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The electromagnetic parameters of the dispersion material and metamaterial are vital in the engineering. The phase unwrapping method is proposed to deal with the phase ambiguity of the transmission and reflection method in electromagnetic (EM) parameters extraction. The computed results demonstrate that the proposed method can give the correct effective parameters. In dealing with scattering parameters with noise, the wavelet transform method is utilized to remove the noise added to the scattering parameters. The simulated results show that the correct material parameters can be obtained by wavelet denoising method. Finally, the proposed method is used to extract the parameters of the photonic crystal. The effective parameter gives a different aspect in explanation to the function for the photonic crystal.

The study of metamaterials with subwavelength basic unit has been a vibrant research topic in the field of THz frequency band. Structuring the metamaterials on a subwavelength scale makes it possible to create electromagnetic media with properties not found in natural materials, while still allowing describing them as effectively continuous media with constitutive parameters such as the electric permittivity and the magnetic permeability. A basic tool in the study of metamaterials is the so-called retrieval method, i.e., the extraction of effective medium parameters corresponding to a metamaterial with given microscopic structure. The effective material parameters are important because they give us the insight between the microscopic response of metamaterials and the macroscopic homogeneous media assumed in many proposed applications. The material parameter extraction methods have recently attracted attention in the literature due to the grown interest towards metamaterials and the need to characterize the electromagnetic properties of the man-made materials.

The subwavelength scale allows the heterogeneous material to be considered as a homogenized effective medium, whereas local resonances lead to new phenomena of the effective medium parameters that are rarely or never observed in nature. The existence of resonance poses a considerable challenge to conventional effective medium theories [

Among the different extraction techniques, there exists a class of methods that are based on measurements (or numerical simulations) of the reflection and transmission coefficients of a planar material sample. The classical Nicolson-Ross-Weir (NRW) technique is part of this class, as well as many other more recent methods [

The phase ambiguity is a key problem for retrieving the electromagnetic parameters with transmission and reflection method. In order to overcome the phase ambiguity, the method that requires the imaginary part of the refraction index larger than zero is presented using physical principle, which can give the uniquely sign of the real part of the index [

In what follows a novel way to overcome the phase ambiguity related to the extracted material parameters is proposed and the performance of the proposed technique is studied with examples. The phase unwrapping method serves to extract the scattering parameters with the sweep frequency. Upon the request of the phase difference of adjacent frequencies, the phase ambiguity can be resolved. Considering the effect of noise in the scattering parameters, the wavelet denoising method is involved with the data preprocessing. The results indicate that the proposed method can give the accurate permittivity and permeability of the dispersion materials. Finally, this method is employed to analyze the photonic crystal which shows resonance in the S11 and S21. The extracted results have many peaks in the permittivity and permeability, which are related to the photonic crystal resonance structure. The effective parameters can help us to understand the photonic crystal in physics and the design in engineering.

In order to retrieve the effective permittivity and permeability of a metamaterial slab, we need to characterize it as an effective homogeneous slab. In this case, we can retrieve the permittivity and permeability from the reflection S11 and transmission S21 data. A typical method is to extract the impedance

The first step in the algorithm is to obtain the impedance

Generally, the extracted parameters are influenced by the noise which may be caused by measurement or simulation. Based on the fact that noise and distortion are the factors that limit the accuracy of the extracted parameters, it is necessary to remove the disturbances before the extraction. Noise is defined as the unwanted signal that interferes with the parameters extraction from the measurement. Here wavelets denoising method is employed, which can increase the accuracy of the extracted parameters. For the scattering parameters S11 and S21, both the real parts and imaginary parts can be regarded as one-dimensional signal and the fast random variation can be considered as noise. Wavelets are characterized by scale and position and are useful in analyzing variations in signals in terms of scale and position. Because of the fact that the wavelet size can vary, it has advantages over the classical signal processing transformations to simultaneously process time and frequency data. The vanishing moments of the wavelet basis can be used as a selection critic. Having

As a validation, a dispersion material is utilized to verify the proposed method on the basis of [

The scattering parameters S11 and S21 are computed as Figure

The Scattering parameters S11 and S21 with noise.

The extracted permittivity and permeability results are presented in Figures

The extracted permittivity from the noised S11 and S21 data.

The extracted permeability from the noised S11 and S21 data.

