# Optical Beamforming Networks for Millimeter-Wave Wireless Communications
## 3 OBFNs with TTD Architectures
### MRR Group Delay Architectures  
With photonic integration technology, a large number of TTDs can be integrated on one substrate, which greatly reduces the size of OBFNs. The MRR group delay architecture, employing multiple MRRs as TTDs, is one approach of such OBFNs within a relatively wide bandwidth, which is typically several gigahertz [26; 47; 57]. Figure 7a plots the structure of the MRR, which comprises one microring and one bus waveguide. The time delays of a MRR with a round-trip time of 20 ps (free spectral range = 50 GHz) are shown in Figure 7b, as calculated using the following equation [31] in the case that the optical loss of the MRR is ignored:  
\[ \begin{array}{c}\tau_{g}(\omega)=(\frac{\gamma^{2}-\gamma\sqrt{1-K}\text{ cos}(\omega\tau_{t})}{\gamma^{2}+1-K-2\gamma\sqrt{1-K}\text{cos}(\omega\tau_{t})} \\ \qquad\qquad\qquad\qquad\qquad+\frac{\gamma\sqrt{1-K}\text{cos}(\omega\tau_{t})- \gamma^{2}(1-K)}{1+\gamma^{2}(1-K)-2\gamma\sqrt{1-K}\text{cos}(\omega\tau_{t} )})\tau_{\tau}\end{array}  \tag{13}\]  
where \(K\) is the coupling coefficient, \(\gamma\) is the optical loss factor, \(\tau_{t}\) is the round-trip time (s), and \(\omega\) is the angular frequency of light (rad/s). When ignoring the optical loss of the MRR, \(\gamma=1\). By changing the \(K\), the variation of the group delay induces different time delays of light. The time delay has a larger variation at the on-resonance wavelength than that at the off-resonance/anti-resonance wavelength. The MRR group delay architectures can adopt the time delays at the on-resonance wavelength and off-resonance wavelength. Figure 8a,b shows two examples of these cases, respectively, with non-coherent optical carriers and a binary tree structure fabricated on the silicon nitride platform [26; 58; 51]. To obtain a large time delay (\(\sim\)one hundred picoseconds), the MRR group delay architecture at the on-resonance wavelength uses one MRR as a basic delay unit, while the MRR group delay architecture at the off-resonance wavelength requires several MRRs, such as three MRRs. The former has a smaller size and much fewer power supplies to control the \(K\) of the MRR than those of the latter. However, the MRR group delay architectures at the off-resonance  
Figure 6: Schematic diagrams of 2D OBFNs based on fiber dispersion delays; (**a**) an OBFN transceiver using an optical frequency comb and fiber dispersion units (reprinted with permission from [51] © The Optical Society), and (**b**) an OBFN transmitter using electro-optic frequency combs and MCF (reprinted with permission from [56] © The Optical Society). wavelength possess a high resolution of time delay, leading to finer beam-sweeping than that of the architecture at the on-resonance wavelength. It is worth noting that these MRR group delay architectures with a binary tree structure have an increasing delay ripple (around several picoseconds) with the number increase of MRRs at the first stage [58]. Moreover, thermal crosstalk is another issue in these architectures to be carefully dealt with during the design stage or the post-processing stage. Furthermore, MRR group delay architectures can also be applied in OBFNs with coherent optical carriers. The main advantage of this kind of OBFN is that the intensity of the noise in the optical signals can be nearly eliminated through the balance PDs (BPD) [59; 60], which reduces the noise of these systems. One example of such an architecture for an OBFN receiver with \(N\) antennas has been proposed [24]; the RF signals received are modulated to the optical carrier by MZMs and phase shifted by the MRR delays, and then upper sidebands of \(N\) signal channels are selected by the optical sideband filter (OSBF) and are combined with optical carrier. Finally, the phase-aligned RF signals are produced by the BPD. The MRR delays have a binary tree structure made up of eight MRRs, seven phase shifters and seven couplers.