0.478/msr-04-004 MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 Methds fr Determinatin f Mean Speckle Size in Simulated Speckle Pattern. Hamarvá, P. Šmíd, P. Hrváth, M. Hrabvský nstitute f Physics f the Academy f Sciences f the Czech Republic, Jint Labratry f Optics f Palacky University and nstitute f Physics f the Academy f Sciences f the Czech Republic, 7. listpadu,, 774, Olmuc, Czech Republic, ivana.hamarva@upl.cz, petr.smid@upl.cz Reginal Centre f Advanced Technlgies and Materials, Jint Labratry f Optics f Palacky University and nstitute f Physics f the Academy f Sciences f the Czech Republic, Faculty f Science, Palacký University, 7. listpadu,, 774, Olmuc, Czech Republic, pavel.hrvath@upl.cz, mirslav.hrabvsky@upl.cz This paper deals with cmputatin f mean speckle size in a speckle pattern generated thrugh a numerical simulatin f speckle after reflectin f a Gaussian beam ff a rugh bject s surface. Within this simulatin varius speckle patterns are btained by means f change in a parameter f the Gaussian beam. The mean speckle size is cmputed thrugh tw appraches using bth the tw-dimensinal and the ne-dimensinal nrmalized autcrrelatin functin in intensity. Additinally, we prpse a distinct ptimizatin f the determinatin f the mean speckle size by reductin f intensity values representing detected speckle patterns. Results f the determinatin f the mean speckle size are cmpared with theretical predictins. Keywrds: Simulatin, speckle, speckle pattern, mean speckle size, autcrrelatin functin. A. NTRODUCTON N OPTCAL speckle effect [] arises when a cherent ptical beam is either reflected frm a rugh surface r prpagates thrugh a medium having scattering centers randmly distributed. n a detectin plane we can bserve a speckle pattern cnsisting f dark and bright speckles. This pattern results frm the interference f multiple cherent spherical waves, emitted by pint surces frming the bject s surface in the case f reflectin r the medium in the case f transmissin []. n general, a speckle parameter such as mean speckle size depends bth n prperties f the light and prperties f the randm surface r medium. Hwever, fr perfectly cherent light the dependence n the randm scatterer is almst negligible if the scatterer intrduces path differences greater than ne wavelength []. The mean speckle size, as a very imprtant parameter f the speckle pattern, is f great imprtance t practical applicatins, e.g., measurement f the rughness f surfaces [], detectin f the scattering center cncentratin in a bilgical fluid [3], the particle aggregatin [4] r determinatin f the ptical thickness and the particle size in the scattering media [5]. Nevertheless, this paper is fcused n determining the mean speckle size influenced purely by the prperties f the light beam. The mean speckle size f a speckle pattern crrespnds t the width f a nrmalized autcrrelatin functin r f intensity bserved in a detectin plane (x, y [], [4], [5]. This paper deals with the calculatin f the nrmalized autcrrelatin functin r and the subsequent estimatin f the mean speckle sizes x and y in bth x - and y -axis directins, while tw appraches f cmputatin are applied and cmpared t each ther. Within the first apprach, which is mre cnventinal than the secnd ne, the mean speckle size is defined as a value where the vertical (hrizntal prfile f the twdimensinal (D nrmalized autcrrelatin functin r f intensity decreases t / [4]. Hwever, substituting the value /e fr / is prpsed in the presented paper. The secnd apprach f the mean speckle size estimatin based n an alternative algrithm is prpsed in [3]. Each vertical (hrizntal intensity prfile f the speckle pattern is extracted as a series f values. Fr each prfile the nedimensinal (D nrmalized autcrrelatin functin f intensity is cmputed. The mean speckle size fr the i-th prfile (the prfile speckle size is defined as a value where the D nrmalized autcrrelatin functin r decreases t /e. The mean speckle size fr the whle detected area is defined as an average f the prfile speckle size. The advantage f the alternative algrithm is that it is faster than the cnventinal algrithm based n the cmputatin f the D autcrrelatin functin f the D intensity signal [3]. Within the apprach based n the cmputatin f the D autcrrelatin