# Focus-Adjustable Head Mounted Display with Off-Axis System
## 2 Materials and Methods  
In paraxial ray tracing, the ray has a height on each surface; therefore, the sag value on each surface can be disregarded [30; 31; 32]. Thus, each surface can be approximated as a flat surface. When this assumption is made, the optical system becomes ideal, without aberration. In an ideal optical system, the ray coincides with the chief ray in the image plane, regardless of its initial height [33; 34; 35]. Therefore, paraxial ray tracing can be used as a design criterion for optical systems. The system matrix can be used in a general rotationally symmetric optical system, as illustrated in Figure 3. The ray slope, \(u\), and the ray height, \(h\), of the ray can be calculated from Equation (1) [36; 37],  
\[\left[ \begin{array}{c}h_{i}\\ n_{i}u_{i}\end{array} \right]=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]\left[ \begin{array}{c}h_{0}\\ n_{0}u_{0}\end{array} \right] \tag{1}\]  
Figure 2: Commercialized head-mounted display (HMD) products. (**a**) See-close type HMD for virtual reality. (**b**) See-through type HMD for augmented reality [18]. where \(n\) is the refractive index and the subscript \(i\) of each variable indicates the surface number. The 1 st orders are calculated using Equations (1)-(5) [25; 38]. \[EFL=-\frac{h_{0}}{u_{i}}=-\frac{n_{i}}{C},\hskip 5.690551 ptBFL=-\frac{h_{i}}{u_{i} }=-\frac{A\cdot n_{i}}{C} \tag{2}\]  
\[H_{2}=EFL-BFL=\frac{n_{i}(1-A)}{C}\hskip 5.690551 pt(\text{from $H_{2}$ to last surface}) \tag{3}\]  
\[EFL=-\frac{h_{i}}{u_{0}}=-\frac{n_{0}}{C},\hskip 5.690551 ptFFL=-\frac{h_{0}}{u_ {0}}=-\frac{n_{0}\cdot D}{C} \tag{4}\]  
\[H_{1}=FFL-EFL=\frac{n_{0}(D-1)}{C}\hskip 5.690551 pt(\text{from first surface to $H_{1}$}) \tag{5}\]  
Equations (2) and (3) yield the effective focal length (EFL), back focal length (BFL), and the second principal surface (\(H_{2}\)) when a collimated beam (solid red line in Figure 3) is sent from left to right in the optical system. Equations (4) and (5) generate the EFL, front focal length (FFL), and first principal surface (\(H_{1}\)) when the collimated beam (red dotted line in Figure 3) is sent from right to left in the optical system. It is observed that the EFL retains the same value regardless of the direction of the collimated beam [36]. The conventional paraxial ray tracing method cannot be applied in off-axis optical systems [29]. Therefore, a rotational transformation is performed such that the axial ray is in a straight line, based on the intersection point between each surface and axial ray, obtaining the intersection point and the refraction or reflection angle on each surface. Figure 4 illustrates the process of straightening the optical axis by rotating it based on the intersection of the optical axis and each surface. Figure 4a depicts the original optical system. Figure 4b demonstrates that the entire optical system is rotated so that the axial ray on the optical axis refracted from the first surface becomes a straight line. Figure 4c indicates the result of similarly rotating the axial ray on the optical axis to a straight line from the second surface to the last surface. In addition, the axial ray and the chief ray are rotationally converted based on the rotation center point and the rotation angle of the axial ray. In Figure 4, the black line represents the optical surface before rotation, and the blue line represents the optical surface after rotation conversion. The 1 st order of the off-axis optical system can be calculated using Equations (2) and (5), obtained from A, B, C, and D for each surface [13]. Here, a smaller axial ray height or chief ray incident angle results in a smaller sag of the optical surface. Therefore, the accuracy of the result was determined by confirming whether the EFL calculated by this method converged. Figure 3: The optical layout for an on-axis optical system. Adjusting the focus moves the image surface as the object distance changes [39, 40]. Figure 5 displays the change in the position of the image surface according to the change in the object distance.