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Abstract
A wide-angle, wideband, and polarization-insensitive metamaterial (MM) absorber was studied based on the theoretical concept of uniaxial perfect matching layer (UPML) in the terahertz range. The MM absorber was designed as a multi-layered anisotropic array structure consisting of a conductive VIA, a bi-layered slot-FSS, and a split-ring resonator (SRR) separated by porous silica spacers. Each component is optimized to approach the required macroscopic uniaxial property for satisfying the reflection-less boundary condition. The SRRs and VIAs were found to play an important role in maintaining large absorption when the angle of oblique incidence increases under transverse electric (TE) and transverse magnetic (TM) polarization, respectively. The absorption achieves 90% in the frequency regime of 0.9 to 10.5 THz, corresponding to a bandwidth of 168% to the central frequency, and retains such high performance up to 60° oblique incidence for both TE and TM waves.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, metamaterial (MM) perfect absorbers and other related devices have attracted much attention due to their potential applications in terahertz communication, [1] and bio-medical images [2–5]. These MM perfect absorbers are usually composed of a periodic frequency selective surface (FSS), a dielectric spacer, and a metal ground plane to form resonant cavities. The attenuation of incident light was mainly originated from the large metallic ohmic losses accompanied with the resonance at the design spectral regime. Such conventional resonant-type absorbers are usually difficult to sustain broadband absorption, leading to a small fractional bandwidth with a typical value less than 20% with respect to the central frequency [6]. For example, various different MM absorbers have been demonstrated such as the cross structure [6], metal-insulator-metal (MIM) mushroom layer [7], free-standing bi-layered structure [8], electric coupled ring resonators [9,10] and variants [11,12] [13]. However, most of these designs suffer from severe efficiency degradation when operating at large oblique incidence, which limits their applications and functionalities.
In order to achieve broadband performance, MM absorbers have been designed to consist of highly-lossy materials and low quality-factor multi-band resonators with several resonant modes close to each other in their absorption spectrum. For example, dual-band and multi-band terahertz absorbers have been demonstrated by stacking multi-layered structures [3] and exploiting multiple different-sized resonators in one unit cell [9,14]. The mechanism of thin Salisbury’s screen was applied to achieve perfect absorption in optical regime [15]. In addition, the absorption bandwidth can be greatly enlarged when the ohmic resistive losses are introduced close to the periodic pattern such as by adding resistive sheets or by directly using resistive ink in the FSS-based periodic pattern [16]. This loss sheet of an FSS-based pattern in a terahertz absorber can be synthesized by depositing titanium or other high-resistive metal particles to form nano-scaled depth thin-film layers [17,18]. The metallic groove grating [19,20] and a cluster of cylindrical particles structure [21] have been demonstrated to support a broadband absorption under particular incident direction. As the absorption spectral response of these broadband absorbers is sensitive to the light polarization when oblique incident angle increases, this leads to a sacrifice of absorption magnitudes and bandwidth under oblique incidence unless the total thickness of absorber is much larger than the wavelength for magnetic-less material absorber.
In this paper, we proposed a wide-angle, wideband, polarization-insensitive terahertz absorptive metasurface consisting of a paired slot-FSS, a vertical rod VIA and a split-ring resonator (SRR) based on the design mechanism of uniaxial perfect match layer (UPML) [22,23]. UPML is an artificial virtual absorber widely applied as boundary conditions in full-wave simulation algorithm to approach infinite open-space computation. We first analyzed the relative permittivity and permeability tensors for a uniaxial medium to achieve perfect absorption. Then, circuit models were applied for designing the components of our terahertz MM absorber satisfying the required macroscopic material anisotropy. By tailoring the resonance of periodic VIAs [24–26] and SRRs [27,28], we can achieve effective anisotropic permittivity and permeability close to UPML. Finally, through full-wave simulations, we demonstrated the reflectance of our MM absorber is below 10% in the frequency range of 0.9 THz to 10.5 THz under normal incidence, corresponding to 168.5% bandwidth to the central wavelength. The absorption retains upon 90% when the incident angle is ranging from 0° to 60°degrees for both transverse electric (TE) and transverse magnetic (TM) waves.
2. Metamaterial absorber design methodology and model
Figure 1 schematically illustrates a general case where a plane wave propagates from an isotropic free space (region 1, z< 0) to a uniaxial medium with its optic-axis along the z axis (region 2, z> 0). The relative permittivity (permeability) tensor for the uniaxial medium can be expressed as,
Fig. 1. The schematic illustration of an incident plane wave and reflection occurred at an interface between an isotropic free space and a uniaxial-anisotropic medium.
