Frequency-dependent Coefficients Research Articles

Diffusion coefficients preserving long-time correlations: consequences on the Einstein relation and on entanglement in a bosonic Bogoliubov system

Abstract We analytically derive the diffusion coefficients that drive a system of N coupled harmonic oscillators to an equilibrium state exhibiting persistent correlations. It is shown that the main effect of the latter consists in a renormalization of the natural frequencies and the friction coefficients of the oscillators. We find that the Einstein relation may be satisfied at low temperatures with frequency-dependent effective friction coefficients provided that the physical constraints are fulfilled. We also investigate entanglement evolution in a bipartite bosonic Bogoliubov system initially prepared in a thermal squeezed state. It is found that, in contrast to what one may expect, strong coupling slows down sudden death of the entanglement, and for initially separable states entanglement generation may occur.

Equivalent Circuit Model of Magnetoelectric Composite Nanoparticles

This study presents an analysis of magnetoelectric nanoparticles (MENPs) through the development of equivalent circuits to predict the frequency-dependent magnetoelectric coefficient, with a focus on the widely utilized CoFe2O4@BaTiO3 core–shell configuration. This approach involves –derivation of phenomenological expressions that capture the dynamic behavior of MENPs under varying magnetic and electric fields. By integrating piezoelectric and magnetostrictive constitutive equations, along with consideration of dynamic effects and bio-load conjugation, a magneto-elasto-electric effect equivalent circuit has been constructed. This circuit model not only facilitates the investigation of longitudinal data in cube-shaped MENPs but also offers insights into fundamental biological processes. The versatility of this model is shown through translation to other core–shell nanoparticles, composite structures, and multiferroic nanostructures. This analysis provides quantitative predictions of the magnetoelectric coefficients, enhancing general understanding of MENP characteristics across a broad frequency range. Furthermore, the study highlights the framework for future refinement to incorporate intrinsic composition-specific resonances, such as ferromagnetic and ferroelectric resonances, to further significantly improve the nanoparticles’ performance. Overall, this work lays the groundwork for future technology to intelligently and wirelessly control biological processes using MENPs, thus paving a way for innovative biomedical applications. This quantitative approach may facilitate further interdisciplinary research and contribute to advancement of magnetoelectric materials and their applications.

Prediction of Frequency-Dependent Optical Spectrum for Solid Materials: A Multioutput and Multifidelity Machine Learning Approach.

The frequency-dependent optical spectrum is pivotal for a broad range of applications from material characterization to optoelectronics and energy harvesting. Data-driven surrogate models, trained on density functional theory (DFT) data, have effectively alleviated the scalability limitations of DFT while preserving its chemical accuracy, expediting material discovery. However, prevailing machine learning (ML) efforts often focus on scalar properties such as the band gap, overlooking the complexities of optical spectra. In this work, we employ deep graph neural networks (GNNs) to predict the frequency-dependent complex-valued dielectric function across the infrared, visible, and ultraviolet spectra directly from the crystal structures. We explore multiple architectures for the spectral multioutput representation of the dielectric function and utilize various multifidelity learning strategies, such as transfer learning and fidelity embedding, to address the challenges associated with the scarcity of high-fidelity DFT data. Additionally, we model key solar cell absorption efficiency metrics, demonstrating that learning these parameters is enhanced when integrated through a learning bias within the learning of the frequency-dependent absorption coefficient. This study demonstrates that leveraging multioutput and multifidelity ML techniques enables accurate predictions of optical spectra from crystal structures, providing a versatile tool for rapidly screening materials for optoelectronics, optical sensing, and solar energy applications across an extensive frequency spectrum.

Rate theory of gas-liquid nucleation: Quest for the elusive quantitative accuracy.

The task of a first principles theoretical calculation of the rate of gas-liquid nucleation has remained largely incomplete despite the existence of reliable results from unbiased simulation studies at large supersaturation. Although the classical nucleation theory formulated by Becker-Doring-Zeldovich about a century ago provides an elegant, widely used picture of nucleation in a first-order phase transition, the theory finds difficulties in predicting the rate accurately, especially in the case of gas-to-liquid nucleation. Here, we use a multiple-order parameter description to construct the nucleation free energy surface needed to calculate the nucleation rate. A multidimensional non-Markovian (MDNM) rate theory formulation that generalizes Langer's well-known nucleation theory by using the Grote-Hynes MDNM treatment is used to obtain the rate of barrier crossing. We find good agreement of the theory with the rate obtained by direct unbiased molecular dynamics simulations-the latter is feasible at large supersaturation, S. The theory gives an experimentally strong dependence of the rate of nucleation on supersaturation, S. Interestingly, we find a strong influence of the frequency-dependent friction coefficient at the barrier top. This arises from multiple recrossings of the barrier surface. We find that a Markovian theory, such as Langer's formulation, fails to capture the rate quantitatively. In addition, the multidimensional transition state theory expression performs poorly, revealing the underlying role of the friction coefficient.

