**4.1. Single-shot parallel four-step phase-shifting Fizeau interferometer**

electronics and the fraction is estimated by electronically sub-dividing the fringe [14, 15]. **Figure 22(a)** shows a configuration of homodyne interferometer. The homodyne interferometer uses a single frequency, 1, laser beam. The beam from the reference is returned to the non-polarized beamsplitter (NPBS) with a frequency 1, but the beam from the moving measurement path is returned with a Doppler-shifted frequency of 1 <sup>±</sup> . These beams interfere in the NPBS and enter the photodetector. **Figure 22(b)** shows a heterodyne interferometer configuration. The output beam from a dual-frequency laser source contains two orthogonal polarizations, one with a frequency of 1 and the other with a frequency of 2 (separated by about 3 MHz using the Zeeman effect). A polarizing beamsplitter (PBS) reflects the light with frequency 1 into the reference path. Light with frequency 2 passes through the beamsplitter into the measurement path where it strikes the moving retro-reflector causing the frequency of the reflected beam to be Doppler shifted by ±. This reflected beam is then combined with the reference light in the PBS and returned to a photodetector with a beat frequency of 2 − 1 <sup>±</sup> . This signal is mixed with the reference signal that continuously monitors the frequency difference, 2 − 1. With a typical reference beat of around 3 MHz, it is possible to monitor values up to 3 MHz before introducing ambiguities due to the beat crossing through zero. The displacement being measured for both homodyne and heterodyne is calculated from this equation = /2, where is a fringe count and *λ* is the wavelength of the incident radiation. Homodyne interferometers have an advantage over heterodyne interferometers because the reference and measurement beams are split at the interferometer

**Figure 22.** Homodyne interferometer configuration (a), and heterodyne interferometer configuration (b).

Phase shifting is an attractive and very robust technique for the analysis of fringe patterns. Since PS takes multiple images over a finite time period, it is sensitive to the time-dependent phase shifts due to vibrations. These vibrations are difficult to correct since the optimum algorithm depends on the frequency and the phase of the vibration. For a given vibration

**4. Fast phase-shifting interferometry**

and not inside the laser.

98 Optical Interferometry

In this section, the common path Fizeau interferometer is combined with a parallel four-step phase-shifting mechanism, thus real-time measurement is achieved [16]. By simultaneously capturing all four interferograms, this system is insensitive to vibration. The schematic diagram of the common path Fizeau interferometer combined with parallel four-step phaseshifting is shown in **Figure 23**. A helium-neon laser beam with vertical polarization passes through a collimating lens was expanded by the beam expander (BE). The collimated beam of the laser light falls upon the beamsplitter and are split into two copies. The transmitted copy from the beamsplitter is incident on the interferometer (the reference and the object) and then reflected from the interferometer with reference wave and object wave carrying the information of the tested curved surface. Note that the sample being tested was mounted as an object, and the quarter-wave plate of *λ*/10 flatness was mounted as a reference in the interferometer. The reflected reference and objects waves are introduced into another quarter-wave plate, whose fast axis is inclined at an angle of 45° relative to the polarization direction of the original reference wave. Thus, the object and reference waves are converted into the perpendicularly circularly polarized lights.

**Figure 23.** Configuration for single-shot parallel phase-shifting Fizeau interferometry, *BE*, beam expander; *L*1-*L*2, achromatic lenses with focal lengths *f*1 = 300 mm, *f*2 = 150 mm.

Two Ronchi phase gratings, *G*1 and *G*2, are located on the paths of the object and the reference waves. The axial distance between the two gratings is *d*, and their grating vectors are perpendicular 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 four frames of (a); (c) 3-D unwrapped phase map of (b).

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 method is efficient, more robust, and highly accurate.

**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 along the black line of (b).
