- Email: [email protected]
Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate
Journal Pre-proof Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate Jiong-Shiun Hsu, Wen-Pin Juan PII:
S0142-9418(19)31970-1
DOI:
https://doi.org/10.1016/j.polymertesting.2020.106376
Reference:
POTE 106376
To appear in:
Polymer Testing
Received Date: 24 October 2019 Revised Date:
8 January 2020
Accepted Date: 23 January 2020
Please cite this article as: J.-S. Hsu, W.-P. Juan, Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate, Polymer Testing (2020), doi: https://doi.org/10.1016/j.polymertesting.2020.106376. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Jiong-shiun Hsu: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Writing - Original Draft, Writing - Review & Editing, Supervision, Project administration, Funding acquisition
Wen-Pin Juan: Software, Validation, Formal analysis, Investigation, Writing - Original Draft
Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate
JIONG-SHIUN HSU*
AND WEN-PIN JUAN
Department of Power Mechanical Engineering National Formosa University Yunlin 632, Taiwan
*Associate Professor Email: [email protected]
Abstract This study proposes an optical methodology to measure the deflection radius of flexible polymeric substrates; to our best knowledge, this is the first report taking into consideration the optical anisotropy effect of these substrates for such measurements. The relation between the deflectioninduced optical phase retardation of two distinct refraction beams and the deflection radius was derived. Full-field phase maps were obtained via a polarization measurement system and the phase-stepping technique. The measurement results for three substrates having nominal deflection radii of 50, 55, and 60 mm were, correspondingly, 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm (error = 2.78%, 4.65%, and 2.15%, respectively); thus, the measurement precision was confirmed.
Keywords: deflection radius, flexible polymeric, optical anisotropy effect, polarization measurement system
1. Introduction Flexible polymeric substrates are widely used for flexible electronics applications. However, the significant difference in mechanical properties between these materials and the films deposited on them may result in severe deflection. Because the flexible electronics are the multilayered structures, the significant stress would generate to maintain the structural safety to satisfy the constitutive relation under such deflection [1-3]. Therefore, developing a precision deflection measurement method has become an important issue for the commercialization of flexible electronics [4-8]. Since flexible polymeric substrates are transparent, the earliest approaches just used a linear actuator or a micrometer for measuring the translation at their end point so to roughly calculate the deflection [9, 10]. Later, several moiré-based optical techniques have been employed. In 2008, Lee et al. [11] used the shadow moiré method to evaluate the radius of an aluminum film deposited on a polymeric substrate and calculate its residual stress. In 2010, Xu and Liechti [12] combined the bulge test and moiré deflectometry to determine the mechanical properties and residual stress of a transparent polyethylene terephthalate (PET) film. Chen et al. [13] used the double beam shadow moiré interferometer to obtain the residual stress of an indium tin oxide film deposited on a PET substrate. In 2014, Huang and Chen [14] adopted the phase shift technique to increase the measurement resolution of their system. In 2017, Chen et al. [15] analyzed the relation between normal and shear stresses of thin films via moiré measurements and the Mohr circle transformation. The fringe patterns of the abovementioned moiré-based measurements [12– 15] result from the overlapping of a real grating and its shadow; this type of optical interference is called mechanical interference. The pitch of real gratings ranges from one hundred micrometers to several millimeters, but the visible light wavelength range is 390–700 nm. Therefore, the precision of
optical measurement methods developed by wavefront interference (i.e., interferenced by the waves of two lights) greatly exceeds that of moiré-based techniques [16, 17]. In 2012, the first author of this paper and his research team proposed an optical technique based on wavefront interference to measure the deflection of a PET substrate [18]; in 2015, they also improved the previously proposed methodology for radius measurements [19]. As a result, significant advances of measurement precision have been made in the previous studies for flexible polymeric substrates. In addition, the optical anisotropic properties of these substrates have been observed [20]. However, the effect of this anisotropy on the deflection measurements has never been considered. Thus, this study proposes an optical method that takes into account the optical anisotropy of the substrates to directly measure the deflection radius; we performed our tests on a polyimide (PI) substrate and assumed its optical property as uniaxial crystal. Furthermore. the optical polarization theory and the phase-stepping technique were used to determine the deflection-induced full-field phase retardation. Based on experimental phase data and the optically anisotropic theory of crystal optics, the deflection radius of a flexible polymeric substrate under deflection can be directly obtained.
