臺灣博碩士論文加值系統

Providing a unique way to examine stress distributions in transparent or translucent materials, photoelasticity has long been an essential aspect of engineering and material science. Nowadays, photoelasticity can be implemented using two types of polariscopes: phase-stepping and fringe-scanning polariscopes. However, both these techniques have their own limitations. The first technique faces two issues: 1) imprecise or undetermined principal angle when its corresponding phase retardation approaches or equals an integer multiple of π, and 2) wrapped results of measured phase retardation and principal angle. The second, which emerged with the advent of digital photoelasticity, can return results with input from an image. However, it is sensitive to the geometrical and optical parameters of the corresponding polariscope and allows only the phase retardation to be determined.
Free from the drawbacks of the phase-stepping and fringe-scanning polariscopes, a white light OPD (optical path difference) scanning polariscope composed of a white light laser, OPD modulator, and circular polariscope is proposed for determining the sample’s whole-field optical path retardation (proportional to the phase retardation) and principal angle. With the uses of the OPD modulator, the intensity of the photoelastic fringe pattern variates with respect to the scanned OPD　introduced by the OPD modulator, and the optical path retardation and principal angle of each sample point can be extracted using two methods: curve-fitting and first-moment methods.
The results confirm the effectiveness and practicality of the proposed polariscope. When using the curve fitting method, the results show a standard deviation of 8 nm for repeated measurements of optical path retardation and 1.21° for repeated measurements of the principal angle, though the polariscope is limited when evaluating fringe orders less than 3.
Coupled with the first-moment algorithm, the polariscope demonstrates a measurement capability for the principal angle from -π/2 to π/2 and precise recognition of the optical path retardation from zero-fringe order up. Repeated measurements of the principal angle and optical path retardation show standard deviations of 0.176° and 6.4 nm, respectively. Additionally, the proposed polariscope is effective in eliminating errors arising from the mismatched wavelength of the quarter-wave plate.
Furthermore, the results demonstrate the versatility of the proposed polariscope, allowing its application to a wide variety of specimens, including those with irregular geometrical shapes.

Providing a unique way to examine stress distributions in transparent or translucent materials, photoelasticity has long been an essential aspect of engineering and material science. Nowadays, photoelasticity can be implemented using two types of polariscopes: phase-stepping and fringe-scanning polariscopes. However, both these techniques have their own limitations. The first technique faces two issues: 1) imprecise or undetermined principal angle when its corresponding phase retardation approaches or equals an integer multiple of π, and 2) wrapped results of measured phase retardation and principal angle. The second, which emerged with the advent of digital photoelasticity, can return results with input from an image. However, it is sensitive to the geometrical and optical parameters of the corresponding polariscope and allows only the phase retardation to be determined.
Free from the drawbacks of the phase-stepping and fringe-scanning polariscopes, a white light OPD (optical path difference) scanning polariscope composed of a white light laser, OPD modulator, and circular polariscope is proposed for determining the sample’s whole-field optical path retardation (proportional to the phase retardation) and principal angle. With the uses of the OPD modulator, the intensity of the photoelastic fringe pattern variates with respect to the scanned OPD　introduced by the OPD modulator, and the optical path retardation and principal angle of each sample point can be extracted using two methods: curve-fitting and first-moment methods.
The results confirm the effectiveness and practicality of the proposed polariscope. When using the curve fitting method, the results show a standard deviation of 8 nm for repeated measurements of optical path retardation and 1.21° for repeated measurements of the principal angle, though the polariscope is limited when evaluating fringe orders less than 3.
Coupled with the first-moment algorithm, the polariscope demonstrates a measurement capability for the principal angle from -π/2 to π/2 and precise recognition of the optical path retardation from zero-fringe order up. Repeated measurements of the principal angle and optical path retardation show standard deviations of 0.176° and 6.4 nm, respectively. Additionally, the proposed polariscope is effective in eliminating errors arising from the mismatched wavelength of the quarter-wave plate.
Furthermore, the results demonstrate the versatility of the proposed polariscope, allowing its application to a wide variety of specimens, including those with irregular geometrical shapes.

