## Abstract

A special configuration of white-light scanning interferometer is described for measuring the absolute air gap thickness between two planar plates brought into close proximity. The measured gap is not located in any interference arm of the interferometer, but acts as an amplitude-and-phase modulator of the light source. Compared with the common white-light interferometer our approach avoids the influence of the chromatic dispersion of the planar plates on the gap thickness quantification. It covers a large measurement range of from approximate contact to tens of microns with a high resolution of 0.1 nm. Detailed analytical models are presented and signal-processing algorithms based on convolution and correlation techniques are developed. Practical measurements are carried out and the experimental results match well with the analysis and simulation. Short-time and long-time repeatabilities are both tested to prove the high performance of our method.

©2009 Optical Society of America

## 1. Introduction

A small-scale air gap formed between two planar plates plays an important role in a lot of research and industrial fields. In optical manufacturing the surface quality of the optical plate is usually characterized by observing the interference fringes produced by the air gap between the surveyed surface and a standard one [1]. In environmental science air gap membrane distillation is well developed to purify the water from the process industry [2]. The well-controlled narrow gaps are also deployed for filtering the cells or molecules in biotechnology, pharmaceutical and food industries [3].

Many novel applications require the gap thickness to be precisely measured and controlled. In the application of rheo-optics, a gap with the thickness of tens of microns was formed by two parallel plates for deriving the viscous force constant for polymer flow [4]. To determine the dielectric properties of genomic and proteomic structures, a 300nm gap capacitor was created by timed etching of the capacitor spacer in the dielectric spectroscopy [5]. In supramolecular nano-stamping, a novel replication technique for DNA microarrays, the gap between the master and substrate needs to be reduced to a small value on the order of 10nm, i.e. close to the situation of ideal contact, to make the single-stranded DNA features duplicated [6]. The nm-µm gap is also helpful in researching the phenomena in the near-field physics, such as radiative heat transfer and quantum-dynamic forces [7].

Many techniques have been successfully developed in the literature for absolute gap thickness quantification. The application of optical geometry combined with X-ray technique was demonstrated in [8] and the measurement by position-sensitive detectors (PSD) was introduced in [9]. In electrochemical machining ultrasound technique was employed to characterize the gap between the tool (the cathode) and workpiece (the anode) [10]. Keyence Co. developed a laser confocal sensor for characterizing multiple-layer structures, which is also suited for gap thickness measurement [11]. In silicon wafer manufacturing the gap between the mask and substrate was scaled and controlled by usage of the diffraction phenomena [12, 13]. Laser interferometer, usually limited to the displacement measurement, was extended to survey the absolute gap thickness by Flanders and Lyszczarz. They used an obliquely incident He-Ne laser to illuminate the gap and obtained its thickness by calculating the spatial frequency of the interference pattern [14]. Johnson group described a white-light phase-shifting interferometry, where the wavelength in the white light source was selected sequentially by a commercial monochromator to illuminate the gap and the gap thickness was calculated according to the phase shift of each wavelength [15]. White-light scanning interferometer was proposed by FOGALE nanotech Inc. to measure the gap thickness (surface spacing) among optical lens, based on the principle that the white-light interference could be only observed when the optical path length in the reference arm matched the positions of the object surfaces in the measuring arm [16].

While the above-mentioned approaches are widely employed, they have various limitations for the micro-scale or nano-scale gap thickness quantification. Methods using optical geometry, ultrasound technique, and PSD are limited by their micron-level resolution and accuracy. The approach with diffraction phenomena can measure the gap as micro as near to contact, but a grating or checkerboard pattern has to be printed on one of the plates forming the gap, which is not convenient for many cases. Flanders and Lyszczarz’s method adopting laser interferometry cannot reach the thickness range less than 20 µm; furthermore, it requires several structural parameters such as incident angle to be measured precisely. The white-light phase-shifting and spectral interferometry can overcome the dispersion of the gap cavity walls; however, their smallest measured thickness, limited by their principles, is usually a quarter of the coherence length of the light source; furthermore, expensive instruments like commercial high-resolution monochromator or spectrometer are usually needed to separate the wavelength from the light source or from the intensity acquired by the detector [17–22].

