## Abstract

We describe a method of absolute xyz-coordinates measurement based on the two-point diffraction interferometer. In this paper we use a new optimization algorithm to the interferometer. Experimental results show that the systematic error of the interferometer is less than 1 μm (peak-to-valley value) within a 60 mm by 60 mm by 20 mm working volume. To extract the systematic error and verify the absolute performance of the interferometer we applied the Fourier self-calibration concept.

©2007 Optical Society of America

## 1. Introduction

There have been many efforts towards absolute distance measurement, which determine the target position by reading interferometric phase values straightway with no need of continuous measurements between points. When interferometric techniques are applied to absolute distance measurements, one of the fundamental ways is to enlarge the equivalent wavelength by grazing incidence [1] or multiple wavelengths synthesis [2, 3]. Tunable sources by frequency modulation [4, 5] are also available for this purpose. White light interferometry [6, 7] can offer better resolutions below a single wavelength within the scanning range. The pulse fringe autocorrelation technique [8, 9] or the wavelength synthesis [10] of femtosecond pulse lasers are applicable to absolute distance measurements. However, it is hard to expand these approaches into absolute coordinate measurement. For absolute three-dimensional coordinates measuring, a technique so-called multilateration [11–14] was proposed. This method determines the absolute three-dimensional coordinates of the retro-reflector target by using a geometric model of the measured diagonal distances between the target and several fixed laser trackers. In this case the geometrical form error and material inhomogeneity of the target can increase the measuring uncertainty.

As part of the effort to overcome the problem, we already proposed the new concept of
the two point-diffraction phase-measuring interferometer [15] and its side version based on the lateral shearing
interferometry [16]. The two point-diffraction phase-measuring interferometer
is intended to directly measure the xyz-coordinates of the target moving in
three-dimensions. The interferometer consists of a fixed photo-detector array and a
movable target. The target is made of two spherical wavefront sources, whose
interference pattern is monitored by the photo-detector array. Phase shifting
technique is applied to obtain the precise phase values. Then the measured phases
are fitted to a geometric model so as to determine the xyz-coordinates of the
target. The two point diffraction sources on the target are given the coordinates as
(*x*
_{1}, *y*
_{1},
*z*
_{1}) and (*x*
_{2},
*y*
_{2}, *z*
_{2}), respectively,
which are to be determined to find out the xyz-location of the target. In this paper
we describe the basic theory, the recently upgraded coordinate estimation algorithm,
and error budget of the interferometer. To check the performance we compare the
readings of our interferometer with the results of conventional coordinate measuring
instruments. In addition, the systematic errors of the interferometer are identified
adopting the concept of self-calibration [17].

## 2. Basic theory and the system configuration

The interferometric intensity field from two spherical wavefronts, referred to as
u_{1} and u_{2}, can be derived as [15]

$$\mathrm{where}\phantom{\rule{.2em}{0ex}}\Pi =\frac{{{U}_{1}}^{2}}{{{r}_{1}}^{2}}+\frac{{{U}_{2}}^{2}}{{{r}_{2}}^{2}},\Gamma =\frac{2{U}_{1}{U}_{2}}{{r}_{1}{r}_{2}},\Phi =\frac{2\pi}{\lambda}\left({r}_{1}-{r}_{2}\right),\mathrm{and}\phantom{\rule{.2em}{0ex}}\Delta \varphi =\left({\varphi}_{1}-{\varphi}_{2}\right).$$

In Eq. (1), *U* is the source strength, λ is
the wavelength, ϕ is the initial phase, and *r* is the
diagonal distance from the source. The subscripts 1 and 2 affixed to the variables
introduced in the above derivation correspond to *u _{1}* and

*u*, respectively. The mean intensity ∏, the visibility Γ, and the phase Φ of the intensity field vary with the diagonal distances, while the initial phase difference Δϕ remains constant. Among the variables, the phase Φ relates to the diagonal distances most simply with the relationship of

_{2}$$\phantom{\rule{.2em}{0ex}}=\frac{2\pi}{\lambda}\left[\sqrt{{\left({x}_{1}-x\right)}^{2}+{\left({y}_{1}-y\right)}^{2}+{\left({z}_{1}-z\right)}^{2}}-\sqrt{{\left({x}_{2}-x\right)}^{2}+{\left({y}_{2}-y\right)}^{2}+{\left({z}_{2}-z\right)}^{2}}\right].$$

Eq. (2) indicates that the six unknowns
(*x _{1}*,

*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) can be determined by solving inverse kinematics if the absolute value of Φ is provided from more than six different locations. For that, a two-dimensional array of photo-detectors is employed to capture the interferometric intensity I at multiple locations. From the measured intensity, the phase Φ+Δϕ of Eq. (1) is computed by applying the well-established phase measuring technique with phase shifting. For description, let us introduce the superscript k so that Φ