Table of Contents

ABSTRACT i
Acknowledgement iii
Table of Contents iv
List of Tables vii
List of Figures viii
Chapter 1. Introduction 1
1.1. Introduction 1
1.2. Phase-stepping techniques (PSTs) 3
1.3. Fringe-scanning (or TFP/RGB) techniques 13
1.4. Summary 18
1.4.1. The phase-shifting in photoelasticity 18
1.4.2. RGB photoelasticity 20
1.5. Motivation and statement of the study. 21
1.6. Thesis structure 22
Chapter 2. Theoretical background. 23
2.1. Polarization 23
2.1.1. The nature of polarized light 23
2.1.2. Polarizers 29
2.1.3. Retarders 37
2.1.4. Photoelastic sample and the stress-optic law 48
2.1.5. The Jones calculation 51
2.2. Photoelasticity 54
2.2.1. Plane polariscope 55
2.2.2. Circular polariscope 58
2.2.3. Polariscope error 59
2.2.4. Light source 60
Chapter 3. White light OPD scanning polariscope with curve-fitting algorithm. 62
3.1. Configuration 62
3.2. Measurement theory 64
3.3. Experimental setup 70
3.4. Experimental results 73
3.5. Discussions 79
Chapter 4. White light OPD scanning polariscope with first moment algorithm. 82
4.1. Introduction 82
4.2. Configuration 83
4.3. Measurement theory 84
4.4. Experimental results 92
Chapter 5. Discussions and future works 101
5.1. Discussions 101
5.1.1. White-light OPD scanning polariscope in general 101
5.1.2. White light OPD scanning polariscope with curve-fitting algorithm 106
5.1.3. White light OPD scanning polariscope with first moment algorithm 108
Chapter 6. Conclusions 112
Chapter 7. Future works 114
7.1. Simplify the OPD modulator 114
7.2. Reflection polariscope and birefringence coating 115
7.3. Half-circular polariscope 116
Appendix 119
Appendix A 119
Appendix B 120
Appendix C 121
Appendix D 127
References 129


List of Tables
Table 1.1. Summary of the intensity equation of the Hecker and Morche’s method. 4
Table 1.2. Summary of intensity equations used in Patterson and Wang’s method 5
Table 1.3. Summary of intensity equations used in three-step PST 7
Table 1.4. Summary of intensity equations used in Asundi et al.’s method 9
Table 1.5. Summary of intensity equations used in Ramji and Ramesh’s ten-step PST method. 12
Table 1.6. Summary of intensity equations to obtain isoclinic in fringe-scanning technique 17
Table 2.1. Jones vectors of polarization states 51
Table 2.2. Jones Matrices of optical elements. 54
Table 2.3. Circular polariscope arrangements. 59

