*3.2.2. Multiple-beam Fizeau interferometer for curved surfaces measurement*

**Figure 16.** (a) Interferogram of the plano-concave surface; (b) phase map of (a); and (c) 2-D surface height of **Figure**

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

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

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

Fizeau interferometry for thin film and curved strongly surfaces measurements.

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

or minimum of each individual fringe by a written program.

**16(b)** along *X*- and *Y*-directions.

94 Optical Interferometry

curved surfaces.

**3.2. Multiple-beam interferometry**

The schematic diagram of the Fizeau interferometer for curved surfaces measurement is illustrated in **Figure 17(b)**. Three curved surfaces of 25.4 mm in size and different radius of curvatures were coated with silver film of reflectivity nearly 90% and mounted parallel and close with the calibrated reference of nominally /50 flatness [7]. The reflectivity of the reference should be the same with the reflectivity of the object to obtain good contrast. **Table 1** shows types of curved surfaces being tested using **Figure 17(b)**.


**Table 1.** The types of curved surfaces with nominal radius of curvatures and the measurement values by Zernike polynomial fitting method.

The three curved surfaces one by one were inserted in the interferometer and adjusted carefully until the inline interferogram is captured. **Figure 19(a**–**c)** shows the three inline interferograms of the corresponding three curved surfaces, large, intermediate, and short radius of curvatures, respectively, after correction with flat fielding. The interferograms were reconstructed by Zernike polynomial fitting to extract the 3-D surface height as shown in **Figure 19(d**–**f)**. In Zernike polynomials fitting, the surface height function ( , ) can be represented by a linear combination of polynomials ( , ) and their weighting coefficients [7]:

**Figure 19.** (a–c) captured interferograms after correction with flat fielding of the three curved surfaces, large, intermediate, and short radius of curvatures, (d–f) the corresponding 3-D surface height.

$$\mathbf{Z}\_r(\mathbf{x}\_r, \mathbf{y}\_r) = \sum\_{j=1}^{M} \mathbf{F}\_j(\mathbf{x}, \mathbf{y}) \mathbf{G}\_j \tag{15}$$

where is the sample index, so it is important to calculate the coefficients to represent the surface.

#### *3.2.3. Testing parallelism degree on standard optical flat using Fizeau interferometer*

Testing parallelism on standard optical flat of 25 mm in size using Fizeau interferometer is shown in **Figure 1(a)**. The optical flat is positioned on the front side as shown in **Figure 20(c)**, the left one. The interference pattern between the reference and the object is obtained and captured, as shown in **Figure 21(a)**, by color charge-coupled device (CCD) camera of frame rate of 15 fps, and pixel area of 2456 (*y*) × 2058 (*x*) μm2 with a pixel size of 3.45 μm. The number of fringes of **Figure 21(a)** over 13 mm is found around 31 fringes or 1 = 31/2. The optical flat is then positioned on the back side as shown in **Figure 20(c)**, the right one. The number of fringes of **Figure 21(b)** over 12 mm is found to be 32 fringes or 2 = 32/2. The number of fringes can be accounted for manually or automatically by writing a small program.

Interferometry and its Applications in Surface Metrology http://dx.doi.org/10.5772/66275 97

**Figure 20.** Testing parallelism degree on standard optical flats over 12 mm in length using (a) Fizeau interferometer and (b) coordinate measuring machine (CMM). Schematic diagram shows the locations of front and back surfaces of the sample in the interferometer (c).

**Figure 21.** Fringe pattern produced at 12 mm on the (a) front side and (b) back side of the standard optical flat.

The change in the angular relationship is 2 − 1 = 1/2. But due to the rotation through 180o , there is a doubling effect. Therefore, the error in parallelism =(2 − 1)/2 = /4 <sup>≅</sup> 160 nm. The wavelength used in the experiment is a He-Ne laser of 632.8 nm, thus the optical flat has an error in parallelism of 160 nm over 12 mm at the middle of 25 mm.

The same size of front and back surfaces of the optical flat has been tested with coordinate measuring machine (CMM) with a suitable tip as shown in **Figure 20(b)** and the average difference between the two surface sides is calculated to be 200 nm.

### **3.3. Displacement interferometry**

respectively, after correction with flat fielding. The interferograms were reconstructed by Zernike polynomial fitting to extract the 3-D surface height as shown in **Figure 19(d**–**f)**. In

**Figure 19.** (a–c) captured interferograms after correction with flat fielding of the three curved surfaces, large, inter-

where is the sample index, so it is important to calculate the coefficients to represent the

Testing parallelism on standard optical flat of 25 mm in size using Fizeau interferometer is shown in **Figure 1(a)**. The optical flat is positioned on the front side as shown in **Figure 20(c)**, the left one. The interference pattern between the reference and the object is obtained and captured, as shown in **Figure 21(a)**, by color charge-coupled device (CCD) camera of frame

of fringes of **Figure 21(a)** over 13 mm is found around 31 fringes or 1 = 31/2. The optical flat is then positioned on the back side as shown in **Figure 20(c)**, the right one. The number of fringes of **Figure 21(b)** over 12 mm is found to be 32 fringes or 2 = 32/2. The number of

fringes can be accounted for manually or automatically by writing a small program.

<sup>=</sup> å (15)

with a pixel size of 3.45 μm. The number

M rrr j j j 1 Z (x ,y ) F (x,y)G =

*3.2.3. Testing parallelism degree on standard optical flat using Fizeau interferometer*

mediate, and short radius of curvatures, (d–f) the corresponding 3-D surface height.

rate of 15 fps, and pixel area of 2456 (*y*) × 2058 (*x*) μm2

( ,

) and their weighting coefficients [7]:

) can be represented by a linear

Zernike polynomials fitting, the surface height function

,

combination of polynomials (

96 Optical Interferometry

surface.

Displacement interferometry is usually based on the Michelson configuration or some variant of that basic design. Displacement measurement is defined simply a change in length. It is usually carried out by counting the number of fringes when either the object being measured or the reference surface is displaced. The fringes are counted by photodetectors and digital 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 and not inside the laser.

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