## Abstract

In this paper, a digital speckle correlation method for coefficient of thermal expansion (CTE) measurement of film is developed, in which CTE is the intrinsic parameter and direct variable. Deformation pattern governed by the CTE and temperature is used to affine transform the image captured after the film is heated. If the values of CTE are properly chosen, the image after affine transformation will have a highest similarity to the original image. This turns CTE measurement into a purely numerical search of an optimal trial CTE. Results of CTEs from this method and conventional DIC methods are compared with the actual CTE, showing an improved accuracy.

© 2011 OSA

## 1. Introduction

Thin films have widespread applications in microelectronics and micro-electro mechanical systems (MEMS). The rapid growth of the MEMS industry has introduced a need for the characterization of thin film properties at all temperatures encountered during fabrication and application of the devices. Thermal expansion is an important mechanical behavior in MEMS. There are several problems that arise from the thermal expansion effect; for example, the mismatch of thermal expansion between the thin films and the substrate may lead to residual stresses in the thin films [1]. Therefore damage or deformation of the micromachined structures may occur. On the other hand, the thermal expansion effect can be exploited to drive the microactuator [2,3]. In order to design micromachined devices properly, it is necessary to characterize the coefficient of thermal expansion for thin film materials [4,5].

An innovative technique for determining CTE of thin films using multilayered cantilever beams has been developed. The technique is based on the thermally induced curvature of the multilayer that results from the difference in thermal expansion coefficients of the layers. The curvature is measured at temperatures of up to 850 °C using an optical curvature measurement system [6]. Determination of property results from comparing the beam response to a numerical model for curvature of multilayers [7]. Full-field optical techniques, such as phase-shifting interferometry [8] and Electronic Speckle Pattern Interferometry (ESPI) [9] have been applied to determine the thermal expansion of thin film under thermal loading. For example, tests conducted by Dudescu et al. [9] successfully determined the CTEs of unidirectional and bidirectional carbon fiber laminates using phase-shifting ESPI.

This fringe based interferometric techniques usually offer high sensitivity and accuracy in displacement and/or deformation measurement [10]. However, since they must use coherent light, interferometric techniques except shearography tend to be prone to environmental disturbances during measurement. This has limited their application to on-site measurement in an engineering environment. Moreover, a subsequent fringe pattern analysis technique (e.g., phase unwrapping) is required to extract the thermal expansion from the obtained fringe patterns, which further increases the complexity of measurement.

Digital speckle/image correlation (DPC/DIC) is a well-known non-interferometric technique, having the capability of measuring the displacement field of an object with an ordinary light source such as white light [11]. Without the need to form an interferometric fringe pattern, the optical set-up of DIC is rather simple. In 2009, Pan etal [12]measured the CTE of pure copper film and PI/SiO2 composite films using DIC. Generally, in order to measure CTE, the strain components accompanying the thermal expansion will be computed by differentiating the displacement fields, but numerical differentiation will amplify the noise contained in the computed displacements. In document [12], in order to obtain the average thermal strain of the test film surface and alleviate the influence of noise, linear planes were used to approximate the computed displacement fields of the test film. However, relatively little of deformation information was used during CTE measurement in the above document [12], a large part of deformation information was not used.

This paper presents a novel Deformation-Pattern-based Digital Speckle Correlation (DPDSC) method for CTE measurement, in which the whole image information before and after deformation is used to CTE analysis in order to improve the measurement precision. Direct displacement measurement and strain calculation is no longer necessary. Thus, the noise will be alleviated and restricted greatly. Instead of using displacement components as the basic variables in conventional digital image correlation, CTE are taken as the direct variables. Each selected CTE corresponds to a specific deformation pattern under different temperature. This deformation pattern as a whole is used to inversely affine transform the digital image captured after heating. Once the CTE are properly chosen, the digital image after inverse affine transformation will have the highest similarity to the digital image taken before the object is heated, or in other words, will recover to the original image. This turns the CTE measurement issue into a pure numerical computational process, i.e., to search for a CTE that will maximise the correlation between the original image and the inverse affine transformed image after heating. This can be implemented through an optimisation procedure. Since the direct variables in such a numerical optimisation process are the CTE, the conventional CTE measurement procedure in photomechanics which involves the complex interpretation of a displacement field is no longer required. All of the calculating process is programmed in software. This will facilitate the development of a compact system for CTE measurement of film with ease of use for the operator.

