## Abstract

We describe a method of absolute distance measurement based on the lateral shearing interferometry of point-diffracted spherical waves. A unique feature is that the distance measurement is not confined only along a single line of the optical axis, but the target is allowed to take movement freely within a volumetric measurement space formed by the aperture angle of point-diffraction. Detailed measurement theory is explained along with experimental verification.

©2006 Optical Society of America

## 1. Introduction

Traditional interferometry is excellent for measurement of incremental displacements but not immediately suitable for determining an absolute distance. Enlargement of the equivalent interference wavelength by way of grazing incidence [1] or two-wavelength synthesis [2] enables to extend the displacement range measurable with no periodic ambiguity. Multiwavelength interferometry by use of multiple light sources [3] or a tunable source with continuous frequency modulation [4] permits an absolute distance to be measured within the range constraint imposed by the number or bandwidth of available wavelengths. White light interferometry [5] allows performing absolute distance measurement to a resolution below a single wavelength within the scanning range. The concept of volumetric interferometer using point-diffracted waves is also a way of measuring absolute distances [6]. Femtosecond ultrashort pulse lasers can be used for measurement of a short absolute distance by pulse fringe autocorrelation [7], or for a long distance by wavelength synthesis [8], or pulse fringe autocorrelation coupled with time-of-flight measurement of pulses [9]. In this paper a new scheme of absolute distance measurement based on the lateral shearing interferometer of point-diffracted spherical waves is presented. In comparison to existing principles, the proposed method provides a unique feature that the distance measurement is not confined only along a single line of the optical axis, but the target is allowed to take movement freely within a volumetric measurement space formed by the aperture angle of point-diffraction.

## 2. System setup

As illustrated in Fig.1, the interferometer system proposed in this investigation is constituted with two main units; a transmitter and a receiver. The main function is to identify the absolute distance of the transmitter with respect to the receiver. The receiver is equipped with multiple photodetectors deployed in a form of evenly-spaced 2-D rectangular array. The transmitter holds four normally cleaved single-mode fibers, each emanating a near-perfect spherical wave by way of point diffraction [10]. For description, the global xyz-coordinate system is set on the receiver plane with its z-axis being normal to the plane. The four spherical waves are aligned to generate two laterally-sheared interferograms; one with an offset S in the horizontal x-axis direction and the other with the same offset in the vertical y-axis direction. The offset S should not be identical for both the x-and y-directions in principle, but it is assumed so for simplicity of explanation.

## 2.1. Lateral shearing interferometry

The four spherical wavefronts arriving at the receiver plane from the transmitter are expressed by adopting a master wavefront W(x,y); in its laterally-shifted forms of W(x-S/2,y), W(x+S/2,y), W(x,y-S/2), and W(x,y+S/2), respectively. The master wavefront W(x,y) has no real existence, but it represents a perfectly spherical wavefront virtually emitted from the location of (x_{c},y_{c},z_{c}) that represents the central point of the four diffraction sources formed by the four fiber emitters housed within the transmitter. To reconstruct the master wavefront W(x,y), a polynomial approximation of W(x, y)=${\sum}_{\mathrm{n}=0}^{\mathrm{k}}$${\sum}_{\mathrm{m}=0}^{\mathrm{n}}$B_{nm}x^{m}y^{n-m} is adopted with k being the degree of polynomials. Then the wavefronts of the two laterally-sheared interferograms are derived, respectively, as

$$\mathrm{where}\phantom{\rule{.2em}{0ex}}{C}_{\mathrm{nm}}=\sum _{j=1}^{\left(k-n+1\right)\u20442}2\left(\begin{array}{c}2j-1+m\\ 2j-1\end{array}\right){\left(\frac{S}{2}\right)}^{2j-1}{B}_{2j-1+n,2j-1+m}$$

and

$$\mathrm{where}\phantom{\rule{.2em}{0ex}}{D}_{\mathrm{nm}}=\sum _{j=1}^{\left(k-n+1\right)\u20442}2\left(\begin{array}{c}2j-1+n-m\\ 2j-1\end{array}\right){\left(\frac{S}{2}\right)}^{2j-1}{B}_{2j-1+n,m}$$

Notice that

Notice that $\left(\begin{array}{c}i\\ j\end{array}\right)\equiv \frac{i!}{\left(i-j\right)!j!}$. Now ΔW_{x} and ΔW_{y} are actually measured by applying the well-established Fourier-transform technique with subsequent phase-unwrapping [11]. Then the coefficients C_{nm} and D_{nm} are computed by fitting the measured data of ΔW_{x} and ΔW_{y} into the above polynomial expressions. Finally, the linear algebraic relations between B_{nm} and C_{nm} in Eq. (1) and B_{nm} and D_{nm} in Eq. (2) allow B_{nm} to be precisely determined, which leads to the final step of reconstructing the master wavefront W(x,y) [12].

## 2.2. Absolute distance determination

With respect to the point-diffraction source (x_{c},y_{c},z_{c}), the reconstructed master wavefront W(x,y) has the Euclidian geometrical relation of

Note that p≡φ_{0}λ/2π where φ_{0} is the initial phase formed at the exit of the point-diffraction source and λ is the wavelength of the source light. The absolute distance of the transmitter to be measured from the receiver is represented as $R=\sqrt{{{x}_{c}}^{2}+{{y}_{c}}^{2}+{{z}_{c}}^{2}}$. To extract R from the reconstructed master wavefront W(x,y), Eq. (3) is approximated as

$$\mathrm{where}\phantom{\rule{.9em}{0ex}}H=\left(-2{x}_{c}x-2{y}_{c}y+{x}^{2}+{y}^{2}\right)\u2044{R}^{2}$$

This is a one-order higher extension of the Fresnel approximation, providing more accurate estimation particularly for the intermediate range where z_{c} is not much larger than x_{c} and y_{c}. Eq. (4) is then rearranged into the form of W=${\sum}_{\mathrm{i}=1}^{15}$A_{i}U_{i}, in which U_{i} are the Zernike polynomials and Ai are their corresponding coefficients that are derived in terms of x_{c}, y_{c}, z_{c}, and R as listed in Table 1. This mathematical manipulation implies that the unknown R can be obtained deterministically once the coefficients A_{i} are computed by transforming the measured W(x,y) into the Zernike form. Specifically, the distance R is obtained using A_{4}, A_{5}, and A_{6} as

This way of determining R allows saving computational time since only three Zernike coefficients need to be calculated, relating to astigmatism and defocus. In addition, the transmitter is allowed to be positioned with flexibility off the z-axis of the receiver.

