Full-field stress analysis by holographic phase-stepping implementation of the photoelastic-coating method

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1 Full-field stress nlysis by hologrphic phse-stepping implementtion of the photoelstic-coting method T Lyubenov 1, E Stoykov 1, B Ivnov 1, W Vn Pepegem, A Degrieck nd V Sinov 1 1 Institute of Opticl Mterils nd Technologies, Bulgrin Acdemy of Sciences Acd.G.Bonchev Str., Bl.101,1113 Sofi, Bulgri Ghent University, Dept. of Mechnicl Construction nd Production, Sint-Pietersnieuwstrt 41, 9000 Gent, Belgium E-mil: t_lubenov@yhoo.com Abstrct. The pper describes system for polriscopic nd hologrphic phse-shifting implementtion of the photolstic-coting method for full-field stress nlysis. The esiest wy to build the combined system is to use lser light source. However, coherent illumintion introduces signl dependent eckle noise which worsens the ccurte phse estimtion nd unwrpping. To nswer the question how it ffects the phse retrievl of isochromtics, isoclinics nd isopchics, we modeled in the presented report the phse-shifting photoelstic mesurement in the presence of eckle noise through clcultion of the complex mplitudes in Mch-Zender interferometer combined with circulr polriscope nd mde denoising of simulted nd experimentl fringe ptterns. The ltter were recorded t pure tensile lod for PhotoStress coted smples with mechnicl stress concentrtor Kw, 4.30.Rx, 4.30.Ms 1. Introduction. A hologrphic phse-shifting reliztion of the photolstic-coting method [1-3] is n effective pproch to seprte the stress components over the tested ecimen due to its esier nd fster implementtion in comprison with the oblique-incidence method, the strip coting method nd the strin gge seprtion method [4-5]. For the purpose, series of six photoelstic fringe ptterns (FPs) re recorded t preliminry known orienttions of the polriztion elements in circulr polriscope to build both isochromtic nd isoclinic phse mps which give the loci of points with constnt difference of the principl stresses nd constnt principl stress direction reectively. In ddition, hologrphic recording of four FPs is performed for retrievl of isopchic fringes which give the sum of the principl stresses. If two-lod phse-shifting technique is pplied for unmbiguous phse retrievl, the number of the required FPs doubles [6]. This strengthens the requirements set on the ccurcy of the phse retrievl nd on the signl-to-noise rtio in the recorded FPs. The esiest wy to perform combined polriscopic nd hologrphic mesurement for full-field stress nlysis is to use lser light source. However, coherent illumintion introduces signl dependent eckle noise which worsens the ccurte phse estimtion nd unwrpping. To nswer the question how it ffects the phse retrievl of isochromtics, isoclinics nd isopchics, we modeled in the presented report the phse-shifting photoelstic mesurement in the presence of eckle noise through clcultion of the complex mplitudes in Mch-Zender interferometer combined with circulr polriscope nd mde denoising of simulted nd experimentl FPs. The ltter were recorded t pure tensile lod for PhotoStress coted smples with mechnicl stress concentrtor.