## 2. Principle of DPDSC for CTE Measurement

The central idea of the DPDSC is to use intrinsic parameters of a measurement task as direct variables in the computation of the correlation coefficient between two images [13]. In this investigation, the CTE *α* is selected as intrinsic parameters, and displacement field is governed by the CTE at a special temperature by Eq. (1).

The DPDSC for CTE measurement developed in this paper is suitable in the elastic range. Conventional approach to CTE measurement is to measure the strain under different temperature expressed in Eq. (1) by measuring the displacement components at several special locations. Considering the inevitable error in each individual measurement, our approach does not follow this normal procedure. We know that the basic concept of DIC is a displacement search process which is based on the criterion of correlation between two images of an object corresponding to before and after its deformation. When a subset of pixels in one image is searched out in the other image, i.e., the correlation between the two subset images reaches the maximum, their coordinate difference is the displacement measured. This concept of correlation search is adopted in our new approach. The difference is that the basic variable in the new search process is the CTE, not the displacement components. In the new approach, the task is to search for intrinsic parameters (CTE). Once they have the CTE values corresponding to the actual mechanical behaviour and are used to inverse affine transform the digital image corresponding to the film heated, the new image will recover to the original image, i.e., the correlation will be a maximum. This can be implemented by an optimisation process which is purely a numerical computation issue.

Therefore in essence, the new approach has taken full advantage of a known deformation pattern, which is related to a specific mechanism, in the current case expanding by thermal loading. Different CTEs correspond to different deformation patterns under the same temperature. By looking at the deformation pattern over the entire region of interest in an image, we can make use of the full information from the images, not just isolated displacements at limited locations or special points.

## 3. Algorithms of DPDSC for CTE measurement

Let (*x, y*) be the coordinate of the optical imaging system, and *F* and *G* the digital images corresponding to before and after the sample is heated, i.e.

*M*and

*N*are the numbers of horizontal and vertical pixels of a digital image, respectively. Based on the deformed image

*G*, an inverse affine transformation is defined as:

*u*and

_{t}*v*are the displacement components by thermal loading, (

_{t}*x”, y”*) are the new coordinate system. In this way, the intrinsic parameters (CTE) are introduced in the inverse affine transform.

After the above inverse affine transformation, a new image *G’* can be formed:

The correlation between this new image *G’* and the original image *F* is calculated as:

*C*|≤1, and

*C*= 1 only when

*F*=

*G’*.

It is obvious that the correlation is an explicit function of the pixel intensities of two images. Displacement field or intrinsic parameters are introduced implicitly by the use of an affine transformation of the image coordinate system. Generally, the correlation is finally a function of the intrinsic parameters (${\alpha}_{x}$,${\alpha}_{y}$) in an implicit sense, i.e.

Different intrinsic parameters CTE (${\alpha}_{x}$,${\alpha}_{y}$) will result in a different correlation value *C*. If the trial intrinsic parameters CTE are equal to the actual CTE, image *G’* will be exactly transformed back to the original image *F*. This can be implemented by an optimization process.

In reality, there are some more issues need to be addressed when using DPDSC to measure CTE. Firstly, in-plane rigid body displacement inevitably occurs during image acquisition before and after the testing sample is heated. It consists of rigid body shift, *u _{0}* and

*v*, along the x and y axes respectively, and in-plane rotation

_{0}*ω*. They are the result of either the accumulated deformation when heating, or the environmental disturbance during image acquisition. Secondly, out-of-plane displacement

*w*often exists too. It is the result of either the movement of the digital camera during image acquisition, or the accumulated deformation of the object. This out-of-plane displacement has a huge effect on the accuracy of strain measurement, because it will result in the change in image magnification during image acquisition. Telecentric lens can be used to resolve this problem. An object-space telecentric lens has a prominent characteristic that its magnification is a constant within a special working distance. The size and shape of an image formed by such a lens is independent of the object's distance or position in the field of view. Object-space telecentric lens creates images of the same size for objects at any distance (within depth of field (DOF)) and has constant angle of view across the entire field of view (FOV). Thus, Out-of-plane rigid body displacement can be ignored if introducing object-space telecentric lens in front of a CCD [14].

