Flexible Generation and Manipulation of Microwave Bottle Beam Using a Reconfigurable Metamirror

Bottle beams have drawn considerable research interest due to their attractive features and applications in beam trapping. Herein, the flexible generation and modulation of bottle beams utilizing a reconfigurable metamirror over a non‐negligible frequency band ranging from 8.5 to 9.5 GHz is investigated. A varactor diode is embedded into each meta‐atom on the metamirror enabling the achievement of a nearly 2π phase coverage by varying the external bias voltage. Based on the caustic theory, several configurations of bottle beam with distinct radius a are realized as proof‐of‐concept validation. Besides, the flexible modulation of the operating frequency is analyzed by judiciously controlling the bias voltages applied to the metasurface. The experimental results are in good agreement with those obtained from numerical simulations. Such a flexible bottle beam generator shows good potential in the microwave regime for near‐field applications, such as imaging and power transmission.


Introduction
A bottle beam is a particular class of hollow beams in which a high-intensity region surrounds a localized minimum field intensity region.Due to such exotic property, many efforts have been devoted to bottle beams in optics and acoustic regimes, which serve as optical tweezers [1] and optical trappers. [2,3]A variety of methods has been proposed to implement bottle beam generation.[6] Meanwhile, a single beam can be exploited to generate a bottle beam using additional functional components, including holograms [7] and spatial light modulators. [8]However, the aforementioned methods suffer from bulky size, complicated design, and low efficiency.
The emergence of metasurface provides a novel route for bottle beam generation.Structured metasurfaces are capable of manipulating the phase, amplitude, and polarization of electromagnetic (EM) illuminations within a subwavelength scale.[16] However, most of them are designed as static platforms for fixed functionalities.[22] Besides, linearly and circularly polarization channels can be regulated simultaneously by combining geometric and propagation phases. [23,24][35] Among them, metasurfaces based on deformable materials are limited by the response speed.Microelectromechanical systems present the drawbacks of low switching speed and complex design.p-i-n diode-loaded metasurfaces are able to give a response in real time but realize only a limited number of discrete phase values.By contrast, varactor diode-loaded metasurfaces can provide a quasicontinuous 2π phase coverage within a wide frequency band in real time.DOI: 10.1002/adpr.202300156Bottle beams have drawn considerable research interest due to their attractive features and applications in beam trapping.Herein, the flexible generation and modulation of bottle beams utilizing a reconfigurable metamirror over a nonnegligible frequency band ranging from 8.5 to 9.5 GHz is investigated.A varactor diode is embedded into each meta-atom on the metamirror enabling the achievement of a nearly 2π phase coverage by varying the external bias voltage.Based on the caustic theory, several configurations of bottle beam with distinct radius a are realized as proof-of-concept validation.Besides, the flexible modulation of the operating frequency is analyzed by judiciously controlling the bias voltages applied to the metasurface.The experimental results are in good agreement with those obtained from numerical simulations.Such a flexible bottle beam generator shows good potential in the microwave regime for near-field applications, such as imaging and power transmission.
In this article, we exploit a reconfigurable metamirror to generate and dynamically modulate bottle beams.A varactor diode is embedded into each meta-atom to control the phase response and provide a nearly 2π phase coverage from 8.5 to 9.5 GHz.The phase profile of different bottle beam generators is imitated from the caustic theory.To investigate the bottle beam generation, both numerical simulations and experimental measurements are performed on the meta-mirror.Four phase profiles are implemented to generate bottle beams with different sizes of the localized minimum intensity (dark) field region parameter.Meanwhile, using the dynamic phase-compensation mechanism capability of the metamirror, the flexible modulation of operating frequency is also verified.The measured results, in good agreement with the numerical ones, reveal that the proposed metamirror is able to generate bottle beams with various configurations, paving the way to potential applications in near-field imaging and wireless power transmission.

