Configuration and design theory

The unit cell of the proposed reflective converter is shown in Fig. 1, including two FSS and three dielectrics, while the last layer is a purely metallic plane. The main reason for using FSSs in different layers is to create different resonant behavior for each and not to destroy each other. A frequency selective structure (FSS) with a circular shape can be assumed as a polygonal structure with many sides, which every side and discontinuity on the patch is assumed as a resonator. Moreover, the current distribution on circular patch is almost stable and uniform in most resonances (i.e., the main and harmonic ones). Thus, a circular patch can present multi-resonance behavior with successive positions. Hence, to create a multi-resonator with high degrees of freedom for flexible design, we introduce a two-element array of connected circular patches as the first FSS (i.e., FSS1) at the $h1$ position. As illustrated in top view, the relevant key parameters are $rc$ and $dc$, which adjust the electrical length of the first reflective resonance, its harmonics, and their number. Hence, the position of the lower edge of the 3 dB-$AR$ band and the number of resonances in the primary and middle bands will be controllable. According to the previous theory, the second FSS (i.e., FSS2) at $h1$+$h2$ height is defined as a two-element array of square patches separated from each other at the two ends of FSS1, so that, due to its single side and very small size, the single resonance in the high frequency of the band is created. The related key parameter is $ls$, which controls the position of the high resonance. Based on these two theories mentioned earlier, we will have two multi-resonance and single-resonance resonators cascaded, and a step-by-step parametric study on them will lead to the creation of a multi-resonance conversion band. Alternatively, we define three dielectric layers with the same dielectric constant but with different thicknesses, $h1$, $h2$, and $h3$ around the FSSs, so that the effective dielectric constant for all reflective resonances can be adjusted. Therefore the quality factor of the resonances can be also controlled. Finally, the frequency bandwidth of each one should be controlled. This number of design freedoms leads to the creation of successive resonances with full overlapping capability and ultimately creates a very wide conversion bandwidth with very low $AR$.

Figure 1. Configuration of the proposed reflective triple-layer dual-FSS-based LP–CP converter. CP = circularly polarized; FSS = frequency-selective structures; LP = linearly polarized.

The initial value of all dielectric constants is equal to 2.2, with a loss tangent of 0.005. As shown in Fig. 1, the proposed cell of the converter is square and each side, as periodicity of the cell in the array, is equal to 10 mm. This value is smaller than the wavelength of the highest desired frequency in the band (i.e., 20 GHz) to ensure most higher-order Floquet modes are evanescent [Reference Kundu, Singh, Singh and Chakrabarty2, Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32].

Circuit modeling, analysis, and discussion

As shown (side view) in Fig. 1, for the analysis and modeling of the cell, as a basic assumption, a TE-polarized (i.e., $y$-axis-directed wave) incident wave is assumed. As illustrated in Fig. 2(a) and considering $\vec E_i^y$ incident wave, the incident and reflected E fields can be readily decomposed into mutually orthogonal $u$ and $v$ components. The proposed dual-FSS cell is made diagonally symmetric to maximally decrease the coupling between the reflected $u$ and $v$ polarized waves. Therefore, the cross-polarized components of the reflected waves can be approached to zero, and then, the admittance matrix study on parallel cascaded ECM (Fig. 2(b)) will be significantly simplified [Reference Kundu, Singh, Singh and Chakrabarty2, Reference Zhang, Zhu, Zhang, Yang, Wang and Liu29, Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32]. This equivalent circuit is based on lumped network theory, and therefore, the ground plane in the last layer is modeled as a short circuit load, and all three dielectrics are modeled as a transmission line with certain characteristic admittance (${Y_d}i$) and length (h_i), where $i = 1,\,2,\,3$. Two important points are included in this modeling, which are: (1) Use of frequency-dependent tensor sheet admittance for multi-resonance FSS1 (i.e., ${Y_1}\left( \omega \right)$) and another tensor sheet admittance for single-resonance FSS2 (i.e., ${Y_2}$). This admittance matrix is calculated by examining the shape and dimensions of the conductor surfaces and their discontinuities [Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32, Reference Lee and Sievenpiper33]; and (2) Defining a mutual-coupling admittance, $Yc$, between FSS1 and FSS2 parallel to the transmission line between them, due to the possibility of their metal surfaces overlapping and changing total reflection characteristics.

