# A Photonically-Excited Leaky-Wave Antenna Array at E-Band for 1-D Beam Steering
## 2 Leaky-Wave Antenna
### Radiating Section  
The radiating section of the array comprises four identical rows. Every row consists of a microstrip line periodically loaded by groups of four stubs plus a patch that truncates it, as shown in Figure 1. The design of the periodic structure is crucial to obtain low values of \(|S_{11}|\), as explained below. One of the difficulties encountered while designing a 1-D periodic LWA is the scan through broadside. At this point, all reflections from the perturbations add in phase back to the source. This can create a bandgap in the \(k\)-\(\beta\) diagram; in this case the wave does not propagate into the structure and is completely reflected, degrading the input impedance matching. Following the discussion in [19], the perturbations of the periodic structure are modeled in a simplified network by shunt impedances connected by segments of TL. At broadside emission, the wave impinges the perturbations in phase, and the equivalent network is a shunt of infinite loads, which produces a short-circuit and the structure reflects the wave completely. On the other hand, if substantial losses are introduced by each perturbation, the current will progressively decay, effectively breaking the infinite periodicity, and a short-circuit will not be achieved. Hence, the latter case will present better matching characteristics. In practice, each perturbation introduces losses and reflections to some degree. They have to be traded-off to obtain the desired radiation and impedance matching characteristics. Since the full-wave simulation of electrically large structures can be a computationally intensive task, it is preferable to use approximate criteria to optimize the period of the structure at the frequency of broadside radiation. To that end, it is assumed that coupling and surface-wave generation are negligible. In [22] it was shown that a spacing larger than \(0.1\lambda_{\text{g}}\) leads to tolerable coupling effects between stubs in a loaded microscrip, where \(\lambda_{\text{g}}\) is the guided wavelength in the microstrip TL. The calculation consists of computing the \(S\)-parameters of an N-periods structure (\(S^{\text{for}}\)), from the scattering matrix of just one. The structure is assumed to be terminated in a matched load, and only the first bounce is considered for reflections, whereas for transmission no bounces are summed up. This is true only if \(|S_{21}|>>|S_{11}|\). In this manner, the following calculation yields expressions that can be readily computed and used for optimization:  
\[S_{11}^{\text{for}}\approx S_{11}\sum_{n=0}^{N-1}S_{21}^{2n} \tag{1}\]  
\[S_{21}^{\text{for}}\approx S_{21}^{N} \tag{2}\]  
The results from Equations (1) and (2), along with the exact computations, are shown in Figure 3 for a 10% and a 20% loss per pass (\(|S_{21}|=\sqrt{0.9}\), \(\sqrt{0.8}\), respectively). A symmetric unit-cell with \(S_{11}=S_{22}\) was assumed to compute the exact values. \(|S_{11}|\) is taken proportional to \(|S_{21}|\) so that the infinite periodic structure presents \(S_{11}^{\text{for}}\) equal to -10 dB (and the bandgap can be considered effectively removed in terms of impedance matching). Figure 3 shows the largest deviation between exact and approximate curves. It occurs when the phase of \(S_{11}\) equals \(\pi\). Second and higher-order even bounces add in phase with the traveling wave, yielding lower transmission losses than the approximate  
Figure 2: Left: Magnitude of isolation, \(|S_{21}|\), for a choke composed of 1, 3, 5, and 7 sections. Center: magnitude of the reflection coefficient, \(|S_{11}|\), for a 7-section choke. Continuous line: calculated by a transmission line (TL) equivalent. Dashed line: simulated. Right: View of the choke. calculation. Additionally, third and higher-order odd bounces add in phase with the first bounce so that \(S_{1I}\) is underestimated in the approximate curves. As \(\left|S_{11}^{Tot}\right|\) decreases, the agreement is better, while it degrades the other way around.