Research Article
Vol. 54, No. 17 / June 10 2015 / Applied Optics
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Direct deflection radius measurement of flexible PET substrates by using an optical interferometry JIONG-SHIUN HSU*
AND
PO-WEI LI
Department of Power Mechanical Engineering, National Formosa University, Yunlin 63208, Taiwan *Corresponding author: [email protected] Received 30 March 2015; revised 15 May 2015; accepted 17 May 2015; posted 18 May 2015 (Doc. ID 237157); published 8 June 2015
The deflection radius is essential in determining residual stress estimations in flexible electronics. However, the literature provides only indirect methods for obtaining a deflection radius. In this study, we present a measurement methodology for directly measuring the deflection radius of a polyethylene terephthalate (PET) substrate (a popular substrate of flexible electronics) by using an optical interferometer. A Twyman–Green optical interferometer was established and phase-shifting technology was used to increase the measurement resolution. Five PET substrates with known deflection radii were prepared to verify the measurement precision of the proposed measurement methodology. The results revealed that the error variance of our proposed measurement methodology is smaller than 3.5%. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.54.005469
1. INTRODUCTION In recent years, because of advances in manufacturing technology, flexible electronics have been gradually used in items such as displays, energy products, sensors, and illumination devices [1–3]. The concept of flexible electronics involves fabricating devices and circuits on a flexible substrate. Three candidates of substrates have been developed, namely those involving flexible transparent sheets (FTSs), metal foils, and ultrathin glass sheets [4–6]. Because the merits of an FTS include excellent transmittance, high resistance to impact, and roll-to-roll manufacturing, it has been characterized as the optimal substrate for use in future flexible electronics [7,8]. Although the FTS has the aforementioned advantages, its mechanical properties are easily influenced by environmental and loading conditions. Nonetheless, no free electron exists in its interior structure, thus it is a nonconductor of electricity. Typically, a transparent conductive oxide (TCO) layer is deposited on the FTS, enabling conductivity. For example, a layer of indium tin oxide is typically deposited on a polyethylene terephthalate (PET) substrate. However, FTSs are a ductile material whereas TCO is more brittle, which constitutes a considerable difference in mechanical properties. Consequently, severe residual stresses may be introduced within the TCO, causing cracking or a failure to degrade its transmittance and electrical properties. Therefore, the problem of residual stress on TCO–FTS structures has been increasingly critical in accelerating the commercialization of flexible electronics [9–12]. 1559-128X/15/175469-06$15/0$15.00 © 2015 Optical Society of America
Most studies on the residual stress of TCO–FTS structures use the Stoney formula [13] to estimate the residual stress of TCO film. Before the residual stress calculation, the radius of the TCO–FTS structure must be experimentally obtained to substitute it into the Stoney formula. Because the radius measurements in previous studies have been indirect [14–17], this study proposed a measurement methodology to directly obtain the radius of the PET substrate by using optical interferometry. The optical interferometer used in our proposed methodology is a Twyman– Green interferometer. First, we derived the relationship between the deflection radius of the PET substrate and optical phase difference. Five aluminum molds with known radii were manufactured; the PET substrates were bonded onto the surfaces of the molds, and their fringe patterns were observed through an optical interferometer. Phase-shifting technology was utilized to obtain a full-field phase map. Finally, a nonlinear regression approach was used to fit the optical phase data and obtain its deflection radius. 2. THEORIES A. Optical Interferometric Theory
A Twyman–Green interferometer was established; its optical arrangement is illustrated in Fig. 1. An object beam was created by placing a PET substrate in one optical path; a second beam without a PET substrate placement served as the reference beam. Let the electrical fields of the reference (E r ) and object (E o ) beams be
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thickness of the PET substrate. In addition, in triangle ABD, an approximate relation is made as follows [19]: t ; (6) d≈ cos θ2
Computer
CCD camera Collimated lens Spatial filter
where d is the optical path passing through the PET sheet. When all of the discussed relationships are applied, d can be further arranged as n2 t (7) d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 : n22 − n21 rt
Mirror
He-Ne laser
NPBS
Polarizer PZT Deflection PET Mirror
Fig. 1. Arrangement of the Twyman–Green interferometer.
E r E r1 e iwt−ϕr ;
(1)
E o E o1 e iwt−ϕr ;
(2)
where E r1 and E o1 are the electrical amplitude of reference and object beams, respectively; w is the circular frequency of the laser beam; t is time; and ϕr and ϕo are the phases of the reference and object beams, respectively. The intensity of the optical interferometric image (I ) is pffiffiffiffiffiffiffiffi (3) I I r I o 2 I r I o cosϕ; where I r E 2r is the intensity of the reference beam, I o E 2o is the intensity of the object beam, and ϕ ϕr − ϕo is the optical phase difference between the reference and object beams [18]. B. Theoretical Derivation of Deflection Radius
Figure 2 illustrates an optical beam passing through a deflection PET substrate, in which the deflection is bent into a circular arc with radius r at center O. By using Snell’s law, we obtain n1 sin θ1 n2 sin θ2 ;
(4)
where n1 and n2 are the refraction indices of the air and PET substrate, respectively; θ1 is the incident angle at point A; and θ2 is the refraction angle at point B. By using triangle ACO, we obtain y sin θ1 ; (5) rt where y is the length between an arbitrary point on the PET substrate, A, and point C; r is the deflection radius; and t is the
PET
1
A
l
E 2 2
t
B D
y
r
1
C
O
Fig. 2. Optical beam passing through a deflection PET substrate.
