International Journal of Antennas and Propagation

Volume 2019, Article ID 8513150, 10 pages

https://doi.org/10.1155/2019/8513150

## Determining the Effective Electromagnetic Parameters of Photonic Crystal by Phase Unwrapping and Denoising Method

^{1}School of Information and Communication Engineering, Communication University of China, Beijing 100024, China^{2}Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854, China

Correspondence should be addressed to Guizhen Lu; nc.ude.cuc@nehziugul

Received 29 March 2019; Accepted 22 May 2019; Published 3 July 2019

Academic Editor: Giuseppe Castaldi

Copyright © 2019 Guizhen Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1].

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 [2–11]. The intrinsic problem of the NRW technique relates to the electrical thickness of the material sample. Namely, the phase of the electromagnetic wave is periodic with a period of 2*π*, which causes ambiguity in the extracted results.

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 [2]. In [3], the double negative material is analyzed by the transfer matrix method, in which the data of various system lengths is collected to calculate the refraction index from the linear fits of and versus the system length . A general homogenization theory is developed to state the difference between the equivalent parameters and the effective constitutive parameters of periodic metamaterials [4]. Considering the relationship between frequency and phase of the transmitted coefficient, the successful phase summing is presented [5]. Since the constitutive parameters of a slab are hard to obtain nearby the total transmission frequency, the cycle shift operator to the periodic multilayer is proposed [6].

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.

#### 2. Method

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 and refraction index from scattering parameters, where and are determined independently. The reflection coefficient at the interface of a semi-infinite material slab and the phase factor of the wave propagating through the measured material slab with thickness is used in the early time. The difficulty with the aforementioned extraction technique relates to the periodicity of the phase factor of the wave propagating through the measured material slab. A method to overcome the phase ambiguity is proposed in [5]; the phase difference between two neighbor frequency points is used to select the correct branch at a given frequency point. However, the phase is obtained by summing all the phase differences before the computed frequency, which may induce errors when the phase difference jumps over 360°. In this paper an improvement is proposed, the phase is unwrapped, which makes the data more robust in the parameter extraction procedure. The proposed technique can be used for material parameter extraction without worrying about whether the solution branch is correct in the measured frequency band.

The first step in the algorithm is to obtain the impedance and transmission coefficient from the scattering parameters S11 and S21.Starting from the phase factor appearing in formula (3)rewrite the natural logarithm of the expression in formula (3) as follows.In order to obtain the corrected branch, the first frequency should be lower enough to insure the slab is electrically thin. The phase at each frequency point is obtained by using the phase unwrap technique, which can be expressed aswhere andwhere the integer is determined from formula (7) as follows.The square bracket in formula (7) is to take the integer. The improved method can overcome the ambiguity when the phase difference is larger than 360°, which is possible in some metamaterials. The cycle shift operator to the periodic multilayer in [6] can also remove the ambiguous phase, but sudden changes near the resonance frequency may cause the cycle shift operator calculation accumulates a lot of errors. Compared to that, the proposed method in this paper can overcome the phase ambiguity in an efficient way while averting the accumulated errors.

#### 3. Removing Noise from the Scattering Parameters by Using Wavelet Method

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 vanishing moment means that wavelet coefficients for -th order polynomial will be zero. That is, any polynomial signal up to order -1st can be represented completely in scaling space. In theory, more vanishing moments mean that the scaling function can represent more signals that are complex accurately; is also called the accuracy of the wavelet. Wavelets that resemble the signal or its properties yield better signal, which can be another selection critic. A new wavelet denoising method is presented for the wavelet basis and level selection [7]. We will use the method to remove the noise from signals of S11 and S21 with noise.

As a validation, a dispersion material is utilized to verify the proposed method on the basis of [5]. Considering a dispersion material slab with thickness mm, the permittivity and permeability are respectively, where GHz, GHz, GHz, and GHz.

The scattering parameters S11 and S21 are computed as Figure 1. The Gaussian noise with mean zero and standard variation 0.025 is added to the S11 and S21 signals.