## Abstract

Abstract: A 3D SEM Moiré Method (SMM) is proposed in this investigation for the first time for 3D shape measurement with nano-scale sensitivity. The geometric model of the 3D SMM has been theoretically established, combining the stereovision technology in an SEM with the existing principles of in-plane SMM. The Virtual Projection Fringe (VPF) generated under different conditions has been analyzed for 3D reconstructions. Two typical applications have been used to experimentally validate the theoretical model. Experimental results, with the height measurement sensitivity less than 10nm, agree well with the theoretical model we proposed. The uncertainty analysis for the method has also been performed by other auxiliary measurements.

© 2013 OSA

## 1. Introduction

Moiré measurement methods, including the geometric Moiré and the Moiré interferometry, have been widely used in macroscopical measurement [1–3].Due to the formation mechanism of Moiré fringes, large field of view to be observed is feasible when the measurement is conducted, considered to be the most attractive characteristic of this age old method. Despite the distinct measurement principles of the above mentioned two methods, one similarity of the two could be found, namely the disability in the measurement at micro-/nano-scales. To overcome this shortcoming, with the wide use of advanced microscopes with ultrahigh resolution such as a Scanning Electron Microscopy (SEM) and a Scanning Probe Microscopy (SPM)and the recent technique of nano-scale grating manufacturing, micro-/nano-Moiré methods were developed [4–9].Based on the imaging manners (mostly the scanning manner) of these microscopes, it is generally recognized that the Moiré pattern formed in electron microscopes results from the geometric interference between the electronic scanning lines and the gratings (or grids) on the surface of tested objects, as shown in Fig. 1. SEM Moiré Method(SMM), developed from the electron Moiré method [10–12], has been investigated by many researchers during the past 20 years, being efficient in measurement for mechanical properties [7,11,13–17] and in inversion for objects with periodic structure [16,18]. However, deformation as well as the surface appearance of actual micro-/nano-objects is mostly three-dimensional, which has been usually simplified into two-dimensional measurements. No article has been found in studying 3D measurement with SMM. This is because the complexity of the imaging mechanism of an SEM and the variety of the factors influencing on the Moiré formation to some extent limit the development of3D SMM. Of extraordinary interest is the study by Yasuhiko Arai *et al*. [19] who proposed a 3D measurement method in an SEM, by machining a micro grating through which the electrons can pass to form the shadow Moiré when electrons emitted from the pole piece of an SEM and scattered from the specimen pass through the grating. Nevertheless, it is difficult to operate in an SEM for the procedures, not only because of the requirement for the special gratings but also of the grating positioning manipulations.

In this study, a novel method used in 3D measurement of micro-/nano-objects with 3D SMM is proposed. The 3D geometric model is established by combing the stereophotography technology of the SEM with traditional in-plane SMM. The Virtual Projection Fringes (VPF) under different conditions is analyzed. Two examples are adopted to experimentally validate the proposed method.

## 2. Theoretical model of the 3D SMM

#### 2.1. Basic principles

Although the imaging mechanism of an SEM is quite different from that of an optical system, the pinhole imaging model is used to approximately describe the imaging system in an SEM. It is comprehensible that the electrons beam is compared to the light beam, on which the SEM stereophotography technology has been mainly based. The signal electrons (such as secondary electrons and backscattered electrons) are collected by a detector, recognized as a key device for imaging. It may be easier to understand that the imaging plane is pictured beneath the sample plane, as pictorialized in Fig. 2(a) and 2(b). Hence, the 3D model of the SEM stereophotography can be established based on the conventional optical model geometrically. Due to this analogy, images on the imaging plane vary with the tilting angle change of the sample plane. Capturing two images at different angles, the acquired image pair can be processed to reconstruct the 3D appearance of the measured object. The well-established geometric model of an SEM can be described mathematically in Eq. (1) which gives the relationship between the height of the sample surface and the projective 2D parallax shift (shown in the SEM monitor) [20]. Accordingly, by acquiring the 2D parallax shift value of the sample surface projected onto the imaging plane, the height (surface shape) of the object is obtained [21].

Inspired by the above principle, grid/grating methods for in-plane deformation measurement can be extended into 3D measurement [22].Images of Objects with Gratings on the Surface (OGSs) under the SEM at different tilt angles are captured and processed to calculate the surface profile value of the OGS, i.e. the 3D shape. Nevertheless, the method from Ref [22]. has an intrinsic shortcoming. At large magnifications, the area that can be observed in the field of view is limited to a relatively small part of the OGS. However, this limitation can be overcome by means of Moiré methods, which can enhance the field of view almost by one order in dimension compared to that achieved by means of the grid/grating methods under the same resolution. Considering the feasibility of 3D measurement in an SEM based on the grid/grating processing technology, SMM is investigated here for its application in 3D shape measurement.

