## Abstract

This research aims to find a new way to get the intensity equations for the phase-shifting model in digital photoelasticity. The procedure is based on the rotation of the analyzer itself. From the intensity equations, the isoclinic and isochromatic equations parameters are deduced by applying a new numerical technique. This approach can be done to calculate how many images allow the resolution of the polariscope. Each image indicates the stress forces in the object. In this study the plane polariscope was used. The amount of images will determinate the number of errors and uncertainties of the study, due to the observation that the veracity of the equations increases considerably with a large amout of images. Several analyses are performed with different amounts of photographic images. The results showed the possibility to measure stress forces with high precision using plane polariscopes.

© 2016 Optical Society of America

## 1. Introduction

In phase-shifting algorithms, the change in phase is achieved by rotation of the optical elements of the polariscope. This requires that all optical elements in polariscope are able to rotate independently. Unfortunately, this is not the case for many commercially available polariscopes widely used in laboratories of industries and universities. The significant advantage of the methodology proposed in this study is that the method only changes the angle of the analyzer in the polariscope and based on that, it is possible to obtain equations for calculating the phase for any number of images in various situations.The physical reason for the proposed numerical model is based on the fact that the measurement uncertainty can be reduced by increasing the number of observations. This new method can be used with any number of photographic measures and in any plane polariscope [1]. Plane polariscope based algorithms give better isoclinic values than the methods that use a circular polariscope [2]. The advantage is its immunity to mismatch with the quarter-wave plate. The isochromatic data evaluated with algorithms based on plane polariscope are accurate, although the unwrapping process of isochromatic data is most difficult [3]. With better results in isoclinic and isochromatic analysis, it is possible to obtain less errors in separation of stresses by the shear difference technique [4]. For this reason, it is important to improve the technique elaborated in [1] for circular polariscopes and extend it for plane polariscopes. In this research, a new numerical model is suggested to overcome the limitation of the commercial polariscopes. A comparative study was made among the theoretical results and the numerical methodologies [1].

## 2. The phase-shifting method

The plane polariscope is one of the simplest optical arrangements possible in photoelastic technique [5]. The Fig. 1 shows the typical arrangement, when *P*, *R* and *A* represent polarizer, retarder (stressed model) and analyzer, respectively. The subscripts written after P and A mean the angle between the polarizing axis and the reference axis x.${P}_{\alpha}{R}_{\theta ,\delta}{A}_{\beta}$indicate that the polarizer is at angle *α* and the analyzer is at an arbitrary angle *β,* both related with the x-axis as well. The subscripts *δ* and *θ* of *R* indicate the retardation added by stressed sample and whose fast axis is at angle *θ* with x-axis.

The output intensity given from the analyzer of the polariscope arrangement ${P}_{\alpha}{R}_{\theta ,\delta}{A}_{\beta}$ is given by:

*I*accounts for the amplitude of vector of the incident light and also the constant of proportionality and

_{o}*I*represents the light intensity.

In the experimental procedure, the diameter and the thickness of the disk are *D* = 50 mm and *t* = 15 mm, respectively. An diametrically opposite load, *P* = 50 N is applied on disk. An epoxy resin with stress fringe value *F* = 4.85 N/mm/fringe was used. In photoelastic analysis, the theoretical value of isochromatic *δ* is related to two principal stress components, *σ*_{1} and *σ*_{2}, as in Eq. (2). The theoretical isoclinic angle *θ* can be calculated using Eq. (3) with stress components *σ*_{x}, *σ*_{y} and *τ*_{xy}.

Thus, using Eqs. (2) and (3), the exact value of *δ* and *θ* can be calculated for each point of the disk. Then, the experimental measurements of *δ* and *θ*, obtained from the proposed method in this research, is compared with its exact values in order to validate the new equations.

## 3. The proposed model

Using the same principle of the mathematical model proposed in [1,5–8], a new model for the phase equations is developed by the comparison with the conventional form for plane polariscopes. The general equation to calculate the photoelastic parameters with any number (*N*) of images is proposed:

*N*is the number of images, b

_{j}, d

_{j}and e

_{j}are the coefficients of numerator, c

_{j}, f

_{j}and g

_{j}are coefficients of denominator, and $j$ is the index of the sum. A mathematical model was obtained as shown in Eq. (4) for isoclinic angle θ and in Eq. (5) for isochromatic parameter

*δ*.

