2.1. Wavefront testing with PDI

finished with the figure error less than 0.2–0.3 nm (RMS), requiring a higher testing accuracy (e.g., 0.1 nm RMS). The optical interferometers, including the Twyman-Green [5] and Fizeau interferometers [6], have been widely used in the testing lenses and mirrors for figure metrology, in which standard lenses are applied to produce the necessary reference wavefront. Due to the fabricating error of the reference optics, the achievable measurement accuracy of the traditional interferometers is generally limited within λ/50 (λ is the operating wavelength). As a novel optical technique, the point-diffraction interferometer (PDI) [7–12] has been developed to overcome the accuracy limitation in traditional interferometers, and it can reach the measurement accuracy in the order of subnanometer. The PDI method employs point-diffraction spherical wavefront as ideal measurement reference, and it can achieve the

provides a feasible way to overcome the accuracy limitation of the reference optics in traditional interferometers. The process of point diffraction determines the reachable accuracy of PDI, thus, a good reproducibility of the measurement accuracy can be achieved with PDI. The ideal spherical wave can be generated from the point-diffraction source such as circle pinhole [13–15] and optical fiber [16–18]. Nikon realized that the high-precision testing of spherical and aspherical surfaces with the pinhole PDI [14, 15], and the Lawrence Livermore National Laboratory measured surface figure with the single-mode-fiber PDI [16]. The diffracted wave in both PDI systems serves as testing wave as well as reference wave. To extend the aperture of the optics, a fiber PDI with two optical fibers serving as point sources was proposed for large optics measurement [18], in which the diffracted wave from one fiber acts as testing wave and that from the other fiber as reference wave. Considering the fact that the diffraction light power from pinhole is poor (transmittance <0.1‰) and the numerical aperture (NA) of diffracted wavefront with single-mode fiber is quite low (<0.20), both approaches limit the measurement range of the existing PDIs. A submicron-aperture (SMA) fiber with cone-shaped exit end has been proposed to obtain both the high diffraction light power and high-NA spherical wavefront [19–21], and it is considered as a feasible way to extend the measurement range of the system. Due to the fact that the achievable testing accuracy of PDI is mainly determined by the sphericity of diffracted wavefront, the analysis of the diffracted wavefront error has become a fundamental way to evaluate the performance of PDI. In our research, both numerical analysis and experimental measurement have been performed to reconstruct the point-diffraction wavefront aberration. Besides, various setups of PDI have been developed to realize the precise surface testing with adjustable contrast [12, 22] and three-dimensional (3D) coordinate measurement [20, 23, 24]. The new PDI for three-dimensional (3D) coordinate measurement avoids the measurement uncertainty introduced by the imperfect target in multilateration and allows the target (made of two point-diffraction sources) to take free movement within a volumetric space over NA of point-diffraction wave. Section 2 presents a brief review of the PDI for testing of wavefront and the optical surface. Section 3 presents the basic theory of the SMA fiber PDI for 3D coordinate measurement. Sections 4 and 5 explain the numerical method and the experimental measurement method for the evaluation of diffracted wavefront sphericity, as well as the

λ. The PDI does not require precise and costly standard parts, and it

accuracy better than 10−<sup>3</sup>

188 Optical Interferometry

corresponding analyzing results.

The PDI was first proposed by Smartt and Strong in 1972 [7]. The early version of the PDI was an interferometer using a PDI plate with partial transmission. Figure 1 shows the principle of the PDI with partially transparent PDI plate. The PDI plate consists of an absorbing metal coating mask on a clear substrate and a tiny pinhole placed near focus to divide the wave into two parts, namely the testing wave and reference wave. The pinhole picks off part of the incident focused light wave and generates the diffracted spherical wavefront as reference wavefront. The test wave passes through the PDI plate and then interferes with diffracted spherical wave. The PDI can be used to measure the wavefront error from imaging optics and flow field, etc. In the design of the PDI shown in Figure 1, it is a key issue to determine the pinhole size and plate transmittance [9]. Due to the reduction in light intensity incident on the tiny pinhole, the controllable reduction in the test wave intensity by mask attenuation is required to get maximum fringe contrast.