From the geometrical relationship in triangle ABE and the trigonometric function formula, the following relationship can be deduced: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t r t2 − y 2 n ty 2 q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l : (8) rt r t2 n2 − n2 y 2 2
1 rt
The optical path difference Δ before and after placement of the deflection PET substrate is " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 t n1 t r t2 − y 2 − Δ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rt n2 − n2 y 2 2
1 rt
# n21 ty 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − y 2 : r t2 n22 − n21 rt
(9)
Therefore, the optical phase difference ϕ before and after placement of the transparent film is " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4π n22 t n1 t r t2 − y2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ − λ rt n2 − n2 y 2 2
1 rt
# n21 ty2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − y 2 : r t2 n22 − n21 rt
(10)
Equation (10) shows that y is an independent variable, radius r is a constant, and the others are known. Finally, a nonlinearfitting approach can be adopted to optimally obtain r. C. Phase-Shifting Theory
Although the fringe patterns provided by the optical interferometer represent the quantitative distribution to be measured, the measurement resolution is limited. A phase-shifting technique can be used to increase the measurement resolution by no longer directly analyzing the fringe patterns but rather calculating the optical phase, which is correlated to the measurement quantity. In the phase-shifting experiment, a piezoelectric ceramics transducer (PZT) connected to a mirror (Fig. 1) is used to introduce four phase shifts. The intensities of the reference and object beams are assumed to be the same, with the corresponding intensities (I 1 , I 2 , I 3 , and I 4 ) undergoing phase shifts of π, π∕2, π, and 3π∕2, as follows: I 1 I b I a cosϕ;
(11)
π I 2 I b I a cos ϕ ; 2
(12)
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I 3 I b I a cosϕ π; 3π ; I 4 I b I a cos ϕ 2
(13) (14)
where I b is the intensity of background, and I a is the amplitude. Therefore, the phase ϕ can be calculated to be I −I (15) ϕ tan−1 4 2 : I1 − I3 The calculated result in Eq. (15), the wrapping phase, distributes within a range of −π∕2 and π∕2. The phase can be further expanded from 0 to 2π by considering the signs of the denominator and numerator in the brackets of Eq. (15). Finally, by using the phase-unwrapping technique, a multiple of 2π is added to a wrapping phase with phase discontinuity to obtain the continuously unwrapping phase [20,21]. D. Fitting Algorithm
We used Microsoft Excel to perform the work of curve fitting. To obtain an optimal curve as well as eliminate the effects of positive and negative deviation, the sum of squares (SS) was utilized to assess the fitting result. The definition of SS can be described as n X SS y data − ycurve 2 ; (16) i1
where n is number of data points, y data is data point, and ycurve is the value of fitting curve at point y data . Furthermore, the generalized reduced gradient algorithm was carried to perform the iterative nonlinear least squares regression such that the best-fitting curve can be obtained. The correlation index (R 2 ) defined as Pn y − y 2 2 R 1 − Pni1 data curve 2 : (17) i1 y data − y mean in which y mean is the mean of data points, was used to evaluate the fitting result. The fitting appropriateness can be assessed from the approaching of R 2 to 1 [22].
A. Optical Interferometer Establishment
A Twyman–Green interferometer was established for this study, as shown in Fig. 3. A 20 mW He–Ne laser (Coherent Inc.) with a wavelength of 633 nm was used as the light source. The laser first passed through a polarizer, which was used to
Mirror
appropriately adjust the transmitted power of the laser beam. The laser beam was then expanded, with noise filtered by the spatial filter. A collimated lens was placed in such a position that the distance from it to the spatial filter was equal to the focal length, thus the expanded laser beam passing through was a collimated beam. The collimated beam was then divided into two beams by using a nonpolarization beam splitter. A deflection PET substrate was placed in the object beam. A CCD camera (XC-ST50 Sony Corp.) was used to capture the optical interferometric fringe patterns. The PZT was supplied with various voltages to introduce phase shifts in the reference beam. Four interferometric fringe patterns under phase shifts of 0, π∕2, π, and 3π∕2 were successively grabbed, and the wrapping phase was calculated by using Eq. (15). Finally, phase-unwrapping technology was used to reconstruct a real phase map. B. Preparation of the Deflection PET Substrates with Known Radii
To verify the measurement precision of the proposed methodology, five aluminum molds with known radii were prepared by using the computer numerical control machine. The general geometry of the aluminum molds is depicted in Fig. 4, in which w 10 mm and a 22.5 mm for all molds. The heights of the molds, h, were 2.5, 3.5, 4.5, 5.5, and 6.5 mm. Subsequently, the radii of each can be calculated using the following equation: h a2 r : (18) 2 2h The calculated radii of the aluminum molds were 102.5, 74.1, 58.5, 48.8, and 42.2 mm. After the aluminum molds were prepared, the PET substrates were bonded onto their surfaces by using doubled-sided tape. The thickness and refraction index of the PET substrate was 125 μm and 1.575, respectively. Finally, the PET substrate bonding on the mold was placed in the Twyman–Green interferometer to capture the fringe patterns. 4. RESULTS AND DISCUSSION A. Optical Interferometric Fringe Patterns
3. EXPERIMENT
He-Ne laser
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The optical interferometric fringe patterns for different phase shifts with a deflection radius of 102.5 mm are shown in Figs. 5(a)–5(d), in which the fringe pattern appears up to third
Mirror
Spatial filter Collimated lens Mirror Polarizer NPBS Mirror Lens CCD camera Computer
PZT Deflection PET
Fig. 3. Twyman–Green interferometer used in this study.