As previously mentioned, the fundamentals, experimental validations of the in-plane SMM have been fully discussed by many other researchers [9–18]. Here, only some indispensable knowledge, concerning the research in this study, is given. The theoretical model and formulas of traditional geometric Moiré method are available for SMM. The in-plane displacement can be conveniently calculated with expressions of geometric Moiré method.

andwhere$u$and$v$are the displacement while${N}_{x}$and${N}_{y}$are the Moiré fringe order number in the$x$and$y$direction, respectively;${p}_{r}$is the spatial frequency of the reference grating.#### 2.2. Virtual projection fringes in an SEM

For the in-plane SMM, the scanning lines of the SEM are recognized as the reference gratings. Regarding the scanning lines of the SEM as stationary lines (here the term *stationary *implies the frequency invariance of the reference grating), the Moiré fringes, if there is any, change their pattern styles when the sample stage moves vertically (either out-of-plane translation or tilt). This variation of the fringe patterns simply results from the rigid body displacement of the specimen, causing the shortening or the lengthening of the projection image of the OGSs, instead of the real deformation, which can be defined as Virtual Projection Fringes (VPF).As the definition suggests, the appeared Moiré fringes in the SMM may be as a consequence partly of the real deformation, partly of the stage movement. Accordingly, to define the Moiré Fringes is of great significance. The following two cases are discussed in detail.

### A. Out-of-plane translation

Sample stage translating slightly up and down may result in remarkable variations in the Moiré Fringes style. As depicted in Fig. 3(a), compared to the initial location of the sample stage, upper and lower location of the sample may influence the imaging in the SEM. Specifically, when the sample stage translates upward with *Δd*, the edge part of the scanning area would be lost in the field of view. In the meantime, the specimen grating becomes sparser. Hence it can be regarded as a virtual stretching deformation. Similarly, when the sample stage translates downward with *Δd*, the specimen grating becomes denser, which is defined as the virtual compression deformation, depicted in Fig. 3(b).Nevertheless, the working distance couples with other parameters of the SEM such as magnification and scanning angle. Besides, tiny adjustment and the self-regulation for the working distance always exist in the SEM when operators attempt to bring the observed sample into focus. Therefore, it is fairly difficult to precisely control the working distance as we are willing to. Fortunately, the change of the working distance has little influence on the generation of VPF, compared to the other factor which is discussed in the following section.

### B. Tilting the stage

Tilting the sample stage can be regarded as a main factor that leads to generate the VPF without any actual deformation. This kind of VPF is more significant, compared with the case we mentioned in section A, for the 3D analysis in an SEM, since the whole SEM stereophotography principle is based on the variation of the SEM images under the tilting manipulation. Figure 4is the schematic for the generation of the VPF by tilting the sample stage. Simulation, as shown in Fig. 4(a), illustrated the formation of the VPF. It is obvious that the projection of the observed object onto the imaging plane has always been compressed whatever the tilt direction is clockwise or anticlockwise. Consequently, unlike the translation case, the tilting manipulation generating the VPF always results in the compressive virtual strain. The above conclusion has great significance, not only for the analysis in the VPF style, but also for the following discussion of the 3D reconstructions.

To quantify the influence of the tilt manipulation on the generation of the VPF, some simple but important geometric deductions are given as follows. The tilting angle$\phi $is a variable which determines the virtual strain, as shown in Fig. 4(b). With the geometric relations, it can be deduced as:

where$L$is the original width of the sample in the field of view (before tilting), $\Delta L$is the virtual compressive displacement in the field of view. More generally analyzing, each point on the sample stage has a corresponding parallax shift when the stage tilts and the parallax shift can be expressed according to Eq. (5).The parallax shift signifies the projection change onto the imaging plane in the equivalent optical system, which can be clearly displayed in the SEM monitor. According to the analysis in 2.1, $\Delta L$can be expressed in the form of the Moiré fringes as:Here, it is notable that the direction$x$in Eq. (6) is perpendicular to the spindle of the SEM sample stage. Obviously, the VPF order number caused by tilting the sample stage is a one-to-one correspondence with the tilting angle, which is demonstrated in the following chapters with designed experiments. In 2.1, we have referred the classical geometric 3D model of the SEM that applies the parallax shift to calculate the height of a random point on the surface of the tested object, achieving the 3D reconstruction. This method, if neglecting other subordinate influencing factors, can be promoted further into a new technique combined with the SMM, namely the 3D SMM.

