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Abstract
We present a generic approach for the generation of pseudo non-diffracting Bessel beams using polarization insensitive metasurfaces with high efficiency. Cascaded unit cells, which are fully symmetric, are designed for the complete 2π phase control in the transmission mode. Based on the topological arrangements of such unit cells, two metasurfaces for the generation of zero-order (i.e., single phase profile) and first-order (i.e., merger of two distinct phase profiles) Bessel beams are designed and characterized. Both numerical simulations and experimental measurements are in agreement with each other, confirming the electromagnetic characteristics of the reported Bessel beams. Owing to the isotropy of the unit cells and the rotational symmetry of the arrangements, the proposed metasurfaces are polarization insensitive, providing a promising avenue for achieving such wave manipulations with any linear or circular polarization.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Bessel beams are paraxial solutions to the free-space Maxwell equations discovered by Durnin in 1987 [1]. They take their name from their transverse intensity profile, which is described by Bessel functions of the first kind. Owing to their non-diffracting nature, Bessel beams have proven themselves useful over a broad spectral domain spanning from microwave to optical frequencies and for various kinds of applications, including particle trapping [2], tractor beams [3], microscopy [4], as well as lithography and laser machining [5,6]. Bessel beams are also beneficial for high-power applications such as microwave drilling [7,8], wireless charging [9], high-resolution radar imaging [10], and the creation of through-the-wall radar [11]. Therefore, it is beneficial to develop simple and practical approaches to generating such beams.
Many techniques have been utilized in the past to realize Bessel beams. These techniques involve axicons [12], computer generated holograms [13], leaky waveguide modes [14], holographic screens [15],near-field plates [16], transverse modes excited in a circular wave guide [8] to name a few. However, these techniques suffer from issues such as low efficiencies, high manufacturing complexities, and bulky sizes. The design proposed in [8] is based on transverse modes excited in the circular waveguide, the electrical size of which is 5.8λ × 5.8λ with a thickness larger than λ. Such a design suffers challenges in fabrication and compatibility issues with the integrated circuits. Albeit numerous investigations have been focused on the generation of zero-order Bessel beams, very few efforts have been made to realize higher-order Bessel beams which have non-diffracting characteristics as well as twisting phase profile. In particular, Comite et. al [17] have proposed the feasibility of Bessel beams carrying twisting waves, i.e., higher order Bessel beam, using the radial line slot array (RLSA) antenna [18]. Circular antenna arrays combined with RLSA [10] were also considered for the generation of Bessel beams carrying an orbital angular momentum (OAM). Despite these demonstrations, realization of efficient and polarization insensitive higher order non-diffracting twisted beams is on open problem.
Metasurfaces provide the freedom of introducing abrupt phase-change along the interface between two media. This unique capability of metasurfaces received significant attention recently due to their low profiles and great flexibilities in tailoring wavefronts [19–21]. The efficient realization of wavefront manipulation with metasurfaces require complete phase control from 0 to 2π along with uniform high amplitude; ideally equal to unity. Such complete phase control can be achieved by tailoring the geometries of artificially engineered subwavelength resonators. Initial demonstration in this regard included subwavelength gold V-shaped antennas arranged on a 2D surface [22]. However, the proposed metasurface exhibited meagre performance due to lower polarization conversation efficiency and substantial ohmic losses associated with the gold. Later on, it was concluded that the polarization conversion efficiency of such metasurfaces are fundamentally limited to 25% [23–25], which is inadequate for practical applications. Reflective type metasurfaces [26–30] are also studied to achieve high generation efficiency. However, since the source antenna is placed in front of the metasurface, it may partially block the generated beam.
