## 1.

## Introduction

The increased demand for light, thin, and flexible electronic devices has caused the flexible electronics industry to grow; examples of such devices are flexible displays,^{1}^{,}^{2} electronic skin,^{3} flexible luminance,^{4}^{,}^{5} flexible solar cell,^{6}^{,}^{7} and biologic sensor.^{8}^{,}^{9} Because flexible substrates on which thin films are fabricated are affected by the stress associated with the fabrication process, the measurement and analysis of thin-film stress are crucial for fabricating and developing flexible electronic devices. Furthermore, in flexible substrates coated with a noncrystalline thin film, the thin-film stress can cause large deformations, and, therefore, conventional testing techniques, such as optical interferometry^{10}11.12.^{–}^{13} and x-ray diffraction method,^{14}^{,}^{15} cannot be used. The moiré method involves a large measurement depth, high stability, and a simple and economical optical configuration, and, therefore, it is considered appropriate for the stress measurement of thin films on flexible substrates. Lee et al.^{16} and Huang and Lo^{17} proposed the methods for measuring the stress in thin films on flexible substrates; the methods involve using the shadow moiré method and analyzing the interval between the moiré fringes. The resolution and accuracy of these methods are not high because only the maximum and minimum of the moiré fringes are measured for determining the fringe contours, and, therefore, the resolution and accuracy can be strongly affected by the uneven spatial distribution of the light intensity. Accordingly, we propose a stress measurement system for thin films on flexible substrates. An expanded and collimated laser light is passed through a linear projection grating to form a self-image of the grating on a flexible test substrate, and the resultant deformed fringes are obtained on a reference grating to generate moiré fringes that are recorded by a CMOS camera. When the projection grating is moved with a constant velocity in the grating plane, every pixel of the CMOS camera records a series of signals that mimic a heterodyne interferometric signal. Therefore, the phase of the signals can be extracted in a manner identical to that of obtaining the phase of heterodyne interferometric signals. The phases of the optimized sinusoidal curves can be calculated using a least-squares sine fitting algorithm. The phase distribution and resultant surface profile of the uncoated and coated flexible substrates can be reconstructed by employing phase unwrapping and a derived equation. Subsequently, polynomial curve fitting is used to determine the curvature radii of the uncoated and coated flexible substrates, and these radii are used in the corrected Stoney formula for obtaining the thin-film stress. This method has the advantages of high stability and high resolution because of the use of the projection moiré method and heterodyne interferometry.

## 2.

## Principle

Figure 1 shows the optical configuration in the proposed method. For convenience, the observation axis of the CMOS camera is considered to be the $z$-axis, and the $y$-axis is directed perpendicular to the plane of the paper. A laser beam of wavelength $\lambda $ is passed through a beam expander (BE) for expanding and collimating the beam, and then it impinges on a projection grating ${\mathrm{G}}_{1}$ at an angle $\theta $ to form a self-image of ${\mathrm{G}}_{1}$ on the flexible test substrate. The self-image distance ${\mathrm{Z}}_{1}$ can be expressed as^{18}

^{19}

## (2)

$$I(x,y)=\frac{1}{4}\{1+\mathrm{cos}[\frac{2\pi}{p}x+\mathrm{\Psi}(x)+{\phi}_{1}]+\mathrm{cos}(\frac{2\pi}{p}x+{\phi}_{2})\phantom{\rule{0ex}{0ex}}+\frac{1}{2}\mathrm{cos}[\frac{4\pi}{p}x+\mathrm{\Psi}(x)+{\phi}_{1}+{\phi}_{2}]+\frac{1}{2}\mathrm{cos}[\mathrm{\Psi}(x,y)+{\phi}_{1}-{\phi}_{2}]\},$$In Eq. (2), the second, third, and fourth terms represent the harmonic noise, and the final term describes the desired moiré fringes formed on the surface of the flexible substrate. Hence, the moiré fringes can be obtained by filtering the harmonic noise, and the intensity of the moiré fringe image can be written as

## (4)

$${I}^{\prime}(x,y)={I}_{0}(x,y)+\gamma (x,y)\mathrm{cos}[\mathrm{\Psi}(x)+{\varphi}_{1}-{\varphi}_{2}],$$## (6)

$$S(x)=\frac{p}{2\pi \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta}\mathrm{\Psi}(x).$$To determine $\mathrm{\Psi}(x)$, Eq. (5) can be rewritten as