Design Principle of Bottle Beam
The desired phase profile applied to metasurface to obtain bottle beam follows the caustic theory and is demonstrated in Figure 1.Taking a 1D array as example, we suppose the trajectory of a circular bottle beam is f(z), corresponding to a desired phase profile φ(x) that will generate the bottle beam as a caustic, which is an envelope of a set of tangent lines.Let us consider the intersection points of arbitrary tangent lines with x-axis and with trajectory (x, 0) and (x 0 , z 0 ), respectively.For arbitrary intersection (x, 0), the caustic of this point can be expressed via the slope of the tangent line θ, with tanθ ¼ Àf 0 ðz 0 Þ ¼ Àdðf ðzÞÞ=dz.
The desired phase profile φ(x) is obtained by integrating the following derivation formula where k is the wave number.Besides, the intersection of the tangent line with the x-axis is calculated by , so that the tangent can be parameterized using Based on the above-mentioned derivation, the curve trajectory x = f(z) can be regarded as an envelope to a set of its tangent lines passing through the x-axis.Thus, the analytical expression of desired phase profile can be obtained from the caustic theory and the geometrical features. [36,37]The bottle beam exhibits a circular low-intensity field region following the trajectory function , where the circular radius is a and the central point is (0, a).Therefore, the phase profile along x-axis on the metamirror can be expressed as Hence, by exploiting the above phase profile, the metamirror is able to convert an incident plane wave illumination into a bottle beam with a circular low-intensity field region, as illustrated in Figure 1.The waves propagate around the dark region and then focus beyond coordinate (0, 2a).For a 2D bottle beam implementation, the phase profile is expanded as where the distance between each meta-atom and the source point set at (0, 0) is r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , with (x, y) being the coordinates of the meta-atom.

Metasurface Design and Characterization
As displayed in Figure 2, the proposed bottle beam generator is implemented by a reconfigurable metamirror driven by an STM32 electronic module.Under an x-polarized EM wave incidence, the metasurface is able to generate a bottle beam.Different radii a = 25, 30, 35, and 40 mm are considered by dynamically adjusting the external direct current (DC) bias voltage applied to each meta-atom.Besides, the phase profiles are also adjusted to maintain the bottle beam features over a non-narrow frequency range.
To achieve such a dynamically-controlled metasurface, the meta-atom embedded with electronically adjustable component is designed, as depicted in Figure 3a.The proposed meta-atom consists of three-layered metallic patches separated by two-layered similar dielectric substrates with relative permittivity ε r = 3.5 and loss tangent tan δ = 0.0037.A MACOM MAVR-011020-1411 varactor diode is embedded into the H-shaped metallic patch on the top layer.In the simulations, the varactor diode is set as RLC series circuit composed of parasitic resistance and inductance and a variable capacitance. [38]The parasitic resistance and inductance are evaluated as 5 Ω and 0.4 nH, respectively.A change in applied DC bias voltage of the varactor diode enables to precisely address the equivalent capacitance of the meta-atom.In the middle layer, a continuous ground plane is designed as a reflective plane, which is further exploited to provide an inductive response so that the metaatom serves as an RLC resonator.
Figure 3b displays the bias network layout of an individual meta-atom.It consists of 15 metallic lines printed on the bottom layer to supply positive DC voltage to the isolated arm of the H-shaped patch via 15 metallic vias.Among them, only one via, represented by the solid circle, is a through one and the rest, represented by hollow circles, are blind vias.The design of blind vias ensures a negligible phase change at different positions of the metasurface.Since the location of each meta-atom is different on the metasurface, the location of the through via connecting the power supply to the varactor diode also changes, which further influences the EM response of the latter meta-atom.As such, the use of blind vias allows to suppress this influence so as to have a negligible phase change when the position of the meta-atom changes on the metasurface.The continuous arm of H-shaped patch is connected to the DC ground.HFSS simulations of the elementary meta-atom with applied periodic boundaries are carried out to investigate the reflection characteristics.Figure 3c,d depicts the simulated reflection coefficients from 7 to 11 GHz.It can be observed that the resonance frequency shifts from below 8 GHz to above 10 GHz when decreasing the capacitance value of the loaded varactor diode.The phase responses show that the proposed meta-atom can provide quasi-full 360°phase coverage from 8.5 to 9.5 GHz by varying the capacitance.
For the construction of the full meta-mirror, four sub-arrays of 15 Â 15 meta-atoms are arranged in a 2 Â 2 matrix to obtain a 30 Â 30 elements array, where each meta-atom can be independently addressed.Figure 3e depicts the details of the printed circuit board (PCB) layout of the metamirror.The blue lines are located on the top face and represent the H-shaped patches with the ground lines connected to their continuous arms.The positive bias lines on the bottom layer, represented by the orange lines, are connected to an STM32 electronic board by flexible printed circuit (FPC) connectors.The middle layer is a quasicontinuous ground plane with 15 ring-shaped slots used for the isolation between the DC voltage and middle metallic plane.The incident wave illuminating the unit cell is reflected by the middle plane without interacting with the feed part.
The fabricated prototype of the reconfigurable metamirror including 30 Â 30 meta-atoms, [35] illustrated in Figure 3f, is utilized to validate the bottle beam generation.In a first step, a lookup table of reflection properties versus bias voltage is acquired from the characterization of the metamirror.The same DC  voltage is applied to each meta-atom of the metamirror and the measured reflection amplitude and phase are acquired with different DC bias voltages from 7 to 11 GHz, as plotted in Figure 3g, h.The measured results reveal that the metamirror exhibits a quasi-full phase coverage (nearly 340°) ranging from 8.5 to 9.5 GHz, which agrees well with the simulated ones shown in Figure 3c,d.

