**3.2. Multiple-beam interferometry**

Multiple beam fringes are extremely sharp. Simple measurements with such fringes can reveal surface micro-topography with a precession close to (*λ*/500). This advantage made multiplebeam interferometers so popular in revealing strongly curved surfaces and steep abrupt edges. Sharp fringes are obtained when the surfaces forming the cavity are coated with higher reflectivity film. The theory of the intensity distribution of Fabry-Perot fringes at reflection from an infinite number of beams collected was dealt with by [11]. Recently, Abdelsalam [12] modified an analytical equation correlate intensity distribution at reflection with a number of beams collected. It is found that 30 number of beams collected produce the same intensity profile as infinity number of beams are collected. In this section, we review multiple-beam Fizeau interferometry for thin film and curved strongly surfaces measurements.

### *3.2.1. Multiple-beam Fizeau interferometer for film thickness measurement*

The schematic diagram of the Fizeau interferometer for film thickness measurement is illustrated in **Figure 17(a)**. Details of the measurement technique are explained by the author in [13]. The fringe pattern is digitized into the computer and then thinned to get the maximum or minimum of each individual fringe by a written program.

**Figure 17.** Schematic diagram of multiple-beam Fizeau interferometer for measurement of (a) film thickness and (b) curved surfaces.

Three different scan directions, *x*-scan, *y*-scan, and *x*-*y*-scan, were tested to obtain the best measurement. The program scans the image row by row for the *x*-scan, or by column in the *y*-scan, or in mixture for the *x*-*y*-scan. In **Figure 18(b)**, the *x*-scan, which scans row by row, was used. At each row, the program finds the pixel coordinate of the minimum intensity of that row. Selected fringes in the middle of **Figure 17(a)** were filtered and processed in the program until these fringes were thinned. The thinning of the selected fringes is shown in **Figure 18(b)**. The isometric plot shown in **Figure 18(c)** shows the average of the measured step height *h*. The average step height value of the film determined from at least 10 readings was 30.6 nm. It was found that the average step height value is very close to the nominal value (31 ± 3 nm).

**Figure 18.** (a) Multiple-beam reflection fringes captured from Fizeau-Tolansky interferometer, (b) thinning of selected fringes in the middle of **Figure 2**, and (c) an isometric plot of (b).
