Optimization Method of Wideband Multilayer Meander-Line Polarizer using Semi-Analytical approach and Application to 6-18GHz Polarizer including test with Horn Antenna

A R T I C L E I N F O A B S T R A C T Article history: Received: 30 May, 2019 Accepted: 28 August, 2019 Online: 24 September, 2019 An optimization method using a semi-analytical approach is detailed in this paper to design a wideband multilayer meander-line polarizer. It consists of the transmission line equivalent circuit study to find the best values of the shunt components that allow the maximum transmission and a 90 phase difference between two orthogonal components over the entire bandwidth, followed by a simulated annealing algorithm to optimize the polarizer’s dimensions by converging the values of the shunt components found earlier, to some rigorous empirical formulas of the equivalent admittances. To validate the proposed method, a wideband (6-18 GHz) four layer’s meander-line polarizer is optimized and manufactured. The polarizer prototype is stacked to a horn antenna operating at the same frequency band and tested. The experimental results verify that the linear waves of the horn antenna alone were converted to circular ones when the multilayer meander-line polarizer is stacked to it.


Introduction
This paper is an extension of work originally presented in the 48th European Microwave Week [1] where a four layer's wideband (6 -18 GHz) meander-line polarizer has been optimized and simulated with different patch antenna for validation.
As well known, the polarizer is a passive broadband device, used to change the polarization properties of an Electro-Magnetic (EM) wave. Combining antennas with those devices to change the polarization of the wave is very useful. A well-designed polarizer permits to change the antenna polarization and to propagate a different polarization without any significant variations in the pattern performance [2]. The Meander-Line (ML) polarizer is the most popular device used to change the Linear Polarized (LP) waves into the Circular Polarized (CP) ones [3]. It is a multilayer structure made of several printed ML sheets spaced by about onequarter wavelength apart. The principle of operation is to convert an incident wave into two equal components at ±45⁰ as summarized in [4]. One component passes through a structure equivalent to a broad-band shunt-inductive filter while the other passes through a broad-band shunt-capacitive filter. the other component, by about 45⁰.
In this paper, the basic principle of the ML polarizer is described in section 2. Section 3 describes the detail of the optimization procedure which includes the TL equivalent circuit and Simulated Annealing (SA) algorithm. Finally, in section 4, the four layer's wideband (6 -18 GHz) meander-line polarizer studied previously in [1] is used as a case study. The optimization study is presented, followed by experimental validation using a wideband horn antenna covering the same bandwidth.

Meander Line Polarizer: Basic Principle
The ML polarizer is a multilayer structure ( Figure 1). Each layer consists of MLs printed on a low-loss dielectric substrate and separated by a spacer respecting one-quarter wavelength apart. The ASTESJ ISSN: 2415-6698 spacer could be air (vacuum) or any kind of low loss material such as foam or dielectric substrate.
The principle of operation is to convert an incident wave into two equal and perpendicular components at ±45 degrees as summarized in [3]. The ML polarizer's layer is placed in the xyplane in such a way that the meander axes are all oriented at an angle Ψ = 45 degrees with respect to the incident electric field along the y-axis ( Figure 2).
The incident electric field will be decomposed into two orthogonal components: one parallel and the other one perpendicular to the ML axis, noted respectively ∥ and ⊥ . As well known, a CP wave can be obtained by designing a ML polarizer that introduces a differential phase shift of 90⁰ between the components ∥ and ⊥ while keeping their magnitudes identical over the entire required frequency bandwidth. Right-hand or left-hand circular polarizations can be easily obtained by switching the orientation angle Ψ from +45⁰ to -45⁰.  The difficulty in the design of a meander-line polarizer is to determine the inductive and capacitive impedances of the grating for the ∥ and ⊥ polarizations with respect to frequency and constitutive dimensions of the meander line.
An optimization method is developed to optimize the ML polarizer's dimensions and described in detail in the below section. It consists of a semi-analytical analysis based on the combination of TL circuit theory and SA algorithm and allows to find the dimensions of the MLs that keep the magnitude of the components ∥ and ⊥ identical with a phase differential shift of 90⁰ over the desired frequency bandwidth.
Our method of optimization has the advantage of avoiding the heavy computations and simulations used in [13] to solve the TL circuit and the unit cell of each layer. This implies a large reduction in the computation time.
• The TL equivalent circuit is solved using the optimization tool of ADS. This tool has the advantage of optimizing the shunt elements of all the layers at one simulation and avoid the heavy computation of the transmission matrices of each layer then the scattering transmission parameters and finally optimize the shunt elements using quasi-Newton algorithm as used in [13].
• SA algorithm is used to optimize the physical dimensions of the MLs. This algorithm optimizes the dimensions and minimize the absolute values of the differences between the theoretical admittances introduced by Chu and Lee in [14], and their lumped-element counterparts optimized by solving the TL circuit using the optimization tool of ADS. This method avoids the full-wave analysis of the unit cell and all the steps described before which are used in [13].