## (7)

$${I}^{\prime}(x,y)=A\text{\hspace{0.17em}}\mathrm{cos}(2\pi ft)+B\text{\hspace{0.17em}}\mathrm{sin}(2\pi ft)+C,$$To obtain the curvature radius of the substrate, the cross-sectional curve (Fig. 2) of the reconstructed surface profile in the $x$-direction and passing through the image center was obtained by using a polynomial fitting method. Let the center of the curve be set as (${x}_{0},{h}_{0}$), and let (${x}_{1},{h}_{1}$), (${x}_{2},{h}_{2}$), and (${x}_{3},{h}_{3}$) be three points on the curve. The three circular equations can then be expressed as

where $R$ is the curvature radius. Let ${x}_{1}-{x}_{2}=L$, ${h}_{2}-{h}_{1}={\delta}_{1}$, ${x}_{2}-{x}_{3}=L$, and ${h}_{3}-{h}_{2}={\delta}_{2}$, where $L$ is an arbitrary value. The curvature radius can then be derived as## (12)

$$R=\sqrt{{\left[\frac{({\delta}_{1}+{\delta}_{2})({\delta}_{1}{\delta}_{2}+{L}^{2})}{2\text{\hspace{0.17em}\hspace{0.17em}}L({\delta}_{1}-{\delta}_{2})}\right]}^{2}+{\left[\frac{(2\text{\hspace{0.17em}\hspace{0.17em}}{L}^{2}+{\delta}_{1}^{2}+{\delta}_{2}^{2})}{2({\delta}_{1}-{\delta}_{2})}\right]}^{2}}.$$^{16}

## (13)

$${\sigma}_{f}=\frac{{({Y}_{s}{t}_{s}^{2}-{Y}_{f}{t}_{f}^{2})}^{2}+4{Y}_{s}{Y}_{f}{t}_{s}{t}_{f}{({t}_{s}+{t}_{f})}^{2}}{6(1+\nu ){Y}_{s}{Y}_{f}{t}_{s}{t}_{f}({t}_{s}+{t}_{f})}\frac{{Y}_{f}^{*}}{(1+\frac{{Y}_{f}^{*}{t}_{f}}{{Y}_{s}^{*}{t}_{s}})}(\frac{1}{R}-\frac{1}{{R}_{0}}),$$## 3.

## Experimental Results and Discussion

To determine the validity of the proposed method, it was applied to a polyimide (PI)-coated flexible substrate. The surface of the PI-coated substrate was coated with a 100-nm thick indium tin oxide (ITO) thin film. The relative parameters of the PI substrate and ITO thin film are shown in Table 1. The experimental setup included a 473-nm diode laser, two linear gratings with a pitch of 0.2822 mm, an imaging lens with a focal length of 200 mm, a motorized translation stage [Sigma Koki/SGSP(MS)26-100] with a resolution of $0.05\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ for generating heterodyne moiré signals with a frequency ($f$) of 1 Hz ($v=0.2822\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}/\mathrm{s}$), and a CMOS camera (Basler/A504k) with an 8-bit gray level and a resolution of $1280\times 1024$. The frame rate of the CMOS camera (${f}_{s}$) was 15 fps, the exposure time ($a$) was 66 ms, and the total time ($T$) taken to record heterodyne moiré signals at different time points was 1 s. Every recorded moiré image was filtered using a $3\times 1$ window through two-dimensional median filtering for eliminating the harmonic noise in the moiré fringes.^{20} Figures 3 and 4 show the experimental results. Figures 3(a) and 4(a) show the moiré patterns on the sample with the uncoated and coated PI substrates. Figures 3(b) and 4(b) show the reconstructed surface profile of the PI substrate before and after thin-film coating. The cross-sectional curve of the reconstructed surface profile in the $x$-direction and passing through the image center is shown. Moreover, the fitting curve of the depicted cross-sectional curve can be obtained through polynomial curve fitting, as shown in Figs. 3(c) and 4(c). The parameters ${R}_{0}$ and $R$ of the uncoated and coated substrates can be calculated as 182.7 and 51.3 cm by using Eq. (12). By substituting these values and the parameters in Table 1 into Eq. (13), the stress in the thin film on the PI substrate can be calculated as 43.8 MPa.

## Table 1

Parameters of the PI substrate and ITO thin film.21,22

PI substrate | ITO thin film | |
---|---|---|

Thickness, $t$ | ${t}_{s}=75\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ | ${t}_{f}=100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ |

Elastic coefficient, $E$ | ${E}_{s}=2.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{GPa}$ | ${E}_{f}=116\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{GPa}$ |

Poisson ratio, $\nu $ | ${\nu}_{s}=0.34$ | ${\nu}_{f}=0.35$ |

According to Eq. (12), the error in the curvature radius in the proposed method can be expressed as

## (14)

$$\mathrm{\Delta}R=\left|\frac{\partial R}{\partial {\delta}_{1}}\right|\mathrm{\Delta}{\delta}_{1}+\left|\frac{\partial R}{\partial {\delta}_{2}}\right|\mathrm{\Delta}{\delta}_{2},$$## (15)

$$\mathrm{\Delta}R=2\left|\frac{\partial R}{\partial {\delta}_{1}}\right|\mathrm{\Delta}{\delta}_{1}=2\left|\frac{1}{2R}\frac{\partial {R}^{2}}{\partial {\delta}_{1}}\right|\mathrm{\Delta}{\delta}_{1}.$$The error in the curvature radius can be calculated as follows by using Eq. (15):

