## Abstract

We propose a multiple height-transfer interferometric technique based on concepts from both multiple wavelength interferometry and wavelength scanning interferometry. Conventional multiple wavelength interferometry requires accurate wavelength information for large step height measurement, while wavelength scanning interferometry is limited by mode-hop-free tuning range. Using the multiple reference heights, it is possible to bypass the wavelength determinations and achieve large step height measurement using relative phase changes. By applying this technique with a proposed multiple height calibration artifact, we experimentally demonstrated accuracy better than 1 micron over 100 mm in a workshop environment.

© 2011 OSA

## 1. Introduction

Optical interferometry has become one of the most popular techniques for distance measurement due to its high accuracy [1]. However, measuring step heights using single wavelength interferometry is typically limited to a single trip optical path difference (OPD) between two measuring points of less than a half-wavelength unless there is additional information to resolve the half-wavelength ambiguity. Multiple wavelength interferometry (MWI) was introduced to extend the unambiguous measurement range, by incorporating two or more different wavelengths into an interferometer to yield a longer wavelength, normally called a synthetic wavelength [2–5]. In recent years, with the development of tunable lasers, commercially available external cavity diode lasers (ECDL), and femtosecond mode-locked lasers have opened up new perspectives in the field of accurate and long range absolute distance measurement [6]. However, the measurement range of MWI depends on the stability and the calibration of the different wavelengths. It means large measurement ranges can be achieved only by using a wavelength calibration device such as an etalon or a wavemeter, which limits its industrial applications due to system cost and complexity.

Wavelength scanning interferometry (WSI) is another technique allowing absolute distances to be measured unambiguously [7–11]. It performs measurement by comparing an interferometer of unknown length with a reference interferometer. The ratio of phase changes, while the tunable laser wavelength is scanning, induced in the two interferometers is proportional to the ratio of interferometer lengths. To implement this technique with high accuracy, the laser requires continuous tuning over a wide wavelength range. However, mode hops can shorten the tuning range and decreases the measurement accuracy. Besides, scanning limits the application in large field-of-view measurements due to CCD camera speed limitations and the measurement accuracy is sensitive to vibration during the scanning.

In this paper, we propose a multiple height-transfer interferometric technique (MHTIT). It preserves the capabilities to determine OPDs unambiguously without accurate wavelength information, and yet does not require the laser to be continuously tuned. This technique provides a complementary way to combine MWI and WSI. MHTIT applies the MWI technique with discrete measurements at several wavelengths instead of a wavelength scanning process, and employs reference heights analogous to WSI to avoid the wavelength accuracy requirement. It also takes advantages of multiple reference heights for phase unwrapping. Synthetic wavelengths are obtained from the ratio of the reference height and corresponding phase differences instead of absolute wavelengths. A modified Fourier Transform (FT) peak finding algorithm is also proposed to calculate object height [5,12]. Section 2 discusses the principle of this technique in detail; section 3 proposes a multiple reference height calibration artifact and provides the experimental results by applying the proposed technique.

## 2. Principle

To understand the principle of MHTIT, consider a Michelson interferometer (Fig. 1 ).

In a common Michelson interferometer, laser light is divided between a reference beam and an object beam. The OPD is measured with respect to the zero path difference plane P. However, the location of plane P may drift leading to measurement errors. In such situations, the OPD can be self-referencing by subtracting two optical paths to minimize this effect, which also allows the OPD to be measured far away from the zero path difference plane. As shown in Fig. 1, a tunable laser illuminates a reference height${h}_{r}$section and a height${h}_{o}$to be measured, both on a common base a distance${h}_{b}$from the zero path position. The unwrapped phase for the base is given by

and the reference and object unwrapped phases are given by subtracting corresponding unwrapped phase from Eq. (1),*s*=

*b*,

*r*or

*o*representing the base, reference or object section, respectively, ${q}_{s,n}$ is an unknown integer fringe order, ${\varphi}_{s,n}$is the measured phase information from the zero path position in the range of $(-\pi ,\pi ]$ [5], and ${\lambda}_{n}$is the wavelength. The measurement, due to the unknown integer fringe order, is limited to ${\Phi}_{s,n}<(-\pi ,\pi ]$and hence severely limits the range to an ambiguity-free-range of${\lambda}_{n}/2$. However, if a second wavelength ${\lambda}_{m}$is used and we subtract the resulting phases, the result is the synthetic wavelength equation

The measurement ambiguity-free-range is then extended to${\Lambda}_{m,n}/2$. Theoretically, two close enough wavelengths are sufficient to measure any height. However, noise comes into play and error amplification leads to poor measurement precision. Thereby more wavelengths are introduced to guarantee a good precision while achieving large measurement range [2]. The wavelength bandwidth determines the nominal height measurement resolution. The measurement ambiguity-free-range depends on the stability and the calibration of the different wavelengths [5]. For this reason, the uncertainty of laser wavelengths limits the measurement ambiguity-free-range of MWI. In order to break through this limitation, in this work we apply both reference and measurement heights at multiple wavelengths.

