## 2 Methods
### Refinement on Calibration Curve  
Practically speaking, however, this is not the end of the story about trackable range. In fact, the calibration for dual-focus optics should be conducted in a smaller range than the trackable range shown in Figure 3C. This is due to the \(I_{det}\) values existing outside the local minimum and local maximum. Since the lookup table for the ratiometric method localizes the axial position of the target point source using \(I_{det}\) value by inverting a graph such as that shown in Figure 3B, identical \(I_{det}\) value that redirects to multiple axial positions of the point source implies the failure of the calibration.  
Figure 4 describes the actual experiment results obtained using the dual-focus optics, for an experiment conducted using 100 nm-diameter beads (Thermo Fisher Scientific, Waltham, MA, USA). As shown in Figure 4A, two different optical paths that were adjusted to have different focal lengths generated two images with different intensity profiles. The changes in \(I_{1}\) and \(I_{2}\) in response to the point source axial position could be obtained in a similar manner to the simulation results, as shown in Figure 4B,C. The calibration curve, which acts as the lookup table for the ratiometric method, could be prepared by inverting the \(I_{det}\) graph over axial position of the point source, as shown in Figure 4D. For example, if the graph shown in Figure 4C is expressed as \(z=f(I_{det})\) where \(z\) represents the axial coordinate and \(f\) indicates the function mapping \(I_{det}\) to \(z\), the calibration curve illustrated in Figure 4D can be described as \(I_{det}=f^{-1}(z)\).  
The reason why the entire region between the local minimum and maximum of \(I_{det}\) cannot be directly utilized as a calibration curve is apparent from Figure 4D. The inverse plot of the \(z-I_{det}\)  
Figure 4: Convertible range of calibration curve acquired by Michelson contrast in dual-focus optics. **(A)** Acquisition of Gaussian peak intensities, \(I_{1}\) and \(I_{2}\), from the images of point source on two different focal planes; **(B)** Changes in \(I_{1}\) and \(I_{2}\) during experiment; **(C)**\(I_{det}\) for each target axial position; **(D)** Calibration curve between \(I_{det}\) and axial coordinate \(z\) position as an inverse plot of **(C)**. The convertible range of the calibration curve is restricted by the area in which the \(I_{det}\) value corresponds to only a single \(z\)-position value in the inverse plot (indicated by the red box), which excludes the blue-colored areas where \(I_{det}\) can have multiple possible axial positions. The dashed line indicates a third-order polynomial fit. \(\Delta z\) can express the resolution in the calibration curve when the image resolution is given as \(\Delta C\) in Equation (1).  
curve does not monotonously change in all sections. When the \(I_{det}\) value is either too small or too large, a single value cannot determine a specific axial position. For example, a value of 0.4 for \(I_{det}\) simultaneously corresponds to two different \(z\) values, i.e., approximately 1000 nm and 1300 nm. Since \(I_{det}\) in the blue-colored areas fits to multiple axial positions, the practically convertible range in calibration curve should exclude such areas, to achieve proper functionality. Therefore, the calibration curve should be defined within the convertible range, where a single \(I_{det}\) value corresponds to one point-source axial position only, as indicated by the red box in Figure 4D.