2 . Experimentl set-up nd light intensity equtions The schemtic digrm of the system we hve built for two-dimensionl hologrphic photoelstic mesurement of isochromtics/isopchics in reflection mode is depicted in figure 1. The light source ws DPSS CW generting lser which emitted verticlly polrized light t 53 nm. We use P, Q, nd A to denote polrizer, qurter-wve (retrdtion) plte, nd n nlyzer, reectively; the subscript indictes the ngle formed by the trnsmission xis of the polrizers or by the fst xes of the retrdtion pltes with the reference xis X. The first qurter-wve plte Q π/4 ws positioned t the output of the lser to produce circulrly polrized light which ws collimted by the lens L fter the bem expnder (lens L 1 ) nd the til pinhole filter (SF). A 10% bem litter formed the two rms of Mch-Zender interferometer with the 90 percents of light used to illuminte the object nd the remining 10% redirected s reference bem. The smple ws illuminted normlly to its surfce nd observed using one nd the sme 50% bem litter. When the shutter in the reference bem ws closed, the system operted s circulr polriscope for photoelstic mesurement of isochromtics nd isoclinics. The phse steps were introduced by rottion of the second qurter-wve plte Q γ nd the nlyzer A β. Accurcy of rottion ws ± 0.1 degree. For the isopchic mesurement the shutter ws open nd the interference ptterns of the object nd reference bems were recorded fter the second 50% bem litter, t different phse shifts, produced by rottion of Q γ nd prllel trnsmission xes of the nlyzer A β nd the polrizer P α. The objective L 3 formed imge plne recording of the fringe ptterns. The whole system ws ssembled on vibrtion insulted hologrphic tble. The loding device ws lso positioned on the hologrphic tble. The fringe ptterns were recorded with Bumer Optics CCD cmer with 78x58 pixels. Tensile lod Photoelstic coting Q A BS 50% L CCD Smple BS 50% P BS 10% Shutter L L Q π/4 DPSS lser M S Figure 1 Opticl set-up for phse-shifting hologrphic photoelstic mesurement of isochromtics, isoclinics nd isopchics. L 1 bem expnder, L collimtor, L 3 objective, SF til filter, BS bem-litter, M mirror, Q qurter-wve plte, P polrizer, A nlyzer. A birefringent smple R whose principl stress σ 1 direction subtends n isoclinic ngle θ with the δ,θ reference xis X is chrcterized with the following Jones mtrix: iδ1 iδ iδ1 iδ cos θ e sin θ ( e e ) sinθ cosθ ( ) iδ1 iδ iδ iδ1 e e sinθ cosθ e cos θ e sin θ e R δ,θ = (1) where δ 1, re the phse retrdtions long the directions of the principl stresses, σ 1,, of the ecimen, reectively. Determintion of the difference nd the sum of both phse retrdtions llows for evlution of the difference nd the sum of the principl stresses, δd = δ1 δ = Ctk( σ 1 σ ) nd δ s = δ1 δ = Dtk( σ1 σ ), where k = π / λ nd λ is the lser wvelength, t is the thickness of the smple, nd C nd D re the opticl constnts of the smple. The prllel, E p, nd norml, E n, to the nlyzer xis components of the light vector cn be described s

3 E E p n = A β Q γ R δ, θqπ / 4Pα E0 where E 0 is the mplitude of the light vector, nd α, β nd γ re the ngles, subtended by the trnsmission xes of the polrizer, the nlyzer nd the fst xis of the qurter-wve plte with the reference xis X. The ngle α tkes only vlues 0 nd π/. The Jones mtrices in () re given by cos β sin β 1 i cos γ i sin γ A β =, Q γ = sin β cos β (3) i sin γ 1 i cos γ The intensity recorded t given pixel of the CCD cmer is complex conjugte. For closed interferometric chnnel we hve I = E p E p () where sterisk denotes in (). The vlues of β nd γ s well s the light intensity equtions re given in [3,7]. From these equtions one redily obtins the formuls for clcultion of the wrpped phse mps of isoclinics, isochromtics nd isopchics: s P π / 1 I 3 5 θ = rctg, sin δ d 0 (4) I6 4 ( I ) sin θ ( I ) cos θ tn δ d = (5) I 1 I = rctn I 7 7 I 8 8 I δ t cos( / ) cos θ δ d (6) where the intensities I 1 I10 correond to different combintions of α, β nd γ. For correct unwrpping of the phse mps (4)-(6) we pplied the two-loding method [6]. To describe nlyticlly the impct of the eckle noise on the recorded FPs, we ssumed tht the complex mplitude fter reflection of the lser bem from the photoelstic smple is rndom phsor sum [8] nd modeled the eckle noise s delt-correlted circulr Gussin process in the loction of the ecimen with the Jones vector of light just fter reflection from the ecimen given by E δ, θ == e iδ ( 11 u0 ( 1 1 b ) i( 1 b ) i( b ) b ) where u ws the mplitude of the lser light field,, b were clculted s N ( 0,0.5) rndom numbers, 0 with (, s ) 11 1 (7) N being n independent Gussin vrible with men nd vrince s, nd the coefficients,, for the cse of isochromtics re given by , 11 1 = sinθ wheres for the cse of isopchics by 11 1 cosθ = sinθ sin Θ ( sin Θ sinθ ) 1 cosθ sin θ ( cosθ cosθ ) = sinθ sin Θ cos θ sin cos θ θ 1 ( sin Θ sinθ ) cosθ cos ) = sinθ ( θ where Θ m = θ m δ d. The rel nd imginery prts of the complex mplitudes t the output of the set-up in figure 1 for isochromtics nd isopchics re given in [3]. From the complex mplitudes one esily obtins the expressions for the intensities which re recorded by the CCD cmer in the presence of eckle noise. In the cse of the interferometric mesurement (FPs I ) one should tke into ccount the reference bem with the complex mplitude ( i ) U 0 exp ϕ 0. 7 I 10 (8), (9)