Consider the following relationship attained by dividing one synthetic wavelength equation by the other.

If we assume that${\Lambda}_{m,n}/2>{h}_{b}>{h}_{o},{h}_{r}$, then the ratio above does not require phase unwrapping or depend on the wavelengths directly. Hence, we can solve for *h _{o}*, since we know

*h*and the measured phases are the same as the unwrapped phase. Also, the variability of the base height is removed. However, noise amplification still limits the practical range. In general, for larger heights a phase unwrapping procedure is required. WSI obtains the phase differences by counting the fringes for both reference and measurement interferometers during the wavelength scan. In this work, we propose two different approaches for phase unwrapping. For the reference phase difference, multiple reference heights are used for phase unwrapping. And for the object phase difference, a modified FT peak finding algorithm is applied to overcome phase wrapping. We now explain these techniques in detail.

_{r}#### 2.1 Reference phase unwrapping

For reference phase unwrapping, considering Eq. (3), if we assume phase difference between two wavelengths is less than 2π, that is, $\Delta \Phi $<2π, then the wavelength difference must satisfy the following equation.

The right side of the expression defines the wavelength acquisition interval. Here, the height *h*, is determined by the maximum wavelength difference without introducing phase ambiguity. Small heights have large acquisition intervals and vice versa. Table 1
shows the wavelength acquisition intervals and corresponding heights centered at 800 nm wavelength.

Consider that the wavelength acquisition interval limits the maximum wavelength difference that can be applied without phase ambiguity. In order to measure *h _{o}* with high accuracy, both small and large reference heights are required. On one hand, a small height needs to be applied for phase unwrapping due to wide wavelength interval/bandwidth, which improves measurement resolution. On the other hand, the final uncertainty of the measurement has two major contributors: uncertainty of calibration of the reference cavity and uncertainty of phase difference. A large height is preferred as the measurement reference to decrease the measurement uncertainty. First, it requires less effort to achieve higher relative calibration accuracy compared with a small height in a practical optical system where noises exist. For instance, if the same calibration uncertainty of the reference height 1e-6 is desirable, the absolute measurement accuracy for 0.5 mm height is 0.5 nm whereas it is 50 nm for 50 mm. Measurement accuracy of 0.5 nm requires about 1/10000 fringe accuracy which is very challenging due to environmental disturbances such as vibration. And it is more practical for an optical system to achieve 50 nm measurement accuracy for 50 mm range. Second, with the help of a large height, uncertainty of phase difference will be decreased since the phase change of a larger height is greater than a small one for the same wavelength scan range.

Therefore, a technique applying multiple step heights named MHTIT is proposed. Small reference heights are used to remove phase ambiguity for large reference heights when wide wavelength intervals are applied. And the longest reference height is employed as the final measurement reference, which provides the best measurement accuracy.

The main challenge of reference phase unwrapping results from uncertainty of laser wavelengths. If two different wavelengths are applied, uncertainty of wavelength difference leads to uncertainty of phase unwrapping for large heights. For example, based on Eq. (3) and the center wavelength is 800 nm, 20 pm uncertainty wavelength difference will cause over 18 radians phase uncertainty for 50 mm height, whereas only 0.18 radian phase uncertainty exists for a 0.5 mm height. Hence, we cannot rely on the wavelength information to do the phase unwrapping since wavelength uncertainty for a typical tunable laser is no better than 100 pm. Instead, we choose the proper shortest reference height so that wavelength uncertainty does not cause fringe order ambiguity. Consequently, the fringe order difference can be determined correctly and thus the phase difference for the shortest reference height can be unwrapped. This unwrapped phase information is then used to unwrap the phase difference for next step height and so on using Eq. (7) motivated by Eq. (5). The process terminates when all phase differences are unwrapped.