Fig. 4. (a) Top view and (b) front view of geometry of the aluminum mold.
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Fig. 5. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 102.5 mm.
order. As shown in Eq. (10), the optical phase was related to the deflection radius. A larger deflection radius represented a smaller deflection, meaning that the appearing order of the interferometric fringe pattern was smaller. Figure 5(e) shows the wrapping phase calculated using Eq. (15). As shown in Figs. 5(a)–5(e), the distribution of the fringe patterns behaved in a winding manner along thickness direction, which was because the behavior of the fringe patterns was easily influenced by the imperfect null field of the optical interferometer under conditions with fewer fringes. In addition, the discontinuities removed by the unwrapping phase in Fig. 5(e) return in Fig. 5(f). The optical fringe patterns, wrapping phases, and unwrapping phases of the molds with deflection radii of 74.1, 58.5, 48.8, and 422 mm are shown in Figs. 6–9. As shown in Figs. 5–9, the amount of fringe increased as the deflection radius decreased. In addition, the fringe patterns became more parallel with a smaller deflection radius, indicating that the imperfect null field of the optical interferometric had less influence on the distribution of the fringe patterns with a larger deflection.
Fig. 6. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 74.1 mm.
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Fig. 7. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 58.5 mm.
Fig. 8. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 48.8 mm.
Fig. 9. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 42.2 mm.
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B. Fitting Function
8r
3 −
8y2
7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: 2 y 2 1 8r 2.48 − 1 2
25 Experimental data Fitting result
20
Phase(Rad)
Substituting the experimental parameters (i.e., λ 633 nm, n1 1, n2 1.575, and t 125 μm) into Eq. (10) yields 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi 1 r −y 0.31 8 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ 19852.14− 2 1 8r 2.48 − 1 y 2
15
10
5
(19)
0
8r
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
Experimental data Fitting result
50
Phase(Rad)
40 30 20 10 0
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
80
Experimental data Fitting reault
70
Phase(Rad)
In this equation, the optical phase ϕ is dependent on the position y and deflection radius r. The term y is an independent variable, ϕ is a dependent variable, and r is a constant to be fitted. We first consider the phase distribution along the line of y 0. When substituted into Eq. (19), the phase of y 0 is irrelevant to the deflection radius and equal to a constant of 1425.6 rad, meaning that this optical phase difference existed before the deflection. Therefore, we subtract this initial phase difference from Eq. (19) for fitting. When the five deflection radii are substituted into Eq. (19) while removing the initial phase difference, their corresponding phase and position relationships (plots shown in Fig. 10) indicate an increasing phase as its position moves from the center to the end. When compared with the optical path along the central line (Fig. 2), the optical path gradually increases, whereas its incident position moves toward the end. Moreover, the optical phase increases as the deflection radius decreases (Fig. 10); this phenomenon is consistent with the observed fringe pattern.
60 50 40 30 20 10
C. Experimental Data Used for Fitting
0
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
120
Experimental data Fitting result
100
Phase(Rad)
The unwrapping phases with different deflection radii are shown in Figs. 5(f)–9(f). We extracted their phase data along the y-axis indicated in Fig. 4(a) for fitting. The resulting plots are illustrated in Fig. 11. As shown in Fig. 11(a), when the deflection radius was 102.5 mm, the phase data exhibited a winding characteristic in a longitudinal direction, which was caused by the imperfect null field of the optical interferometer, as discussed. A similar phenomenon occurred with a deflection radius of 74.1 mm [Fig. 11(b)]. Furthermore, as indicated by
80 60 40 20 0
180
-20 -15 -10
-5
0
5
10
15
20
Position(mm) 160
r=102.5 mm r= 74.1 mm r= 58.5 mm r= 48.8 mm r= 42.2 mm
140
160 Experimental data Fitting result
140 120
100
Phase(Rad)
Phase (Rad)
120
80 60
100 80 60 40
40 20
20
0 -20 -15 -10
-20
-10
0
10
0
5
10
15
20
20
Position (mm)
Fig. 10. radii.