4 3. Denoising of fringe ptterns We simulted the combined polriscopic-hologrphic mesurement for n epoxy resin disk with rdius 1. mm, thickness 6 mm, C = m N -1 nd D = m N -1 under concentrted dimetrl compression. The wvelength ws 53 nm. The pplied two lods were 350 N nd 355 N. The simulted imge size in the plne of the smple ws pixels. The free propgtion fter the smple to the second qurter-wve plte ws modeled in Fresnel pproximtion [9] with complex mplitude of the light flling on the qurter-wve plte given by 1 U ( x, y) = FFT { H FFT U ( x y) ]}, where U s ( x, y) is the complex mplitude fter the smple, H QW s, [ is the coherent trnsfer function of the free ce nd FFT[..] denotes the Fourier trnsform. We modeled lso the integrtion by the CCD cmer within window pixels, so the size of the recorded imge ws pixels. The cing of the interferometric pttern ws 0.6 mm. In the simultion we kept the rtio 9:1 between the bem illuminting the object nd the reference bem. We pplied the following denoising lgorithms: 1) medin filtering, ) Jmes-Stein filter fter homomorphic trnsformtion of the signl [10], nd 3) filtering with n nnul filter in the frequency domin fter normliztion of the fringe pttern [11]. The first two filters yielded smooth profiles of the recorded intensity if the size of the verging window ws greter thn 1x1 pixels. The second filter provided better results. The drwbck of using lrge filtering windows is tht the estimtes of the intensities I 1 I 10 re negtively or positively bised. However, since the bises in different estimtes re more or less correlted nd the input dt for (4)-(5) re not the intensities but their differences, one cn chieve stisfctory ccurcy of phse retrievl of isochromtics nd isoclinics. To illustrte this sttement, we present in figure one-dimensionl profiles of the differences I 1, I3 5 nd I6 4 for the noiseless fringe ptterns nd the eckled ptterns fter verging with Jmes-Stein filter in window of 15x15 pixels. The profiles re tken long line 9 mm below the top of the disk to be comprtively fr wy from the point of force ppliction. Since the simultion hs been mde for concentrted force pplied to point, σ 1 ± σ experiences rpid increse in its vicinity, mking determintion of δ s, d close to it unrelible. Figure exhibits good coincidence between the filtered profiles nd the noiseless ones. Figures 3 nd 3 c show the profiles of δ d nd θ tken long the sme line s in figure from the noiseless nd filtered unwrpped isochromtic nd isoclinic phse mps. Figures 3 b nd 3 d present the histogrms of the differences between the noiseless nd filtered vlues. As it cn be seen, the error for estimtion of isochromtics is red between rd nd 0.01 rd wheres for the isoclinics it is within the intervl (-0.05, 0.05) rdins. The expression for δ s contins the intensities I 7 I 10 both in the nomintor nd denomintor, nd some of them re included with opposite signs. We obtined much better result if denoising ws pplied directly to I7 8 I 9 10 nd I7 I thn to ech of the FPs seprtely. Nevertheless, the error in determintion of isopchics occupies greter intervl (figures 3 e nd 3 f) in comprison with determintion of δ nd θ. d Figure One-dimensionl distributions of intensity differences long the X xis t 9 mm below the top of the disk under unidirectionl compression; X xis is norml to the direction of the pplied force (grey line noiseless profile, blck line filtered profile).

() (b) (c) (d) (e) (f) Figure 3 One-dimensionl unwrpped distributions of δ d, θ nd δ s (,c,e) long the X xis t 9 mm below the top of the disk under unidirectionl

the filtered FPs. 4. Filtering of experimentl fringe ptterns.

The studied smples were with length 35 mm nd width 0 mm. The thickness of the tested film ws 0.1 mm. We used thick PhotoStress coting [1] to induce birefringence.

We mesured the ptterns for film with hole with dimeter of 6 mm in the middle s stress concentrtor.

