**3.1. Two-beam interferometry**

It has been noted that measuring the feature height of asphere surfaces and strongly curved surfaces using two-beam interferometers is very complicated due to the higher fringe density [7], so it has been shown that it is possible to measure the asphere form using multiple beam fringes. This is because two-beam interferometers suffer from the fact that they produce cos2 intensity distributions. **Figure 9(a)** and **(b)** compare precontour fringes of asphere surface obtained by two-beam Fizeau interferometry and multiple-beam Fizeau interferometry, respectively. As seen from **Figure 9(b)**, the multiple-beam Fizeau interferometry has the ability to resolve very small irregularities compared to common two-beam Fizeau interferometry. In this chapter, we review some of two- and multiple-beam interferometers for surface microtopography measurement.

**Figure 9.** Precontour fringes of asphere surface obtained by (a) two-beam Fizeau interferometry and (b) multiple-beam Fizeau interferometry.

### *3.1.1. Twyman-Green interferometer*

The light source used in a Twyman-Green interferometer is a quasi-monochromatic point source that is collimated by a collimating lens. This collimated light is incident on a beamsplitter which divides the beam into two copies: a reference beam and a test beam. The interferometer is used here for testing a spherical optical flat. The reference beam is incident on the known reference optical flat and returns to the beamsplitter. The test beam is incident on the unknown test part and also returns to the beamsplitter. The beams from the reference and the object interfere at the beamsplitter and constitute an interferogram relayed by an imaging lens to the observation plane. **Figure 10** shows a Twyman-Green interferometer for testing a curved surface in reflection [8, 9]. The reference is an optical flat of 1 inch in size and flatness of *λ*/20 nm. The object being tested is a curved surface of radius of curvature around. The sample is mounted carefully and four inline interferograms (no tilt between the reference and the object) are captured with phase shift between images of π/2 as shown in **Figure 11**. The intensities in the four fringe patterns can be expressed as follows:

**Figure 10.** Optical schematic of the Twyman-Green interferometer.

**3.1. Two-beam interferometry**

90 Optical Interferometry

topography measurement.

Fizeau interferometry.

*3.1.1. Twyman-Green interferometer*

the four fringe patterns can be expressed as follows:

It has been noted that measuring the feature height of asphere surfaces and strongly curved surfaces using two-beam interferometers is very complicated due to the higher fringe density [7], so it has been shown that it is possible to measure the asphere form using multiple beam fringes. This is because two-beam interferometers suffer from the fact that they produce cos2 intensity distributions. **Figure 9(a)** and **(b)** compare precontour fringes of asphere surface obtained by two-beam Fizeau interferometry and multiple-beam Fizeau interferometry, respectively. As seen from **Figure 9(b)**, the multiple-beam Fizeau interferometry has the ability to resolve very small irregularities compared to common two-beam Fizeau interferometry. In this chapter, we review some of two- and multiple-beam interferometers for surface micro-

**Figure 9.** Precontour fringes of asphere surface obtained by (a) two-beam Fizeau interferometry and (b) multiple-beam

The light source used in a Twyman-Green interferometer is a quasi-monochromatic point source that is collimated by a collimating lens. This collimated light is incident on a beamsplitter which divides the beam into two copies: a reference beam and a test beam. The interferometer is used here for testing a spherical optical flat. The reference beam is incident on the known reference optical flat and returns to the beamsplitter. The test beam is incident on the unknown test part and also returns to the beamsplitter. The beams from the reference and the object interfere at the beamsplitter and constitute an interferogram relayed by an imaging lens to the observation plane. **Figure 10** shows a Twyman-Green interferometer for testing a curved surface in reflection [8, 9]. The reference is an optical flat of 1 inch in size and flatness of *λ*/20 nm. The object being tested is a curved surface of radius of curvature around. The sample is mounted carefully and four inline interferograms (no tilt between the reference and the object) are captured with phase shift between images of π/2 as shown in **Figure 11**. The intensities in

**Figure 11.** Intensity images of a curved object with a phase shift of 0*π*, 0.5*π*, 1*π*, and 1.5*π*, for (a–d), respectively.

$$I\_{\cdot}(\mathbf{x}, \mathbf{y}) = I\_{\mathbf{o}} + I\_{\mathbf{n}} + 2\sqrt{I\_{\mathbf{o}}I\_{\mathbf{n}}} \cos(\boldsymbol{\varphi} + (\boldsymbol{j} - \mathbf{l})\boldsymbol{\pi} / 2), \tag{13}$$

where *I*O and *I*<sup>R</sup> are the intensities of the object and the reference waves, respectively, is the phase encoded in the intensity distribution, and *j* = 1, 2, 3 and 4 (four frames) is the number of the phase-shifted frames.

Using the four-phase step algorithm, the phase distribution can be expressed as follows:

$$\varphi = \tan^{-1} \left( \frac{(I\_4 - I\_2)}{(I\_1 - I\_3)} \right). \tag{14}$$

The evaluated phase is wrapped between −*π* and *π* due to arctangent function. The wrapped phase map resulted from the four frames in **Figure 11** is shown in **Figure 12(a)**.

**Figure 12.** (a) Wrapped phase map resulted from the four frames of **Figure 10**; (b) 3-D unwrapped phase map of (a); and (c) two-dimensional height at the middle of (b) in the *x*-direction.

The wrapped phase map is then unwrapped to remove the 2π ambiguity and the unwrapped phase map is shown in **Figure 12(b)** and profile along **Figure 12(b)** is shown in **Figure 12(c)**.