_{2}^{k}refers to the computed value of Φ at the location of (

*x*,

^{k}*y*,

^{k}*z*). All the measured values of Φ

^{k}^{k}are processed to be unwrapped, starting from a particular reference principal phase value that is for convenience designated as Φ

^{0}at location (

*x*,

^{0}*y*,

^{0}*z*). Then, we define a new geometric model Λ

^{0}^{k}as

Now, as the final step for absolute distance measurements, using the relationship of
Eq. (3), the six unknowns (x_{1}, y_{1},
z_{1}) and (x_{2}, y_{2}, z_{2}) are determined so
as to minimize the cost function that is defined as

where the summation over k is performed all over the photo-detectors to be
considered, and Λ̑ represents the actually measured value of
Λ^{k}. The cost function E is so highly nonlinear in terms of
the unknowns (*x _{1}*,

*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) that no explicit solutions exist. But the function E is convex with six unknowns, because ∇

_{2}^{2}E(

*x*,

_{1}*y*,

_{1}*z*,

_{1}*x*,

_{2}*y*,

_{2}*z*) is positive definite throughout the domain of

_{2}*x*,

*y*, and

*z*. Note that this condition theoretically means that the function E has a global minimum and there exists one true solution set. Thus, numerical technique is used to search for the global minimum of the cost function. In this paper we used a new optimization algorithm based on the simulated annealing technique [18] instead of the previous BFGS (suggested by Broyden

*et al*.) method [15] because of its robustness. To check both algorithms, we generated random noise sets whose rms (root mean square) values were 1, 5, and 10 nm and then added each sets on the two spherical wavefronts. Finally we estimated the coordinates by using both algorithms. We repeated each procedure 100 times.

The simulation result shows that the new optimization algorithm gives us an 81 %
probability of convergence while the previous algorithm offer a 53 % when the noise
level is 5 nm as show in Table 1. This means the new algorithm is more robust than the
previous one. If the convergence is failed, we should repeat the whole measurement
again. Note that the probability of convergence depends on the stopping criterion,
*E _{c}*, that decide whether the cost function is
successfully converged or not. (When the cost function

*E*in Eq. (4) is smaller than

*E*, the optimization process is stopped.) A tight

_{c}*E*makes a poor probability but the estimated coordinates will be very close to the real solutions. On the other hand a loose

_{c}*E*increases the probability of convergence but we cannot avoid sacrificing the uncertainty. For a good balance between the probability of convergence and the uncertainty, we used 0.006 μm

_{c}^{2}

*E*, which allows the error budget listed in Table 2.

_{c}In Table 2, we considered the worst cases that could possibly
occur over the whole working volume. The expanded uncertainty [20] was 418 nm for x-, 548 nm for y-, and 9.7 nm for
z-directions when we use a 640 × 480 CCD camera (The coverage factor
*k* = 1). In addition the genetic algorithm may be another
applicable solution, but we suppose that the genetic algorithm takes longer than 60
sec for one point calculation.

Figure 1 illustrates the optical configuration of the
two-point diffraction phase-measuring interferometer. A beam of stabilized HeNe
laser (wavelength: 632.8 nm) is fed into one fiber through an input coupler, which
is then connected to a 2×2 coupler that divides the input beam by 50:50
into two output branches. Each branch fiber is wound around a tube type PZT that
elongates the length of the fiber to induce phase shift. Only one of the two PZT
extenders is operated for phase shifting, while the other is inserted as a dummy
just for matching fiber length. The target is made of two single-mode optical
fibers, whose exit ends are polished and aligned side by side with an inter-axis
distance of nearly 125 μm. So the six unknowns
(*x _{1}*,

*y*,

_{1}*z*) and (

_{1}*x*,

_{2}*y*,

_{2}*z*) have the physical relationship such as

_{2}where the constraint d is considered a fixed value, so that we compute the value once for all. By this method we solve only five unknowns instead of six, thereby accelerating its convergence. For the purpose of checking the phase shifting process, back-reflected intensity from the two fiber ends is monitored by a single photo-detector connected to an input end of the 2×2 coupler. An array of photo-detectors is constituted by a two-dimensional CCD with 640 × 480 size, whose spacing is 8.44 μm in the x direction and 9.78 μm in the y-direction in terms of mean values.

## 3. Performance test

#### 3.1 One dimensional test

We set up the interferometer on a precision one-axis stage and repeatedly measured the target coordinates 35 times at various different positions within the working volume [15]. The repeatability in terms of ±2σ value was less than 0.05 μm for the x-coordinate, 0.1 μm for the y-coordinate and 0.01 μm for the z-coordinate. One-dimensional uncertainty was verified through a comparison test with a commercial heterodyne HeNe laser interferometer. The maximum deviation was measured 0.26 μm over a travel of 80 mm.