Another PDI plate with double apertures has been applied in the PDI for the evaluation of wavefront error in imaging optics [25], as shown in Figure 2. A beam splitter such as the

Figure 1. Principle of PDI with PDI plate. (a) PDI plate and (b) PDI operation.

Figure 2. PDI plate with double aperture.

transmission grating is used to divide the test wave into two waves with a small angular separation. The PDI plate consists of one tiny pinhole and one large window on an opaque mask, and both of them are placed at the respect focal points of two beams. The pinhole is applied to generate the spherical reference wavefront by diffraction and the window transmits the test wave. The intensity of the diffracted wave relative to that of the conventional PDI is increased by several orders of magnitude. With the application of double-aperture PDI plate, there is no need to further attenuation in the test wave to match the intensities of interfering waves. Besides, the beam division enables the potential to introduce the various phase shifts between the interfering waves.

### 2.2. Optical surface testing with PDI

Figure 3 shows the typical configuration of PDI for optical surface testing. Either the pinhole (Figure 3a) [13–15] or the single-mode optical fiber (Figure 3b) [16–18] can be used as the point-diffraction source to generate the required spherical wavefront. The diffracted wavefront is separated to two parts, i.e., the test and reference wavefronts. The test wave travels toward the spherical surface under test, and the pinhole (exit end of fiber) is positioned at the curvature center of the test surface, so that the reflected wave from the test surface converges at the pinhole mirror (semitransparent metallic film on the output end of fiber) and then is reflected at the pinhole mirror (semitransparent metallic film on the output end of fiber). The test wave combines and interferes with the reference wave after reflection. By translating the test surface with a precise PZT scanner, the test surface error can be measured with the phase-shifting method. The selection of beam polarization state in the system can significantly influence the measurement. The polarization would affect the sphericity of the diffracted wavefront, and the reflection at mirror over a high NA can also introduce polarization-dependent phase shifts. In the PDI system, the polarization state of diffracted spherical wave is generally adjusted to be circularly polarized [26], in which the effect of polarization on the measurement precision is negligible.

Figure 3. Configuration of PDI for optical surface testing. (a) Pinhole PDI [13], (b) single-mode fiber PDI [16].

In the PDI with pinhole method, the circular pinhole of submicron (or even smaller) diameter is adopted to obtain high measurable NA. Figure 4(a) shows a scanning-electron microscope (SEM) picture of the pinhole fabricated by etching the chromium film with the focused ionbeam etching (FIBE) method [22], in which the metallic layer is sputtered onto the silica substrate. A nearly perfect circular pinhole can be obtained with the FIBE method. The high measurable NA can be achieved with the pinhole method; however, the nonadjustable fringe contrast would limit the measurement accuracy in the testing of low-reflectivity surface, due to the poor fringe contrast and difficulty in fringe processing.

transmission grating is used to divide the test wave into two waves with a small angular separation. The PDI plate consists of one tiny pinhole and one large window on an opaque mask, and both of them are placed at the respect focal points of two beams. The pinhole is applied to generate the spherical reference wavefront by diffraction and the window transmits the test wave. The intensity of the diffracted wave relative to that of the conventional PDI is increased by several orders of magnitude. With the application of double-aperture PDI plate, there is no need to further attenuation in the test wave to match the intensities of interfering waves. Besides, the beam division enables the potential to introduce the various phase shifts

Figure 3 shows the typical configuration of PDI for optical surface testing. Either the pinhole (Figure 3a) [13–15] or the single-mode optical fiber (Figure 3b) [16–18] can be used as the point-diffraction source to generate the required spherical wavefront. The diffracted wavefront is separated to two parts, i.e., the test and reference wavefronts. The test wave travels toward the spherical surface under test, and the pinhole (exit end of fiber) is positioned at the curvature center of the test surface, so that the reflected wave from the test surface converges at the pinhole mirror (semitransparent metallic film on the output end of fiber) and then is reflected at the pinhole mirror (semitransparent metallic film on the output end of fiber). The test wave combines and interferes with the reference wave after reflection. By translating the test surface with a precise PZT scanner, the test surface error can be measured with the phase-shifting method. The selection of beam polarization state in the system can significantly influence the measurement. The polarization would affect the sphericity of the diffracted wavefront, and the reflection at mirror over a high NA can also introduce polarization-dependent phase shifts. In the PDI system, the polarization state of diffracted spherical wave is generally adjusted to be circularly polarized [26], in which the effect of polarization on the measurement precision is

Figure 3. Configuration of PDI for optical surface testing. (a) Pinhole PDI [13], (b) single-mode fiber PDI [16].

between the interfering waves.

