4.1. Simulation of pinhole diffraction

Figure 11 shows the amplitude and phase distribution in the near-field diffraction, respectively, corresponding to the pinholes of 0.5 μm and 1 μm diameters, respectively. According to Figure 11, the phase distribution within Airy disk range is close to an ideal sphere, and the angle of Airy disk increases with the decrease of pinhole diameter.

Figure 10. Procedure for simulation of point-diffraction wavefront based on FDTD method.

of the PDI system in practical application, including the detector size and light power, though there is no theoretical limit in principle. The PDI system can be further improved by increasing CCD detector size, lateral off set of two SMA fibers and source power. Compared to the traditional single-mode fiber, the SMA fiber provides a feasible way to obtain the ideal spherical reference wavefront, whose aperture angle almost covers half space, and the lateral mea-

As one of the most important elements in the PDI, the point source for the diffracted reference wavefront determines the achievable accuracy in the measurement. Purely empirical design of point source parameters is both time consuming and costly. The numerical method based on diffraction theory is a feasible way for the analyzing point-diffraction wavefront. The scalar diffraction theory is valid only when the pinhole size is several times larger than the operating wavelength. For the high-NA spherical wavefront emerging from a tiny aperture with the size comparable with or less than operating wavelength, the vector diffraction theory (that is a nonapproximate method) is required to realize the accurate estimation of diffracted wavefront error. In this section, numerical analysis based on finite difference time domain (FDTD) method (that is a vector diffraction theory) [27] is presented. Figure 10 shows the flow diagram for the simulation of point-diffraction wavefront based on FDTD method. Due to limitations of computer memory capacity and runtime, FDTD cannot be directly applied to calculate the farfield distribution of pinhole diffraction. In the first step, the near-field distribution of point diffraction is analyzed with the FDTD method, and then the near-to-far field transition based on Huygens' principle is performed to obtain the far-field distribution of point-diffraction wavefront at the position under study. Finally, the sphericity evaluation is carried out to get

Figure 11 shows the amplitude and phase distribution in the near-field diffraction, respectively, corresponding to the pinholes of 0.5 μm and 1 μm diameters, respectively. According to Figure 11, the phase distribution within Airy disk range is close to an ideal sphere, and the

surement range is greatly extended.

196 Optical Interferometry

4. Numerical analysis of point-diffraction wavefront

the departure of point-diffraction wavefront from an ideal sphere.

angle of Airy disk increases with the decrease of pinhole diameter.

Figure 10. Procedure for simulation of point-diffraction wavefront based on FDTD method.

4.1. Simulation of pinhole diffraction

Figure 11. Near-field distribution of pinhole-diffraction wavefront. (a) Amplitude and (b) phase distribution with 0.5 μmdiameter pinhole; (c) amplitude and (d) phase distribution with 1 μm-diameter pinhole [27].

Figure 12. Simulation results for pinhole-diffraction wavefront. (a) Diffracted wavefront error over various NAs for different pinhole diameters, and (b) diffracted wavefront error under various film thicknesses [27].

To analyze the wavefront error where the test surface with 500 mm curvature radius is placed, far field is positioned at 500 mm away from pinhole. The wavefront errors over various NAs of diffracted wavefronts corresponding to different pinhole diameters are shown in Figure 12(a). According to Figure 12(a), the point-diffraction wavefront error grows both with the pinhole size and NA range, and the wavefront error RMS for the 1 μm pinhole diameter over 0.35 NA is smaller than 0.53 nm. Thus, it can be taken as ideal reference wavefront and applied to realize the measurement accuracy reaching to RMS λ=1000 (λ = 532 nm). Besides, larger diffraction angle could be obtained with smaller pinhole, and it allows much higher measurable NA. The diffraction angle corresponding to 1 μm pinhole diameter is about 75°, and that for 3 μm pinhole diameter is about 30°. Thus, the small pinhole is required in the testing of surfaces with high NA. However, the small pinhole size would result in significant reduction of diffraction light intensity. The pinhole size could be optimized according to the requirement of testing accuracy and achievable diffracted wavefront precision corresponding to various pinhole diameters. The pinhole in PDI plays not only the role as a filter to remove the aberration in diffracted wavefront, but also as waveguide. Figure 12(b) shows the numerical results about the effect of film thickness on diffracted wavefront for different pinhole diameters, with the thickness ranging from 150 nm to 450 nm. The mean values of wavefront error RMS corresponding to 300 nm, 400 nm, and 500 nm pinhole diameters are about 0.02 nm, 0.08 nm, and 0.36 nm, respectively, and the corresponding fluctuation ranges are within 10%. Thus, the effect of variation in film thickness ranging from 150 nm to 450 nm is negligible.