-5
Position(mm)
0
Relationship between phase and position of deflection
Fig. 11. Phase data used for fitting and the fitting results of deflection radii of (a) 102.5 mm, (b) 74.1 mm, (c) 58.5 mm, (d) 48.8 mm, and (e) 42.2 mm.
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the smooth phase data for the deflection radii decreasing from 58.5 to 42.2 mm [Figs. 11(c)–11(e)], the imperfect null field has less influence on the experimental results of a small deflection radius (i.e., large deflection). D. Radius Calculation
In calculating the radius, Eq. (19) was used to optimally fit the extracting phase data along a longitudinal direction. In this study, Excel was used to perform the nonlinear fitting process. The fitting results with different deflection radii were plotted (Fig. 11). The fitting results and measurement resolution of the deflection radius were 104.7 (119.2 μm), 72.3 (44.0 μm), 56.4 (18.7 μm), 48.1 (11.5 μm), and 42.6 mm (7.9 μm), respectively. Also, their corresponding error variances and correlation indices were 2.1% (1.00), 2.3% (0.97), 3.5% (1.00), 1.4% (0.99), and 1.1% (0.97), respectively. All of the correlation indices are close to 1, showing the goodness of the fitting result. The uncertainty of error contribute from reasons including the phase-shifting process, nonlinear fitting uncertainty, and PET bonding. In this paper, the elongation resolution of the PZT used to introduce phase shifts is rather small, i.e., 0.15 nm. The above fitting result also revealed that the relative error resulting from nonlinear fitting process is not obvious. The main reason causing the error is PET bonding configuration. It is known that the geometries of PET bonding on aluminum molds are all symmetric. It means that the fringe patterns grabbing by CCD camera should be horizontally parallel while PETs were perfectly bonded on the aluminum molds. However, some nonhorizontal fringe patterns appear in Figs. 5–9 due to unsymmetrical PET bonding, causing the dominate source of error. Finally, from the close agreement between the real radius and fitting results, it indicated that the proposed measurement methodology can accurately and directly measure the deflection radius of an FTS.
5. CONCLUSION In this paper, we present a measurement methodology for directly measuring the radius of a PET substrate by using a Twyman–Green interferometer. A relationship between the optical phase difference and deflection radius was derived. A previous analysis revealed that the optical phase increases as the deflection radius decreases. In addition, five aluminum molds with known deflection radii were prepared to verify the measurement accuracy of the proposed methodology. The results indicate that the error variance of our proposed measurement methodology is less than 3.5%, confirming its measurement precision. In the future, the presented measurement methodology will be extremely valuable for providing a direct deflection radius measurement in residual stress calculations applied to flexible electronics. Ministry of Science and Technology, Taiwan (100-2221-E150-017, 102-2221-E-150-017, 103-2221-E-150-018).
REFERENCES 1. C. Jang, K. Kim, and K. C. Choi, “Flexible photoluminescent display fabricated with low-temperature process using PET substrates,” J. Disp. Technol. 8, 250–255 (2012). 2. B. Muhsin, R. Roesch, G. Gobsch, and H. Hoppe, “Flexible ITO-free polymer solar cells based on highly conductive PEDOT:PSS and a printed silver grid,” Sol. Energy Mater. Sol. Cells 130, 551–554 (2014). 3. W. Y. Chang, T. H. Fang, and Y. C. Lin, “Physical characteristics of polyimide films for flexible sensors,” Appl. Phys. A 92, 693–701 (2008). 4. J. S. Hsu, B. J. Wen, and P. Y. Chen, “Full-field deflection measurement of the flexible transparent sheets,” Polym. Test. 31, 1105–1114 (2012). 5. P. Dutta, M. Rathi, N. Zheng, Y. Gao, Y. Yao, J. Martinez, P. Ahrenkiel, and V. Selvamanickam, “High mobility single-crystalline-like GaAs thin films on inexpensive flexible metal substrates by metal-organic chemical vapor deposition,” Appl. Phys. Lett. 105, 092104 (2014). 6. A. Plichta, A. Weber, and A. Habeck, “Ultra thin flexible glass substrates,” MRS Online Proc. Libr. 769, H9.1.1–H9.1.10 (2003). 7. Y. L. Chen, H. C. Hsieh, W. T. Wu, B. J. Wen, W. Y. Chang, and D. C. Su, “An alternative bend-testing technique for a flexible indium tin oxide film,” Displays 31, 191–195 (2010). 8. G. Singh and D. S. Mehta, “Measurement of change in refractive index in polymeric flexible substrates using wide field interferometry and digital fringe analysis,” Appl. Opt. 51, 8413–8422 (2012). 9. Y. C. Lee, T. S. Liu, C. I. Wu, and W. Y. Lin, “Investigation on residual stress and stress-optical coefficient for flexible electronics by photoelasticity,” Measurement 45, 311–316 (2012). 10. C. K. Chao, R. C. Chang, and Y. C. Li, “Effects of different interlayers on indium-tin oxide thin films sputtered on flexible PET substrate,” J. Chin. Inst. Eng. 34, 549–557 (2010). 11. R. C. Chang, T. C. Li, and W. C. Lin, “Influence of various thickness metallic interlayers on opto-electric and mechanical properties of AZO thin films on PET substrates,” Appl. Surf. Sci. 258, 3732–3737 (2012). 12. H. Machinaga, E. Ueda, A. Mizuike, Y. Takeda, K. Shimokita, and T. Miyazaki, “Effects of annealing temperature on mechanical durability of indium-tin-oxide film on polyethylene terephthalate substrate,” Thin Solid Films 559, 36–39 (2014). 13. G. G. Stoney, “The tension of metallic films deposited by electrolysis,” Proc. R. Soc. B 82, 172–175 (1909). 14. S. Grego, J. Lewis, E. Vick, and D. Temple, “Development and evaluation of bend-testing techniques for flexible-display application,” J. Soc. Inf. Disp. 13, 575–581 (2005). 15. J. L. Ni, X. F. Zhu, Z. L. Pei, J. Gong, C. Sun, and G. P. Zhang, “Comparative investigation of fracture behavior of aluminum-doped ZnO films on a flexible substrate,” J. Phys. D 42, 1–7 (2008). 