Optical materials often exhibit stress-induced birefringence^[1–5], which is a critical factor for ensuring the reliability of optical devices. Photoelasticity, a powerful stress analysis method, measures induced birefringence to detect inner stress in these materials^[6–8]. Using either linearly or circularly polarized light enables the measurement of δ (phase difference between polarization states) and corresponding τmax (maximum shear stress). Circularly polarized light, commonly employed in photoelasticity, typically uses a phase-shifting technique^[9] for τmax and θ (retardation angle) calculation. However, challenges arise in dynamic samples due to the need to rotate the detecting wave-plate and detecting polarizer to six pairs of angles, requiring a minimum of 10 s for data acquisition. This complicates efficient stress detection in dynamic samples using the classic phase-shifting method. The simultaneous use of six cameras in photoelasticity promises real-time data acquisition^[10,11]. However, it introduces errors in assessing minute stresses due to differing view angles. Additionally, a multi-wavelength method^[12] enhances high-speed photoelasticity, employing circularly polarized light beams with slightly different wavelengths. Sequential illumination captures two polarized intensity images for each wavelength. A developed computing algorithm calculates δ from four frames, and by replacing mechanical rotation with LED switching, data acquisition is achieved within milliseconds. Quasi-real-time photoelasticity with a large view field is realized, though the speed remains insufficient for rapidly changing dynamic samples. The highest sensitivity of the photoelastic measurement device^[13] achieved is about 0.1 nm phase retardation, corresponding to about 1/6000 wavelength, the spatial resolution is about several millimeters depending on the optical imaging systems applied, and temporal resolution of the video rate was reported^[14]. Photoelasticity was successfully applied to reveal biomechanics such as the bur-row extension of annelid worms^[15], the single-leg force produced by cockroaches^[16], the slithering motion of snakes^[17], and the development of soft human tissue surrogates^[18]. Typical research also includes^[19] the utilization of the natural birefringence property of human eye cornea in imaging the whole field shear distribution and evaluating its stress concentration factor, the employment of photoelastic stress analysis in characterizing UV-cross linkable polyurethane for the repair of rotator cuff tear^[20], and the analysis of strain distribution in ligaments^[21,22]. Some new techniques including Brillouin scattering microscopy have been developed recently for^[23–25] higher resolution, higher accuracy, and higher sensitivity in measuring the inner stress in materials and biological samples.

In this study, we propose a structured illumination-based multi-wavelength photoelasticity method to expedite data acquisition. This involves modulating two light beams with slightly different wavelengths using perpendicular gratings. Consequently, two distinct-wavelength beams simultaneously illuminate the sample, enabling the capture of four intensity images. This reduces overall data acquisition time to match the camera’s frame rate, reaching time on the order of tens of microseconds. This technological advance not only enhances efficiency but also makes it feasible to detect inner stress in dynamic samples efficiently using the photoelasticity method.

The classic phase-shifting photoelasticity method is illustrated in Fig. 1(A). Circularly polarized light is generated by the polarizer P1 and quarter-wave plate Q1, directed onto the birefringent sample. Transmitted light passes through another quarter wave-plate Q2 and polarizer P2 before detection by a detector^[26–28], $$\begin{gathered} I=Ib+I02[1−sin2(β−φ2)cos(2πδλ) \\ +sin2(θ−φ2)cos2(β−φ2)sin(2πδλ). \end{gathered}$$(1)

Using angles φ2 and β for the fast-axis of Q2 and the polarization axis of P2 with respect to the x-axis, the intensity images received by the detector can be determined by Eq. (1). Here θ represents the angle of the direction of σ1 with respect to the x-axis.

To determine the value of δ in Eq. (1), we rotate the quarter-wave plate Q2 and the polarizer P2 through six sets of angles φ2 and β, capturing six different intensity images (I1 to I6). Then, we use Eqs. (2) and (3) to calculate the angles θ and δ. These angles help us understand how stress is distributed in the photoelastic material we are studying, θ=0.5×tan−1(I3−I5I6−I4),(2)δ=λ2π×tan−1[(I3−I5)sin2θ+(I6−I4)cos2θI2−I1].(3)

In the multi-wavelength photoelasticity setup depicted in Fig. 1(B), a light source sequentially emits two distinct wavelengths, λ1 and λ2. A polarizer P1 and a wide-band quarter-wave plate Q1 collectively transform the incident beams into circularly polarized light. Notably, the fast axes of Q1 and Q2 are aligned. A polarization camera is employed to simultaneously capture dark-field and bright-field images that are vertically polarized to each other. Under the illumination of λ1, two images captured by the polarization camera are expressed as I1=Ib1+0.5I01[1−cos(2πδλ1)] and I1′=Ib1+0.5I01[1+cos(2πδλ1)]. A similar intensity expression can be written for λ2 as I2 and I2′. The phase retardation of δ can then be computed as δ={λ12πcos−1(I1′−I1I01),I1′−I1I01<I2′−I2I02−λ12πcos−1(I1′−I1I01),I1′−I1I01≥I2′−I2I02.(4)