190 Optical Interferometry

negligible.

2.2. Optical surface testing with PDI

A pinhole PDI with adjustable fringe contrast can be adopted for the testing of high-NA spherical surfaces with low reflectivity [22]. The polarizing elements are applied to transform

Figure 4. Point diffraction sources in PDI. (a) Pinhole [22], and (b) submicron-aperture fiber [21].

Figure 5. Configuration of pinhole PDI with adjustable fringe contrast [22].

the polarization states and adjust the relative intensities of the interfering beams, by which the adjustable fringe contrast can be achieved. Figure 5 shows the optical configuration of the pinhole PDI with adjustable fringe contrast. A quarter-wave plate (QWP2) with special structure (consisting of a thin waveplate film and a plano-convex substrate) is placed at the test path, with the fast axis oriented at −45° to horizontal. The diffracted wave is adjusted to be circularly polarized. The test wave W<sup>1</sup> travels toward the test surface and passes through QWP2 twice, respectively, before and after the reflection at the test surface, then the test wave W<sup>1</sup> becomes opposite circularly polarized with respect to the reference wave W2. The relative intensities of the test and reference beams can be adjusted by rotating the transmission axis of the analyzer, realizing the adjustable fringe contrast. Due to the fact that the diffracted wave is divergent, QWP2 in the test path would introduce different phase retardations in various directions. To minimize the effect of QWP2 on the divergent test wave, a true zero-order waveplate, which has advantages of less sensitive to variation in angle of incidence and wavelength, is employed in the system. In the true zero-order waveplate, a thin waveplate film is cemented on the glass substrate. To minimize the aberrations introduced by the glass substrate in the case of divergent waves, a plano-convex substrate is used in the true zeroorder waveplate QWP2.

Figure 6 shows the testing results about a spherical surface with the reflectivity 4%, NA 0.40 and aperture diameter 137.7 mm. Figure 6(a) is the surface error measured with the adjustable-contrast pinhole PDI system shown in Figure 5, and Figure 6(b) is that obtained with the ZYGO interferometer. According to Figure 6, a good agreement between the PDI result and that from the ZYGO interferometer is achieved, and the PV and RMS differences between the testing results are about 0:0214λ and 0:0025λ, respectively. Thus, high-accuracy testing of the surface with low reflectivity is realized with the adjustable-contrast PDI system.

The light transmission through the pinhole is quite low (<0.1‰), and it significantly limits the achievable measurement range of the pinhole PDI. In the PDI with single-mode fiber, the

Figure 6. Measured surface error of test spherical surface with (a) the adjustable-contrast pinhole PDI and (b) the ZYGO interferometer [22].

adjustable fringe contrast is easy to realize and high light transmittance (>10%) can be obtained; however, the measurable NA in the fiber method is limited by the NA of the fiber, which is commonly less than 0.2. Besides, the measurable aperture of the test surface is approximately half that of the diffraction wave, both for the pinhole PDI and single-mode fiber PDI shown in Figure 3. A PDI system with two optical fibers as point-diffraction sources was developed for full use of the diffracted wavefront [18], in which the diffracted wave from one fiber serves as reference wavefront and that from the other fiber as test wave. Besides, a novel submicron-aperture (SMA) fiber with cone-shaped exit end, as shown in Figure 4(b), has been proposed to obtain both the high diffraction light power and high-NA spherical wavefront [19–21]. The SMA fiber taper surface is coated with metallic film and the exit aperture is formed from the polished tip, it is formed with the same processing technology as manufacturing of fiber-based probes for the scanning near-field optical microscopy. With the SMA fiber, the measurable range can be almost extended to a half space, and the corresponding light transmittance over 50% is obtained [20]. Thus, it is considered as a feasible way to extend the measurement range of the system.