Figure 2. (a) Incident and reflective $u$ and $v$ orthogonal E field components of y-polarized incident wave. (b) Equivalent circuit model of the unit cell (sh. cir.: short circuit load). (c) Simplified model.

In the analysis of the model based on Fig. 2(b), the first step is to calculate ${Y_{eq}}1$ using the expression $ - j{Y_d}1\,cot\left( {{\beta _1}{h_1}} \right)$, where ${Y_d}i = {Y_0}/\sqrt {{\varepsilon _r}i} $ and ${\beta _i} = 2\pi \sqrt {{\varepsilon _r}i} /\lambda $ ($i = 1,2,3$) [Reference Balanis34]. ${Y_{eq}}1$ is assumed constant with respect to $u$ and $v$ polarizations, but ${Y_1}\left( \omega \right)$ is defined as a variable. These admittances are parallel, and their equivalent matrix is presented in Fig. 2(c) with the title of $Y_{in,1}^{uu/vv}$. Therefore, $Y_{in,1}^{uu} = - j{Y_d}1\,cot\left( {{\beta _1}{h_1}} \right) + {Y_1}^{uu}\left( \omega \right)$ and $Y_{in,1}^{vv} = - j{Y_d}1\,cot\left( {{\beta _1}{h_1}} \right) + {Y_1}^{vv}\left( \omega \right)$ are defined as final loads for each of the $u$ and $v$ polarizations, independently. Before these loads, the admittance of the second transmission line, ${Y_d}2$, is added to the polarization-dependent mutual-coupling admittance, $Y_C^{uu/vv}$, and results in an equivalent matrix as shown in Fig. 2(c). Here, the sum of these two admittances is defined as an equivalent characteristic admittance for each polarization, individually. Therefore, for the mentioned equivalent line and final loads, the input admittance ${Y_{eq}}2$ for each polarization is defined as follows [Reference Balanis34]:

(1)\begin{align} Y_{eq}^{uu/vv}2= &\left( {{Y_d}2 + Y_C^{uu/vv}} \right)\nonumber\\ &\times\frac{\begin{array}{c} - j{Y_d}1\,cot\left( {{\beta _1}{h_1}} \right) + {Y_1}^{uu/vv}\left( \omega \right) \\+ j\left( {{Y_d}2 + Y_C^{uu/vv}} \right).tan\left( {{\beta _2}{h_2}}\right)\end{array}}{\begin{array}{c} {Y_d}2 + Y_C^{uu/vv} \\+ j\left( { - j{Y_d}1\,cot\left( {{\beta _1}{h_1}} \right) + {Y_1}^{uu/vv}\left( \omega \right)} \right).tan\left( {{\beta _2}{h_2}} \right)\end{array}}\end{align}

According to Fig. 2(c), $Y_{eq}^{uu/vv}2$ is parallel to $Y_2^{uu/vv}$ corresponding to FSS2. Hence, $Y_{in,2}^{uu/vv} = Y_{eq}^{uu/vv}2$ + $Y_2^{uu/vv}$ is defined as the terminal loads of the third transmission line for each polarization independently. Again, we use the input admittance formula of a transmission line with characteristic admittance of ${Y_d}3$ with terminal load of $Y_{in,2}^{uu/vv}$ to calculate $Y_{in}^{uu/vv}$ as follows:

(2)\begin{equation}Y_{in}^{uu/vv} = {Y_d}3\,\frac{{Y_{in,2}^{uu/vv} + j{Y_d}3\,tan\left( {{\beta _3}{h_3}} \right)}}{{{Y_d}3 + jY_{in,2}^{uu/vv}\,tan\left( {{\beta _3}{h_3}} \right)}}\end{equation}