#### 3.2 Two-dimensional test

For the two-dimensional performance test the interferometer has been set up on a two-dimensional precision stage as illustrated in Fig. 2.

The target emitting two spherical wavefronts was fixed on the granite table and the CCD photo-detector array was attached to the moving stage. This setup arouses another error due to the tilting motion of the CCD. The rms values of the tilt were measured as 0.49 nrad along y-and 0.50 nrad along x-axis. These two terms brings the increase of uncertainty about 210 nm. However if we fix the CCD on the granite table and attach the target on the stage, the tilting effect might be greatly reduced because the spherical wavefront source is not affected by tilt. Nevertheless we used this experimental setup because of the expansion-possibility to three-dimensional test (The CCD is hard to setup on the small block gauge.). In Fig. 2, the readings of the two-dimensional optical linear scale installed on the measurement axes of the stage were taken as the initial values of the numerical search of the interferometer, so the convergence to the true xyz-coordinates can be accelerated. The stage developed for a wire-bonding process of the semiconductor has the 0.2 μm resolution within 63mm × 63mm working area. Figure 3 shows the differences between the readings of the optical scale and the results of the interferometer at each point with 10 mm space.

Test results were obtained within 60mm × 60mm working area, in which the maximum deviations were measured 0.482 μm for x- and 1.308 μm for z-direction as shown in Fig. 3. However this deviation map does not mean the true error of the interferometer because it includes the error from misalignment between two metrology frames, the optical scale’s systematic error, the scale factor, the random measurement noise and so on.

#### 3.3 Fourier self-calibration

To extract the true error of the proposed interferometer, the employment of a standard specimen is considered. We used the artifact plate of 7 × 7 mark array as a standard specimen as illustrated in Fig. 4, which was made by an electron beam lithography system. Each mark consists of 8 lines. To obtain the center position (x- and z- coordinates) of the mark, we used a microscope as shown in Fig. 2 and a center of gravity algorithm [21]. However the quantified accuracy of the center position of each mark is not enough to verify the interferometer, because the uncertainty of the specimen is on the same level of the interferometer errors. Furthermore the center detection algorithm also brings some errors.

Fourier self-calibration [17] in precision metrology aims to extract/remove the systematic errors of the instrument without relying upon externally calibrated artifacts. The stage self-calibration is the procedure of calibrating the metrology frame of the stage by an artifact plate whose mark positions are not exactly known. By assuming rigidness of the plate, this method extracts the error map of the stage metrology frame from comparison of three different measurement views of the plate as illustrated in Fig. 5. We applied the Fourier self-calibration algorithm to our interferometer. When we obtained the coordinates of each mark of Fig. 4 for calibration, we used averaging technique to reduce the random noise effect. Figure 6(a) shows the systematic error of the two-dimensional optical scale, in which the absolute values of the maximum error were measured 1.472 μm for x- and 1.347 μm for z-direction. On the other hand, Fig. 6(b) shows the systematic error of the interferometer itself, in which the absolute values of the maximum error were measured 0.997 μm for x- and 0.999 μm for z-direction. Through a repeat test of self-calibration, we confirm that the reconstructed systematic error has reasonable repeatability.

Moreover we expanded the test region to three-dimension. 10 mm thickness block gauges were installed to lift up and down the target along y-direction as shown in Fig. 7(a).

Figure 7(b) shows three test planes; the y-coordinate of the target is about −21.12 mm, − 11.12 mm, and −1.12 mm. These coordinate values were obtained by our interferometer. To apply the self-calibration, we measured the three views at each plane. Thus the whole measured volume was 60 mm × 60 mm × 20 mm. Figure 8 shows the self-calibration results at the second and the third planes (The result of the first plane is already displayed in Fig. 6). Figure 6 and Fig. 8 demonstrate that the proposed interferometer is capable of measuring the xyz-coordinates with maximum 0.999 μm errors in terms of the absolute value, while the optical scale has maximum 2.005 μm errors.

## 4. Conclusion

For absolute coordinates measurement in three-dimension, a two-point diffraction phase-measuring interferometer had been developed and its one-dimensional performance had also been tested in the previous work [15]. In this paper we applied new optimization algorithm based on the simulated annealing technique to the interferometer. By using this algorithm we greatly increased the probability of convergence. Our next goal is to improve this probability to 95 % without any sacrificing the uncertainty. We also compared the result of the interferometer with the reading of a two-dimensional optical scale, and expanded the test to three-dimension. The systematic errors of the interferometer were identified adopting the concept of self-calibration. The maximum error was 0.999 μm within a test volume of 60 mm × 60 mm × 20 mm.

## Acknowledgments

The authors would like to thank professor Seung-Woo Kim (KAIST) for insightful discussions and suggestions concerning the direction of this research.

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