Tensor tomography of the residual stress field in graded-index YAG’s single crystals

Alfred Puro; Egor Marin

doi:10.1515/jiip-2021-0047

Article

Tensor tomography of the residual stress field in graded-index YAG’s single crystals

Alfred Puro and Egor Marin

Published/Copyright: August 25, 2023

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From the journal Journal of Inverse and Ill-posed Problems Volume 31 Issue 6

Abstract

This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection.

Keywords: Tomography; non-destructive testing; inhomogeneous optical media; cristal optics; birefringence

MSC 2020: 35R30; 35Q60; 92C55

A Inversion of the ray integral H 1

We write H 1 in polar coordinates

H 1 ⁢ ( S ) = 2 ⁢ π 44 ⁢ ∫ s 1 σ r ⁢ z ⁢ sin ⁡ ( ν ) ⁢ R ⁢ ( r ) ⁢ d ⁢ r R 2 ⁢ ( r ) - S 2 ⁢ ( s ) = 2 ⁢ π 44 ⁢ ∫ s 1 σ r ⁢ z ⁢ S ⁢ ( s ) R ⁢ ( r ) ⁢ R ⁢ ( r ) ⁢ d ⁢ r R 2 ⁢ ( r ) - S 2 ⁢ ( s ) = 2 ⁢ π 44 ⁢ ∫ S R ⁢ ( 1 ) σ r ⁢ z ⁢ d ⁢ r d ⁢ R ⁢ S ⁢ d ⁢ R R 2 - S 2 .

This is the Abel integral equation and its solution can be written as

σ r ⁢ z ⁢ [ R ⁢ ( r ) ] = - 1 2 ⁢ π 44 ⁢ d ⁢ R d ⁢ r ⁢ ∫ R ⁢ ( r ) R ⁢ ( 1 ) d d ⁢ S ⁢ H 1 ⁢ ( S ) ⁢ d ⁢ S S 2 - R 2 ⁢ ( r ) .

B Inversion of the ray integral H b

The properties of integrals of this type have been studied in detail earlier [10], although an explicit analytical solution was not given there. We implement the method used in [25] to inversion H b . We transform H b by using polar coordinates R ⁢ ( r ) , φ

H 02 ⁢ ( S ) = ∫ 0 l σ ⁢ ( r ) ⁢ cos ⁡ ( 4 ⁢ α + 2 ⁢ ν ) ⁢ 𝑑 l = 2 ⁢ ∫ S R ⁢ ( 1 ) σ ⁢ [ r ⁢ ( R ) ] ⁢ L ⁢ ( R ) ⁢ cos ⁡ [ 4 ⁢ α ⁢ ( R , S ) + 2 ⁢ ν ] ⁢ R ⁢ d ⁢ R R 2 - S 2 , L ⁢ ( R ) = d ⁢ r d ⁢ R .

Bearing in mind that

ν = a ⁢ sin ⁡ ( s ⁢ n ⁢ ( s ) r ⁢ n ⁢ ( r ) ) = π 2 - arccos ⁡ ( S R ) and cos ⁡ [ 4 ⁢ α ⁢ ( R , S ) + 2 ⁢ ν ] = - cos ⁡ [ 4 ⁢ α ⁢ ( R , S ) - 2 ⁢ arccos ⁡ ( S R ) ] ,

we can write H b as

(B.1) H b ⁢ ( S ) = - 2 ⁢ ∫ S R ⁢ ( 1 ) cos ⁡ [ E ⁢ ( R , S ) ] ⁢ σ ⁢ ( R ) ⁢ L ⁢ ( R ) ⁢ R ⁢ d ⁢ R R 2 - S 2 ,

where

E ⁢ ( R , S ) = 4 ⁢ α ⁢ ( R , S ) - 2 ⁢ arccos ⁡ ( S R ) = ∫ S R S ⁢ [ 4 ⁢ D ⁢ ( K ) - 2 ] ⁢ d ⁢ K K ⁢ K 2 - S 2 , arccos ⁡ ( S R ) = ∫ S R S ⁢ d ⁢ K K ⁢ K 2 - S 2 .