2. Optical theory 2.1 Plane polarization In this study, the effect of the optical anisotropy of a flexible polymeric substrate was considered and, then, the theories of optical polarization and crystal optics were used. The direction of a beam impinging on a crystal and its refractive index ellipsoid are schematized in Fig. 1, where α and β are the angles, respectively, between the x and x’ axes and between the incident beam and the z or z’ axis. For this experiment, PI substrates were used as
specimens. First, to determine the orientation of the in-plane axes of the refractive index ellipsoid, an undeformed substrate was placed in the plane of the polarization measurement system, as shown in Fig. 2(a); the polarized axes of the polarizer and analyzer were mutually perpendicular, i.e., a dark field arrangement was considered. According to the Jones matrix calculation, the electrical fields along and
normal to the polarized axis of the analyzer (E and E , respectively) are
given by E E
=
cos − isin cosγ −isin sin2γ
−isin sin2γ
0 ke cos + isin cosγ 1
(1)
where δ is the optical phase retardation due to deflection, γ is the angle between the in-plane slow axis and the horizontal axis, k =
, λ, and ω are,
respectively, the wave number, wavelength, and frequency of the light source,
and t is time. Therefore, the light intensity captured by a complementary
metal-oxide-semiconductor (CMOS) camera can be expressed as [21–23] I = I$ sin
sin 2γ
(2)
where I$ is the amplitude of intensity. In the experiment, the rotation of the
undeformed PI substrate was controlled by a rotation stage and images at different rotation angles (γ) were captured by the CMOS camera. At each
integer γ, the gray-level values of all the pixels of an image were summed
and, then, divided by the pixel number to calculate the average gray-level value; this value was used to identify the orientation of the in-plane axes of the refractive index ellipsoid of the substrate.
2.2 Elliptical polarization After determining the in-plane axes orientation of the refractive index ellipsoid, the PI substrate was cut into a rectangle along one in-plane axis and, then, bonded to an aluminum mold having a known radius. The resulting deflected substrate was placed in the elliptical polarization measurement system, as illustrated in Fig. 2(b). The experimental details are described later. In this case, according to the Jones matrix calculation, E and E are given by E E
(
=
1 cosψ& 2 −sinψ&
sinψ& 1 − icos2ψ' cosψ& −sin2ψ'
cos − isin cos2ψ* )
)
−isin sin2ψ* )
−isin cos2ψ* )
−isin2ψ' 1 + icos2ψ'
cos + isin cos2ψ* )
)
+
1 − icosψ −sin2ψ
−isin2ψ 1 + icos2ψ (3)
where ψ, , ψ , ψ* , ψ' , and ψ& are the angles between, respectively, the
polarized and horizontal axes, the slow axis of the first plate and the horizontal axis, the slow axis of the PI substrate and the horizontal axis, the slow axis of the second quarter-wave plate and the horizontal axis, and the polarized axis of the analyzer and the horizontal axis, and φ is the optical
phase retardation by deflection. In the optical arrangement adopted in this study, ψ, = , ψ = '
*
'
, and ψ* = π. The intensity of the interferometric fringe
pattern captured by the CMOS camera (I) can be calculated as [24–26]
I=
,
I$ 1 + ,
2sin23ψ& − ψ' 4cosφ − sin23π − ψ' 4cos23ψ& − ψ' 4sinφ56 (4)
cosψ, Ae sinψ,
2.3 Phase-stepping technique To derive the deflection-induced optical phase retardation from Eq. (4), the four-step phase-stepping technique was adopted. This technique was performed by rotating the second quarter-wave plate and the analyzer shown in Fig. 2(b), i.e., by changing ψ' and ψ& . The angle schemes of (ψ' , ψ& ) *
under different phase shifts were (0, ' ), ( ' ,
*
'
), (0,
*
'
), and ( ' , ' ); thus, the
intensities of the corresponding interferometric fringe patterns (I, , I , I* , and I' ) are given as follows:
I, =
78
31 + cosφ4
(5)
I =
78
31 − sinφ4
(6)
I* =
78
31 − cosφ4
(7)
I' =
78
31 + sinφ4
(8)
As a consequence, φ can be calculated as [27–29] φ = tan:,
7; :7< 7= :7>
(9)
2.4 Relation between deflection radius and deflection-induced optical phase retardation
The φ value from Eq. (9) carries
information about the substrate
deflection. Its optical anisotropy results in the production of two refraction beams: the ordinary beam (o-beam) and the extraordinary beam (e-beam). The geometrical relation between the incident and refracted beams is depicted in Fig. 3. According to the Snell’s law, we have n, sinθ, = n@ sinθ@
(10)
n, sinθ, = nA sinθA
(11)
where n, is the refractive index of air, θ, is the incident beam angle, n@ and
θ@ are the refractive index and refraction angle of the o-beam, and nA and θA
are the refractive index and refraction angle of the e-beam. From ∆CDA and ∆BDA, it gives
d@ = tsecθ@
(12)
dA = tsecθA
(13)
where d@ and d A are the distances of the refraction paths of the o- and ebeams, respectively, and t is the substrate thickness. In ∆DOE, sinθ, = HI
(14)
where y is the distance between an arbitrary point on the substrate and the
horizontal axis and R is the deflection radius of the substrate. Eqs. (10)–(14)
yield
dL =
MN
< < OMN < :P= Q RST
(15)
and
dA =
MU
< < OMU < :P= Q RST
(16)
Since the optical phase difference between the o- and e-beams (∆) is given by ∆= n@ d@ − nA dA
(17)
the deflection-induced optical phase retardation can be calculated as follows: φ=
X \ M
(18)
2.5 Crystal optics theory The PI substrate investigated in this study was assumed to be a uniaxial crystal having the optical axis normal to its surface, as represented in Fig. 3.
