*3.1.2. Fizeau interferometers*

Fizeau interferometers are most commonly used for testing surface figure, flatness, and parallelism of optical components. **Figure 13** shows a schematic diagram of a commercial laser phase-shifting Fizeau interferometer equipped with a tunable laser. Two-beam Zygo interferometer of type VeriFireMST and wavelength 632.467 nm at PTB, Germany, was used to measure the flatness of a high-flatness optical flat of size 60 mm (photograph of the optical flat being tested is shown in **Figure 13**). The reference of the interferometer is a transmission optical flat of 100 mm in size and flatness of /500 nm. Zygo's advanced phase analysis methods are coupled with fast Fourier transformation to separate each of these individual frequencies [10]. When the optical flat is carefully positioned on the interferometer, four different frequency patterns are obtained due to reflections from the surfaces as shown in the schematic diagram in **Figure 13**. The reflections are from the transmission flat (*S*1), plate front surface (*S*2), plate back surface (*S*3), and reference surface (*S*4).

**Figure 13.** Familiar transmitted wavefront test geometry using a commercial laser Fizeau interferometer.

Because the object (optical flat) is relatively thin (*G*2 = 15 mm), the gap *G*<sup>1</sup> is the intermediate thickness of 100 mm, and *G*<sup>3</sup> the thickest (115 mm) as shown from the three peaks of the OPD spectrum plot in **Figure 14(a)**. Each peak in the spectrum corresponds to the OPL of a particular elemental cavity. The spatial phase variation for the interferometer cavity is calculated according to the OPL. The interferogram in a four-surface cavity is shown in **Figure 14(b)**. The horizontal fringes correspond to the interference of the front side *S*2 of the optical flat and the reference, while the second inclined dark fringes correspond to the interference of the back surface of the object *S*3 and the reference. The software analyzes each fringe pattern and the surface from of the optical flat was obtained. **Figure 14(c)** shows the phase map of the front side of the sample, and profiles through *x*- and *y*-directions are shown in **Figure 15**, respectively. The measurement has been calculated at a temperature of 20.5°C with uncertainty in the measurement of 15 nm.

**Figure 12.** (a) Wrapped phase map resulted from the four frames of **Figure 10**; (b) 3-D unwrapped phase map of (a);

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)**.

Fizeau interferometers are most commonly used for testing surface figure, flatness, and parallelism of optical components. **Figure 13** shows a schematic diagram of a commercial laser phase-shifting Fizeau interferometer equipped with a tunable laser. Two-beam Zygo interferometer of type VeriFireMST and wavelength 632.467 nm at PTB, Germany, was used to measure the flatness of a high-flatness optical flat of size 60 mm (photograph of the optical flat being tested is shown in **Figure 13**). The reference of the interferometer is a transmission optical flat of 100 mm in size and flatness of /500 nm. Zygo's advanced phase analysis methods are coupled with fast Fourier transformation to separate each of these individual frequencies [10]. When the optical flat is carefully positioned on the interferometer, four different frequency patterns are obtained due to reflections from the surfaces as shown in the schematic diagram in **Figure 13**. The reflections are from the transmission flat (*S*1), plate front surface (*S*2), plate

**Figure 13.** Familiar transmitted wavefront test geometry using a commercial laser Fizeau interferometer.

and (c) two-dimensional height at the middle of (b) in the *x*-direction.

back surface (*S*3), and reference surface (*S*4).

*3.1.2. Fizeau interferometers*

92 Optical Interferometry

**Figure 14.** (a) Peaks of the OPD spectrum; (b) interferogram in a four-surface cavity; and (c) phase map of the front side of the sample.

**Figure 15.** (a) 2-D surface height of **Figure 14c** along *x*-direction; and (b) along *y*-direction.

Another type of plano-concave surface of radius of curvature of 12744.1 mm at a focal length of −6000 mm has been tested using two-beam Zygo interferometer of type VeriFireMST and a wavelength of 632.467 nm. Circular fringes of the curved surface being tested are shown in **Figure 16(a)**; as shown from **Figure 16(a)**, the number of fringes/12 mm is around 15 fringes, which means the surface is nearly strong. The phase map of **Figure 16(a)** is shown in **Figure 16(b)**. Two-dimensional surface height of **Figure 16(b)** along *X*- and *Y*-directions is shown in **Figure 16(c)**.

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