Introduction

The absorption of electromagnetic energies is of great importance in a wide range of applications1,2,3,4, including electromagnetic compatibility5,6, radar cross-section reduction and energy harvesting7,8,9, etc. Traditional pyramidal and multilayer dielectric absorbers are broadly used in microwave engineering for scattering and radiation measurements10. However, due to their poor performance at large impinging angles and electrically large size, these absorbers usually require the conventional anechoic chambers to be adequately large, e.g., tens to several thousands of wavelengths10,11,12,13. Metamaterials offer the prospect of implementing compact absorbers with ultra-subwavelength thicknesses. Since the advent of the first metamaterial absorber (MMA) in 200814, the design and implementation of MMAs have been a hot topic till date1,15,16,17,18,19,20. However, in most previous works, high absorption efficiencies can only be obtained at a single or several specific illumination angles21,22. The realization of omnidirectional absorption with high efficiency has been widely acknowledged as a challenging task due to the intrinsic angular selective impedance match23. Although the well-known uniaxial perfectly matched layer (UPML), previously defined mathematically in computational electromagnetics24,25, have been theoretically demonstrated to allow omnidirectional absorption, the requirements on gain elements render them difficult to implement practically. To the best of our knowledge, the experimentally implemented UPML absorbers are all with reduced parameters23,26, and thus can optimally work only with specific incident angles. Similarly, the absorptive parity-time symmetric metamaterials27,28,29 and transformation optical media30,31 face the same challenge owing to the requirements on gain elements.

In 2015, Horsley et al. theoretically demonstrated that a class of isotropic, one-dimensionally (1D) inhomogeneous susceptibility profiles can be utilized to realize omnidirectionally perfect absorption without involving any gain elements, provided that they satisfy the spatial Kramers-Kronig (KK) relation32,33,34,35,36,37. Recently, it has been shown that, apart from achieving polarization-independent broadband omnidirectional and direction-dependent invisibility38, the spatial KK relation can also be extended to X-ray nanoimaging and chiral phase modulation39,40, showing great promises for all-optical functional devices in quantum information processing and optical communication networks. So far, the spatial KK media for the absorption of TE-polarized waves have been demonstrated experimentally by designing gradient mesoscopic structures with specially tailored electric responses41,42,43. To the best of our knowledge, simultaneous perfect absorption for both TE and TM polarizations has yet to be realized experimentally, probably owing to the difficulty in implementing isotropic spatial KK profiles in practice.

In this work, we report the experimental realization of a polarization-independent omnidirectionally impedance-matched absorber utilizing the spatial KK medium. To realize the required electric and magnetic responses simultaneously, we propose a matryoshka metamaterial, whose effective permittivity and permeability can be independently tuned to the spatial KK profile with minimized crosstalk. To extend the absorption bandwidth, the inner truncation boundary is set at a position far away from the spatial Lorentz resonance. As a result, the metamaterial at the truncation boundary has constant permittivity and permeability over a broadband frequency range. To demonstrate the polarization-independent omnidirectional absorption performance, a cylindrical anechoic chamber, 6.83-wavelength in diameter, is fabricated and implemented with the designed annulus-shaped spatial KK absorber. Analytical solutions, full-wave simulations, and experimental measurements all confirm the polarization-independent omnidirectionally reflectionless absorption feature. Our design strategy can be easily extended to higher frequencies, paving a way to implement polarization-independent absorbers for applications in photovoltaics, bolometry, photodetection, stealth technology, mechanical manipulation, etc.44,45,46.