The mathematical model is easy to be solved for the fact that it involves a linear programming and a maximization. Both can be obtained using the Simplex Methods. The processing time to solve those equations, even using personal computers, is considerably fast, taking a few seconds to be completed.

The shift from obtaining equations for calculating the phase analytically to obtain them numerically is a significant innovation. It breaks a paradigm that was hitherto used by several authors [9–16]. After several attempts, the following mathematical model was identified to solve the problem numerically:

The mathematical models above maximize the coefficients of numerator (*b _{j}*,

*d*,

_{j}*e*) and the coefficients of denominator (

_{j}*c*,

_{j}*f*,

_{j}*g*) in both equations, thus the obtained values for them are large enough to be significant in Eqs. (4) and (5). The Step represents the increasing angle of the analyser and has values greater than two.

_{j}*N*is the number of images and an integer number between three and the value of the step.

The constraints 1 in Eqs. (6) and (7) are made so that all coefficients generate correct values for the calculation of *θ* and *δ*. To ensure that one has a hyper-restricted problem [1], suggest that the number of greater restrictions must have be least equal with the number of variables. The constraints 2 and 3 in Eqs. (6) and (7) indicate that the coefficients are real numbers with values between one and minus one. With these procedures, the error propagation is avoided. For the evaluation phase, these limiting factors will increase the values of intensity *I _{o}* that contains errors due to noise and improve discretization in pixels and in shades of gray 1. In the model, the values of

*Io*is limited between 0 and 255,

*θ*is limited between –π/4 and π/4 and

*δ*is limited between 0 and π. The analyzer angle

*β*is an integer number which varies from 0° to 90°. The

*step*defines the distance between the angles adjusted in the analyzer, for instance, a value

*step*= 4 gives four images with

*β*equally spaced from 0° to 90°, if

*N*= 4 the four images are used, if

*N*= 3 only the three first images are used. Table 1 shows the coefficients for

*N*= 4,

*step*= 4,

*α*= 90° and the angles of the analyzer (

*β*) as follows: 0°, 30°, 60°, 90°. Table 2 shows the coefficients for

*N*= 3,

*step*= 4,

*α*= 90° and the angle

*β*as follows: 0°, 30°, 60°.

To verify the new equations for the calculating phase, a computer program that generated random values of *I _{o}*,

*θ*and

*δ*was created. With Eq. (1) are calculated

*N*values of

*I*to each value of

_{j}*β*

_{j}, then the new phase equations were applied on the values of

*I*to determine whether they produced correct values of

_{j}*θ*and

*δ*. The calculations of the accuracy were performed thousands of times (at least 100000 times) for each equation in phase calculation. At least 99.999% of the times, we obtain the accuracy (│

*θ*-

^{random}*θ*││

*δ*-

^{random}*δ*│) ≤ 10

^{−6}. The models of Eqs. (4) and (5) were tested until

*step*and

*N*equal to 2000, the value at which the increment Δ

*β*would be 0.05°. This numerical test was performed to evaluate the capability of Eqs. (4) and (5) calculate correct values for

*θ*and

*δ*from intensitity values.

To evaluate the performance of the Eqs. (4) and (5) in photoelastic analysis, an experiment was performed using a disk under diametral compression, a pixel number of M = 1024 x 1024 in the digitization, a grey level between 0 and 255 and a white light source. Figure 2 shows a flowchart of processing to phase-shifting method applied.

The new equations were tested in the experimental procedures using a disk under diametral compression whose exact solution is known. Thus, using Eqs. (9) and (10), it was possible to evaluate the average relative error by comparing the experimental and exact results.

where E_{θ} and E_{δ} represent the average relative error for θ and δ, respectively. M is the number of pixels of the image and ${\theta}_{i}^{exact}$ and ${\delta}_{i}^{exact}$ are the exact values, calculated by the theory of elasticity. ${\theta}_{i}^{}$and ${\delta}_{i}^{}$ are calculated using Eqs. (4) and (5). This comparison process was made for a number of images between 3 and 46.

## 4. Results and discussion

The results calculated with the Eqs. (4) and (5) were compared with the analytical solution for the disk under compression using the Eqs. (9) and (10). We used fourty-four images (*N* from 3 to 46) and *step* values of 4, 5, 6, 7, 10, 11, 13, 16, 19, 21, 31, 37 and 46 were considered. The results obtained from Eqs. (9) and (10) are plotted in Fig. 3. The data in Fig. 3 show that the average error decreases when the number of images increases. It was also noted that for a same number of images, the average error increases when the variation of the angle *β* between the images decreases. The possible origins of this trend is unknown.