When a beam impinges at an arbitrary angle, two refraction beams, o-beam and e-beam, are generated; n@ remains constant, while nA varies with the
incident angle. Fig. 4 shows the relation between these refractive indices and the principal ones, where n@ and nA are given by [30] n@ = n$ = n]
(19)
and
nA = O
M<8 M<^
M<8 _ M< `U IM
(20)
By substituting Eqs. (19),(20) into Eq. (18), we can express the deflectioninduced optical retardation as
φ=
X M
where θ@ = sin:,
M=
MN
\ [ [ < P Q< P< P< Y < < 8 a< < : = < [ P8 cdP eU SPa aNc eU 3RST4 Z < P< 8 Pa < e SP< aNc< e P< cdP U U 8 a
sinθ, and θA = sin:,
M=
MU
sinθ, .
3. Experimental details 3.1 In-plane axes orientation of the refractive index ellipsoid
(21)
For the experiment, an undeformed PI substrate was cut and attached to the
platform of a rotation stage; its principal refractive indices n$ , n] , and nb
were 1.74, 1.74, and 1.63, respectively. This substrate/stage combination was placed in the plane polarization measurement system (Fig. 5(a)) to determine the orientation of the in-plane principal axes of the refractive index ellipsoid. A sodium light of wavelength 589 nm was used as the light source. The stage rotation degree was controlled by the LabVIEW program and images were captured at each integer degree by a CMOS camera (acA1300-30gc, Basler) equipped with an image acquisition card (PCIe-8236, National Instruments); the LabVIEW program also calculated the average gray-level value at each integer degree. 3.2 Deflection radius measurements The elliptical polarization measurement system was arranged as shown in Fig. 5(b). Three aluminum molds (Fig. 6) with different known radius (50, 55, and 60 mm) were manufactured with a computer numerical control machine. The PI substrate was cut into a rectangle along one of the in-plane principal axes of the refractive index ellipsoid to obtain a realigned substrate; then, it was bonded on a mold surface, and placed in the experimental setup (Fig. 5(b)) to measure their deflection radius. The second quarter-wave plate and the analyzer were rotated to achieve the four distinct phase shifts mentioned previously. The CMOS camera captured the interferometric fringe patterns and the LabVIEW program was used for digital image processing and phase reconstruction.
4. Results and discussion The images of the original and realigned PI substrates at different rotation angles are presented in Fig. 7; the regions enclosed by the red lines indicate where the beam passes through the substrate and each one include 57,489
pixels. For the original substrate (Fig. 7(a)), before starting the rotation, i.e., at γ = 0°, the image was not the darkest or brightest of the series. This implies
that the in-plane principal axes of the refractive index ellipsoid were not along the horizontal or vertical direction of the substrate but at a specific angle, which must be measured herein. Then, the intensity of the interferometric image increased when γ was increased from 0° to 30°, gradually decreased
when further increasing γ to 75°, and increased again between 75° and 90°. The corresponding average gray-level values are plotted in Fig. 8(a). Some
differences were observed between the lower part of the plot and the square of the sinusoidal function indicated in Eq. (2). In addition, the value distribution at each valley was truncated, because the gain value of the image capturing system was relatively small. It was intended to use the first peak in Fig. 8(a) to determine the angle measured. Therefore, an appropriate gain value was adjusted to avoid overexposure and to fully record the gray level variation around peaks. Furthermore, this first peak appeared at 28°, indicating that the fast axis of the substrate was at 28° to the horizontal axis; this value is also the angle which the substrate must be cut along to obtain the realigned PI substrate. To verify the precision of the abovementioned measurements, part of the realigned substrate was cut along 28° with the horizontal axis and, then, bonded on the rotation stage to repeat the experiment. Fig. 7(b) shows the images of the realigned substrate at different angles. The darkest image was obtained before starting the rotation, while the greatest intensity was observed when increasing γ from 0° to 45° . When further increasing γ , the image
gradually became darker until returning to the lowest intensity value at 90°. The relation between the average gray-level value and the rotation degree is
plotted in Fig. 8(b); the first peak appeared at 45°, confirming the measurement precision for the realigned substrate.
Then, the PI substrate was cut into three rectangles along 28° with the horizontal axis and each of them was bonded on an aluminum mold having a different radius. Their measurement results are shown in Figs. 9-11 in which the fringe patterns reveal skewed This occurred because the substrate was actually a biaxial crystal and its three principal refractive indices differed rather than being the same for both of in-plane principal refractive indices as assumed in this study. Therefore, unlike the assumed generation of an o-beam and an e-beam, when the beam impinged on the PI crystal, two e-beams were produced, both having the refractive indices dependent on the incident beam angle. This led to skewed fringe patterns. In addition, the fringe order shown in Fig. 9(a) is larger than that in Fig. 10(a) that, in turn, is larger than that in Fig. 11(a). This resulted from the deflection order corresponding to the three mold radii: 50 mm > 55 mm > 60 mm. The fringe patterns at the aforementioned four steps and their corresponding wrapping phases from Eq. (9) were calculated. The distribution of the arctan function was limited between −π/2 and π/2; after considering the numerator and denominator signs, the range was merely constrained between −π and π. Thus, when π was added to the obtained phase, the wrapping phase ranged from 0 and 2π. The wrapping phases are expressed in gray scale in Figs. 9(e)–11(e). LabVIEW was used to reconstruct the real and full-field phase maps [23– 25], removing the discontinuity of the wrapping phases shown in Figs. 9(e)– 11(e); the resulting unwrapping phases are presented in Figs. 9(f)–11(f). The data were collected along the central line and are represented by black curves in Fig. 12. Finally, nonlinear fitting [31] was performed by incorporating these values in Eq. (21) to obtain the deflection radius; the fitting curves are shown in red in Fig. 12. The experimental phase data were consistent with the fitting curves, confirming the rationality of the nonlinear fitting process. We conducted 10 measurements on the PI substrates having three different nominal deflection radii, namely, 50, 55, and 60 mm, as mentioned above; the
respective deflection radii measured were 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm.