16. H. C. Chen, K. T. Huang, K. T. Huang, and Y. M. Lo, “Measurement of residual stress for ITO/PET substrates by the double beam shadow moiré interferometer,” Appl. Opt. 51, 1566–1571 (2012). 17. K. T. Huang and H. C. Chen, “Automatic measurement and stress analysis of ITO/PET flexible substrate by shadow moiré interferometer with phase-shifting interferometry,” J. Disp. Technol. 10, 609–614 (2014). 18. A. Ghatak, Optics, 3rd ed. (McGraw-Hill, 2005). 19. M. Medhat, N. I. Hendawy, and A. A. Zaki, “Fringes of equal tangential inclination by curvature-induced birefringence,” Opt. Laser Technol. 35, 31–35 (2003). 20. D. Malacara, ed., Optical Shop Testing (Wiley, 2007). 21. T. Yoshizawa, ed., Handbook of Optical Metrology: Principles and Applications (CRC Press, 2008). 22. A. M. Brown, “A step-by-step guide to non-linear regression analysis of experimental data using a Microsoft Excel spreadsheet,” Comput. Methods Prog. Biomed. 65, 191–200 (2001).
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Direct deflection radius measurement of flexible PET substrates by using an optical interferometry JIONG-SHIUN HSU*
AND
PO-WEI LI
Department of Power Mechanical Engineering, National Formosa University, Yunlin 63208, Taiwan *Corresponding author: [email protected] Received 30 March 2015; revised 15 May 2015; accepted 17 May 2015; posted 18 May 2015 (Doc. ID 237157); published 8 June 2015
The deflection radius is essential in determining residual stress estimations in flexible electronics. However, the literature provides only indirect methods for obtaining a deflection radius. In this study, we present a measurement methodology for directly measuring the deflection radius of a polyethylene terephthalate (PET) substrate (a popular substrate of flexible electronics) by using an optical interferometer. A Twyman–Green optical interferometer was established and phase-shifting technology was used to increase the measurement resolution. Five PET substrates with known deflection radii were prepared to verify the measurement precision of the proposed measurement methodology. The results revealed that the error variance of our proposed measurement methodology is smaller than 3.5%. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.54.005469
1. INTRODUCTION In recent years, because of advances in manufacturing technology, flexible electronics have been gradually used in items such as displays, energy products, sensors, and illumination devices [1–3]. The concept of flexible electronics involves fabricating devices and circuits on a flexible substrate. Three candidates of substrates have been developed, namely those involving flexible transparent sheets (FTSs), metal foils, and ultrathin glass sheets [4–6]. Because the merits of an FTS include excellent transmittance, high resistance to impact, and roll-to-roll manufacturing, it has been characterized as the optimal substrate for use in future flexible electronics [7,8]. Although the FTS has the aforementioned advantages, its mechanical properties are easily influenced by environmental and loading conditions. Nonetheless, no free electron exists in its interior structure, thus it is a nonconductor of electricity. Typically, a transparent conductive oxide (TCO) layer is deposited on the FTS, enabling conductivity. For example, a layer of indium tin oxide is typically deposited on a polyethylene terephthalate (PET) substrate. However, FTSs are a ductile material whereas TCO is more brittle, which constitutes a considerable difference in mechanical properties. Consequently, severe residual stresses may be introduced within the TCO, causing cracking or a failure to degrade its transmittance and electrical properties. Therefore, the problem of residual stress on TCO–FTS structures has been increasingly critical in accelerating the commercialization of flexible electronics [9–12]. 1559-128X/15/175469-06$15/0$15.00 © 2015 Optical Society of America
Most studies on the residual stress of TCO–FTS structures use the Stoney formula [13] to estimate the residual stress of TCO film. Before the residual stress calculation, the radius of the TCO–FTS structure must be experimentally obtained to substitute it into the Stoney formula. Because the radius measurements in previous studies have been indirect [14–17], this study proposed a measurement methodology to directly obtain the radius of the PET substrate by using optical interferometry. The optical interferometer used in our proposed methodology is a Twyman– Green interferometer. First, we derived the relationship between the deflection radius of the PET substrate and optical phase difference. Five aluminum molds with known radii were manufactured; the PET substrates were bonded onto the surfaces of the molds, and their fringe patterns were observed through an optical interferometer. Phase-shifting technology was utilized to obtain a full-field phase map. Finally, a nonlinear regression approach was used to fit the optical phase data and obtain its deflection radius. 2. THEORIES A. Optical Interferometric Theory
A Twyman–Green interferometer was established; its optical arrangement is illustrated in Fig. 1. An object beam was created by placing a PET substrate in one optical path; a second beam without a PET substrate placement served as the reference beam. Let the electrical fields of the reference (E r ) and object (E o ) beams be
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thickness of the PET substrate. In addition, in triangle ABD, an approximate relation is made as follows [19]: t ; (6) d≈ cos θ2
Computer
CCD camera Collimated lens Spatial filter
where d is the optical path passing through the PET sheet. When all of the discussed relationships are applied, d can be further arranged as n2 t (7) d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 : n22 − n21 rt
Mirror
He-Ne laser
NPBS
Polarizer PZT Deflection PET Mirror
Fig. 1. Arrangement of the Twyman–Green interferometer.