Given the determination of $Y_{in}^{uu/vv}$at the position connected to air, it is necessary to calculate the reflection coefficients for both polarizations using ${\Gamma ^{uu/vv}} = ({Y_0} - Y_{in}^{uu/vv})/({Y_0} + Y_{in}^{uu/vv})$ [Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32]. As discussed in [Reference Balanis34], the condition ${\Gamma ^{vv}} = \mp j{\Gamma ^{uu}}$ must be established to produce CP in reflected waves, where the upper or lower sign is used to have right-hand circularly polarization (RHCP) or left-hand circularly polarization (LHCP), respectively. To simplify the analysis, we assume the total losses to be close to zero, and then we can model the FSSs with pure imaginary shunt admittances as: ${Y_1}^{uu} = j{B_1}^{uu}$, ${Y_2}^{uu} = j{B_2}^{uu}$, ${Y_1}^{vv} = j{B_1}^{vv}$, and ${Y_2}^{vv} = j{B_2}^{vv}$. On the other hand, according to the direction defined for both FSSs (i.e., $v$ axis), ${B_1}^{uu}$, ${B_2}^{uu}$, and $Y_C^{uu}$ are generally very small due to minimal effective width along $u$-direction. Therefore, they are omitted for simplicity. In addition, considering the equality of dielectric constants for all layers, we have ${Y_d}1 = {Y_d}2 = {Y_d}3 = {Y_d}$ and ${\beta _1} = {\beta _2} = {\beta _3} = \beta $. Finally, given that the third dielectric is considered only a thin and protective layer and its behavior toward both polarizations is the same, we can disregard it, and then, equation (2) becomes $Y_{in}^{uu/vv} = Y_{in,2}^{uu/vv}$. Based on these assumptions and simplifications and applying all mentioned relations in the CP condition (i.e., ${\Gamma ^{vv}} = \mp j{\Gamma ^{uu}}$), the required $v$-directed admittances of the FSSs will be [Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32]:

(3)\begin{equation}{B_2}^{vv} = \left[ {\frac{{Y_{eq}^{uu}2\, \mp \,j{Y_0}}}{{{Y_0}\, \mp \,jY_{eq}^{uu}2}} - Y_{eq}^{vv}2} \right]\,\,{\text{and}}\;{B_2}^{uu} \cong 0,\end{equation}

where

(4)\begin{equation}Y_{eq}^{uu}2 = j{Y_d}\left[ {\frac{{tan\left( {\beta {h_2}} \right) - cot\left( {\beta {h_2}} \right)}}{{1 + tan\left( {\beta {h_2}} \right)cot\left( {\beta {h_2}} \right)}}} \right]\,\end{equation}

and

(5)\begin{align}{B_1}^{vv} & = {Y_d}cot\left( {\beta {h_1}} \right) - j({Y_d} + Y_C^{vv})\frac{{Y_{eq}^{vv}2 - j({Y_d} + Y_C^{vv})tan\left( {\beta {h_2}} \right)}}{{({Y_d} + Y_C^{vv}) - jY_{eq}^{vv}2\,tan\left( {\beta {h_2}} \right)}}\nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \;{\text{and}}\;{B_1}^{uu} \cong 0.\end{align}

According to the dielectric characteristics of the second layer, equation (4) can be calculated, and to determine equation (3), we need to determine the unknowns of equation (5). As another technique, if one value (e.g., ${B_2}^{vv}$) is taken as reference, then it will be possible to use equations (4) and (3) simultaneously to compute the corresponding required value of $Y_{eq}^{vv}2$. In the following, using equation (5), ${B_1}^{vv}$ is calculated. The method of obtaining certain values for ${B_1}^{vv}$and ${B_2}^{vv}$, based on the tensor surface impedance matrix and the corresponding physical parameters, has been reported in [Reference Doumanis, Goussetis, Gomez-Tornero, Cahill and Fusco32] and [Reference Lee and Sievenpiper33]. To determine $Y_C^{vv}$ in equation (5), another reference can be assumed based on the overlap of FSSs (i.e., the increase rate of $rc$ and $ls$ simultaneously) and the characteristics of the second dielectric.