List of Figures
Figure 1.1. Optical arrangement of the Hecker and Morche’s method. 3
Figure 1.2. Patterson and Wang’s six step PST method 4
Figure 1.3. Sarma et al.’s three step PST method 6
Figure 1.4. Asundi et al.’s method 8
Figure 1.5. Ramji and Ramesh’s method 10
Figure 1.6. Calibration specimens: (a) four-point-bent, (b) eccentrically loaded tensile, (c) eccentrically loaded C-shaped. Simulated fringes are embedded at the region of interest to illustrate the maximum fringes obtainable for the same given load. The relevant loading and geometric parameters used are: Fσ = 12 N/mm/fringe, P = 700 N, a = 25 mm, c = 100 mm, d = 10 mm, h = 50 mm, and thickness = 6 mm. 14
Figure 2.1. (a) The E-field linearly polarized in the first and third quadrants, (b) the same oscillating field seen from the front. (c) light linearly polarized in the second and fourth quadrants. 23
Figure 2.2. The electric field, which has a constant amplitude, rotates clockwise. 26
Figure 2.3. Elliptical light. The endpoint of the electric field vector sweeps out an ellipse as it rotates once around. 28
Figure 2.4. A wire-grid polarizer. The grid eliminates the E-field's horizontal component (the one parallel to the wires) while passing through the vertical component. 30
Figure 2.5. Dichroic crystal with the E-field parallel to the optic axis is transmitted without diminishment. 31
Figure 2.6. (a)Double image formed by a calcite crystal, (b) polarization splitting in calcite. 33
Figure 2.7. (a) Rochon prism, (b) Wollaston prism 34
Figure 2.8. Unpolarized light being scattered by a molecule 35
Figure 2.9. Reflected light becomes polarized 36
Figure 2.10. Illustration of the polarized light pass through a retarder. 38
Figure 2.11. Principal of the half-wave plate 39
Figure 2.12. Principal of the Quarter-wave plate. 41
Figure 2.13. (a) Quarter-wave plate Fresnel rhomb retarder, (b) Half-wave plate Fresnel rhomb retarder 42
Figure 2.14. Babinet compensator. 44
Figure 2.15. Soleil compensator. 45
Figure 2.16. Twyman-Green modulator itroducing the optical path diferrence of Λ. 47
Figure 2.17. Common-path modulator using the Wollaston prism. 47
Figure 2.18. Illustrated of the optical path retardation Λ_S and principal angle θ of the birefringence material. 48
Figure 2.19. The incident beam passes through (a) an optical element, (b) a series of optical elements. 52
Figure 2.20. Stressed photoelastic specimen placed in a dark-field plane polariscope 55
Figure 2.21. Photoelastic sample placed in a dark-field plane polariscope with (a) transmission axis of analyzer at 0°, (b) transmission axis of analyzer at +22.5°. 56
Figure 2.22. The stressed sample with its isoclinic and isochromatic fringes placed between the dark-field plane polariscope using the white light source. 57
Figure 2.23. Stressed photoelasticity specimen in a dark-field circular polariscope. 58
Figure 2.24. Isoschromatic fringes of the sample placed in dark-field circular polariscope. 59
Figure 3.1. Schematic of the proposed polariscope. T: transmission axes of the polarizer and analyzer; S: slow axes of the BHWP, sample, QWP1, and QWP2; each optical element's specified angle is measured from its intersection with the x-z plane. 62
Figure 3.2. Part of the OPD modulator and the routes along each of which the corresponding beam component makes a round trip to M1. 63
Figure 3.3. Simulation of the intensity correlograms of (a) I(Λ_P), (b) J(Λ_P). 66
Figure 3.4.Top view photographs of the (a) OPD modulator, (b) dark-field circular polariscope. 71
Figure 3.5. (a) Obtained I(P); (b) calculated H(P) and its fitting curve; (c) retrieved J(????????); (d) calculated G(????????) and its fitting curve. 73
Figure 3.6. Obtained θ andΛ_Sof the quartz plate. 74
Figure 3.7. Photographs of the (a) PET bottle, (b) second sample, and (c) third sample; color fringe patterns of the (d) second sample and (e) third sample captured using a dark-field circular polariscope. 75
Figure 3.8. (a) Obtained whole-field θ; (b) obtained whole-field Λ_S;(c) reconstruction fringe pattern utilizing the equation of DCP; (d) reference fringe pattern was recorded using a DCP; (e) reconstruction fringe pattern utilizing the equation of DPP; (f) reference fringe pattern was recorded using a DPP. The orange color indicates the zones with invalid data; the white-line boundaries surround the valley walls of the sample with low-contrast fringes. 76