Under oblique incidence, the wave vector in region 2is ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over \beta } _2} = \hat{y}{\beta _{2y}} + \hat{z}{\beta _{2z}}$, and thus the wave equation can be derived as
To satisfy Maxwell’s curl equation at the interface, one can deduce the reflection coefficient for TE and TM polarizations, respectively, with field-continuous boundary condition [23],
Due to phase-matching condition at z = 0 interface, the y- and z- component of the wave vector in region 2 are
Figure 2 illustrates the unit cell structure of our terahertz MM absorber consisting of a multi-layered structure with four components. The lattice constant of the unit cell is 68 μm. The top layer is resistive chromium SRR. The second and third layers are a resistive titanium bi-layered square ring-slot. The bottom layer is a solid metal ground plane. The spacer of the MM absorber is made of porous silica with refractive index of 1.05. The thickness of the porous silica spacer separated between the ground and bi-layered FSS layers are 25 μm and 12.5 μm, respectively. A resistive titanium vertical rod [25] was used to connect the ground plane and the middle layer. As a metallic thin film with a nano-scaled thickness can be characterized by high-impedance sheet resistance in optical band [29], the conductivity of the 4 nm chromium SRR was set as σ=3.3×105 S/m in our simulation [30]. The conductivities of the metal films were set to be constant as they have small variation in THz frequency range [30]. The thicknesses of the top- and bottom-layered slot FSS are 9 nm and 4.5 nm, respectively. The sheet resistance of 9 nm thick titanium slot-FSS and VIA [17,25] were set to be 250 Ω/□, which corresponds to a conductivity of σ=4.45×105 S/m [17,29,30]. Both 4.5 nm and 9 nm thick titanium slot-FSS have identical conductivity, and therefore the sheet resistance of 4.5 nm pattern is the double of 9 nm thick slot-FSS [29]. As graphene has been employed in THz MM design recently as well [31], such tunable conductive material can be an alternative choice to reach the proposed conductivity in demand [18,19,32,33].
Fig. 2. The unit cell of THz MM absorber with the periodicity LP of 68 μm. The total thickness (t2) of MM absorber from the ground plane to the top SRR is 50μm. The thickness t1 is half of t2, and t3 is quarter of t2. The pink part is a SRR with radius R1=31 μm, R2=20 μm, width w2=10 μm, and split-gap Wg=1 μm. The yellow parts in diagram are metallic VIA and slot-FSS patterns, respectively. The diameter of VIA(Dv) is equal to 20 μm. The slot-FSS has the geometric parameters of w1 = 7 μm, L1=62 μm, and L2=48 μm.
In this multi-layered structure, the periodic VIA is an array of rods with the effective permittivity showing Drude-like dispersion along the z direction as displayed in Fig. 3(a) [34], which is designed to account for the element ɛa in Eq. (1) of the dielectric tensor. The plasma frequency can be dictated by the radius (rvia) and periodicity (Lp) of the rod array [34] and can be expressed as,
The Drude-like dispersion of ɛa can be written asFig. 3. UPML condition with ɛa=μc. (a) The metal rod array shows Drude-like dispersion along the z direction. (b) The real part of ɛa, μc and μd. The values of ɛa and μc are closed to each other from 3 to 4.5 THz. The green dashed line indicates the permeability μd in vacuum.
To satisfy UPML condition, the frequency response of permeability element μc should have a Drude-like profile as well. However, a magnetic Drude-like dispersive material is absent in natural at high-frequency THz region. We thus utilized the SRR structure to approximate the magnetic Drude dispersion. Generally, the effective permeability of SRR is a Lorentz-like profile (i.e., ${\mu _c} = 1 - \frac{{\alpha {\omega ^2}}}{{({\omega ^2} - \omega _0^2) + j\omega \gamma }}$). The magnetic plasma frequency of SRR is determined by its average radius (rsrr), the split gap (Wg), and the periodicity (Lp) [35],
If the plasma frequency of VIA array and the magnetic plasma frequency of SRR are designed to correspond to the same value, the Lorentz and Drude-profile can be approximated to each other after the plasma frequency. Figure 3(b) shows the plasma frequency of VIA and the magnetic plasma frequency of SRR are designed to be the same at 0.75 THz. Thus, a good correspondence between ɛa and μc can be found in the frequency regime of 3 to 4.5 THz.
