## 1 Introduction  
Intracellular imaging is one of the most important techniques in biomedical optics, and offers immediate visual information on the specific positions and movements of subcellular organelles [1, 2]. In a practical imaging microscopy system for a living cell, a fully stable, single focal plane or simultaneous imaging at multiple focal planes are frequently required, depending on the movement range of the intracellular organelle of interest. For example, an imaging of the comparably slow motion of a cell requires only a single image plane with fine stability [3]. However, to track a target organelle traveling a long axial range, such as in the case of endocytosis where a vesicle navigates from the cell membrane to the nucleus periphery, multiple images taken at different focal planes are demanded for acquiring the accurate geometric information, in order not to miss or misconceive the actual dynamics of the target organelle. This is because an organelle in the cytoplasmic area can be transported by cytoskeletal networks in three dimensions [4, 5], which refers that the conventional imaging system with a single focal plane cannot chase the fast-changing axial position of the organelle of interest, once it leaves the focal plane. Figure 1 illustrates the multifocal imaging concept and the three-dimensional movement of an intracellular organelle. Therefore, three-dimensional localization imaging techniques have been developed with a wide range of techniques to acquire biological information, as [PERSON] et al. summarized recently [6]. For example, engineered point-spread function (PSF) methods and multifocal plane microscopy are the representative methods for localization of intracellular particle in three dimensions. These advances in imaging microscopy have led to the faster and more accurate detection and tracking. Particularly, multifocal plane imaging acquires the position information of the target from multiple image planes by splitting the optical paths, which can extend the range of detection as installing additional lenses and cameras [7; 8]. Although the developed techniques have enabled us to localize the target of interest in the imaging microscopy with high precision and speed, those methods might be often accompanied by burdens such as complex computations and preparation of additional instruments which can hinder the direct adaptation to the practical system. Bifocal plane imaging microscopy, also known as dual-focus optics, is the simplest type of multifocal plane imaging method that minimizes the above concerns [9; 10; 11]. In dual-focus optics, two focal planes at different axial positions can be imaged on a single camera image sensor, and the intensity information detected from each image can be exploited to extract the three dimensional coordinates of the target organelle [11; 12; 13]. The major disadvantage of the dual-focus optics is that the axial detectable range is limited by the number of the image planes and the fixed distance between the two focal planes. In addition, because of the nonlinear profile of the calibration curve fit utilized for axial position detection, the localization accuracy is non-uniform for the entire convertible region [14]. Here, we propose an extended dual-focus imaging microscopy system by combining the conventional bifocal plane imaging method and an objective lens oscillation system, to achieve longer trackable range with high detection accuracy, offering a detailed method for detection resolution calculation and the determination of convertible range in calibration: First, the axial localization reliability of the dual-focus optics with ratiometric method is examined with simulation. In the simulation, the distance between the two focal planes and the differences in the axial positions of the consecutive objective lenses are determined.