Metasurfaces based on Huygens principle were also proposed to control the 2π phase range of the co-polarized components which have the potential for achieving a 100% efficiency in theory [31]. Such a metasurface introduces both electric and magnetic discontinuities to control the electromagnetic field. It was proposed that the metasurface efficiency can be improved by increasing the thickness of the metasurface [23]. Recently, a few works have been reported on achieving higher transmission efficiency using metasurfaces made of multilayer structures [32–35]. These designs employ the conversion of spin to OAM based on the photon spin Hall effect, where the cross-polarized component of electromagnetic wave could be utilized efficiently. Later on many cascaded metasurfaces are proposed to control the 2π phase range with good efficiency [36–38]. However, these designed metasurfaces are typically limited to a single polarization because of their asymmetric geometry. The direct comparison between the recently reported planar transmissive metasurfaces with context to the realized efficiency are listed in Table I.
Table 1. Efficiency comparisons for transmissive metasurfaces realized in literature, where LP for linear polarization and CP for circular polarization
Metasurface lenses with both polarization insensitiveness and high conversion efficiency are highly demanded. To the best of our knowledge, the only reported metasurface lens of such type was proposed by Khorasaninejad et al. [42] and was made of standing dielectric resonators. However, since such a lens is not a planar structure, its fabrication becomes very difficult and costly. In this paper, we present highly symmetric planar metasurfaces with high efficiency for generating Bessel beams, which can be easily manufactured using the standard printed circuit board technique. Cascaded unit cells are employed in the design as they help to achieve high efficiency. The unit cells have equal dimensions of (λ0/3.75) × (λ0/3.75) × (λ0/11.97), which are designed to achieve the full phase control from 0 to 2π with a weighted average efficiency of 71%. As examples, we present the designs of two metasurfaces for the generation of the zero-order and first-order Bessel beams. The measured results are in good agreement with the simulated ones, confirming clearly the electromagnetic characteristics of the Bessel beams of different orders.
The rest of this paper is organized into the following sections. Section 2 introduces the design of the cascaded unit cells for the full 0–2π phase control and presents the discussion of the high transmission amplitudes and the physics behind their resonance modes. Section 3 describes the strategy for designing metasurfaces that are capable of generating Bessel beams and presents simulation results on the generation of the fundamental and first-order Bessel beams. The results of experimental measurements are given in Section 4 finally, the conclusions of our work are given in Section 5.
2. Unit cell theory and design
The unit cell under consideration is shown in Fig. 1. It consists of four circular copper patches, of which two middle patches have holes of radius r = 1 mm. The radii R of the four patches are all alike. The three dielectric spacers are made of Rogers RO4003C, which has dielectric permittivity of 3.48 and thickness t = 0.813 mm. The period of the unit cell is p = 8 mm, which equals 1/3.75 of the operation wavelength at 10 GHz. The amplitude and phase of the wave transmitted by the metasurface consisting of such unit cells can be flexibly adjusted by changing R. It is assumed that the incident plane wave propagates along the z direction, with the electric field vector along the y axis and the magnetic field vector along the x axis.
Fig. 1 Unit cell on top and two designed metasurfaces at bottom. The geometrical parameters are r = 1 mm, p = 8 mm, and t = 0.813 mm of the substrate. (The thickness shown is modified for better visualization)
The reflection coefficients of the metasurfaces made of periodically patterned unit cells are plotted in Fig. 2. It is seen that there are three resonance dips in the reflection curves over the frequency band of interest. It should be noted that in our case it is easier to show the resonance behaviors in the reflection spectra, because the transmission is quite high within the entire frequency band of interest. The reflection dips correspond to the resonance peaks of the transmission spectrum. All the dips are seen to shift to higher frequencies, with the second and third resonances becoming much closer to each other, as the radius of the patches decreases. The two high-frequency resonances merge together for R = 3.53 mm. We will further analyze how these resonances enable one to achieve the full phase control with high efficiency.
As electromagnetic fields are continuous naturally, the discontinuity in phase is achievable via the introduction of a surface with electric and magnetic currents. These circulating electric and magnetic currents result from resonances which can ultimately be controlled in metasurfaces. At our operation (design) frequency of 10 GHz, the first resonance appears when the patch radius is about R = 3.53 mm. As the patch radius increases, the second and third resonances shift close to the design frequency while the first resonance shifts towards lower frequencies. The maximum of three resonances corresponding to the number of the dielectric spacers are produced around the designed frequency. By sweeping these resonances through the designed frequency, the whole phase range from 0 to 2π can be obtained.
