**5. Conclusion**

dicular with each other. After passing through the gratings *G*1 and *G*2, both the object and the reference waves are diffracted into different orders. A polarizer array is mounted in front of the CCD camera to perform the polarization phase shifting. Thus, we can achieve the singleshot recording of the interferogram containing the information of the four phase-shifted interferograms whose phases of the reference (a quarter wave plate) wave are constant and the phases of the object were shifted. **Figure 24(a)** shows the four phase-shifted interferograms with π/2 rad generated from the proposed setup. Using the four-phase step algorithm [17], the phase distribution is wrapped between −π and π due to arctangent function. The wrapped phase map is shown in **Figure 24(b)**. The wrapped phase map is then unwrapped [18] to

remove the 2π ambiguity and the unwrapped phase map is shown in **Figure 24(c)**.

**Figure 24.** Experimental results of the single-shot, four-step phase-shifting using on-axis Fizeau interferometer; (a) intensity images of a spherical object with phase shift of 0, π/2, π, and 3π/2; (b) wrapped phase map resulted from the

Another sample of step height has been tested using this technique. **Figure 25(a)** shows the four phase-shifted interferograms with π/2 rad generated from the proposed setup. The 3-D phase map is shown in **Figure 25(b)** and two-dimensional profile along the middle of **Figure 25(b)** is shown in **Figure 25(c)**. The step height has been measured again as shown in **Figure 23(c)** to confirm our method. As shown from **Figure 25(c)**, the proposed

**Figure 25.** Experimental results of the single-shot, four-step phase-shifting using on-axis Fizeau interferometer; (a) intensity images of a step height object with phase shift of 0, π/2, π, and 3π/2; (b) 3-D phase map of (a); (c) 2-D profile

four frames of (a); (c) 3-D unwrapped phase map of (b).

100 Optical Interferometry

along the black line of (b).

method is efficient, more robust, and highly accurate.

In conclusion, we have presented new frontiers in interferometry carried out by the author for surface characterization. In this chapter, the fundamentals of interferometry and its ability to investigate the shape of surfaces with focus on denoising and impact of noise on phase unwrapping are presented. Also, limitations of optical instruments and optical aberrations measurement are discussed. Finally, we have described a fast phase-shifting technique, namely single-shot parallel four-step phase-shifting Fizeau interferometer for surface characterization. Experimental results are presented to verify the principles.
