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Abstract
We propose a three-wavelength elliptically polarized light photoelasticity method for high efficiency and low-cost stress measurement. By illuminating the sample with two different forms of elliptically polarized light for each wavelength sources, twelve images are acquired. From these images, phase delay and the principal internal stress difference are precisely computed using developed algorithms. Our proposed method based on an LCD panel has the unrivalled advantage that elliptically polarized light can be automatically adjusted, which reduces the mechanical rotation of the system, in contrast to the traditional six-step phase-shifting photoelasticity method, which requires manual rotation for circularly polarized light. In addition, the system has the potential to theoretically expand the area of illumination infinitely, thereby expanding the measurement area. The viability of the suggested methods is confirmed with numerical simulation and stress measurement.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Stress-induced birefringence is an important characteristic that is widely observed in optical materials such as silicon, plastic, and glass [1–3]. To investigate the form of stress, various methods have been developed based on effects of stress on materials and components [3–7]. A majority of these techniques are primarily based on stress-strain and photoelasticity measurements [8] and also incorporates a variety of other approaches such as x-ray diffraction [9,10], Stoney curvature [11–13], and micro-Raman spectroscopy [14–16]. The development of photoelasticity approaches, such as the striped glass slide [17], RGB photoelasticity [18], and phase-shifting [19–24] procedures, has sparked renewed interest due to the accessibility of cutting-edge computational methods and advanced image processing techniques. Some methods for measuring large samples appear as the sample size rises, such as the deformation loading method [25], Layer-by-layer milling method [26], and large diameter linear polarization measuring instrument [27]. However, these large-sample measurement approaches result in spoiling the sample or low accuracy. Phase-shifting, as a non-destructive stress measurement method, has a straightforward measuring process and small relative error. A phase-shifting approach to measuring stress in large samples would be useful in a range of commercial applications.
Traditional 2D stress models consider two major stresses to be orthogonal on the measurement plane, with the difference between them representing the level of stress at a given location [28,29] as
C denotes the stress optical coefficient of the sample, which is determined by the model material and the wavelength λ of the illuminating light. The thickness of the sample perpendicular to the direction of light propagation is represented by d, and $({{\sigma_1} - {\sigma_2}} )$ is the principal stress difference at the given point.
Phase shifting is a technique that involves introducing a specific phase shift by rotating a wave plate, which in turn modulates the phase information in the domain. Phase-shifting techniques, such as six-step phase-shifting [30], eight-step phase-shifting [31], and others, are currently mature technologies. In the majority of the existing phase-shifting photoelasticity investigations, a circularly polarized light is produced by two quarter-wave plates, which is then used to illuminate the sample and take six intensity images. The principle of classical six-step phase-shifting method is schematically shown in Fig. 1. The light system is symmetrical and circularly polarized light generated by a polarizer ${P_1}$ and a quarter-wave plate ${Q_1}$ illuminates the sample, and the linearly polarized light carrying stress information is captured by a CCD after passing through the quarter-wave plate ${Q_2}$ and polarizer ${P_2}$. The intensity at the detection plane is derived from the Jones matrix as
Fig. 1. Traditional photoelasticity measurement method based on circularly polarized light. ${P_1}$, ${P_2}$: Polarizers; ${Q_1}$, ${Q_2}$: Quarter-wave plates; $s$: Sample.
Table 1. The principle expression of classical six-step phase-shifting
In practical applications, classical phase-shifting methods are required to be illuminated with circularly polarized light, which requires a quarter-wave plate in front of the sample. As a result, circularly polarized light is obtained by rotating the plate. And in order to measure large samples, the aperture of the plate will increase, resulting in a sharp increase in the cost of the system.
This study proposes a simple and adaptable optical system for a stress assessment using an elliptically polarized light photoelasticity approach. By using liquid crystal display (LCD) instead of quarter-wave plates, the system can automatically adjust the form of elliptically polarized light, greatly reducing costs and providing a theoretical basis for large sample measurement. The illumination comprises of three closer wavelengths and an LCD panel directs two forms of elliptically polarized light towards the sample in a vertical position. A CCD then records images of linearly polarized light that has been modulated by a quarter wave plate and a polarizer. The principal stress difference $({{\sigma_1} - {\sigma_2}} )$ is then precisely determined from a set of 12 phase shifted images using developed algorithms. The experimental results with a polycarbonate sample show that the photoelasticity method of elliptically polarized light is in good agreement with the stress calculation results of the traditional six-step phase-shifting photoelasticity method. Moreover, this approach demonstrates twice the efficiency. The photoelasticity method using elliptically polarized light delivers greater precision than typical procedures, with an error value of only 0.36 × 105, according to numerical simulation results.
