**4. PWPA and OWPA results and discussion**

#### **4.1 PWPA (plane wave patch antenna)**

PWPA has been widely used in communications because it is one of the efficient radiators, with easy and low-cost fabrication and smooth integration with microwave circuits, and it supports linear and circular polarizations. In this work, the patch

antenna is designed at frequency *f* ¼ 2*:*45 GHz for demonstration and comparison purposes on Rogers RT5880 substrate having dielectric constant (*ε<sup>r</sup>* ¼ 2*:*2), thickness (*t* ¼ 0*:*787 *mm* ¼ 30 *mils*), and loss tangent (*δ* ¼ 0*:*0009). The antenna has a coaxial probe feed from the ground plane, and it is located 6.4 mm in X-direction from the center point (0, 0, 0), as shown in **Figure 9**.

The designed procedure outline is based on the well-established cavity model formulation with a simulation model of circular microstrip patch antenna for the dominant mode *TMz lp*0. The radius ð Þ*r* of the circular patch is determined for the desired resonant frequency at a given value of dielectric constant *ε<sup>r</sup>* [65].

$$r = \frac{c}{2\pi\sqrt{\varepsilon\_r}} \left(\frac{\chi\_p'}{f\_{lp0}}\right) \tag{22}$$

where *χ*<sup>0</sup> *lp* represents the zeros of the derivative of the Bessel function *Jl*ð Þ *x* . A circular patch antenna has one degree of freedom, i.e., increasing the patch radius will decrease the resonant frequency and vice versa.

The CST Studio Suite® is utilized for the analysis of the patch antenna. 3D CAD is analyzed using a time-domain solver (FIT) and frequency-domain solver (FEM). The convergence of the S-parameter result is ensured from both the numerical techniques so that the simulated results match with measured results. The FIT solver technique on hexahedral mesh with Perfect Boundary Approximation® (PBA), to achieve a conformal meshing of the patch without having staircase approximation of mesh in 3D, is used for fast and accurate computation of fields in the patch, and the FEM solution is based on the tetrahedral mesh. The FIT solver is typically suited for electrically large and broadband problems as the requirement of hardware memory scale linearly with the number of mesh cells. In contrast, the FEM solver is suited for electrically small, narrowband, and resonant problems as the memory scale-up quadratically with the number of mesh cells [66].

The simulated and measured reflection coefficients of PWPA are plotted in **Figure 10**. PWPA is simulated with both FIT and FEM, which suggest a nearly perfect agreement with measurements. All the values are well below �30 dB, as depicted in **Figure 10**. The 1D polar plot and 3D radiation pattern plot of realized gain (dBi and linear) are depicted in **Figure 11**. All the plots show 7 dBi of realized gain in simulation, whereas 5 dBi in measurement, offset of merely 2 dB in measurement versus

*Orbital Angular Momentum Wave and Propagation DOI: http://dx.doi.org/10.5772/intechopen.104477*

**Figure 10.** *Simulated and measured reflection coefficient of PWPA.*

**Figure 11.**

*(a) 1D polar plot of simulated and measured realized gain for E-plane and H-plane realized gain in dBi, and (b) realized gain in linear. (c) Plot of simulated 3D radiation pattern realized gain in dBi, and (d) realized gain in linear.*

simulation. This offset is attributed to surface wave loss, dielectric loss, and exclusion of the connector model in simulation. On a linear scale, the realized gain in simulation is 5, while the measurement is 3.5.

#### **4.2 OWPA (OAM wave patch antenna)**

In literature, the OAM wave antenna has generally been designed with a UCCA and complex feed structure. In this work, we designed an unsymmetrical patch antenna [67, 68] by identifying a particular feed location where the patch will produce the OAM wave. The OWPA parameters are the same as those of PWPA. The red dot indicates the position of coaxial feed at X=8.75 mm, Y=21 mm from the center point of the antenna, as shown in **Figure 12**. Before a full-wave analysis of this antenna, characteristic mode analysis (CMA) is performed to examine the distribution of electric field and surface current on the antenna and excite the particular mode such that it is set to generate the desired OAM wave. The modal approach has been widely popular in the research community to design novel antennas for various applications, whereas CMA is gaining its popularity because it provides insight for improved

**Figure 12.** *Front and back view of simulated OWPA in CST studio suite®.*

antenna design. CMA is a modal analysis (without excitation) for radiating structures or infinite open structures, whereas eigenmode analysis is for cavity structure or cavity filter design. The main advantage of CMA is to provide a better physical understanding of the surface current and field behavior of an antenna, which helps improve antenna design or postulate new antenna ideas. CMA provides valuable information such as resonant frequencies of inherent modes, far-field modal radiation patterns, modal surface currents on the analyzed structure, and modal significance at given frequencies. CMA can provide insight into the physical phenomena of an antenna of arbitrary shape and thus facilitate the analysis, synthesis, and optimization of antennas.