_{0}To ensure the new approach works effectively, in-plane rigid body displacement (*u _{0}*,

*v*,

_{0}*ω*) should be taken into account in the correlation search process. In this investigation, in order to check the effect of using telecentric lens, out-of-plane rigid body displacement is also calculated. Therefore, the affine transformation becomes:

During the correlation search process, those trial rigid body displacements and CTE that maximise the correlation are regarded as the final actual value for the measurement.

In practice, the whole search process is implemented by a two stage approach. The first stage involves only the detection of the bulk of in-plane rigid body displacement field, i.e., only integer pixel rigid body shifts are considered. Once the bulk of rigid body displacement is obtained, it is compensated for by simply shifting the image *G* backwards accordingly. This paves the way for the next stage of a fine search process, focusing on the minute scale differences between the two images. These minute differences consist of both the remainder (fractional pixel) of the rigid body displacement field and the result of CTE, deformation, and/or environmental disturbances. Once this second stage is done, CTE measurement is completed.

## 4. Optimisation scheme for the correlation search process

Since there are six variables in the fine search process, the key to success is to design a proper optimization scheme such that the possibility of being trapped in local maxima is small while the computational cost of finding the global maxima of correlation is not too high to be carried out with a conventional computer. In order to minimise the possibility of a false optimisation result, a bounded exhaust search scheme [13] is adopted. The bound for in-plane displacement (*u*, *v*) is within ±0.1 pixel, ±1% for in-plane rotation *ω* and ±2% for CTE (CTE). The search steps for each variable vary, which follows a coarse-to-fine search procedure, such that the total number of correlation search is not too large (1,000 to 20,000 times) for a conventional computer to carry out the whole process within a reasonable period of time [13].

## 5. Validation tests for CTE measurement

#### (A) Test material and film sample

Two types of polymer films were used. One is Polyimide Film, with a thickness of 50 micron, which is usually used as a dielectric substrate for flexible printed circuits and high density interconnects. The other is Polyetherimide film with a thickness of 250 Micron, which has a broad range of electrical and electronic applications. The two types of film material were cut into specimens with size of 10×10mm^{2}. The surface of the sample was sprayed with black and white high-temperature paint in order to form speckle-like images. Three specimens were fabricated for each type of material. Figure 1
is a speckled image of one of the film specimens’ surface.

#### (B) Experimental set-up

The experimental set-up consists of a high temperature furnace and a digital image capturing system. The high temperature furnace can heat the sample from room temperature to 300 °C using the electric resistance wire imbedded under the bottom of the furnace. The temperature within the furnace is measured and controlled by a separate temperature controller. The upper part of the furnace is covered by a quartz glass plate, which serves for thermal insulation and reducing the optical distortion. When the intended temperature is achieved, the corresponding sample surface image is recorded by the digital image capturing system for subsequent DPDSC and DIC analysis. The image capturing system consists of a CMOS camera with resolution of 1600 x 1200 pixels at 256 gray levels and an object-space telecentric lens with a working distance of 120mm mounted in front of the camera.

#### (C) Experimental procedure

Measurement of CTE of a film sample by the DPDSC method follows the following procedures. At first, the test film sample with artificial speckles was placed horizontally on a polytetrafluoroethylene plate in the high temperature furnace without any restraint. A white light source was used to illuminate the sample surface during thermal expansion. The camera axis should be placed perpendicular to the test sample surface. In order to decline the influence of the residual stresses in the films during manufacture on CTE calculation, only samples after their first exposure to elevated temperatures were used.

