Array Of Split-ring Resonators Research Articles

Design and analysis of ultra-wideband microstrip patch antenna with various conductive materials for terahertz gap

The paper explores the design and analysis of a wideband microstrip patch antenna with a metallic patch and a 3 × 3 split ring resonator (SRR) array operating in the 0.1–5 THz frequency range. The antenna's structure incorporates different conductive materials such as gold, silver, and graphene as a metallic patch. The dimensions of the metallic patch and SRR are calculated to achieve wideband operation within the desired THz range. The SRR array enhances electromagnetic resonance, thereby improving bandwidth and radiation characteristics for medical imaging. The study discusses the equivalent circuit and design equations for the microstrip patch antenna and SRR unit cell. For designing and analyzing the proposed antenna, CST Microwave Studio 2019 software have been used. Performance parameters such as return loss, bandwidth, gain, efficiency, directivity, VSWR, and radiation pattern have been evaluated. The advantages and limitations of each conductive material are evaluated to determine their suitability for THz-based medical imaging applications. The goal is to maximize the antenna's bandwidth, gain, and image resolution for medical imaging purposes. The findings highlight the performance characteristics of gold, silver, and graphene as conductive materials for medical imaging applications, facilitating the development of high-resolution, non-invasive imaging systems with improved diagnostic capabilities.

All‐metal wide‐angle and polarization‐independent microwave metamaterial absorber

AbstractAn innovative all‐metal self‐support microwave metamaterial absorber (MMA) based on a continuous complementary split ring resonator (CSRR) array frequency selective surface (FSS) is constructed. The CSRR‐based FSS has a selected frequency of 23 GHz and is proved to be polarization insensitive. And the all‐metal MMA has the same response frequency as the FSS, showing convenience in the geometrical scaling process. Additionally, in both transverse electric and transverse magnetic modes, the all‐metal MMA shows stable response at oblique incidences with an incident angle as large as 50°. Along with lightweight, easy fabrication and high cost‐performance ratio, the new proposed all‐metal FSS‐based MMA has good potential to serve in many practical applications.

Compact terahertz harmonic generation in the Reststrahlenband using a graphene-embedded metallic split ring resonator array

Harmonic generation is a result of a strong non-linear interaction between light and matter. It is a key technology for optics, as it allows the conversion of optical signals to higher frequencies. Owing to its intrinsically large and electrically tunable non-linear optical response, graphene has been used for high harmonic generation but, until now, only at frequencies < 2 THz, and with high-power ultrafast table-top lasers or accelerator-based structures. Here, we demonstrate third harmonic generation at 9.63 THz by optically pumping single-layer graphene, coupled to a circular split ring resonator (CSRR) array, with a 3.21 THz frequency quantum cascade laser (QCL). Combined with the high graphene nonlinearity, the mode confinement provided by the optically-pumped CSRR enhances the pump power density as well as that at the third harmonic, permitting harmonic generation. This approach enables potential access to a frequency range (6-12 THz) where compact sources remain difficult to obtain, owing to the Reststrahlenband of typical III-V semiconductors.

Terahertz MIMO antenna array for future generation of wireless applications

Abstract The paper proposes a high gain, low ECC, and high isolation THz MIMO antenna array for future generation wireless applications to accommodate a growing population of mobile users. A THz MIMO antenna array is designed with dimensions of 1200 × 2200 × 191.29 µm3. It employs the array of double-slit complementary split-ring resonators (D-CSRRs) on the bottom layer to improve the gain, return loss, and isolation. The gain of the proposed antenna array is 11.3 dBi with 66.45 % radiating efficiency at 0.65 THz. The −10 dB impedance bandwidth lies from 0.62–0.66 THz. The parameters of the MIMO antenna are evaluated such as isolation, envelope correlation coefficient (ECC), diversity gain (DG), and channel capacity loss. The ECC and DG value observed are 2.84 × 10−6 and 9.98 dB, respectively, at 0.65 THz. The channel capacity loss and isolation of the proposed MIMO antenna array are below 0.4 bps/Hz and (|S 21| &gt; 25 dB), respectively, in the complete operating band. The suggested THz MIMO antenna array can be utilised for THz high-speed wireless communication, video-rate imaging systems, sensing, the medical field for cancer imaging, security scanning, and the detection of illicit goods.