E r E r1 e iwt−ϕr ;
(1)
E o E o1 e iwt−ϕr ;
(2)
where E r1 and E o1 are the electrical amplitude of reference and object beams, respectively; w is the circular frequency of the laser beam; t is time; and ϕr and ϕo are the phases of the reference and object beams, respectively. The intensity of the optical interferometric image (I ) is pffiffiffiffiffiffiffiffi (3) I I r I o 2 I r I o cosϕ; where I r E 2r is the intensity of the reference beam, I o E 2o is the intensity of the object beam, and ϕ ϕr − ϕo is the optical phase difference between the reference and object beams [18]. B. Theoretical Derivation of Deflection Radius
Figure 2 illustrates an optical beam passing through a deflection PET substrate, in which the deflection is bent into a circular arc with radius r at center O. By using Snell’s law, we obtain n1 sin θ1 n2 sin θ2 ;
(4)
where n1 and n2 are the refraction indices of the air and PET substrate, respectively; θ1 is the incident angle at point A; and θ2 is the refraction angle at point B. By using triangle ACO, we obtain y sin θ1 ; (5) rt where y is the length between an arbitrary point on the PET substrate, A, and point C; r is the deflection radius; and t is the
PET
1
A
l
E 2 2
t
B D
y
r
1
C
O
Fig. 2. Optical beam passing through a deflection PET substrate.
From the geometrical relationship in triangle ABE and the trigonometric function formula, the following relationship can be deduced: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t r t2 − y 2 n ty 2 q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l : (8) rt r t2 n2 − n2 y 2 2
1 rt
The optical path difference Δ before and after placement of the deflection PET substrate is " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 t n1 t r t2 − y 2 − Δ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rt n2 − n2 y 2 2
1 rt
# n21 ty 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − y 2 : r t2 n22 − n21 rt
(9)
Therefore, the optical phase difference ϕ before and after placement of the transparent film is " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4π n22 t n1 t r t2 − y2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ − λ rt n2 − n2 y 2 2
1 rt
# n21 ty2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − y 2 : r t2 n22 − n21 rt
(10)
Equation (10) shows that y is an independent variable, radius r is a constant, and the others are known. Finally, a nonlinearfitting approach can be adopted to optimally obtain r. C. Phase-Shifting Theory
Although the fringe patterns provided by the optical interferometer represent the quantitative distribution to be measured, the measurement resolution is limited. A phase-shifting technique can be used to increase the measurement resolution by no longer directly analyzing the fringe patterns but rather calculating the optical phase, which is correlated to the measurement quantity. In the phase-shifting experiment, a piezoelectric ceramics transducer (PZT) connected to a mirror (Fig. 1) is used to introduce four phase shifts. The intensities of the reference and object beams are assumed to be the same, with the corresponding intensities (I 1 , I 2 , I 3 , and I 4 ) undergoing phase shifts of π, π∕2, π, and 3π∕2, as follows: I 1 I b I a cosϕ;
(11)
π I 2 I b I a cos ϕ ; 2
(12)
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I 3 I b I a cosϕ π; 3π ; I 4 I b I a cos ϕ 2
(13) (14)
where I b is the intensity of background, and I a is the amplitude. Therefore, the phase ϕ can be calculated to be I −I (15) ϕ tan−1 4 2 : I1 − I3 The calculated result in Eq. (15), the wrapping phase, distributes within a range of −π∕2 and π∕2. The phase can be further expanded from 0 to 2π by considering the signs of the denominator and numerator in the brackets of Eq. (15). Finally, by using the phase-unwrapping technique, a multiple of 2π is added to a wrapping phase with phase discontinuity to obtain the continuously unwrapping phase [20,21]. D. Fitting Algorithm
We used Microsoft Excel to perform the work of curve fitting. To obtain an optimal curve as well as eliminate the effects of positive and negative deviation, the sum of squares (SS) was utilized to assess the fitting result. The definition of SS can be described as n X SS y data − ycurve 2 ; (16) i1
where n is number of data points, y data is data point, and ycurve is the value of fitting curve at point y data . Furthermore, the generalized reduced gradient algorithm was carried to perform the iterative nonlinear least squares regression such that the best-fitting curve can be obtained. The correlation index (R 2 ) defined as Pn y − y 2 2 R 1 − Pni1 data curve 2 : (17) i1 y data − y mean in which y mean is the mean of data points, was used to evaluate the fitting result. The fitting appropriateness can be assessed from the approaching of R 2 to 1 [22].