Effect of hetero-valence ions on structural and electrical properties of La2Ni(Mn0.5Ti0.5)O6 ceramics

The study of powder X-ray diffraction of La2Ni(Mn0.5Ti0.5)O6 (0.5LNMT) confirms a multi-phasic composition. The monoclinic P21/n phase content dominates. Occurrence of minor rhombohedral R3‾c phase and precipitates of MnO2 and La2MnO4 were detected. The morphology was analyzed using a SEM test and ceramics showed a porous structure and variation in the grain size below 4 μm. Dielectric permittivity of marked magnitude varies from ∼30 to ∼100 000 when temperature increases. Dielectric loss coefficient frequency dependence exhibits well-defined dielectric relaxation peaks, which were found to obey the Arrhenius law with the activation energy of 0.29 eV, below the presumed phase transition. Two sets of relaxation peaks in the imaginary part of electric modulus spectra relate to relaxations of different activation energies of 0.28 eV and 0.38 eV. An optical gap magnitude of 0.8 eV was determined from the diffuse reflectance spectrum in the Vis-NIR range. It compares with the band gap estimated from the X-ray photoelectron spectroscopy test. We also compare the influence of doping ions on the crystal lattice of the surface and the bulk ceramics. Electrical conductivity relaxation processes were attributed to the electron hopping mechanism between multi-valence Mnk+ and Nin+ ions’ states, which were determined from XPS analysis. The dc resistivity shows thermally generated dependence. Activation energy Edc magnitude varies between 0.27 and 0.37 eV which indicates the occurrence of defects and structural disorder. Such disorder correlates to the variable range hopping of small polarons involved in electrical conductivity. The density of states near Fermi energy was estimated N(EF) = 2.1 × 1025 eV−1 m−3. The 0.5LNMT application prospects were discussed.

Effect of retarded friction and added mass on the swimming speed of a vibrating two-sphere

A theoretical expression is derived for the mechanical contribution to the mean swimming speed of a vibrating two-sphere, consisting of two spheres connected by an elastic spring, and immersed in a viscous incompressible fluid. The spring provides a harmonic potential for oscillations about a mean distance between centers. The system is made to oscillate at a chosen frequency by activating forces which sum to zero. The mechanical contribution to the resulting mean swimming velocity is calculated from the mechanical equations of motion and the corresponding impedance matrix of linear response. The frequency-dependent pair friction coefficients are found from approximate expressions derived earlier. The mechanical contribution is calculated to second order in the amplitude of stroke as a function of the scaling number, a dimensionless combination of size, frequency, and kinematic viscosity. Retarded friction and added mass determine the functional behavior.

Study on the electronic structures and NLO properties of C6H6F6 doped superalkalis X3M (X=Li, na, K; M=O, S, Se)

To explore novel high-performance nonlinear optical (NLO) materials, this paper focuses on studying and discussing the geometric, electrical, and NLO properties of superalkali-doped C6H6F6. Geometrical and thermodynamic properties have been calculated by density-functional theory (DFT) calculations at the ωB97XD/6-31 + G (d,p) level of theory. Electronic properties related to charge transfer between the superalkali and C6H6F6 were obtained through calculations such as NBO charge analysis and frontier molecular orbitals (FMO). The electronic properties are rationalized by the density of states analysis, and in this paper, using the two-energy level model to analyzed the factors that affect the first hyperpolarizability. The frequency-dependent hyperpolarisation coefficients are analyzed to understand the dynamic NLO response. To obtain the NLO response of these new complexes experimentally, the Hyper-Rayleigh scattering (βHRS) coefficients of the complexes were also investigated. The dynamic and static hyperpolarizability properties of these complexes show good NLO response. It is also hoped that this work may provide theoretical help for the future synthesis of new efficient NLO materials.