#### 2.3. The model of the 3DSMM and the height measurement sensitivity

For the realization of the shape measurement in an SEM, which differs from the deformation measurement, the parallax shift in the monitor is the main measured value according to Eq. (1) or Eq. (2). In other words, any 3D measurement based on optics methods is ultimately processed with2D calculation. The logic of the 3D SMM method can be described as follows: Tilting the sample stage generates the VPF, representing the in-plane parallax shift. To process the VPF, the parallax shift can be obtained, which is next substituted into the 3D model Eq. (Eq. (1) or Eq. (2)). The 3D reconstruction is consequently achieved. Combining the Moiré expression (Eq. (3) and Eq. (4)) with Eq. (1) and Eq. (2) respectively, the height expressions can be obtained as:

The parameters in Eq. (7) and Eq. (8) areas previously defined. Furthermore, according to Eq. (8), the relation between the magnification of the SEM and the spatial frequency of the reference gratings is linear, which has been calibrated in the experiments, expressed as:

Let$M=350$,$N=1$,$\phi ={10}^{\circ}$and ${p}_{r}=830nm$, the case that a 1200lines/mm grating has been selected, the height value$h$calculated with Eq. (8) is approximately 6.8$nm$and the height resolution is 1.1$nm$when a 3000lines/mm grating is used. Hence we know that the theoretical sensitivity of the method we proposed reaches to the scale less than10$nm$. Theoretically, the increase of the spatial frequency of the specimen grating can further enhance the sensitivity of the method. This enhancement depends heavily on the development of advanced micro fabrication technologies for high frequency gratings. Although, logically, it is reasonable for the above derivation, the calculation from Eq. (7) to Eq. (9) should be under one important assumption that the tilting manipulation must be eucentric tilting, which can be satisfied in most new-style SEMs [23]. Another key point is that, like any other Moiré method, the formation of Moiré fringes is conditioned geometrically. When the tilting angle is not as large as it should be, there will not be any VPFs. Some discussion, therefore, has been requisite, given in the following section.

Assume that the minimum parallax shift$\Delta u$is 1 pixel, (only larger than this minimum value, will there be the Moiré fringes), expressed as$\Delta {u}_{\mathrm{min}}=L(1-\mathrm{cos}\phi )={p}_{r}=1pixel$. Let the lateral resolution of the SEM monitor be 1024 pixels, and obviously the minimum level of the VPF is that the distance between two fringes is 1024 pixels. The minimum tilting angle is obtained by the following Eq.:

Therefore, the tilting angle should be larger than 2.53°to form the least Moiré fringes. Similarly, there also exists a maximal tilting angle, below which the generated VPF can be used in the measurement. According to Ref [22], an empirical value of 10 degree is commonly adopted.

#### 2.4. The order number of the fringes

The acquisition for the displacement field, according to Eq. (3) and Eq. (4) relates to the order number of the Moiré fringes using any Moiré methods. The Moiré fringes, essentially the isolines of displacement, denotes the larger displacement of the varying (either real deformation or the virtual deformation) OGSs with the increase of the fringe order. In the 3D SMM model of the SEM, defining the fringe order facilitates the 3D reconstruction. Considering the tilting manipulations, the order of the fringe farthest from the eucentric tilting spindle is selected as$N=0$. Distinct from the traditional geometric Moiré methods, the definition proposed here for the fringe order number does not represent the position with zero displacement physically. The fringe with the order number$N=0$is defined as a datum fringe. The displacement values of other fringes have been calculated by contrasting with this datum fringe.

Besides, it is notable that the displacement value obtained before tilting the stage is generated by the mismatch between the reference grating (scanning lines) and the sample grating. This means that the displacement is not the consequent of a true movement or deformation. This initial displacement is usually defined as a carrier wave displacement used in the following calculation for surface height of a tested object.

## 3. Experimental analysis

To fully demonstrate the method, two cases were analyzed experimentally, in which different materials and loads were applied. The optimized nanoimprint lithography [24] was applied to fabricate the gratings on the tested object surface. Before any experimental validation, the calibration for the spatial frequency of the reference grating (the scanning lines of the SEM) has been performed, shown in Fig. 5.All the experiments were performed using an FEI Quanta 450SEM.

The first task is to validate the VPF theory we proposed in the above sections. Standard gratings fabricated using photolithography technique with the spatial frequencies of 3000lines/mm and 1200lines/mm were selected in the experiment. Figure 6shows the VPF formed by tilting the sample stage with the tilting angle of 6 degree. In Figs. 6(a) and 6(b), two standard gratings with various spatial frequencies are tested. The distance variation between the two fringes results from the tilting, which is coupled with the tilting angle according to Eq. (5) and Eq. (6). By calculating the displacement variation, the tilting angle can be obtained,$L(\mathrm{cos}{\phi}_{2}-\mathrm{cos}{\phi}_{1})=\left|L-L\text{'}\right|=\Delta L$, where${\phi}_{1}$and${\phi}_{2}$are the angles relative to the horizontal plane at the first position (before the tilt) and the second one (after the tilt) of the sample stage, respectively. The length$L$and${L}^{\prime}$have both been marked in Fig. 6. By calculating with the above equations, the angle has been determined to be, approximately, the value of6.3°, which is consistent with the actual value (6°).Achieving the 3D reconstruction depends not only on the theory model, but also on the calculational software packages such as MATLAB to process the Moiré fringe images and to solve the equations. The fringe processing methodology has been seen in a series of articles [25–29], which is not discussed here in detail.

