}$. Hence, for active arrays, the difference between diffraction grating and phased-array approaches is of a practical nature in essence. In diffraction gratings, a periodic physical configuration stipulated by the incidence and reflection angles is chosen so that the design domain is reduced to a single supercell. In phased arrays, the cell size (i.e., the element spacing) is determined a priori (usually ${\lambda} / {2}{)}$ based on the required beam scanning capability. The functionality of creating one propagating wave is performed by element excitations following a spatially linear phase profile.
The fundamental difference between the two design approaches is in realizing scanning capability. For diffraction gratings, the period of the array should be adjusted for each desired reflection angle, which, in practice, requires enough small subwavelength unit cells. For phased arrays, the period is fixed, which, in principle, allows the use of ${\lambda} / {2}$-sized unit cells. In this case, the required current distribution over the plane is, in general, aperiodic. This issue does not create practical problems for active phased arrays, which are all of a transmitting type. However, the realization of the required linear phase profile for wide-angle scanning by adjusting the reactive bulk loads of passive reflectarray antenna elements is rather difficult.
Let us stress that there is a very important simplifying assumption in the root of the phased-array antenna theory. Namely, it is assumed that the current distribution over one array element does not depend on the currents on the other elements. The current density on the first element can be written as ${I}_{0}{f}{(}{x}{)}$, where ${f}{(}{x}{)} = {0}$ at ${x}\,{<}\,{0}$, with ${x}\,{>}\,{d}$, and ${I}_{0}$ is the current amplitude fixed by external sources. Only under this assumption, the current of the Mth element is ${I}_{M}{f}{(}{x}{-}{Md}{)}$, where f is the same function. This assumption is an approximation. It holds very well for active resonant antenna elements (when the current distribution is approximately fixed by the resonant mode) and arrays of horn antennas (where the current, or aperture field, is very close to the fundamental mode of the horn). We stress that the current amplitudes ${I}_{M}$ depend on external voltages applied to all elements. By knowing the impedance matrix of the array, we can always find such voltages to realize the desired distribution of current amplitudes over the antenna plane.
Finally, let us discuss the role of diffraction grating theory in understanding phased arrays. Although the current and field distribution over a reflectarray aperture is not periodic, the array is “geometrically periodic,” as it is formed by identical antennas separated by a ${\lambda} / {2}$ distance. Moreover, the current distribution is Floquet periodic, defined by a periodic modulation function ${F}{(}{x}{)}$. Another aspect is that an array without periodicity of loads and induced current can be treated as a periodic array with an infinite period ${(}{D} = {\infty}{)}$. In this case, there are infinitely many propagating “Floquet harmonics” of the diffraction gratings theory. Generally speaking, the diffraction grating method has limited applicability; however, some general conclusions remain valid.
In contrast to active antenna arrays, where each antenna element is fed by an external controllable source, currents on elements of reflectarrays and metasurfaces are excited by incident waves (and fields scattered from all other array elements). The usual design aim is to synthesize the induced current distribution ${J}{(}{x}{)}$ to create a constant amplitude linear phase profile. However, unlike in the active phased-array case, we do not have full control over the magnitude and phase of ${J}{(}{x}{)}$ by using a passive surface subject for the illumination of incident waves. The coupling of array elements and interference of incident and reflected waves need to be carefully taken into account.
Let us first consider the design approach based on the diffraction grating theory. That is, we fix the array period as defined by the incidence angle and desired reflection direction and then design a surface to perform the required operation. In the following discussion, we consider the same example functionality as in the examples in the preceding: the reflection of a normally incident plane wave into a plane wave in an arbitrary direction. We can consider two cases:
Neither case is satisfactory. Researchers work on finding other solutions that use only passive (as small a dissipation as possible) array elements. The only possibility is to allow the excitation of other field modes beside the only plane wave in the desired direction. If all these other modes are evanescent Floquet harmonics, they do not propagate into the far zone, and the perfect performance of the device is not compromised. There can be several ways to approach this task:
Let us next discuss the design approach based on the phased-array theory. That is, we fix the unit cell size, usually to ${\lambda} / {2}$. We position geometrically identical passive antenna elements in each cell and find such loads connected to the elements so that the induced current distribution has the required linear phase profile [with a periodic modulation, as in (22)].