Enhancing the brain MRI at ultra-high field systems using a meta-array structure.

The main advantage of ultra-high field (UHF) magnetic resonance neuroimaging is theincreased signal-to-noise ratio (SNR) compared with lower field strength imaging. However, the wavelength effect associated with UHF MRI results in radiofrequency (RF) inhomogeneity, compromising whole brain coverage for many commercial coils. Approaches to resolving this issue of transmit field inhomogeneity include the design of parallel transmit systems (PTx), RF pulse design, and applying passive RF shimming such as high dielectric materials. However, these methods have some drawbacks such as unstable material parameters of dielectric pads, high-cost, and complexity of PTx systems. Metasurfaces are artificial structures with a unique platform that can control the propagation of the electromagnetic (EM) waves, and they are very promising for engineering EM device. Implementation of meta-arrays enhancing MRI has been explored previously in several studies. The aim of this study was to assess the effect of new meta-array technology on enhancing the brain MRI at 7T. A meta-array based on a hybrid structure consisting of an array of broadside-coupled split-ring resonators and high-permittivity materials was designed to work at the Larmor frequency of a 7 Tesla (7T) MRI scanner. When placed behind the head and neck, this construct improves the SNR in the region of the cerebellum,brainstem and the inferior aspect of the temporal lobes. Numerical electromagnetic simulations were performed to optimize the meta-array design parameters and determine the RF circuit configuration. The resultant transmit-efficiency and signal sensitivity improvements were experimentally analyzed in phantoms followed by healthy volunteers using a 7T whole-body MRI scanner equipped with a standard one-channel transmit, 32-channel receive head coil. Efficacy was evaluated through acquisition with and without the meta-array using two basic sequences: gradient-recalled-echo (GRE) and turbo-spin-echo (TSE). Experimental phantom analysis confirmed two-fold improvement in the transmit efficiency and 1.4-fold improvement in the signal sensitivity in the target region. In vivo GRE and TSE images with the meta-array in place showed enhanced visualization in inferior regions of the brain, especially of the cerebellum, brainstem, and cervical spinal cord. Addition of the meta-array to commonly used MRI coils can enhance SNR to extend the anatomical coverage of the coil and improve overall MRI coil performance. This enhancement in SNR can be leveraged to obtain a higher resolution image over the same time slot or faster acquisition can be achieved with same resolution. Using this technique could improve the performance of existing commercial coils at 7T for whole brain and other applications.

Topology optimization of array of split-ring resonators in two-dimensional acoustic waveguide for low-frequency transmission problems

This work focuses on the topology optimization of an array of split-ring resonators to achieve nearly zero transmission in the low-frequency regime ranging from 500 Hz to 1000 Hz in 2D waveguide, under the condition of normal-incident plane wave and based on a homogenization scheme known as the Effective Medium Approach (EMA). The EMA transforms each individual resonator defined by its outer radius (rout), neck thickness (h), and the slit width (d), into a homogenous medium characterized by frequency-dependent physical parameters. The transmission coefficient (Tcoeff) of the array can thus be easily evaluated without relying on computationally expensive Finite Element (FE) software. Two optimization algorithms, namely “Sequential Quadratic Programming (SQP)” and “Genetic Algorithm (GA),” have been implemented to the abovementioned topological parameters with sum of Tcoeff being the objective function to minimize. Accuracy of the results has also been validated using software COMSOL Multiphysics® and the satisfactory level of accuracy has proven this approach to be a more efficient and time-saving method to conduct topology optimization of split-ring resonators in low-frequency regime. Moreover, it is noteworthy that the optimized dimensions of resonators are over 20 times smaller than the wavelength in the low frequency range where almost-zero transmission is achieved.

Origin of plasmonic Fano resonance in metal-hole/split-ring-resonator metamaterials disclosed by temporal coupled-mode theory.

