Ultrawideband chromatic aberration-free meta-mirrors

Chromatic aberration-free meta-devices (e.g., achromatic meta-devices and abnormal chromatic meta-devices) play an essential role in modern science and technology. However, current efforts suffer the issues of low efficiency, narrow operating band, and limited wavefront manipulation capability. We propose a general strategy to design chromatic aberration-free meta-devices with high-efficiency and ultrabroadband properties, which is realized by satisfying the key criteria of desirable phase dispersion and high reflection amplitudes at the target frequency interval. The phase dispersion is tuned successfully based on a multiresonant Lorentz model, and high reflection is guaranteed by the presence of the metallic ground. As proof of the concept, two microwave meta-devices are designed, fabricated, and experimentally characterized. An achromatic meta-mirror is proposed within 8 to 12 GHz, and another abnormal chromatic meta-mirror can tune the reflection angle as a linear function. Both meta-mirrors exhibit very high efficiencies (85% to 94% in the frequency band). Our findings open a door to realize chromatic aberration-free meta-devices with high efficiency and wideband properties and stimulate the realizations of chromatic aberration-free metadevices with other functionalities or working at higher frequency.


Introduction
The manipulation of electromagnetic (EM) waves with highefficiency and wideband is of great importance to increase the accuracy and efficiency of integrated optics components, especially for chromatic aberration-free meta-devices (e.g., achromatic meta-devices and abnormal chromatic meta-devices).
In this paper, we propose a multiresonant Lorentz model to control both the phase dispersion and reflection amplitude, guided by the transfer-matrix-method (TMM). [44][45][46] Realistic three-resonant meta-atoms are designed and optimized to achieve desirable phase dispersion and high reflection distributions, yielding the realizations of chromatic aberration-free meta-devices with high-efficiency and wideband properties. As proof of the concept, five frequency-dependent meta-mirrors with different chromatisms are designed, which show the wideband chromatic property (same phase gradient within the working band, see details in Sec. H of the Supplemental Materials), achromatic property (same deflection angle within the working band, see the design of the THz and optical achromatic metamirrors in Sec. G of the Supplemental Materials), and abnormal chromatic property (deflection angle changes as a linear function within the working band), respectively. More importantly, all of these devices can cover a broadband (microwave metamirror within 8 to 12 GHz, THz case in 0.5 to 0.8 THz, optical case within 1150 to 1875 nm), and realize very high working efficiencies (microwave meta-mirror in the range of 85% to 94%, THz case in the range of 75% to 84%, and optical case in the range of 85% to 91%). Our findings provide a solid step in solving the major problem of integrating the efficiency and bandwidth in meta-devices.

Materials and Methods
In our previous works, 31,32 we have proposed the highperformance criteria for reflective and transmissive metasurfaces working at a target frequency of f 0 by satisfying high amplitudes and required phase distributions at certain positions. We now describe our strategy to design chromatic aberrationfree meta-devices with high-efficiency and ultrawideband properties. As we know, the EM functionality of the meta-device can be determined by the total phase φ tot ðr; λÞ ¼ φ m ðr; λÞ þ φ p ðr; λÞ, 17 with φ m ðr; λÞ being the phase imparted by the metasurface, φ p ðr; λÞ being the phase accumulated by propagation length in free space, and λ being the wavelength at frequency of f. 17 Here, φ p ðr; λÞ ¼ 2π λ lðrÞ, with lðrÞ being the distance between the interference at r and the desired wavefront. Therefore, to ensure high efficiency and wideband properties of chromatic aberration-free meta-devices, we need not only to control the phase dispersion φ m ðr; λÞ as a function of frequency within an ultrabroad band but also to control the corresponding amplitudes (reflection or transmission) of the metasurface close to unit. Here, we adopt a reflective system working at the X band (8 to 12 GHz) with global mirror symmetries to control the dispersion efficiently. 31,32,[44][45][46] The reflective amplitude keeps close to unit because of the presence of a metallic ground layer, which can naturally reflect all incoming waves back. In this way, the key step is to control the distributions of φ m ðr; λ i Þ at local positions.