As a result, it is proved that the degrees of freedom for the proposed design are high for great control over the number of reflection resonances, their overlap, the upper and lower edges of the conversion band, the conversion bandwidth, and the quality of the LP–CP conversion. The conversion quality is evaluated by the $AR$ parameter of the reflected waves for two polarizations, which is calculated as follows [Reference Balanis34]:

(6)\begin{equation}AR = {\left( {\frac{{{{\left| {{r_{xx}}} \right|}^2} + {{\left| {{r_{yx}}} \right|}^2} + \sqrt {XY} }}{{{{\left| {{r_{xx}}} \right|}^2} + {{\left| {{r_{yx}}} \right|}^2} - \sqrt {XY} }}} \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.} \!\lower0.7ex\hbox{$2$}}}},\end{equation}

where

\begin{equation*}XY = {\left| {{r_{xx}}} \right|^4} + {\left| {{r_{yx}}} \right|^4} + 2{\left| {{r_{xx}}} \right|^2}{\left| {{r_{yx}}} \right|^2}\cos \left( {2\Delta {\varphi _{xy}}} \right)\end{equation*}

Here, ${r_{xx}}$ and ${r_{yx}}$ are the reflection coefficients of mutually orthogonal components, and $\Delta {\varphi _{xy}}$ is the phase difference between them. The condition of CP is $\left| {{r_{xx}}} \right| \cong \left| {{r_{yx}}} \right|$ and $\Delta {\varphi _{xy}} = {\varphi _{xx}} - {\varphi _{yx}} = 2m\pi \pm \frac{\pi }{2}$ simultaneously, where $m$ is an integer. In this paper, to have a high-quality conversion, the references are selected in such a way that we reach the target $AR$ below 2 dB.

Study on FSS1 and relevant dielectric

According to the theory mentioned above, FSS1 should have a multi-resonance behavior and provide a major contribution in covering a wide conversion band. Figure 3(a) and (b) shows the AR graphs of the proposed converter for different values of $rc$ and $dc$ in FSS1, respectively. It is noteworthy that FSS2 was not used in this study. From Fig. 3(a), it is quite clear that increasing the radius of the circular patches or $rc$ leads to the lowering of the initial resonance and finally reduces the lower edge of the band, according to 3 dB reference. Here, the lower edge of the band drops from 11 GHz to 6.2 GHz, which is very valuable and provides a significant increase in bandwidth. In addition, with the increase of $rc$, the number of resonances in higher positions increases and can be a factor in reducing the level of $AR$ in the high band. As shown in Fig. 3(b), if the distance of the circular patches (or $dc$) is large enough, the electrical length of the first resonance increases, and the lower edge of the band decreases significantly. Also, with the increase of $rc$, the frequency distance of band nulls ($AR$ with high value) increases from each other, and then, the bandwidth and conversion quality in the middle band increases. Based on Fig. 3, the strong influence of $rc$ and $dc$ on the conversion characteristics is clear. The next key parameter is the thickness of the FSS1 substrate, which will have a significant effect on the effective dielectric constant and the quality factor of most resonances. The result of the study is presented in Fig. 4. By controlling $h1$, the resonance of the first band moves down to some extent, and most importantly, the overlapping quality of the resonances in the middle band increases significantly. The relative reduction of the lower edge of the band and very high improvement of $AR$ in the middle band are the results of this change. Simultaneous optimization of these parameters is required.

Figure 3. $AR$ graphs (normal incident) of the proposed LP–CP converter (without FSS2) for different values of (a) $rc$ (when $dc$=7 mm and $h1$=1.55 mm) and (b) $dc$ (when $rc$=2.5 mm and $h1$=1.55 mm) in FSS1. AR = axial ratio; CP = circularly polarized; FSS = frequency-selective structures; LP = linearly polarized.

Figure 4. $AR$ graphs (normal incident) of the proposed LP–CP converter (without FSS2) for different values of $h1$ in FSS1 (when $dc$=7 mm and $rc$=2.5 mm). AR = axial ratio; CP = circularly polarized; FSS = frequency-selective structures; LP = linearly polarized.