2. Theory
The principle of the proposed elliptically polarized light photoelasticity is illustrated in Fig. 2. The system is illuminated by an LED light source that emits three wavelengths (${\lambda _1},\; {\lambda _2},\; \; and\; {\lambda _3}$), while a Liquid Crystal Display (LCD) panel modulates the linearly polarized light from polarizer ${P_1}$ to an elliptically polarized light. Q is a wide-band quarter-wave plate, which modulates elliptically polarized light back to linearly polarized light, with the fast-axis of Q parallel to the major axis of elliptically polarized light. Polarizer ${P_2}$ is used to construct bright and dark fields. Since the wavelengths of illuminations are closer, respective refractive indices are almost equal. The outer polarization film must be removed from the LCD in the system in order to guarantee that elliptically polarized light is modulated. Liquid Crystals (LCs) possess both the fluidity of a liquid and the anisotropy of crystalline materials. The ordered arrangement of liquid crystal molecules results in an intermediate state having qualities of both crystal and liquid. Particularly cholesteric liquid crystals have exceptional optical rotation, making them perfect for producing elliptically polarized light [32–34]. These liquid crystal molecules collectively rotate when a voltage is applied, with the angle of rotation being affected by the voltage [35,36]. By altering the degree of rotation of the liquid crystal molecules, one can produce a variety of elliptically polarized light, each with a unique angle that places its major axis differently than linearly polarized light. The elliptically polarized light emerging from the LCD can be expressed as:
${e_x}$ and ${e_y}$ are components along major and minor axis of elliptically polarized light with a and b as respective amplitudes. w is the vibration frequency of the light wave and t is the time. The x-axis of the coordinate system is set to overlap with the major axis of elliptically polarized light. The sample is transparent and the stress induced birefringence modulates elliptically polarized light to carry the stress information which is expressed byFig. 2. Schematic of elliptically polarized light photoelasticity measurement. $LED({{\lambda_1}/{\lambda_2}/{\lambda_3}} )$: LEDs with ${\lambda _1} = 591nm$, ${\lambda _2} = 613nm$, ${\lambda _3} = 631nm$; ${L_1}$, ${L_2}$: Lens; LCD: 7-inch LCD panel with 1 piece of polarization film on the side facing ${L_1}$; $s$: Sample; Q: Wide-band quarter-wave plate; P: Polarizer; CCD: Charge coupled device.
$\theta $ represents the angle between principal stress direction and the x-axis. The orientation of analyzer ${P_2}$ is responsible for a bright or dark field illumination on the CCD plane. In particular, when the transmittance direction of polarizer ${P_2}$ is parallel to the linearly polarized light, a bright field is produced, and when it is perpendicular to the linearly polarized light, a dark field is produced. The intensities of bright and dark field images captured by CCD are represented by
In general stress models [37], the principal stress difference $({{\sigma_1} - {\sigma_2}} )$ is often regarded as the most crucial parameter, as it represents the magnitude of stress. Since C, d, and $\lambda $ are all known in Eq. (1), the phase delay $\delta $ now becomes an important parameter to determine. By using light sources of three different wavelengths for lighting and measurement, the determined value of $\delta $ becomes more accurate and Table 2 lists the final intensity calculation based on Eq. (11) for each wavelength.
Table 2. Calculated intensity of each wavelength with elliptically polarized light obtained by LC rotation angles 0° and 45°
LC rotation angle, which represents the deflection angle of the major axis of the elliptically polarized light passing through the liquid crystal, is determined by calibrating the system prior to the experiment.
Intensities ${I_1}$, ${I_3}$, and ${I_5}$ are obtained with LC rotation angle of 0° (major axis of elliptically polarized light is parallel to the transmittance axis of the polarizer) and ${I_2}$, ${I_4}$, and ${I_6}$ are obtained with LC rotation angle of 45° (major axis of elliptically polarized light makes an angle of 45° with the polarizer transmittance axis).
