- Photonics Research
- Vol. 11, Issue 6, 1085 (2023)
Abstract
1. INTRODUCTION
The meta-atom is the basis of metamaterial design; thereby the electromagnetic properties of the meta-atom determine the functional upper limit of the devices. In the past decades, metamaterial has progressed tremendously and gradually extended from microwave to acoustics, optics, and wider regions. In addition to deeper fields studies, people also tend to improve the functional properties of devices and promote operating efficiency. The working bandwidth has also experienced the development from narrowband to wideband. Initially, researchers designed many narrowband absorbers based on the strong loss at the anomalous dispersive point [1–4]. After that, wideband absorption of multi-point resonant superposition was also achieved [5–10]. In addition, the low-loss wideband devices were also reported a lot. In the optics region, different groups proposed some achromatic devices. The reported metalenses are operated over a continuous bandwidth in the visible by dispersion engineering of wavelength-dependent phase shifters [11–19]. But they still suffer from inefficiency due to loss which is related to Kramers–Kronig relations of permittivity [20–22]. The wide bandwidth always seems to be accompanied by larger loss and lower efficiency, which needs to be solved.
Different from previous studies that have balanced bandwidth and loss, in this work, we first explore the low-loss wideband condition in the view of the reflective scheme and obtain the wideband anomalous dispersion requirement. Then we find the leaky cavity modes (LCMs) make the permittivity inversely proportional to frequencies. To improve the efficiency of LCMs, we customize the transverse plasmonic mode to match with Fabry–Perot (F-P) resonance. In this way, the specific th resonance can be promoted to the main mode which can multiply the permittivity by a factor of , as is exhibited in Fig. 1. In this process, the LCMs which are enhanced enable larger phase accumulation, thereby forming a nonlinear phase distribution. As a proof of concept, we demonstrate an ultrathin weak chromatic-aberration focusing reflector (WCAFR) which has high-efficiency focusing above 80% from 16.0 to 21.0 GHz within thickness for right-handed circularly polarized (RHCP) incidence. The WCAFR is experimentally operated by the nonlinear phase distribution and still has focal diffraction phenomenon. But compared to the diffraction limits, its focal spots are mostly centered around the same location. Briefly, the proposed WCAFR might provide alternatives to fields of electromagnetic communication, antenna design, and radar imaging, etc. The concept can be readily extended to higher frequencies such as terahertz, infrared, and optical frequencies.
2. LOW-LOSS WIDE BANDWIDTH CONDITION FOR REFLECTIVE SCHEME
In principle, the standing wave model (SWM) can be used to explain most meta-atoms in the reflective scheme, which are composed of upper metallic patterns, dielectric substrates, and metal backgrounds. The illustrated parts objectively produce a cavity system and enable the incident wave to oscillate several times, which produces a standing wave.
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Suppose that the incident wave , , propagates in the direction through medium 1 and strikes the boundary of medium 2 at [shown in Fig. 2(a)]. The reflected wave, returning in the direction, combines with incidence to yield the resulting field strength. By introducing the reflection coefficient at the boundary , we write the equation as [23]
is the propagation constant of the electromagnetic wave, and in other patterns, . is the phase factor, and is the real part attenuation coefficient of dielectric material. For the reflective meta-atom, the reflection coefficient . Considering the real field components of amplitude, Eq. (1) can be expressed as
Replace the wavelength variable with frequency, and further make a transform as follows:
Figure 1.Schematic of coupling meta-atom and its realistic light deflection of WCAFR. All lights which have impact on the metasurface will deflect and intersect at the focal point. In the transverse dimension, the surface current distribution shows the order of the plasmonic mode on the meta-atom pattern. In the longitudinal dimension, there is a sketch map of diffraction mechanism and multiple reflections effect for reflective Fabry–Perot resonators. Every single unit can be regarded as an F-P resonator, inside which the wave oscillates back and forth
Figure 2.(a) Normal standing wave performance formed on the boundary between two media. (b) Ideal operating bandwidth of the reflective meta-atom [
To further analyze the permittivity solutions, we make a comparison between the metallic dipole meta-atom and later designed wideband meta-atom. Figure 3 shows the dispersion trends and operating bandwidths of the two structures. The metallic dipole is the typical Lorentz model that has a narrow bandwidth [25]. At the resonant point, the electric susceptibility shows anomalous dispersive property with strong loss. The latter designed meta-atom has a wide bandwidth and wide anomalous dispersion in the operating band. The commonality between the two meta-atoms proves the conclusion of the above results.
