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Abstract
Airy beams, accelerating optical beams with exotic properties of self-bending, self-healing and non-diffraction, are essential for a wide range of photonics applications. Recently, metasurfaces have provided an efficient platform for generating desired Airy beams within a thin thickness, but they suffer from the narrow bandwidth, especially for two-dimensional (2D) Airy beams. Here, we propose an amplitude-tailorable polarization-converting metasurface to enable ultra-wideband 2D Airy beam generation. The amplitude and phase profiles for the 2D Airy beam can be realized by tuning only the orientation of the multi-resonant meta-atom, which can operate in the range of 6.6 GHz to 23.7 GHz, or fractional bandwidth of 113%. An exemplary prototype is measured to validate the design principle, which is in agreement with the simulation results. The proposed method holds great promise for wavefront shaping, and may facilitate the uses of Airy beam for practical applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Airy wave packet was first discovered by Berry and Balazs in the field of quantum mechanics and then introduced to the optics community by analogizing the Schrödinger Equation and the paraxial Helmholtz Equation [1–3]. Airy beam is the optical beam with the fundamental form of Airy wave packet, which possesses the properties of non-diffraction, self-accelerating and self-healing [4–7]. These peculiar characteristics make Airy beam a research hotspot in the recent decades [8,9]. Airy beams can be classified as one-dimensional (1D) and two-dimensional (2D) according to the spatial dimensions satisfied by their Airy wave packets [10]. Both the 1D and 2D Airy beams have found various applications, such as high-resolution light-sheet microscopy [8], routing surface plasmon polaritons [11], and optical micromanipulation [12,13]. The first experimental implement of Airy beams was in 2007 [4,5], where the cubic phase distribution is obtained from the Fourier transform of the Airy function based on the combination of Fourier lens and spatial light modulator. However, the traditional methods for Airy beam generation suffer from the bulky size of the equipment and the optical elements used for the necessary mathematical operations, for example, the Fourier-transform lens, which makes such devices difficult to be integrated with other optical components [14]. Moreover, the traditional methods also face the possible deterioration of beam quality due to the wavelength-comparable pixel sizes of the optical components.
Recently, metasurfaces composed of subwavelength-scale building blocks have emerged as a new platform for modulating the light fields due to their flat low profiles with reduced loss [15]. By judiciously designing the physical parameters of meta-atoms and arranging them into spatially-varying distribution, the field discontinuities across the metasurface can be manipulated at will [16], which have promoted the development of many meta-devices, such as chromatic aberration-free meta-mirrors [17], broadband meta-gratings [18], Huygens meta-lens [19], arbitrary polarization converter [20], meta-hologram imager [21–25], invisibility cloak [26,27], omni-beam generator [28,29], and vortex meta-multiplexer [30–33], etc. Due to their powerful ability of amplitude and phase control, metasurfaces have also been utilized for implementing Airy function of electromagnetic wave across the whole spectrum [9,10,34–44]. Compared with 1D Airy beam that still spreads in the orthogonal axis when propagates in free space, 2D Airy beam shows non-diffraction properties in both axes which is more favored by practical applications, for example, the long-distance wireless power transfer in microwave region [10]. Nevertheless, generating 2D Airy beams require more complex amplitude and phase profiles than that of 1D ones. Though some methods, such as C-aperture [10,42], Huygens’ dipolar structure [45], and silicon nanopillars [34], can produce high-efficiency Airy beam, the operation bandwidth is still limited due to the dispersion effect of the resonant structures. In microwave region, the single-layer metasurfaces composed of C-shaped aperture or single metallic patch can effectively producing 2D Airy beam with simple fabrication process, but their operation bandwidths are less than 40% while the maximum amplitude of the meta-atom is less than 0.5 [10]. Therefore, while 2D Airy beams have potential uses in many applications, such as directional communication and long-distance high-efficiency wireless power transfer [10,46], it is still challenging to implement via metasurfaces with the merits of simple configuration, ultra-broad bandwidth and high operation efficiency.