However, this existing method involves the sequential activation of two illumination beams (λ1 and λ2) for capturing intensity patterns, requiring several milliseconds for data acquisition. In contrast, our proposed method, as depicted in Fig. 2, utilizes gratings G1 and G2 illuminated with λ1 and λ2, respectively, which are concurrently imaged onto the sample through lenses L1 and L2, resulting in a streamlined process and a significant reduction in the time required for data acquisition. The overall illumination on the sample can be expressed as I0=I01[Acos(k1x)+1]+I02[Bcos(k2y)+1].(5)

The images captured by the polarization camera are represented as {I1=Ib1+0.5I01[Acos(k1x)+1](1−cos2πδλ1)+Ib2+0.5I02[Bcos(k2y)+1](1−cos2πδλ2)I2=Ib1+0.5I01[Acos(k1x)+1](1+cos2πδλ1)+Ib2+0.5I02[Bcos(k2y)+1](1+cos2πδλ2).(6)

Spatial filtering of the Fourier transform of the above equation yields the following four terms: {a1=0.5I01Aδ(k−k1,0)⊗(1−cos2πδλ1)^a2=0.5I02Bδ(0,k−k1)⊗(1−cos2πδλ2)^a1′=0.5I01Aδ(k−k1,0)⊗(1+cos2πδλ1)^a2′=0.5I02Bδ(0,k−k1)⊗(1+cos2πδλ2)^.(7)

The inverse Fourier transform of Eq. (7) yields four intensity images: {I1=0.5AI01(1−cos2πδλ1)I2=0.5BI02(1−cos2πδλ2)I1′=0.5AI01(1+cos2πδλ1)I2′=0.5BI02(1+cos2πδλ2).(8)

Utilizing the relationships, cos2πδλ1=I1′−I1I1′+I1, cos2πδλ2=I2′−I2I2′+I2, the principle of the multi-wavelength photoelasticity method allows for the calculation of δ: δ={λ12πcos−1(I1′−I1I1′+I1),I1′−I1I1′+I1<I2′−I2I2′+I2−λ12πcos−1(I1′−I1I1′+I1),I1′−I1I1′+I1≥I2′−I2I2′+I2.(9)

While the computation of internal stress is achieved through the analysis of four frames of images captured by a polarization camera, the overall data acquisition time is solely determined by the frame rate of the polarization camera. This characteristic makes the multi-wavelength photoelasticity approach significantly faster than the conventional two-wavelength method.

To validate the proposed method, a proof-of-principle experiment was conducted [Fig. 3(A)]. Grating G1, with vertical pitches, was illuminated by 528 nm light via lens L1 and imaged onto the sample using beam splitter BS. Simultaneously, grating G2, with horizontal pitches, was illuminated by 508 nm light via lens L2 and imaged onto the sample through the same beam splitter BS. The sample used in the experiment is a circular PMMA plate with a thickness of 1 mm and a diameter of 10 mm. Pressure was applied at two opposite points along its diameter to measure inner pressure and related birefringence using the proposed method. The loaded sample is depicted in Fig. 3(B), and the intensity of the illumination on the sample surface is shown in Fig. 3(C), where clear cross fringes are visible. In our experiments, a class round plastic plate was loaded at two opposite points along a diameter with two screws. One screw is fixed on the mechanic base, and another screw was pressed by a piezometer, which can show the value of the pressure loaded on the round plastic plate. In our experiments, the pressured loaded is 100 N.

Vertical fringes are formed by light of 528 nm, while horizontal fringes are formed by light of 508 nm. The image of the sample was captured by a polarization camera via a quarter-wave plate Q2, with its fast axis aligned with the x-axis. The pixels of the polarization camera were categorized into four groups for this experiment. The first group featured a polarizer oriented in the x-direction, the second group had a polarizer oriented in the y-direction, the third group employed a polarizer set at a 45 degree angle to the x-axis, and the fourth group utilized a polarizer set at a −45 degree angle to the x-axis. In the course of this experiment, we specifically utilized the third and fourth groups of pixels to capture two frames of images.

Upon simultaneously activating both LEDs, the first image captured by the third group of pixels of the polarization camera is presented in Fig. 4(A1), referred to as the dark-field image. Mathematically, this can be expressed as the first equation of Eq. (6). The second image, captured by the fourth group of pixels, is shown in Fig. 4(A2), referred to as the bright-field image and expressed mathematically as the second equation of Eq. (6). Cross fringes in both Figs. 4(A1) and 4(A2) are distinctly visible. Figures 4(B1) and 4(B2) represent the Fourier transforms of Figs. 4(A1) and 4(A2) in a logarithmic scale, respectively. Bright spots on the horizontal axis denote spatial components of λ1, while bright spots on the vertical axis represent spatial components of λ2.

