# Integrated Optical Phased Arrays for Beam Forming and Steering
## 2 Principles
### Beam Steering Based on WGAs
#### 2.3.2 Steering Based on Phase Differences & Wavelengths  
For this kind of OPAs, each channel contains a diffractive WGA with inherent wavelength-angle dispersion characteristics. Along the axis of the WGA, the outcoupling angle \(\theta\) shown in Figure 3b is governed by the grating equation [7,8], which is given by  
\[\sin\theta=\frac{\Lambda n_{eff}-\lambda_{0}}{n_{ct}\Lambda}, \tag{6}\]  
where \(\Lambda\) is the grating period, \(n_{eff}\) is the effective index, \(n_{ct}\) is the refractive index of the cladding, \(\lambda_{0}\) denotes the wavelength. The beam steering of \(\theta\) angle can be achieved by wavelength tuning. In addition, the outcoupling angle \(\theta\) can also be changed by effective index changing [7,51,64] based on thermal or electric optical modulations.  
Ultra-large numerical apertures and nearly diffraction-limited far-field beamwidths were achieved by millimeter-scale WGA-based OPAs [65,66]. The azimuthal and polar beam steering can be formed by phase modulators and tunable lasers, respectively [11,12]. For a \(N\times M\) array, only \(N\) phase controls and one wavelength control are required [12]. The simplified control circuits greatly boost the development and application of OPAs.  
Figure 3: (**a**) Half-wavelength element spacing OPA with integrated TO heaters for beam steering from left to right. Reproduced from [61] with the permission of AIP Publishing. (**b**) Schematic view of the cross section of a WGA. Reprinted with permission from [8] © The Optical Society. (**c**) Schematic image of the proposed 2D beam steering device at a fixed wavelength. Reprinted with permission from [62] © The Optical Society. (**d**) Schematic of a serpentine OPA. Reprinted with permission from [63] © The Optical Society. (**e**) A 2 \(\times\) 2 OPA receiver. Reprinted with permission from [43] © The Optical Society.  
2.3.3. 2D Steering Based on Optical Switches at a Fixed Wavelength  
2D beam steering at a fixed wavelength can be achieved by optical switches and phase controllers. As illustrated in Figure 3c, the beam steering in polar direction is accessed by actively switching among various WGAs with different diffraction angles, while the beam is azimuthally steered via phase differences. 2D beam steering over an FOV of 3\({}^{\circ}\)\(\times\) 17.6\({}^{\circ}\) was demonstrated using a silicon nitride (Si\({}_{3}\)N\({}_{4}\)) integrated OPA at a fixed wavelength of 905 nm [62]. In this case, optical switches were used to enable beam manipulation and dynamic control. However, the performance and application of the implemented OPA are limited by the FOV, especially for the polar direction. This approach also results in a relatively large chip size.  
OPAs consisting of electro-optic (EO) p-i-n phase shifters and TO tunable n-i-n WGANs can be used to demonstrate 2D beam steering at a fixed wavelength. As mentioned above, the effective refractive index \(n_{eff}\) in Equation (6) can be changed by the currents in heaters. The lateral radiation can be steered by phase differences based on the electro-optical effect. A 16-element Si OPA enables 2D beam steering of 45.4\({}^{\circ}\)\(\times\) 10.0\({}^{\circ}\) with a beamwidth of 3.2\({}^{\circ}\)\(\times\) 5.8\({}^{\circ}\) at 1.55 \(\upmu\)m. The tuning efficiency is 0.016\({}^{\circ}\)/mW in the polar beam steering [64]. The 64-element OPA containing EO p-i-n phase shifters, and TO tunable n-i-n grating radiators covers the FOV of 46.0\({}^{\circ}\)\(\times\) 10.2\({}^{\circ}\) with a beamwidth of 0.7\({}^{\circ}\)\(\times\) 0.9\({}^{\circ}\) at 1.532 \(\upmu\)m [51]. This provides a solution to remove the need for a tunable source or WGAs with different periods as in [62]. Nevertheless, the polar beam steering efficiency is low. The efficiency of OPA is limited by power-hungry TO shifters. The thermal crosstalk raises another problem, especially in the case of large-scale integration.