Wavelet shrinkage methods provide effective signal denoising with minimum computational complexity. In the wavelet domain, the signal is coherent and has concentrated “energy” residing in just a few high magnitude coefficients, whereas incoherent noise is represented by a large number of coefficients with small magnitude. This sparsity of wavelet coefficients representing the signal is exploited by wavelet shrinkage methods to separate noise from signal coefficients. In the denoising by wavelet method, the selection of wavelet basis and decomposition level is very crucial. In general, standard wavelets that resemble the signal or its properties yield better signal and noise separation as well as sparsity. The Meyer wavelet here is more suitable to the problem, so the Meyer wavelet basis is selected. The decomposition level is selected as 7 from the wavelet decomposition as shown in Figures

The wavelet coefficient distribution, the approximation and detail component.

The wavelet coefficient distribution, the detail component for levels 8 and 9.

The peak-to-sum ratio of the wavelet detail coefficient for S11 real part.

The S11 and S21 signals with noise removed are shown in Figure

The S11 and S21 signals with noise removed.

The extracted and denoised permittivity and permeability results are shown in Figures

The comparison between the extracted permittivity and original permittivity.

The comparison between the extracted permeability and original permeability.

The results illustrate that the noise effects of the scattering parameters can be removed by using wavelet denoising method. Selection of the wavelet basis is based on the resemblance between the signal and the wavelet basis. The decomposition level is determined by using the ratio of peak-to-sum for the detail coefficient of wavelet transform.

Photonic crystals are comprised of periodic, dielectric structures. In its forbidden bands, the electromagnetic wave cannot travel through the photonic crystal. These disallowed bands of frequencies are called photonic band gaps. The fabrication of optical photonic crystals is quite complex. There are several methods to calculate the dispersion relation and thereby the range of the band gaps. Reference [

Considering a photonic crystal as shown in Figure

A unit cell of photonic crystal structure for numerical simulation.

The numerical method FEM is used to compute the S11 and S21 parameters in the frequency band 1.4 THz to 2.8 THz, which is used to extract the effective permittivity and permeability of the photonic crystal slab. In the simulation, the incidence wave direction is normal to the slab surface. The reference planes of the two ports are selected as the slab top surface and bottom surface, respectively, which is important in extracting the correct effective permittivity and permeability. The wavelet denoising method is also utilized in the photonic crystal results, and the noise is also eliminated effectively by the method proposed in this paper. The simulated S11 and S21 parameters noise-added and denoised results are shown in Figure

The S11 and S21 results of the photonic crystal.

The extracted effective parameters of the photonic crystal are shown in Figure

The effective permittivity and permeability for photonic crystal.

As a validation, the S11 and S21 are computed from the effective homogeneous medium with the extracted permittivity and permeability shown in Figure

Scattering parameters comparison between the photonic crystal and effective medium.

The refraction index reflects the effect of the permittivity and permeability for the wave propagation phase, and the refraction index and impedance of the photonic crystal calculated from the scattering parameters are shown in Figures

The refraction index calculated from the scattering parameters.

The normalized impedance calculated from the scattering parameters.

The impedance of the photonic crystal appears as average value, just like the mixture of the different materials far from the resonance position. In the resonance position, the real part of the impedance gives a lower or higher value. For the imaginary part of the impedance, there are large negative value or large positive value, which corresponds to the stored capacitive energy and inductive energy. Hence, in conclusion, only a small part of the energy is propagated along the propagation direction at the resonance frequency, and most of the energy is stored in the photonic crystal, which can also explain the negative imaginary value of the refraction index.

The dispersion electromagnetic parameters such as permittivity and permeability abide by the Kramers-Kronig relation. The effective parameters of photonic crystal have the dispersion properties which may have negative properties in some frequencies; it is helpful to validate the Kramers-Kronig relation for the photonic crystal [

The comparison between the extracted permittivity and Kramers-Kronig results.

The comparison between the extracted permeability and Kramers-Kronig results.

Our results indicate that the proposed technique is efficient when applied to material parameters extraction for realistic material samples from the S-parameters. The phase ambiguity may cause the multivalue problem and give the wrong results in the material parameters extraction. The proposed phase unwrapping method can solve the phase ambiguity problem. Correctness of the effective parameters depends on the S11 and S21 parameters which are measured or computed by numerical method. The noise coming from the measurement and simulation may induce the errors. In order to address this issue, the wavelet denoising method is employed to eliminate the noise. The results demonstrate that the wavelet denoising method can improve the noised S11 and S21 signals significantly in parameters extraction. Finally, from the aspect of the effective medium, the effective homogeneous medium parameters of the photonic crystal slab are computed, which are validated by comparing the S parameter from numerical computing results and effective homogeneous medium parameter. The explanation for the effective homogeneous medium parameters is given, which can help us to understand the function of photonic crystal slab and the design in engineering.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Science Foundation of China Grant no. 61701447.