Take a meta-mirror as an example, the dispersion φ m should compensate at each frequency as where θ i ðf i Þ is the predesigned deflection angle, λ i is the wavelength at f i , and φ 0 is an initial phase. For a conventional chromatic meta-mirror, [9][10][11][12][13][14][15][16] the phase difference between adjacent meta-atoms (Δφ m ) is a constant at target frequency f 0 [see Fig. 1(a)] or, within a frequency interval from f 1 to f 2 [see Fig. 1(b)], the incident waves are bent to angles linked with wavelengths, and the deflection angle at the low frequency of f 1 is larger than that at the high frequency of f 2 . When θ i ðf i Þ equals a constant angel θ 0 within the frequency band [see Fig. 1(c)], an achromatic meta-mirror can be obtained. [17][18][19][20][21][22][23][24][25][26] In this case, φ m is linear against frequency while having different slopes for meta-atoms at different locations, and an abnormal chromatic meta-mirror can be achieved if θ i ðf i Þ changes as a function of frequency [see Fig. 1(d)]. Next, we use the effective medium theory to manipulate the dispersion distributions of a reflective meta-atom. [44][45][46][47] A typical reflective meta-atom is a sandwich structure, composed of a metallic patch resonator with size a 1 × b 1 and a continuous metal sheet, separated by a 1-mm-thick F4B spacer, as shown in the inset of Fig. 2(c), which can be well described by a Lorentz model. The F4B spacer has been considered as the dielectric losses by assuming the dielectric constant as ε r ¼ 2.65 þ 0.01i, and the metal is considered a perfect electric conductor since the meta-atom is working in the microwave regime. Figure 2(b) shows the effective-medium model of such a reflective metasurface. For a single resonant meta-atom, the Lorentz model can be described as where ω 0 , ω p , and γ are the resonance, plasma, and damping frequencies. 45,47 The reflection coefficient can be calculated using the 2 × 2 TMM as r ¼ −Q 21 ∕Q 22 , with Q being the transfer matrix. 45,47 We can tune ω 0 and ω p to obtain different reflection phase spectra by optimizing the patch structural parameter b 1 and a 1 in a realistic meta-atom. High agreements between TMM and the finite-difference-time-domain (FDTD) simulations in Fig. 2 2Þ GHz, f p ¼ ð60; 60; 60Þ GHz, and b 1 ¼ ð9.4; 8; 6.5Þ mm, respectively. The phase undergoes continuous −180 deg to 180 deg as the resonant frequency f 0 decreases gradually, as the reflection phase map shown in Fig. 2(d). However, the obviously resonant curves cannot satisfy the requirement of arbitrary dispersion calculated in Eq. (1), and the phase variation range is limited within −180 deg to 180 deg. Therefore, the single resonant meta-atom is unable to design chromatic aberration-free meta-devices. We solve the issues by constructing a multiresonant model. The reflection phase variation range can be enlarged by mutual interactions among multiresonant systems, 32,48,49 and the phase spectra slope can be modulated by carefully tuning these resonances. A Lorentz model with n resonances can be expressed as With ω i , ω ip , and γ i being the i'th resonance, plasma, and damping frequencies. We demonstrate this model by the dual-resonant meta-atom composed of a double-layered patch structure. Referring to Fig. 2(e), we can see that the reflection phase spectra calculated by FDTD can also coincide with those TMM results by properly tuning f 1 (ω 1 ) and f 2 (ω 2 ). To show clearly the effect of resonant frequency on φ m , Fig. 2(f) plots the dependence of the reflection phase spectra on f 1 . We can see that the phases change significantly at the low-frequency band while keeping nearly consistent in the high-frequency range, since the phase at low frequency is predominantly determined by f 1 . Oppositely, the phase spectra are quite insensitive at the low-frequency bands but are sensitive in the highfrequency range, as we only vary f 2 ; see details in Sec. A of the Supplemental Materials. By interchanging f 1 and f 2 , we can obtain a reflection phase map considering all dual resonances, which is summarized in Sec. A of the Supplemental Materials. From this map, we can successfully find some linear phase spectra at the frequency band of 8 to 12 GHz, while the phase variation range is still limited, inconvenient for the design of ultrawideband chromatic aberration-free meta-devices. Continuing to increase the number of resonances can further improve the phase variation range, while the realistic structure becomes more complex. Therefore, it is a tradeoff between the Typical scattering patterns of (b) wideband chromatic metamirror, (c) achromatic meta-mirror, and (d) abnormal chromatic meta-mirror with high performance within a wide bandwidth. The reflection angle at low frequency f 1 is larger than that at high frequency f 2 for (a) and (b) for the same phase gradients within the frequency interval from f 1 to f 2 . The reflection angle for the achromatic meta-mirror is the same as shown in (c). While the reflection angle satisfies a linear function for the abnormal chromatic meta-mirror in (d), the larger frequency f 2 exhibits a larger reflection angle. The frequencies satisfy f 2 > f 0 > f 1 .