By using results of [25], we will show that the inversion of integral H b can be written as

(B.2) σ ⁢ ( r ) ⁢ n ⁢ ( r ) = 1 π ⁢ r ⁢ { ∂ ∂ ⁡ r ⁢ [ ∫ R R ⁢ 0 c ⁢ h ⁢ [ B ⁢ ( S , R ) ] ⁢ H 02 ⁢ ( S ) ⁢ S S 2 - R 2 ⁢ 𝑑 S ] + ( 4 r - 2 ⁢ d ⁢ l ⁢ n ⁢ [ R ⁢ ( r ) ] d ⁢ r ) ⁢ ∫ R R ⁢ 0 s ⁢ h ⁢ [ B ⁢ ( S , R ) ] ⁢ H 02 ⁢ ( S ) ⁢ 𝑑 S } ,

where

B ⁢ ( S , R ) = ∫ R | S | S ⁢ [ 4 ⁢ D ⁢ ( K ) - 2 ] ⁢ d ⁢ K K ⁢ S 2 - K 2 .

We will not give all the proof; we will indicate only those changes that are directly related to this solution. At first, we insert (B.1) into to solution (B.2) and will have then the equation as in [25, equation (6)]. In our case, the integral G 1 (see [25, (equation (9)]) transforms to

G 1 = Re ⁢ ∫ R K c ⁢ h ⁢ [ I ⁢ ( S , R , K ) ] K 2 - S 2 ⁢ S 2 - R 2 ⁢ S ⁢ 𝑑 S , I ⁢ ( S , R , K ) = ∫ R K [ 4 ⁢ D ⁢ ( T ) - 2 ] ⁢ d ⁢ T T ⁢ 1 - ( T S ) 2 = ∑ m = 0 ∞ a m ⁢ S - m .

Here,

a 0 = ∫ R K [ 4 ⁢ D ⁢ ( T ) - 2 ] ⁢ d ⁢ T T = ∫ R K { 4 ⁢ d ⁢ l ⁢ n ⁢ [ t ⁢ ( T ) ] d ⁢ l ⁢ n ⁢ ( T ) ⁢ T - 2 T } ⁢ 𝑑 T = 4 ⁢ ∫ r ⁢ ( R ) k ⁢ ( K ) d ⁢ t t - 2 ⁢ ∫ R K d ⁢ T T = ln ⁡ { [ k ⁢ ( K ) r ⁢ ( R ) ] 4 ⁢ [ R K ] 2 } .

Thus, the integral G 1 is equal to

G ⁢ 1 = π 4 ⁢ [ ( k ⁢ ( K ) r ⁢ ( R ) ) 4 ⁢ ( R K ) 2 + ( r ⁢ ( R ) k ⁢ ( K ) ) 4 ⁢ ( K R ) 2 ] .

The integral G ⁢ c 1 (see [25, equation (11)]) transforms to

G ⁢ c 1 = 1 4 ⁢ Re ⁢ ∮ s ⁢ h ⁢ [ I ⁢ ( S , R , K ) ] K 2 - S 2 ⁢ 𝑑 S = π 4 ⁢ [ ( k ⁢ ( K ) r ⁢ ( R ) ) 4 ⁢ ( R K ) 2 - ( r ⁢ ( R ) k ⁢ ( K ) ) 4 ⁢ ( K R ) 2 ]

and (B.2) transforms to

σ ( r ) n ( r ) = - 2 π ⁢ r { ∂ ∂ ⁡ r [ ∫ R R ⁢ 0 π 4 [ ( k ⁢ ( K ) r ⁢ ( R ) ) 4 ( R K ) 2 + ( r ⁢ ( R ) k ⁢ ( K ) ) 4 ( K R ) 2 ] σ ( K ) L ( K ) K d K ]

+ ( 4 r - 2 ⁢ d ⁢ l ⁢ n ⁢ [ R ⁢ ( r ) ] d ⁢ r ) ∫ R R ⁢ 0 π 4 [ ( k ⁢ ( K ) r ⁢ ( R ) ) 4 ( R K ) 2 - ( r ⁢ ( R ) k ⁢ ( K ) ) 4 ( K R ) 2 ] σ ( K ) L ( K ) K d K } .

After differentiation, we obtain the identity. In the case of straight ray tomography ν = 0.5 ⁢ π - α , the solution (B.2) coincides with Cormack’s inversion formula for the second angular harmonics of the Radon transform. Using elementary transformations, the reconstructing algorithm can be written in the more familiar form

σ ⁢ ( r ) = d ⁢ R ⁢ ( r ) π ⁢ d ⁢ r ⁢ { [ ∫ R R ⁢ 0 c ⁢ h ⁢ [ B ⁢ ( S , R ) ] S 2 - R 2 ⁢ d d ⁢ S ⁢ H 02 ⁢ 𝑑 S ] + 4 ⁢ ∫ R R ⁢ 0 s ⁢ h ⁢ [ B ⁢ ( S , R ) ] ⁢ H 02 ⁢ ( S ) S 2 - R 2 ⁢ ∫ R S D ′ ⁢ ( K ) ⁢ d ⁢ K S 2 - K 2 ⁢ 𝑑 S } .

