3. PDI for 3D coordinate measurement

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 zero-

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

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

order waveplate QWP2.

192 Optical Interferometry

system.

interferometer [22].

In this section, a fiber PDI with submicron aperture [20], which is based on the single-mode fiber with a narrowed exit aperture, for the absolute 3D coordinate measurement within large angle range is described, and the system configuration is shown in Figure 7. After passing through the half-wave plate (HWP1), the frequency-stabilized laser beam is separate into p-polarized and s-polarized beams by the polarized beam splitter (PBS), and they are coupled into the single-mode fibers SF1 and SF2, respectively. The exit ends of the fibers SF1 and SF2 are integrated into a target with certain lateral offset, and the output point-diffraction waves W<sup>1</sup> and W<sup>2</sup> interfere on the CCD detector. By translating the PZT scanner, the point-diffraction interference field can be retrieved with phase-shifting method, and the corresponding 3D coordinates of target can be obtained with numerical iterative reconstruction algorithm.

To increase the NA of diffracted wavefront with optical fiber, a single-mode fiber with submicron aperture (as shown in Figure 4b) is applied as point-diffraction source. Both the exit ends

Figure 7. System configuration of the SMA fiber PDI for 3D coordinate measurement [20].

of two single-mode fibers are integrated in a target with certain lateral offset. According to the PDI system shown in Figure 7, the absolute 3D coordinates of target can be measured from the phase distribution in interference field corresponding to the optical path difference (OPD), as shown in Figure 8.

Denoting the plane of CCD detector as the xy plane and the central position o, with the distances r<sup>1</sup> and r<sup>2</sup> between an arbitrary point Pðx, y, zÞ on the CCD detector and the exit apertures of two fibers on the target, as shown in Figure 8, we have the phase difference ϕðx, y, zÞ,

$$\varphi(\mathbf{x}, y, z) = k[r\_1(\mathbf{x}\_1, y\_1, z\_1; \mathbf{x}, y, z) \neg r\_2(\mathbf{x}\_2, y\_2, z\_2; \mathbf{x}, y, z)],\tag{1}$$

where k ¼ 2π=λ, ðx, y, zÞ is the known 3D coordinate of the point P, ðx1, y1, z1Þ and ðx2, y2, z2Þ are those of two fiber apertures in the target under measurement. According to the one-to-one correspondence relationship of the phase ϕ at position P (that is the ith pixel on CCD) and 3D coordinates of two fiber apertures in the target, the phase difference can be written as follows:

$$f\_i(\mathbf{OP}) = k(\varphi^i \neg \varphi^0) \hat{-\xi}\_i,\tag{2}$$

where the vector Φ indicates the coordinates of two fiber apertures under measurement that is Φ ¼fðx1, y1, z1Þ;ðx2, y2, z2Þg; ξ \_ <sup>i</sup> is the measured phase difference. The phase differences ϕ<sup>i</sup> and ϕ<sup>0</sup> can be written as follows:

$$\begin{cases} q^i(\mathbf{x}^i, y^i, z^i) = k[r\_1(\mathbf{x}\_1, y\_1, z\_1; \mathbf{x}^i, y^i, z^i) - r\_2(\mathbf{x}\_2, y\_2, z\_2; \mathbf{x}^i, y^i, z^i)],\\ q^0(\mathbf{x}^0, y^0, z^0) = k[r\_1(\mathbf{x}\_1, y\_1, z\_1; \mathbf{x}^0, y^0, z^0) - r\_2(\mathbf{x}\_2, y\_2, z\_2; \mathbf{x}^0, y^0, z^0)],\end{cases} \tag{3}$$

where <sup>ð</sup>xi , yi , zi Þ is the 3D coordinate of the ith pixel on CCD. From Eq. (2), various equations can be established corresponding to the pixel positions on the CCD, and we have the matrix equations,

Figure 8. Model for 3D coordinate reconstruction [20].

$$f(\mathbf{O}) = \{f\_i(\mathbf{O})\} = \begin{cases} k(q^1 - q^0) - \widehat{\boldsymbol{\xi}}\_1 \\ \vdots \\ k(q^j - q^0) - \widehat{\boldsymbol{\xi}}\_i \\ \vdots \\ k(q^m - q^0) - \widehat{\boldsymbol{\xi}}\_m \end{cases},\tag{4}$$

where the subscript m is the number of the pixels selected for coordinate reconstruction. For the 6 unknowns in Eq. (2), at least 6 pixels are needed to obtain the 3D coordinates of two fiber apertures. To realize the accurate measurement of 3D coordinates with the PDI system, over 6 pixels could be applied to reconstruct the 3D coordinates, and a quadratic function corresponding to Eq. (4) can be obtained as:

of two single-mode fibers are integrated in a target with certain lateral offset. According to the PDI system shown in Figure 7, the absolute 3D coordinates of target can be measured from the phase distribution in interference field corresponding to the optical path difference (OPD), as