In plasmonic Fano resonance, the interaction between a discrete plasmonic mode and a continuum of plasmonic mode gives rise to an asymmetric line shape in the scattering or absorption spectrum, enabling a wide range of applications such as sensing, switching, and slow light devices. Here, we establish a theoretical solution in the framework of temporal coupled-mode theory (TCMT) to study the three-dimensional (3D) and two-dimensional (2D) Fano resonances induced by strong coupling between metal hole (MH) and split ring resonator (SRR) array. We first separately analyze the transmission spectra of the MH array and SRR array under different polarized light excitation. We further investigate the electromagnetic field and charge density distribution corresponding to the resonant modes at the peak or valley wavelength of the transmission spectrum and figure out the electric/magnetic dipole feature of these resonance modes. We then establish a theoretical solution by TCMT for Fano resonances arising from the coupling of these modes. The calculated transmission spectrum is closely matching with the numerically simulated transmission spectrum for these Fano resonances in the MH-SRR array, which effectively elucidates that the asymmetry of the Fano resonances is caused by the coupling between bright and dark plasmonic modes involved in the two structures. Our results can help to understand the profound physics in such coupled plasmonic systems.

Mutual Coupling Reduction in MIMO DRA through Metamaterials.

A single negative metamaterial structure with hexagonal split-ring resonators (H-SRRs) is inserted within a two-port multiple-input multiple-output (MIMO) dielectric resonator antenna (DRA) in order to achieve a reduction of mutual coupling between closed multiple antenna elements. Between closed, tightly coupled, high-profile antenna elements, the single negative magnetic inclusions (H-SRRs) are embedded. By incorporating magnetic structures within antenna elements, the mutual coupling is significantly diminished. Mutual coupling reduction is attained by inserting an array of hexagonal split-ring resonators between the inter-spacing elements. An operative approach for the reduction of the mutual coupling between two × two MIMO DRAs initially operating at 5.2-GHz band is provided. To make the simulated design replica of the fabricated prototype, an air gap is introduced between the substrate, DRs, and H-SSRs. The addition of the air gap shifts the simulated results to 5.9 GHz, which closely resembles the measured values. The mutual coupling reduction is realized by integrating a meta-surface amid the two × two MIMO DRAs, which are settled in the H-plane. The meta-surface embraces an array of hexagonal split-ring resonator (H-SRR) cells that are unified along the E-plane. The H-SRR structure is designed to offer band-stop functionality within the antenna bandwidth. The proposed design has an overall dimension of 40 × 58.3 × 4.75 mm3 (1.5λ × 1.02λ × 0.079λ). By stacking the DRA with a one × three array of H-SRR unit cells, a 30 dB reduction in the mutual coupling level is attained without compromising on the antenna performance. The corresponding mutual impedance of the MIMO DRA is better than 30 dB over 5.9-6.1 GHz operating bandwidth. The proposed design has a DG of 10 db, ECC < 0.02, CCL < 0.02 bits/s/Hz, and an MEG of 0 dB. The overall design has a promising performance, which shows its suitability for the target wireless application.

DNG Metamaterial-Based Superstrate for Performance Enhancement of Compact Size Wideband Patch Antenna

Recently, electromagnetic metamaterials are of great interest to researchers because of their unusual electromagnetic properties. The proposed work presents a metamaterial as a superstrate for enhancing the performance parameters of a conventional patch antenna. The metamaterial unit cell is examined using classical waveguide theory to confirm its metamaterial properties such as permittivity and permeability at the resonant frequency. The proposed meta-structure comprises of a 5 × 5 array of circular-shaped split ring resonators backed with wire strips and placed above an inset-fed rectangular patch antenna that resonates at 9.96 GHz. The metamaterial superstrate improves the gain, impedance bandwidth, and radiation efficiency by 2.4 dBi, 770 MHz, and 18.4%, respectively and simultaneously reduces reflection coefficient by 5 dB. Several benefits such as compact size, simpler structure, low profile, and low cost have been obtained from this design. EM simulation of the proposed composite structure has been carried out with the help of the HFSS EM simulator and measurement has been done for validation.

