## 3 Results
### Error Analysis  
It is necessary to verify the validity of the calculated EFLs in Table S4, as well as the level of measurement precision. Manufacturing and assembly errors are inevitable when parts are mass-produced. Thus, errors within the limit of the performance boundary of the product are allowed, in general, and such allowed errors are referred to as the tolerance [14; 15]. The validity of the measurement method used in this study was verified by checking whether the measured EFL fell within the tolerance range [16]. The EFL of a system can be expressed as a function of the curvature (\(R\)), thickness (\(d\)), and refractive index (\(n\)) of the lens material as follows:  
\[EFL=f(R_{1},d_{1},n_{1},R_{2},d_{2},n_{2},R_{3},\cdots,R_{i},d_{i},n_{i}) \tag{5}\]  
where the subscript \(I\) stands for the surface number. Re-expressing Equation (5) using a Taylor series to identify the infinitesimal changes in the function for each tolerance item gives:  
\[EFL=EFL_{0}+\frac{\partial f}{\partial R}\Delta R_{1}+\frac{\partial f}{ \partial d}\Delta d_{1}+\frac{\partial f}{\partial n}\Delta n_{1}+\cdots \tag{6}\]  
where \(EFL_{0}\) is the initial value of the focal length and \(\Delta R\),\(\Delta d\), and \(\Delta n\) refer to errors in the radius of the curvature, thickness, and refractive index, respectively. In this case, only the first-order term needs to be considered because the higher-order terms give little contribution [17]. The coefficients of the errors in the radius of the curvature, thickness, and refractive index in Equation (5) are calculated by differentiating Equation (5). Equation (5) is not analytic, however, and the coefficients of Equation (6) must be calculated by entering the EFL change for the given tolerance into the optical design software. Table S5 lists the EFL sensitivities calculated for a typical tolerance error of an optical system. If the standard deviations of the curvature, thickness, and refractive index follow a normal distribution of the tolerance error, the standard deviation for the overall changes in the EFL (\(f\)) for the system may be calculated by summing the squares of each design variable's standard deviation. This is because the change in the EFL of the system is expressible as a linear sum of the EFL changes in relation to the errors in each design variable, as shown in Equation (6). Moreover, the standard deviations would be larger if the statistical distribution of each design variable did not follow a normal distribution, but instead followed a triangular or a uniform distribution. Even if the errors are not normally distributed, as long as there are plentiful sample data within the distribution, the standard deviation for the EFL changes may be interpreted equally as the normal distribution based on the central limit theorem [18]. The standard deviation of the triangular distribution with poor error management and that of the uniform distribution, at worst, can be approximately calculated as \(\sqrt{2}\)-\(\sqrt{3}\) greater than the standard deviation of the normal distribution [19]. Table S6 lists the calculated standard deviations of the statistical distribution of the design variables when they follow triangular, uniform, and normal distributions. Under the assumption that each design variable is normally distributed, the standard deviation for the distribution of EFL changes in the optical system calculated from Table S5 would be \(\sim\)4.84 \(\upmu\)m. Thus, the difference between the measured and theoretical EFL is a factor of \(\sim\)2.2. Moreover, there is a 99.7% probability that the EFL would be distributed within a factor of 3 of the standard deviation because the EFL changes are normally distributed. The statistical distribution of the Newton ring, which indicates the error of curvature at the lens manufacturing stage, and the thickness are prone to be shifted towards a greater mean value [20; 21]. The EFL distribution of the manufactured optical system would be shifted slightly more than the design value.