## 1 Introduction  
Single fluorescent nanoparticle tracking is a method to measure the motion trajectory of nanoparticulate fluorescent probes in real time, which has been widely used in the fields of biology, biomechanics, and rheology [1; 2; 3; 4; 5]. Particularly, the single particle tracking of upconverting nanoparticles currently attracts a large amount of interest for bio-imaging and medical applications [6; 7; 8; 9]. Originally, this method could only measure the two-dimensional trajectory of fluorescent nanoparticles, namely fluorescent nanoparticle motions along the X and Y coordinate axes were obtained, and the motion along the Z coordinate axis was lacking. Therefore, a complete analysis of particle probe trajectories in three-dimensions was not possible [10; 11].  
Recently, several three-dimensional (3D) tracking methods have been developed, including a differential interference microscopic imaging method with a trapezoidal prism [12], a tracking method based on focusing beam intensity measurements [13], multi-plane imaging technology [14], stimulated emission depletion-4\(\pi\) microscopy [15], and a 3D structured lighting imaging method [16]. However, these methods require complex optical systems or can only image one nanoparticle. In addition, there are various types of 3D microscopy [17; 18; 19; 20; 21; 22], where scanning microscopy, epifluorescence microscopy, and total internal reflection microscopy are conventional and popular. These 3D microscopies can be categorized into two kinds of techniques: \"z-scanning\" and \"wide-field\" schemes. There are advantages and disadvantages to each of the microscopies, which can be found in the literature. [PERSON] et al. [9] has already summarizedthe pros and cons of scanning microscopy (confocal microscopy), epifluorescence microscopy, and total internal reflection microscopy.  
An earlier method that added a cylindrical lens to the optical path of the detector of a standard microscope [23; 24; 25] was adopted. By observing nanoparticle mobilities along the X and Y coordinate axes, the nanoparticle motions along the Z coordinate axis are also obtained in real time through the astigmatism of the cylindrical lens. This method of adding a cylindrical lens to a microscope is simple, easily implemented, and can simultaneously image the 3D trajectories of many nanoparticles. When using this method, different sizes and types of fluorescent nanoparticle probes can be selected according to the tested specimen. Therefore, the appropriate magnification should be adopted to obtain a sufficient number of nanoparticle probes that are in motion and can provide accurate imaging at each tracking sampling location in the tested specimen. High magnification not only causes the noise to increase, but also decreases the number of nanoparticle probes collected, leading to an inaccurate mean square displacement (MSD) of the obtained trajectory. Meanwhile, low magnification causes a low spatial-sampling frequency, leading to inaccurate imaging, which directly affects the sampled displacement accuracy of each nanoparticle probe. Therefore, this astigmatic microscopic imaging technique should be able to realize variable magnification microscopic measurements. To achieve accurate fluorescent nanoparticle tracking of this variable astigmatic microscopic imaging method, it is not only necessary to obtain accurate values of different magnifications, but also to ensure the coaxial alignment of the micro-imaging elements after varying the magnification.  
In this paper, a minimal optical imaging configuration for fluorescence particle tracking is proposed. The micro-imaging system of the proposed method comprises only a spherical lens, a cylindrical lens, and a charge coupled device (CCD). By integrating this minimal structure with a dual-spot position detection unit, a variable magnification 3D microscopic tracking system was achieved, which had the self-calibrating functions of coaxiality and magnification.