# Design of a Double-Matched Cross-Polar Single Antenna Harmonic Tag
## 3 Antenna Design
### Rationale of the Antenna Design Procedure  
An annular ring microstrip antenna (ARMSA) is shown in Figure 4, with \(r_{o}\) and \(r_{i}\), the outer and inner radii, respectively. The resonance frequency of the ARMSA is given by [39]  
\[f_{mm}=\frac{cX_{nm}}{2\pi r_{i}\sqrt{\epsilon_{e}}} \tag{1}\]  
where \(c\) is the speed of light, \(\epsilon_{e}\) is the effective dielectric constant, and \(X_{nm}\), \((n,m)\in\{(\mathbb{N},\mathbb{N}^{+})-(0,1)\}\), are the roots of a characteristic equation involving derivatives of Bessel functions. The values of \(X_{nm}\) depend on the ratio \(\frac{r_{o}}{r_{i}}\), as well as the order \(n\) of the Bessel function and the index \(m\) of the root. An ARMSA supports a number of modes: if the ratio \(\frac{r_{o}}{r_{i}}\) is not too large, \(X_{21}\) is about double \(X_{11}\), so that two modes having resonant frequency, one double the other, are possible. As an example, in the case of \(\frac{r_{o}}{r_{i}}=2\), \(X_{11}=0.6773\) while \(X_{21}=1.3406\). Consider now Figure 4a, where the annular ring is fed by a microstrip line (to make the description easier, in the following, we name the direction of that line'vertical'). The mode \(TM_{11}\) (Figure 4a) is characterized by a couple of surface currents that flow over the ring and are distributed symmetrically with respect to a symmetry line passing through the feeding point and the point (shown as 'x' in Figure 4) of the null (or minimum) magnitude of current. That current distribution allows it to radiate a linear polarized field with maximum directivity at the boresight direction (i.e., the direction orthogonal to the plane of the ring)  
Figure 4: Schema of annular ring microstrip antenna fed by a microstrip line; \(r_{o}\) and \(r_{i}\) are outer and inner radius, respectively. Red arrows represent currents, crosses position of nulls (or minima) of current, circles maxima of current. On right side, the corresponding radiation patterns. (**a**) \(TM_{11}\) mode; (**b**) \(TM_{21}\) mode. and a beam similar to that of a half-wavelength circular patch antenna (i.e., cardioid-like), shown on the right side of Figure 4a. In \(TM_{21}\) mode, currents are again symmetric with respect to the symmetry line but are anti-symmetric (i.e., 180\({}^{\circ}\) out of phase) with respect to the line orthogonal to the symmetry line (Figure 4b) (for simplicity, we name it anti-symmetric line). For this reason, we can identify four currents that radiate fields that are 180\({}^{\circ}\) out of phase with respect to the anti-symmetry line. The superposition of fields radiated by these four currents gives a null of field at the boresight direction, and the beam results are conical, as shown on the right side of Figure 4b. Since we are interested to have similar radiation patterns at both fundamental and second harmonic frequencies, the field radiated by \(TM_{21}\) mode is not suitable for our purposes. Moreover, it would be beneficial for harmonic tags to use an orthogonal polarization in replying to an interrogation, in order to mitigate the environmental clutter. Perturbation methods can be used in ARMSA to change polarization using slits or protrusions similar to stubs [38]. Figure 5 shows a schema of a possible model of perturbed ARMSA for radiating linear polarization (aligned along the anti-symmetry line) in second harmonic and preserving the maximum of directivity at the boresight direction. The feeding line is 90\({}^{\circ}\) rotated with respect to that of \(TM_{11}\) mode (we name it 'horizontal'), while a stub (protrusion) is connected to the ring in the opposite direction along the anti-symmetry line. At the frequency of the second harmonic, the \(TM_{21}\) current distribution is preserved over the ring but an additional in-phase current is introduced over the stub (protrusion). This additional current radiates a not-null field in the boresight direction having polarization along the anti-symmetry line.