5. Conclusion We propose a method to directly measure the deflection radius of a flexible polymeric substrate by taking the effect of its optical anisotropy into account. The relation between the deflection-induced optical phase retardation and the deflection radii of the generated o- and e- beams was also derived. First, a plane polarization measurement system was designed to experimentally determine the orientation of the in-plane principal axes of the substrate refractive index ellipsoid; next, the deflection radius of the substrate was evaluated via an elliptical polarization measurement system and the four-step phase-stepping technique was used to determine the deflection-induced optical phase retardation. Then, nonlinear regression was performed to fit the experimental phase data with the derived phase relationship. To verify the measurement precision of the proposed methodology, three PI substrates with different nominal deflection radii, namely, 50, 55, and 60 mm, were prepared for the experiment; the corresponding measured deflection radii were 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm, with errors of 2.78%, 4.65%, and 2.15%, respectively, confirming the precision of our method. The interference mechanism of the proposed technique is based on wavefront interference and, therefore, provides the advantages of full-field and high resolution. Moreover, since both the optical polarization measurement systems used are of the common-path type, they are less sensitive to environmental disturbances. Finally, since the proposed method can directly determine the deflection radius rather than the deflection of the flexible polymeric substrate, it has great potential for future applications in the residual stress estimation for flexible displays.
Acknowledgments The authors are very grateful to Prof. Ju-yi Lee of National Central University for his valuable discussion on optical polarization during the research period. This paper was supported in part by the Ministry of Science and Technology of the Republic of China (Grant no. MOST 105-2221-E-150016-MY2 and 104-2221-E-150-061).
Disclosures The authors declare no conflicts of interest. Data Availability The processed data required to reproduce these findings are available to download
from
http://dx.doi.org/10.17632/b9gz7j3szw.1#file-3b250700-
3732-4223-905c-f7bf7cb8921c.
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Fig. 1 Schematic of the incidence direction of a beam and the ellipsoid of the refractive index of the crystal
1
(a) Plane polarization measurement system
(b) Elliptical polarization measurement system Fig. 2 Illustration of plane and elliptical polarization measurement systems
2
Fig. 3 Geometrical relation between the incident and refracted beams
3
(a) o-beam
(b) e-beam Fig. 4 Relation between refractive indices of o-beam and e-beam and in-plane principal refractive indices 4
(a) Plane polarization measurement system
(b) Elliptical polarization measurement Fig. 5 Photographs of plane and elliptical polarization measurement systems
5
Fig. 6 Geometry of the aluminum mold with known radius
6
0°
0°
15°
15°
30°
30°
45°
45°
60°
60°
75°
75°
90°
90°
(a) original (b) realigned Fig. 7 Images of original and realigned PI substrates at different angles of the rotation stage
7
120
Averaged gray-level value
100
80
60
40
20
0 0
60
120
180
240
300
360
240
300
360
Degree
(a) original 120
Averaged gray-level value
100
80
60
40
20
0 0
60
120
180
Degree
(b) realigned Fig. 8 Average gray-level value of interferometric images of original and realigned PI substrates
8
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 9 Experimental result of the PI substrate under a deflection radius of 50 mm
9
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 10 Experimental result of the PI substrate under a deflection radius of 55 mm
10
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 11 Experimental result of the PI substrate under a deflection radius of 60 mm
11
10
Phase data Fitting curve 8
Phase
6
4
2
0 -20
-10
0
10
20
Position(mm)
(a) 50mm 8
Phase data Fitting curve
7 6
Phase
5 4 3 2 1 0 -20
-10
0
10
20
Position(mm)
(b) 55 mm 6
Phase data Fitting curve
5
Phase
4
3
2
1
0 -20
-10
0
10
20
Position(mm)
(c) 60 mm Fig. 12 Data of unwrapping phase along the central line and the obtained fitting curve for the different deflection radii of the PI substrate
12
1. This study proposed an optical methodology to measure the deflection radius of flexible polymeric substrates. 2. The optically anisotropic effect of flexible-polymeric substrate was first taken into consideration. 3. The phase steeping technique was used to obtain the phase. 4. The plane and elliptically polarization systems were respectively used to measure. 5. The proposed measurement methodology can be used to the residual stress estimation applied in flexible electronics.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0142-9418(19)31970-1
DOI:
https://doi.org/10.1016/j.polymertesting.2020.106376
Reference:
POTE 106376
To appear in:
Polymer Testing
Received Date: 24 October 2019 Revised Date:
8 January 2020
Accepted Date: 23 January 2020
Please cite this article as: J.-S. Hsu, W.-P. Juan, Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate, Polymer Testing (2020), doi: https://doi.org/10.1016/j.polymertesting.2020.106376. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Jiong-shiun Hsu: Conceptualization, Methodology, Software, Formal analysis, Investigation, Resources, Writing - Original Draft, Writing - Review & Editing, Supervision, Project administration, Funding acquisition
Wen-Pin Juan: Software, Validation, Formal analysis, Investigation, Writing - Original Draft
Optical polarization measurement for measuring deflection radius of the optically anisotropic flexible-polymeric substrate
JIONG-SHIUN HSU*
AND WEN-PIN JUAN
Department of Power Mechanical Engineering National Formosa University Yunlin 632, Taiwan
*Associate Professor Email: [email protected]
Abstract This study proposes an optical methodology to measure the deflection radius of flexible polymeric substrates; to our best knowledge, this is the first report taking into consideration the optical anisotropy effect of these substrates for such measurements. The relation between the deflectioninduced optical phase retardation of two distinct refraction beams and the deflection radius was derived. Full-field phase maps were obtained via a polarization measurement system and the phase-stepping technique. The measurement results for three substrates having nominal deflection radii of 50, 55, and 60 mm were, correspondingly, 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm (error = 2.78%, 4.65%, and 2.15%, respectively); thus, the measurement precision was confirmed.