Utilizing discarded face masks to fabricate sustainable high-performance panels for enhanced building thermal and acoustic comfort

This study aims to design, fabricate, and optimize fibrous panels made of discarded polypropylene face masks (FMs) for application in sound absorption and thermal insulation. The discarded FMs were shredded and transformed into a fibrous mass using a Garnet machine. The resulting fibers, after being mixed with a bio-based binder, were molded based on an experimental design using response surface methodology in conjunction with the central composite experiment design (RSM-CCD) with five different thicknesses, densities, and blending ratios of spunbond to meltblown fabrics. The average of the sound absorption coefficients for the twelve one‐third‐octave bands from 200 to 2500 Hz (SAA) and effective thermal conductivity (Keff) of the panels were measured as the dependent parameters. These values were calculated within the ranges of 0.28–0.53 and 0.031–0.056 W/(mK), respectively. Additionally, the interactive effects of input variables on SAA and Keff of the manufactured panels were discussed. The quadratic and two-factor interaction models were proposed for optimizing the values of SAA and Keff, respectively. The optimal conditions were determined to achieve the maximum SAA and minimum Keff in a thickness of 26.7 mm, density of 135 kg/m3, and blending percentage of 32.4 %. Under these conditions, the measured values of SAA and Keff were 0.534 and 0.0279 W/mK, respectively. Finally, three fibrous panels were manufactured under optimal conditions, and their acoustic and thermal properties were measured. It was observed that excellent agreement exists between the model results and the experimental data with a 95% confidence level. The bending properties of the optimally designed panels were also investigated and the panels exhibited an impressive flexural strength of 2.1 MPa. Additionally, the fire performance of the panels was examined and all panels showed self-extinguishing properties. These findings suggest that panels made of discarded FMs hold promise as an eco-friendly and sustainable construction material for manufacturing acoustic and thermal insulation panels. Moreover, the frequency-dependent absorption coefficients of the panels were predicted using the Johnson-Champoux-Allard model.

Dielectric Response and Linear Absorption Spectroscopy of Ionic Systems.

Time-dependent electric fields applied to ionic systems can induce both a dielectric and a conductive response, leading to the generation of macroscopic polarization and current, respectively. It has long been recognized that it is not possible to determine the two types of responses separately. However, this aspect is often not adequately accounted for in dielectric and absorption spectroscopies of ionic systems. To clarify this, we theoretically investigate the dielectric and conductive responses of ionic systems containing polyatomic ions based on linear response theory. We derive general expressions for the frequency-dependent dielectric functions, conductivity, and absorption coefficient, including those measured experimentally. Furthermore, we show that the dielectric and conductive responses cannot be uniquely distinguished even at the theoretical level and, therefore, cannot represent experimentally measured quantities. Instead, dielectric and absorption spectra of ionic systems should be expressed in terms of the generalized dielectric function that encompasses both dielectric and conductive responses. We propose a computational method to calculate this generalized dielectric function reliably. Model calculations on concentrated aqueous solutions of NaCl, a monatomic salt, and LiTFSI, a polyatomic salt, show that the dielectric and linear absorption spectra of the two systems based on the generalized dielectric function are significantly different from purely dielectric counterparts in the far-IR, terahertz, and lower-frequency regions. Moreover, the spectra are mainly determined by the autocorrelations of total dipole and total current, but dipole-current cross-correlation can also significantly contribute to the spectra of the LiTFSI solution. The present theoretical approach could be extended to nonlinear spectroscopy of ionic liquids and electrolyte solutions.

A promising exponentially-fitted two-derivative Runge–Kutta–Nyström method for solving [formula omitted]: Application to Verhulst logistic growth model

Explicit exponentially-fitted two-derivative Runge–Kutta–Nyström method with single f-function and multiple third derivatives is proposed for solving special type of second-order ordinary differential equations with exponential solutions. B-series and rooted tree theory for the proposed method are developed for the derivation of order conditions. Then, we build frequency-dependent coefficients for the proposed method by integrating the second-order initial value problem exactly with solution in the linear composition of set functions eλt and e−λt with λ∈R. An exponentially-fitted two-derivative Runge–Kutta–Nyström method with three stages fifth order is derived. Linear stability and stability region of the proposed method are analyzed. The numerical tests show that the proposed method is more effective than other existing methods with similar algebraic order in the integration of special type of second-order ordinary differential equations with exponential solutions. Also, the proposed method is used to solve a famous application problem, Verhulst logistic growth model and the result shows the proposed method still works effectively for solving this model.

