# Optical Beamforming Networks for Millimeter-Wave Wireless Communications
## 4 OBFNs with Phase Shifter Architectures
### Matrix Architectures  
Matrix architectures, including the Butler matrix, Blass matrix and Nolen matrix are popularly adopted in electric beamforming networks [103; 104; 105; 106]. A distinguished advantage of these architectures is that multi-beam beamforming can be straightforwardly realized when injecting signals into different input ports, in addition to the fact that these architectures have compact structures. There are few works that have adopted these architectures in OBFNs [29; 91; 96; 107; 108; 109; 110], in which the optical Butler matrix architecture has the superiorities of footprint and optical loss. Figure 20a shows a 4 \(\times\) 4 optical Butler matrix which consists of four 3 dB couplers, two phase shifters and a waveguide crossing [111], which is similar to the electrical counterpart. With this structure, each optical signal from one of the input ports will be split into four output ports with a linear phase relationship and an even power. For example, the phases of four optical outputs, when launching from the first input port (In\({}_{1}\)), are \(\varphi_{1}\), \(\pi/4+\varphi_{1}\), \(\pi/2+\varphi_{1}\), and \(3\pi/4+\varphi_{1}\), separately. Using the similar Butler matrix, several implementations for OBFN transmitters and receivers were established [91; 107; 108; 109; 110]. For example, a Butler matrix architecture for an OBFN receiver has been demonstrated with an 8 \(\times\) 8 Butler matrix fabricated on a lithium niobate (LiNbO\({}_{3}\)) platform with a footprint of \(\sim\)32 mm \(\times\) 0.9 mm [91]. Employing this Butler matrix in an OBFN receiver has advantages over using other OBFN receivers since each output of the matrix can combine a high-power LO for increasing the receiving power of the system. Figure 19: Schematic diagrams of OBFNs based on phase shifter array with (**a**) thermo-optic phase shifters [22] (Copyright 2002, IEEE, #5491100730358), and (**b**) electro-optic phase shifters [28] (Copyright 2020, IEEE, #5491100991991). The typical Butler matrix shown above cannot tune the amplitude of the output ports, in order to control the side lobe and grating lobe of the beam pattern formed by the PAA. This issue can be solved by the Blass matrix and Nolen matrix, although these architectures have a larger footprint than that of the Butler matrix. Figure 20b presents a 4 \(\times\) 4 Blass matrix, in which a tunable coupler together with a phase shifter is served as one node [29]. The phase and amplitude of lights at four outputs can be tuned simultaneously, and each output has the ability to export four optical signals from the four inputs. Based on this principle, an \(M\times N\) Blass matrix is designed for feeding an \(N\)-element PAA [96]. This architecture adopts the self-heterodyne detection technique to produce RF signals. The main operation is as follows. \(M\) parts of the optical carrier are modulated by \(M\) independent RF signals; then, the upper sidebands are selected by the optical sideband filter and tuned by the \(M\times N\) Blass matrix, and are coupled with the optical carrier at the couplers and mixed back to the RF signals at PDs. Finally, RF signals are fed to \(N\) antennas. In this architecture, \(M\times N\) phase shifters and tunable couplers are utilized, which occupy a relatively large footprint and need a great number of electric control units. One approach to remediate this issue is to use a Nolen matrix, which reduces the number of nodes to half of that in the Blass matrix. The Nolen matrix has a triangular structure which can also be seen in Figure 20b by ignoring the section below the dashed line. A 144 \(\times\) 36 Nolen matrix has been reported, and its beamforming pattern has been theoretically investigated [95]. It is worthy to note that the Nolen matrix architecture for OBFN has a footprint that is three orders of magnitude smaller than that of its electric counterpart.