## Abstract

A three-dimensional (3-D) residual stress detection technique is proposed to detect and evaluate the residual stress occurring in optical components due to repairs carried out at laser induced damage sites. It is possible with a cross-orthogonal reflective photo-elastic setup to obtain complete 3-D information of the residual shearing stress around the damage site. The damaged volume of the optical component is numerically sliced into multilayers for this purpose and reflected light intensity is recorded from each layer. The shearing stress from the reflected light intensity is then calculated based on photo-elasticity theory. The validity of the approach is also verified in experiments where it could measure 3-D residual stress with an axial resolution of 10 µm along the light path.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

High power lasers have become highly indispensable in current research that requires power outputs that are of the order of 10^{13} W [1]. This is made possible by incorporating several thousands of large scale optical elements in the driver setup. However, any minor defect in it can cause serious undesirable modulation of laser beam because of strong optical nonlinear effects - leading to small-scale foci of ultra-high intensities that can damage downstream optical elements [2]. Furthermore, new defects may arise from laser induced damages that are capable to generate more subsequent damages on other optical elements [3]. Such damages can rapidly degrade the performance of laser driver. These damages are to be detected and repaired as soon as it occur and becomes an integral part of routine maintenance of laser driver. Laser damage repair is commonly realized by evaporating tiny fragments or melting cracks with focused laser beam to restore the transparency of the damaged region [4]. Since CO_{2} laser has high absorption and suitable thermo-mechanical property in fused-silica, it is the most commonly applied laser for such practices [5]. However, residual stress may arise around those repaired spots and can cause new cracks, if the induced stress is too strong. This may become further reason for subsequent damages on the downstream optical elements. Challenges in dealing with a strong residual stress that is evolved from CO_{2} laser repair techniques limits its wider application in the field of high power laser. Any technique that can timely measure and evaluate the extent of residual stress around any damaged spot before and after its repair can definitely enhance the quality of the repair and can further guide us towards further technological improvement. However, an accurate 3-D stress detection is still not realized because of the lack of proper tools.

Nanoindentation technique is widely used for residual stress detection because of simplicity in its setup and operation, however, it can only analyze the residual stress in superficial layers [6]. Hydrofluoric acid etching approach provides a 3-D information of residual stress by causing an irreversible destruction to the sample detected [7]. Non-destructive techniques such as X-ray diffraction [8], Raman spectroscopy [9] and photoelasticity methods [10,11] also have limitations. X-ray diffraction method is not suitable for measurement of amorphous quartz glass, and Raman spectroscopy cannot provide 3-D information in most of cases. Photo-elastic method can provide only an average value of residual stress of laser damage by detecting the stress induced birefringence of the damaged volume. In photo-elastic methods with double principle stress model [12], the light beam illuminating the optical element is divided into two components along directions of two orthogonal principle stresses *σ*_{1}(*x*, *y*) and *σ*_{2}(*x*, *y*), which records its phase difference *φ*(*x*, *y*) after passing through damaged volume as φ(*x*, *y*) =*dC*[*σ*_{1}(*x*, *y*)-*σ*_{2}(*x*, *y*)], where *d* is the average thickness of damage volume and C is a constant for a given material. According to photo-elasticity theory, *σ*_{1}(*x*, *y*)-*σ*_{2}(*x*, *y*) is the maximal inner shearing stress *τ*(*x*, *y*). For uniform birefringence plate with a known thickness *d*, its inner average *τ*(*x*, *y*) can be obtained as *τ*(*x*, *y*) = *φ*(x, y)/*d*C [13–15]. However, for laser repaired damage spots, the shearing stress *τ*(*x*, *y*, *z*) is a fast varying function of coordinates (*x*, *y*, *z*), while what is obtained in common photo-elasticity device is the integration of *τ*(*x*, *y*, *z*) in *z* direction ∫*τ*(*x*, *y*, *z*)*dz*. Hence, such a measurement of laser repaired damage with common photo-elasticity device is of little help to understand the stress generation mechanism and consequent technological improvement.

A 3-D photo-elastic measurement technique is proposed to make an accurate measurement of *τ*(*x*, *y*, *z*) on laser induced damage sites and a proof-of-principle experiment is carried out to verify its feasibility. The proposed 3-D stress measurement technique can measure *τ*(*x*, *y*, *z*) with a *z*-axial resolution of 10 µm, making it a first non-destructive technique to achieve such a *z*-axial resolution inside an optical element.