After obtained unwrapped values of *θ* and *δ* [3,17], it was applied in the shear difference technique for whole field stress separation. Both normal and shear stress were estimated. The Fig. 4 shows the results obtained with the application of the new phase calculation equations, using N = 4 and step = 4.

## 5. Conclusion

In this study a simpler numerical model for plane polariscope using the phase-shifting technique with numerical equations for the calculation of phase was developed. It was concluded that increasing the number of images, the average error decreases,which is an advantage over the conventional method. The main advantage of this numerical approach over the conventional method is its versatility, due to the fact that it could be used in any commercially available polariscope, eliminating the need for dedicated equipment to determine experimental photoelastic parameters. Furthermore, the new method facilitates the automation of the analysis; now only the analyzer is rotated to collect the photographic images. The results of the research indicates that the method provides small errors, being applicable to experimental stress or strain analysis. It was observed that by increasing the number of images there is a linear rate tendency of reduction of relative error in stress measurements.

## References and links

**1. **C. A. Magalhães and P. A. A. Magalhães Júnior, “New numerical methods for the photoelastic technique with high accuracy,” J. Appl. Phys. **112**(8), 083111 (2012). [CrossRef]

**2. **M. Ramji, V. Y. Gadre, and K. Ramesh, “Comparative study of evaluation of primary isoclinic data by various spatial domain methods in digital photoelasticity,” J. Strain Anal. Eng. **41**(5), 333–348 (2006). [CrossRef]

**3. **P. Pinit and E. Umezaki, “Absolute fringe order determination in digital photoelasticity,” JSMME **2**(4), 519–529 (2008).

**4. **M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – Issues, implementation and application,” Opt. Lasers Eng. **46**(3), 257–271 (2008). [CrossRef]

**5. **K. Ramesh, *Digital Photoelasticity: Advanced Techniques and Applications* (Springer-Verlag, 2000).

**6. **M. Ramji and K. Ramesh, “Whole field evaluation of stress components in digital photoelasticity – issues, implementation and application,” Opt. Lasers Eng. **46**(3), 257–271 (2008). [CrossRef]

**7. **K. Gallo and G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” J. Opt. Soc. Am. B **16**(2), 267–269 (1999). [CrossRef]

**8. **J. A. Quiroga and A. González-Cano, “Method of error analysis for phase-measuring algorithms applied to photoelasticity,” Appl. Opt. **37**(20), 4488–4495 (1998). [CrossRef] [PubMed]

**9. **C. Fang, Y. Xiang, K. Qi, C. Zhang, and C. Yu, “An 11-frame phase shifting algorithm in lateral shearing interferometry,” Opt. Express **21**(23), 28325–28333 (2013). [CrossRef] [PubMed]

**10. **F. Liu, Y. Wu, and F. Wu, “Phase shifting interferometry from two normalized interferograms with random tilt phase-shift,” Opt. Express **23**(15), 19932–19946 (2015). [CrossRef] [PubMed]

**11. **L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express **22**(25), 30910–30923 (2014). [CrossRef] [PubMed]

**12. **O. Medina, J. C. Estrada, and M. Servin, “Robust adaptive phase-shifting demodulation for testing moving wavefronts,” Opt. Express **21**(24), 29687–29694 (2013). [CrossRef] [PubMed]

**13. **Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express **20**(15), 16471–16479 (2012). [CrossRef]

**14. **H. Guo and B. Lü, “Phase-shifting algorithm by use of Hough transform,” Opt. Express **20**(23), 26037–26049 (2012). [CrossRef] [PubMed]

**15. **C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, G. Rodriguez-Zurita, and V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express **21**(14), 17228–17233 (2013). [CrossRef] [PubMed]

**16. **F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express **23**(8), 10794–10807 (2015). [CrossRef] [PubMed]

**17. **N. Plouzennec and A. Lagarde, “Two-wavelengh method for full-field automated photoelasticity,” Exp. Mech. **39**(4), 274–277 (1999). [CrossRef]

**18. **M. Ramji and R. G. R. Prasath, “Sensivity of isoclinic data using various phase shifting techniques in digital photoelasticity towards generalized error sources,” Opt. Lasers Eng. **49**(9-10), 1153–1167 (2011). [CrossRef]