Study on FSS2 and relevant dielectric

As discussed in the theory section, FSS2 is used to create high resonance and thus increase the bandwidth and improve the conversion quality from the upper edge of the band. The position of the square patches of FSS2 is fixed relative to the corners of the cell, and its dimensions (i.e., $ls$) increase toward the center of the cell; the results of which are presented in Fig. 5(a). It is important to note that altering $ls$ can impact the field coupling between FSS2 and FSS1, potentially leading to changes in the quality factor and bandwidth of high resonances. It is quite clear that $ls$ controls the high resonance position. By choosing values around 1.5 mm, not only the high resonance is at its highest position at around 20.8 GHz, but also the $AR$ quality at the top of the band from 15 GHz onward is greatly increased. In the best case, the range of 6.2–21.2 GHz is covered with an $AR$ of approximately less than 1.8 dB, which is the outcome of the simultaneous use of two FSSs, multi-resonance and single-high-resonance ones, at independent heights and positions.

Figure 5. $AR$ graphs (normal incident) of the proposed LP–CP converter (with FSS1) for different values of (a) $ls$ and (b) $h1$+ $h2$ in FSS2 (when $dc$=7 mm, $rc$=2.5 mm, and $h1$=1.55 mm). AR = axial ratio; CP = circularly polarized; FSS = frequency-selective structures; LP = linearly polarized.

The thickness of the substrate below FSS2 is equal to $h1$+ $h2$. As illustrated in Fig. 5(b), this parameter can be effective partly in the position of high resonance and relevant $AR$ quality, which should be involved in the optimization process.

Parametric study on angular stability, dielectric constant analysis, and optimal results

At first, the incident angle is changed from 0° to 45°, and the 3 dB-$AR$ bandwidths are calculated for the changes in the three key parameters of the converter. The relevant data are summarized in Table 1 for better comparison and study based on angular stability. It should be noted that the related graphs are not presented due to their large number and volume so as not to occupy more space. This table implies that the angular stability is maintained for the optimal values of the parameters more than the other values. In other words, the amount of bandwidth loss for increasing the incident angle is less when the parameter is at its optimal value. This result also exhibits the optimal number of parameters in terms of angular stability.

Table 1. 3dB-$AR$ bandwidths for different values of the parameters ($rc$, $dc$, and $ls$) versus different incident angles (BW: bandwidth)

AR = axial ratio.

As a second study, the dielectric constant of all dielectrics changes within a reasonable range (1–6), so that its effects on the effective dielectric constant, the electrical length of the resonances, and the bandwidth are determined. The related $AR$ results are shown in Fig. 6. Their properties are summarized in Table 2. According to Fig. 6, it is known that by increasing the dielectric constant and adjusting the quality factor of the resonances, the bandwidths decrease and the resonances overlap in a more limited range at lower frequencies. In addition, as the dielectric constant increases, the lower edge of the band is significantly reduced from 7.2 to 3.65 GHz. This ultimately results in a wider bandwidth of over 116% and, importantly, covers a portion of the standard S band. It is expected that by increasing the dielectric constant to 8, the entire S band (2–4 GHz) will be covered.

Figure 6. $AR$ graphs for different dielectric constants for all layers simultaneously. AR = axial ratio.

Table 2. AR properties of the proposed converter for different values of dielectric constants (according to Figure 6)

AR = axial ratio.