Denoting the plane of CCD detector as the xy plane and the central position o, with the distances r<sup>1</sup> and r<sup>2</sup> between an arbitrary point Pðx, y, zÞ on the CCD detector and the exit apertures of two fibers on the target, as shown in Figure 8, we have the phase difference

where k ¼ 2π=λ, ðx, y, zÞ is the known 3D coordinate of the point P, ðx1, y1, z1Þ and ðx2, y2, z2Þ are those of two fiber apertures in the target under measurement. According to the one-to-one correspondence relationship of the phase ϕ at position P (that is the ith pixel on CCD) and 3D coordinates of two fiber apertures in the target, the phase difference can be written as follows:

where the vector Φ indicates the coordinates of two fiber apertures under measurement that is

, yi , zi

<sup>ϕ</sup><sup>0</sup>ðx<sup>0</sup>, <sup>y</sup><sup>0</sup>, <sup>z</sup><sup>0</sup>Þ ¼ <sup>k</sup>½r1ðx1, <sup>y</sup>1, <sup>z</sup>1; <sup>x</sup><sup>0</sup>, <sup>y</sup><sup>0</sup>, <sup>z</sup><sup>0</sup>Þ−r2ðx2, <sup>y</sup>2, <sup>z</sup>2; <sup>x</sup><sup>0</sup>, <sup>y</sup><sup>0</sup>, <sup>z</sup><sup>0</sup>Þ�,

can be established corresponding to the pixel positions on the CCD, and we have the matrix

−ϕ<sup>0</sup> Þ−ξ \_

f i

Þ ¼ <sup>k</sup>½r1ðx1, <sup>y</sup>1, <sup>z</sup>1; xi

\_

<sup>ð</sup>ΦÞ ¼ <sup>k</sup>ðϕ<sup>i</sup>

ϕðx, y, zÞ ¼ k½r1ðx1, y1, z1; x, y, zÞ−r2ðx2, y2, z2; x, y, zÞ�, (1)

<sup>i</sup> is the measured phase difference. The phase differences ϕ<sup>i</sup>

<sup>Þ</sup>−r2ðx2, <sup>y</sup>2, <sup>z</sup>2; xi

Þ is the 3D coordinate of the ith pixel on CCD. From Eq. (2), various equations

, yi , zi Þ�,

<sup>i</sup>, (2)

and

(3)

shown in Figure 8.

194 Optical Interferometry

Φ ¼fðx1, y1, z1Þ;ðx2, y2, z2Þg; ξ

ϕi ðxi , yi , zi

Figure 8. Model for 3D coordinate reconstruction [20].

ϕ<sup>0</sup> can be written as follows:

ϕðx, y, zÞ,

where <sup>ð</sup>xi

equations,

, yi , zi

$$F(\mathbf{O}) = \frac{1}{2} f(\mathbf{O})^T f(\mathbf{O}) = \frac{1}{2} \sum\_{i=1}^{m} f\_i^2(\mathbf{O}). \tag{5}$$

Thus, the space coordinates of two fiber apertures can be determined from the global minimum Φ� of the function F (that is the least-square solution of Eq. (4)), with a single true solution set.

An experimental SMA-fiber PDI system with aperture size about 0.5 μm, has been set up for the measurement of 3D coordinates. For comparison, the target is installed on the probe of a CMM with the positioning accuracy about 1.0 μm, and the 3D coordinates measured with CMM is taken as the nominal value. Figure 9 shows the measurement results about the 3D coordinate deviations in x and z directions corresponding to the initial target position (−0.375, 15, and 200) mm. According to Figure 9, a good agreement between the CMM results and those from the SMA-fiber PDI system is achieved, and the coordinate measurement errors in x and z directions are 0.47 μm and 0.68 μm RMS, respectively. Due to the decreasing in the light intensity and enlargement in fringe spacing, the measurement error also grows with the increase of measurement distance. Several hardware factors can limit the measurement range

Figure 9. Measurement error in 3D coordinate measurement experiment. Measurement errors in (a) x direction and (b) z direction.

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 measurement range is greatly extended.