TL Equivalent Circuit Model
As summarized in [4], the incident electric field will be decomposed into two orthogonal components ∥ and ⊥ . One component passes through a structure equivalent to a broad-band shunt-inductive filter while the other passes through a broad-band shunt-capacitive filter. The phase shift through either filter should have almost the same slope, so that if the differential phase shift is 90⁰ at one frequency in the common passband, it remains close to 90⁰ everywhere in the common passband.
The TL equivalent circuit of the N-layers ML polarizer excited by an incident wave along y-axis ( Figure 1) can be represented as shown in the Figure 3 where 0 is the characteristic impedance of the vacuum ( 0 = √ 0 / 0 ) with 0 and 0 the permeability and the permittivity of vacuum, respectively. The parameters hd1, hd2, …, hdN and 1 , 2 , …, are respectively the thickness and the relative permittivity of the 1 st , 2 nd , …, N th layer. The parameters hs1, hs2, …, hsN and 1 , 2 , …, are respectively, the thickness and the relative permittivity of the 1 st , 2 nd , …, N th separator. The 1 , 2 , …, and 1 , 2 , … , are respectively, the lumped inductance and capacitance equivalents of the 1 st , 2 nd , …, N th periodic strip lines on each layer.
The input parameters of the TL model of ML polarized are: the frequency bandwidth (center frequency f0), the number of layers and the relative permittivity and thickness of each dielectric substrate and separator. This model must be solved by optimizing the shunt components 1 , 2 , …, and 1 , 2 , …, which verify the maximum transmission coefficients (S21 and S43) in term of magnitude (~0dB) while

Simulated Annealing Algorithm
Simulated Annealing is a mathematical equivalent of the thermodynamic annealing. The energy of the particle in thermodynamic annealing process can be compared with the cost function to be minimized in optimization problem. The particles of the solid can be compared with the independent variables used in the minimization function. The Simulated Annealing Algorithm is detailed in [15].
The MLs dimensions must be optimized to verify that the impedance equivalent of each layer is equal to the one found by solving the TL circuit model.
The optimization procedure is summarized in Figure 5.

Case Study: 6-18GHz Four Layers ML Polarizer
A four layers ML polarizer covering the bandwidth from 6 GHz to 18 GHz is studied in this section.

Polarizer Optimization
The optimization procedure starts by defining the input parameters which are the frequency bandwidth , center frequency f0 = 10.4GHz, number of layers N = 4, relative permittivity and thickness of the dielectric substrates ( , hdi) and separators ( , hsi) for = {1, … , 4}.
To simplify the TL equivalent model, the equivalent circuit was studied with one-quarter wavelength apart ( /4) air separators and without considering the substrate effect since a very thin substrate (ℎ = 0.127 ) with low permittivity and low loss (Rogers RT/duroid 5880 ( = 2.2, = 0.0009)) is used.
The choice to have the same MLs dimensions (i.e. same Leq and Ceq) for each layer was also made to make the system integration simple and flexible. The TL equivalent circuit of the four layers ML polarizer is shown in the Figure 6.
The optimization tool of ADS is used to find the best values of the equivalent inductance and capacitance to verify the maximum transmission coefficients (S21 and S43) in term of magnitude (~ 0dB) and 90⁰ differential phase shift between them The ML dimensions (a, b, h, and w) are optimized using the SA algorithm, as detailed previously, by making the admittances values (1/ and ), found using the TL circuit model (Figure 6), as close as possible to the ones given by Chu and Lee (1). The optimum values found are a = 2.17mm, b = 13.08mm, w = 0.446mm and h = 5.151mm.