After the first exposure, the original temperature was set as 30 °C and the corresponding images were recorded as reference images. After that, the temperature within the furnace was elevated to 100°C and then 150°C, two images under each temperature were captured as deformed images. These images were analyzed by the DPDSC technique to extract the CTE directly. The test process was repeated three times on one sample. When the test on one sample was finished, this process was repeated on another two samples of the same type. Each type of material was tested nine times with three samples at a special temperature difference (from 30°C to 100°C and from 100°C to 150°C), and three times for each sample to obtain the average value at the fixed temperature difference.

It is noted that hot airflow around the furnace may cause the density and the refractive index of the air non-uniform, and finally result in abnormal distortion in local area of the images captured. Thus, the influence of the distorted images used on the CTE calculated cannot be ignored. So it is of very importance to restrain the image distortion to get a good result during heating. Three strategies were applied to resolve this problem effectively in this investigation. (a)At each temperature, the temperature is maintained for at least 30minutes to make sure that the nearby hot airflow is stable and the sample is subjected to a uniform thermal loading. (b) No actions were taken that would affect the airflow, such as moving quickly or, opening or closing the door quickly, etc. when testing and capturing images. (c) When the airflow was stable, a pair of images were captured at specific temperature at a fixed time interval (about 5 minutes), calculating the CTE simply using the image pairs. If the calculated virtual CTE was nearly zero or the correlation coefficient was larger than 0.990, the two images were applied to the final CTE calculation, otherwise, they were recaptured when the hot airflow was stable. After this assessment of image distortion, the images captured were used for the formal CTE calculation.

#### (D) Experimental results and discussion

After the heating tests, the CTE calculating procedure is as follows. (1) A pair of images, one before and the other after heating, are loaded into the CTE program edited on the basis of the DPDSC principle. The image scale factor is calculated automatically by measuring the size of width or length of the sample in pixel which is then converted to the actual dimension. (2) To select a region of interest (ROI). This is done by adjusting the radius of a circle of selection or side length of a square of selection. The pixels of the ROI are used in the subsequent affine transformation to calculate the CTE. (3) The button on the menu is pressed to start the correlation optimisation process to calculate directly the CTE. It takes several minutes to complete the whole computational process, depending on the total number of pixels of the ROI selected and the optimisation strategy used.

The result of calculation is the in-plane rigid body displacement (*u*, *v*) and rotation *ω*, out-of-plane rigid body displacement *w*, and the CTE components (${\alpha}_{x},{\alpha}_{y}$). In this investigation, the film materials used are isotropic one, so${\alpha}_{x}={\alpha}_{y}=\alpha $.

Extensive validation measurements have been made to test the accuracy, sensitivity, reliability and repeatability of the proposed method. By selecting images taken at different temperatures, we can form various image pairs which correspond to various temperature differences. These image pairs were loaded into the measurement system, and CTEs were calculated and compared to their actual values, taken from the materials’ supplier’s website [15,16]. In this way, the sensitivity and repeatability of the measurement system can be evaluated. In addition, the CTE also were measured using the conventional DIC method, that is, firstly to measure the displacement, secondly to calculate the strain and then to calculate the CTE. The accuracy of this DPDSC method developed for CTE measured can be discussed by comparing the CTE values using DPDSC, DIC methods and the actual values. Figure 2 is the calculating window with a measured CTE result using one of Polyetherimide specimens at the temperature ranges of 30-100°C. From Fig. 2, it can be seen that the in-plane rigid body displacement calculated is −0.247 pixel in the horizontal direction and 0.237 pixel in the vertical direction. The rigid body rotation is −0.070 deg(clockwise direction). When the absolute value of the out-of-plane rigid body displacement is less than 0.01pixel, indicating that the out-of-plane rigid body displacement is no longer necessary to be concerned, in this case, it is directly set to 0 in this program for unambiguity. In fact, in most of the tests, the out-of-plane rigid body displacement is usually very close to zero, which is attributed to the telecentric lens used here. So, in this investigation, the influence of the out-of-plane rigid body displacement on CTE doesn’t need to be taken into account. The total expansion is 3.6359/1000 for the temperature difference of 70°C, so the CTE is 51.94 ppm/°C.