Herein, we propose a single-layer reflective amplitude-tailorable metasurface operating in microwave region for generating high-efficiency 2D Airy beam with ultra-broadband performance. The proposed meta-atom is composed of an H-shaped metallic patch, a thin dielectric substrate and a metallic ground. The meta-atom exhibits maximum cross-polarized amplitude over 0.9 in the frequency band of 6.6 to 23.7 GHz, reaching a fractional bandwidth of 113%. The reflection amplitude and phase responses of the meta-atom stem from the polarization-converting effect, which can be flexibly manipulated by its in-plane geometric orientation. As the proof-of-concept, we design an ultra-broadband 2D Airy beam meta-generator operating in cross-polarized channel that shows non-diffraction, self-accelerating and self-healing properties in both x- and y-directions. As schematically depicted in Fig. 1(a), ultra-wideband x-polarized 2D Airy beam is generated under the illumination of y-polarized wave. Notably, the main lobe has increasing offsets in both x- and y-directions as the propagation distance increases, which is clearly shown in Fig. 1(b). Experiments are conducted by measuring the near-field distributions of the output beam, which are in agreement with the theoretical predictions.
Fig. 1. Conceptual illustration of the designed ultra-wideband 2D Airy beam generator. (a) Schematic view of the Airy beam generator. Inc. y-pol. and Ref. x-pol represent incident y-polarized and reflected x-polarized waves, respectively. (b) Field distributions at 12 GHz in xoz plane and yoz plane where the main lobe is located.
2. Strategy and meta-atom design
To construct the meta-generator with 2D Airy function envelope, the key step is to design meta-atom with flexible and independent control of amplitude and phase responses. First, we consider an arbitrary anisotropic metasurface structure with mirror self-symmetry in both x-direction and y-direction. When illuminated by a linearly polarized plane wave, the reflection Jones matrix R of such a structure can be written as
The first and the second subscripts represent the reflected and incident polarization states, respectively. Due to the structural mirror symmetry along the x- and y-directions, cross-polarization coefficients (rxy and ryx) in the matrix should be 0, which means that no cross-polarized components in the reflected field. If the structure is rotated with an in-plane angle α, then the reflection matrix can be further represented as
By combining Eqs. (1)-(3), we can further obtain the specific form of the Jones matrix as
As seen, a positive correlation between the polarization conversion and the sine of twice the rotation angle appears in the cross-polarized channel. Once rxx and ryy are fixed, the cross-polarization coefficients are only determined by sin2α. The variation of sin2α provides an opportunity to flexible control the amplitude and the phase, which is one of the basic conditions for constructing metasurface Airy beam generator [39].
Then, we consider a suitable structure to meet the above equations for manipulating the amplitude and phase responses of the cross-polarized output. Figure 2(a) shows perspective view of the proposed meta-atom structure. The H-shaped metallic pattern is deposited on a thin dielectric substrate. In order to ensure a reflection operation, a metallic ground plane is used as the bottom layer to block the wave transmission. All the metallic patterns are copper with thickness of 0.018 mm while F4B medium with thickness of h = 3 mm is selected as the dielectric substrate, and it possesses a dielectric constant of 2.2 and loss tangent of 0.001. The period of the meta-atom is set as p = 9.4 mm. In our previous work, the H-shaped resonator has been demonstrated to offer wideband response in cross-polarization channel due to its multi-resonance mode, with a reflection bandwidth of 9-22 GHz (or 84% fractional bandwidth) [39]. The cross-polarized amplitude is directly determined by the rotation angle (α) of the resonator, offering a flexible method to control the amplitude profile of the metasurface. Here, we introduce additional electromagnetic resonances to largely expand the bandwidth at high frequencies by adding the H-shaped structure with two thin metallic strips, and the operation bandwidth is 6.6-23.7 GHz (or 113% fractional bandwidth). Figure 2(b) is the top view of the meta-atom in the case of