Figure 3.(a) and (d) The normalized electric susceptibility of dipole under liner-polarization incidence and S-pattern under circular-polarization incidence (
3. REALIZATION TO WIDEBAND ANOMALOUS DISPERSION
In fact, the reflective cavity is not completely closed. Each round trip of the resonant wave through the cavity produces a partial leakage wave which is called leaky cavity modes in wave optics [26]. As shown in Fig. 1, the F-P resonance diagram, after the wave is incident on the F-P resonator, it will diffract with different orders and the trend presents a continuous and exponential decline due to dielectric loss. In this process, each order resonant wave escapes from the cavity with different efficiencies after passing through the dielectric interface. In the fixed thickness cavity, because the order of high-frequency diffracted wave is higher than that of low frequency, the high-order resonant mode increases the optical path of the corresponding order diffracted wave. That results in its loss also getting amplified times of the related order. Therefore, under the joint impact between the low-order diffraction in the low-frequency band and the high-order diffraction in high-frequency band, the LCMs get more severe with reducing wavelength so that they can make the F-P resonator show an anomalous dispersion trend.
To further analyze the coupling between the transverse plasmonic modes and longitudinal F-P resonances, we consider the high-order F-P diffraction within the meta-atom. As Fig. 1 shows, after the wave experiences a round trip in the cavity, it will produce a phase difference [26]:
Due to the specific F-P resonance having been promoted to the main mode, the th-order diffraction multiples the phase difference with the factor of . At the same time, we can plug the effective index of the meta-atom into Eq. (5) and write the phase distribution as Eq. (6):
As a functional verification, we design a realistic ultrathin WCAFR with its architecture consisting of meta-atoms. The verified WCAFR with at the frequency is a regular diffractive reflector. Its phase profile was produced by the S-pattern meta-atom using the geometric phase for RHCP incidences. Its focal length is set as 100 mm, and the focal spot is located at the center of the section.
The WCAFR was simulated by CST Studio within open-add space along the and directions and with a plane wave as the incentive port in the axis. Experimentally, the focal lengths at different frequency points were obtained by measuring their intensity profiles along the propagation direction ( axis) in steps of 2 mm, as shown in Fig. 4. The coordinate corresponding to the peak intensity indicates the focal length for a given frequency point.
Figure 4.Performances of WCAFR. (a) Focal lengths and shifts of diffraction limits and WCAFR under different illuminated frequencies. (b) The calculated intensity distribution of diffraction limits in the
The former researchers preferred to divide the phase into basic phase and compensating phase [11,14]. Similarly, here we use the geometric phase as the basic one and the difference between the nonlinear and linear distribution impact as the compensating phase. (More details can be obtained in Appendix B [25,27–32].) Therefore, the proposed WCAFR cannot be completely achromatic, it is still slightly diffracted, and the focal point shifts back with increasing frequencies. But compared with the diffraction limits, its focal spots are almost centered around the same location.
We characterized the performance of the proposed WCAFR in terms of its focal length and focal spot profiles. They were measured at the focal plane under the RHCP incidence in an anechoic chamber. The normalized results show a weak trend of growing focal length as the frequencies increase but centering around 100 mm along the propagation axis within 10% error range. The spot profiles intercept the energy result () at to show itself. To further evaluate the ability of the WCAFR, focus efficiencies within its operating band are calculated in terms of the following equation [33]:
is the efficiency. and represent the reflection power and entire plane incident power, respectively. is the focus power. Here, the powers of the focal spot and total plane are integrated to obtain the power values. is the reflection coefficient of co-polarization under RHCP incidence. The efficiencies within 16.0 to 21.0 GHz are all over 80% (Fig. 5). Our design shows that the focal length shift of the WCAFR still presents a weak backward trend as the frequency increases, which objectively conforms to the traditional diffraction characteristic. Moreover, the proposed theory skillfully fits in the focusing theme and the planar WCAFR enables it to possibly achieve the weak chromatic aberration focusing across the microwave region from 16.0 to 21.0 GHz within the thickness.
Figure 5.Efficiencies of five observed frequency points.
4. NUMERICAL SIMULATIONS AND MEASUREMENTS
All numerical simulations were performed using CST Studio. The WCAFR was settled with open-add space along the , , and directions and a plane wave was configured as the incentive port in the axis. The field monitors were utilized to obtain the near-field results.
All measurements were performed in a microwave anechoic chamber to avoid possible interference from the environment. The focal lengths at different frequency points were obtained by measuring their intensity profiles along the propagation direction ( axis) in steps of 2 mm, as shown in Fig. 9. The details can be garnered in Appendix C.