The white-light scanning interferometer possesses the high resolution and accuracy, and its structure is not as complicated as the white-light phase-shifting or spectral interferometry. However, its applications in gap thickness measurement is limited by the dispersion of the cover plate of the gap, i. e. the air cavity’s wall through which the white light transfers, especially when the cover plate is much thicker than the gap. To overcome this disadvantage a compensate plate with identical thickness and refractive index must be inserted into the reference arm of the interferometer; otherwise, the residual dispersion will influence significantly or even destroy the measurement. This specific requirement is hard to meet in many practical applications, for it needs the thickness and refractive index of the cover plate to be precisely measured simultaneously.

The goal of this paper is to develop a special white-light scanning interferometer which can characterize the nano-scale gap thickness, and more importantly, can overcome the chromatic dispersion of the plates including the gap, making the compensate plate not necessary, nor the quantification of the thickness and refractive index of the cover plate. A special configuration of white-light interferometer was introduced to measure the thickness of transparent films, in which the measured film was not located in any interference arm, but acted as a modulator of the light source [23]. Unfortunately, the advantage of this configuration was not totally disclosed in that application for the measurement was still limited by the dispersion of the film material and thus the minimum measured thickness was tens of microns, making it not attract much attention or find any further applications. Here, we extend this configuration into gap thickness measurement, design a new signal processing algorithm for it and find further advantages. With this progress it not only can be used for air gap thickness measurement, but also provides a new approach for the quantification of the covered (or buried) features in various micro devices.

The rest of this paper is organized as follows. In Section 2 we introduce the experimental configuration and theoretical concept of our method, and derive the fundamental relations. In Section 3, the signal processing algorithm is described and simulated, including the usage of convolution/correlation techniques. Practical measurements are presented in Section 4, where the experimental results are compared with the theoretical simulation for proving the validity of our approach and the short-time and long-time repeatabilities are both examined. In Section 5 the precision analysis and future work are discussed. Section 6 concludes the whole paper.

## 2. Methodology

#### 2.1 Experimental configuration

The experimental apparatus used in this approach is shown schematically in Fig. 1 (a). White light from a quartz halogen lamp, covering the spectrum from 400nm to 1100nm, is collected into a light guide and collimated by an aperture and a lens. The parallel beam is reflected by a beamsplitter and incident on the gap formed between the two optical plates; here multiple-beam interference takes place. The gap-modulated light is then guided into a Michelson interferometer, with a fixed mirror placed in the reference arm and a PZT (Piezoelectric Transducer) driven mirror in the moving arm. The beams reflected from the two arms are recombined and form the interference intensity, which is measured with a photodetector and transferred via a coaxial cable to a computer for data acquisition, processing, and storage.

As compared with common white-light scanning interferometers, the main difference in this configuration is that the measured gap does not act as one interference arm, but a modulator of the light source by the multiple-beam interference. The consequent advantages will be explained in detail in the following section.

For the plates surrounding the air gap are aligned parallel and the transmitted light beam is collimated, there will be an even light intensity distribution *I* on the photodetector plane, without any interference pattern. When the moving mirror is driven by the PZT so that the length difference between two interference arms Δ is scanned, the continous variation of the *I* will be acquired by the photodetector to form an *I*—Δ curve, from which the gap thickness will be eventually derived.

#### 2.2 Basic principle

The multiple beam interference occurring at the gap is illustrated in Fig. 1 (b), where the air gap is surrounded by planar plate 1, the cover plate, and plate 2. Let *E _{input}* be the complex amplitude of the input beam, and

*E*,

_{1}*E*,

_{2}*E*, and

_{3}*E*the sequentially reflected beams from four different surfaces. A series of

_{4}*r*and

*t*denote the reflection and transmission amplitude coefficients at the corresponding surfaces, and their values can be easily calculated by Fresnel equations.

*h*,

*h*and

_{1}*h*are the thickness of the gap and the two plates, and

_{2}*n*,

_{a}*n*and

_{1}*n*their refractive indexes. According to the practical applications of micro gaps

_{2}*h*is much smaller than

*h*and

_{1}*h*. The incident angle is represented by α; and the refractive angles in two plates by β and γ.