functin r f intensity, we prpse ptimizatin f the determinatin f the mean speckle size by reducing intensity values representing the detected speckle pattern. Hence, the main advantage f the ptimized methd rests nt nly n less cmputer time cnsuming algrithm, but als n the pssibility f determining the mean speckle size frm a smaller amunt f the detected intensity values. This paper uses the simulatin mdel (Fig.. accrding t [6-8] and the way f reducing intensity values (decimatin presented in [6]. The aim f the paper [6] is t present an extensin f the ne-dimensinal speckle crrelatin methd, which is primarily intended fr determinatin f ne-dimensinal bject s translatin, fr detectin f general in-plane bject s translatin. The results presented in [6] shw that the use f decimatin enables the prpsed D crss-crrelatin methd t successfully replace the time-cnsuming D crss-crrelatin methd. Unlike [6], this paper presents that the decimatin can als be useful in evaluatin f the mean speckle size and it prvides 77
MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 reasnable results. Hwever, D autcrrelatin functin needs t be cmputed in several rws and clumns (nt in all rws as in [3] while in [6] D crss-crrelatin functin is cmputed nly nce fr each axis directin frm D signal acquired after numerical prcessing f D speckle pattern. Fr simulatin f the speckle effect the numerical simulatin mdel [7], [8] is used. The main part f the prgram cde is cmprised f the Fresnel-Kirchhff diffractin integral calculatin thrugh the rectangle methd. t cmputes the distributin f the cmplex amplitude f speckle field at a distance frm the bject s surface. The bject s surface is represented by a matrix f values defining randm variable surface rughness. The presented numerical mdel enables t select initial parameters f simulatin independently, including nnzer angle f bservatin [8]. Within the presented numerical mdel the bject generating the speckle effect is illuminated by a Gaussian beam prpagating frm its waist situated at a distance L s frm the bject. A beam radius ω in the bject plane (x, y varies in cnsequence f cntrlled variatin f a beam radius ω at the beam s waist. Then several speckle patterns with different speckle sizes are simulated, since the mean speckle size depends n the size f the beam spt in the bject plane []. Numerical results f the determinatin f the mean speckle size fr six different simulated speckle patterns fr the Gaussian beam radii ω = 40 µm, 50 µm, 60 µm, 70 µm, 80 µm and 90 µm are cmpared with the results btained frm theretical relatins.. SUBJECT & METHODS As mentined abve, the mean speckle size is defined as width f the nrmalized autcrrelatin functin r f intensity f the speckle pattern bserved in the detectin plane (x, y. Fr the D nrmalized autcrrelatin functin r f intensity ne can write []-[3] (, (, (, (, (, y r (, =. ( Let us dente by x Di and y Dj the values at which functins ( and (3 are equaled t /e, i.e. r ( x Di = /e and r ( y Dj = /e. The values f x Di and y Dj define the prfile speckle sizes (radii frm the i-th rw and j-th clumn f a matrix detectr. Then averages and m D m i= Di = (4 n D n j = D j = (5 cmputed fr m rws and n clumns f the matrix detectr (m n define the mean speckle sizes fr the whle matrix detectr [3]. Let us dente this methd by the methd f D crrelatin. Nevertheless, we can ptimize the methd f D crrelatin by decimatin [9] f the D intensity signal representing the speckle pattern, which is based n reductin in reslutin f the D intensity signal by skipping f a certain amunt f the intensity values within rws and clumns f the matrix detectr. Let us dente by the factr the distance between rws (clumns f the matrix detectr, which are used within the ptimized methd f D crrelatin. Then the mean speckle sizes x D and y D cmputed frm a decimated D intensity signal can be expressed as and m D s D( i + =, m s =Flr(m/ - (6 ms i= 0 n D s D( j + y = y, n s =Flr(n/ -, (7 ns j= 0 where Flr(x gives the greatest integer less than r equal t x. Let us dente by x D and y D the mean speckle sizes (radii defined as values where bth the hrizntal and the vertical prfiles f the functin ( decrease t /e, i.e. r ( x D, 0 = /e and r (0, y D = /e. Let us dente this methd by the methd f D crrelatin. By analgy with (, the D nrmalized autcrrelatin functins r f intensity frm a selected rw and clumn f the matrix detectr can be written as ( ( ( ( ( x r ( =, ( ( ( ( ( ( y r ( =. (3 Fig.. The gemetrical arrangement fr detectin f the speckle pattern used within the numerical simulatin. 78
MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 n this paper values x D and y D, x D and y D f the mean speckle sizes in the simulated speckle patterns are determined by means f the abve-mentined methds (the methd f D crrelatin and D crrelatin, respectively. Fig.. shws the gemetrical arrangement fr detectin f the speckle pattern used within the numerical simulatin. n additin, stated values are simultaneusly cmpared with theretical mean speckle sizes x and y btained frm theretical relatins derived as fllws. Let us cnsider that the nrmalized autcrrelatin functin r ( f intensity bserved in the detectin plane (x, y can als be expressed by means f intensity distributin P(x, y f an illuminating beam in the bject plane (x, y [], [5] r (, = P π λl ( x, y exp i ( x + y P ( x, y dxdy dxdy, (8 where L is the distance between the bject plane (x, y and the detectin plane (x, y and λ is the wavelength f the light. n the case f illuminatin by the Gaussian beam the intensity distributin P(x, y in the bject plane (x, y is [0] ω x + y P, (9 ( x, y = exp ω ω where ω and ω are the radii f the Gaussian beam at its waist and in the bject plane (x, y at a distance L s frm the waist, respectively (Fig... The radius ω is determined as [0] ω = ω + ω. (0 After substituting (9 int (8 the nrmalized autcrrelatin functin r f intensity can be written as r ( (, exp π ω = + λ L. ( f the theretical mean speckle sizes x and y are defined as the x -axis and the y -axis values where the nrmalized autcrrelatin functin r f intensity decreases t /e, i.e. then exp π ω ( + = exp( λ L, ( λl λl λl + ω λl + ω Fr the next purpse, let us replace (3 and (4 by λl π csθ ω λl, + ω (3. (4, (5 λl λl, (6 + ω where ω x and ω y represent prjectins f the Gaussian beam radius ω int the x -axis and the y -axis, respectively. The prjectins are defined by virtue f a nnzer angle θ f bservatin, as is specified in the next sectin. 3. RESULTS Speckle patterns generated by an bject f the size 4 mm 4 mm after illuminatin by the Gaussian beam with the radii ω = 40 µm, 50 µm, 60 µm, 70 µm, 80 µm and 90 µm at its waist situated at the distance L s = 0. m frm the bject are simulated. The speckle pattern is detected by a matrix detectr f the size 3 mm 3 mm at the distance L = 0.4 m frm the bject. The bject cnsists f m i n i = 400 400 pints and the intensity f the speckle pattern is detected at m n = 300 300 pints. The wavelength f the light is λ = 63.8 nm. n the simulated experimental setup (Fig.., the angle between the x-axis in the bject plane (x, y and the x -axis in the detectin plane (x, y is θ = 30, whereas the y-axis is parallel t the y -axis. Hence, the speckle pattern is detected at the nnzer angle f bservatin θ = 30. Then the prjectins ω x and ω y f the Gaussian beam radius ω int the x -axis and y -axis are ω x = ω cs 30 and ω y = ω, respectively. The achieved numerical results are summarized int the fllwing graphs (Fig.. - Fig.9.. Firstly, let us explain the results illustrated in Fig.. and Fig.3. Figs.. and 3. shw the stated mean speckle size as a functin f the Gaussian beam radius ω. The theretical mean speckle sizes x and y (square marks are cmputed by means f (5 and (6 after substituting the input parameters f simulatin. Values f x D, y D and x D, y D are stated by means f the methd f D crrelatin (circle marks and the methd f D crrelatin (triangle marks, respectively. 79
MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 relative errrs ρ ( x D and ρ ( y D are higher than 3 % (Fig.4.. The reasn is that a large number f speckles is mre necessary fr the applicatin f the methd f D crrelatin than the methd f D crrelatin. The speckle patterns with larger speckles ( x, y 40 µm bviusly d nt fit the cnditin well. Fig.. Mean speckle sizes x (determined frm thery, x D and x D (stated by the D and D crrelatin methds as a functin f the Gaussian beam radius ω at its waist. The size f the matrix detectr sampled by 300 300 pints is 3 mm 3 mm. Fig.4. Relative errrs ρ ( x D, ρ ( y D, ρ ( x D and ρ ( y D f stated mean speckle sizes determined fr the size 3 mm 3 mm f the matrix detectr. Fig.3. Mean speckle sizes y (determined frm thery, y D and y D (stated by the D and D crrelatin methds as a functin f the Gaussian beam radius ω at its waist. The size f the matrix detectr sampled by 300 300 pints is 3 mm 3 mm. Fr illustratin f differences amng all stated values f x D, y D and x D, y D and the theretical mean speckle sizes x, y, Fig.4. is presented. Fig.4. shws hw relative errrs ρ ( x D, ρ ( y D and ρ ( x D, ρ ( y D f the stated mean speckle sizes depend n the theretical mean speckle sizes x, y fr the size 3 mm 3 mm f the matrix detectr. The graphs in Fig.., Fig.3., and Fig.4. shw that the mean speckle sizes x D and y D stated by means f the methd f D crrelatin are apprximately equal t the theretical mean speckle sizes x and y in the presented range f ω. Except fr the values f x D and y D fr ω = 90 µm (Fig.. and Fig.3., the stated relative errrs ρ ( x D and ρ ( y D d nt exceed 3 % (Fig.4.. On the ther hand, the mean speckle sizes x D and y D stated thrugh the methd f D crrelatin are fr higher theretical values f the mean speckle sizes ( x, y 40 µm apparently lwer than the theretical mean speckle sizes x and y. Fr x, y 40 µm the This fact can be explained as fllws. Let us cnsider ne individual speckle f circular shape situated at the matrix detectr. Then nly the central prfile speckle size x Di crrespnds t real size (radius f the speckle, whereas the ther prfile speckle sizes x Di are evidently smaller than the real speckle size. Subsequently, the resultant mean speckle size x D cmputed as an average f all prfile speckle sizes x Di is smaller as well. Nevertheless, as the number f speckles increases, the number f central prfile speckle sizes increases, thus resultant mean speckle size x D crrespnds better t the right value. T testify the abve-mentined cnsideratin, the speckle patterns detected at a larger area f the detectr recrding a larger number f speckles are simulated and subsequently analyzed. Figs.5. and 6. shw the mean speckle size as a functin f the Gaussian beam radius ω stated frm the detectr f the size 5 mm 5 mm. n rder t keep the same distance between neighbring pints in the matrix detectr as in the previus case, the number f pints f the detectr is 570 570. The ther parameters f the simulatin remain the same. Fig.7. shws hw the relative errrs ρ ( x D, ρ ( y D and ρ ( x D, ρ ( y D depend n the theretical mean speckle sizes x, y fr the size 5 mm 5 mm f the matrix detectr. As results frm Fig.5., Fig.6., and Fig.7., the stated mean speckle sizes x D and y D are apprximately equal t the theretical mean speckle sizes x and y within a larger range f speckle sizes than in the previus case illustrated in Fig.., Fig.3., and Fig.4. n this case the stated values f x D and y D significantly differ frm the theretical mean speckle sizes x and y fr x, y 80 µm (the relative errrs ρ ( x D, ρ ( y D > 3 %. 80
MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 Fig.5. Mean speckle sizes x (determined frm thery, x D and x D (stated by the D and D crrelatin methds as a functin f the Gaussian beam radius ω at its waist. The size f the matrix detectr sampled by 570 570 pints is 5 mm 5 mm. Fig.8. Stated mean speckle size x D as a functin f the factr. The size f the matrix detectr sampled by 570 570 pints is 5 mm 5 mm. Fig.6. Mean speckle sizes y (determined frm thery, y D and y D (stated by the D and D crrelatin methds as a functin f the Gaussian beam radius ω at its waist. The size f the matrix detectr sampled by 570 570 pints is 5 mm 5 mm. Fig.9. Stated mean speckle size y D as a functin f the factr. The size f the matrix detectr sampled by 570 570 pints is 5 mm 5 mm. Further, let us fcus n results acquired by means f the ptimizatin f the D crrelatin methd prpsed in sectin. The fllwing graphs (Fig.8. and Fig.9. shw behavir f the mean speckle sizes x D and y D cmputed by (6 and (7 as a functin f the factr. The theretical mean speckle sizes x and y are represented by dashed lines. As can be seen, values f x D and y D d nt change dramatically with the factr within the interval [,0]. Hence, the minimum number f rws (clumns f the matrix detectr (m n = 570 570, which ne can evaluate the mean speckle sizes x D (y D frm, is m/ = 570/0 9 (n/ = 570/0 9. Then gd results are achieved frm relatively small amunt f intensity values selected frm the whle matrix detectr. 4. CONCLUSONS ρ (x D, ρ (y D, ρ (x D ρ (y D Fig.7. Relative errrs and f stated mean speckle sizes determined fr the size 5 mm 5 mm f the matrix detectr. n this paper tw appraches f cmputatin f the mean speckle size using bth the D and the D nrmalized autcrrelatin functin r f intensity are applied n the 8
