# Detecting the Extremely Small Angle of an Axicon by Phase-Shifting Digital Holography
## 3 Experimental Result
Within this range, the phase of the axicon with a base angle of 1\({}^{\circ}\) changes by approximately eight cycles, while that of the axicon with the base angle of 0.5\({}^{\circ}\) varies by approximately four cycles, which is half of the former. Figure 4b,d,e presents the corresponding experimental results, thereby indicating that the simulation and experimental results are consistent. Figure 4: Phase of axicons with different base angle. (**a**) Theoretical simulation of axicon with base angle of 0.5\({}^{\circ}\). (**b**) Experimental measurement of axicon with base angle of 0.5\({}^{\circ}\). (**c**) Theoretical simulation of axicon with base angle of 1\({}^{\circ}\). (**d**) Experimental measurement of one axicon with base angle of 1\({}^{\circ}\). (**e**) Experimental measurement of another axicon with base angle of 1\({}^{\circ}\). Figure 3: solved phase with different phase-shifted holograms. (**a**) Four phase-shifted holograms; (**b**) five phase-shifted holograms; (**c**) eight phase-shifted holograms. The phase change curve along the radial distance (along the black line in Figure 4b,d,e) shows the details of the error between theoretical simulation and experimental measurements, as presented in Figure 5. The experimental results of the three axicons show consistent results with the theoretical simulation. After calculation in radial range from 0 mm to 0.63 mm, the axicon with a base angle of 0.5\({}^{\circ}\) changed 3.924 periods in simulation and 4 periods in the experiment. Therefore, the error is 1.94%. The first axicon with the base angle of 1\({}^{\circ}\) (Figure 4d) shifts 7.849 periods in the simulation and 8.197 periods in the experiment, with the error of 4.43%; the second 1\({}^{\circ}\) axicon (Figure 4e) shifts 7.977 periods in the experiment, and the error is 1.63%. The errors may be caused by the following factors. First, axicon is not well-manufactured. As shown in Figure 3b, the position of the axicon tip is not at the center of the phase diagram, and the phase of the central spot shows insignificant change. Second, the phase-shift introduced by SLM may deviate from the required value. We solve the phase of an axicon by phase-shifting digital holography, and obtain the base angle of the axicon by the phase map. If the digital holography is used instead of digital-phase shifting holography, only one hologram is recorded. In the hologram, the axicon phase is embedded with the phase of the reference beam. Some possible methods can be considered to solve the phase of the axicon. One is traditional digital hologram. For this method, the reconstruction of axicon's phase should introduce another beam to illuminate the hologram in simulation. To perfectly qualify for reconstruction, this beam should possess the same properties as the reference beam, which is difficult to realize. Also, the statistic property of the hologram should be satisfied. Two other methods can be considered to eliminate the influence of reference beam. One is using a plane wave as a reference, then the phase of the reference can be treated as a constant. However, an ideal plane wave is difficult to achieve in experiment. The other one is detecting the phase of reference, which requires additional experimental measurement. Moreover, all these methods will introduce additional error, which affects the detection of the angle, especially for axicons with extremely small angle (smaller than 1 degree). Figure 5: The phase profile along the radial direction. The base angle of the axicon is (**a**) 0.5 degree; (**b**) the first 1 degree; (**c**) the second 1 degree.