References

[1] H. Aben, Integrated Photoelasticity, McGraw–Hill, New York, 1979. Search in Google Scholar

[2] H. Aben, J. Anton and A. Puro, Residual stress measurement in glass articles of complicated shape using integrated photoelasticity, International conference on residual stresses, TIB, Leipzig (1997), 834–839. Search in Google Scholar

[3] H. Aben and E. Brossman, Integrated photoelasticity of cubic single crystals, VDI Ber. 313 (1978), 45–51. Search in Google Scholar

[4] H. Aben and J. Josepson, On the precision of integrated photoelasticity for hollow glasswar, Opt. Lasers Eng. 22 (1995), no. 2, 121–135. 10.1016/0143-8166(94)00019-7Search in Google Scholar

[5] H. Aben, B. R. Krasnowski and J. T. Pindera, Nonrectilinear light propagation in integrated photoelasticity of axisymmetric bodies, Trans. Can. Soc. Mech. Eng. 8 (1984), no. 4, 195–200. 10.1139/tcsme-1984-0029Search in Google Scholar

[6] J. A. Adam and M. Pohrivchak, Evaluation of ray-path integrals in geometrical optics, Int. J. Appl. Exp. Math. 1 (2016), 10.15344/ijaem/2016/108. 10.15344/ijaem/2016/108Search in Google Scholar

[7] L. Ainola and H. Aben, Hybrid mechanics for axisymmetric thermoelasticity problems, J. Thermal stresses 23 (2000), 685–697. 10.1080/01495730050130066Search in Google Scholar

[8] M. Born and E. Wolf, Principles of Optics, Pergamon Press, New York, 1968. Search in Google Scholar

[9] P. L. Chu, Nondestructive measurement of index profile of an optical–fibre preform, Electronics Lett. 13 (1977), no. 24, 736–738. 10.1049/el:19770520Search in Google Scholar

[10] M. V. de Hoop and J. Ilmavirta, Abel transforms with low regularity with applications to x-ray tomography on spherically symmetric manifolds, Inverse Problems 33 (2017), no. 12, Article ID 124003. 10.1088/1361-6420/aa9423Search in Google Scholar

[11] E. F. C. Dreyer, Fabrication and Characterization of Fiber Waveguides from Single-Crystal Er3+-Doped YAG, CLEO: Applications and Technology 2015, Optica Publishing, Washington (2015), https://doi.org/10.1364/CLEO\_␣AT.2015.JTh2A.30. 10.1364/CLEO_AT.2015.JTh2A.30Search in Google Scholar

[12] A. A. Fuki, Y. A. Kravtsov and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media, Gordon and Breach Science, Amsterdam, 1998. Search in Google Scholar

[13] J. A. Harrington, Single–crystal fiber optics: A review, Proc. SPIE 8959 (2014), 10.1117/12.2048212. 10.1117/12.2048212Search in Google Scholar

[14] S. Idnurm, Determination of stresses in cubic single crystals of cylindrical form by the Abel inversion, Proc. Acad. Sc. Est. SSR. 35 (1986), 172–179. 10.3176/phys.math.1986.2.08Search in Google Scholar

[15] D. Karov, Polarization tomography of stresses in graduated index structures, Ph.D. thesis, St. Petersburg Polytechnic University, 2012. Search in Google Scholar

[16] D. Karov and A. Puro, Integrated photoelasicity of residual stresses in axysimmetric GRIN optical structures, Proceedings of the 2018 IEEE International Conference on Electrical Engineering and Photonics (EExPolytech 2018), IEEE Press, Piscataway (2018), 194–198. 10.1109/EExPolytech.2018.8564365Search in Google Scholar

[17] 4 D. Karov, A. Puro, A. Fadeev and A. Kuzmina, Optical tomography of residual stresses in GRIN rod lenses with transverse and longitudinal translucence, J. Phys. Conf. Ser. 1236 (2019), Article ID 012038. 10.1088/1742-6596/1236/1/012038Search in Google Scholar

