## 2 Micro-Imaging System
### Analysis of the Micro-Imaging System
Whereas, during calibration, because the adopted nanoparticle probe is fixed between the cover glass and the glass slide with fixative, \(\rm d_{0}\) is constant, and \(\rm d_{var}\) varies with \(\rm d_{1}\). Therefore, when analyzing the two aspects of this micro-imaging system mentioned above, it is also necessary to verify whether the same \(\rm d_{var}\) value has an identical influence on each of the two aspects. In addition, there are noises (e.g., Poisson noise and background noise) that affect the two aspects in an actual micro-imaging process, and so de-noising processing is needed, where there are many effective de-noising methods used to reduce the noises in actual measurements. Here, the effect of noises on micro-imaging is outside the scope of this study; analysis of the micro-imaging below is stated under the condition of non-noise, which does not contain the analysis of errors caused by noises. The entrance pupil of one point on the fluorescent nanoparticle is determined by four marginal rays that are tangent to the spherical lens, as shown by Figure 2a. In Figure 2b, light exits from the cylindrical surface of the cylindrical lens. The width of the valid imaging spot along the X-axis is equal to the distance between the incident points of the right and left marginal rays, and the width along the Y-axis is equal to the distance between the incident points of the upper and lower marginal rays. To ensure the established optical matrix of Equation (1) is modeled accurately, it requires surface size of the curved lens participating in micro-imaging to be significantly smaller than its radius of curvature, and this allows the lenses to be more accurately modeled in the paraxial approximation. In Figure 1,  
Figure 2: Valid pupil area determined using a cone of light propagating from one point on the surface of a nanoparticle: (**a**) valid entrance pupil, and (**b**) corresponding valid exit pupil. it is easy to make the cylindrical lens size far smaller than its radius of curvature, but the spherical lens size cannot meet the requirement. Obtaining a true nanoparticle image intensity distribution is not the desired outcome; rather, analyzing the two aspects stated above is our aim. The four marginal rays determining the angular aperture of the micro-imaging system can be used to analyze characters of the two aspects using the optical matrix of Equation (1); therefore, the rays determining the central part of the angular aperture also have the same function, namely the proportional variation of the angular aperture has no effect on the analysis of the two aspects. Reducing the angular aperture determined by the four marginal rays on each object plane with one same proportion is done to confirm the central part of the angular aperture, and to get four rays of the central angular aperture (namely, four central rays of the angular aperture) on each object plane. The scaled-down proportion can be chosen (here ten times is used), and the micro-imaging size of spherical lens determined by the four central rays is able to reach the requirement for running the optical matrix of Equation (1). The four central rays instead of original four marginal rays are used to implement the analysis with the optical matrix of Equation (1). The four marginal rays stated below are the four central rays. According to diffraction theory, in the paraxial approximation, the image intensity distribution of a point light source is identical if the size of the entrance pupil (or the size of the exit pupil) remains the same. Figure 3 shows that irrespective of the distance \(\Delta\) (along the X-axis or Y-axis) the point source (fluorescent nanoparticle) covers on an object plane, the size of the entrance pupil determined by the four central rays is constant, namely the nanoparticle image intensity distribution is constant for an object plane.