# Variable Curvature Displays: Optical Designs and Applications for VR/AR/MR Headsets
## 3 Design Algorithm  
The analytical procedure described above is used for an initial assignment of the image surface curvature in different configurations. This idea underlies the proposed design and optimisation  
\begin{table}
\begin{tabular}{c l c c c c} \hline \hline
**Distance, m** & **Criterion** & **R, m** & **Spot X, μm** & **Spot Y, μm** & **Spot RMS, μm** \\ \hline
1.5 & Seidel Sum Tang. & 128.55 & 12.7/46.2 & 12.7/27.9 & 9.0/27.5 \\  & Seidel Sum Sag. & 75.12 & 26.6/25.4 & 26.6/43.2 & 15.2/28.7 \\  & Marginal ray & 151.87 & 9.8/50.6 & 9.8/25.1 & 8.8/28.4 \\  & RMS & 117.44 & 14.6/43.4 & 14.6/29.9 & 9.4/27.2 \\  & None & inf. & 21.1/74.6 & 21.1/23.0 & 15.9/38.0 \\ \hline
5 & Seidel Sum Tang. & 129.50 & 12.1/46.4 & 12.1/ 26.6 & 9.0/27.2 \\  & Seidel Sum Sag. & 74.23 & 26.3/25.1 & 26.3/41.9 & 15.1/27.9 \\  & Marginal ray & 140.67 & 10.6/48.7 & 10.6/25. & 8.9/27.7 \\  & RMS & 113.59 & 14.8/42.5 & 14.8/29.2 & 9.5/26.7 \\  & None & inf. & 21.5/74.3 & 21.5/24.1 & 16.0/38.0 \\ \hline inf. & Seidel Sum Tang. & 129.89 & 11.9/46.5 & 11.9/26.1 & 9.0/27.2 \\  & Seidel Sum Sag. & _73.87_ & 26.2/25.0 & 26.2/41.4 & 15.0/28.0 \\  & Marginal ray & 136.53 & 10.9/47.8 & 10.9/25.1 & 9.0/27.4 \\  & RMS & 112.08 & 14.9/24.1 & 14.9/29.0 & 9.6/26.5 \\  & None & inf. & 21.6/74.1 & 21.6/24.4 & 16.2/38.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Field curvature computations for demonstrative example (FoV center/corner).  
algorithm. The design algorithm is presented as a simplified block diagram in Figure 2. If the application implies zooming or refocusing of the optical system, after choice of the initial optical design (1), the design is split into a number of zoom configurations (2), having, e.g., different object positions. At the next step, the initial focal surface curvature for each configuration is computed through Equations (1)-(7) (3). Afterwards, the optical design is optimised for each configuration. It is practically useful to perform two optimisation loops with different quality criteria, e.g., geometrical aberrations (4a) and ensquared energy (4b). In both cases, the variables include all the radii, thicknesses and the aspheric coefficients of the optics. The boundary conditions include maintaining the image and object size as well as limitations on the center and edge thicknesses. At the first stage (4a), the merit function is based on the root mean square radii in the spot diagram used with equal weight coefficients. At the second stage (4b), we use the fraction of energy ensquared in one pixel and the weight coefficient for the edge of the FoV is doubled. The obtained optical system is subject to a detailed performance analysis (5), which includes definition of the energy concentration and some secondary indicators such as the distortion and relative illumination of the image. The main path assumes a varying image surface curvature separately for each zoom position and is marked as \"**CV**\" for \"curved, variable\". For comparison, we also consider the cases of static curvature (\"**CS**\" for \"curved, static\"), when the same value is assigned to all configurations, and flat surface (\"**FS**\" for \"flat, static\").  
In the following section, we apply this algorithm to design a refocusable lens suitable for VR applications.