_{2}The complex amplitude of each reflected beam is:

$${E}_{2}={E}_{\mathrm{input}}{t}_{1}{r}_{2}{t}_{1}\text{'}{e}^{i{\phi}_{1}};$$

$${E}_{3}={E}_{\mathrm{input}}{t}_{1}{t}_{2}{r}_{3}{t}_{2}\text{'}{t}_{1}\text{'}{e}^{i\left(\delta +{\phi}_{1}\right)};$$

$${E}_{4}={E}_{\mathrm{input}}{t}_{1}{t}_{2}{t}_{3}{r}_{4}{t}_{3}\text{'}{t}_{2}\text{'}{{t}_{1}\text{'}e}^{i\left(\delta +{\phi}_{1}+{\phi}_{2}\right)}$$

where the phase change δ, φ_{1} and φ_{2} are:

Neglecting the reflections in higher orders due to their little intensity, the sum of the complex amplitude of the reflected beams is written as:

$$={E}_{\mathrm{input}}({r}_{1}+{t}_{1}{r}_{2}{t}_{1}\text{'}{e}^{i{\phi}_{1}}+{t}_{1}{t}_{2}{r}_{3}{t}_{2}\text{'}{t}_{1}\text{'}{e}^{i\left(\delta +{\phi}_{1}\right)}+{t}_{1}{t}_{2}{t}_{3}{r}_{4}{t}_{3}\text{'}{t}_{2}\text{'}{{t}_{1}e}^{i\left(\delta +{\phi}_{1}+{\phi}_{2}\right)})$$

Apparently, both the amplitude and phase of the input light are modulated by the gap. In case of vertical incidence, α=β=γ=0.

According to the characterization of Michelson interferometer, the half transmission of each beamsplitter and the wavelength range covered by the white light source, the final intensity received by the photodetector can be described by formula (4). In this formula *a*, *b*, *c*
*d* and *A* are positive constants, and *C _{1}* to

*C*are functions of Δ. The negative sign in

_{7}*C*,

_{2}*C*,

_{3}*C*, and

_{4}*C*is due to the 180 phase shift introduced by reflection at the interface from a low refractive index media to a high one. $I=\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}{I}_{\mathrm{gap}}\mathrm{cos}\left(1+\genfrac{}{}{0.1ex}{}{4\pi \Delta}{\lambda}\right)d\lambda =A+{C}_{1}+{C}_{2}+{C}_{3}+{C}_{4}+{C}_{5}+{C}_{6}+{C}_{7}$ with

_{7}$${C}_{2}=-\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}bc\phantom{\rule[-0ex]{.2em}{0ex}}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left({n}_{a}h+\Delta \right)}{\lambda}d\lambda -\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}bc\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left({n}_{a}h-\Delta \right)}{\lambda}d\lambda $$

$${C}_{3}=-\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{ab}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [{n}_{1}\left(\lambda \right){h}_{1}+\Delta ]}{\lambda}d\lambda -\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{ab}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left[{n}_{1}\right(\lambda \left){h}_{1}-\Delta \right]}{\lambda}d\lambda $$

$${C}_{4}=-\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{cd}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [{n}_{2}\left(\lambda \right){h}_{2}+\Delta ]}{\lambda}d\lambda -\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathit{cd}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left[{n}_{2}\right(\lambda \left){h}_{2}-\Delta \right]}{\lambda}d\lambda $$

$${C}_{5}=\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{ac}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [({n}_{1}\left(\lambda \right){h}_{1}+{n}_{a}h+\Delta ]}{\lambda}d\lambda +\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathit{ac}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left[{n}_{1}\left(\lambda \right){h}_{1}+{n}_{a}h-\Delta \right]}{\lambda}d\lambda $$

$${C}_{6}=\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{bd}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [({n}_{2}\left(\lambda \right){h}_{2}+{n}_{a}h+\Delta ]}{\lambda}d\lambda +\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathit{bd}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left[{n}_{2}\left(\lambda \right){h}_{2}+{n}_{a}h-\Delta \right]}{\lambda}d\lambda $$

$${C}_{7}=-\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{ad}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [({n}_{1}\left(\lambda \right){h}_{1}+{n}_{a}h+{n}_{2}\left(\lambda \right){h}_{2}+\Delta ]}{\lambda}d\lambda $$

$$-\genfrac{}{}{0.1ex}{}{1}{4}{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}\mathrm{ab}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi [({n}_{1}\left(\lambda \right){h}_{1}+{n}_{a}h+{n}_{2}\left(\lambda \right){h}_{2}-\Delta ]}{\lambda}d\lambda $$

The elements of *C _{1}* to

*C*denote different components in the

_{7}*I*—

*Δ*signal, which is acquired by the photodetector when the PZT drives the moving mirror continuously. A simulated