In the CMA theory, a generalized eigenvalue problem is solved. The generalized eigenvalue problem is given as follows [69]:

$$X\!\!\!\!/\_n = \lambda\_n \!\!\!R\!\!/\_n\tag{23}$$

where R & X are real and imaginary Hermitian parts. *Jn* and *λ<sup>n</sup>* are real eigenvectors and eigenmodes of the nth order mode. There are two main parameters that should be investigated while performing the CMA, which is given in the following:

Model significance (MS): MS is an intrinsic property that signifies the coupling capability of each characteristic mode with external sources. Individual mode contribution for the EM response of a given source is pointed by modal significance. It is easier to use MS rather than eigenvalue to examine the resonance of a structure. For perfectly radiating mode, MS = 1 for *λ<sup>n</sup>* = 0.

$$\text{MS} = \left| \frac{\mathbf{1}}{\mathbf{1} + j\lambda\_n} \right| \tag{24}$$

Characteristic angle (*αn*): It sets forth the comprehension of the mode behavior near resonance. Mathematically, *<sup>α</sup><sup>n</sup>* <sup>¼</sup> <sup>180</sup><sup>∘</sup> � arctan ð Þ *<sup>λ</sup><sup>n</sup>* .


In general, for a resonant mode, the following three criteria should be satisfied as

$$
\lambda\_n = 0, \mathsf{MS} = \mathbf{1}, a\_n = \mathbf{180^\circ}.
$$

In this work, CMA is performed for five modes tracking in CST Studio Suite® between 2 and 3 GHz, and it is found that mode 1 (*f* = 2.3345 GHz) and mode 3

#### **Figure 13.**

*Simulated (a) eigenvalue, (b) modal significance, (c) characteristic angle, and (d) MS vs. S-parameter using CMA in CST–MWS, (e) surface current of mode 2, mode 4, and mode 5.*

(*f* = 2.4505 GHz) are the primary contributing modes for radiation. Mode 2, mode 4, and mode 5 have MS approaching 1 but do not satisfy all three resonant mode criteria. Moreover, the surface current of mode 2, mode 4, and mode 5 does not have null energy at the center, as shown in **Figure 13(e)**; the desired characteristic for twisting wave generation, henceforth, has been removed from further analysis. In addition, among mode 1 to mode 5, which are the most contributing modes, are determined by studying the nature of surface currents, electric fields, and far-field radiation behaviors of different CMA modes. Two orthogonal CMA modes are combined to form a full-wave analysis mode. The modal radiation quality factor is calculated as in Eq. (25). **Table 2** shows all five modes sorted at 2.45 GHz, resonance frequencies, Q-factors, and MS. Upon performing CMA, the plot of the significance of the modes sorted over frequency, in other words, which mode is contributing as the strongest to the radiation from the antenna at a given frequency (*f* = 2.45 GHz) is obtained in **Figure 13(b)** [70].

$$Q\_n = \frac{\alpha}{2} \left| \frac{d\lambda\_n}{d\alpha} \right| \Big|\_{\alpha = w\_m} \tag{25}$$

We can also directly obtain the surface current distribution, electric and magnetic field, and far-field plot for all the modes at a given frequency of interest at 2.45 GHz using CMA.

In addition, overlaying the full-wave S-parameter curve on the modal significance plot as depicted in **Figure 13(d)** immediately shows which CMA mode is contributing to the resonances observed in the driven S parameter curve. In fact, for a given feed point, we can see a strong correlation between the driven mode currents and the CM currents. It is worth noting that when we drive the antenna, there is a contribution from all the modes at this frequency. Therefore, we cannot expect a perfect correlation. CMA is performed with metallic and dielectric together, giving better information than the classical approach of calculating CMA without lossy dielectric substrate. **Figure 14** clearly signifies the fullwave mode at *f* = 2.45 GHz is a combination of two CMA modes, mode 1 (*f* = 2.3345 GHz) and mode 3 (*f* = 2.4505 GHz). Overall, this gives rise to zero surface current, electric field, and far-field radiation at the central axis of the antenna. Besides, it generates a spiral motion of the electric field, which is the desired characteristic for twisting waves. With this understanding through CMA, we determine the location of the feed of an antenna, which will contribute to mode 1 and mode 3. We choose the feed as a common location for the surface current of both mode 1 and mode 3 of an elliptical patch antenna.

In this way, we will tame a simple plane wave patch antenna to an OAM wave patch antenna. Simulated and measured reflection coefficients of OWPA are shown in **Figure 15**. OWPA are simulated with FIT and FEM, showing a nearly perfect agreement with measurements. All the reflection coefficient values are well below �30 dB,


**Table 2.** *Parameters of CMA in CST studio suite®.* *Orbital Angular Momentum Wave and Propagation DOI: http://dx.doi.org/10.5772/intechopen.104477*

**Figure 14.**

*Simulated (a) surface current, (b) electric field, (c) 3D radiation pattern for full-wave analysis, CMA mode 1 and mode 3 analysis.*

as shown in **Figure 15**. The 1D polar plot of realized gain (dBi and linear) is depicted in **Figure 16**. The plots show 3.8 dBi of realized gain in simulation, whereas 1.8 dBi in measurement offset of merely 2 dB in simulation and measurement. On a linear scale, the realized gain in simulation is 2.5, whereas in measurement is 1.5.
