# A Photonically-Excited Leaky-Wave Antenna Array at E-Band for 1-D Beam Steering
## Appendix A Method for Probe De-Embedding  
\(S\)-parameter measurements at mm-wave frequencies with coplanar probes comprise the challenge of probe and interconnection de-embedding. Indeed, the ground signal ground (GSG) probe interconnection with the device under test (DUT), as well as the probe itself, can introduce a relevant contribution to the overall measurement, and thus its effect has to be removed for an accurate determination of \(S\)-parameters. Besides, for one-port components, only one-port probe measurements might be available. In this section, we describe the procedure employed to de-embed the probe and the interconnection from \(S_{11}\) measurements. The difficulty of fabrication of 50-\(\Omega\) loads at mm-wave frequencies in the same substrate as the DUT makes the use of the standard short-open-load-through (SOLT) calibration impractical [31]. Another of the standard methods, through-reflection-line (TRL), requires the use of two probes. Finally, one finds open-short (OS) or the short-open (SO) methods, which include the modeling of parasitic lumped elements introduced by the probes plus interconnection. These methods, however, rely on a specific probe plus interconnection, and it was shown in [31] that beyond 30 GHz substantial discrepancies existed between both for a reverse-biased Schottky diode on a silicon substrate. The method presented herein is a variant from [31] based on simulation plus measurements. Instead of imposing \(S_{11}=S_{22}\), as it is done in [31], an additional load is measured to obtain a fourth relation. It consists of a set of four measurements, three on test loads whose values are assumed, plus another one on the DUT. The three measurements plus the reciprocity condition yield four relations from which the ABCD parameters of the interconnection can be retrieved analytically. It is also assumed that its ABCD matrix is maintained through measurements. This implies that the same interconnect has to be used, in addition to landing the probes at the same positions. The measurement schematic is shown in Appendix A Figure 11.  
The \(S_{11}\) parameter at a calibrated reference plane of an arbitrary structure plus load, \(Z_{L}\), is given by Equation (A1), where A, B, C, and D are the ABCD-matrix elements of the structure between the load and the reference plane:  
\[S_{11}=-\frac{Z_{c}Z_{L}C+Z_{c}D-AZ_{L}-B}{Z_{c}Z_{L}C+Z_{c}D+AZ_{L}+B}\] (A1)  
A short-circuit (S) and an open circuit (O) are loads that can be relatively simply manufactured in the same conditions as the DUT. Substituting in Equation (A1), the corresponding relations are found for S and O:  
\[D = \frac{B\left(1-S_{11}^{S}\right)}{Z_{c}\left(1+S_{11}^{S}\right)},\] (A2) \[C = \frac{A\left(1-S_{11}^{o}\right)}{Z_{c}\left(1+S_{11}^{o}\right)}.\] (A3)If the probe plus interconnect structure are composed of passive, isotropic materials, one can enforce reciprocity and obtain the following relation:  
\[\text{AD}-\text{BC}=1 \tag{10}\]  
Finally, a third load is measured to obtain from Equation (11) the fourth relation. It consists of a translated short (TS) or translated open (TO). The surge impedance of the transmission line, \(Zc\), and the propagation constant, are obtained from full-wave simulations. These are parameters that can be easily simulated for standard low-loss 50-\(\Omega\) transmission lines. The translated length of the load is chosen as \(\lambda\)/8 to distribute evenly the three loads on the Smith chart. Note that if broadband measurements are performed, different windows of frequencies will require different translated lengths. The manufactured O, S, and TS are shown in Appendix A Figure 10a-c, as well as one DUT (two-row array) in Figure 10d. To minimize the effect of the GSG pads, and to adapt the probe pitch to the G-CPW, a tapered interconnection is optimized (Appendix A Figure 10e). Additionally, the short and open structures are optimized using full-wave simulations to obtain nominal \(S_{II}\) parameters.