The intensity equations in Table 2 are combined to eliminate the term related to the principal stress direction θ and obtain the phase retardation term as $\cos \delta $. For example, $cos{\delta _1}$ is obtained by combining equations of ${I_1}$ and ${I_2}$ and applying double angle formula as
Similar steps are followed to obtain $cos{\delta _2}$ from ${I_3}$ and ${I_4}$ and $cos{\delta _3}$ from ${I_5}$ and ${I_6}$ which are further used to obtain accurate stress difference $({{\sigma_1} - {\sigma_2}} )$ [38]. Let ${\delta _1} = 2\pi Cd({{\sigma_1} - {\sigma_2}} )/{\lambda _1}$, ${\delta _2} = 2\pi Cd({{\sigma_1} - {\sigma_2}} )/{\lambda _2},$ λ1<λ2, and ${\delta _1}({x,y} )- {\delta _2}({x,y} )< 2\pi $. Wavelength dependence of $\delta $ contributes to a small difference in the calculated values of ${\delta _1}$, ${\delta _2}$ and ${\delta _3}$. If $cos({{\delta_1}} )\ge cos({{\delta_2}} )$, $co{s^{ - 1}}[{\cos ({{\delta_1}} )} ]$ is a decreasing function in the range of [0, π], that is, ${\delta _1} ={-} co{s^{ - 1}}[{\cos ({{\delta_1}} )} ]$; on the contrary, $co{s^{ - 1}}[{\cos ({{\delta_1}} )} ]$ is an increasing function in the range of [0, π], that is, ${\delta _1} = {\cos ^{ - 1}}[{cos({{\delta_1}} )} ]$. Similarly, stress difference can also be obtained from ${\delta _2}$ and ${\delta _3}$. The specific formula is shown below
Unwrapping the phase retardation and combining the results of above two calculations, the accurate stress difference $({{\sigma_1} - {\sigma_2}} )$ can be obtained by the following formula
Flow-chart of elliptically polarized light photoelasticity measurement is schematically shown in Fig. 3. Two types of elliptically polarized light with three wavelengths are used to acquire a total of 12 images in bright and dark fields. Intensities in Table. 2 are calculated from Eq. (11), with the background light removed which is further used to obtain $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$. $\frac{{{\delta _1}{\lambda _1}}}{{2\pi Cd}}$ and $\frac{{{\delta _2}{\lambda _2}}}{{2\pi Cd}}$ are obtained after phase unwrapping. The principal stress difference $({{\sigma_1} - {\sigma_2}} )$ is determined by the larger value of $\frac{{{\delta _1}{\lambda _1}}}{{2\pi Cd}}$ and $\frac{{{\delta _2}{\lambda _2}}}{{2\pi Cd}}$.
3. Simulation
The proposed method is verified in a numerical simulation using a virtual polycarbonate sample in the setup in Fig. 2. The sample has a diameter of 25 mm and it is loaded at two opposite points on the same horizontal plane and the two principal stress components (${\sigma _1}$ and ${\sigma _2}$) and phase difference ($\delta $) are expressed using the following formula [39,40]
Numerical simulation assumed the following values: ${\lambda _1} = 591nm$, ${\lambda _2} = 613nm$, ${\lambda _3} = 631nm$, $P ={-} 150N$, $d = 1.5mm$, $C ={-} 400 \times {10^{ - 13}}P{a^{ - 1}}$. Phase difference for three wavelengths $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ are calculated from 12 captured images and are presented in Figs. 4(a1)-4(a3), which corresponds to Step 3 in Fig. 3. The two wrapped images calculated by Eq. (12) and Eq. (13) are shown in Figs. 4(b1) and 4(b2), which corresponds to Step 4 in Fig. 3. The reconstructed $({{\sigma_1} - {\sigma_2}} )$ measured by elliptically polarized light photoelasticity is shown in Fig. 4(d), which corresponds to Step 6 in Fig. 3.