## 2. Principle

Studies on laser induced damage suggests the distribution of residual stress around laser damage site as shown in Fig. 1 [16,17]. A view along the *z*-axis is shown in Fig. 1(a) with a radial principal-stress ${\mathrm{\sigma }_{{ /{/} }}}({x,y,z} )$ and a circumferential principal-stress ${\mathrm{\sigma }_ \bot }({x,y,z} )$ at the damaged site. In Fig. 1(b), the stress induced birefringence material around damage site is numerically sliced into multilayers along the z-axis. $n_k^{{ /{/} }}({x,y} )$ and $n_k^ \bot ({x,y} )$ are the two refractive indices of *k ^{th}* layer for light components polarized in radial and circumferential directions, respectively. A light beam illuminates the material in Fig. 1(b) from the top. Since the change in refractive index caused by the stress is always much less than a few thousandth of original refraction index

*n*[18,19], the intensity of the laser beam illuminating each layer is assumed to be the same as that of incident light on the first layer. If the illumination has an intensity of E

_{0}_{0}and a linear polarization angle of

*θ*with respect to

*x*-axis, the radially and circularly polarized components reflected from the interface of

*k*and

*k*+1 layers can be approximated by Eq. (1).

The expression for intensity is

*(*

_{k}*x*,

*y*) is the difference between shearing stresses of

*k*and

*k*+1 slices, we can obtain stress τ

*(*

_{k}*x*,

*y*) of the

*k*

^{th}slice in Fig. 1(b) with ${\tau _k}({\textrm{x},\; \textrm{y}} )= \mathop \sum \limits_{i = 1}^{k - 1} \mathrm{\Delta }{\tau _i}({x,\; y} )$ after measuring the reflected light intensity

*I*(

_{k}*x*,

*y*)|

_{k }_{= 1,2…}from each interface and calculating Δτ

*(*

_{k}*x*,

*y*) with Eq. (7). However, Eq. (7) gives only the absolute value of Δτ

*(*

_{k}*x*,

*y*) and we should find a way to determine its sign.

Figure 2 schematically shows the change in reflected intensity with the depth of reflecting interface. According to Fresnel reflection formula, light energy reflected from interface with equal refractive index on both sides is equal to zero, and the sign of Δ*τ* will change at this interface. Thus, in Fig. 2, Δ*τ* will change its sign at the depth of *z *= *z*_{k}, where the reflected energy reaches its minimum *I*_{min}. Since the inner stress grows step by step with the increasing depth from z = 0, Δ*τ* should take positive value in the range of (0, *z _{k}*) and negative value for z >

*z*. If there are several local minima $I_{\min }^1$, $I_{\min }^2$, $I_{\min }^3$, $I_{\min }^4$, ⋯, $I_{\min }^n$ at z = z

_{k}_{1},z

_{2},z

_{3}, z

_{4}⋯ z

_{n}, the sign of Δ

*τ*can be determined as Δ

*τ>*0 in the range of (z

*, z*

_{k}

_{k}_{+1})|

_{k }_{= 0,2,4…}and Δ

*τ*<0 in the range of (z

*, z*

_{k}

_{k}_{+1})|

_{k }_{= 1,3,5…}.

It is worth to point out that the discrete slicing in Fig. 1 may seem to be unreasonable at the first glance, since the refractive index always change continuously around the damage site, and thus the light intensity reflected exactly from the *k ^{th}* interface at z =

*k*Δz is approximately equal to zero. However, since there is a finite z-axis resolution of

*δz*for any practical measurement device, the received intensity

*I*(

_{k}*x*,

*y*) is essentially the integration of reflected light energy within the range of (

*k*Δz-δz/2,

*k*Δz+δz/2), thus

*I*(

_{k}*x*,

*y*) will have a non-zero value in practice, and the refractive indices $n_k^{{ /{/} }}$ and $n_k^ \bot $ the

*k*

^{th}slice essentially take their average values from the same range. In other words, the discrete slicing in Fig. 1(b) is a reasonable one for the proposed method.