## (16)

$$\mathrm{\Delta}R=\frac{1}{R}[\frac{A{\delta}_{1}}{{C}^{2}}-\frac{{A}^{2}}{{C}^{3}}+\frac{B{D}^{2}}{2\text{\hspace{0.17em}\hspace{0.17em}}{L}^{2}{C}^{2}}+\frac{{B}^{2}D}{2\text{\hspace{0.17em}\hspace{0.17em}}{L}^{2}{C}^{2}}-\frac{{B}^{2}{D}^{2}}{2\text{\hspace{0.17em}\hspace{0.17em}}{L}^{2}{C}^{3}}]\mathrm{\Delta}{\delta}_{1},$$## (21)

$$\mathrm{\Delta}{\delta}_{1}=\left|\frac{\partial {\delta}_{1}}{\partial p}\mathrm{\Delta}p\right|+\left|\frac{\partial {\delta}_{1}}{\partial \beta}\mathrm{\Delta}\beta \right|+\left|\frac{\partial {\delta}_{1}}{\partial {\theta}_{1}}\mathrm{\Delta}{\theta}_{1}\right|+\left|\frac{\partial {\delta}_{1}}{\partial {\theta}_{2}}\mathrm{\Delta}{\theta}_{2}\right|,$$^{23}Considering the related experimental conditions and the visibility of the moiré fringes with a value of 0.3, the phase error $\mathrm{\Delta}\theta $ can be evaluated with a value of 0.32 deg. Substituting the values of $\mathrm{\Delta}p,\mathrm{\Delta}\beta $, and $\mathrm{\Delta}\theta $, the experimental conditions, and the results into Eqs. (16)–(21), yielded a $\mathrm{\Delta}R$ estimate of $5.81\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$.

Furthermore, the error in the thin-film stress $\mathrm{\Delta}{\sigma}_{f}$ originating from $\mathrm{\Delta}R$ can be derived as

## (22)

$$\mathrm{\Delta}{\sigma}_{f}=\left|\frac{\partial {\sigma}_{f}}{\partial R}\right|\mathrm{\Delta}R+\left|\frac{\partial {\sigma}_{f}}{\partial {R}_{0}}\right|\mathrm{\Delta}{R}_{0}\phantom{\rule{0ex}{0ex}}=\frac{{({Y}_{s}{t}_{s}^{2}-{Y}_{f}{t}_{f}^{2})}^{2}+4{Y}_{s}{Y}_{f}{t}_{s}{t}_{f}{({t}_{s}+{t}_{f})}^{2}}{6(1+v){Y}_{s}{Y}_{f}{t}_{s}{t}_{f}({t}_{s}+{t}_{f})}\frac{{Y}_{f}^{*}}{(1+\frac{{Y}_{f}^{*}{t}_{f}}{{Y}_{s}^{*}{t}_{s}})}\xb7[\left|\frac{1}{{R}^{2}}\right|\mathrm{\Delta}R+\left|\frac{1}{{R}_{0}^{2}}\right|\mathrm{\Delta}{R}_{0}],$$## 4.

## Conclusion

This paper proposes a stress measurement system for thin films on flexible substrates by using a projection moiré method and heterodyne interferometry. The phase of the optimized heterodyne moiré signal is determined using a least-squares sine fitting algorithm, and the surface profile of the flexible test substrate is then obtained. The thin-film stress is obtained by representing the cross-sectional curve of the substrate by using a polynomial fitting method, estimating the resultant curvature radii of the uncoated and coated substrates, and using these two radii in the corrected Stoney formula. This method offers the advantages of high accuracy, high stability, high resolution, and high capacity for substrates with high flexibility and a large measurement depth.

## Acknowledgments

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract Nos. MOST 103-2221-E-035-031 and MOST 104-2221-E-035-062.

## References

## Biography

**Kun-Huang Chen** received his BS degree from the Physics Department of Chung Yuan Christian University, Taiwan, in 2000 and his PhD from the Institute of Electro-Optical Engineering of National Chiao Tung University, Taiwan, in 2004. In 2004, he joined the faculty of Feng Chia University, where he is currently a professor with the Department of Electrical Engineering. His current research activities are optical metrology and optical sensors.

**Jing-Heng Chen** received his BS degree from the Physics Department of Tunghai University, Taiwan, in 1997 and his MS and PhD degrees from the Institute of Electro-Optical Engineering, National Chiao Tung University, Taiwan, in 1999 and 2004, respectively. In 2004, he joined the faculty of Feng Chia University, where he is currently a professor with the Department of Photonics. His current research interests are optical testing and holography.

**Hua-Ken Tseng** received his BS and MS degrees from the Department of Electrical Engineering of Feng Chia University, Taiwan, in 2012 and 2014, respectively. Currently, he is working at Taiwan Semiconductor Manufacturing Co., Ltd. His current research interests are biosensors and optical testing.

**Wei-Yao Chang** received his MS degree from the Department of Electrical Engineering of Feng Chia University, Taiwan, in 2009 and his PhD from the Institute of Electro-Optical Engineering of National Chiao Tung University, Taiwan, in 2015. Currently, he is working in Chroma Ate Inc. His current research interests are biosensors and optical testing.