Theoretical stress difference obtained with the parameters used in numerical simulation is shown in Fig. 4(e). Figures 4(c1)-4(c6) show images captured by classical six-step phase-shifting method according to Table 1, and the reconstructed $({{\sigma_1} - {\sigma_2}} )$ is shown in Fig. 4(f). Figure 4(g) and 4(h) records the difference between theoretical $({{\sigma_1} - {\sigma_2}} )$ and the results obtained in numerical simulation and classical method. The errors are of the order of 10−5 (0.36 × 10−5 for elliptically polarized light photoelasticity and 0.8 × 10−5 for classical six-step phase-shifting) which concludes that the elliptically polarized light system has better measurement accuracy than the classical six-step phase-shifting method. The grid distribution of elliptically polarized light output from the LCD panel as illumination will have an impact, even if the method has shown its ability to increase efficiency in a number of stress measurements. This is addressed in the experiment section.
Fig. 4. Numerical simulation of elliptically polarized light photoelasticity and classical six-step phase-shifting photoelasticity. (a1)-(a3) $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$; (b1)-(b2) wrapped ${\delta _1}$ and wrapped ${\delta _2}$; (c1)-(c6) intensity images of classical six-step phase-shifting photoelasticity; (d) reconstructed $({{\sigma_1} - {\sigma_2}} )$ by elliptically polarized light photoelasticity; (e) theoretical $({{\sigma_1} - {\sigma_2}} )$; (f) reconstructed $({{\sigma_1} - {\sigma_2}} )$ by classical six-step phase-shifting photoelasticity; (g) Difference of (d) and(e); (h) Difference of (f)-(e).
4. Experiments
The proposed elliptically polarized light stress measurement system is shown in Fig. 5(a) along with classical six-step phase-shifting photoelasticity system in Fig. 5(b). The light source has three LEDs with wavelengths 591 nm, 613 nm, and 631 nm. The system also includes a CCD camera (Allied Vision, Germany), LCD panel (7 “3-in-1/HD, Xianglan, China) with polarization film removed and a computer to modulate elliptically polarized light. The polarizer and wide-band quarter-wave plate (Thorlabs, United States) are located on the optical axis of the system. A polycarbonate plate with a diameter of 25 mm and a thickness of 3 mm is used as a sample. Symmetrical point forces are applied at either end of the plate's horizontal diameter. The custom made three wavelength LED light sources are connected by a three-in-one optical fiber with a diameter of 1 mm. A specific controller is used to select the desired wavelength that is coupled to the fiber.
Fig. 5. Optical system for photoelasticity measurement. (a) elliptically polarized light photoelasticity and (b) classical six-step phase-shifting photoelasticity. ${\lambda _1} = 591nm$, ${\lambda _2} = 613nm$, ${\lambda _3} = 631nm$; The arrangement includes LCD (liquid crystal display), ${P_1}$, ${P_2}$, $P$(polarizers), ${Q_1}$, ${Q_2}$, $Q$(wideband quarter-wave plate), $s$(sample), ${L_1}$, ${L_2}$, $L$(lens), $FH$(fiber holder) and CCD camera.
Stress-induced birefringence of the polycarbonate sample is measured by modulating different types of elliptically polarized light as outlined in Table 2, which corresponds to Step 2 in Fig. 3. To evaluate the accuracy of the elliptically polarized light photoelasticity system, the same sample is also measured using the classical six-step phase-shifting photoelasticity method. Figures 6(a1)-6(a3) show Step 3 where $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ are calculated by elliptically polarized light photoelasticity. Wrapped ${\delta _1}$ and ${\delta _2}$ in Step 4 are shown in Fig. 6(b1) and 6(b2). Figures 6(c1)-6(c6) show intensity images obtained by classical six-step phase-shifting photoelasticity. Figure 6(d) and Fig. 6(e) are the reconstructed $({{\sigma_1} - {\sigma_2}} )$ via proposed elliptically polarized light photoelasticity and classical six-step phase-shifting photoelasticity, respectively. Both of them are evenly distributed and in good agreement with the numerical simulation results, as shown in Fig. 4 (d) and (f). Figure 6(f) shows the difference among reconstructed $({{\sigma_1} - {\sigma_2}} )$ obtained by either methods, where the maximum absolute difference is 2 Pa. Larger variations are concentrated on the edge of the sample, which may be caused by the aberration of the system or the grid formation of liquid crystal molecules.