A. Optical Interferometer Establishment
A Twyman–Green interferometer was established for this study, as shown in Fig. 3. A 20 mW He–Ne laser (Coherent Inc.) with a wavelength of 633 nm was used as the light source. The laser first passed through a polarizer, which was used to
Mirror
appropriately adjust the transmitted power of the laser beam. The laser beam was then expanded, with noise filtered by the spatial filter. A collimated lens was placed in such a position that the distance from it to the spatial filter was equal to the focal length, thus the expanded laser beam passing through was a collimated beam. The collimated beam was then divided into two beams by using a nonpolarization beam splitter. A deflection PET substrate was placed in the object beam. A CCD camera (XC-ST50 Sony Corp.) was used to capture the optical interferometric fringe patterns. The PZT was supplied with various voltages to introduce phase shifts in the reference beam. Four interferometric fringe patterns under phase shifts of 0, π∕2, π, and 3π∕2 were successively grabbed, and the wrapping phase was calculated by using Eq. (15). Finally, phase-unwrapping technology was used to reconstruct a real phase map. B. Preparation of the Deflection PET Substrates with Known Radii
To verify the measurement precision of the proposed methodology, five aluminum molds with known radii were prepared by using the computer numerical control machine. The general geometry of the aluminum molds is depicted in Fig. 4, in which w 10 mm and a 22.5 mm for all molds. The heights of the molds, h, were 2.5, 3.5, 4.5, 5.5, and 6.5 mm. Subsequently, the radii of each can be calculated using the following equation: h a2 r : (18) 2 2h The calculated radii of the aluminum molds were 102.5, 74.1, 58.5, 48.8, and 42.2 mm. After the aluminum molds were prepared, the PET substrates were bonded onto their surfaces by using doubled-sided tape. The thickness and refraction index of the PET substrate was 125 μm and 1.575, respectively. Finally, the PET substrate bonding on the mold was placed in the Twyman–Green interferometer to capture the fringe patterns. 4. RESULTS AND DISCUSSION A. Optical Interferometric Fringe Patterns
3. EXPERIMENT
He-Ne laser
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The optical interferometric fringe patterns for different phase shifts with a deflection radius of 102.5 mm are shown in Figs. 5(a)–5(d), in which the fringe pattern appears up to third
Mirror
Spatial filter Collimated lens Mirror Polarizer NPBS Mirror Lens CCD camera Computer
PZT Deflection PET
Fig. 3. Twyman–Green interferometer used in this study.
Fig. 4. (a) Top view and (b) front view of geometry of the aluminum mold.
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Fig. 5. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 102.5 mm.
order. As shown in Eq. (10), the optical phase was related to the deflection radius. A larger deflection radius represented a smaller deflection, meaning that the appearing order of the interferometric fringe pattern was smaller. Figure 5(e) shows the wrapping phase calculated using Eq. (15). As shown in Figs. 5(a)–5(e), the distribution of the fringe patterns behaved in a winding manner along thickness direction, which was because the behavior of the fringe patterns was easily influenced by the imperfect null field of the optical interferometer under conditions with fewer fringes. In addition, the discontinuities removed by the unwrapping phase in Fig. 5(e) return in Fig. 5(f). The optical fringe patterns, wrapping phases, and unwrapping phases of the molds with deflection radii of 74.1, 58.5, 48.8, and 422 mm are shown in Figs. 6–9. As shown in Figs. 5–9, the amount of fringe increased as the deflection radius decreased. In addition, the fringe patterns became more parallel with a smaller deflection radius, indicating that the imperfect null field of the optical interferometric had less influence on the distribution of the fringe patterns with a larger deflection.
Fig. 6. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 74.1 mm.
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Fig. 7. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 58.5 mm.
Fig. 8. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 48.8 mm.
Fig. 9. Experimental results of the optical interferometer for (a) phase shift 0, (b) phase shift π∕2, (c) phase shift π, (d) phase shift 3π∕2, (e) the wrapping phase, and (f) the unwrapping phase with a deflection radius of 42.2 mm.