Keywords: deflection radius, flexible polymeric, optical anisotropy effect, polarization measurement system
1. Introduction Flexible polymeric substrates are widely used for flexible electronics applications. However, the significant difference in mechanical properties between these materials and the films deposited on them may result in severe deflection. Because the flexible electronics are the multilayered structures, the significant stress would generate to maintain the structural safety to satisfy the constitutive relation under such deflection [1-3]. Therefore, developing a precision deflection measurement method has become an important issue for the commercialization of flexible electronics [4-8]. Since flexible polymeric substrates are transparent, the earliest approaches just used a linear actuator or a micrometer for measuring the translation at their end point so to roughly calculate the deflection [9, 10]. Later, several moiré-based optical techniques have been employed. In 2008, Lee et al. [11] used the shadow moiré method to evaluate the radius of an aluminum film deposited on a polymeric substrate and calculate its residual stress. In 2010, Xu and Liechti [12] combined the bulge test and moiré deflectometry to determine the mechanical properties and residual stress of a transparent polyethylene terephthalate (PET) film. Chen et al. [13] used the double beam shadow moiré interferometer to obtain the residual stress of an indium tin oxide film deposited on a PET substrate. In 2014, Huang and Chen [14] adopted the phase shift technique to increase the measurement resolution of their system. In 2017, Chen et al. [15] analyzed the relation between normal and shear stresses of thin films via moiré measurements and the Mohr circle transformation. The fringe patterns of the abovementioned moiré-based measurements [12– 15] result from the overlapping of a real grating and its shadow; this type of optical interference is called mechanical interference. The pitch of real gratings ranges from one hundred micrometers to several millimeters, but the visible light wavelength range is 390–700 nm. Therefore, the precision of
optical measurement methods developed by wavefront interference (i.e., interferenced by the waves of two lights) greatly exceeds that of moiré-based techniques [16, 17]. In 2012, the first author of this paper and his research team proposed an optical technique based on wavefront interference to measure the deflection of a PET substrate [18]; in 2015, they also improved the previously proposed methodology for radius measurements [19]. As a result, significant advances of measurement precision have been made in the previous studies for flexible polymeric substrates. In addition, the optical anisotropic properties of these substrates have been observed [20]. However, the effect of this anisotropy on the deflection measurements has never been considered. Thus, this study proposes an optical method that takes into account the optical anisotropy of the substrates to directly measure the deflection radius; we performed our tests on a polyimide (PI) substrate and assumed its optical property as uniaxial crystal. Furthermore. the optical polarization theory and the phase-stepping technique were used to determine the deflection-induced full-field phase retardation. Based on experimental phase data and the optically anisotropic theory of crystal optics, the deflection radius of a flexible polymeric substrate under deflection can be directly obtained.
2. Optical theory 2.1 Plane polarization In this study, the effect of the optical anisotropy of a flexible polymeric substrate was considered and, then, the theories of optical polarization and crystal optics were used. The direction of a beam impinging on a crystal and its refractive index ellipsoid are schematized in Fig. 1, where α and β are the angles, respectively, between the x and x’ axes and between the incident beam and the z or z’ axis. For this experiment, PI substrates were used as
specimens. First, to determine the orientation of the in-plane axes of the refractive index ellipsoid, an undeformed substrate was placed in the plane of the polarization measurement system, as shown in Fig. 2(a); the polarized axes of the polarizer and analyzer were mutually perpendicular, i.e., a dark field arrangement was considered. According to the Jones matrix calculation, the electrical fields along and
normal to the polarized axis of the analyzer (E and E , respectively) are
given by E E
=
cos − isin cosγ −isin sin2γ
−isin sin2γ
0 ke cos + isin cosγ 1
(1)
where δ is the optical phase retardation due to deflection, γ is the angle between the in-plane slow axis and the horizontal axis, k =
, λ, and ω are,
respectively, the wave number, wavelength, and frequency of the light source,
and t is time. Therefore, the light intensity captured by a complementary
metal-oxide-semiconductor (CMOS) camera can be expressed as [21–23] I = I$ sin
sin 2γ
(2)
where I$ is the amplitude of intensity. In the experiment, the rotation of the
undeformed PI substrate was controlled by a rotation stage and images at different rotation angles (γ) were captured by the CMOS camera. At each
integer γ, the gray-level values of all the pixels of an image were summed
and, then, divided by the pixel number to calculate the average gray-level value; this value was used to identify the orientation of the in-plane axes of the refractive index ellipsoid of the substrate.