Results and discussion

Theoretical analysis

Previous literature shows that a spatial KK profile can be achieved utilizing a spatially inhomogeneous dispersive metamaterial, whose relative permittivity εr (or permeability μr) is characterized by an r-dependent Lorentz resonance, in the form of43,47

$${\varepsilon }_{{{{{{\rm{r}}}}}}}(\omega ,r)=1-\frac{{\omega }_{P}^{2}}{{\omega }^{2}+i\gamma \omega -{({\omega }_{L}-qr)}^{2}}.$$
(1)

Here, ωL—qr, γ, and ωP are the resonance, damping, and plasma frequencies, respectively. q determines the gradient of this r-dependent resonance frequency. Such a spatially inhomogeneous profile spans over the whole space (from r = 0 to r = +∞), and thus has to be truncated for the practical implementation. We highlight that an infinite spatial KK medium can achieve perfect light absorption at any frequency, whereas the truncated one can only absorb light over a finite spectrum. Hence, the truncation boundaries must be properly designed to cover the frequency range of interest. There are in general two criteria: (1) at the inner truncation boundary, εr(ω, r) → 1 (i.e., the KK medium is impedance matched to air) over the whole frequency range of interest; (2) at the outer truncation boundary, ωL—qr < 0 such that light propagating from the inner to the outer truncation boundary can experience the spatial Lorentz resonance to get absorbed without any reflection. These criteria require the two truncation boundaries to be set far away from the spatial Lorentz resonance.

To give a quantitative illustration, we consider a spatial Lorentz-profile medium with ωP = 0.4ω0, ωL = 2.35ω0, γ = 0.02ω0, and q = 0.1ω0/λ0 (λ0 is the wavelength in vacuum for frequency ω0). Figure 1a plots the real and imaginary parts of εr(ω, r) as a function of ω and r. To give more details, Fig. 1b plots the εr(ω, r) as a function of r at three different frequencies, ω = 0.9ω0, ω = 1.0ω0 and ω = 1.1ω0, respectively. Note that when the inner truncation boundary is set at r = 10λ0, the permittivities at 0.9ω0 (εr = 1.17), 1.0ω0 (εr = 1.19), and 1.1ω0 (εr = 1.24) remain a constant around 1.2 with negligible imaginary parts. In other words, setting r = 10λ0 can readily minimize the reflection at the inner truncation boundary from 0.9ω0 to 1.1ω0, if the background has a permittivity of 1.2. On the other hand, to ensure the total absorption, the outer truncation boundary is set at r = 15λ0, (i.e., the truncated spatial permittivity Lorentz-profile medium has a thickness of 5λ0). Considering practical applications, it is necessary to make the spatial permittivity Lorentz-profile medium impedance matched to air. To this end, we can introduce a thin layer with the permittivity linearly varying from 1 to 1.2 within r = 9λ0 and r = 10λ0 (1λ0 in thick). This linear-profile medium can be easily obtained in a broad frequency band using dielectric-air composites, according to the Maxwell Garnett mixing principle48. It should be noted that this linear-profile medium improves the impedance matching and extends the operating bandwidth, especially for larger incident angles. Detailed discussions on the impact of the linear-profile medium can be found in Supplementary Note 1. In our previous work, such a linear-profile medium is not required, because the proposed spatial KK profile involves two typical Lorentz resonances, whose relative permittivity is equal to 1 at a special position42. The inner truncation boundary is set at this position, and the KK medium is then impedance matched to free space. Finally, to show the absorption behavior of the whole KK medium (comprised of 5λ0-thick spatial Lorentz-profile medium and 1λ0-thick linear-profile medium), we plot in Fig. 1c absorptance A as functions of the incident angle θ and working frequency ω. The algorithm used in the calculation is the same as our previous work49. Apparently, over a broad frequency band from 0.85ω0 to 1.15ω0, the absorptance A > 99% is obtained when the incident angle varies from 0° to 78°, confirming the broadband nearly omnidirectionally reflectionless absorption.

Fig. 1: Space-frequency KK permittivity profile.
figure 1

a Real (I) and imaginary (II) parts of the permittivity profile defined by Eq. (1), with ωP = 0.4ω0, ωL = 2.35ω0, γ = 0.02ω0, and q = 0.1ω0/λ0. b The r-dependent spatial permittivity profile at ω = 0.9ω0, ω = 1.0ω0 and ω = 1.1ω0. c Calculated absorptance A as functions of the incident angle and working frequency for the air-matched KK medium slab. d Simulated amplitude distributions while placing a line source with a z-polarized electric field in the air region surrounded by the cylindrical air-matched KK medium. Panels I, III show the electric field amplitude distributions at ω = 0.95ω0 when the source is located at (0, 0) and (7λ0, 0), respectively. Panels II, IV show the corresponding results at ω = 1.05ω0.