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B. Fitting Function
8r
3 −
8y2
7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5: 2 y 2 1 8r 2.48 − 1 2
25 Experimental data Fitting result
20
Phase(Rad)
Substituting the experimental parameters (i.e., λ 633 nm, n1 1, n2 1.575, and t 125 μm) into Eq. (10) yields 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi 1 r −y 0.31 8 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ 19852.14− 2 1 8r 2.48 − 1 y 2
15
10
5
(19)
0
8r
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
Experimental data Fitting result
50
Phase(Rad)
40 30 20 10 0
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
80
Experimental data Fitting reault
70
Phase(Rad)
In this equation, the optical phase ϕ is dependent on the position y and deflection radius r. The term y is an independent variable, ϕ is a dependent variable, and r is a constant to be fitted. We first consider the phase distribution along the line of y 0. When substituted into Eq. (19), the phase of y 0 is irrelevant to the deflection radius and equal to a constant of 1425.6 rad, meaning that this optical phase difference existed before the deflection. Therefore, we subtract this initial phase difference from Eq. (19) for fitting. When the five deflection radii are substituted into Eq. (19) while removing the initial phase difference, their corresponding phase and position relationships (plots shown in Fig. 10) indicate an increasing phase as its position moves from the center to the end. When compared with the optical path along the central line (Fig. 2), the optical path gradually increases, whereas its incident position moves toward the end. Moreover, the optical phase increases as the deflection radius decreases (Fig. 10); this phenomenon is consistent with the observed fringe pattern.
60 50 40 30 20 10
C. Experimental Data Used for Fitting
0
-20 -15 -10
-5
0
5
10
15
20
Position(mm)
120
Experimental data Fitting result
100
Phase(Rad)
The unwrapping phases with different deflection radii are shown in Figs. 5(f)–9(f). We extracted their phase data along the y-axis indicated in Fig. 4(a) for fitting. The resulting plots are illustrated in Fig. 11. As shown in Fig. 11(a), when the deflection radius was 102.5 mm, the phase data exhibited a winding characteristic in a longitudinal direction, which was caused by the imperfect null field of the optical interferometer, as discussed. A similar phenomenon occurred with a deflection radius of 74.1 mm [Fig. 11(b)]. Furthermore, as indicated by
80 60 40 20 0
180
-20 -15 -10
-5
0
5
10
15
20
Position(mm) 160
r=102.5 mm r= 74.1 mm r= 58.5 mm r= 48.8 mm r= 42.2 mm
140
160 Experimental data Fitting result
140 120
100
Phase(Rad)
Phase (Rad)
120
80 60
100 80 60 40
40 20
20
0 -20 -15 -10
-20
-10
0
10
0
5
10
15
20
20
Position (mm)
Fig. 10. radii.
-5
Position(mm)
0
Relationship between phase and position of deflection
Fig. 11. Phase data used for fitting and the fitting results of deflection radii of (a) 102.5 mm, (b) 74.1 mm, (c) 58.5 mm, (d) 48.8 mm, and (e) 42.2 mm.
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the smooth phase data for the deflection radii decreasing from 58.5 to 42.2 mm [Figs. 11(c)–11(e)], the imperfect null field has less influence on the experimental results of a small deflection radius (i.e., large deflection). D. Radius Calculation
In calculating the radius, Eq. (19) was used to optimally fit the extracting phase data along a longitudinal direction. In this study, Excel was used to perform the nonlinear fitting process. The fitting results with different deflection radii were plotted (Fig. 11). The fitting results and measurement resolution of the deflection radius were 104.7 (119.2 μm), 72.3 (44.0 μm), 56.4 (18.7 μm), 48.1 (11.5 μm), and 42.6 mm (7.9 μm), respectively. Also, their corresponding error variances and correlation indices were 2.1% (1.00), 2.3% (0.97), 3.5% (1.00), 1.4% (0.99), and 1.1% (0.97), respectively. All of the correlation indices are close to 1, showing the goodness of the fitting result. The uncertainty of error contribute from reasons including the phase-shifting process, nonlinear fitting uncertainty, and PET bonding. In this paper, the elongation resolution of the PZT used to introduce phase shifts is rather small, i.e., 0.15 nm. The above fitting result also revealed that the relative error resulting from nonlinear fitting process is not obvious. The main reason causing the error is PET bonding configuration. It is known that the geometries of PET bonding on aluminum molds are all symmetric. It means that the fringe patterns grabbing by CCD camera should be horizontally parallel while PETs were perfectly bonded on the aluminum molds. However, some nonhorizontal fringe patterns appear in Figs. 5–9 due to unsymmetrical PET bonding, causing the dominate source of error. Finally, from the close agreement between the real radius and fitting results, it indicated that the proposed measurement methodology can accurately and directly measure the deflection radius of an FTS.
5. CONCLUSION In this paper, we present a measurement methodology for directly measuring the radius of a PET substrate by using a Twyman–Green interferometer. A relationship between the optical phase difference and deflection radius was derived. A previous analysis revealed that the optical phase increases as the deflection radius decreases. In addition, five aluminum molds with known deflection radii were prepared to verify the measurement accuracy of the proposed methodology. The results indicate that the error variance of our proposed measurement methodology is less than 3.5%, confirming its measurement precision. In the future, the presented measurement methodology will be extremely valuable for providing a direct deflection radius measurement in residual stress calculations applied to flexible electronics. Ministry of Science and Technology, Taiwan (100-2221-E150-017, 102-2221-E-150-017, 103-2221-E-150-018).
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