2.2 Elliptical polarization After determining the in-plane axes orientation of the refractive index ellipsoid, the PI substrate was cut into a rectangle along one in-plane axis and, then, bonded to an aluminum mold having a known radius. The resulting deflected substrate was placed in the elliptical polarization measurement system, as illustrated in Fig. 2(b). The experimental details are described later. In this case, according to the Jones matrix calculation, E and E are given by E E
(
=
1 cosψ& 2 −sinψ&
sinψ& 1 − icos2ψ' cosψ& −sin2ψ'
cos − isin cos2ψ* )
)
−isin sin2ψ* )
−isin cos2ψ* )
−isin2ψ' 1 + icos2ψ'
cos + isin cos2ψ* )
)
+
1 − icosψ −sin2ψ
−isin2ψ 1 + icos2ψ (3)
where ψ, , ψ , ψ* , ψ' , and ψ& are the angles between, respectively, the
polarized and horizontal axes, the slow axis of the first plate and the horizontal axis, the slow axis of the PI substrate and the horizontal axis, the slow axis of the second quarter-wave plate and the horizontal axis, and the polarized axis of the analyzer and the horizontal axis, and φ is the optical
phase retardation by deflection. In the optical arrangement adopted in this study, ψ, = , ψ = '
*
'
, and ψ* = π. The intensity of the interferometric fringe
pattern captured by the CMOS camera (I) can be calculated as [24–26]
I=
,
I$ 1 + ,
2sin23ψ& − ψ' 4cosφ − sin23π − ψ' 4cos23ψ& − ψ' 4sinφ56 (4)
cosψ, Ae sinψ,
2.3 Phase-stepping technique To derive the deflection-induced optical phase retardation from Eq. (4), the four-step phase-stepping technique was adopted. This technique was performed by rotating the second quarter-wave plate and the analyzer shown in Fig. 2(b), i.e., by changing ψ' and ψ& . The angle schemes of (ψ' , ψ& ) *
under different phase shifts were (0, ' ), ( ' ,
*
'
), (0,
*
'
), and ( ' , ' ); thus, the
intensities of the corresponding interferometric fringe patterns (I, , I , I* , and I' ) are given as follows:
I, =
78
31 + cosφ4
(5)
I =
78
31 − sinφ4
(6)
I* =
78
31 − cosφ4
(7)
I' =
78
31 + sinφ4
(8)
As a consequence, φ can be calculated as [27–29] φ = tan:,
7; :7< 7= :7>
(9)
2.4 Relation between deflection radius and deflection-induced optical phase retardation
The φ value from Eq. (9) carries
information about the substrate
deflection. Its optical anisotropy results in the production of two refraction beams: the ordinary beam (o-beam) and the extraordinary beam (e-beam). The geometrical relation between the incident and refracted beams is depicted in Fig. 3. According to the Snell’s law, we have n, sinθ, = n@ sinθ@
(10)
n, sinθ, = nA sinθA
(11)
where n, is the refractive index of air, θ, is the incident beam angle, n@ and
θ@ are the refractive index and refraction angle of the o-beam, and nA and θA
are the refractive index and refraction angle of the e-beam. From ∆CDA and ∆BDA, it gives
d@ = tsecθ@
(12)
dA = tsecθA
(13)
where d@ and d A are the distances of the refraction paths of the o- and ebeams, respectively, and t is the substrate thickness. In ∆DOE, sinθ, = HI
(14)
where y is the distance between an arbitrary point on the substrate and the
horizontal axis and R is the deflection radius of the substrate. Eqs. (10)–(14)
yield
dL =
MN
< < OMN < :P= Q RST
(15)
and
dA =
MU
< < OMU < :P= Q RST
(16)
Since the optical phase difference between the o- and e-beams (∆) is given by ∆= n@ d@ − nA dA
(17)
the deflection-induced optical phase retardation can be calculated as follows: φ=
X \ M
(18)
2.5 Crystal optics theory The PI substrate investigated in this study was assumed to be a uniaxial crystal having the optical axis normal to its surface, as represented in Fig. 3.