The parameters of the meta-atom are as follows: , , , , , and . The component materials include F4B: , ; copper: ; and resistor: . The meta-atom has a metallic ground plate with a thickness of 0.2 mm.
5. CONCLUSIONS
In summary, a low-loss wideband meta-atom is achieved both in theory and fabrication, which shows the necessity of wideband anomalous dispersion. In this process, we first obtained the anomalous dispersion condition through deriving the reflective meta-atom in the view of SWM. After that, to promote the efficiency of LCMs, we employed the high-order plasmonic mode to match with F-P resonance so that the high efficiency LCMs enable higher anomalous dispersive properties. At the same time, the promoted meta-atom enlarges a higher phase accumulation and therefore presents nonlinear phase distribution both in the longitudinal and transverse dimensions. As a functional verification, an ultrathin WCAFR was proposed, which has high-efficiency focusing above 80% from 16.0 to 21.0 GHz within thickness for RHCP incidences. Compared with diffraction limits, the nonlinear phase difference enables the focal spot centered around 100 mm. Briefly, the proposed weak chromatic aberration focusing devices might provide alternatives to fields of electromagnetic communication, antenna design, and radar imaging, etc. And the concept can be readily extended to higher frequencies such as terahertz, infrared, and optical frequencies.
APPENDIX A: METHOD TO OBTAIN THE EFFECTIVE PERMITTIVITY OF META-ATOM
For normal incidence of the reflective metasurface, the calculation and extraction of effective permittivity are difficult. That is because the available data can only be plugged into a set of equations and cannot get the unique solution. Here, we skillfully obtain the effective permittivity by considering the continuous cavity thickness variation, which can indicate the wavelength in the media.
For the convenience of extraction permittivity, we set a lumped resistor at the middle position of the meta-atom, which can only produce the absorption impact for left-handed circularly polarized (LHCP) incidence [
In Fig.
Figure 6.Relations among the cavity thickness (
Figure 7.(a) The phase for a WCAFR at arbitrary frequency of
It is known from the propagation of light in the medium that the cavity refractive index is equal to the wavelength of the wave vacuum divided by the wavelength in the cavity. Meantime, the nonmagnetic resonator has magnetic field insensitivity and thereby the effective permeability is 1 [
APPENDIX B: THE ACHROMATIC PHASE REQUIREMENT FOR WIDEBAND
Taking the case of the one-dimension focusing lens, the phase profiles can be written as
The achromatic condition for planar lens means that the focal length should have no relation to the frequency. Therefore, we should eliminate the effect of frequency variable on focal length in the above equation set. Considering these, the following conditions should be established:
As the previous research indicated [
For the circular incidence scheme, we added a turning factor to the dipole model and made a further derivation. Supposing a light field that is normally incident on a dipole, the correctional drive equation is expressed as
But for the circular incidence scheme, the meta-atoms are always placed with different rotation angles to obtain the original phase. The phase is also called the Pancharatnam–Berry phase. Therefore, the above drive equation should be corrected with an angle (Fig.
Figure 8.Circular incidence scheme for dipole with the
Figure 9.Near-field demonstration for the WCAFR metasurface. (a) Schematic of near-field measuring for the proposed WCAFR metasurface. (b) The surroundings and relevant details for the measuring experiment.
Therefore, the position vector is solved as
Thus, the equivalent electric polarizability is calculated as
The real part of is
The plasma frequency of metal is extremely high. Therefore, in the microwave band, [
And we can further obtain the effective permittivity:
The order of magnitude of item is much smaller than the previous item (). Hence, the effective refractive index and final phase expression can be listed:
The phase distribution of the proposed meta-atom also verifies the above derivation, and it is shown in Fig.
APPENDIX C: THE DETAILS OF NUMERICAL SIMULATION AND EXPERIMENTALLY MEASURED CONDITIONS FOR THE VERIFIED METASURFACE
The FDTD numerical simulation results are shown in Figs.
Experimentally, the verified metasurface is fabricated with the printed circuit board process and measured in the anechoic chamber. We place a circular polarization horn antenna as the excitation port but with enough setup distance (1 m) to guarantee the equivalent plane wave. The near-field data are collected through scanning the region by probe. Restricted by the experimental conditions, we only scan a limited region of which is enough to present the WCAFR metasurface’s wideband focusing ability. In the paper, we have reduced the shown area of the result so that the short edge has the same width as the metasurface edge. In the process of scanning the region to collect data, we set the probe to move at a step of 3 mm in the direction and 2 mm in the direction.
For near-field simulation, the FDTD is chosen here, which can provide an intrinsically fast speed, and the CST Microwave Studio, which is much easier to operate than other software.
References
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