α = 45° that the meta-atom can achieve the maximal cross-polarized reflection. Other parameters of the meta-atom are: wp = 0.2 mm, lp = 3.49 mm, rx = 1.68 mm, ry = 6.66 mm, lc = 2.7 mm, and wc = 0.4 mm. The simulated results calculated by the commercial software are shown in Fig. 2(c). Under the illumination of linearly y-polarized wave, the amplitude of the cross-polarized output can be maintained above 0.9 in the range of 6.6-23.7 GHz. The wideband response can be attributed to the multi-resonant modes of the metallic pattern, which can be manifested by the four resonant peaks in the cross-polarization spectrum or indicated by the co-polarization dips. The different resonant modes and their corresponding electric field distributions are shown in Sec. S1 of Supplement 1. To illustrate the capability of amplitude and phase control in the wideband range, we analyze the simulated and theoretical relationship at three frequencies between amplitude/phase response and the rotation angle in Figs. 2(d)-(i). rxy and ryx are exactly consistent due to the reciprocal theorem, for which only rxy is analyzed here. At 7 GHz, 15 GHz, and 23 GHz, the relationships of the simulated amplitude versus α are in good agreement with |sin2α| which can realize continuous amplitude modulation from 0 to 1. At the same time, a phase difference of 180° is observed for all these cases when the rotational angle α is changed from positive value to negative value, as shown in Figs. 2(g)-(i). At 7 GHz, the binary phase states have some slight deviation with the variation of α, which is caused by the coupling effect between adjacent meta-atoms and more analysis can be found in Sec. S2 of Supplement 1. Therefore, a free combination of amplitude and binary phase states can be achieved by changing the rotation angle α of the meta-atom, immune from the frequency change, which is workable in the wide operation range from 6.6 GHz to 23.7 GHz.
Fig. 2. Configuration and performance of the proposed meta-atom. (a) Perspective view of the meta-atom. (b) Top view of the meta-atom. (c) Reflection amplitude spectra in the cross-polarized and co-polarized channel under y-polarized incidence. The cross-polarized reflection amplitude versus rotation angle at (d) 7 GHz, (e) 15 GHz, and (f) 23 GHz. The cross-polarized reflection phase versus rotation angle at (g) 7 GHz, (h) 15 GHz, and (i) 23 GHz.
3. Design of the 2D Airy beam generator
As the 2D Airy wave packet can be extended from the 1D one, we start with analyzing the design process of 1D Airy beams. Ideal Airy beams cannot be realized because of their infinite energy, but this barrier can be broken by introducing an exponential function eas to provide a decay factor [4,5]. For 1D finite-energy Airy beams, they propagate based on the Airy solution of the paraxial diffraction equation, which can be written as
Here, s is the dimensionless transverse scale that can be specifically defined as s = x/w. x and w are the argument of the Airy function and the scaling length, respectively. Ai is the Airy function. a is the truncation factor, which is usually a small positive variable used to ensure the infinite Airy tail. q = zλ/2πw2 represents normalized propagation length, z is the length along the propagation direction. Particularly, when the propagation length is set as 0, Eq. (5) can be reduced as
This new function contains an Airy function term that oscillates between the positive maxima and negative minima with an exponential decay term, which can be viewed as the initial state when Airy beam is generated from the metasurface. Correspondingly, we can realize the positive/negative values in Eq. (6) by means of meta-atoms with 0/180° phase state and arbitrary amplitude response. Hence, the requirement of generating a specific Airy beam is reduced to that as desired amplitude and phase profile on the metasurface aperture. When these meta-atoms are arranged along a single direction and periodically extended in the other orthogonal direction, the 1D Airy beams can be effectively generated. Differently, 2D Airy beam generation requires field oscillation and exponential decay in both orthogonal directions (e.g., x- and y-directions). Therefore, if we consider an Airy beam generator in the xoy plane, Eq. (6) should be written to
Similar to Eq. (6), the initial amplitude and phase profiles to generate Airy beam in Eq. (7) can be realized via the proposed meta-atoms.
