# Optical Beamforming Networks for Millimeter-Wave Wireless Communications
## 3 OBFNs with TTD Architectures
### Fiber Dispersion Delay Architectures  
The fiber dispersion delay architectures are the most mature ones in TTD architectures, owing to adopting commercially mature fibers and devices. The time delay difference between adjacent beam angles or adjacent channels is produced by the chromatic dispersion of different wavelengths and can be given by the following [43; 51]:  
\[\Delta\tau=LD(\lambda_{2}-\lambda_{1}) \tag{11}\]  
where \(D\) is the chromatic dispersion coefficient (ps/nm/km), \(L\) is the length of the fiber (km), and \(\lambda_{1}\) and \(\lambda_{2}\) represent the adjacent wavelengths (nm). As shown in Figure 5a,b, the fiber dispersion delay architecture with \(N\) channels (equal to the number of antennas) can be established by one tunable laser, one electro-optic modulator, \(1\times N\) splitter, \(N\) different fiber delay lines and \(N\) PDs, namely the \(1\lambda\times N\) framework [42], while it can also be built by \(N\) tunable lasers, one multiplexer, one electro-optic modulator, one fiber, one de-multiplexer and \(N\) PDs, namely the \(N\lambda\times N\) framework [52]. The \(1\lambda\times N\) framework uses one tunable laser and \(N\) fiber channels with different lengths of dispersive fibers and non-dispersive fibers, causing progressive time delays for adjacent antenna elements, whereas the \(N\lambda\times N\) framework use only one dispersive fiber, such as a common single-mode fiber (SMF), to afford various delays for \(N\) channels. These frameworks realize 1D beam sweeping by changing the laser wavelength, which has the advantage of wide bandwidth that is only limited by the bandwidth (\(\sim\)100 GHz) of optical devices such as modulators and PDs [53; 54]. Moreover, the \(N\lambda\times N\) framework can achieve very low loss, due to the low transmission loss of commercial SMFs. This advantage can also be obtained in the \(1\lambda\times N\) framework if SMFs with different lengths are utilized in this architecture, instead of using relatively high-loss dispersion fibers. Although the \(N\lambda\times N\) framework can largely reduce the cost of the fiber, the increase in number of the tunable laser may be counter-productive in terms of the cost of whole system. Nevertheless, the wavelength multiplexing applied in the \(N\lambda\times N\) framework is one of paramount advantages in OBFNs compared to electrical beamforming networks. Note that, a PAA with a large number of antenna elements needs high-resolution scanning with a wide coverage, which further demands a tunable laser with very high  
Figure 4: Timeline of TTD architectures proposed in the past [17; 18; 23; 24; 42; 43; 44; 45; 46; 47; 48; 49; 50]. tuning precision. For example, as calculated by Equation (12) [52], where \(n\) is the bit number corresponding to the resolution of the PAA, a PAA with a six-bit scanning resolution and a RF of 30 GHz requires a tunable laser with a wavelength resolution of less than 10 p.m.  
\[\Delta\lambda=1/(2^{n+1}LDf_{\text{RF}}) \tag{12}\]  
For practical implementation, 2D beamforming is a basic requirement for 2D planar PAAs, which can exhibit the superiority of OBFNs thanks to their multi-dimensional multiplexing abilities such as wavelength multiplexing and spatial multiplexing. As illustrated in Figure 6a, an OBFN architecture for 2D PAAs has been proposed [51], using an optical frequency comb, fiber dispersion unit, programmable optical filter and microwave photonic filter (MPF) to control the delay of each channel. The optical frequency comb can simultaneously produce multiple optical carriers with various wavelengths and replace multiple lasers presented in the \(N\lambda\times N\) framework, reducing the size of OBFNs.