PECULIARITIES OF CYLINDER DIAMETER DETERMINATION BY DIFFRACTION METHOD

Transcription

1 XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 PECULIARITIES OF CYLINDER DIAMETER DETERMINATION BY DIFFRACTION METHOD Y.V. Chugui ad A.A. Pavlov Laboratory for Techical Visio Techological Desig Istitute of Scietific Istrumet Egieerig Siberia Brach of the Russia Academy of Scieces 4 Russkaya str., Novosibirsk, 60058, Russia Federatio Abstract: The detailed aalysis of diffractio patter i far zoe for circular absolutely reflectig metal cylider with regard to the ifluece of wave reflected from its surface is preseted. The method of statioary phase was used whe calculatig the reflected compoet of the iput field. The behaviour of diffractio patter miima for wide rage of agles is studied. The formula for agular positio of miima i case of small observatio agles was foud. Three algorithms of cylider diameter recostructio usig its diffractio patter were proposed. They allow theoretically to iease appreciably the accuracy of the dimesioal measuremets. Keywords: Frauhofer diffractio, cylider, optical ispectio, laser metrology INTRODUCTION The metallic bodies (or their fragmets) with reflectig cylidrical surfaces are frequetly used as objects for dimesioal geometrical parameters measuremet. While precisio measurig the geometrical parameters of such exteded objects (with error about µm) by shadow ad diffractio methods as opposed to plae objects (zero thickess) it arises the ecessity of takig ito accout the diffractio pheomea peculiarities o D bodies. Sice the strict solutios of light diffractio o metallic circular cylider with R radius [] are very cosumig for egieerig applicatios ad kow works [ - 4] i this field are devoted to the exclusively experimetal ivestigatios of diffractio o cylider, a urget demad i the costructive theory of Fraughofer pheomea o such bodies appears. This theory must be simple ad at the same time sufficietly strict, aalogous to the proposed early for the exteded object of costat thickess [5]. It is essetial, that for the cyliders with perfectly reflectig surfaces the mai cotributio to far zoe field (whe the wavelegth λ << R) gives the compoet, caused by the icidet wave reflectio from the frot part of cylider [ - 4]. This field compoet is the additioal to the kow oe, correspodig to diffractio o plae see the same size D = R (plae aalogue of cylider). It brigs about the shift of positio of diffractio patter miima. As a result i determiatio (recostructio) of cylider diameter by the same algorithms as for the plae objects, deviatios take place betwee the experimetal data ad expected oes. I so doig the measuremet error of cylider diameter by diffractio method (versus D ad spatial badwidth) ca be foud from uits to tes mios [, ]. I our opiio the absece of the above metioed costructive theory of diffractio pheomea o cyliders gives o way to receive the satisfactory explaatio to the experimetally observed effects. Prelimiary results of ivestigatios of diffractio pheomea o circular perfectly reflectig cyliders were preseted by us i [6]. The aim of this work is more detailed study of peculiarities for Frauhofer diffractio pheomea o perfectly reflectig circular cylider, icludig calculatio ad aalysis of ifluece of reflected compoet o its diffractio patter (spectrum), as well as its compariso with the case of plae see, study of behaviour of the spectrum miima ad the the search for more precise ad costructive (applicable for egieerig calculatios) algorithms for diameter determiatio by its diffractio patter. DIFFRACTION PATTERN FOR ABSOLUTELY REFLECTING CYLINDER Let us cosider the light diffractio i far zoe (Frauhofer diffractio) by metal circular cylider, illumiated by moochromatic plae wave with legth λ (Fig. ). Suppose that the mai cotributios ito the diffractio field Å(θ), observed by agle θ, give two compoets: Å ref (θ) ad Å tras (θ),