*I*—

*Δ*signal by Matlab is presented in Fig. 2, where each

*C*produces one or two local curves and each curve includes a sharp peak with its side lobes, as shown in Fig. 2 (a) and (b). For instance, for

*C*only as Δ equals zero all the cos(

_{1}*4*πΔ/λ) functions obtain the maximum 1 for every wavelength, so a peak will occur at the position of Δ=0, i.e. the position where the PZT translates to make the length of two interference arms equal. The same goes for

*cos*[

*4*π(Δ+

*n*)/λ] and

_{a}h*cos*[

*4*π(Δ-

*n*)/λ] in

_{a}h*C*, so that two symmetrical downward peaks can be observed at Δ=

_{2}*n*and Δ=-

_{a}h*n*, respectively. As a result, the spacing between the peaks in the local curves denoted by

_{a}h*C*and

_{1}*C*is exactly the value of

_{2}*n*, and therefore the gap thickness can be determined by measuring this spacing, which forms the basic principle of our method. In normal situation the air refractive index

_{a}h*n*can be considered a consistent of 1.00028 [24], whose variation can be negligible in our experimental environments.

_{a}#### 2.3 The influence from dispersion of the plates

(*a*) *For common white-light scanning interferometers*

In the common white-light scanning interferometer the measured gap is located in the measuring arm of the interferometer, and a PZT driven moving mirror in the reference arm. Now the intensity acquired by the photodetector will be:

$$+{\int}_{{\lambda}_{1}}^{{\lambda}_{2}}{M}_{4}\mathrm{cos}\genfrac{}{}{0.1ex}{}{4\pi \left[{n}_{1}\left(\lambda \right){h}_{1}+{n}_{a}h-\Delta \right]}{\lambda}d\lambda +.....$$

where *M _{1}* to

*M*are constant values; the third and fourth items in formula (5) result in two local curves in the

_{4}*I*—Δ signal, and the spacing between their peaks is expected to be a measure for the gap thickness. One practical case is simulated in Fig. 3, where a 20 µm air gap is included by a 200-µm-thick cover plate and another 300 µm plate and the cover plate is made of BK7 glass. The two local curves before and after consideration of the dispersion of the cover plate are presented in Fig. 3 (a) and (b) for comparison. In (a)

*n*is given by a constant, the refractive index of BK7 glass for the central wavelength of the light source. In (b)

_{1}*n*is a function of λ due to the actual dispersion as in Fig. 3 (d), and now no certain Δ can make the cosine functions in formula (5) get simultaneously the maximum for all the wavelengths. As a result, the widths of the local curves are extended, amplitudes of the peaks are reduced and their positions deviate from the ideal ones. If the cover plate is thicker or the dispersion of the material is more significant, the negative influence of the dispersion will be more evident to make the gap thickness measurement impossible.

_{1}(*b*) *For our proposed method*

For our presented approach the variation of *n _{1}* and

*n*with λ will not cause any change to the values of

_{2}*C*and

_{1}*C*in Eq. 4, nor the peak spacing denoting the gap thickness which is completely governed by

_{2}*n*. Even with significant dispersion in the cover plate, the Central Curve and Side Curve are the same as in Fig. 2 (a) and (b). Therefore, the dispersion of the air cavity’s walls does not affect our measurement at all.

_{a}h## 3. Signal processing methodology

In micro gap thickness measurement, Δ will be scanned by the PZT in a limited range of tens of microns near Δ=0; therefore, only the three local curves represented by *C _{1}* and

*C*can be included in the practically obtained

_{2}*I*—Δ signal. Other local curves, too far away from the location of Δ=0, will not be considered in the following analysis. Now the three local curves generated by

*C*and

_{1}*C*are named by Central Curve and Side Curves, and the three peaks dominating the gap thickness measurement are Central Peak and Side Peaks, respectively.

_{2}#### 3.1 Signal processing for gap thickness larger than 10 µm

Gap thickness larger than 10 µm can be quantified directly by the spacing between the Central Peak and Side Peak. In practical measurements these peaks are evident enough to be indentified by the maximum or minimum value in the acquired *I*—Δ signal. (A spare algorithm by calculating the centroid of the local curve to obtain the peak position is also developed.)

Random noise, inevitable in practical experiments, may result in errors in determining the positions of the peaks. Assuming the light intensity incident on the gap *I _{input}* to be 1W, the random noise with a substantial amplitude of 4mW is simulated and added into the signal, as shown in Fig. 4 (a). The Central and Side Curves are zoomed in (b) and (c), where the noise makes the peak positions hard to indentify precisely.