To implement Eq. (7) concretely in the form of a meta-generator, the continuous amplitude and phase profiles are discretized into pixels by considering the period of the meta-atom (p = 9.4 mm). Herein, the parameters in Eq. (7) are set as w1 = w2 = 0.05 and a = 0.032 and more calculated amplitude profiles with different w and a can be found in Sec. S3 of the Supplement 1. We consider a sampling range of -352.5 mm to 117.5 mm in the xoy plane which totally contains 50 × 50 = 2500 meta-atoms. Figures 3(a)-(b) show the required amplitude and phase profiles on the metasurface aperture that is discretized based on the pixel size of the meta-atom. To realize such profiles on the aperture, we sweep the parameters of the meta-atoms and search the meta-atoms with the most appropriate electromagnetic responses. The final orientation distribution of meta-atoms is shown in Fig. 3(c). Then, the metasurface prototype is manufactured by the standard printed circuit board (PCB) technology. As clearly seen in Fig. 3(d), the meta-atoms in the main lobe region have rotation angles around 45°, offering high reflection amplitude in cross-polarized channel. For positions with lower amplitudes, the rotation angle of the meta-atom is closer to 0° or 90°.
Fig. 3. Design of the prototype and the measurement setup. Theoretical evaluation of the (a) amplitude, (b) phase, and (c) rotation angle distributions of the 2D Airy beam generator. (d) Photograph of the fabricated prototype. (e) Schematic of the measurement setup.
Figure 3(e) depicts the schematic of the measurement setup. The metasurface is surrounded by a microwave-absorbing screen to construct absorption boundaries. A wideband horn antenna with operation frequency band of 8-18 GHz is selected as the transmitting antenna (y-polarization), which is placed at a distance of 2 m from the metasurface to provide an illumination condition close to the plane wave. An x-polarized dipole probe is fixed on a 3D scanning platform, which is driven by a personal computer (PC) so that the probe can be automatically moved in a specified plane to detect the reflected field distribution. The transmitting antenna and the dipole probe are connected to two ports of the vector network analyzer (Agilent N5244A PNA-X), respectively.
The simulated and measured field distributions of the prototype at different frequencies are shown in Fig. 4. The 2D scanning area and the step resolution are set as 470 × 470 mm2 and 10 mm, respectively. Figures 4(a), (c), and (e) present the results in two observation planes (50 mm and 350 mm away from the metasurface) at 8 GHz, 12 GHz and 18 GHz, respectively. As seen, the simulated and measured results clearly show the typical Airy profile and its self-acceleration behavior along both x- and y-directions. To analyze the location offset of the main lobe due to the self-acceleration in details, we further offer 1D intensity results in xoz and yoz planes, as shown in Figs. 4(b), (d) and (f). The left panels depict the results in xoz plane while the right panels depict the results in yoz plane. Taking 8 GHz as an example, we simulate and measure the 1D amplitude profiles at z = 100 mm, 200 mm, 300 mm and 400 mm in the xoz (yoz) plane where the main lobe is located. The insets in Fig. 4(b) show the schematics for the detected xoz plane (y = 0 mm) and yoz plane (x = 0 mm). In the xoz plane, the position of the main lobe is continuously distributed in the positive region of the x-axis as the propagation distance increases. This phenomenon is also observed in the yoz plane, which is consistent with the self-bending property of the 2D Airy wave packet. When the propagation distance in the z direction varies from 50 mm to 350 mm, the full width at half maximum (FWHM) of the main lobe in these two planes are all around 50 mm, which well verifies the quasi-diffraction-free property of the 2D Airy beam.
Fig. 4. Simulated (Sim.) and measured (Mea.) intensity distributions of the reflected cross-polarized field. Simulated and measured (a) 2D and (b) 1D results at 8 GHz. Simulated and measured (c) 2D and (d) 1D results at 12 GHz. Simulated and measured (e) 2D and (f) 1D results at 18 GHz.
The self-accelerating and quasi-diffraction-free properties can be well maintained in a broadband range, as illustrated in Figs. 4(c)-(f). Notably, metasurface has certain chromatic dispersion especially for ultra-wideband beam shaping due to the change of the operating wavelength. Hence, the generated ultra-wideband Airy beam is observed with dispersive change, for example, the shift of main lobe in x-direction from z = 50 mm to z = 350 mm is 20 mm at 8 GHz, but this indicator is reduced to 5 mm at 18 GHz. Besides, when the frequency increases, the quality of the generated Airy beam slightly decreases, because the relative meta-atom size compared with wavelength becomes larger which reduce the resolution of both amplitude and phase manipulations. Considering the experiment and assembly tolerance, the measured results agree with the simulated ones, both of which validate that the reflective prototype has an ultra-wide frequency band, even though the measured results are only up to 18 GHz due to limitations of the experimental setup. The final measured efficiency of the generated 2D Airy beam at different frequencies is 1.57% at 8 GHz, 3.55% at 12 GHz and 3.99% at 18 GHz, respectively. Particularly, the theoretical maximum efficiency is 4.9% for a same-sized 2D Airy beam generator using amplitude-phase manipulation [39]. More simulated results at lower/higher frequencies and the calculation detail of efficiency can be found in Sec. S4 and Sec. S5 of Supplement 1, where 2D Airy beam can be clearly observed with 2D self-bending and quasi-diffraction-free properties in the ultra-wideband range.