## 3. Experiments

White light interferometry is used to set up the proposed photo-elastic experiment in reflective mode and is shown in Fig. 3. A linearly polarized beam from a super continuum laser source (SC-PRO, YSL photonics, China) with a power of 0.8 W is split into two beams after expansion, one illuminates the damaged sample from the other side of damaged site, and the other one illuminates the reference object. Light that is reflected from both arms are combined by a beam splitter and then collected by a lens. A detector is placed at the common imaging plane of both sample and reference object. The polarizer and analyzer are orthogonal to each other. Since the central wavelength of the laser source is 850 nm and full width of its spectrum at half maximum is 100 nm, the coherence length of the illumination is about 7.2 µm. Thus only the light reflected from the *k*^{th} interface, whose optical path length is equal to that of reference object (or optical path length difference smaller than 7.2 µm), can interference with reference light to form regular interference fringes. Thus, by scanning the reference object along the optical axis step by step, intensities of light reflected from all interfaces in Fig. 1(b) can be measured separately, and then Δτ* _{k}*(

*x*,

*y*) at all depths. A refractive index matching oil is used on the back surface of the optical element to eliminate the reflection from glass-air interface so that it does not spoil the signal light reflected from the birefringence volume of the sample. Additionally, a grating is used as a reference object rather than a simple mirror because the first order diffraction of grating can generate a phase ramp without introducing additional optical path difference. The quarter wavelength plate in Fig. 3 adjusts the polarization incident on reference grating and the intensity of the reference light reaching the detector can be adjusted by rotating the quarter wavelength plate. The sample is a piece of fused-silica glass with a laser induced damage on its surface. It was produced by focused pulsed laser with a power of 5.3 J, wavelength of 1064 nm in a time duration of 1 ms. The left image of the top left insert is the photo of the laser induced damage used.

Figure 4(a) shows a set of recorded interferograms where we can see both damage pit and regular interference fringes, and the red broken lines shown the edges of damage caves. The two bright spots in Fig. 4(a) marked with tiny green arrows are reflected light from tiny fragments around the edge of damage cave, which can be found in the upper left insert images of Fig. 3. Since only reflected light with optical path length equal to that of reference beam can generate interference fringes, the fringe visibility is quite low. Figure 4(b) shows the measured intensities *I _{k}*(

*x*,

*y*) which is reflected from interfaces at varying depth. The residual shearing stress τ

*(*

_{k}*x*,

*y*) of

*k*

^{th}layer can be obtained by calculating $\tau ({x,\; y} )= \mathop \sum \limits_{i = 1}^{k - 1} \mathrm{\Delta }{\tau _k}({x,\; y} )$ using Δτ

*(*

_{i}*x*,

*y*)|

_{i}_{=1,2,3…}computed with Eq. (7). Some of the obtained τ

*(*

_{k}*x*,

*y*) were shown in Fig. 4(c), where we can find that the residual shearing stress was almost rotationally symmetric at each depth, and became stronger as we go from the surface to deeper into the optical element and has reached the maximum at a depth of about 60 µm. It then became weaker with increasing depth and approaches to zero at a depth of about 110 µm.

Since no other method is available now to realize similar 3-D stress measurement, it was not possible to compare the measurement result in Fig. 4(c) with any previous measurements obtained through other experimental methods. However, the inner stress around the damage cave produced by long laser pulse can be well predicted theoretically and numerically for rotationally symmetric laser beam and uniform materials [17,20]. We adopted Finite Element Method (FEM) to calculate the inner stress around laser induced damage cave with parameters same as those used in experiment for Fig. 4(c). To be specific, the laser wavelength, the pulse energy and the pulse duration were assumed to be 1064 nm, 5.3 J and 1ms, respectively. The material was assumed as fused silica, and the cross energy profile of the laser beam was assumed to be Gaussian function with a waist diameter of 350 µm. The shearing stresses are calculated at the same depths and are shown in Fig. 4(d). A comparison of both results show that the general trend of change in stress with increasing depth matches with the experimental results obtained in Fig. 4(c).

For further comparison and analysis, residual stress is calculated for 12 slices and are plotted three-dimensionally in Fig. 5(a). The average stress along the radial and depth directions are plotted in Figs. 5(b) and 5(c), respectively. Solid red line represents the average shear stress at an unrepaired site and it peaks at a depth of 60 µm as shown in Fig. 5(c). The average radial distribution at the depth 60 µm is plotted with solid red line in Fig. 5(b), where the maximum stress of 14 MPa appears at a radial distance of about 380 µm, and another smaller peak at 600 µm. Further, the residual stress has decreased exponentially. The 3-D simulated shearing stress is shown in Fig. 5(d), which is quite similar to Fig. 5(a). Average shear stress obtained in simulations (dotted line in Figs. 5(b) and 5(c)) follow the same trend as that of the experimental results represented by solid red line. Both curves peaks approximately at the same position. This verifies the validity of the suggested method as it is in good agreement with the simulation results.