Numerical and Experimental Results
In the following, we consider the generation of bottle beams using the 2D phase profile in Equation (3).Four different radii a (a = 25, 30, 35, and 40 mm) are considered at 9 GHz.The desired phase profile that needs to be implemented in the reconfigurable metamirror for different scenarios is calculated and presented in Figure 4a.Based on the desired phase profiles φ(x), the reflective electric field is reconstructed by the scalar Rayleigh-Sommerfeld diffraction integral, [39] as Uðξ, 0, zÞ ¼ À 1 2π where Uðξ, 0, zÞ represents the complex electric field at point ðξ, 0, zÞ and R is the distance between the source point (x, 0, 0) and each field point.Furthermore, experiments are conducted on the metamirror with implemented desired phase profiles.The fabricated prototype is experimentally measured using the near-field scanning setup in an anechoic chamber, as schematically illustrated in Figure 5a.A 2-18 GHz broadband horn antenna is employed to emit the EM waves illuminating the metasurface.An all-dielectric fiber-optic active antenna probe is used as receiver to collect both the amplitude and phase of the electric field in the scanning region.The probe, having a very small all-dielectric head of 6.6 mm Â 6.6 mm, [40] has no influence on the illumination field.Both the horn antenna and monopole probe are connected to an Agilent vector network analyzer (VNA).The probe is fixed on computer-controlled displacement rails to scan the field over a desired scanning region.The experimental tests are performed in xoz-, yoz-, and xoy-planes.Two steps are required for the measurement of the reflection response.A reference measurement is first done by measuring the incident field generated by the horn antenna without the presence of the metasurface.The other step consists of measuring the electric field in the presence of the metasurface, which involves collecting both the incident field and the field reflected by the metasurface.The post-processing procedure, illustrated in Figure 5b, is then performed, such that only the reflected electric field is extracted by subtracting the incident field collected in the reference measurement from the total field (incident and reflected) collected in the second measurement.
The measured results in the xoz-, yoz-, and xoy-planes are depicted in Figure 6.As expected, the measured results for different a at 9 GHz presented in Figure 6b show a bottle beam after the illuminating wave is reflected by the metamirror, which is in good agreement with the theoretical calculations.The measured radii a are 22, 26, 28, and 32 mm, which are consistent with the calculated ones.The results illustrate that the reflected fields can be focused to any desired bottle beam when a judiciously calculated phase distribution is applied.It is necessary to mention that the radii a in calculations and experiments deviate slightly away from the preset ones mainly due to the limited aperture size of the metamirror.
Moreover, the flexible modulation of the operating frequency of the bottle beam is achieved by judiciously controlling the phase profiles via the bias voltages applied to the metamirror.Hence, the experimental measurements are also performed at 8.5 and 9.5 GHz and the electric field magnitude distributions are plotted in Figure 6a,c.A 2D bottle beam with different radius a can be observed at each frequency, which reveals that the reconfigurable metamirror is able to achieve bottle beam in a non-narrow frequency band using different phase profiles.The electric field distributions in the xoy-plane are also presented in Figure 6, where low-intensity and high-intensity fields are measured at z = z 1 and z = z 2 , respectively.The different radii a of the generated bottle beams generally follow the variation trend of the theoretical ones.
To further investigate the performance of generated bottle beams, Figure 7 illustrates the statistical plot of the variation of the focal length with radius and frequency, where the dashed patterns denote the simulated data and solid blocks denote the measured data.The results indicate that the focal length increases with the radius a at each considered frequency.For a fixed radius, while the simulated focal length does not change within the considered 8.5-9.5 GHz frequency band, the measured data shows a slight increase in focal length, which can be due to the fabrication and measurement imperfections.To quantitatively evaluate the performances of the proposed metamirror for bottle beam generation and modulation, the total efficiency η is calculated as where P out is the reflected power and P in is the incident power, respectively, and S represents the size of the detection area.Thus, the calculated measurement efficiency for all bottle beam generation is around 60% for a = 25 mm, 63% for a = 30 mm, 77% for a = 35 mm, and 75% for a = 40 mm.