Prediction of surface admittance impulse responses from frequency-dependent sound absorption coefficients

In numerical room acoustic modeling certain material properties are required as inputs for setting the boundary conditions for the calculations. In finite element or finite difference models, locally reactive boundaries are often represented as FIR or IIR filters. However, most building material manufacturers provide sound absorption coefficient or equivalent absorption area data only, in accordance with the relevant standards, so a method to convert sound absorption to admittance or impedance impulse responses is required. Since there is a loss of information when admittance is converted to sound absorption coefficients, the conversion in the opposite direction has multiple solutions. We derive an iterative method of determining either the impedance or admittance impulse response from arbitrary, frequency-dependent diffuse sound absorption coefficients and validate it with the transfer matrix method. We present the implementation of the method in the soundy.ai application.

A finite element study of absorption coefficient measurement at low frequencies

The frequency-dependent absorption coefficients of an acoustically absorbent material sample can be measured using the reverberation chamber method, defined by the ISO standard 354:2003. The lower frequency bound for the measurement defined in the standard is 100 Hz, since measurement data below this frequency may be unreliable. However, the ability to measure absorption coefficients at low frequencies is becoming increasingly more important, to facilitate the enhanced control of low frequency sound fields. This paper presents a finite element method study of the reverberation chamber measurement method at frequencies below 100 Hz, and the predicted performance of the ISO standard method is reported. Additionally, a recently proposed method that makes use of finite element-based eigenvalue analyses to estimate the locally reacting complex-valued impedance of a sample at modal frequencies is applied to the problem. It is shown that the proposed method can be used to estimate frequency-dependent absorption coefficients at modal frequencies below 100 Hz.

A Contribution to Experimental Identification of Frequency-Dependent Dynamic Coefficients of Tilting-Pad Journal Bearings with Centered and Off-Centered Pivot

Linearized dynamic bearing parameters and models are of essential interest for rotordynamic analyses in machine design. This paper experimentally studies the impact of pad preload and pivot offset on the frequency-dependent characteristics of dynamic stiffness (K) and damping coefficients (C) of a KC-model for tilting-pad journal bearings. For this purpose, two four-pad test bearing configurations (preload m=0.17, pivot offset 0.5 and preload m=0.47, pivot offset 0.6) that differ highly with respect to the pad design parameters are investigated. Contributing effects on the results due to geometric differences are excluded as far as possible, as only one aligning ring and one pivot support design is used. The tests are conducted on a high-performance test rig for surface speeds up to 140 m/s and excitation frequencies of 500 Hz. Significant deviations between the two bearings are identified that generally match theoretically predicted differences and, therefore, contribute to the validation of dynamic bearing modeling.

Determining Attenuation Q-Factor Values for Selected Seismic Stations in Iraq

The coda wave decay (CWD) method was used to estimate the quality factor from the coda wave amplitude decay rate for the broadband seismic stations (IBDR, MSL, and NSR4) using a single backscattering model. This research aims to calculate the attenuation coefficient and the frequency-dependent coefficient (α) for three stations and compare them to comprehend the details of Earth's crust structure in those stations. Then, to test if they are homogenous or not. Also, investigate how sensitive extracted values to tectonic differences in study stations. The attenuation properties are important for providing information about the medium that seismic waves pass through and can also be applied to areas with seismic activity as initial disaster mitigation. The study used data from the Mesopotamian Network (MP) and the Iraqi Meteorological Organization and Seismology (IMOS) from 2011 to 2021. The selected earthquakes have less than < 100 km epicentral distances, and magnitudes ranging from 3.5 to 5.6. Forty-six earthquakes were recorded by three stations (IBDR, MSL, and NSR4). These earthquakes were analyzed at eight center frequencies (1, 2, 3.5, 6, 8, 10, 12, and 16 Hz) and a 30-sec length window by using the SEISAN Earthquake Analysis Software. The QC values are computed by the frequency-dependent relationship QC= Q0 f α, where Q0 is QC at 1Hz and α denotes frequency dependence degree. The attenuation relationship of the station IBDR is obtained as QC =98 f 1.07, QC = 99 f 1.09 in the MSL station, and QC = 77 f 0.98 in the NSR4 station. The results showed that low Q0 corresponds to high attenuation. The comparison of the estimated QC for these three stations indicates that attenuation is highest at NSR4, followed by other stations.