In addition, according to Eqs. (4) and (5), the element ɛb should be the inversion of ɛa. As ɛa preserves a Drude-like dispersion profile, ɛb should be a Lorentz profile with an inverse phase difference [Fig. 4(a)] to satisfy reciprocal relation,
Fig. 4. UPML condition with ɛa=ɛb−1. (a) The inverse of related ɛa as a function of frequency, showing a Lorentz profile with an inverse phase difference. (b) The calculated admittance spectra of a bi-layered slot-FSS by full-wave simulation, where the blue (red) curve is the real (imaginary) part of the admittance. The inset shows the equivalent circuit model of the bi-layer slot-FSS. (c) The corresponding effect permittivity of the slot-FSS, showing the required Lorentz profile with inverse phase after the resonance at 0.75 THz. (d) The real part of ɛa, ɛb−1 and μc. The values of ɛa and ɛb−1 show a good correspondence after 2 THz. The green dashed line indicates the permeability μd in vacuum.
Figure 4(b) shows the calculated admittance (Y) spectra of a bi-layered slot-FSS via the full-wave simulation, while the inset shows its circuit model as a twin RLC parallel circuit. The terminal boundary strip can be considered as an inductance, and the square-ring slot is represented by a series capacitance paralleled with an inductance. The intrinsic ohmic loss of the slot-FSS is taken into account through the resistances. Since the admittance (currents) and oscillated charged dipoles are out of phase with each other (J ∝ jωP), the differentiation of admittance with respect to frequency is proportional to the polarization vector with an opposite sign (-dY/df ∝ P). We can thus deduce the element ɛb by substituting the admittance differentiation into the polarization vector term in the following equation,
Finally, the reflection-less condition also requires that μd should be the inverse of μc. Since we implemented the SRR structure with Lorentz-like dispersion to approach the required matching condition between μc and ɛa, the inverse of μc leads to a Drude-like profile with an inverse phase for μd. However, a magnetic Drude-like permeability with inverse phase is only existed in some ferrite material sat low frequency around 1 to 10 MHz. Due to the lack of magnetic materials in high frequency regime, μd= μc−1 can be achieved only at a finite-band region within finite incident angles. Therefore, the value of μd for our proposed absorber is close to one without dispersion as shown in Figs. 3(b) and 4(d). Due to the development of MM in recent year, perfect magnetic mirror/absorber structure [15] gives us inspiration to approach the required μd, and we will conduct a compressive study in the future work.
Next, through substituting the deduced effective dispersive tensor elements ɛa, ɛb, μd, and μc into reflection coefficients [Eqs. (4) and (5)], the theoretically predicted reflectance spectra as a function of incident angles for TE and TM waves are displayed in Figs. 5(a) and 5(b), respectively. As expected, the absorber functionalized as a UPML maintains a low reflectance below 10% from 0° to 60° oblique incidence in the frequency range of 0.9THz to 10THz for both polarizations. While the material conductivity is increased in the order from 105 to 106, both polarized response will be degraded at low-frequency with large incident angle shown in Figs. 5(c) and (d). This can be attributed to the fact that the macroscopic anisotropic property of our design matches the tensor components of UPML well after 0.9THz [(Fig. 3(b) and Fig. 4(d)].
Fig. 5. Theoretically predicted reflectance spectra as a function of incident angles for (a) TE and (b) TM waves, respectively, through substituting effective dispersive ɛa, ɛb and μd, μc tensor elements into reflection coefficients. (c) and (d) show when the damping loss of each ɛa, ɛb and μc are reduced to one-tenth of (a) and (b), the absorption performance will be degraded at low-frequency regime with large incident angles. The white dash curves and black dash-dot curves indicate the gradient contours for 10% and 50% reflectance, respectively.