Given the large number of degrees of freedom in the design, a meticulous optimization process was conducted in the final stage, simultaneously considering all effective parameters within their logical ranges. Table 3 provides the optimal values, and then the total thickness of the cell (i.e., $h1$+$h2$+$h3$) is calculated to be equal to 3.2 mm (i.e., 0.06λ @ 6 GHz). The graphs of the optimal cell are presented in Fig. 7. According to Fig. 7(a), 3 dB-$AR$ bandwidth is 111.2% (6.12–21.42 GHz), while an excellent $AR$ of less than 1.7 dB is obtained from 6.3 to 21 GHz (BW = 107%). As clearly seen from Fig. 7(b), the amplitude condition for having CP mode (i.e., $\left| {{r_{xx}}} \right| \cong \left| {{r_{yx}}} \right|$) is met from 6 to 21 GHz. As expected, their phase difference, $\Delta {\varphi _{xy}}$ (i.e., ${\varphi _{xx}} - {\varphi _{yx}}$) is shown in Fig. 7(c), which is around $ - \frac{\pi }{2}$ or $ + \frac{{3\pi }}{2}$, indicating the production of only RHCP mode in the entire band. Another qualitative parameter is the polarization conversion efficiency (PCE), which is calculated as follows [Reference Lin, Guo, Lv, Wu, Ma, Liu and Wang3, Reference Zhang, Zhu, Zhang, Yang, Wang and Liu29]:

(7)\begin{equation}PCE = \,\frac{{{{\left| {{r_{RHCP - j}}} \right|}^2}}}{{{{\left| {{r_{RHCP - j\,}}} \right|}^2} + {{\left| {{r_{LHCP - j\,}}} \right|}^2}}}\;\left( {\,j = x\,,\,y} \right)\end{equation}

Table 3. Optimal values of the parameters

Figure 7. Optimal results including: (a) $AR$, (b) reflection coefficients, (c) reflection phase difference, (d) $PCE$, and (e) polarization mode. AR = axial ratio; PCE = polarization conversion efficiency.

where the expressions related to right- and left-handed circular polarizations are

(8)\begin{equation}\{ \begin{array}{*{20}{c}} {{r_{RHCP - j}} = \frac{{\sqrt 2 \left( {{r_{yj}} + i{r_{xj}}} \right)}}{2}} \\ {{r_{LHCP - j}} = \frac{{\sqrt 2 \left( {{r_{yj}} - i{r_{xj}}} \right)}}{2}} \end{array}\,\,\left( {\,j = x,y} \right).\end{equation}

Based on equations (7) and (8), PCE, RHCP, and LHCP are calculated, which are presented in Fig. 7(d) and (e), respectively. An excellent conversion efficiency of over 95% is achieved from 6.2 to 21.2 GHz. As shown in Fig. 7(e), there is a high purity between RHCP and LHCP modes. Moreover, RHCP amplitude ripple is very low in the entire band, which proves low conversion loss. It should be noted that due to the diagonal symmetry of FSSs, the results will be the same for x- and y-polarized incident waves, and so it is not necessary to repeat the results.

Current distribution and CP validation

To fully comprehend the LP to CP conversion mechanism, it is crucial to investigate the surface current distributions on both the top and bottom surfaces of the cell at the desired frequencies with minimum $AR$. Figure 8 illustrates the results for four specific resonances. It is worth noting that for all resonances and on both upper and lower surfaces, all current vectors are aggregated, and the equivalent current vector (${\text{Jeq}}$) is provided, as shown in the figure. It is evident that the equivalent currents in the upper and lower surfaces are perpendicular to each other, which is the primary condition for creating CP, while simultaneously stimulating a 90° phase difference between them. Based on Fig. 8(a) and (b), long and uniform currents are generated along the FSS1 diagonal, which is a contributing factor to creating high-efficiency reflective resonances at low frequencies. As depicted in Fig. 8(c), at 15.3 GHz, the edges of the circular patches are also activated, which creates a shorter electrical length and slightly tilts the direction of ${\text{Jeq}}$. According to Fig. 8(d), two patches of FSS2 are activated, perpendicular to ground current. It should be noted that at this frequency, the current distribution on FSS1 is highly irregular and therefore will not be effective in reflecting the wave. We have not included the related graph for brevity. Lastly, it is important to mention that we examined the current distributions for both $x$- and $y$-directed incident waves, and the results were quite similar. Therefore, we do not present them here for brevity.

Figure 8. Current distributions on top (FSS1 and FSS2) and bottom (ground plane) layers at resonances with minimum $AR$: (a) 6.5 GHz, (b) 10.7 GHz, (c) 15.3 GHz, and (d) 20.1 GHz. AR = axial ratio; FSS = frequency-selective structure.