structure complexity and manipulation capability of φ m . Finally, a Lorentz model with three resonances is selected, realized by a three-layered composite meta-atom. Figure 2(a) shows the schematic of a typical meta-atom composed of four metallic layers separated by three F4B dielectric spacers (total thickness of 3.5 mm).  For demonstration, Fig. 2(g) shows three representative linear phase spectra obtained by the TMM model and the corresponding realistic meta-atoms. The desirable φ m and high reflection amplitude (near unit) make a promise toward high-efficiency and ultrawideband chromatic aberration-free meta-devices. The bandwidth can be further extended by introducing more resonators. However, it is limited by the modulation freedom of the phase dispersion and also the meta-atom size, and the ratio of the up and low frequencies is about 2∶1. 53 In the following two sections, we will discuss two examples to illustrate our strategy and demonstrate the flexibility of dispersion manipulation for the meta-atoms.

Achromatic Meta-Mirror
We first design an achromatic meta-mirror with high-efficiency and ultrawideband properties using the proposed threelayered meta-atoms. To this end, the beam deflection angle at the frequency interval is set to be 30°, with Eq. (1) calculated as with N i being the i'th number of meta-atoms, and the total number is set as N ¼ 20. Figure 3(a) shows the calculated reflection phase distributions as a function of element number and frequency. Completely different from the reported beam deflectors, [9][10][11][12] there are no supercells in our design, and thus the phase can be tuned to vary as a linear function against the frequency with different slopes. To satisfy the unique requirement of phases, we retrieved the resonant frequencies in Eq. (3) from the reflection phase map in Fig. S3 in the Supplemental Materials. Then, we fixed the structural size of b 1 , b 2 , b 3 and a 1 , a 2 , a 3 of all meta-atoms to simultaneously satisfy these resonances, with the detailed parameters recorded in Sec. D of the Supplemental Materials. To validate our design, we fabricated an achromatic sample containing 20 × 20 metaatoms and occupying a total area of 200 mm × 200 mm.

Figures 3(b)-3(d)
show the photographs of the fabricated sample at different layers. The fabricated meta-atom reflection phase, phase differences compared with the theoretical case, and also the amplitude distribution are estimated in Sec. D of the Supplemental Materials. We note that the phase profiles of the fabricated meta-device match well with Eq. (4); meanwhile, the reflection amplitudes exhibit high values (jr xx j > 0.94).
With the fabricated sample in hand, we experimentally demonstrate its achromatic and high-efficiency performances. The achromatic property is considered first by measuring the scattering patterns within the operating frequency band. Illuminating anx-polarized microwave (7 to 13 GHz) normally onto our meta-device, we detected the angular distributions of scattered waves at the reflection side of the metasurface. As shown in Fig. 3(f), within our target frequency band (8 to 12 GHz), most incident EM waves are reflected to an identical angle 30°, as predesigned in Eq. (4), with the angle further confirmed by the measured and simulated radiation patterns shown in Fig. 3(g). More importantly, other undesired modes are totally suppressed, leaving only the anomalous reflection mode to survive, which implies high working efficiency of the designed achromatic meta-mirror. The FDTD simulations are performed to well reproduce almost all measured results, which are shown in Fig. 3(e). While outside the operating band, undesired modes, especially the specular reflections, increase significantly, 31,32,40 and the deflection angle shifts slightly from the theoretical case. To understand the physics of the achromatic property, we measure the electric field distributions on the xoz plane at the reflection space. As shown in Fig. 4(a), the electric field, including amplitude and phase information, is recorded by a self-designed monopole antenna (∼20 mm long) as our meta-device shined by anx-polarized plane wave. To see clearly the beam-bending effect, we deduce the incident field (obtained by repeating the measurement by moving away the meta-mirror 28,32 from the measured results, thus we can get only the scattered field). Figures 4(b)-4(f) show the Re½Ẽ distributions at the frequency interval of 8 to 12 GHz in the step of 1 GHz. The very clean beam bending signature and the exact deflection angles of 30°reinforce our notion of achromatism and high working efficiencies in a wide bandwidth.