Full size image

Such a finite air-matched cylindrical KK medium is an excellent candidate to realize miniaturized anechoic chamber. As an example, we use this air-matched permittivity profile to construct a cylindrical anechoic chamber. To verify its absorption behavior, we perform full-wave simulations by placing a line source with a z-polarized electric field at (0, 0) and (7λ0, 0) within this anechoic chamber. To illustrate the broadband responses, simulations are conducted at two different frequencies 0.95ω0 (panels I, III of Fig. 1d) and 1.05ω0 (panels II, IV of Fig. 1d). The results clearly show that, at both frequencies, light impinging upon the boundary r = 9λ0 smoothly goes into the KK medium without any observable reflection and eventually gets dissipated at the positions of the corresponding spatial Lorentz resonances. To quantitatively evaluate the reflection, we define Rmax/Rmin as the maximum standing wave radio. Here, Rmax and Rmin are the maximum and minimum field amplitudes normalized to the line source field in free space (note that Rmax/Rmin = 1 corresponds to zero reflection in the chamber). The calculated Rmax/Rmin in panels I, II, III, and IV of Fig. 1d are only 1.01, 1.02, 1.13 and 1.11, respectively, confirming the source-independent absorption behavior of the anechoic chamber designed above.

Design and simulation

Our purpose is to realize a cylindrical spatial KK medium to absorb unpolarized light. It requires both the axial permittivity and permeability profiles to satisfy Eq. (1). In order to design a unit cell with desired magnetic and electric Lorentz resonances with negligible crosstalk, we introduce a matryoshka structure composed of metallic split rings and deformed I-shaped metallic patterns50,51,52,53. The split rings (top right corner of Fig. 2a) and I-shaped patterns (top left corner of Fig. 2a) are responsible to absorb TM (with a z-polarized magnetic field) and TE (with a z-polarized electric field) polarized waves, respectively. Such unit cells are aligned periodically along the z and azimuthal φ directions. The radial width d is fixed as 2.5 mm for all unit cells. It is smaller than λ0/10 at the operating frequencies from 10 to 10.5 GHz. Hence, the reflection resulting from the structural discretization is minimized and the whole structure can be considered as homogeneous42,43. To retrieve the effective permittivity and permeability, we perform full-wave simulations on this unit cell using periodic boundary condition. The detailed descriptions about the simulation setup are given in Supplementary Note 2. As a special example, Fig. 2b plots the effective permittivity and permeability of the unit cell using the retrieval algorithm of reference54. Interestingly, such a unit cell supports electric and magnetic resonances at the same frequency with negligible crosstalk.

Fig. 2: Design of the unit for the polarization-independent KK profile.
figure 2

a Geometry of the matryoshka structure composed of metallic split rings and deformed I-shaped metallic patterns. b Retrieved complex constitutive parameters along different directions, with d = 2.5 mm, W0 = 6.5 mm, W1 = 2 mm, W2 = 2.47 mm, W3 = 1 mm, W4 = 0.7 mm, W5 = 0.25 mm, L1 = 1.225 mm, L2 = 1.82 mm, L3 = 0.135 mm, L4 = 0.175 mm, L5 = 0.1 mm, R0 = 0.2 mm and h = 1.2 mm.

