# Detecting the Extremely Small Angle of an Axicon by Phase-Shifting Digital Holography
## 3 Experimental Result  
The experimental setup is shown in Figure 1. A laser beam is split into two beams after transmitting through the beam expander consisting of two lenses. One of the beams is reflected onto the SLM by BS2. Given that SLM can only modulate light with polarization along the horizontal axis, a polarizer is inserted in front of the SLM. Another beam directly passes through the axicon, which is placed close to BS3 to reduce the distance that the object light travels before it interferes with the reference beam. Then, two beams are combined and interfered by BS3, whose results are recorded by a CCD. In the experiment, the phase shift is introduced by the SLM. According to the interference holograms with different phase shifts, the base angle of the axicon can be measured. Based on the experimental setup, we can record the interference patterns, as shown in Figure 2. The upper four figures are recorded with the presence of the axicon. Figure 2a-d shows the interference results with the phase shifts of 0, \(\pi\)/2, \(\pi\), and 3\(\pi\)/2, which correspond to \(I_{11}\), \(I_{12}\), \(I_{13}\), and \(I_{14}\) in Equation (3). Considering the line-focusing feature of the axicon, a focal spot surrounded by rings is generated. Removing the axicon from the experimental setup, the interference patterns of the reference light and the object light without axicon can be recorded, as presented in the lower part of Figure 2. These two beams can be approximately regarded as plane waves, thereby resulting in a series of oblique parallel lines. When introduced with different phase shifts, the fringes translate accordingly. With a phase shift of \(\pi\), the fringes move roughly half the fringe's width, which is consistent with the theoretical results. Using three phase extracting methods, we measure the angle of an axicon experimentally. The phase distribution solved by four interference holograms is a pattern with poor symmetry (Figure 3a) because the method assumes the background phase as 0, indicating the background phase is assumed to be constant. However, in the experiment, the background phase may change, which results in errors in the solved phase. A phase distribution with significant noise is solved using five interference holograms  
Figure 1: Experiment setup of phase-shifting digital holography for measuring the base angle of axicon. SLM: Spatial light modulator; CCD: Charge coupled device. Figure 2: (**a–d**) Interference patterns of the light passing through the axicon interfering with the reference light with different phase shift. (**e–h**) Interference fringes of the plane wave interfering with the reference light with different phase shift. (**a,e**) Phase shifts of 0; (**b,f**) phase shifts of \(\pi\)/2; (**c,g**) phase shifts of \(\pi\); (**d,h**) phase shifts of \(3\pi\)/2. (Figure 3b). To solve the phase, we introduce a filter to eliminate the high-frequency noise in this method. Meanwhile, the introduction of the filter will result in an error at the position where the phase changes significantly. A better result is obtained by using eight interference holograms, as shown in Figure 3c. The phase distribution has a good circular symmetry, which agrees with the feature of the axicon. Using eight interference holograms, we measure the phase distribution of three axicons, that is, one with a base angle of 0.5\({}^{\circ}\) and the other two are 1\({}^{\circ}\). Then we compare the experimental results with the theoretical data. Axicon has a tapered structure, whose thickness varies with radial distance. Therefore, the beam phase also presents periodic circular symmetric distribution as radial distance changes. Figure 4a,c show the theoretical simulations of axicons with the base angles of 0.5\({}^{\circ}\) and 1\({}^{\circ}\). Phase changes periodically with increasing radial distance. In the figure, the selected radial range is 0.63 mm.