Convolution calculation [25] is applied to address this problem. The simulated Central Peak is specially extracted as in Fig. 4 (d), and the convolution between this peak and the whole signal is calculated. This process, acting as a signal filtering, reduces dramatically the influence of the noise and makes the peak positions much easier to determine. The calculation result is shown in Fig. 4 (e), which almost forms the same curve as the ideal signal without any noise. From the peaks enlarged in (f) and (g), the gap thickness can be precisely achieved as 10 µm.

#### 3.2 Signal processing for gap thickness between 1µm and 10 µm

The signal simulated for a 2 µm gap is shown in Fig. 5 (a). Now the Central Curve and the Side Curves are close enough to combine partly with one another, and thus the side lobes are added to the peaks to make them deviate from original shapes and shift from ideal locations, especially for the two Side Peaks. In this case calculating the gap thickness only through peak spacing becomes difficult.

To overcome the combination of the Central and Side Curves, an algorithm is designed to subtract the Central Curve from the signal, to make only the side ones left. The concept is as follows. When the gap is very thick so that the spacing between the three local curves is quite large, the Side Curves will move out of the scanning range of the PZT. As a result, only the Central Curve exists in the scanned *I*—Δ signal and thus it can be recorded separately, as shown in Fig. 5 (b). In practical measurements this procedure only needs to be operated for one time before measuring the gaps, thereafter this data can be used as a reference and subtracted from the obtained signals for gaps thinner than 10 µm. The signal in Fig. 5 (a) is processed and the data after subtracting the Central Curve is shown in Fig. 6, in which the two Side Peaks return to their ideal locations, 2 µm and -2 µm. Now the gap thickness can be characterized accurately by calculating the half spacing between these two downward peaks. If heavy noise exists in the signal, the convolution calculation introduced in the former section can still be employed.

#### 3.3 Signal processing for gap thickness less than 1µm

For the gap thickness less than 1 µm the Central Curve is first of all subtracted from the *I*—Δ signal. But even after that the Side Peaks still deviate from their ideal locations because now even the two Side Curves are close enough to combine with one another. The simulated curve of a 400 nm gap after subtracting the Central Curve is illustrated in Fig. 7.

Signal matching technique, i.e. the correlation calculation between the acquired signal and the simulated ones in a database, is applied to process this case. The detailed process is as follows. In numerical simulations, we include two Side Curves and vary their distance, i.e. the spacing between two Side Peaks, from 1 nm to 1 µm to obtain their corresponding signals, forming a database for gap thickness from 1 nm to 1 µm. The practically obtained signal, after subtracting the Central Curve, will be compared with the signals in this database by correlation calculation, and the one matching the practical signal best shows the gap thickness.

To establish the database the Side Curve data must be obtained first. From Eq. 4 we know that the Central Curve and Side ones, produced by *C _{1}* and

*C*, have completely the same shape; the only difference exists in their amplitude and orientation (upward or downward). Therefore, if the acquired Central Curve in Fig. 5 (b) is placed upside down, it can be taken as the Side Curve by a ratio; and the influence of this ratio will be removed in the correlation calculation because correlation only cares about the shape of curves.

_{2}The *I*—Δ signal for a 400 nm gap with a 2mW peak-to-peak random noise is simulated in Fig. 8 (a), and the results of the correlation calculation between it and the curves in the database are illustrated in (b). The gap thickness is determined as 400 nm exactly, from which one can see that the correlation calculation helps finding the gap thickness from the database in a high accuracy, even for cases with heavy noise.

## 4. Experiments

#### 4. 1 Experimental setup

The experimental setup is shown in Fig. 9 (a), in which 1 is the light guide collecting the white light from the light source (Dolan-Jenner DC-950H); an aperture with selectable diameter is placed in front of the guide to form a point light source; 2 denotes a achromatic lens collimating the light, which also reduces dramatically the chromatic and spherical aberration; 3 is the second aperture dominating the illuminated area of the plates; 4 and 5 are cube beamsplitters with the reflecting surface orientated at 45° angle with the incident beam; 6 the fixed mirror; 7 and 8 are two planar glass plates with nominal flatness 1/20 wavelength; by using screw-adjusted kinematic mounts the gap included by the two plates can be adjusted to different thickness values for measurement. 9 is the high-speed, high-sensitivity and low-noise photodetector (SD112-42-11-231) from Advanced Photonix, Inc.