Investigation of Giant Nonlinearity in a Plasmonic Metasurface with Epsilon-Near-Zero Film

Plasmonic metamaterials can exhibit a variety of physical optical properties that offer extraordinary nonlinear conversion efficiency for ultra-compact nanodevice applications. Furthermore, the optical-rectification effect from the plasmonic nonlinear metasurfaces (NLMSs) can be used as a compact source of deep-subwavelength thickness to radiate broadband terahertz (THz) signals. Meanwhile, a novel dual-mode metasurface consisting of a split-ring resonator (SRR) array and an epsilon-near-zero (ENZ) layer was presented to boost the THz conversion efficiency further. In this paper, to explore the mechanism of THz generation from plasmonic NLMSs, the Maxwell-hydrodynamic multiphysics model is adopted to investigate complex linear and intrinsic nonlinear dynamics in plasmonics. We solve the multiphysics model using the finite-difference time-domain (FDTD) method, and the numerical results demonstrate the physical mechanism of the THz generation processes which cannot be observed in our previous experiments directly. The proposed method reveals a new approach for developing new types of high-conversion-efficiency nonlinear nanodevices.

Negative group-delay assisted anomalous propagation in stacked Split Ring Resonator array

This paper explores a novel method to achieve negative group delay-assisted anomalous propagation without significant signal attenuation using stacked Split-Ring Resonator (SRR) array in the microwave regime. A sub-wavelength stacking thickness between the SRR layers creates transverse magnetic dipole moments due to the excitation of anti-parallel currents on these layers resulting in increased transmission amplitude. Multipole scattering theory has been utilized to extract the reason behind anomalous propagation by evaluating the scattered power from the principal electric and magnetic dipole moments excited on the structure. Time-domain analysis are performed on the samples, and negative group delay behavior is confirmed by observing the precedence of the output electromagnetic Gaussian pulse within the anomalous dispersion regime. The coupling effects are studied numerically using full-wave solvers and verified in experiments by performing transmission measurements inside an anechoic chamber.

Split ring resonator inspired defected ground plane dual operating band antenna for wireless LAN and RFID applications

Defected ground structure (DGS) using an array of the three split ring resonators inspired antenna for the Wireless Local Area Network(WLAN) and RFID sensor is proposed in this manuscript. The complete physical measurement for the proposed antenna is 50 X 40 mm2 (at lower operating frequency). The defected ground plane structure along with the inclusion of the square split ring resonators (SRRs) in the ground plane helps in gain and bandwidth improvement. 1 × 3 split ring resonator array is used to make the existing ground plane to be defected. The cost effective FR – 4 materials is used as a substrate for the fabrication with the standard thickness of 1.6 mm. The resonance frequencies are at 2.4 and 5.8 GHz having the average gain 2.65dBi. The impedance bandwidths for both the frequencies are 5.0 % (2.31 to 2.49 GHz) and 3.9% (4.9 to 6.1 GHz). The dipole and omni directional radiation patterns, overall gain and efficiency of around 75% make the antenna most practically possible for the RFID and WLAN devices.

Series-Fed Coupled Split-Ring Resonator Array Antenna With Wide Fan-Beam and Low Sidelobe Level for Millimeter-Wave Automotive Radar

This paper proposes a millimeter-wave series-fed coupled split-ring resonator array (SCSRRA) antenna with a wide fan-beam pattern and low sidelobe level (SLL) for automotive radar. The proposed array antenna has the advantages of low cost, low profile, and low manufacturing difficulty. The proposed array antenna uses a split-ring resonator as a radiating element coupled from a microstrip line. The split-ring resonator has the advantage of suppressing cross-polarization due to feed phase errors. To reduce the surface wave through the ground edges and enhance the beamwidth, we added the mushroom-shaped electromagnetic bandgap (EBG) structure. Amplitude tapering to reduce the SLL was implemented by adjusting the distance between each element and the microstrip line. The prototype SCSRRA antennas operating at 79 GHz were fabricated, and the characteristics of the wide fan-beam pattern and low SLL were experimentally verified.