While the two resonances are sufficient for achieving the full phase control from 0 to 2π [43–45], maintaining the high efficiency would be difficult. By fully sweeping the radius of patches from 3.59 to 3.99 mm to adjust the second and third resonances, it is possible to obtain a full 2π range of phase control. However, the magnitude decreases significantly when R is sufficiently large. Particularly, the efficiency for R = 3.99 mm is only about 16%. For this reason, we have included the contribution from another resonance at R = 3.53 mm (i.e., the first resonance) to achieve the full 0–2π phase control with high efficiency. The magnitude and phase responses of the designed unit cells are shown in Fig. 3. The proposed design is based on a frequency selective surface acting as a low pass filter. The response of patch radii below the first resonance (i.e., R < 3.53 mm) acts as a non-resonant transparent surface where the transmission magnitude is high but the transmission phase variation is very slow. The variation starts to increase at resonances. This allows us to use smaller radii for better transmittance at these particular phase values. The weighted average efficiency throughout the whole 0 – 2π range is around 71%, with the highest and lowest magnitudes of the transmission coefficients around 95% and 65%.
Fig. 3 Phase/magnitude of co-polarized transmission co-efficient (S21) versus radius R of the patches at 10 GHz.
The design frequency of the proposed unit cell depends on the periodicity p, the substrate thickness t, and the hole radius r. By tuning these parameters, the operating frequency can be shifted as desired. Apart for shifting the design frequency, the thickness t and holes in middle patches have additional effects on the transmission magnitude and phase. By increasing the thickness of the substrate, the transmission magnitude can be improved. However, a tradeoff is necessary to be made between higher transmittance and lower profile. The periodic unit cells can achieve phase control in the range of 0 – 2π. However, the cases with large phase values usually suffer due to low transmittance as a result of sharp cut-off of the stop band. To maintain relatively high transmittance at these high phase values, minor tuning is performed by decreasing the capacitance via drilling holes of r = 1 mm in the two middle patches.
In Fig. 4, we show the field distributions of the unit cell at the resonance frequencies for better understanding the physics behind these resonances. Magnetic fields around and inside the unit cell are shown in Figs. 4(a)–4(c) and their equivalent models based on boundary conditions [46] are shown in Figs. 4(d)–4(f). These fields and models are shown for three different resonances at 10 GHz for three radii R = 3.53 mm, 3.94 mm, and 3.98 mm. Since the thickness of the entire unit cell is subwavelength (λ0/11.97), it is reasonable to treat the whole surface as a single interface. For the patch radius of 3.53 mm, the magnetic field inside all three dielectric spacers are directed in the +x direction and the currents induced on the middle sheets cancel each other. The circulating currents outside the unit cell form the +x-directed magnetic fields and, hence, the unit cell at the first resonance acts as a magnetic dipole. The second and third resonances have contributions from both electric and magnetic resonances, because these resonances are close to each other. For the second resonance, the electric response is dominant due to strong outer circulating magnetic fields and the weak inner magnetic field. This electric response is not a pure one because of the weaker inner magnetic field that gives contribution to the magnetic response. For the third resonance, two oppositely directed magnetic fields generate an electric dipole, while the third magnetic field contributes to a magnetic resonance. Both electric and magnetic dipoles have nearly equal contributions to the third resonance.
Fig. 4 Magnetic field distribution and equivalent current distribution for three different resonances at 10 GHz: (a) Hy field and (b) equivalent current distribution for R = 3.55 mm; (c) Hy field and (d) equivalent current distribution for R = 3.94 mm; (e) Hy field and (f) equivalent current distribution for R = 3.98 mm.