3. Full-wave simulation results
The MM absorber is further simulated and optimized using the commercial full-wave simulation software (Ansys HFSS), which is based on the finite element method in the frequency domain. Periodic boundary conditions and a floquet port are utilized to simulate the infinite periodic cell. Since the MM absorber substrate is metal-ground backed with negligible transmission, the absorption can be determined precisely by using only the reflection response. The reflection coefficient S11 is used to characterize the absorption property of the absorber, which can be expressed as A=1-|S11|2. The simulated reflectance spectra (Fig. 6(a) and Fig. 6(b)) show a good agreement with the theoretical prediction. The reflectance of the MM absorber is below 10% in a wide incident angles ranging from 0° to 60° for both TE and TM polarizations within the frequency regime of 0.9 to 10.5 THz, corresponding to 168% bandwidth to the central wavelength. In addition, although the high-loss SRR layer possesses C-2 symmetry, the absorber is still insensitive to the light polarizations as the working region of the absorber is beyond the resonance of SRR at normal incidence. In addition, the sheet resistance of a metal thin film is intrinsically determined by its thickness, we had tested the sheet resistance variation of SRR layer from the original design value of 4 nm [30] to a variation of +/- 25%. The reflectance of proposed absorber is still well below 10% in general as the film thickness of SRR varies from 3 to 4.5 nm. Due to the development of various deposition methods, the aforementioned thickness tolerance of metal films in our design should be acceptable [36,37]. As from the aspect of fabrication, the challenge usually lies in how to well define the patterns instead of the control of film thickness. The line-width of pulse laser irradiation processed pattern can range from 103 nm to 420 nm [37], which is less than our required 1 μm.
Fig. 6. Full-wave simulated reflectance spectra of the proposed structure as function of incident angles for (a) TE polarization and (b) TM polarization. Because ɛa and ɛb−1 are identical for most frequencies, the TM performance with large incident angle is better than TE. The white dash curves and black dash-dot curves indicate the gradient contours for 10% and 50% reflectance, respectively.
To study the origin of absorption enhancement, we analyzed the surface current distributions at the central frequency of 1.5 THz under normal and oblique incidence of 45°, respectively. Under normal incidence, surface currents can be observed to be mainly distributed on the middle square-slot resistive metallic layer [Fig. 7(a)]. For TE-polarized excitation under oblique incidence, the z-component of the magnetic field will induce a surface current on the SRR, leading to the coupling of the induced current in the lossy split circuit-loop by mutual-inductance between the inner split-ring and the outside split-circuit-loop. This gives rise to induced circular currents both on double concentric resistive SRR and on the middle slot-FSS layer [Figs. 7(b) and 7(c)], resulting in a magnetic dipole moment that supports an effective permeability μc along the optical-axis z direction. The electric field distribution on the top surface of SRR layer is shown Fig. 7(d), where the E-field is concentrated at split-gap of SRR. On the other hand, for TM-polarized wave under oblique incidence, the z-component of the incident electric field will be guided to the middle slot-layer by the VIA. The VIA will attenuate the magnitude of Ez field and result in radially flowed surface currents from central VIA toward slot-FSS [Fig. 7(e) and (f)].
Fig. 7. (a) The surface current distribution of multi-layered structure under normal incidence. For oblique 45° TE polarization, circular surface currents are formed (b) on the middle slot-FSS layer and (c) on double concentric resistive SRR. (d) The E-field distribution on the top SRR layer, which is concentrated on split-gap of SRR. For oblique 45° TM polarization, (e) surface currents and (f) electric field are radially distributed in the surrounding of VIA.
It is known that for a given frequency response, the total thickness of a nonmagnetic absorber cannot be less than a theoretical limit, which depends on the bandwidth and reflection coefficient [12,38]. The constraint relation of these parameters should satisfy the following equation
4. Conclusion
We demonstrated a wideband and wide-angle terahertz absorber based on the concept of UPML, where the reflectance can achieve below 10% from 0.9THz to 10.5THz, corresponding to 168% of the central frequency under 0° to 60° incidence, for both TE and TM waves. The SRR, a bi-layered slot-FSS, and VIA structures were optimized to achieve equivalent electric and magnetic plasma frequencies and similar dispersion profiles to approach the effective permittivity and permeability tensor of an ideal UPML. In addition, the SRR and VIA were found to play an important role in field attenuation especially for oblique incidence. The lossy vertical conductor VIA causes the decay of Ez field under oblique TM incidence, while the SRR structure weakens the Hz field under oblique TE incidence. This wideband, wide-angle, and polarization-insensitive THz absorber is promising for the applications in THz imaging, black-body and bolometric detector. The design concept can be extended to arbitrary frequency ranges, such as radio and infrared optical frequencies.
Funding
Ministry of Science and Technology, Taiwan (MOST 104-2221-E-002-058, MOST 107-2112-M-003-013-MY3, MOST 107-2221-E-002-153-MY2, MOST 107-2627-M- 002-007).
Disclosures
The authors declare no conflicts of interest.
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