[18] C. Killer, Abel inversion algorithm, 2016 https://au.mathworks.com/matlabcentral/fileexchange/43639-abel-inversion-algorithm. Search in Google Scholar

[19] M. Lancry, E. Regnier and B. Poumellec, Fictive Temperature measurements in silicabased optical fibers and its application to rayleigh loss reduction, Optical Fiber New Developments, IntechOpen, London (2009), 7–137. 10.5772/7575Search in Google Scholar

[20] J. F. Nye, Physical Properties of Crystals, Clarendon Press, Oxford, 1957. Search in Google Scholar

[21] L. Pagnotta and P. Poggialini, Measurements of residual internal stresses in optical fiber preforms, Exp. Mech. 43 (2003), 69–76. 10.1007/BF02410486Search in Google Scholar

[22] Y. Park, Measurement method for profiling the residual stress of an optical fiber: Detailed analysis of off-focusing and beam-deflection effects, Appl. Opt. 42 (2003), no. 7, 1182–1190. 10.1364/AO.42.001182Search in Google Scholar

[23] G. Pretzier, Comparison of different methods of abel inversion using computer simulated and experimental side-on data, Z. Naturforsch. 47a (1992), 10.1515/zna-1992-0906. 10.1515/zna-1992-0906Search in Google Scholar

[24] A. Puro, Polarization tomography of residual stress in single crystals of YAG, Optics Spectr. 122 (2017), no. 2, 322–328. 10.1134/S0030400X17020230Search in Google Scholar

[25] A. Puro, Tomography in optically axisymmetric media, Optics Spectr. 124 (2018), no. 2, 278–284. 10.1134/S0030400X18020145Search in Google Scholar

[26] A. Puro and H. Aben, Tensor field tomography for residual stress measurement in glass articles, Proceedings of the 7th European Conference on Non-Destructive Testing, Portuguese Society of NDT, Lisbon (1998) 2390–2397. Search in Google Scholar

[27] A. Puro and D. Karov, Inverse problem of thermoelasticity of fiber gratings, J. Thermal Stresses 39 (2016), no. 5, 500–512. 10.1080/01495739.2016.1158606Search in Google Scholar

[28] A. Puro and D. Karov, Polarization tomography of residual stresses in cylindrical gradient-index lenses, Optics Spectr. 124 (2018), no. 5, 735–740. 10.1134/S0030400X1805017XSearch in Google Scholar

[29] A. Puro and K.-J. Kell, Complete determination of stress in fiber performs of arbitrary cross section, J. Light Wave Technol. 10 (1992), no. 8, 1010–1014. 10.1109/50.156839Search in Google Scholar

[30] W. A. Ramadan, Two-dimensional refractive index and birefringence profiles of a graded index bent optical fibre, Optical Fiber Technol. 36 (2017), 115–124. 10.1016/j.yofte.2017.03.005Search in Google Scholar

[31] M. Schmidt, Numerical simulation and analytical description of thermally induced birefringence in laser rods, IEEE J. Quantum Electron. 36 (2000), no. 5, 620–626. 10.1109/3.842105Search in Google Scholar

[32] J. Teichman, Measurement of gradient index materials by beam deflection, Technical Report, Institute for Defense Analyses Alexandria, 2013. 10.1117/1.OE.52.11.112112Search in Google Scholar

[33] M. J. Weber, Handbook of Optical Materials, CRC Press, Boca Raton, 2003. Search in Google Scholar

[34] J. Wu, C. S. Bohun and H. Huang, Semi-analytic solutions for a thermoelastic problem with cubic anisotropy, J. Crystal Growth. 310 (2008), 4373–4384. 10.1016/j.jcrysgro.2008.07.014Search in Google Scholar

[35] A. D. Yablon, Recent progress in optical fiber refractive index profiling, Optical Fiber Communication Conference, Optica Publishing, Wahington (2011), https://opg.optica.org/abstract.cfm?uri=ofc-2011-OMF1. 10.1364/OFC.2011.OMF1Search in Google Scholar

Received: 2021-07-21

Revised: 2022-11-23

Accepted: 2022-12-02

Published Online: 2023-08-25

Published in Print: 2023-12-01

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Articles in the same Issue

https://doi.org/10.1515/jiip-2021-0047

Keywords for this article

Tomography; non-destructive testing; inhomogeneous optical media; cristal optics; birefringence