Conclusion
In summary, we elaborate on an electronically controlled reconfigurable metamirror to tailor arbitrary bottle beams at microwave frequencies.By loading varactor diodes in the meta-atoms, the proposed meta-mirror can provide a nearly 2π phase coverage from 8.5 to 9.5 GHz.Based on the caustic theory, the corresponding phase profiles are calculated and applied to realize distinct bottle beams with radii a = 25, 30, 35, and 40 mm.The circular low-intensity region followed by a high-intensity focusing region along the propagation path can be clearly observed from the electric field magnitude distributions, which indicates that bottle beam is generated successfully.
To exploit the flexible beam manipulation, each bottle beam configuration is analyzed at different operating frequencies by adjusting the bias voltages applied to the metasurface.The experimental results are in good agreement with those obtained from the numerical calculations.Such a flexible bottle beam generator has good potentials in the microwave regime particularly for near-field applications, such as imaging and power transmission.

Figure 1 .
Figure 1.Schematic illustration of the trajectory of the bottle beam x = f(z) (blue line) and its tangent lines (red line).

Figure 2 .
Figure 2. Schematic illustration of bottle beam generation using a reconfigurable metamirror.

Figure 3 .
Figure 3. Features and characterization of the reconfigurable metamirror.a) Constituting meta-atom embedding a voltage-controlled varactor diode.The optimized geometric parameters are as follows: p = 6 mm, w = 5.8 mm, l 1 = 3.6 mm, h 1 = 1 mm, h 2 = 0.25 mm, l 2 = 0.25 mm, l 3 = 1.1 mm, d 1 = 0.73 mm, and d 2 = 0.16 mm.b) Bias layer of the meta-atom, where the solid circle represents the through via and the hollow ones represent blind vias.c) Simulated reflection magnitude.d) Simulated reflection phase.e) Layout of the top (shown in blue) and bottom (shown in orange) faces of the metamirror.f ) Photograph of the fabricated sample.g) Measured reflection phase.h) Measured reflection magnitude.
Figure 4b presents the normalized electric field magnitude distributions of the generated bottle beams with a = 25, 30, 35, and 40 mm in the xozand xoy-planes.From the calculated results in the xoz-plane, we clearly observe that the reflected field from the metasurface forms a low-energy region close to the metasurface and then a focusing region with high-intensity electric field, which verifies the main feature of bottle beams.To provide an illustrative view on the generated 2D bottle beam, field distributions at different distances above the metamiror in the xoy-plane are displayed.All results in the z = z 1 plane show a clear ring and a focusing spot can be

Figure 4 .
Figure 4. Generation of bottle beams at 9 GHz.a) The phase profiles are calculated for different configurations of bottle beams (a = 25, 30, 35, and 40 mm).b) The calculated normalized electric field magnitude distributions in the xoz plane show the energy distributions for the designed bottle beams.The electric field distributions in the xoy plane highlight a circular low-intensity region at z = z 1 and a focusing region at z = z 2 , where z 1 = a, z 2 = 2a þ 20 mm.

Figure 5 .
Figure 5. a) Schematic illustration of the experimental setup.The inset shows the all-dielectric fiber-optic active antenna, used as field sensing probe.b) Representation of the procedure process used to extract only the reflected field from the metasurface.The incident field from illuminating horn antenna is subtracted from the total (incident and reflected) to extract only the reflected field.

Figure 6 .
Figure 6.Measured normalized electric field magnitude distributions showing energy distributions for designed bottle beam with different a at a) 8.5 GHz, b) 9 GHz, and c) 9.5 GHz.The xoy-plane shows low-intensity field region at z = z 1 and high-intensity field region at z = z 2 , where z 1 = a, z 2 = 2a þ 20 mm.

Figure 7 .
Figure 7.Comparison between calculated and measured focal lengths of the bottle beam with radius a = 25, 30, 35, and 40 mm.