Here, ${h}_{1},\text{\hspace{0.05em}}{h}_{2},\mathrm{...}{h}_{M}$ are multiple heights in ascending order, ${q}_{n1}$ is the integer fringe order change from wavelength n to wavelength 1, and $\Delta {\varphi}_{n1}$ is the wrapped phase difference from wavelength n to wavelength 1.

#### 2.2 Object phase unwrapping

After the phase difference for the longest height ${h}_{M}$is determined, Eq. (8) is given at different wavelengths for the object height${h}_{o}$.

By recognizing that the wrapped phase $\Delta {\varphi}_{n}$ can be taken into account by using the periodicity of a sinusoid and that the measured phase corresponds to starting phase, then simply adding the sinusoids together will give a peak signal at the object height generated via the following equation.

An example plot of normalized PHRF in 10 mm range is shown in Fig. 2 .

#### 2.3 MHTIT measurement procedure

The detailed measurement procedure of MHTIT is shown in Fig. 3 .

There are 4 steps for MHTIT measurements. First, OPDs of reference heights ${h}_{1},\text{\hspace{0.17em}}{h}_{2},\text{\hspace{0.17em}}\mathrm{...}{h}_{M}$ in the system have to be calibrated accurately. Second, applying different wavelengths, wrapped phases ${\varphi}_{1},\text{\hspace{0.17em}}{\varphi}_{2},\text{\hspace{0.17em}}\mathrm{...}{\varphi}_{M}$ are measured by phase shifting technique. The third step requires reference phase unwrapping using multiple reference step heights. Finally, a modified FT algorithm is proposed for object height measurement.

In MWI, variable synthetic wavelengths are generated by combinations of the varied wavelengths and the initial wavelength. The object height is thus measured by synthetic wavelength as reference. In MHTIT, object height is measured using reference height and corresponding phase differences as the basis of the measurement. In theory, the ratio of reference heights and total phase differences in MHTIT is the equivalent synthetic wavelengths used in MWI. MHTIT then provides an alternative way to find synthetic wavelengths for interferometry.

## 3. Experiments and results

#### 3.1 Apparatus

Experiments of applying MHTIT are conducted in a holographic measuring system applying MHTIT. The measuring system is a Coherix ShaPix® system which employs multiple wavelengths of a tunable laser as part of measuring object surface shapes [5]. A schematic diagram of the Coherix ShaPix unit is illustrated in Fig. 4 . It is a holographic interferometric metrology instrument that directly measures the shape of 300 mm by 300 mm surfaces and larger with stitching. The light source is a tunable laser of about 15 nm tuning range. The wavelength of the light is varied under computer control. Linearly polarized light is fed into a polarization maintaining (PM) fiber which feeds a variable beam splitter that splits and inserts the light into object and reference PM fibers.

The object light is used to flood illuminate the object with a collimated wave via a parabolic mirror, which in turn collects the scattered object light and sends it to a digital camera. The camera lens images the object onto the camera detector array. The reference light is combined with the light scattered from the object via a beam splitter to form an inline interferogram detected by the camera and fed to the computer for processing. The reference wave is set to diverge from a point that is equal distant to that of the focal point of the parabolic mirror. Thus, the reference wave and object wave both appear to diverge from the same plane. The reference PM fiber has extra length to compensate for the free space propagation length of the object wave such that the zero path length difference is slightly above the object surface. The phase of the light is stepped with a computer controlled phase shifter incorporated into the variable beam splitter box. A number interferograms are acquired, each with its own unique wavelength and step phase. These interferograms are used to form digital holographic images that lead to generating fine resolution 3D shape maps of the object.

As an implementation of the technique described in this paper, an array of calibrated fiducial reflectors in the instrument’s field-of-view is used as multiple reference height calibration device [14]. This array could be a set of flat surfaces composed of an array of height gages or some other calibrated multi-surfaced machined block. We investigate using a linear array of staggered retroreflectors as the fiducials. The placement of the array is illustrated in Fig. 4. The light path from the input focus point for the parabolic mirror to the array and then back to offset output focus position for the parabolic mirror are shown. The dash line represents the path of the light for multiple reference heights. The insert shows the interferogram obtained from the system for a reference array composed of five retroreflectors. Note that since the object illumination source and the receiving aperture are displaced from each other, there is typically laser light along one side that is not used and camera pixels along the other side that are not used. Thus by tapping off the otherwise unused light with a mirror and directing it at the array and then using the otherwise unused pixels to view the light from the array via a beam splitter, these previously unused resources can be utilized.