Research on the forward-modeling method of viscoelastic Chapman extension in partially saturated rocks

The propagation of seismic waves in fractured rocks is greatly influenced by the fracture system and fluid content. Seismic anisotropy varies with frequency and fluid saturation. Previous theories on frequency-dependent anisotropy (FDA) are mostly limited to the assumption of single-phase fluid, whereas almost all reservoirs are usually partially or fully saturated with one or more fluids. The inelasticity, heterogeneity, and anisotropy exhibited by real earth media seriously challenge the traditional theory of uniform fully elastic media. Therefore, based on the Chapman elastic theory, we consider the viscoelastic model, the relative mobility of saturated fluids, and the coupled effects of commonly independently considered squirt flow and patch effects on the frequency-dependent anisotropic seismic wave propagation in viscoelastic media. By calculating the frequency-dependent anisotropic elastic coefficients of two-phase immiscible fluid-saturated fractured rocks, a new viscoelastic Chapman extension model for FDA in partially saturated rocks, which includes the unified seismic wave propagation effects of squirt flow and patch effects, is constructed. The effect of relative permeability is significant, and compared to the case of complete saturation, the effective fluid mobility in partially saturated rocks may be lower, which may lead to nonmonotonic changes in elastic modulus and water saturation. Numerical experiments are conducted on our viscoelastic Chapman extension model to discuss the coupling effects of squirt flow and patch effects on the FDA of viscoelastic media in cracked rocks with partially saturated reservoirs, under the conditions of no fractures and the existence of fractures. The simulation results of the P-wave modulus, P-wave velocity, and anisotropy parameters confirm the validity of our method and model. We link the anisotropy of viscoelastic media with fluid flow parameters in fractures, which is conducive to improving the understanding of frequency-dependent seismic anisotropy in partially saturated rocks with fractures and the combination of seismology and reservoir engineering.

Mathematical model of spatial and temporal processing of broadband signals in satellite radio systems of broadband access and radio navigation

To effectively receive a useful broadband signal, it is necessary to compensate for distortions caused by the complex frequency-dependent coefficient of the receiving and transmitting antenna, multiplicative interference and noise from the heterogeneity of the propagation medium, additive interference and noise. Using only correlation processing under certain conditions will not allow receiving a useful broadband signal with the required quality. The article develops a mathematical model of spatio-temporal processing of broadband signals in satellite broadband access systems, showing that spatio-temporal processing should be carried out in two stages. At the first stage, space-time processing is carried out in the channel adaptively to the antenna array based on the formation of a complex frequency-dependent vector of weighting coefficients, which should change with a change in the signal-interference situation in real time, taking into account the change in the direction of radiation sources. At the second stage, correlation processing of the broadband signal is performed based on the use of long-length code sequences. This model is the basis for the development of methods for the formation and correlation (temporal) processing of broadband signals and methods of spatio-temporal processing of broadband signals under conditions of intentional and unintentional exposure. Based on the presented mathematical model of spatio-temporal processing of broadband signals, the main directions of increasing the noise immunity of radio-electronic systems are determined.

Efficient Frequency-Dependent Coefficients of Explicit Improved Two-Derivative Runge-Kutta Type Methods for Solving Third-Order IVPs

This study aims to propose sixth-order two-derivative improved Runge-Kutta type methods adopted with exponentially-fitting and trigonometrically-fitting techniques for integrating a special type of third-order ordinary differential equation in the form u^''' (t)=f(t,u(t),u^' (t)). The procedure of constructing order conditions comprised of a few previous steps, k-i for third-order two-derivative Runge-Kutta-type methods, has been outlined. These methods are developed through the idea of integrating initial value problems exactly with a numerical solution in the form of linear composition of the set functions e^ѡt and e^(-ѡt)for exponentially fitted and e^iѡt and e^(-iѡt) for trigonometrically-fitted with ѡ ϵ R. Parameters of two-derivative Runge-Kutta type method are adapted into principle frequency of exponential and oscillatory problems to construct the proposed methods. Error analysis of proposed methods is analysed, and the computational efficiency of proposed methods is demonstrated in numerical experiments compared to other existing numerical methods for integrating third-order ordinary differential equations with an exponential and periodic solution.