2 XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 determied by diffractio of the wave, reflected from the frot part of cylider, ad usual diffractio of wave i trasmissio light by plae see with the size D = R, i.e.: x θ z Figure. Ray cofiguratio for light diffractio by metal circular cylider of radius R. Å(θ) = Å tras (θ) + Å ref (θ), Takig ito cosideratio the peculiarities of icidet wave reflectio from cylider surface [4], oe ca obtai i optogeometrical approximatio the followig equivalet iput distributio f(x) (i diametral plae of cylider): x f ( x ) = rect β exp j kr ( x R ) Y( x R ) R [ ], where k = π/λ wave umber, Y(x) is the step Heaviside fuctio, β is some coefficiet. To receive i aalytical form the field i far zoe Å(θ), which is Fourier trasformatio of the iput distributio f(x), we used uder R/λ >> the statioary phase method whe calculatig the reflected compoet Å ref (θ). As a result, the ormalized field itesity i far zoe (whe θ 0) is desibed by the expressio: I( θ) = sic ( k θ R) + 0.5β λ sic( k θ R) θ cos( k θ R where sic( z ) = si z / z. λ θ + 0.5β R π R /8+ k θ R ), 4 Except the first term, I 0 (θ)=sic (kθr), correspodig to the usual diffractio by plae see (plae see spectrum), there are two additioal terms, due to the ifluece of wave, reflected from the frot part of cylider. It is substatially, that their ifluece o distributio I(θ) appreciably ieases at agles θ θ. = λ / R, whe volume effects are strog eough, i.e. whe kr θ / 8 = π / 4. As it will be idicated below the most essetial cotributio to the field gives the last iterferece compoet, which brigs to the appearace of low-frequecy oscillatios i spectrum amog the highfrequecy oes. As a result, istead of equidistat located miima i spectrum (that is valid for plae aalogue of cylider) oe ca observe more complicated depedece. CYLINDER DIFFRACTION PATTERN ANALYSIS Let s study the peculiarities of behaviour of distributio () ad compare it with plae see spectrum I 0 (θ)=sic (kθr). () I(θ) 0.00 θ θ θ θ, rad Figure. The typical diffractio patters for the circular cylider (curve ) ad plae see of the same size D = R spectrum (curve ) at large agles (R = 60 µm ad λ = 0,6 µm). I Fig. (curve ) the typical diffractio patter for the circular cylider with R = 60 µm ad λ = 0,6 µm at agles θ > θ is show. Oe ca see that as compared to plae see spectrum (curve ) uder θ > θ there is the itesity ieasig due to the ifluece of reflected compoet i diffractio patter. I so doig, simultaeously with the high-frequecy modulatio (like Frauhofer type), caused by the iterferece of two fields from opposite cylider parts, there is the low-frequecy modulatio (by the ifluece of reflected compoet Å ref (θ)), the depth of which is deeasig for θ > θ.

3 XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 From the optical ispectio poit of view the positio of diffractio miima is the mai characteristic for the determiig the cylider radius R by diffractio method. The behaviour of cylider diffractio patter miima positios ad their deviatios from plae object s case were ivestigated i details. As metioed above, for plae see the miima of its spectrum are equidistat, i.e. ( 0 ) ( 0 ) ( 0 ) θ = θ+ θ = λ / D. I case of small observatio agles (θ << θ ) it was foud [6] that for absolutely reflectig metal cyliders the agular positio of -th diffractio miima θ (uder β = ) is determied by the followig expressio: θˆ / = θ θ = λ / D 0.5( λ / D), (4) I Eq. (4) first term θ = λ D correspods to the light diffractio o plae see, ad the secod / / oe θ = 0.5( λ / D) desibes the cotributio of reflected compoet. For example, I(θ) a), θ θ, rad I(θ) θ 0 4 b) θ, rad Figure. The diffractio patters of the circular cylider with diameter D (dash lie) ad plae see with the size D (solid lie) i the viciity of the itical agles θ (a) ad θ (b) at R = 60 µm, λ = 0.6 µm. For simplicity the positios ad the scales of dash plots were chaged. at R = 60 µm, λ = 0.6 µm, = 5 the value θ θ is 0.5λ D 0.%. As the observatio agle ieases θ > θ, the differece i miima positio for the plae see ad the cylider is growig. Moreover, at the certai agle θ = θ this differece is substatial so the positio of miimum of distributio I(θ) coicides with the positio of maximum of fuctio I 0 (θ) = sic (kθr) (Fig. a). The agle θ As a result, it equals to ca be obtaied from the followig coditio: Ψ = kθ R 8 π 4 = π. θ = λ R. Uder the give parameters R ad λ the value θ is about While further ieasig the observatio agle up to θ = θ umber of miima of iterferece patter for plae object ad cylider differs by oe, i.e. the fuctio sic (kθr) (the plae object case) has N miima ad the distributio I(θ) icludes N+ miima (Fig. b). As for the itical agle θ., it equals to 7λ / R (Ψ=/π) ad for the same parameters R ad λ correspods to θ 4 0.

4 XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 I practice it s more coveiet to deal rather with the umber of diffractio miima tha with agular miimum positio. Let s defie itical parameter N as the umber of diffractio miima of distributio I 0 (θ) (with agular distace betwee them θ 0 =λ/d), icluded i a agular rage 0 θ θ. It is equal to N θ θ 7( R λ) equals to N = 8. =. Critical umber for the same λ ad R 0 = 4 CYLINDER DIAMETER RECONSTRUCTION ALGORITHMS While recostructio the cylider diameter D by its diffractio patter let s take as a zero approximatio the stadard algorithm applied to the case of plae see. Accordig to this algorithm the recostructed cylider diameter equals to D = 0 λ / θ (θ is the positio of -th miimum i cylider diffractio patter). We have ivestigated i detail the behaviour of error, takig place i this case: = D D = λ / θ 0 D, (5) I Fig.4 there is the depedece of error D versus umber. Oe ca see that the behaviour of DD, mm 4 I I Figure 4. Error D() versus umber of diffractio miimum uder the cylider diameter (D) recostructio usig the stadard algorithm applied to the plae see with the size D, zoes I, ad I are spectrum regios for the recostructio (λ=0,6 µm, D =0 µm). error D() strogly depeds o the observed regio. So, i regio of small agles (I) the error, accordig to Eq. (4) ieases proportioally to. I case of middle agles (regio ) this error is a ~ costat value. Fially, i the regio of large agles (I) the error is. Three algorithms of parameter D determiatio by cylider diffractio patter were developed. The first algorithm uses some first miima of diffractio patter ad is based o Eq. (4). By solvig this equatio regardig to D (with kow θ ad ) oe may receive the followig expressio for the cylider diameter D evaluatio: D = λ / θ 0. 5λ (6) I I such a way the error D of diameter D is desibed by the simple equatio: λ (7) D I = 0.5 ad, for example, at = 4 equals to 0,5λ. Accout of error D accordigly to Eq.(7) allows to improve the measuremet accuracy. The allowable upper limit for agle θ I, used for the recostructio D equals to θ 0.θ = 0. 7λ R. I Accordig to the secod algorithm of diameter recostructio while usig the diffractio agles i the middle regio for which D = cost, oe ca assume from the physical poit of view that the error D value is the followig: ~ λ D. It was established that proportioal costat value is 0,. As a result, oe may receive the followig expressio for the error value:

5 XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 D = 0. λ D0 I Fig. 5 there is a plot of fuctio (8) depedig o the diameter recostructio error. The differece of approximated fuctio, for example, at R = 60 µm, from the real error value λ = 0.6 µm is about µm. The agle rage, i which this approximatio fuctio is valid, is determied by iequality 0.θ θ 0.5θ. Oe ca see that the methodical error of give algorithm is egligible small ad, i our opiio, the mai cotributio to error will give oes of miima agular positio determiatio., µm, µm D Figure 5. Error of diameter recostructio () versus diffractio umber i the middle agular regio while usig the stadard algorithm applied to plae see (dashed plot).approximatio fuctio is show by solid plot. Th-e third algorithm of cylider diameter recostructio is based o usig its diffractio patter uder large observatio agles (regio I). Cylider spectrum aalysis allows to approximate error D by the quadratic fuctio, amely: I = D 0 θ 4, (9) where proposed agle rage for recostructio D is equal to θ θ θ. I Fig. 6 the plot of fuctio (9) ad depedece of error recostructio by the third algorithm are preseted. It is evidet, that the depedece (9) sufficietly exactly desibes the error i give rage., µm.6, µm D I I (8) Figure 6. Error of diameter recostructio () versus diffractio umber i the large agular regio while usig the stadard algorithm applied to plae see (dashed plot). Approximatio fuctio () is show by solid plot. I

6 The calculated maximal fuctio XVI IMEKO World Cogress Measuremet - Supports Sciece - Improves Techology - Protects Eviromet... ad Provides Employmet - Now ad i the Future Viea, AUSTRIA, 000, September 5-8 Λ D deviatio from the real error D I i this rage is about ~ 0. µm. It should be, however, oted, that give calculatios are based o equatio, which true for small observatio agles. 5 CONCLUSION The obtaied results of light diffractio by circular metal cylider with regard to the ifluece of the wave reflected from its surface are preseted i this work. The field calculatio was made by statioary phase method uder cylider radius R >> λ. It is show that the cylider diffractio patter is appreciably differet from the patter for the plae see (plae aalogue of cylider) with the same size D = R, for agles θ θ. = λ / R. As a result of aalysis of the diffractio miima behaviour the equatio for their agular positios (case of small observatio agles) was foud. The three algorithms of cylider diameter recostructio usig its diffractio patter i various agle rages were proposed. These algorithms allow theoretically to iease appreciably (tes of times) the accuracy of the dimesioal measuremets for metal circular cylider, that requires, however, the experimetal testig. The obtaied results could be applied to optical dimesio ispectio, optics ad laser metrology. REFERENCES [] P.B. Vagaov, B.Z. Katseelebaum. Diffractio theory basis. Moscow, Nauka, 98. [] S. Schmidt, Ei Beitag zur Erklarug der Lichtbegug am metallische kreiszglider, PTB- Mitteluge, 976, 86, ¹ 4. [] M. Kochsiek, H. Kuzma ud J. Tatau. Awedug beugugsoptischer Methode zur Messug der Durchmesser vo klaie Welle. PTB-Mitteluge, 976, 86, ¹ 4. [4] G.D. Dew. The applicatio of spatial filterig techiques to profile ispectio, ad a associated iterferece pheomeo. Optica Acta, 970, vol. 7, ¹ 4, pp [5] Yu. V. Chugui. Costructive theory of formatio ad filterig the optical images ad Frauhofer diffractio patters of D opaque objects of costat thickess i coheret light // Proc. SPIE P.67. [6] Yu. V. Chugui, A. A. Pavlov. Proc. of -th It. Cof. "Measuremet-99", Slovak Republik, Slovak Academy of Scieces, Istitute of Measuremet Sciece 999. pp AUTHORS: Prof. Yuri CHUGUI ad Ato PAVLOV. Laboratory for Techical Visio, Techological Desig Istitute of Scietific Istrumet Egieerig, 4, Russkaya str., Novosibirsk, 60058, Russia, Phoe It , Fax It , chugui@tdi.sk.su or chugui@tdisie.sc.ru ad ato@uic.su.ru