Finally, the self-healing property of the generated 2D Airy beam is also verified by full-wave simulations and experimental measurements. In order to effectively block the propagation of the main lobe, we place a metallic block with dimensions of 50 mm × 50 mm × 10 mm at position (x, y) = (0, 0). As shown in Fig. 5(a), the metallic block is placed 30 mm away from the metasurface. The slightly asymmetric simulation results in x and y-directions are due to the different diffraction capacities of x-polarized waves in these two directions. Figure 5(b) illustrates the simulated results of the main lobe in xoz plane (y = 0). Clearly, the main lobe reappears after a propagation distance though it is first blocked by the metallic obstacle. For high-frequency operation, the obstacle size is larger than a wavelength, so the self-healing performance is reduced compared to that of low frequency. Since the designed prototype is diagonally symmetric in xoz plane, the main lobe has the same propagation properties to that in the yoz plane, so here we only present the simulated results of xoz plane. Moreover, the simulated and measured field distributions in xoy plane at z = 50 mm and z = 350 mm are presented in Fig. 5(c). Taking 8 GHz, 12 GHz and 18 GHz as examples, the main lobe is completely blocked in the broadband range, as shown in the top row (z = 50 mm). When the observed plane is 350 mm away from the prototype, the main lobe will reappear with a displacement with respect to the initial position of (x, y) = (0, 0), as shown in the middle row of Fig. 5(c). To validate these properties, we used the near-field scanning system to detect the intensity of the cross-polarized output electric fields, and the measured results are shown in the bottom row of Fig. 5(c). Within the allowable fabrication and measurement errors (including the blocking effect of incident field by detection equipment), the measured results are in agreement with the simulation ones. The above results can firmly validate the robust self-healing property of the generated 2D Airy beam across an ultra-wide bandwidth. Compared with the previous works, the proposed reflective Airy beam meta-generator features obvious advantages of ultra-wide bandwidth [9,10,36,38,40–43,45,47,48].
Fig. 5. Verification of the self-healing property of the ultra-wideband 2D Airy beam. (a) Experiment setup and photograph of the barrier. Cross-polarized intensity distributions in the (b) xoz plane and (c) xoy plane at 8 GHz, 12 GHz and 18 GHz.
4. Conclusion
In summary, we have proposed and experimentally demonstrated the ultra-wideband generation of 2D Airy beam by employing a single-layer reflective metasurface based on H-shaped resonator, which can provide high-efficiency cross-polarized amplitude control from 6.6 GHz to 23.7 GHz. We theoretically show that a continuous amplitude manipulation with binary phase responses can be easily obtained by controlling the orientation of the resonator. A 2D Airy beam generator is configured via this type of resonator with spatially-varying distribution. In the experimental measurement, the typical characteristics of quasi-non-diffraction, self-accelerating, and self-healing are verified for the generated 2D Airy beam, as well as its ultra-wideband performances. The proposed methodology for 2D Airy beam generation may trigger potential applications in wireless communications, wireless power transfer, etc.
Funding
National Natural Science Foundation of China (62071215, 62271243, 91963128); National Key Research and Development Program of China (2017YFA0700201); The Joint Fund of Ministry of Education for Equipment Pre-research (8091B032112); Priority Academic Program Development of Jiangsu Higher Education Institutions; Fundamental Research Funds for the Central Universities; Jiangsu Provincial Key Laboratory of Advanced Manipulating Technique of Electromagnetic Wave.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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