## 3 Analysis  
### Spherical Mode Decomposition  
To examine the radiating principle of the proposed array with low pattern correlation, we conduct spherical mode decomposition for the radiation patterns of the two antenna elements using a data-fitting method with the following steps. The first step obtains each active element pattern in the directions of \(\theta\) and \(\varphi\) (\(361\times 30\) points). Then, in the far-field region, each active pattern is expanded using the orthonormal spherical harmonics with even and odd modal coefficients [43]. \[Y_{mm}^{e}(\theta,\phi) = \bigg{[}\varepsilon_{m}\frac{(2n+1)}{2\pi}\frac{(n-m)!}{(n+m)!} \bigg{]}^{1/2}P_{n}^{m}(\cos\theta)\cos(m\phi), \tag{4}\] \[Y_{mm}^{o}(\theta,\phi) = \bigg{[}\varepsilon_{m}\frac{(2n+1)}{2\pi}\frac{(n-m)!}{(n+m)!} \bigg{]}^{1/2}P_{n}^{m}(\cos\theta)\sin(m\phi),\] (5) \[E_{tot} = \sum_{n=\ 0}^{N}\sum_{m=\ 0}^{n}[a_{mm}Y_{mm}^{e}(\theta,\phi)+b_{ mm}Y_{mm}^{o}(\theta,\phi)]. \tag{6}\]  
The even and odd orthonormal harmonics of \(Y_{mm}^{e}\) and \(Y_{mm}^{o}\) contain the associated Legendre function according to the angles of \(\theta\) and \(\varphi\), where \(\varepsilon_{m}\) is Neumann's number (1 for \(m=0\), and 2 for \(m>0\)). The total electric field \(E_{tot}\) is defined by the combinations of the even and odd coefficients of \(a_{mm}\) and \(b_{mm}\) with the harmonics (\(n=0\), 1, 2... \(N\) and \(m=0\), 1, 2... \(n\)). The maximum mode number \(N\) is set to 12, and spherical mode decomposition for each active element pattern is conducted to obtain the unknown coefficients \(a_{mm}\) and \(b_{mm}\) using the nonlinear least square method. In both element patterns of the proposed array, the coefficient \(a_{mm}\) is observed as much larger than \(b_{mm}\); therefore, the even mode coefficients should be examined more carefully to observe the modal characteristics of the active element patterns. In particular, among the resulting coefficients (\(a_{mm}\)) for both elements, the coefficients with an index of \(m=0\) predominantly determine the shape of the radiation patterns. Figure 12 shows the coefficient values of \(a_{n0}\) versus the mode index \(n\) for the monopole and patch elements. The monopole element has a zero mode as the dominant mode, while a null pattern on the \(z\)-axis can be generated with negative coefficients in the second or higher modes. By contrast, most modal coefficients for the patch element are positive and significantly different from those of the monopole element. We propose the MDM to observe the modal difference between two antennas in a quantitative value. Herein, all mode coefficients of each antenna are normalized by the maximum  
Figure 11: Array radiation patterns of the proposed array antenna: (**a**) \(zx\)-plane (H-plane); (**b**) \(zy\)-plane (E-plane). coefficient value of the two antennas as written in Equation (7). Then, we calculate the MDM based on the average deviation of the normalized mode coefficients between the two antennas, as shown in Equation (8). \[M_{nm}^{1}\ =\ \frac{\left|a_{nm}^{1}\right|}{\max(\left|a_{nm}^{1}\right|, \left|a_{nm}^{2}\right|)},\ M_{nm}^{2}\ =\ \frac{\left|a_{nm}^{2}\right|}{\max(\left|a_{nm}^{1} \right|,\left|a_{nm}^{2}\right|)}, \tag{7}\]  
\[MDM\ =\ \frac{2}{(N+1)(N+2)}\sum_{n=0}^{N}\sum_{m=0}^{n}\left|M_{nm}^{1}-M_{nm}^{2 }\right|, \tag{8}\]  
where the superscript 1 and 2 indicate the number of the antenna ports. The MDMs of 0.704, 0.307, and 0.219 are obtained for the proposed array, the patch array, and the monopole array, respectively. Note that, the high MDM means an obvious difference exists in the radiating pattern modes of the two antennas. Therefore, the results demonstrate that the low pattern correlation can be maintained for the proposed two-element array, even with an extremely narrow interelement spacing. To confirm the pattern correlation between the two elements, we examine the ECCs of the proposed array compared to the conventional two-element patches and monopoles in terms of the interelement spacing.