3. Generation of Bessel beam with metasurfaces: numerical simulations
In the previous section, unit cells were designed for the full 2π phase control with high efficiency against geometric parameters. The next step is to design metasurfaces for realizing Bessel beams of different orders with the predesigned unit cells. As examples, two metasurfaces are designed for both the zero-order and first-order Bessel beams, which have dimensions of 304 × 304 mm2 along the x and y axes. The electric far field of the Bessel beam generated by the metasurface at an arbitrary direction b̂ in space can be expressed as
where p is the polarization of the incident electric field, m and n are the center coordinates of the unit cell on the 2D metasurface, r⃗mn is the radius vector of the unit cell, Fp is incident plane wave, A is scattering pattern of the unit cell, b̂0 is the desired beam direction, and ϕ is the desired phase profile of the generated Bessel beam. Due to the isotropic nature of the unit cells, such a metasurface is independent of the incident beam polarization.3.1. Zero-order Bessel beam
The phase profile of Bessel beams is described by Bessel functions of the first kind. The phase distribution for the zero-order (J0) beam is given by [47]
where x and y are the Cartesian coordinates on the metasurface, x2 + y2 = r2, and NA is the numerical aperture of the metasurface. The designed metasurface for the zero-order Bessel beam with NA = 0.7 is shown in Fig. 1 (bottom left side). One can see from Eq. (2), that the phase profile for the zero-order Bessel beam varies in the radial direction. The metasurface is distributed into annular strips and each strip has the same phase profile. Each annular strip has its specific unit cells, whose phase profiles are calculated using Eq. (2). Such a distribution is not ideally given in square lattices. The circular strips are arranged in such a way that the distances between adjacent cells are nearly equal to the periodicity and the relative differences are below 4.46% which are even lower for outer circular strips. Such small differences would not affect significantly on the performance of the metasurface lens.The performance of the metasurface is assessed numerically using full-wave simulator CST Microwave Studio. An incident plane wave polarized along the x axis is used to illuminate the metasurface and the electric field at the other side of the metasurface is recorded. The metasurface is assumed to be placed in plane z = 0. In Figs. 5(c) and 5(d), we show the magnitude and phase profiles of the x component of the electric field in plane z = 100 mm. It can be seen that the most of the power is contained in the main lobe whereas the side lobes are sufficiently weaker. The phase profile matches well with the analytical phase profile in Fig. 5(b) except a minor asymmetry around the radial distance of 90 mm. Such an asymmetry is mainly caused by the diffraction of incident waves from the edges of the metasurface lens due to its limited size. It is worth noting that an ideal Bessel beam should be generated from an infinitely large lens, which is impossible to be implemented practically. However, using finite metasurface lens, it is possible to approximate the Bessel profile to few rings as have been demonstrated here. The magnitude curve for y = 0 in this cut-plane, plotted in Fig. 5(e), clearly shows a profile similar to the Bessel function. The electric magnitude distribution of the Bessel beam in the yz plane is also shown in Fig. 5(f). It can be observed that the Bessel beam contains a high-energy region in the form of a cone. The focal depth of the Bessel beam is given by
For D = 304 mm and NA = 0.7 the non-diffracting region of the Bessel beam is found from this equation to be 153 mm, which nearly coincides with the simulated focal depth of 155 mm. The full width at half maximum (FWHM) of the zero-order Bessel beam J0 is defined as the waist of this beam where the intensity of its center bright spot is at half of its peak value, This equation gives the zero-order Bessel beam waist of 15 mm, which correlates well with the simulated value of 17 mm.Fig. 5 theoretical zero-order Bessel profile; (a) magnitude and (b) phase; Simulation results of the zero-order Bessel beam; (c) magnitude and (d) phase distributions of electric field in plane z = 100 mm; (e) magnitude of electric field for y = 0 and z = 100 mm; (f) electric field magnitude in plane x = 0.