When a beam impinges at an arbitrary angle, two refraction beams, o-beam and e-beam, are generated; n@ remains constant, while nA varies with the
incident angle. Fig. 4 shows the relation between these refractive indices and the principal ones, where n@ and nA are given by [30] n@ = n$ = n]
(19)
and
nA = O
M<8 M<^
M<8 _ M< `U IM
(20)
By substituting Eqs. (19),(20) into Eq. (18), we can express the deflectioninduced optical retardation as
φ=
X M
where θ@ = sin:,
M=
MN
\ [ [ < P Q< P< P< Y < < 8 a< < : = < [ P8 cdP eU SPa aNc eU 3RST4 Z < P< 8 Pa < e SP< aNc< e P< cdP U U 8 a
sinθ, and θA = sin:,
M=
MU
sinθ, .
3. Experimental details 3.1 In-plane axes orientation of the refractive index ellipsoid
(21)
For the experiment, an undeformed PI substrate was cut and attached to the
platform of a rotation stage; its principal refractive indices n$ , n] , and nb
were 1.74, 1.74, and 1.63, respectively. This substrate/stage combination was placed in the plane polarization measurement system (Fig. 5(a)) to determine the orientation of the in-plane principal axes of the refractive index ellipsoid. A sodium light of wavelength 589 nm was used as the light source. The stage rotation degree was controlled by the LabVIEW program and images were captured at each integer degree by a CMOS camera (acA1300-30gc, Basler) equipped with an image acquisition card (PCIe-8236, National Instruments); the LabVIEW program also calculated the average gray-level value at each integer degree. 3.2 Deflection radius measurements The elliptical polarization measurement system was arranged as shown in Fig. 5(b). Three aluminum molds (Fig. 6) with different known radius (50, 55, and 60 mm) were manufactured with a computer numerical control machine. The PI substrate was cut into a rectangle along one of the in-plane principal axes of the refractive index ellipsoid to obtain a realigned substrate; then, it was bonded on a mold surface, and placed in the experimental setup (Fig. 5(b)) to measure their deflection radius. The second quarter-wave plate and the analyzer were rotated to achieve the four distinct phase shifts mentioned previously. The CMOS camera captured the interferometric fringe patterns and the LabVIEW program was used for digital image processing and phase reconstruction.
4. Results and discussion The images of the original and realigned PI substrates at different rotation angles are presented in Fig. 7; the regions enclosed by the red lines indicate where the beam passes through the substrate and each one include 57,489
pixels. For the original substrate (Fig. 7(a)), before starting the rotation, i.e., at γ = 0°, the image was not the darkest or brightest of the series. This implies
that the in-plane principal axes of the refractive index ellipsoid were not along the horizontal or vertical direction of the substrate but at a specific angle, which must be measured herein. Then, the intensity of the interferometric image increased when γ was increased from 0° to 30°, gradually decreased
when further increasing γ to 75°, and increased again between 75° and 90°. The corresponding average gray-level values are plotted in Fig. 8(a). Some
differences were observed between the lower part of the plot and the square of the sinusoidal function indicated in Eq. (2). In addition, the value distribution at each valley was truncated, because the gain value of the image capturing system was relatively small. It was intended to use the first peak in Fig. 8(a) to determine the angle measured. Therefore, an appropriate gain value was adjusted to avoid overexposure and to fully record the gray level variation around peaks. Furthermore, this first peak appeared at 28°, indicating that the fast axis of the substrate was at 28° to the horizontal axis; this value is also the angle which the substrate must be cut along to obtain the realigned PI substrate. To verify the precision of the abovementioned measurements, part of the realigned substrate was cut along 28° with the horizontal axis and, then, bonded on the rotation stage to repeat the experiment. Fig. 7(b) shows the images of the realigned substrate at different angles. The darkest image was obtained before starting the rotation, while the greatest intensity was observed when increasing γ from 0° to 45° . When further increasing γ , the image
gradually became darker until returning to the lowest intensity value at 90°. The relation between the average gray-level value and the rotation degree is
plotted in Fig. 8(b); the first peak appeared at 45°, confirming the measurement precision for the realigned substrate.
Then, the PI substrate was cut into three rectangles along 28° with the horizontal axis and each of them was bonded on an aluminum mold having a different radius. Their measurement results are shown in Figs. 9-11 in which the fringe patterns reveal skewed This occurred because the substrate was actually a biaxial crystal and its three principal refractive indices differed rather than being the same for both of in-plane principal refractive indices as assumed in this study. Therefore, unlike the assumed generation of an o-beam and an e-beam, when the beam impinged on the PI crystal, two e-beams were produced, both having the refractive indices dependent on the incident beam angle. This led to skewed fringe patterns. In addition, the fringe order shown in Fig. 9(a) is larger than that in Fig. 10(a) that, in turn, is larger than that in Fig. 11(a). This resulted from the deflection order corresponding to the three mold radii: 50 mm > 55 mm > 60 mm. The fringe patterns at the aforementioned four steps and their corresponding wrapping phases from Eq. (9) were calculated. The distribution of the arctan function was limited between −π/2 and π/2; after considering the numerator and denominator signs, the range was merely constrained between −π and π. Thus, when π was added to the obtained phase, the wrapping phase ranged from 0 and 2π. The wrapping phases are expressed in gray scale in Figs. 9(e)–11(e). LabVIEW was used to reconstruct the real and full-field phase maps [23– 25], removing the discontinuity of the wrapping phases shown in Figs. 9(e)– 11(e); the resulting unwrapping phases are presented in Figs. 9(f)–11(f). The data were collected along the central line and are represented by black curves in Fig. 12. Finally, nonlinear fitting [31] was performed by incorporating these values in Eq. (21) to obtain the deflection radius; the fitting curves are shown in red in Fig. 12. The experimental phase data were consistent with the fitting curves, confirming the rationality of the nonlinear fitting process. We conducted 10 measurements on the PI substrates having three different nominal deflection radii, namely, 50, 55, and 60 mm, as mentioned above; the
respective deflection radii measured were 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm.