### 4.2. Simulation of SMA fiber diffraction

Figure 13(a) shows the simulation results about the full wavefront aperture angle for the SMA fibers with various exit apertures [20]. According to Figure 13(a), the full aperture angle of diffracted wave obviously increases with the decreasing of fiber aperture, and that corresponding to the 0.5 μm fiber aperture are about 160°, providing the necessary conic boundary to extend the measurement range almost within a half space. Figure 13(b) shows the wavefront errors over various NAs of diffracted wavefronts corresponding to different SMA fiber apertures [20]. From Figure 13(b), the increase in NA range and fiber aperture could lead to the growth of diffracted wavefront error. The wavefront error over 0.70 NA range for the 0.5 μm fiber aperture is better than λ=1000 RMS, and it can be applied as ideal spherical reference wave in the PDI system.

Figure 14 shows the analyzing result about light transmittance in the point diffraction both with the SMA fiber method and pinhole method. According to Figure 14, the light transmittance in both the SMA fiber diffraction and pinhole diffraction grows with the increase in aperture size. It can be seen from Figures 12(a) and 13(a) that the similar aperture angles can be obtained with the same diffraction aperture size, however, the light transmittance in the SMA fiber diffraction is far larger than that in pinhole diffraction according to Figure 14. The light transmittance corresponding to the 0.5 μm aperture in the SMA fiber method and pinhole method is about 67% and 6%, respectively. Thus, the SMA fiber provides a feasible way to

Figure 13. Simulation results for SMA-fiber-diffraction wavefront. (a) Aperture angle of diffracted wave and light transmittance for various fiber apertures, and (b) diffracted wavefront error within various NA ranges for different fiber apertures [20].

Figure 14. Simulation results for the light transmittance in pinhole diffraction and SMA fiber diffraction.

aberration in diffracted wavefront, but also as waveguide. Figure 12(b) shows the numerical results about the effect of film thickness on diffracted wavefront for different pinhole diameters, with the thickness ranging from 150 nm to 450 nm. The mean values of wavefront error RMS corresponding to 300 nm, 400 nm, and 500 nm pinhole diameters are about 0.02 nm, 0.08 nm, and 0.36 nm, respectively, and the corresponding fluctuation ranges are within 10%. Thus, the effect of variation in film thickness ranging from 150 nm to 450 nm is negligible.

Figure 13(a) shows the simulation results about the full wavefront aperture angle for the SMA fibers with various exit apertures [20]. According to Figure 13(a), the full aperture angle of diffracted wave obviously increases with the decreasing of fiber aperture, and that corresponding to the 0.5 μm fiber aperture are about 160°, providing the necessary conic boundary to extend the measurement range almost within a half space. Figure 13(b) shows the wavefront errors over various NAs of diffracted wavefronts corresponding to different SMA fiber apertures [20]. From Figure 13(b), the increase in NA range and fiber aperture could lead to the growth of diffracted wavefront error. The wavefront error over 0.70 NA range for the 0.5 μm fiber aperture is better than λ=1000 RMS, and it can be applied as ideal spherical

Figure 14 shows the analyzing result about light transmittance in the point diffraction both with the SMA fiber method and pinhole method. According to Figure 14, the light transmittance in both the SMA fiber diffraction and pinhole diffraction grows with the increase in aperture size. It can be seen from Figures 12(a) and 13(a) that the similar aperture angles can be obtained with the same diffraction aperture size, however, the light transmittance in the SMA fiber diffraction is far larger than that in pinhole diffraction according to Figure 14. The light transmittance corresponding to the 0.5 μm aperture in the SMA fiber method and pinhole method is about 67% and 6%, respectively. Thus, the SMA fiber provides a feasible way to

Figure 13. Simulation results for SMA-fiber-diffraction wavefront. (a) Aperture angle of diffracted wave and light transmittance for various fiber apertures, and (b) diffracted wavefront error within various NA ranges for different fiber

4.2. Simulation of SMA fiber diffraction

198 Optical Interferometry

reference wave in the PDI system.

apertures [20].

obtain the high light intensity required in the optical testing, enabling the extension of measurement range with PDI system.