The laser induced damage was measured again with the same setup after it was repaired with CO_{2} laser. The photo of the repaired damage site was shown as the inserted figure of Fig. 3, where we can find that the damage site has become quite smooth. Three dimensionally measured residual stress is shown in Fig. 5(e), where the measured residual stress is much weaker than that was shown in Fig. 5(a). The average stress that is changing with depth is shown with a solid green line in Figs. 5(b) and 5(c) and its maximum still appears at a depth of 60 µm. The residual stress of repaired damage is remarkably weaker than that of original laser damage, but it is still quite strong.

In order to check the accuracy of this proposed 3-D stress measurement technique, we used it to measure the stresses of four additional laser induced damage samples, which were produced with laser pulses of different energies. The measured results are shown in Fig. 6, where Fig. 6(a), Fig. 6(b), Fig. 6(c) and Fig. 6(d) corresponded to laser pulse energies of 4.9 J, 4.4 J, 3.7 J and 3.0 J, respectively. We can find that the stresses reduced obviously with decreasing laser energy, and these results reasonably match with our expectation. To check the accuracy of these measurements quantitatively, we have added all these measured stresses together along the optical axis and compared them with the measured results with common transmission photo-elasticity method [14], which can only measure the integral of shearing stress along the optical axis. Both calculated integral of our suggested method and the measured results with classic photo-elasticity are shown in 7^{th} and 8^{th} columns, respectively, where we could find that they matched quite well. The red curves in the 9^{th} column show the measurements of the suggested method along with dotted lines in the 7^{th} column. The green lines show the measurements of common transmission photo-elasticity method. The difference between red curves and green curves were always much less than 10%, which satisfactorily verifies the validity of the proposed method.

The z-resolution of above experiments was tested by using a resolution target (USAF1951) as sample and keeping all other parameters of the optical setup unchanged. Since resolution target was not a birefringence sample, its holograms were captured by rotating the analyzer about one degree to obtain some reflected light from it. A series of low coherence holograms were captured while changing the position of the reference mirror along the optical axis and keeping the sample unhanged. Figure 7 shows all the reconstructed amplitudes (square root of the reconstructed intensity) from these recorded holograms. The curve in Fig. 7 shows the average amplitude of reconstructed image changing with the position of reference mirror. When the reference mirror is moved step by step along the optical axis from the origin, the amplitude of reconstructed image also increased from zero until the mirror reached the position of z_{3}=21.5 µm, where the optical path of the reference mirror to detector was exactly equal to that of the resolution target, and the amplitude of the reconstruction reached the highest value. The reconstructed amplitude then decreased while moving the reference mirror. Since the resolution target is an infinitely thin 2-D sample, and the obtained image had a full width at half maximum of 14 µm in the direction optical axis, the imaging system in Fig. 3 had a z-resolution of 14 µm in air. Essentially, this z-resolution is mainly determined by the coherence length of laser source and is almost the same as that of common OCT (optical coherence tomography). For fused silica sample used in experiments demonstrated above, the z-resolution inside the sample should be 10 µm (14 µm divided by the refraction index of 1.4 for fused silica). Though the above analysis was on the z-resolution of amplitude image, this is also applicable to the measurement system in Fig. 3 since the stress is proportional to the square root of reflected intensity in Eq. (7).

## 4. Summary

The proposed 3-D photo-elasticity based detection technique is capable to obtain the residual stress information with a resolution that meets the requirement of current applications that makes use of high energy laser drivers. The experimental results show that the evaluated stress around the laser induced damage before and after its repair matched with the theoretical predictions. This proposed measurement technique has the advantage of 3-D detection capability, high accuracy and high axial resolution and can be adopted to measure residual stress of similar samples with only minor modifications.

## Funding

National Natural Science Foundation of China (61705092, U1730132); Fundamental Research Funds for the Central Universities (JUSRP51721B); Natural Science Foundation of Jiangsu Province (BK20170194).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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