Full size image

The polarization-independent spatial KK medium requires that the electric and magnetic resonances both have the same r-dependent resonance frequency ωL—qr. To realize such a response, we line up 30 layers along the radial direction from r2 = 61d to r3 = 90d, (d = 2.5 mm) and gradually increase W1 from 2.0 mm to 3.07 mm and W2 from 2.47 mm to 3.70 mm (other dimensions remain unchanged). Figure 3a depicts the retrieved axial permittivity and permeability of the designed spatial Lorentz-profile medium dependent on the frequency (ω) and the radius (r). The detailed dimensions of W1 and W2 in each layer from r2 to r3 are shown in Supplementary Note 3. Through proper numerical fittings, such space-frequency dependences can be described by,

$${\varepsilon }_{{{{{{\rm{r}}}}}}z}(\omega ,r)=1.2-\frac{{(2\times {10}^{9})}^{2}}{{\omega }^{2}+i\omega \times 70\times {10}^{6}-{(14.78\times {10}^{9}-0.055\times {10}^{9}\times \frac{r}{d})}^{2}}$$
(2)

and

$${\mu }_{{{{{{\rm{r}}}}}}z}(\omega ,r)=1-\frac{{(2.1\times {10}^{9})}^{2}}{{\omega }^{2}+i\omega \times 60\times {10}^{6}-{(14.78\times {10}^{9}-0.055\times {10}^{9}\times \frac{r}{d})}^{2}},$$
(3)

in accordance with our theoretical analysis above.

Fig. 3: Actual design of the polarization-independent spatial KK medium.
figure 3

a Retrieved axial relative permittivity (I, II) and permeability (III, IV) of the spatial Lorentz-profile medium, dependent on the frequency and the space variable r. b Entire strip of unit cells along the radial direction to construct the cylindrical air-matched spatial KK medium. In the spatial Lorentz-profile medium (r2 ≤ r < r3), the parameters W1 and W2 for each unit (Detailed geometric dimensions they denote can be found in Fig. 2a) change along the radial direction to form the desired spatial profile. Similarly, the parameters Ws and Wa, denoting the lengths of the dielectric strips and metallic wires, respectively, also change along the radial direction to constitute the linear-profile medium (r1 ≤ r < r2) (Details can be found in Supplementary Note 4). dL and d denote the periods of units to construct the Linear-profile medium and the Lorentz-profile medium along the radial direction, respectively. c r-dependent axial permittivity (I) and permeability (II) of the designed cylindrical anechoic chamber using the air-matched KK medium (gray regions) at 10 and 10.5 GHz. d, e Calculated absorptance of a slab with the spatial KK profiles shown in (c), under TE (d), and TM (e) wave incidences, respectively. f Simulated z-polarized electric field distributions while placing a line source inside the anechoic chamber composed of the designed KK medium. Panels I and II show the results at 10 GHz when the line source is located at (0, 0), and (9 cm, 0), respectively. Panels III and IV show the corresponding results at 10.5 GHz. g Simulated magnetic field distributions when the line source is with a z-polarized magnetic field.

Full size image

At the inner truncation boundary (r = r2) of the spatial Lorentz profiles, the constitutive parameters from 10 GHz to 10.5 GHz are constants: εrz ≈ 1.3, εrφ ≈ 2.2, and μrz ≈ μrφ ≈ 1. As analyzed above, a linear-profile medium is introduced to achieve impedance matching with air. In particular, the required linear-profile medium is nonmagnetic but anisotropic. In our implementation, this linear-profile medium is divided into 2 parts: the dielectric-air composite (with εrz linearly varying from 1 to 1.3 and εrφ linearly varying from 1 to 1.5), and the cut-wire metamaterial (with εrz = 1.3 and εrφ linearly varying from 1.5 to 2.2)55. Figure 3b shows the entire strip of unit cells along the radial direction, which consists of the linear-profile medium (r1 ≤ r < r2) and the designed spatial Lorentz-profile medium (r2 ≤ r < r3). The linear-profile medium is composed of 50 layers (each layer is 1 mm thick) along the radial direction, where the first 20 layers are the dielectric-air composite and the last 30 layers are the cut-wire metamaterial. The detailed design and geometric dimensions of the linear-profile medium is given in Supplementary Note 4.