Figure 3.9. (a) Obtained whole-field θ; (b) obtained whole-field Λ_S; (c) reconstruction fringe pattern utilizing the equation of DCP; (d) reference fringe pattern was recorded using a DCP; (e) reconstruction fringe pattern utilizing the equation of DPP; (f) reference fringe pattern was recorded using a DPP. 78
Figure 3.10. (a) Reconstruction fringe pattern utilizing the equation of DCP; (b) reference fringe pattern was recorded using a DCP. 81
Figure 3.11. (a) Reconstruction fringe pattern utilizing the equation of DCP; (b) reference fringe pattern was recorded using a DCP. 81
Figure 4.1. Schematic of the proposed polariscope. 83
Figure 4.2. Plot of (a) S_H (Λ_P) and (b) S_G (Λ_P). 88
Figure 4.3. (a) Obtained I(Λ_P); (b) obtained J(Λ_P); (c) calculated H(Λ_P) and its mirror mapping H_M (Λ_P); (d) calculated G(Λ_P) and its mirror mapping G_M (Λ_P); (e) determined SH (ΛP) and its parameters AH, QH, and T; (f) determined SG (ΛP) and its parameters AG, QG.; 93
Figure 4.4. Obtained θ and Λ_S of the quartz plate. 94
Figure 4.5. The results obtained θ and Λ_S the Soliel-Babinet compensator, (a) set Λ_S=0.335μm, (b) set Λ_S=0.670μm, (c) set ΛS=1.34μm, (d) set ????S =2.01μm. 95
Figure 4.6. Acrylic bar under three-point bending load (a) Color fringe patterns; (b) obtained whole-field θ; (c) obtained whole-field Λ_S;(d) reconstruction fringe pattern utilizing the equation of DCP; (e) reference fringe pattern was recorded using a DCP; (f) reconstruction fringe pattern utilizing the equation of DPP; (g) reference fringe pattern was recorded using a DPP. 97
Figure 4.7. The complex compression caused interaction between three acrylic discs; (a) obtained whole-field θ; (b) obtained whole-field Λ_S; (c) reconstruction fringe pattern utilizing the equation of DCP; (d) reference fringe pattern was recorded using a DCP; (e) reconstruction fringe pattern utilizing the equation of DPP; (f) reference fringe pattern was recorded using a DPP. 99
Figure 5.1. The fringe drifting for 10 minutes when (a) an electric fan running on the optical table, (b) the optical table's anti-vibration system turned off. 102
Figure 5.2. (a) Obtained full-field θ; (b) obtained full-field Λ_S. The orange color indicates the zone (also the zone containing the triangle-tooth shape boundary) with invalid data. 103
Figure 5.3. Three round trips to M1. 104
Figure 5.4. (a) Obtained whole-field θ; (b) obtained whole-field Λ_S; (c) reconstruction fringe pattern utilizing the equation of DCP; (d) reference fringe pattern was recorded using a DCP; (e) reconstruction fringe pattern utilizing the equation of DPP; (f) reference fringe pattern was recorded using a DPP. The orange color indicates the zone (also the zone of the sample’s groove walls) with invalid data; the red-line boundaries encircle the sample’s groove walls with low-contrast fringes. 106
Figure 5.5. (a) Reconstruction fringe pattern utilizing the equation of DPP; (b) reference fringe pattern was recorded using a DPP; (c) reconstruction fringe pattern utilizing the equation of DPP (the mirror mapping algorithm was used). 108
Figure 5.6. PET film; (a) obtained whole-field θ; (b) obtained whole-field Λ_S; (c) reconstruction fringe pattern utilizing the equation of DCP; (d) reference fringe pattern was recorded using a DCP; (e) reconstruction fringe pattern utilizing the equation of DPP; (f) reference fringe pattern was recorded using a DPP. 109
Figure 7.1. Schematic of the reflection polariscope. 115
Figure 7.2. Schematic of the half-circular polariscope. 116
Figure C.1. Calibration the zero-point of PZT-stage, i.e., Λ_P = 0, (a) data acquisition, (b) data analysis flowchart. 121
Figure C.2. Calibration the zero-point procedure (a) acquire the data following step c)(listed in section 3.2), (b) fitted the obtained H(i,j)(ΛP). 122
Figure C.3. (a) Data acquisition, (b) data analyses from the obtained full-field of I, and (c) data analyses from the obtained full-field of J and compute the full-field of θ and Λ_s. 122
Figure C.4. Calibration the zero-point of PZT-stage, i.e., Λ_P = 0, (a) data acquisition, (b) data analysis flowchart using first-moment algorithm. 124
Figure C.5. (a) Data acquisition, (b) data analysis flow chart from the calculated average full-field of I, (c) data analysis flow chart from the calculated average full-field of J and compute the full-field of θ and Λ_s. 124

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