Then, we quantitatively characterize the working efficiency of our achromatic meta-mirror. The efficiency is defined as the ratio between the power taken by the desired anomalously deflected mode and the incident power at different frequencies. 31,32 We experimentally obtain the power carried by the desired mode by integrating the scattered wave over an appropriate angle region at each frequency, while the incident power is calculated by integrating over the angle region of the specularly reflected mode when the metasurface is replaced by a metallic plate of the same size. Figure 3(h) shows the experimental and simulated efficiencies as a function of frequency. At the target frequency band of 8 to 12 GHz, the absolute efficiencies of our meta-device are within the range of 85% to 91% retrieved from our experimental results, and it is 85% to 94% based on the FDTD simulations, where the missing power is taken away by the dielectric loss (∼2% to 4% in both simulations and measurements) and other undesirable reflection. The slight difference between the simulation and measurement is attributed to inevitable fabrication errors and imperfections of the incoming wavefronts generated by our microwave horns. Outside the band, the efficiency decreases rapidly due to the uncontrolled dispersion φ m , which is similar to the reported cases. 9-33,54

Abnormal Chromatic Meta-Mirror
As the second example, we design an abnormal chromatic metamirror by manipulating the bending angles as a linear function of frequency. To not lose generality, the deflection angles θ i ðf i Þ are set as θ i ðf i Þ ¼ 8 þ 2f i , with the unit of f i in GHz and θ i in degrees. Therefore, the designed meta-device should exhibit the following phase distributions: To the best of our knowledge, it is the first time for controlling the deflection angle within broadband using a passive approach, because the dispersion of the passive materials is very difficult to control as desirable. 50,51 Here, we break the fixed relation of the deflection angle within the working band using the Lorentz model and extend the freedom of wavefront manipulation capability significantly for the meta-mirrors. The meta-atom number is fixed as N ¼ 16. Figure 5(a) plots the theoretical reflection phase spectrum for each meta-atom as a function of frequency. We note that the phase profiles are similar to that of the achromatic beam deflector except the phase slope, which can be used to examine the flexibility of our proposed strategy. We carefully optimize each meta-atom with its detailed structural parameters shown in Sec. E of the Supplemental Materials, together with the designed phase profiles, phase differences compared with those calculated by Eq. (5), and the amplitude distributions against frequency. The high reflection and desirable φ m indicate high performance of our meta-device. For practical applications, we fabricate a meta-device sample composed of three layers, with their pictures shown in Fig. 5(b). The sample consists of 16 × 20 meta-atoms with a total area of 160 mm × 200 mm.
The wave-manipulation performances of this meta-mirror are also experimentally examined. We first demonstrate its ultrabroadband beam bending property. Using an identical experimental configuration with that of the achromatic meta-device, we measure the angular distributions of scattered waves under anx-polarized incidence at different frequencies, with the results shown in Fig. 5(c). Within the frequency interval of 8 to 12 GHz, most incident power is reflected to an oblique angle as predesigned [solid pink star in Fig. 5(b)]. We perform FDTD simulations to calculate the scattered field distributions in Sec. F of the Supplemental Materials, which agree well with the measurements. We note clearly that the zero-order reflection mode almost disappears at the frequency band, indicating highperformance of our meta-mirror. The resultant linear beam deflection angles distinguish completely with the normal chromatic meta-mirrors, which have the same phase gradients, and their deflection angles have the fixed relation determined by the generalized Snell's law [solid white circles in Fig. 5(b)]. [9][10][11][12] In fact, the normal chromatic meta-mirror with wideband and high efficiencies can also be realized by our meta-atoms [see Fig. 1(b)], with the results shown in Sec. H of the Supplemental Materials. The designed normal chromatic meta-mirror can achieve beam bending effects within 8 to 12 GHz, and the efficiencies are in the range of 90% to 95%, which is better than the reported cases. 31,32,40 Second, we characterize the working efficiency of the designed meta-mirror. The working efficiency is calculated by integrating the anomalous reflection mode and the total power against frequencies. Figure 5(e) shows the simulated and measured results, indicating that the efficiencies are better than 85% (86%) for measurements (simulations) within the target 8 to 12 GHz, while it degrades significantly outside the band, which is due to the successfully suppressed undesired modes within the band, and the loss of the meta-device is <3.5% in the target frequency interval. We make the comparison of performances between the previous research and this work, as the results shown in Table S1 in Sec. J of the Supplemental Materials. It is obvious that the proposed chromatic aberrationfree meta-mirrors achieve desirable improvements both in very high working efficiency and bandwidth enhancement. Finally, we measure the Re½Ẽ distributions by recording the electric fields at the xoz plane at different frequencies, with the results shown in Fig. 5(d). Gradually increased deflection angles against increased frequencies demonstrate the feasibility of our design for a second time, and the deflection angles are retrieved, which is totally consistent with Eq. (5). Moreover, the parallel wavefront at each frequency is highly desired, indicating the efficient phase and dispersion manipulation capability of our meta-mirror.