Figure 3(a) and (b) are the displacement field U and V measured by conventional DIC using the same speckled image pairs as Fig. 2. It can be seen that rigid body motion is included besides thermal expansion deformation. After eliminating the in-plane rigid body displacement using the calculated value of DPDSC above(−0.247 pixel in the U field and 0.237 pixel in the V field), a resultant displacement vector is formed and shown in Fig. 3(c). Figure 3(c) shows a uniform thermal expansion deformation, but rigid body rotation still present. After eliminating the rigid body rotation, an exact uniform thermal expansion deformation field is formed and shown in Fig. 3(d). From these images, it can be concluded that these measured values of in-plane rigid body motion using DPDSC are correct.

After eliminating the in-plane rigid body motion, the resultant displacement field measured using DIC can be applied to calculate strain and CTE according to the common process [12].

The tests were finished with six samples according to above process, the obtained CTE values by DPDSC and DIC for Polyetherimide and Polyimide Film specimens are listed in Table 1 , and compared with the actual CTEs. The actual value of CTE of Polyetherimide film is 52 ppm/°C according to the test standard IPC-TM-650 [15], while for Polyimide Film, it is 17 at the temperature range of 30-100°C,and 32 at 100-150°C according to the test standard ASTM D-696-91 [16]. The mean of the measured values and standard deviation are also listed.

For comparison convenience, the CTE of measured, mean and actual values are also shown in Fig. 4 (a) and (b) . From these figures, it can be seen that the measured values using DPDSC are closer to the actual values than those using DIC. The absolute error of DPDSC for CTE is less than 2%. That is, the DPDSC method for CTE has higher precision than that of conventional DIC.

In general, the higher the temperature are, the larger of the standard deviation is. DPDSC has a smaller standard deviation. In this investigation, the standard deviation of DPDSC for CTE is less than 1.34 ppm/°C, while DIC is larger than 1.59 ppm/°C, so DPDSC has a better repeatability than DIC. It seems that the result from DPDSC approach is better than conventional DIC, which demonstrates the benefit of taking into account more of the available information during CTE calculations.

The displacement field is always the direct focus in conventional DIC for CTE measurement. DPDSC for CTE, however, has shifted that focus from displacement to CTE. In addition, the current developed approach has turned a CTE measurement task to an optimisation process based on the criterion of correlation. This is equivalent to looking at the overall deformation pattern associated with a specific thermal expansion behaviour that is represented by a set of intrinsic parameters CTE, rather than making isolated displacement measurements at limited locations within the field of view in DIC. The intrinsic parameter CTE governs the deformation pattern of an object under investigation, and thus is the prime concern of a measurement task. By employing an optimisation strategy, the task of CTE measurement can be turned to a correlation search process. That is, to find a set of intrinsic parameters that, after being used to perform an affine transformation, will best recover the deformed digital image to its original one. Apparently, the direct output of the proposed approach is no longer the displacement field but the intrinsic parameters CTE. Once the intrinsic parameters are found, the associated deformation patterns are already determined.

## 6. Conclusions

A novel DPDSC method for CTE measurement is proposed. It uses the intrinsic parameters CTE as the direct variables in a correlation computation. Each set of such intrinsic parameters and parameters of rigid body motion corresponds to a specific deformation pattern, which as a whole is used to affine transform the digital image captured after the object is deformed by heating. This turns the CTE measurement issue into a pure numerical computation process, i.e., to search for a set of intrinsic parameters that will maximise the correlation between the inverse affine transformed image and the actual original image.

The proposed approach is implemented through an optimisation procedure. Since the direct variables in such a numerical optimisation process are the intrinsic parameters CTE, the conventional strain measurement procedure in CTE which involves the complex displacement and strain calculation is no longer required. Validation tests have proved the viability of the new approach, and the accuracy and reproducibility of CTE measurement has been improved compared to the conventional DIC method.

## Acknowledgements

This work was supported by the Marie Curie International Incoming Fellowship (Project No.221623) of the European Commission, the National Natural Science Foundation of China under grant No. 11072033.

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