Fig. 6. Experimental verification of elliptically polarized light photoelasticity. (a1) -(a3) $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$; (b1) -(b2) wrapped ${\delta _1}$ and wrapped ${\delta _2}$; (c1) –(c6) intensity images of classical six-step phase-shifting photoelasticity; (d) reconstructed $({{\sigma_1} - {\sigma_2}} )$ via elliptically polarized light photoelasticity; (e) reconstructed $({{\sigma_1} - {\sigma_2}} )$ via classical six-step phase-shifting photoelasticity; (f) difference of (d) and (e)
The effectiveness of the suggested method for larger stress measurement is investigated, and the findings are displayed in Fig. 7. The difference obtained in reconstructed $({{\sigma_1} - {\sigma_2}} )$ by either method indicates that the proposed method also works well in larger stress measurement, where the maximum absolute difference is 3 Pa.
Fig. 7. Experimental verification of elliptically polarized light photoelasticity in higher pressure. (a1) -(a3) $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$; (b1) -(b2) wrapped ${\delta _1}$ and ${\delta _2}$; (c1) –(c6) intensity images of classical six-step phase-shifting photoelasticity; (d) reconstructed $({{\sigma_1} - {\sigma_2}} )$ via elliptically polarized light photoelasticity; (e) reconstructed $({{\sigma_1} - {\sigma_2}} )$ via classical six-step phase-shifting photoelasticity; (f) difference of (d) and (e)
In common LCD panels, liquid crystals exist in small units, resulting in a grid-like distribution. The distance between the LCD panel and the sample, and the aberrations in the system form different morphologies of the grid, with a square grid being the most common form. The existence of such grids affects the transmittance of LCD panel and hinders the measurement of background light that needs to be measured before the sample is formally measured. Considering that the grid is a high frequency spatial information, an ideal low-pass filter is used to solve the above problem. Figures 8(a1)-8(a3) show $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ without low-pass filtering, and Figs. 8(b1)-8(b3) show $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ after low-pass filtering. Without a frequency domain filtering, LC grid reduces fringe sharpness and contrast, which results in an error in stress calculation, as shown in Fig. 8(c) and Fig. 8(d). The diameter of the elliptically polarized light is determined by the size of the LCD panel and hence the system can theoretically achieve large aperture illumination of the sample.
Fig. 8. The importance of frequency domain filtering. (a1) -(a3) $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ without frequency domain filtering; (b1) –(b3) $cos{\delta _1}$, $cos{\delta _2}$, and $cos{\delta _3}$ after frequency domain filtering; (c) Reconstructed $({{\sigma_1} - {\sigma_2}} )$ without frequency domain filtering; (d) Reconstructed $({{\sigma_1} - {\sigma_2}} )$ after frequency domain filtering.
To demonstrate the suggested method's potential for industrial applications, a quantitative study is performed using a quarter-wave plate and a half-wave plate with a central wavelength of 633 nm. The internal phase delay of both plates was evenly distributed. Figure 9 (a) represents the measurement result of the half-wave plate, while Fig. 9 (b) illustrates the measurement result of the quarter-wave plate. Only 613 nm single wavelength was used to measure the above phase delay, and the mean values are 3.4221 rad and 1.6470 rad, respectively, while the theoretical results were 3.3618 rad and 1.6195 rad, respectively. The phase errors of the two plates were 1.8% and 1.7%, respectively. The photoelasticity approach using elliptically polarized light has the flexibility to replace the polarizer and CCD combo with a polarizing camera. Because the wave plate requires only two revolutions, the photoelasticity measurement efficiency is boosted by a factor of two when compared to the typical six-step phase-shifting photoelasticity.
Fig. 9. Internal phase delay measured for (a) half wave plate and (b) quarter wave plate using the proposed method.
5. Conclusion
In this paper, an elliptically polarized light photoelasticity measurement system is proposed, which employs that elliptically polarized light for illumination generated by LCD panel. The principal stress difference $({{\sigma_1} - {\sigma_2}} )$ is calculated from twelve intensity images obtained by illumination with two types of elliptically polarized light with three wavelengths. This system has an efficient modulation mode and greatly reduce the system cost, which can effectively reduce the mechanical rotation, and make it suitable for many applications. Results obtained in numerical simulation and experiments demonstrate the accuracy in the measurements. With advanced electronic control modulation technology of liquid crystals, large scale stress measurements can find more practical applications in engineering and material science.
Funding
National Natural Science Foundation of China (61827816).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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