5 () (b) (c) (d) (e) (f) Figure 3 One-dimensionl unwrpped distributions of δ d, θ nd δ s (,c,e) long the X xis t 9 mm below the top of the disk under unidirectionl compression (grey line clcultion from noiseless FPs, blck line clcultion from the filtered FPs); histogrms of errors in determintion of δ d, θ nd δ s (b,d,f) from the filtered FPs. 4. Filtering of experimentl fringe ptterns. The performed photoelstic experiment included mesurement of isochromtics/isopchics of window security film with different stress concentrtors t pure tensile loding. The studied smples were with length 35 mm nd width 0 mm. The thickness of the tested film ws 0.1 mm. We used thick PhotoStress coting [1] to induce birefringence. Instlltion of the coting on the smple ws mde using dhesives of the producer. We mesured the ptterns for film with hole with dimeter of 6 mm in the middle s stress concentrtor. () (b) (c) (d) Figure 4 (,b) 8-bit imges of the noisy nd filtered FP ; (c ) intensity profiles long the line AB for the noisy (blck line) nd the filtered (grey line) FPs; (d) 8 bit imge of the unwrpped isochromtics phse mp. I Figure 4 shows the effect of denoising of n experimentl FP correonding to recorded t loding 0 N with the Jmes-Stein filter with the size of verging window 1x1. Figure 4 c gives the intensity I

Filtering ws performed fter normliztion of the fringe ptterns in the frequency domin [11]. The filter filed to remove the line noise observed in the recorded imges I 7 I10 (see figure 5b).

The result of this procedure is shown in figures 5 c nd d which exhibit substntil improvement in the qulity of the filtered imge.

6 profile long the line AB. Successful retrievl of the isochromtic phse mp is presented in figure 4 d. The obtined distribution of σ1 σ is smooth nd consistent with the theory. Figure 5 depicts the FP recorded for the sme smple nd sme loding for clcultion of isopchics fter filtering with I 7 the Jmes-Stein filter within window of pixels. Filtering ws performed fter normliztion of the fringe ptterns in the frequency domin [11]. The filter filed to remove the line noise observed in the recorded imges I 7 I10 (see figure 5b). To remove this noise we pplied nother pproch normliztion nd filtering with n nnul filter in the frequency domin with inner nd outer rdii 4 nd 70 pixels. The result of this procedure is shown in figures 5 c nd d which exhibit substntil improvement in the qulity of the filtered imge. () (b) (c) (d) Figure 5 () 8 bit imge of the normlized nd filtered with the Jmes-Stein filter FP ; (b) intensity profile long line AB in figure (); (c) 8 bit imge of the normlized nd filtered with the nnul filter in the frequency domin FP ; (d) intensity profile long line AB in figure (c). I 7 In summry, we proved by simultion the possibility for ccurte mesurement of isochromtics/isopchics using combined hologrphic polriscopic system with coherent light source. Accurcy of phse retrievl of isochromtics nd isoclinics from properly filtered nd normlized FPs is completely cceptble. Processing of the experimentl dt confirmed this conclusion. Retrievl of isopchics is more vulnerble to eckle noise nd requires dditionl refinement of the filtering procedure. I 7 References [1] Yoneym S, Morimoto Y, nd Kwmur M 005 Mes. Sci. Technol [] Stoykov E, Lyubenov T, nd Sinov V 011 Proc. SPIE N-1-10 [3] Dulieu-Brton J, nd Quinn S 008 Appl. Mech. Mt [4] Solguren-Besco Fernández M, Alegre Clderon J, Brvo Diez P, nd Cuest Segur I 010 J. Strin Anl. Eng. Des. 45, 1-17 [5] Tong L, Asundi A, nd Boy C 000 Opt. Eng [6] Lei Z, Yun H, Yun D, nd Kng Y 007 Opt. Ls. Eng [7] Goodmn J 008 Speckle Phenomen in Optics: Theory nd Applictions (USA: Roberts&Compny Publishers) [9] Equis E, nd Jcquot P 006 Proc. SPIE [10] Terrillon J-C 1996 Appl.Opt [11] Ocho N, nd Silv-Moreno A 007 Opt. Commun. 70, () [1] Tech Note TN-700,

β 1 = 2 π and the path length difference is δ 1 = λ. The small angle approximation gives us y 1 L = tanθ 1 θ 1 sin θ 1 = δ 1 y 1

β 1 = 2 π and the path length difference is δ 1 = λ. The small angle approximation gives us y 1 L = tanθ 1 θ 1 sin θ 1 = δ 1 y 1 rgsdle (zdr8) HW13 ditmire (58335) 1 This print-out should hve 1 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 001 (prt 1 of ) 10.0 points