# Variable Curvature Displays: Optical Designs and Applications for VR/AR/MR Headsets
## 2 Theoretical Background  
To obtain a good starting approximation for the optical design and to demonstrate the field curvature dependence on the system configuration, we consider Seidel aberration theory [12]. The general layout of a projection lens is shown in Figure 1. Note that hereafter we assume that the raytracing procedure is fully reversible. The figure shows tracing of two paraxial rays used for definition of the Seidel sums and the corresponding notation.  
The image surface shape is determined by two Seidel sums corresponding to astigmatism and field curvature:  
\[S_{III}=\sum_{i=1}^{q}h_{i}P_{i}\big{(}\frac{\beta_{i+1}-\beta_{i}}{\alpha_{i+ 1}-\alpha_{i}}\big{)}^{2}, \tag{1}\]  
\[S_{IV}=-\sum_{i=1}^{q}\frac{1}{h_{i}}\frac{\alpha_{i+1}n_{i+1}-\alpha_{i}n_{i} }{n_{i}\alpha_{i}}, \tag{2}\]  
where  
\[P_{i}=\bigg{(}\frac{\alpha_{i+1}-\alpha_{i}}{1/n_{i+1}-1/n_{i}}\bigg{)}^{2} \bigg{(}\frac{\alpha_{i+1}}{n_{i+1}}-\frac{\alpha_{i}}{n_{i}}\bigg{)}, \tag{3}\]  
where \(n_{i}\) is the index of refraction after the \(i\)th surface.  
In addition, in the aberration equations, the Lagrange invariant is used in the following form:  
\[I=-n_{1}\alpha_{1}(S-S_{P})\beta_{1}, \tag{4}\]  
Figure 1: General raytracing scheme used to derive the field curvature.  
Then, the longitudinal aberration can be derived for the rays propagating in tangential and sagittal planes:  
\[\Delta z_{t}=\frac{y^{2}B}{2n_{q}^{\prime}(S-S_{P})\alpha_{1}a_{q}^{\prime}\beta_ {1}^{2}}\big{(}3S_{III}+I^{2}S_{IV}\big{)}, \tag{5}\]  
\[\Delta z_{s}=\frac{y^{2}B}{2n_{q}^{\prime}(S-S_{P})\alpha_{1}a_{q}^{\prime} \beta_{1}^{2}}\big{(}S_{III}+I^{2}S_{IV}\big{)}, \tag{6}\]  
where \(B\) is the linear magnification.  
The corresponding radii of curvature are:  
\[R_{t(s)}=-\frac{\Delta z_{t(s)}^{2}+y^{\prime 2}}{2\Delta z_{t(s)}}. \tag{7}\]  
One can see that the image surface curvature changes with the change of both distance to the object \(S\) and magnification \(B\). To demonstrate this effect and to estimate the accuracy of this theoretical approximation, we perform a series of computations for a simple singlet lens [13]. The lens is scaled to have a focal length \(f^{\prime}=36\) mm and \(F/\#=12.9\). The linear FoV is \(\pm 12.7\) mm. Since Seidel theory works for monochromatic aberrations, we consider a single wavelength \(\lambda=780\) nm, which is used in the original patent. Three positions of the objects were considered: infinity, 5 m and 1.5 m. For each computation, the image surface curvature was calculated with Equation (7). For comparison, it was also computed from the marginal ray intersection coordinate and by numerical optimisation of the root mean square (RMS) radius in the spot diagram. Finally, a calculation with a flat image surface in the nominal focus position was used to compare with. For each calculation, the lens ray aberrations were computed. The results are summarised in Table 1.  
This simple example clearly shows three important points. Firstly, the image quality can be significantly improved by using a curved image surface. Secondly, the curvature changes notably with a change in object position. Thirdly, the value computed with the Seidel sum for rays in the tangential plane provides a good approximation for the curvature and can be computed analytically for any optical system.