#### 3.2 Calibration of the reference array

A MWI setup was used to calibrate the OPDs for an array composed of 5 identical solid retroreflectors. The 7.2 mm diameter BK7 retroreflectors with silver coated reflective surfaces were mounted on a super invar base to minimize spacing variation with temperature changes. A diagram of the calibration setup is shown in Fig. 5 .

In this calibration setup a commercial tunable laser was used to tune 14 wavelengths over about a 15 nm range. A commercial Burleigh WA1500 wavemeter was used to measure the wavelengths with relative accuracy of 2e-7. In addition, a Tropel spectrum analyzer was used as a frequency drifting monitor with a resolution of 3 MHz to make sure the laser frequency is stable during the calibration process for each wavelength. A digital 4 mega-pixel camera was used to capture the interferogram frames.

Twelve temporal phase shifting frames were captured at each wavelength and used in a least square fitting (LSF) algorithm to determine the phase for each pixel on the retro reflector surface [13]. After wrapped phase was extracted, a reliability-guided phase unwrapping algorithm was applied on the data to get unwrapped phase information. To reduce the phase measurement error, statistical data analysis methods were used. First, we select reference pixels with high modulation value trying to reducing the photon noise. Pixels with modulation depth less than a certain threshold were removed as background noise; mean value *µ* and standard deviation *σ* of the remaining pixels are computed; those reference pixels must have modulation depth falling into the range (*µ-σ, µ + 2σ*) for all 14 wavelengths. Second, apply a LSF algorithm to valid reference pixel phase map generating a LSF plane on the tilted wavefronts to remove outliers assuming reflecting retro reflector surface was perfectly flat (residual errors are negligible). Finally, the phase value of the center pixel of the LSF plane chosen from remaining valid phase map and the corresponding phase difference of given pixels were calculated. Since most fringe area was used to calculate this phase difference, it is a high signal-to-noise ratio result that is not significantly influenced by poor quality pixels.

The array difference OPDs centered on retro reflectors were then calculated from the FT peak finding algorithm based on the phase and wavelength information, and used in subsequent calculations as the reference height values [5, 12]. Following the above calibration procedure, 4 step fiducial heights are calibrated with approximated height differences of 0.33 mm, 1.6 mm, 8 mm, and 40 mm. Calibration relative measurement uncertainty of 1.2e-6 for the largest height difference was achieved in a workshop environment.

#### 3.3 Experimental results and uncertainty analysis

The height differential between steps for a calibrated GSG 4-inch Z-AXIS stainless steel gage block GX1, shown in Fig. 6 , was measured in a ShaPix system. With 0.1 nm uncertainty of the commanded laser wavelengths, the system could only measure up to 5 mm height differentials with micron level accuracy. By applying MHTIT with calibrated fiducial OPDs, the 4-inch gage block was successfully measured. The experimental result is shown in Table 2 .

Where, the average distance between two separated surfaces among 100 × 100 pixels area are given as mean value and STD represents the standard deviation of 10000 independent measurements. As shown in Table 2, a large range of height measurement is achieved successfully by applying MHTIT with sub-micron accuracy. The system measurement range is increased from 5 mm to over 100 mm without requiring accurate wavelength information.

There are many factors that add uncertainty to our measurement such as temperature variation along optical paths, uncertainty in the reference heights calibration and phase measurement error. A detailed analysis is currently underway and will be the topic of a future publication. For this paper, preliminary uncertainty estimates for several important parameters have been carried out for 4-inch step height measurement with 40 mm reference fiducial height shown in Table 3 .

## 4 Conclusions

A multiple height-transfer interferometric technique has been developed that increases the absolute distance measurement capability with a tunable laser. By making use of multiple accurately calibrated reference heights, this technique relaxes the requirement of knowing accurate wavelength information for multiple wavelength interferometry while maintaining its advantages. Synthetic wavelengths used in classical metrology system are instead replaced by the ratio of reference heights and corresponding phase differences. This new technique reduces the sensitivity to certain types of noises, allowing it to be more suitable for full surface measurements with large height difference in industrial environments. This improved extension for multiple wavelength interferometry can benefit precision manufacturing leading to improving product quality and reducing warranty cost.

## Acknowledgments

This work was supported by the National Institute of Standards and Technology Advanced Technology Program “High Definition Metrology and Processes: 2 Micron Manufacturing” contract number 70NANB7H7041.

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