To implement the optical alignment of the system a He-Ne laser is employed as a reference. The laser spots reflected from the two surfaces of the gap, or from the reference and moving mirrors, are received at a screen far away, and the complete superposition of the spots indicates the parallelism of the air gap walls and the perpendicularity between the two interference mirrors. To guarantee the performance all the measurement system is set up on a vibration isolated optical table, and the environmental light is shielded to the fullest extent.

Fig. 9 (b) shows the moving mirror driven by a PZT, which is held in a flexure based mechanism manufactured by abrasive water jet machining technique. 1 represents the PZT from PI Company; 2 is the capacitance senor (ADE Tech Probe 2805) to measure on-line the displacement of PZT, providing a feedback signal to form the close loop control to assure the high accuracy and repeatability of the PZT moving. 3 is the collar holding the mirror, which is connected to 4, the flexure.

#### 4. 2 Experimental Data for gaps with different thickness

The method we propose for white light scanning interferometry requires the moving mirror to be scanned in a range of microns and at a sub-nanometer resolution. To this end, we use a piezoelectric scanner in tandem with a flexural mechanism and sensor for closed-loop feedback around the motion scanning stage. Flexural mechanisms driven by piezoelectric actuators are well-known for achieving sub-nanometer resolutions. Piezoelectric actuators with 0.1 nm resolution and 1 nm repeatability are commercially available [26]. While these benefits are in open-loop operation, many closed-loop control algorithms have been developed for improving robust stability and performance. The control aspects of achieving sub-nanometer resolution are beyond the scope of this paper. For these details, the reader is referred to results reported by us in [27, 28]. Closed-loop control for tracking 0.2nm steps of a PZT-driven flexural scanner stage with a capacitance probe is achieved in [28], and therein the details of the control algorithm used are provided. With low-pass filtering of capacitance probe output data, the resolution is demonstrated to be limited to 0.1nm by the filtered RMS-noise of the probe electronics.

In this work, we proceed with the same flexure-based PZT scanning stage used at a closed-loop position resolution of 0.1 nm, and this dictates the measurement resolution of our approach. The measurement range depends on the scanning length of the applied PZT, currently 16 µm; if larger range is needed an amplification mechanism can be selected, or an electromechanical actuator such as the voice coil motor can be selected.

The experimental signal acquired for a practical gap is shown in Fig. 10 (a-1), in which abscissa axis shows the PZT translation and axis of ordinates the intensity received by the photodetector. Based on our signal processing technique including the convolution calculation, the gap thickness is estimated as 10.1258 µm. For this case the PZT scanning range 16 µm cannot cover all the three local curves, so only the Central Curve and one Side Curve are included in the acquired curve.

In the practically obtained curve the difference between the intensity of the peak and of its side lobes appears not so significant, because the light source does not emit all the wavelengths from 400 nm to 1100 nm by an even ratio, and the same goes for the transmission of the lens and beam splitters, as well as the detection of the photodetector. Taking into account the practical parameters of these components, the theoretical curve for a 10.1258 µm gap is simulated by Matlab and illustrated in Fig. 10 (a-2), from which we can see the high similarity between the practical and simulated results. The correlation between these two signals is calculated as 0.957, verifying the validity of our approach.

By adjusting the fine screws in the kinematic mount and aligning the parallelism of the two plates, two thinner gaps are formed for quantification. Their acquired signals are shown in Fig. 10 (b-1) and (c-1), from which one can see evidently the gap turn thinner gradually. Now the Central Curve and Side ones are close enough to combine partly with each other, especially for the case in Fig. 10 (c-1), so the subtraction of the Central Curve introduced in Section 3.2 is applied, and the gap thicknesses are determined to be 6.3733 µm and 1.6514 µm. The theoretical signals for the two gap thicknesses are also simulated in Fig. 10 (b-2) and (c-2), and the correlation ratios between the experiments and simulations are 0.921 and 0.936, respectively.

Fig. 10 (d-1) shows the acquired signal for gap thickness less than 1 µm, for which both the subtraction of the Central Curve and signal matching technique are applied. The results of the correlation calculation between the acquired signal and the simulated ones in the database are illustrated in (d-2), from which the gap thickness is determined as 494.3 nm.