3.2. First-order Bessel beam
The phase profile of higher-order Bessel beams is similar to that of the zero-order Bessel beam except that an additional phase term lφ is introduced,
where φ = arctan(y/x) is the azimuthal phase and l is the Bessel beam order. In addition to the non-diffracting nature, higher-order Bessel beams have helical phase profiles and can be regarded as non-diffracting OAM beams. The twist of the phase profile is determined by the order l of the Bessel beam, which also coincides with the OAM mode number. For the first-order (J1) beam, i.e., l = 1, the phase profile undergoes rotation from 0 to 2π through the azimuthal direction.To realize the first-order Bessel beam, we designed a 304 × 304 mm2 metasurface with a numerical aperture of 0.7 as shown in Fig. 1 (bottom right side). Such a metasurface consists of a 38 × 38 array of unit cells. Each unit cell is capable of adding an additional phase shift to the incident wave given by Eq. (5).
Next, the performance of the designed metasurface is analyzed using numerical simulations similar to how it was done earlier. The electric field distributions in the transmitted wave are examined for a normally incident wave at 10 GHz. Figs. 6(c) and 6(d) show the magnitude and phase of the x component of the electric field at the cut-plane z = 100 mm. Unlike the zero-order Bessel beam, the first-order Bessel beam has a ring-like magnitude distribution and a helical phase profile with a topological charge of l = +1. The field profile along the line y = 0 in the same cut-plane is plotted in Fig. 6(e). The first-order beam is seen to have zero intensity at the origin. The electric field at the cut-plane x = 0 is shown in Fig. 6(f). It is worth noting that unlike analytical profiles, obtained Bessel profiles for both the zero-order and the first-order modes does not form perfect ring and are of oval shape. This arises due to the non-unity NA as compared to analytical calculations, also pointed out in a recent paper [47]. Moreover, a minor effect could also be due to non-uniform transmittance of unit cells, which contributes to the non-uniform field distributions along the angle direction.
Fig. 6 theoretical first-order Bessel profile; (a) magnitude and (b) phase; Simulation results of the first-order Bessel beam; (c) magnitude and (d) phase distributions of electric field in plane z = 100 mm; (e) magnitude of electric field for y = 0 and z = 100 mm; (f) electric field magnitude in plane x = 0.
The first-order beam is non-diffracting until a certain distance. The non-diffracting region of the first-order Bessel beam is around 150 mm according to the simulations and is nearly the same as predicted by theory, which is 153 mm for NA = 0.7 and D = 304 mm. The FWHM for the first-order Bessel beam J1 is twice the distance from the center dark spot to the closest point on the ring at the half of the maximum intensity,
The waist of the first-order Bessel beam calculated from Eq. (7) is 13 mm, which is in good agreement with the numerically obtained value of 15 mm.Unlike periodic surfaces, the metasurfaces designed for real applications as presented here require different phases at subwavelength resolution over the whole surface. As a result of this discontinuous phase distribution, different subwavelength particles are required violating the periodic conditions. However, the phase distribution calculated according to Eqs. (2) and (5) are uniform and changes are gradual from one cell to another and relative for consecutive cells. This leads to a consistent relative phase shift and does not affect much. This is also confirmed as the obtained efficiencies are quite good. Moreover, the obtained magnitude and phase profiles are quite matched for the zero order and the first order Bessel beams to their respective analytical profiles.
It is worth noting that the FWHM and the focal depth are directly related to the numerical aperture of the metasurface. With the increase in NA, it is possible to narrow the waist of the Bessel beam at the expense of smaller focal depth. Such a trade-off between the length and the width of the beam should be considered depending on the specific application of the metasurfaces.
4. Generation of Bessel beam with metasurfaces: experimental results
In order to experimentally assess the performance of our designs, two metasurfaces for the zero-order and first-order Bessel beams are fabricated using the standard printed circuit board technique as shown in Figs. 8(a0) and 8(a1). The experimental setup is shown in Fig. 7. A horn antenna connected to a vector network analyzer is used for generating quasi-plane waves, and a waveguide probe of cross section area of 25 ×15 mm2 is used for detecting the transmitted electric fields. The probe gives the field value averaged over the entire cross section. The horn antenna generates x-polarized quasi-Gaussian beam with conical shape beam. The designed metasurfaces are placed at a sufficient distance away from the horn antenna to receive approximately plane waves. The probe is set to be x-polarized and the transmitted electric field is recorded at the plane which is 100 mm away from the designed metasurface with a step size of 3 mm along x and x directions.