Terahertz Vibrational Modes of Sodium Magnesium Chlorophyllin and Chlorophyll in Plant Leaves

The low-frequency (terahertz) vibrational spectroscopy has been investigated experimentally for two chlorophyll species, Chl-a and one of its magnesium derivatives (Chl-Mg-Na). The combination of terahertz time-domain spectroscopy and Fourier transform infrared spectroscopy has enabled a broad frequency range to be covered (0.2 to 18 THz). For Chl-Mg-Na, the terahertz spectra show clear and well-marked features at 1.44, 1.64, and 1.83 THz dominated by intermolecular interactions. The frequency dependent refractive index and absorption coefficient of Chl-Mg-Na were both determined using the Fit@TDS software. Below 1.0 THz, a refractive index of 2.09 was measured. In order to acquire further understanding of the observed vibrational modes, a detailed study of the temperature dependence of the line positions of the lowest modes in Chl-Mg-Na was performed. As the temperature was increased from 88 K to 298 K, the feature at 1.83 THz experienced a notable red shift of frequency and line shape broadening, whereas the feature at 1.44 THz showed little change. These results suggest that the 1.83 THz feature is dominated by intermolecular motions occurring over the crystalline unit cell of the Chl-Mg-Na molecular crystal. Finally, terahertz time-domain spectroscopy was used to acquire the spectra of an ornamental plant bearing yellow-green variegated leaves (ivy, Aureomarginata variety), the yellow sectors having lower chlorophyll content compared to the green sectors. In dehydrated green tissue, the chlorophyll molecules showed well-marked intermolecular vibrational modes at 1.86 THz, indicating that chlorophyll molecules are prone to packing with an ordered molecular arrangement. These results demonstrate the potential application of THz spectroscopy in the field of agronomy.

Optimization and modeling of the sound absorption behavior of polyurethane composite foams reinforced with kenaf fiber

In this study, hybrid composites made of polyurethane (PU) matrix and natural kenaf fiber filler were designed and fabricated. The Response Surface Methodology (RSM) optimization approach coupled with a Central Composite Design (CCD) was used to design the experiments and optimize the parameters affecting the acoustic performance of the samples in the low and mid-frequency ranges. The acoustic performance of the samples was investigated using the impedance tube method. Additionally, the acoustic performance was predicted using a developed Quadratic model as well as the Delany-Bazley (DB) and Johnson-Champoux-Allard (JCA) models. The results showed that the acoustic absorption of neat PU foam is remarkably improved at all frequencies with the addition of kenaf fibers as the filler. It was found that the acoustic performance of the composite samples is optimal when the added amount of kenaf fibers to the polymer matrix and fiber length are 1.2 wt% and 8 mm, respectively. The sound absorption average (SAA) of the optimized composite and neat PU foam was calculated as 0.65 and 0.48, respectively i.e., an increase of 35.4 %. The Quadratic model showed very high accuracy for the prediction of the SAA of the optimized sample. The JCA model provided higher accuracy for the prediction of the frequency-dependent sound absorption coefficient of the composites as compared with the DB model.

Shedding light on static and dynamic hyperpolarizabilities of thia[7&8]circulenes, toward their NLO applications.

Herein, we examined the nonlinear optical properties of thia[7&8]circulenes (1-18). Circulenes are the building blocks of various nanomaterials such as graphene, nanotubes, and fullerenes. Many studies on circulenes have focused on the aromaticity of circulenes, but less attention has been paid on optoelectronics properties. Carbon atoms of the [7&8]circulenes are replaced with multiple sulfur atoms to designed thia[7&8]circulenes (1-18). These circulenes (1-18) are thermodynamically, kinetically, and chemically stable. Nonlinear optical (NLO) response is evaluated through static and frequency-dependent first and second hyperpolarizability values. The static first hyperpolarizability (βo) of these compounds ranges between 0.00 and 496 au. The frequency-dependent coefficients for all thia[7&8]circulenes show remarkable enhancement at 532 and 1064nm, respectively. The nonlinear refractive index is increased up to 1.13 × 10-14 au for circulene 9 at 532nm. These findings successfully demonstrated that nonlinear optical response of thia[7&8]circulenes can be increased by decorating multiple sulfur atoms. The unsymmetrical distribution of sulfur atoms is more effective in enhancing nonlinear optical response.