MEASUREMENT SCENCE REVEW, Vlume 4, N. 3, 04 simulated speckle patterns. The btained numerical results are cmpared with the results btained by the theretical relatins. t is shwn, that within the presented range f speckle size accuracy f determinatin f the mean speckle sizes x D, y D acquired by means f the D crrelatin methd des nt dramatically depend n the size f the matrix detectr. Reasnable results are achieved fr bth smaller and larger size f the matrix detectr (3 mm 3 mm and 5 mm 5 mm. On the ther hand, in the case f the D crrelatin methd, mre accurate results f determinatin f the mean speckle sizes x D, y D are btained by using the larger area f the detectr (5 mm 5 mm. The reasn is that the larger number f speckles in the speckle pattern is detected, and therefre, mre meaningful evaluatin f the speckle size thrugh the D crrelatin methd is perfrmed. The range f determinatin f the mean speckle sizes x D, y D by means f the D crrelatin methd increases apprximately t 60 µm, as the area f detectr increases t 5 mm 5 mm (Fig.5. - Fig.7.. Further, the ptimizatin f the methd f D crrelatin is prpsed. The ptimizatin is based n decimatin f the D intensity signal representing the detected speckle patterns by the factr. t is shwn that t get reasnable results f evaluatin f the mean speckle size, ne can select nly several rws (clumns frm the whle matrix detectr. Hence, the main advantage f the ptimized methd f D crrelatin rests n the pssibility f determining the mean speckle size frm a relatively small amunt f detected intensity values, which can psitively influence the data prcessing speed. Mrever, since sme types f matrix detectrs are able t decimate the D signal frm the area f the detectr, then lwer amunt f data frm the detectr can be transferred t the cmputer, which can additinally imprve the effectiveness f the data prcessing. ACKNOWLEDGEMENT The authrs gratefully acknwledge the Grant f the Czech Science Fundatin N. 3-30S and the Grant f the Ministry f Educatin, Yuth and Sprts f the Czech Republic N. LG3007. REFERENCES [] Dainty, J.C. (ed. (984. Laser Speckle and Related Phenmena (nd ed.. Springer-Verlag. [] Lehmann, P. (999. Surface-rughness measurement based n the intensity crrelatin functin f scattered light under speckle-pattern illuminatin. Applied Optics, 38 (7, 44-57. [3] Chicea, D. (007. An alternative algrithm t calculate the bispeckle size in cherent light scattering experiments. Rmanian Jurnal f Physics, 54 (-, 47-55. [4] Piederriere, Y., Meur, J.L., Cariu, J., Abgrall, J.F., Bluch, M.T. (004. Particle aggregatin mnitring by speckle size measurement; applicatin t bld platelets aggregatin. Optics Express, (9, 4596-460. [5] Piederriere, Y., Cariu, J., Guern, Y., Jeune, B.L., Brun, G.L., Ltrian, J. (004. Scattering thrugh fluids: Speckle size measurement and Mnte Carl simulatin clse t and int the multiply scattering. Optics Express, (, 76-88. [6] Hamarvá,., Šmíd, P., Hrváth, P., Hrabvský, M. (04. A simulatin analysis f an extensin f nedimensinal speckle crrelatin methd fr detectin f general in-plane translatin. The Scientific Wrld Jurnal, article D 704368. [7] Hamarvá,., Hrváth, P., Šmíd, P., Hrabvský, M. (0. The simulatin f the rigin and prpagatin f the speckle field generated thrugh a plane wave and Gaussian beam and its verificatin by speckle crrelatin methd. Optik, 3 (5, 404-408. [8] Hamarvá,., Hrváth, P., Šmíd, P., Hrabvský, M. (00. Cmputer simulatin f the speckle field prpagatin. n Prceedings f SPE 7746. SPE, 77460M--77460M-8. [9] Gnzales, R.C., Wds, R.E. (008. Digital mage Prcessing (3th ed.. Pearsn Educatin nc. [0] Saleh, B.E.A., Teich, M.C. (99. Fundamentals f Phtnics. Jhn Wiley & Sns. Received August 9, 03. Accepted June 0, 04. 8