By selecting the (0, 1)th order from Fig. 4(B1) and performing an inverse Fourier transform, the resulting intensity is depicted in Fig. 4(C1), corresponding to I1 in Eq. (8). Likewise, by choosing the (1, 0)th order from Fig. 4(B1) and applying an inverse Fourier transform, the resulting intensity is shown in Fig. 4(D1), corresponding to I1′ in Eq. (8). Similarly, Figs. 4(B2) and 4(D2) can be obtained by selecting the (0, 1)th and (1, 0)th orders from Fig. 4(B2) and performing inverse Fourier transform on them, respectively. Figures 4(D1) and 4(D2) correspond to I2 and I2′ in Eq. (8), respectively.

From the four obtained images of I1, I1′, I2, and I2′, we computed the values of cos(I1′−I1I1′+I1) and cos(I2′−I2I2′+I2), presented in Figs. 4(E1) and 4(E2), respectively. The phase retardation induced by inner stress can be determined by Eq. (9) and is depicted in Fig. 4(F). After performing phase unwrapping on the obtained phase in Fig. 4(F) and multiplying the unwrapped phase by the photo-strain constant C of PPMA, we obtain the measured maximum shearing stress, as shown in Fig. 4(G). Since all required raw images were acquired with the single exposure of the detector, data acquisition can be finished in less than 1 ms when a strong enough light beam is applied for illumination. Compared to the original multi-wavelength photoelasticity^[12], where several frames of raw images were recorded sequentially while LED light sources were switched on/off, the data acquisition speed of this proposed method was improved by about 10 times.

To validate the accuracy of the measured maximum shearing stress in Fig. 4(G), the inner stress of the same sample was measured using the classic phase-shifting photoelastic method by obtaining six frames of images as shown in Figs. 5(A)–5(F). Using Eqs. (2) and (3), the wrapped phase retardation in Fig. 5(G) and the corresponding maximum shearing stress in Fig. 5(H) are obtained by performing phase unwrapping and multiplying by a photo-strain coefficient.

Like in other photoelastic methods, the generated wrapped phase map of this proposed method is quite clear and includes little speckle noise, and thus the common phase unwrapping method based on the transport of the intensity equation was applied in our experiments^[29]. The disparity between the proposed and the classical measurements is illustrated in Fig. 5(I). Remarkably, the differences observed are relatively minor, and the most significant discrepancies are found in positions where phase values undergo abrupt transitions from −π/2 to π/2, a characteristic of the two-wavelength method that can be corrected by adopting a third wavelength. In Fig. 5(J), the shearing stress is depicted along two dotted lines in Figs. 5(H) and 4(G). The classic phase-shifting method measurements are represented by pink curves, while our proposed method measurements are depicted by blue lines. We can find that the difference between the measured values of the proposed method to those of the conventional method is smaller than 6.4%, substantiating the viability of our approach. The phase accuracy of the conventional phase shifting method was about Δφ=0.03rad in the experiments, which has been illustrated in the literature^[30]. As discussed in Ref. [12], the phase accuracy of two-wavelength photoelasticity was determined by the tiny peaks in Fig. 5(I), which is at the scale of about Δφ=0.3rad. By multiplying Δφ by material optical-strain coefficient C, the precisions of shearing stress Δδ for the conventional method and our proposed method are 0.08 Mpa and 0.4 Mpa, respectively, which match the experimental result in Fig. 5(I) well. Unlike conventional multi-wavelength methods, our method excels in facilitating online measurements of dynamic samples. To validate this capability, we loaded the sample with a pressure of 150 N and then rapidly unloaded it to generate rapid pressure changes, inducing creep and a gradual reduction in inner stress. Utilizing our optical setup to capture images at the video rate, we computed the inner stress for each frame, resulting in a seamlessly evolving video illustrating the dynamic shearing stress. For a comprehensive demonstration, refer to Visualization 1. The observed video underscores the remarkable smoothness, providing substantial evidence of the method’s efficacy in measuring dynamic samples.

In the above experiments, the power of the light source is 3 W because the light is incoherent and radiates in 4Π cubic angle, the light reaching the sample is less than 1 W, and the light reaching the detector after passing the sample, polarizers, and wave plates is less than 1 mW essentially. Since about 7 s of time was required for the inner stress to roughly disappear from the rapid unloading of the sample, and our method was applied to record the changing of the inner stress during this period to show its capability in measuring dynamic samples, we do not need to load and unload the sample repeatedly.

In conclusion, we introduce an innovative approach to enhance the speed of the multi-wavelength photoelasticity method. By modulating the intensities of two illumination beams with slightly different wavelengths using distinct orientation gratings and simultaneously recording dark-field and bright-field images with a polarization camera, the proposed method enables the computation of inner stress with a single detector exposure. Proof-of-principle experiments validate the method’s feasibility, demonstrating a measurement error of less than 6.4% compared to the classic phase-shifting technique. The results also highlight the high measurement speed, showcasing successful measurements of dynamic samples at video frequency. Overall, these findings underscore the potential and practicality of this approach in advancing the efficiency and applicability of multi-wavelength photoelasticity. It is worth pointing out that, since there were multiple images of different spatial carrying frequencies with the single exposure of the detector, the contrast of each image was much lower than that of traditional methods. This leads to the low signal-to-noise ratio of the proposed method, and we need to obtain the recorded data with high-contrast 16-bit CCD cameras.