## 2.3. Measurement range

The measurement range of the proposed method is of no theoretical limit in principle but practically restricted by several hardware factors pertaining to the transmitter and the receiver as well. Firstly, the spherical wave diffracted from a single-mode fiber is confined within the conic boundary defined by the aperture angle θ, which is ~7° for our fibers constructing the transmitter. The lateral measurement range is consequently restricted by x_{c}<z_{c}tanθ and y_{c}<z_{c}tanθ. Secondly, the longitudinal range regarding z_{c} is associated with the fringe sampling capability of the receiver. The average spacing of the shearing fringes observed in ΔW_{x} and ΔW_{y} becomes dense with the relation of λz_{c}/S as z_{c} decreases. Therefore, the lower bound of z_{c} is imposed by the Nyquist sampling limit such as z_{c}>2dS/λ, in which d is the spatial resolution of the photodetector array comprising the receiver. Similarly, the upper bound of z_{c} is given as z_{c}<2DS/λ in which D denotes the overall size of the photodetector array so that the minimum measurable fringe spacing is limited to 2D. Another factor to be considered for the upper bound of z_{c} is the total power of the source, of which irradiance reduces, in proportion to 1/R^{2}, below the level of electric noise when z_{c} reaches a certain threshold. The overall performance is expected to improve gradually if the size of the photodetector array in the receiver is further increased or the shearing offset between the fibers in the transmitter is made larger.

## 3. Experiments and discussions

#### 3.1. Wavefront reconstruction using Fourier-transform

Experimental validation was carried out using a HeNe laser of 632.8 nm as the source light. The transmitter was fabricated with single-mode fibers of an effective core diameter of 2 µm. The lateral offset S was given 2.0 mm. The receiver was constructed conveniently by adopting a 2-D photodetector array of 1000 x 1000 pixels, each pixel having a dimension of 9.0 µm×9.0 µm. Figure 3(a) shows a typical fringe pattern of lateral shearing interferometry sampled at a distance of 700 mm, in which two orthogonally-sheared interferograms ΔW_{x} and ΔW_{y} are observed simultaneously being overlapped. Figure 3(b) depicts the Fourier-transformed frequency spectrum, in which the two peaks indicated as A and B correspond to ΔW_{y} and ΔW_{x}, respectively. Each peak is isolated by a low-pass filter of finite width and then inverse Fourier-transformed with subsequent phase determination of ΔW_{x} and ΔW_{y} as illustrated in Fig. 3(c). Finally, W(x,y) is reconstructed following the computational procedure of Eq. (1) and Eq. (2), of which result is drawn in Fig. 3(d).

## 3.2. Experiment results

Figure 4(a) shows an experimental result obtained by moving the transmitter along a straight line, off the z-axis, starting from (30 mm, 30 mm, 400 mm) to (30 mm, 30 mm, 1200 mm) with steps of 50 mm. The measurement error represented by the standard deviation of repeatability among 25 consecutive measurements is no more than 0.01 mm, which tends to increase as the distance increases. Figure 4(b) shows another test result measured from (30 mm,-50 mm, 1000 mm) to (30 mm, 50 mm,1000 mm) with steps of 5 mm. In this case, the measurement repeatability turned out to be rather uniformly less than 0.01 mm as there is no significant change in the measured distances between the transmitter and receiver. A commercial heterodyne laser interferometer was used for the validation of both the measurement errors, which provides an overall uncertainty of one part in 10^{5} without compensation of temperature, pressure, and humidity. The main causes of the measurement errors are considered the temperature variation in air, electrical noise encountered in interferograms sampling, and external vibration. These sources of errors tend to be more significant as the distance increases.

To avoid systematic errors, it is necessary to identify the exact value of the shearing offset S in the x-and y-direction separately. For the purpose, the transmitter is located at a known position with respect to the receiver. Then, turning on the two shearing interferograms sequentially, in the x-direction first and then in the y-direction, allows each offset value to be determined precisely with subsequent fitting the mathematical models of Eq. (1) and Eq. (2) to the resulting interferograms. Another issue is to balance the power of two sources so that the two interferograms yield the same intensity level. This power balancing helps avoid unwanted computational errors in measuring the phase values of interferograms caused by the nonlinearity in intensity sampling of the CCD camera in use.

## 4. Conclusion

The proposed method of lateral shearing interferometry of point-diffracted spherical waves is found theoretically feasible as a means of determining absolute distances. The measurement range is not confined along a single line as the target is allowed to take movement freely within a volumetric measurement space formed by the aperture angle of point-diffraction. The longitudinal measurement range is adjustable by selecting the spacing and size of the 2-D photodetector array used as the receiver. Practically achievable precision is in the level of 10^{-5} when measuring a distance up to 1200 mm, being disturbed by the presence of temperature fluctuation, electrical noise, and vibration which deteriorate the temporal and spatial stability of the spherical wavefronts emitted from the transmitter fibers.

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