Discussion
Our strategy to design chromatic aberration-free metadevices with high efficiencies and ultrawideband properties can work not only at the microwave but also at the high frequency band. To show this ability, we have designed a THz achromatic meta-mirror and an optical achromatic meta-mirror in reflection geometries by tuning both the material dispersion and phase properties. Similar with the microwave case, the THz achromatic meta-atom realizes the multiresonances using a composite structure consisting of four gold layers separated by three polyimide spacers. The meta-mirror can work at a broad bandwidth (0.5 to 0.8 THz) with high efficiencies (in the range of 75% to 84%), and the metal loss increases to 5% to 13% within the target frequency interval. At the optical region, we realize our strategy to tune both reflection phase and dispersion based on multiresonant dielectric meta-atoms. Three kinds of dielectric materials with different thicknesses are optimized to reach the required phase distributions. As a result, an optical achromatic meta-mirror over a continuous wavelength region from 1150 to 1875 nm is realized, and the efficiency is in the range of 85% to 91%. The details of meta-mirror design at the high frequency region are shown in Sec. G of the Supplemental Materials. Moreover, our strategy can also help us to design other chromatic aberration-free meta-devices in the high frequency range based on the multiresonant systems, which show great superiority over those reported achromatic devices. 17  We propose a strategy to design reflective chromatic aberrationfree meta-mirrors with ultrawideband and high-efficiency properties by tuning both the phase dispersion and reflection amplitude. Our strategy can be easily transferred to the transmission geometry with a similar criterion except that we tune the transmission phase dispersion and transmission amplitude in this case. The reflective chromatic aberration-free meta-mirrors are analyzed using the multiresonant Lorentz model theoretically and thus achieved in realistic structures using the proposed three-layered composite meta-atom. Two kinds of microwave meta-mirrors are designed/fabricated with very high efficiencies in the frequency interval of 8 to 12 GHz (in the range of 85% to 94%). One is the achromatic meta-mirror, which can achieve the same deflection angle at different frequencies. The other one is the abnormal chromatic meta-mirror, which can deflect the incident waves as a linear function of frequency. Our results establish a solid platform to efficiently manipulate EM waves within the broad band, which can stimulate further studies related to wideband meta-devices, such as achromatic meta-lens and frequency-controlled multifunctional meta-devices, and can stimulate the realization of high-performance optical components.

Appendix: Experimental Setup
All microwave samples were fabricated with a standard printedcircuit-board method. We perform both near-field and far-field experiments in an anechoic chamber to avoid interferences from the environment. In our near-field measurements, a horn antenna and a monopole antenna (with the length of ∼20 mm) are connected to a vector-field network analyzer (Agilent E8362C PNA) to record both the amplitude and phase information of the electric field with a pixel of 0.5 mm. The monopole antenna is digitally controlled to move in a spatial step of 0.5 mm; see Figs. 4(b)-4(f) and 5(d) for the measured Re½Ẽ distributions. In our far-field measurements, we fixed the meta-device and source antenna on a table that can rotate freely to emit the scattered signal in different directions and then used another identical antenna to receive the scattered-wave power. The receiver was placed 10 m away from the sample to guarantee that the measurement is carried out in a far-field approximation, and the results are shown in Figs. 3(f) and 5(c). More detailed information on the experimental setup can be found in Sec. I of the Supplemental Materials.