Flexible terahertz Metamaterial sensor for sensitive detection of imidacloprid

In this paper, a flexible terahertz (THz) Metamaterial (MM) with high sensitivity is proposed for the detection of the pesticide imidacloprid. The MM sensor is composed of a polyimide (PI) substrate and a periodic aluminum (Al) split ring resonator array. The sensing performance of the MM is verified by both simulation and experiment. The simulation results show that the refractive index (RI) sensitivity of the proposed MM sensor is 220.7 GHz/RIU, and the figure of merit (FOM) of this MM is 1.52. Then, the sensing performance of the MM sensor is verified by using imidacloprid solutions with different concentrations. The experimental results show that the detection sensitivity of the MM sensor is 12.3 GHz/(mg/ml), and the limit of detection (LOD) is 0.0112 mg/ml. The THz MM sensor proposed in this paper is flexible, simple to make, cost-effective, highly sensitive and has a very broad application prospect in the concentration detection of pesticides such as imidacloprid.

Battery-Free, Artificial Neural Network-Assisted Microwave Resonator Array for Ice Detection

Ice detection has developed into an integral part of aerospace and wind turbine sectors with the aim of preventing hazards and component breakdown. This article presents an ice detection system consisting of a battery-free, chip-less wirelessly interrogated resonator array, and an artificial neural network for enhanced detection robustness. The designed array of split-ring resonators (SRRs), operating at 3.05 GHz, was a narrowband frequency-selective structure with a ground plane reflector which shielded resonance from the effect of installation material. For an incident wave on the array’s surface, the reflection coefficient changed with the electrical properties of the ice and water, consequently affecting the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{11}$ </tex-math></inline-formula> parameter of an interrogator antenna. The array had a surface area of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$13\times6.5$ </tex-math></inline-formula> cm2 and a substrate thickness of 0.79 mm and was wirelessly interrogated from a distance of 33 cm by a standard gain horn antenna. A custom LabView program was utilized for time-based data acquisition of the antenna’s reflection coefficient [ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{11}$ </tex-math></inline-formula> (dB)], with results demonstrating a resonant frequency shift of 150 MHz when 30 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{L}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim$ </tex-math></inline-formula> ) of ice was formed on the main split of SRRs. The artificial neural network then classified the reflection coefficients of the interrogator antenna to enhance ice/water differentiation. The neural network improved the classification of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{11}$ </tex-math></inline-formula> (dB) raw data to 94.67% while achieving an accuracy of 93.33% for a noisy simulated data set. The proposed battery-free artificial neural network-assisted ice sensor can be implemented in various sizes (depending on the footprint requirements of installation) with applications in aerospace and wind turbine industry.

Efficient parallel strategy for molecular plasmonics – A numerical tool for integrating Maxwell-Schrödinger equations in three dimensions

An efficient parallelization approach to simulate optical properties of ensembles of quantum emitters in realistic electromagnetic environments is considered. It relies on balancing computing load of utilized processors and is built into three-dimensional domain decomposition methodology implemented for numerical integration of Maxwell's equations. The approach employed enables directly accessing dynamics of collective effects as the number of molecules in simulations can be drastically increased. Numerical experiments measuring speedup factors demonstrate the efficiency of the proposed methodology. As an example, we consider dynamics of nearly 700,000 diatomic molecules with ro-vibrational degrees of freedom explicitly accounted for coupled to electromagnetic radiation crafted by periodic arrays of split-ring resonators and triangular nanoholes. As an application of the approach, dissociation dynamics under strong coupling conditions is scrutinized. It is demonstrated that the dissociation rates are significantly affected near polaritonic frequencies.