Figure 3c plots the entire axial permittivity (left panel) and permeability (right panel) of this air-matched layered KK medium (gray region: r1 ≤ r < r3), as a function of r at two different frequencies 10 GHz and 10.5 GHz. At both frequencies, the layered KK medium (1) supports spatial electric and magnetic Lorentz resonances simultaneously, and (2) is impedance matched to free space at r = r1 for both TE and TM polarizations. In other words, the layered KK medium designed above is expected to achieve polarization-independent absorption over a bandwidth of at least 0.5 GHz from 10 to 10.5 GHz. To illustrate its absorption performance, we calculate the absorptance A with respect to the frequency and incident angle based on the retrieved parameters. The results given in Fig. 3d, e show that, in the frequency range from 9.8 to 11.0 GHz (9.7−10.9 GHz), the absorptance is over 99% for TE (TM) polarizations when the incident angle is less than 78o (74o). It is seen that the absorption of TE wave is better than that of TM wave, especially for large incident angles (around 75°− 85°). It comes from the different frequency dispersions of permittivity and permeability, as shown in Fig. 2b. The relative permittivity approaches to 1.2 at low frequencies while the relative permeability approaches to 1. The different asymptotic values of ε and μ at low frequencies originate from the nonmagnetic substrate, of which εs > 1 and μs = 1. As a result, the permeability of the designed KK medium experiences a sharper change at r2 than the permittivity. Finite element simulations based on CST Microwave Studio have also been performed to analyze the absorption performance of the real structure. The results given in Supplementary Note 5 reveal an omnidirectional absorption over 90% from 9.95 to 11.01 GHz and 9.65 to 10.69 GHz for TE and TM polarizations, respectively.

To realize a miniaturized anechoic chamber, we roll up the layered KK medium (consisting of 5 cm thick linear-profile medium and 7.5 cm thick Lorentz-profile medium) into an annulus structure with an inner radius of 10 cm and an outer one of 22.5 cm. Full-wave simulations are performed while placing a line source inside the anechoic chamber. Figure 3f, g depict the results for TE and TM polarizations, respectively. In panels I and II, the line source is placed at (0, 0) and (9 cm, 0) and the incident frequency is set to 10 GHz. Similar results under 10.5 GHz incidence are plotted in panels III and IV. As expected, neither TE nor TM incidence leads to observable standing wave or phase distortion, e.g., the maximum standing wave ratios Rmax/Rmin for panels (I)−(IV) in Fig. 3f, g are only 1.040 (1.012), 1.260 (1.122), 1.041 (1.052), and 1.294 (1.124), respectively. Similar to Fig. 1d, the incoming radiations at different frequencies are significantly dissipated around the corresponding spatial Lorentz resonances. The 1D plots of the fields along the x-axis given in Supplementary Note 6 show that the field intensities are depleted by more than 60 dB and 90 dB within the KK medium for TE and TM polarizations, respectively. These results confirm that the layered KK medium can perform omnidirectional polarization-independent absorption, similar to the homogeneous one studied in Fig. 1.

Experimental measurements

To verify the polarization-independent absorption behavior, we perform the measurement of the field distributions within the fabricated anechoic chamber, as shown in Fig. 4a. Fabrication details can be found in the Methods section. For the TE polarization, a 2-dimensional (2D) omnidirectional monopole antenna is placed inside the anechoic chamber to serve as the source, whilst another monopole antenna is used as the probe to measure the local electric fields. The source and probe antennae are connected to the input and output ports of a network analyzer (Agilent E8361A), respectively. For the TM polarization, the monopole antennae are replaced by omnidirectional loop antenna and the local magnetic fields are measured using a similar procedure. Figures 4b and c plot the measured amplitude and phase (in the form of cos (phase)) distributions of the z-polarized electric fields under TE incidence. Corresponding results under TM incidence are shown in Fig. 4d, e. The measurements are performed at two frequencies, 10 GHz (panels I and II) and 10.5 GHz (panels III and IV), and for two source locations, (0, 0) (panels I and III) and (9 cm, 0) (panels II and IV). Apparently, for all cases, the amplitude of the field varies smoothly and decreases slowly away from the source antenna without obvious reflection, with successfully compressed phase distortion.

