# A Photonically-Excited Leaky-Wave Antenna Array at E-Band for 1-D Beam Steering
## 2 Leaky-Wave Antenna
### Radiating Section
In the latter case, the effects of stationary waves should be included computing multiple bounces. Finally, it is worth mentioning that as the losses per pass increase, \(\left|S_{1I}\right|\) can be proportionally larger that \(\left|S_{2I}\right|\) to achieve the same \(S_{11}^{Tot}\) threshold (the sum in Equation (1) approaches 1). Hence, increasing the losses per period relaxes the demand on \(S_{1I}\) for adequate input impedance matching, in line with the earlier discussion. Note also that truncating the structure helps to decrease the reflection coefficient. Finally, for frequencies other than that of broadside directed emission, (1) and (2) can be adapted to introduce the phase change per period in \(S_{2I}\). As an example of the usefulness of (1) and (2), we simulated the unit-cell reported in [23] (Figure 10) that removed the bandgap at broadside. The structure can be well described by a traveling wave, following our assumptions. \(S_{11}^{Tot}\), \(S_{21}^{Tot}\) were computed for \(N=32\) and the approximate calculations yielded a return loss of 14.5 dB and a transmission loss of 16.3 dB, whereas simulation of the complete structure (using waveports at both ends) yielded 15.1 dB and 18.5 dB, respectively, with a CPU simulation time 18 times larger than the single unit-cell. This indicates that Equations (1) and (2) constitute a useful tool for preliminary optimization of this kind of structure. The unit-cell used in our design was first reported in [22], and it is shown in Figure 4a. The stub length (\(l\)) is approximately \(\lambda\)/2, presenting high input impedance, to introduce a small perturbation in the TL, whereas its width (\(w\)) controls the amount of radiated power [22]. In turn, the stubs form groups of four: two towards the right and two towards the left, equispaced at distance \(s\). This configuration allows one to increase the return loss if \(\Delta arg(S_{2I})\) per stub plus inner separation is approximately \(\pi\)/2. In addition, it introduces more losses per period, so that the design can be made more compact. On the other hand, the stubs do not radiate in phase but in quadrature. This results in a pattern that points off-broadside in the plane perpendicular to the row; this effect is conveniently eliminated by the array factor of the complete four-row structure. The simulated and calculated \(S\)-parameters using Equations (1) and (2) are shown in Figure 4b for broadside emission. The periodic structure presents \(S_{1I}<-12\) dB, and the calculated curve is in agreement with simulated values. The calculated values for \(S_{2I}\) are slightly overestimated due to the excluding of the higher-order bounces in the calculation, as also occurred in Figure 3. Once the group of four stubs is optimized (Figure 4b), a segment of transmission line is added to obtain the required length, \(p\), for a total phase change of \(2\pi\) in \(S_{2I}\) per period at the central frequency of operation. To do so, the \(S\)-matrix of the four stubs is retrieved from full-wave simulations, and the corresponding segment of TL is calculated. When several periods are simulated, the length of TL needs to be re-tuned. Its final value changes by about 4%. Hence, effects associated with surface waves and coupling introduce only a small perturbation. To show  
Figure 3: \(S\)-parameters as a function of the number of periods in the structure. Continuous line: exact calculation. Dashed line: approximate calculation from (1) and (2). The percentage indicates the power loss per pass of one period of the structure. the suppression of the open-stopband, Figure 4c represents the dispersion diagram and attenuation constant of the optimized unit-cell within the band of interest (solid black line). Dispersion and attenuation are also plotted for the same unit-cell geometry with \(w=50\) um (solid grey line), and for a unit-cell with just a \(50\) um wide stub (dashed grey line). In the latter cases, \(p\) is tuned to obtain \(\Delta arg(S_{21})\) equaling \(2\pi\) approximately at 81 GHz.