Under the simplifying assumption that the current distribution over each antenna element is the same for any incident field distribution and any loads connected to all the elements, we can use the impedance matrix method to find the load impedances for realizing the desired phase distribution. Unfortunately, we run into the same problem as in the diffraction grating design: these loads have active/lossy behavior [35]. Thus, one needs to impose an additional constraint on the load impedances (zero real parts) and design some optimization procedure for finding reactive loads that approximate the required amplitude and current distribution in the best possible way. Basically, we end up with the same problem of engineering near fields so that the power is properly channeled from “virtually lossy” to “virtually active” antenna elements (engineering spatial dispersion). It is not clear if just one control element per ${\lambda} / {2}$ will be enough to reach the goal, and we may need to use subwavelength elements, similar to the metasurface scenario.
The main difficulty in the phased-array (reflectarray) approach is that the array of antennas (including the loads) is no longer a periodic structure. This makes the number of ports for connecting load impedances infinite, and all the loads are, in general, different. In contrast, the periodic metasurface approach requires the design and optimization of only one period, and usually only a few parameters need to be optimized. We stress that this is not a problem in a conventional practical phased-array antenna in either transmission or receiving mode because it corresponds to a transmissive device. A transmissive device does not have the interference problem between the incident and scattered waves, which is the reason for the active/lossy nature of the required surface impedance of an ideal boundary anomalously performing reflection.
In recent literature on metasurfaces for the control of reflected waves, there have been many publications on the suggested use of reconfigurable anomalous reflectors for the engineering and optimization of the propagation environment. The vast majority of these works are based on the locally periodic approximation of the array response. That is, a locally defined reflection coefficient is used for calculating the reflected fields, and it is assumed that the fields can be somehow controlled by changing the parameters of the unit cell at each position. This is the conventional reflectarray antenna design assumption. It is known that this local design gives acceptable performance for moderate tilt angles (moderate deviations from the reflection law for uniform mirrors). However, for envisaged applications, it is required that beams can be directed in any desired direction, which needs more advanced designs. The majority of the current research in this field uses variations of periodic metasurfaces (diffraction grating design methods). Here, impressive results have been achieved in designing and realizing anomalous reflectors for fixed sets of incidence and reflection angles. The main problem is the realization of electrical reconfigurability: in this design approach, the period of the array is defined by the incidence and reflection angles, which demands the change of the array period by adjusting the tunable components of small unit cells.
On the other hand, the phased-array approach uses a set of periodically arranged antennas, and the geometrical period is fixed for all scan angles. However, the distribution of controllable loads is not periodic in this case, which requires the global optimization of the whole array for each scan angle. Moreover, it is not known if the optimization of loads of a conventional ${\lambda} / {2}$-spaced array can lead to acceptable performance for the requirement of an extremely wide scan angle range. Approaches for optimizing the reactive loads of reflectarray antennas can be found, e.g., in [35] and [36], but it appears that more research on this design approach is needed.
This work was supported, in part, by the National Natural Science Foundation of China, under grant 12274339, the European Integrated Training Network Meta Wireless and the U.S. Army Research Office, under grant W911NF-19-2-0244. Fu Liu also acknowledges financial support from Xi’an Jiaotong University and Shaanxi Province, China.
Fu Liu (fu.liu@xjtu.edu.cn) is with the School of Electronic Science and Engineering, Faculty of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China. His research interests include complex artificial electromagnetic materials (metamaterials and metasurfaces), transformation optics, and wireless power transfer. He is a Member of IEEE.
Do-Hoon Kwon (dhkwon@umass.edu) is with the Department of Electrical and Computer Engineering, University of Massachusetts Amherst, Amherst, MA 01003 USA. His research interests include antenna and array bandwidth properties, microwave metamaterials and metasurfaces, and cloaking. He is a Senior Member of IEEE.
Sergei Tretyakov (sergei.tretyakov@aalto.fi) is with the Department of Electronics and Nanoengineering, Aalto University, 02150 Espoo, Finland. His research interests include electromagnetic field theory, complex media electromagnetics, metamaterials, and microwave engineering. He is a Fellow of IEEE.
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