Fig. 4: Experimental setup and measured results.
figure 4

a Photograph of the fabricated sample and experimental setup. Top two insets show the antennae exciting and receiving TE, and TM waves, respectively. b, c Measured amplitude (b) and phase (c) distributions of the z-polarized electric field in the air region, respectively, where the phase distributions are plotted in the form of cos (phase). Panels I and II show the results measured at 10 GHz when the line source is located at (0, 0), and (9 cm, 0), respectively. Panels III, IV show the corresponding results at 10.5 GHz for two source locations. d, e Corresponding measured amplitude (d) and phase (e) distributions of the z-polarized magnetic field.

Full size image

We also assess quantitatively the quiet zone of the fabricated anechoic chamber by measuring the perturbation of S11. Here, S11 denotes the reflection coefficient of the probe antenna, which was firstly calibrated by placing the antenna in free space (as a “matched load”). Thus, the probe antenna can work with the highest sensitivity to the adjacent scatterers. Then, we placed the probe inside the designed miniaturized anechoic chamber and measured S11 point by point. In such a case, |S11| distributions can be used to evaluate the reflection from the anechoic chamber. The smaller the |S11 | , the better the anechoic chamber. Detailed measurement process and experimental results are presented in Supplementary Note 7. Supplementary Fig. 6 demonstrates that the perturbed |S11 | (dB) is less than −29 dB and −33 dB for TE and TM polarization, respectively, a clear proof that the miniaturized anechoic chamber designed with the KK medium is an artificial analogue of free space for both TE and TM polarizations. We highlight that the tiny fluctuations of the measured S11 result from the discontinuous edges of the KK medium along the z-direction, which can be further suppressed by increasing the height of the sample.

Conclusions

In conclusion, a polarization-independent KK medium is experimentally realized in this work. The key to our design is a special matryoshka metamaterial that can support both magnetic and electrical resonances at the same frequency with negligible crosstalk. This design not only provides a feasible solution to the longstanding problem, i.e., to achieve broadband, omnidirectionally matched absorption of unpolarized wave, but also adds a new perspective on the metamaterial design to overcome the polarization limitation. An annulus-shaped anechoic chamber with a diameter of 6.83 wavelengths was fabricated. It can achieve omnidirectional polarization-independent absorption for both TE and TM polarizations, as confirmed by both numerical simulations and experimental measurements.

Considering that the designed spatial resonance positions get farther away from the inner truncation boundary as the frequency decreases, as shown in Fig. 3c, the operating frequencies can be further extended to lower ones only if we add more gradient units along the r direction to cover the resonance regions. In such a case, the operating bandwidth can be flexibly controlled by tuning the outer truncation positions of the space-frequency KK medium. Meanwhile, it should be noted that the designed KK medium is actually only for 2D case, due to the anisotropic building units. If the resonant units are deliberately designed to be 3D isotropic, the proposed approach can also be easily extended to the full 3D case, such as designing all-dielectric spherical or cubic resonators56. Certainly, they can also be designed using 3D symmetrical metallic structures but with more complicated shapes57. Finally, this method can also be extended to higher frequencies such as near-infrared and optical regimes, using all-dielectric Mie resonators instead of metallic structures, which have been widely used to construct optical metamaterials56,58. This approach provides a new route to realize ideal absorbers, and shows great application prospect in stealth technology, electromagnetic compatibility, and photodetection.

Methods

In the practical implementation, we fabricate the layered KK medium using PCB printing. Figure 4a depicts the fabricated annulus-shaped KK medium, with all geometric dimensions the same as those in Fig. 3f, g. Microwave-transparent foam plates with a thickness of 4.8 mm are placed between each PCB layer along the z-axis to support the sample. A total of 20-layer PCB samples are stacked in the z-direction. Finally, the miniaturized anechoic chamber with a 10 cm inner radius, 22.5 cm outer radius, and 12 cm height is obtained.