# Variable Curvature Displays: Optical Designs and Applications for VR/AR/MR Headsets
## 5 Variable Curvature Display Feasibility  
The image surface curvature in the designed lens varies from \(-110\) to \(-138.7\) mm when refocusing from infinity to \(1.5\) m. The obtained ratio of the display size and curvature is close to the current limit of the curving technology [18], defined by the die breakage limit. Thus, additional modelling was performed in order to demonstrate the design feasibility.  
We estimated the feasibility of variable curvature displays parameters relying on the theoretical basement described in [19] and using finite element analysis (FEA) according to the method presented  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Distance, m** & **Distortion, \%** & **Min. rel.illum.** \\ \hline \multicolumn{3}{c}{**CV**} \\ \hline
1.5 & 13.2 & 62.0 \\
5 & 13.7 & 60.6 \\ inf. & 13.9 & 63.6 \\ \hline \multicolumn{3}{c}{**CS**} \\ \hline
1.5 & 12.1 & 79.0 \\
5 & 12.1 & 78.3 \\ inf. & 12.1 & 83.4 \\ \hline \multicolumn{3}{c}{**FS**} \\ \hline
1.5 & 9.2 & 57.8 \\
5 & 9.3 & 59.0 \\ inf. & 9.3 & 69.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Auxiliary imaging performance indicators.  
Figure 4: Energy ensquared in one (11\(\mu\)m \(\times\) 11 \(\mu\)m) and four (22\(\mu\)m \(\times\) 22 \(\mu\)m) pixels for different designs and different object positions.  
in [20; 21]. From this study, to reach the required radii of curvature prescribed from the optical design above, we calculated that the root mean square surface error we can reach is lower than 20 \(\upmu\)m. The difference between the desired and actual surface shape along the diagonal is shown in Figure 5. It is compared to the depth of focus, which equals \(\pm\)68.2 \(\upmu\)m for the lens with \(F/\#=6.2\) and \(11\times 11\)\(\upmu\)m pixel. Thus, the calculated display surface shape can be reproduced with a satisfactory precision, although it may be improved by means of a proper mechanical design.  
The corresponding amount of von Mises Stress generated in the display is lower than 130 MPa, while the breakage limit of silicon is at least 560 MPa [22]. Thus, the safety factor is about 4.3\({}^{\times}\).  
In addition, we performed a simple test to estimate the required precision and stability of the curving force. We computed the tolerances on the design parameters, including the display shape. Assuming that a 10% increase of the aberrations is acceptable, we found the tolerance on the display radius equal to \(\pm\)1.2 mm. It corresponds to the precision of the curving force within \(\pm\)1.1%. Thus, the display actuator should have a high accuracy and the display should be well isolated from all other external loads. The latter point should be accounted for in the optomechanical design, which is out of the subject of this study.  
This indicates that the proposed design with variable focal plane curvature can be implemented with the known curving technique. However, we must note that this modelling is preliminary and it is intended only for estimation of the safety factor and the precision requirements. Development of an actual curving setup would require some advanced simulations and additional studies, including nonlinear effects and consideration of technological limitations.