Polarizer test with 6-18GHz Horn Antenna
To verify the performance of ML polarizer, a wideband horn antenna (6 to 18 GHz) shown in figure 7(a) is used. The four ML layers of the polarizer were fabricated. A mechanical support was designed and realized using a 3D printer to maintain the quarter wave length distance between each layers and the antenna. The picture of the polarizer attached to the antenna is shown in figure 7(b). Figure 7  The measurements were performed for the horn antenna alone and with the ML polarizer excited in vertical polarization. The reflection coefficients of the horn antenna alone (without the ML polarizer) and with the ML polarizer are plotted in Figure 8.
Both reflection coefficients are very close, which proves that the ML polarizer doesn't mismatch the antenna except at the around 6GHz, where the reflection coefficient is measured around -6.5dB. In figure 9, the and components of horn antenna alone and with ML polarizer are plotted at 6 GHz in both planes ( = 0 0 and = 90 0 ). The results seem to show that the horn antenna initially excited for a vertical polarization (Figure 9 (a) and (b)) is radiating a circular polarized wave when it is stacked with the ML polarizer (Figure 9 (c) and (d)). At the boresight, the difference between the co and cross components of the horn antenna measured alone is more than 28 dB while it reduces to 1dB when the horn antenna is measured with the ML polarizer. The measured Axial Ratio (AR) of the horn antenna with the ML polarizer at this frequency is around 6dB (Figure 16). Therefore, the ML polarizer is converting the linear polarization waves into elliptical polarized ones. The measured insertion losses are around 2.5dB ( Figure 17) which is due to the mismatch of the antenna with the ML polarizer at this frequency (Figure 8).
The and components of the horn antenna alone and with ML polarizer are plotted in figure 10 -14 for the frequencies from 8GHz to 16GHz. The vertical polarized waves of the horn antenna alone are perfectly converted to circularly polarized ones when it is stacked to the ML polarizer with the maximum AR of The measured values of the AR at boresight are reported in figure 16 versus frequency for the horn antenna with the ML polarizer. The AR is less than 3dB at the frequency band between 8 GHz and 16 GHz. Therefore, the ML polarizer converts the linear polarization of the horn antenna to a circular one. In both ends of the frequency band (6 GHz and 18 GHz), the ML polarizer converts the linear polarization of the horn antenna to an elliptical one.
The total electrical fields (Etot) values at boresight are reported in figure 17 versus frequency for the horn antenna alone and with the ML polarizer. This figure shows clearly that the Etot of the horn antenna with the ML polarizer is following the same allure as the Etot of the horn antenna alone variation with some insertion losses.  In almost the entire bandwidth, except at 6 GHz, the insertion losses are less than 1.5dB with the minimum of 0.24 dB at 14 GHz. The maximum insertion losses (2.5dB) are measured at 6 GHz which is due to the mismatch of the horn antenna with the ML polarizer.

Conclusion
An optimization Method of Wideband Multilayer Meander-Line Polarizer using a Semi-Analytical approach is described in this paper. The originality of this method lies in the optimization of: i) values of the shunt equivalent inductance Leq and capacitance Ceq, determined by the ADS software -ii) ML dimensions (a, b, h and W) using the SA algorithm. This method was applied to optimize a four layers wideband (6 -18 GHz) ML polarizer as study case.
The optimized polarizer was manufactured and a wideband Horn antenna covering the same bandwidth was used for the testing. The test consists of the measurements of the horn antenna vertically polarized without and with the ML polarizer. Good measured results were observed where the linear polarization of the antenna was converted to a purely circular one in almost the entire frequency band (8 GHz -16 GHz, corresponding to an octave). In both ends of the frequency band (6 GHz and 18 GHz), an elliptical polarization was achieved with a maximum axial ration of around 6dB.
At 6GHz, the axial ratio is around 1dB but the insertion losses are around 2.5dB which is due to the mismatch of the horn antenna with the ML polarizer at this frequency. At 18 GHz, the axial ratio is around 5dB and the insertion losses are around 0.6 dB.