#### 4. 3 Repeatability of our experiments

To test the short-time repeatability of our approach an air gap with the estimated thickness about 500 nm is measured eight times within one minute. The results from the eight measurements are listed in the first column of Tab. 1 and the SD (Standard Deviation) of the measurements is calculated as 5.8 nm. The long-time repeatability is examined by quantifying the same gap in one hour with the time spacing of 5 minutes, as shown in the second column in Tab. 1, and the SD is determined as 7.3nm.

Another gap thickness about 5.88 µm is characterized in the same manner. The measurement results are also shown in Tab. 1, with the short-term and long-term SDs calculated to be 7.4 nm and 19.7 nm, respectively. The high repeatability of our approach makes it competent for the practical applications.

## 5. Discussion

The measurement errors mainly come from four sources. The first one is the influence from the environmental noise. Fortunately, in our approach the gap thickness is determined not directly by the intensity but by the position of the peaks, and convolution/correlation algorithm is developed to process the signal, so the influence of the noise has been dramatically reduced. The second source is the asymmetry of the acquired local curves. The Central Curve and the Side one are not perfectly symmetric about their own middle lines, especially the Central Curve; and on the left of the Central Peak there are more side lobes than on the right. This asymmetry, showing that the light transmitting in the system is not a perfect parallel beam, may induce error in identifying the peak positions. There are several possible sources causing this asymmetry. (1) The collimating lens does not remove all the chromatic and spherical aberrations from the incident beam, making it include convergent or divergent rays. (2) The fixed mirror or the moving one is bent when it is fixed in the mount, making it deviate from a planar surface. The accuracy error of the PZT translation, e.g. error in the repeatability and linearity, and the thermal expansion of the components are other two elements which can also introduce error into the eventual measurement result.

In theory we can characterize gap thickness as small as close to the resolution of our method; however, limited by the practical experimental conditions, e.g. the dust particle concentration in the laboratory and the actual flatness and cleanness of the two plates forming the gap, we did not reach this scale.

The laser interferometry may be a better approach if the gap thickness Δ is limited within half wavelength of the applied laser, for it can achieve sub-nanometer accuracy measurement. For example, for a He-Ne laser interferometer, λ/2 is 316.4 nm. When the Δ goes beyond this limit there will occur 2π phase ambiguity, i.e. the laser interferometer, when measuring the absolute gap thickness, can not identify Δ from Δ+*n* λ/2, (*n*=1, 2, 3…). The measurement range required in many applications is often much larger than hundreds of nanometers, but often tens or hundreds of microns. For these cases the laser interferometer is not suitable. Our approach can cover the gap thickness quantification from approximate contact to tens of microns; if larger range is needed, we just need to use an electromechanical actuator such as the voice coil motor to replace the PZT, with nothing else changed in the setup.

The current verification of our measurement accuracy and the repeatability is based on the high performance of the PZT translation and our data processing; the better approach is to use some reference metrology method, e.g. laser interferometer, or standard sample. Now we are combining our approach into a novel hot-embossing technique, to measure and control the gap between the tool and the sample. In this coming work the accuracy of our approach will be verified by using a laser interferometer, Model SP120 from SIOS Mebtechnik GmbH. This progress will be presented in the future.

In future work we will improve the performance of our interferometer by ameliorating the working environment and updating hardware in our experiments. Currently the gap thickness is adjusted by the three screws in the kinematic mount. A new flexure based mechanism is in development to move the two plates and align their orientation, to realize high-speed, high-resolution and on-line control for the gap thickness.

As part of future work we will apply this approach into the practical hot-embossing machine for manufacturing microfluidic devices, where the gap thickness at three different positions will be characterized on the gap surface simultaneously, for the real-time control of the parallelism of the tool and sample.

## 6. Conclusion

We successfully measured nano-scale gap thickness by a special configuration of white-light scanning interferometer, in which the measured gap acts as a modulator of the light source. It overcomes the shortage in the common white-light interferometer, avoids influence of the chromatic dispersion in the plates surrounding the gap, and provides a new approach for the characterizing the covered features such as steps in a variety of micro devices. In this implementation we do not employ any expensive instrument like commercial spectrometer or monochromator, whose prices range from several thousands to tens of thousands

## Acknowledgments

The authors would like to thank the Singapore-MIT Alliance Program for supporting this research. We are indebted to Dr Li Shiguang in Nanyang Technological University, who gave us much help in this work.

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