Fig. 8 (a) Fabricated metasurfaces, (b) magnitude and (c) phase distributions of electric field in plane z = 100 mm. Subscripts 0 and 1 mark panels representing the zero-order and first-order Bessel beams.
The results of our experimental assessment of the fabricated metasurfaces presented in Fig. 8 are in good agreement with the results of numerical simulations. The phase profiles of the beams indicate that there is no azimuthal phase rotation for the zero-order Bessel beam and that the first-order Bessel beam has a twisting phase profile. The experimental results clearly validate the simulated results except for a few discrepancies such as that the magnitude profile of the first-order Bessel beam is not a uniform ring. This is not the case for the zero-order Bessel beam, because the corresponding metasurface has only radial phase variations and is symmetrical in nature, whereas the metasurface for the first-order Bessel beam has variations in both the radial and azimuthal directions. The deviation between the simulated and the measured results could be accounted for the misalignment of the source horn antenna which causes shift of source focus away from the metasurface center. This generates asymmetry for the first-order Bessel beam and focuses more energy towards one side of the ring as shown in Fig. 9. A minor discrepancy could also be due to fabrication limitation.
Fig. 9 Comparison between theory, simulation and measured results: zero-order Bessel beam(a) x = 0, (b) y = 0; first-order Bessel beam (c) x = 0, (d) y = 0.
The FWHM of the zero-order Bessel beam is found to be 15 mm, which is slightly smaller than the value predicted by numerical simulations. On the other hand, the FWHM of the first-order Bessel beam is 17 mm and is slightly larger than the theoretical prediction. There are two reasons for this discrepancy. One is the limitation in the fabrication of the minimum dimension and another is a comparatively large cross section area of the waveguide probe, which makes the measured response being averaged over the entire waveguide cross section.
We have thus successfully demonstrated the feasibility of generating the zero-order Bessel beam and the non-diffracting OAM beam by both simulations and experiments. Also noteworthy is that the proposed metasurfaces are polarization insensitive and provide a promising solution for achieving Bessel beams with linear polarization in any direction or even circular polarization.
As a concluding remarks, we have analyzed the efficiency of our designed Metasurfaces. The transmission efficiency, η, of the designed metasurfaces can be estimated using the following equation,
where E⃗t and H⃗t are the electric and magnetic fields at the transmission side, E⃗i and H⃗i are on the incident side of the metasurface and S is the integration plane of 200×200mm2 size, 100mm away from the metasurface. The simulated efficiency for the J0 beam metasurface is obtained to be 76% while for the J1 beam metasurface is 83%. The difference of the two obtained efficiencies is mainly due to the different distributions of the unit cells required for the two different beams. The detailed discussion on achieved efficiencies for the transmissive metasurfaces based on periodic units cell have been included in section I. However, to the best of authors knowledge the transmission efficiency for the whole metasurfaces generating Bessel beams is not discussed before. Moreover, converging OAM Beam (first order Bessel beam) in this work achieved higher efficiency of 83% compared to prior works [14,32,48], which is less than 50%. It is worth noting that the proposed metasurfaces are specifically designed for electromagnetic waves of normal incidence. While for oblique incidences, additional phase delays should be considered in the design of such metasurfaces.5. Conclusion
In summary, we have presented the design, fabrication, and characterization of planar high-efficiency metasurface for generating Bessel beams. Cascaded unit cells were designed for the full phase control in the transmission mode, where the underlying resonance modes were analyzed based on the electric/magnetic dipole theory. As two examples, metasurfaces for the generation of the zero-order and first-order Bessel beams were designed and manufactured. The functionalities of the proposed metasurfaces were examined in both simulations and measurements, which clearly confirmed the effectiveness in generating Bessel beams of different orders.
Funding
National Natural Science Foundation of China (61701303); Natural Science Foundation of Shanghai (17ZR1414300); Shanghai Pujiang Program (17PJ1404100).
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