# Non-Mechanical Beam Steering with Polarization Gratings: A Review
## 2 Non-Mechanical Beam Steering with PGs  
### PG Fundamentals from a Beam Steering Perspective  
Polarization gratings consist of a patterned birefringent material that exhibits a continuous in-plane rotation of the birefringence director (Figure 1). When patterned as a linear grating, such a structure produces a geometric phase delay across the aperture leading to diffraction given by  
\[\sin\theta_{d}=\frac{m\lambda}{\Lambda}+\sin\theta_{in} \tag{1}\]  
and [14]  
\[\eta_{\pm 1}=\frac{1}{2}\big{(}1\mp S^{\prime}{}_{3}\big{)}\sin^{2}\bigg{(} \frac{\pi\Delta nd}{\lambda}\bigg{)} \tag{2}\]  
Equation (1) is the familiar grating diffraction equation where \(\theta_{d}\) is the diffraction angle, \(m\) is the diffracted order, \(\lambda\) is the wavelength of incident light, \(\Lambda\) is the period of the in-plane birefringence rotation, and \(\theta_{in}\) is the incident angle. In Equation (2), \(\eta_{\pm 1}\) is the diffraction efficiency into the \(\pm 1\) orders, \(S^{\prime}{}_{3}\) is the normalized Stokes vector indicating the ellipticity of the incident light polarization, and \(\pi\Delta nd/\lambda\) is the retardance of the birefringent layer. Equation (2) shows that PGs offer the unique property of being able to couple 100% of incident light into either the \(+1\) or \(-1\) diffraction order when illuminated with circularly polarized light and the retardance is equal to a half wavelength. Moreover, PGs offer the important ability to select between these orders by controlling the handedness of incident polarization. Fortunately, such high diffraction efficiency is not merely a mathematical fiction and PGs with diffraction efficiency > 99.5% are routinely produced in real manufacturing environments (Figure 2a). Figure 1: Schematic representations of a PG showing (**a**) in-plane director rotation and pitch and (**b**) orientation of steering. Such high diffraction efficiency is shared by volume Bragg gratings (VBG), however PGs possess the additional properties of being relatively thin and having a wide angle of acceptance. Specifically, typical polymer PGs are roughly 1-5 \(\upmu\)m thick compared to 0.5-25 mm for VBGs [15]. With regard to acceptance angle, PGs exhibit peak diffraction efficiency at the angle of minimum deviation and a gradual roll-off of diffraction efficiency as incidence angle increases in either direction [13; 16; 17]. This roll-off is sensitive to the ratio of wavelength to grating pitch, and thus to angle, wherein gratings with larger diffraction angles experience greater roll-off. This roll-off and its asymmetry has been attributed to the change in effective retardance with increasing angle of incidence (AOI) [14]. For example, Figure 2a shows the angular response of a polymer PG designed to steer 1550 nm light with a diffraction angle for three different diffraction angles. In the \(\pm 6^{\circ}\) PG case, the diffraction efficiency exceeds 97% over \(\pm 15^{\circ}\) angle of incidence (AOI). By contrast, typical transmissive VBGs have acceptance angles of up to 10 mrad [15]. Other PG variations have been demonstrated that hybridize traditional PGs and VBGs to provide large diffraction angles. Depending on the details of design, these gratings include Polarization Volume Gratings (PVGs) [18], Bragg Polarization Gratings (BPGs) [19], and Dual-Twist Pancharatnam Phase Deflectors (DTPPDs) [20]. All of these PG variations rely on using chiral dopant to induce twist in the birefringence director along the optical axis. The twist provides an additional angled periodicity within the PG film that enhances diffraction efficiency at large angles. These approaches are promising for beam steering applications, as their efficiency at large diffraction angles (e.g., >40\({}^{\circ}\)) exceeds what can be realized with simple PGs. However, these approaches possess a preferred steering direction, instead of steering to both +1 and \(-1\) orders, and tend to exhibit sharp decreases in efficiency at certain AOI.