DESIGN OF MINIATURE CIRCULAR MICROSTRIP PATCH ANTENNA BASED ON METAMATERIAL FOR Wi-Fi 5 AND Wi-Fi 6 APPLICATIONS

The objective of this work is to design a miniature circular shape patch antenna simulated by the CST microwave studio (CST MWS) software. This antenna based on metamaterials is to be used for wireless communication applications in Wi-Fi in the &amp;#91;5.15-5.875&amp;#93; GHz and &amp;#91;5.925-7.125&amp;#93; GHz frequency bands for Wi-Fi-5 and Wi-Fi-6 resonant frequencies, respectively. The miniaturization was achieved through periodic arrays of square-shaped complementary split ring resonators inserted at the ground plane of the structure. A miniaturization of the active zone of the radiating element was achieved up to 43.88&amp;#37;. For the purpose of increasing the bandwidth and gain of the miniature antenna, we subsequently applied the technique of modifying the ground plane geometry to have frequency bands around 450 MHz and 760 MHz with gains of the order of 5 dB and 3.34 dB for the two resonant frequencies 5 GHz and 6 GHz, respectively. The simulation results are discussed and commented in terms of return loss, bandwidth, gain, and radiation patterns.

Complementary Vanadium Dioxide Metamaterial with Enhanced Modulation Amplitude at Terahertz Frequencies

One route to create tunable metamaterials is through integration with ``on-demand'' dynamic quantum materials, such as vanadium dioxide (${\mathrm{VO}}_{2}$). This enables modalities to create high-performance devices for historically challenging applications. Indeed, dynamic materials have often been integrated with metamaterials to imbue artificial structures with some degree of tunability. Conversely, metamaterials can be used to enhance and extend the natural tuning range of dynamic materials. Utilizing a complementary split-ring resonator array deposited on a ${\mathrm{VO}}_{2}$ film, we demonstrate enhanced terahertz transmission modulation upon traversing the insulator-to-metal transition (IMT) at approximately $340$ K. Our complementary metamaterial increases the modulation amplitude of the original ${\mathrm{VO}}_{2}$ film from 0.42 to 0.68 at 0.47 THz upon crossing the IMT, corresponding to an enhancement of 62%. Moreover, temperature-dependent transmission measurements reveal a significant redshift of the resonant frequency in a narrow temperature range where phase coexistence is known to occur. Neither Maxwell-Garnett nor Bruggeman effective medium theory adequately describes the observed frequency shift and amplitude decrease. However, a Drude model incorporating a significant increase of the effective permittivity does describe the experimentally observed redshift. Our results highlight that symbiotic integration of metamaterial arrays with quantum materials provides a powerful approach to engineer emergent functionality.

Ultraviolet and visible double Fano resonances of three-dimensional metamaterials by plasmonic coupling with aluminium surface

We theoretically investigate double Fano resonances (FRs) from the ultraviolet to visible range by combining the array of aluminium 3D split-ring resonators (Al 3D-SRRs) and an Al mirror. The interaction between the first-order (1st) magnetic plasmon (MP) mode of the 3D-SRRs and the 1st surface plasmon polariton (SPP) mode propagating on an Al substrate, and the interaction between the second-order (2nd) MP mode and the 2nd SPP mode generate the double narrow FRs. The 2nd ultraviolet FR of the proposed metamaterial has a high refractive index sensitivity of 240 nm RIU−1 and a figure of merit of 22, suggesting promising applications in ultraviolet sensing.

Ultraviolet Sharp High-Order Magnetic Plasmon Based Fano Resonance of 3D Metamaterials

We theoretically study the ultraviolet sharp Fano resonances (FRs) produced through the optical diffraction coupling between the narrow collective surface mode supported by the array periodicity and the broad high-order magnetic plasmon (MP) mode in the three-dimensional (3D) metamaterials consist of aluminum vertical split-ring resonator (Al 3D-SRR) array. The linewidth of the second-order MP induced ultraviolet FR is as narrow as 7.3 nm due to the diffraction coupling feature of the magnetic fields. The intensity and position of the ultraviolet sharp FR are easily tuned by varying the array period as well as the base and arm length of the Al 3D-SRR structure. The conventionally defined performance index of optical sensors, sensitivity ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> ) and figure of merit ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FoM</i> ) of our designed 3D metamaterial in the ultraviolet range, possess quite large values of 88 nm/RIU and 12, respectively.