To validate the theory model we proposed in the previous sections, some applications are given in the following sections.

#### 3.1. The 3D shape measurement for a polydimethylsiloxane (PDMS)sheet(sample 1)

A PDMS sheet with nano-scale gratings on the surface was prepared in our experiments, combing the optimized nanoimprint lithography. The PDMS sheet was solidified in a 2000lines/mm nanoimprint stamper. Then the PDMS sheet plate was compressed by a mechanical clamp to generate a 3D deformation. Figure 7 shows the PDMS sample. It can be noticed that there appears the Moiré fringes on the surface of the PDMS sheet at the tilt angle of 0°.This is due to the in-plane deformation, which can be neglected when 3D shape reconstruction becomes the main purpose of the measurement. This is, to some extent, difficult to comprehend physically. Actually, initial fringes have always been arising even though the reference grating (scanning lines) fits the specimen grating very well. It is acceptable that the initial fringes exist, since the variation of the fringes parallax shift between the two different tilting angles is the decisive factor to the 3D reconstruction. Consequently, the initial 2D Moiré fringes can be recognized as the carrier wave displacement field. To apply the previously mentioned 3D SMM model in the experiment, some calculations are indispensable using MATLAB software package. The 3D reconstruction with the calculation was depicted in Fig. 8.

#### 3.2. Deformation of the grating sample (sample 2)

Gratings fabricated with lithography may deform when they were in humid environments for a long time, since the residual developing liquid reacts with metal films. The 3D shape of the deformed gratings used in our experiment is the blister shape, as shown in Figs. 9(a) and 9(b). The scanning Moiré fringes have been shown in Figs. 9(c) and 9(d). Different tilting directions may generate different styles of fringes. The displacement distribution can be obtained at angle 1 and angle 2, respectively, as shown in Fig. 9(e).

## 4. Results discussion

As presented in the previous paragraph, two typical experiments were conducted. Both of the samples have been measured using the 3D SMM, by which the 3D shape can be obtained. To experimentally validate the feasibility and the correctness as well as analyze the error of the results, a Laser Scanning Confocal Microscope (LSCM) and a 3D digital microscopy were used. The LSCM and the 3D microscopy used in our experiments are made by Lasertec Corporation and Keyence Corporation, respectively. The measurements of the samples have been listed in Table1 and Table2, depicted in Fig. 10 and Fig. 11. The measured profile of the samples used in the first experiment (sample 1) and the second experiment (sample 2) are marked in Fig. 8 and Fig. 9(a) with section line$A-B$ (the purple dotted line).

According to the results, the measurement obtained by 3D SMM coincides, in the shape trend, with that by the LSCM and the 3D digital microscopy. It can also be determined quantitatively that the average relative uncertainty of 3D SMM, based on the result data, to the results by LSCM and 3D digital microscopy are less than, 7.5% and 16.7%, respectively. The accuracy of the LSCM measurement is higher than that of the 3D digital microscopy. Hence the former relative uncertainty, namely 7.5%, is more receivable.

It is noted that the experimental uncertainty is influenced by various factors. The most significant one is that the scanning lines of the SEM are not geometrically perfect. The non-linearity of the scanning lines induces image distortion and the minor change of the frequency of the reference grating, which may limit the accuracy and the extensive application of the proposed 3D SMM. The distortion elimination in SMM will play an important role in improvement of the measurement accuracy, and the scanning lines/rasters calibration can be performed according to distortion elimination principle of SEM images reported in [30, 31].

## 5. Conclusion

We have, for the first time, proposed a 3D shape measurement method for micro- or nano-objects. The technology combines the SEM stereophotography technique with the in-plane SEM Moiré method. By analyzing the virtual projection fringes formed in the tilting manipulation of the sample stage, the 3D geometrical model of the SEM Moiré method can be established. Experiments for different samples validate the feasibility of the method in 3D shape measurement. Commercial measuring devices are used in our study to obtain data compared with the results by the proposed method. It has been proved that the relative uncertainty of the method was no more than 7.5%.

## Acknowledgments

The authors are grateful to the financial support from the National Basic Research Program of China (“973” Project) (Grant No.2010CB631005, 2011CB606105), the National Natural Science Foundation of China (Grant Nos. 11232008, 91216301, 11227801, 11172151, 11072033), Tsinghua University Initiative Scientific Research Program, Program for New Century Excellent Talents in University (Grant No.NCET-12-0036), Natural Science Foundation of Beijing, China (Grant No. 3122027).

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