5. Conclusion We propose a method to directly measure the deflection radius of a flexible polymeric substrate by taking the effect of its optical anisotropy into account. The relation between the deflection-induced optical phase retardation and the deflection radii of the generated o- and e- beams was also derived. First, a plane polarization measurement system was designed to experimentally determine the orientation of the in-plane principal axes of the substrate refractive index ellipsoid; next, the deflection radius of the substrate was evaluated via an elliptical polarization measurement system and the four-step phase-stepping technique was used to determine the deflection-induced optical phase retardation. Then, nonlinear regression was performed to fit the experimental phase data with the derived phase relationship. To verify the measurement precision of the proposed methodology, three PI substrates with different nominal deflection radii, namely, 50, 55, and 60 mm, were prepared for the experiment; the corresponding measured deflection radii were 51.39 ± 0.37, 57.56 ± 1.00, and 58.71 ± 0.85 mm, with errors of 2.78%, 4.65%, and 2.15%, respectively, confirming the precision of our method. The interference mechanism of the proposed technique is based on wavefront interference and, therefore, provides the advantages of full-field and high resolution. Moreover, since both the optical polarization measurement systems used are of the common-path type, they are less sensitive to environmental disturbances. Finally, since the proposed method can directly determine the deflection radius rather than the deflection of the flexible polymeric substrate, it has great potential for future applications in the residual stress estimation for flexible displays.
Acknowledgments The authors are very grateful to Prof. Ju-yi Lee of National Central University for his valuable discussion on optical polarization during the research period. This paper was supported in part by the Ministry of Science and Technology of the Republic of China (Grant no. MOST 105-2221-E-150016-MY2 and 104-2221-E-150-061).
Disclosures The authors declare no conflicts of interest. Data Availability The processed data required to reproduce these findings are available to download
from
http://dx.doi.org/10.17632/b9gz7j3szw.1#file-3b250700-
3732-4223-905c-f7bf7cb8921c.
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Fig. 1 Schematic of the incidence direction of a beam and the ellipsoid of the refractive index of the crystal
1
(a) Plane polarization measurement system
(b) Elliptical polarization measurement system Fig. 2 Illustration of plane and elliptical polarization measurement systems
2
Fig. 3 Geometrical relation between the incident and refracted beams
3
(a) o-beam
(b) e-beam Fig. 4 Relation between refractive indices of o-beam and e-beam and in-plane principal refractive indices 4
(a) Plane polarization measurement system
(b) Elliptical polarization measurement Fig. 5 Photographs of plane and elliptical polarization measurement systems
5
Fig. 6 Geometry of the aluminum mold with known radius
6
0°
0°
15°
15°
30°
30°
45°
45°
60°
60°
75°
75°
90°
90°
(a) original (b) realigned Fig. 7 Images of original and realigned PI substrates at different angles of the rotation stage
7
120
Averaged gray-level value
100
80
60
40
20
0 0
60
120
180
240
300
360
240
300
360
Degree
(a) original 120
Averaged gray-level value
100
80
60
40
20
0 0
60
120
180
Degree
(b) realigned Fig. 8 Average gray-level value of interferometric images of original and realigned PI substrates
8
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 9 Experimental result of the PI substrate under a deflection radius of 50 mm
9
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 10 Experimental result of the PI substrate under a deflection radius of 55 mm
10
(a) 1st step
(b) 2nd step
(c) 3rd step
(d) 4th step
(e) wrapping phase
(f) unwrapping phase Fig. 11 Experimental result of the PI substrate under a deflection radius of 60 mm
11
10
Phase data Fitting curve 8
Phase
6
4
2
0 -20
-10
0
10
20
Position(mm)
(a) 50mm 8
Phase data Fitting curve
7 6
Phase
5 4 3 2 1 0 -20
-10
0
10
20
Position(mm)
(b) 55 mm 6
Phase data Fitting curve
5
Phase
4
3
2
1
0 -20
-10
0
10
20
Position(mm)
(c) 60 mm Fig. 12 Data of unwrapping phase along the central line and the obtained fitting curve for the different deflection radii of the PI substrate
12
1. This study proposed an optical methodology to measure the deflection radius of flexible polymeric substrates. 2. The optically anisotropic effect of flexible-polymeric substrate was first taken into consideration. 3. The phase steeping technique was used to obtain the phase. 4. The plane and elliptically polarization systems were respectively used to measure. 5. The proposed measurement methodology can be used to the residual stress estimation applied in flexible electronics.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: