ACTIVE SPECKLE PHOTOGRAPHY METHOD USING FOURIER TRANSFORM FOR MEASURING THE THICKNESS OF A TRANSPARENT PLATE

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1 Internatonal Jornal of Physcs and Research IJPR ISSN Vol 3 Isse 3 Ag TJPRC Pvt Ltd ACTIVE SPECKLE PHOTOGRAPHY METHOD SING FORIER TRANSFORM FOR MEASRING THE THICKNESS OF A TRANSPARENT PLATE NASSER A MOSTAFA Department of Physcs Faclty of Scence Helwan nversty An Helwan Caro Egypt ABSTRACT Speckle photography offers a smple and fast technqe for measrng the thckness of a transparent plate It s a two-stage process In the frst stage two states of the transmtted waves throgh the transparent plate one before and one after the rotaton of the plate are recorded In the second stage the two recorded mages are added and by applyng a Forer transform a frnges can be obtaned and processed to estmate the thckness of the plate ncertanty of thckness measrement s nvestgated The method s nvestgated and ts vablty s demonstrated by epermental reslts KEYWORDS: Speckle Photography Forer Transform Thckness Measrement INTRODCTION Mcrometers and dal gages are sefl for measrng the thckness of a plate to an accracy of 00 m; 0 m accracy s achevable wth nterferometry A common technqe for measrng the thckness of a transparent plate s to employ a spectrometer sch as a Mchelson nterferometer to measre transmtted lght as a fncton of wavelength The resltant spacng of nterference frnges may be smply related to the so-called optcal thckness Francon has presented an nterferometrc technqe not reqrng a spectrometer by measrng the anglar dameters at nfnty of the frst and n th frnges A Forer transform spectrometer or varant thereof s typcally sed to make broadband measrements of the reflecton and transmsson of a transparent plate 3 By lookng at the spacng of nterference frnges verss wave nmber the optcal thckness can be estmated Measrng frnges verss both angle and wave nmber then provdes a way to estmate both the absolte nde of refracton and thckness of a plate Sch an approach may be complementary to the technqe of Okada et al 4 who measre srface shape and refractve nde nhomogenety sng a tnable dode laser and a Twyman-Green nterferometer Speckle photography whch was orgnally developed as a method for measrng the lateral or n-plane dsplacement 5-7 and refractve nde of a transparent plate 8 s a smpler technqe than other havng the advantages that t does not reqre sch a hgh degree of stablty of the optcal system It can therefore be more readly sed otsde the laboratory The prpose of ths work s to etend the applcaton of Forer transform method to speckle photography to measre the thckness of the transparent plate and detect the major ncertanty components for mprovng the performance of the thckness estmaton 9- A relaton between the thckness of a transparent plate and the angle of ncdence of the collmated lght s derved theoretcally and epermentally confrmed THEORETICAL ANALYSIS Let frst the transmsson of a plane wave throgh a transparent plate of refractve nde and thckness d

2 6 Nasser A Mostafa srronded by free space Fgre Let yd and y0 are the comple ampltdes of the plane wave after and before transmsson respectvely Therefore the rato ty = yd / y0 represent the comple ampltde transmttance of the plate j nd y d ae e plates Where a and φ are the ampltde and phase of the ncdent wave respectvely n s the nmber of the transmttance The transmtted wave llmnates a scatterng srface les n the ζη plane Any pont on the observaton plane XY s placed at a dstance n front of ths srface wll receve contrbtons from all ponts on the srface Assme that P s s any pont on the scatterng srface plane P o s any pont on the observaton plane and r os s the dstance between P s and P o makng an angle θ wth Accordng to Fresnel prncple of dffracton 3 we can wrte wavefeld across the XY plane as follows ros j nd e p ae e ps cosds j r 0 os Where cos r os Ths eqaton can be wrtten as: ros j nd e X Y ae e dd 3 j r os Where r os X Y 4 Let m be a nmber less than nty and consder the eqaton m m m 5 8 Then r os r os X Y 6 X Y 7 X / Y / e j nd X Y ae e dd j X / Y / 8

3 Actve Speckle Photography Method sng Forer Transform for Measrng the Thckness of a Transparent Plate 63 ae j e X / X Y 9 j nd X Y e e Y / dd The terms neglected then X / and Y / are very small and ther effect on the wave ampltde can be X Y X Y j nd X Y ae e e e e e dd j 0 If >> ξ +η / then X Y X Y j nd X Y ae e e e e dd j THICKNESS MEASREMENT To measre the thckness of a transparent plate sng the theory of refracton of the ncdent wave The plate s ptted on a gradated rotatable dsc Spatally coherent lght transmtted throgh the plate and llmnated a rogh srface The obtaned speckle s recorded twce one before the rotaton of the transparent plate and other after the rotaton and ncreasng the thckness of the plate Then the two mages are combned dgtally added and the resltant mage wll contan a par of dentcal speckle pattern separated by a dstance The dsplacement j 45 s dsplaced n the form of frnge pattern by applyng FFT to the resltant mage So a brght spot srronded by a speckle pattern modlated by a cosnsodal frnges can be observed The brght central spot s formed by the n-dffracted lght Whle the cosnsodal frnges are formed becase each par of correspondng speckles acts as a par of dentcal sorces of coherent lght whch form Yong s frnges These frnges are separated by a dstance dependng on the vale of the rotaton and thckness of the transparent plate sng eqaton and pt X Y and y X Y j nd X Y ae e e e j y e dd j The feld X before the rotaton of the transparent plate can be represented by: Y X Y j nd X Y ae e e e F{ } 3 j Where F{ } s the Forer transform fncton If the transparent plate rotated wth an angle the ncdent waves refracted wth an angle The comple ampltde nsde the plate s now proportonal to ep nd/ cos So that the comple ampltde transmttance of the plate:

4 64 Nasser A Mostafa t y ep nd/ cos 4 The feld X after rotaton can be represented by: Y nd X Y j cos j y X Y ae e e e e dd j 5 nd X Y j cos j j y X Y ae e e e e e dd j 6 Assme that the dsplacement s n one drecton -as Hence; jnkd X Y j cos j X Y ae e e e e F{ } 7 j By addng the two Eqatons 3 and 7 X Y j nd j cos X Y X Y ae e e F{ } e e e 8 j The ntensty can be wrtten as nd I I nd j nd cos F { }{ e e } 9 knd F { }{ cos knd } 0 cos At mamm ntensty knd knd m cos Let m = we get: nd cos Pt nd cos 3 Where λ s the wavelength of the sed lght and represent the frnge spacng n the -drecton

5 Actve Speckle Photography Method sng Forer Transform for Measrng the Thckness of a Transparent Plate 65 From the Snell s law / sn = sn where / = s the refractve nde of the ar and s the refractve nde of the glass plate By sbsttton from the Snell s law nto Eqaton 3 The thckness of a transparent plate can be gven by: d n cos sn sn 4 EXPERIMENTAL ARRANGEMENT PROCEDRES AND RESLTS The optcal setp sed to obtan the thckness of the transparent plate s shown n Fgre A 7 mw He-Ne Laser =0638 m was sed for the eperments The laser lght was llmnatng a transparent rogh srface wth rms 96 m For estmatng the thckness of the glass plate the glass plate s ptted on a gradated rotatable dsc n front of the collmated lght Havng a verner the smallest gradaton can be read to 0 degree Dfferent spatal carrers can be accomplshed by the non-rotaton and rotaton of the transparent plate whch have a defnte thckness The plate was tlted wth an angle 05 degree to be constant drng the eperment A frst eposre s made by llmnatng the rogh object n case of the non-rotaton of the transparent plate and receved sng a webcam camera attached to a compter A second eposre s made by llmnatng the rogh srface n case of the rotaton of the transparent plate wth an angle 05 degree the thckness of the plate was 58 mm The nt to be measred on montor s pel So we mst calbrate the magnfcaton before the test The calbraton process of the eperment made by pttng the rogh srface on a travellng mcrometer and move t aally and measrng the vales of 3 shows the calbraton crve for measrng the vale of n vale of and and wth the correspondng vales of the mcrometer readngs Fg m Fg4 shows the calbraton crve for measrng the n m The vales and 6 whch are shown n the eqatons of the calbratons of are consdered as constants dependng on the calbraton process From the recorded two mages and by sng the software program Image J Program we can obtan the vale of Combnng the two mages nto one and comptng ther two-dmensonal Forer transform we get the frnge mage n the spectral feld Fg 5 for angle of rotaton 05 degree and thckness of 58 mm Scannng Fg 5 along the -as =0 we can get the dstrbton of the lght ntensty so the vale of can be measred After measrng the vale of y and we sbsttte n Eq 4 we can get the thckness of the glass plate Fgre 6a b show the frnge mages n the spectral feld for dfferent thcknesses of vales 6 mm and 406 mm respectvely Fgre 7 shows the dependence of the rato rato of the dstance between the two dentcal speckle pattern and the frnge spacng n the -drecton on the angle of ncdence From the fgre we can conclde that the rato appromately have the same vale for dfferent thcknesses at the angle of ncdence 05 When the angle of ncdence ncrease the rato for the dfferent thcknesses gettng away from each other and t ncreases gradally wth ncreasng the angle of ncdence as shown n the fgre Ths mean that the hgh accracy

6 66 Nasser A Mostafa of measrements can be obtaned at the small angle of ncdence when the angle of ncdence ncreases the accracy of measrements decreases Fgre 8ab show the mages n the spectral feld for dfferent thckness of vales 6 mm and 406 mm respectvely and the angle of ncdence was o For the nvestgaton of the dependence of the measred ncertanty on the changng of wavelengths dfferent flters wth wavelengths and m respectvely were sed Fgre 9ab show the mages n the spectral feld for wavelengths and m respectvely The angle of rotaton was 05 degree and the thckness was 6 mm As we can see the vsblty decrease wth decreasng the sed wavelengths MEASREMENT NCERTAINTIES OF METHOD The ncertanty of d was estmated by combnng the standard ncertantes of the parameters n and Generally the reslt of measrement s determned from the relaton between the reslt d and the vales of the npt parameters and can be epressed by a model: n f d 5 where n and represent model npt parameters The ncertanty of the reslt d depends on the ncertanty of the npt parameters and s descrbed by the eqaton: N d d 6 Where N represent model npt parameters n are the standard ncertantes of the npt parameters and d s the senstvty coeffcent The senstvty coeffcent descrbes how the measrement reslt vares wth changes n the vale of npt estmates Eq 6 s vald for measrement where there s no correlaton between npt parameters Accordng to eqatons 5 and 6 d can be epressed n the form: cos cos cos sn cos cos 4 n n n d 7 To calclate the epanded ncertanty of the reslt of measrement at the 95% confdence level the reslt for combned ncertanty was mltpled by a coverage factor of

7 Actve Speckle Photography Method sng Forer Transform for Measrng the Thckness of a Transparent Plate 67 Relatve ncertanty contrbtons are sed to llstrate the relatve mpact of dfferent ncertanty components The relatve contrbton r of an ncertanty component to the combned standard ncertanty s defned here as: r d 8 d Where d s the model eqaton d f N are the npt parameters n and d s the combned ncertanty calclated accordng to Eq 6 The vales npt estmates wth ther respectve standard ncertantes senstvty coeffcents relatve contrbtons and types of standard ncertanty evalatons are gven n Table Fgre 0 shows the dependence of the ncertanty of measrement on the changng of wavelengths for dfferent thcknesses As we can see the ncertanty decrease wth ncreasng the wavelength of the sed lght and decreasng the thckness of the transparent plates More accrate reslts can be obtaned by sng a lght of wavelength 0638 μm and angle of rotaton 05 o The present method can be more actve for the measrement of the small thcknesses of the transparent plates CONCLSIONS A smple dgtal method for processng speckle photography s sed for measrng the thckness of a transparent plate by applyng FFT fncton to the doble eposre speckle pattern Forer transform s appled to determne the reslts and theoretcal analyss show good agreement wth the epermental reslts ncertanty of measrements s evalated For more accrate and precse measrement t shold be sed a small angle of rotaton 05 o and a laser lght of wavelength 0638 μm REFERENCES D Malacara Optcal Shop Testng P 78 John Wley & Sons New York 99 M Francon Optcal Interferometry p58 Academc Press New York W J Moore Errors n flm refractve-nde determnatons from nterference frnges Appl Opt K Okada H Saktta T Ose and J Tsjh Separate measrements of srface shapes and refractve nde nhomgenety of an optcal element sng tnable-sorce phase shftng nterferometry ApplOpt A E Ennos Speckle nterferometry n Laser Speckle and Related Phenomena JCDanty Ed Sprnger- VerlagBerln 975 pp R K Erf Specalzed metrologcal applcaton of speckle n Speckle Metrology RKErfed Academc New York 978 Chap 9 pp M Francon Informaton processng sng speckle patterns n Laser Speckle and Related Phenomena JCDanty ed Sprnger-Verlag Berln 984 Chap 5 pp NA Mostafa Forer transform n speckle photography for measrng the refractve nde for a transparent plate Materal Scence Research Inda Vol 6 No pp99-04 Jne 009

8 68 Nasser A Mostafa 9 Andreja Drolc Mlenko Ros Evalaton of measrement ncertanty n the determnaton of total phosphoros sng standard spectrometrc method ISO 6878 Acta Chm Slov Cosme Frlong Jeffery S Yokm and Ryszard J Prytnewcz Estmaton of epermental ncertanty n qantatve optoelectronc holography technqes for absolte shape measrements Proc Int Conf on measrements n advanced materals and systems Mlwakee WI0- Jne 00 pp Raner GDorsch Gerd Häsler and Jürgen MHermann Laser tranglaton: fndamental ncertanty n dstance measrement Appled Optcs Vol 33 No7 March 994 M Angano Morales Amala MartìnezJARayas Raúl RCordero and Fernando Labbe ncertanty analyss of whole-feld phase-dfference rereved from ESPI frnge patterns by sng the Forer transform method FTM OptComm 8 pp Joseph WG Introdcton to Forer Optcs nd edn McGraw- Hll Sngapore Bedermann K and EKL A recordng and dsplay system for hologram nterferometry wth low resolton magng devces Jornal of Physcs E: Scentfc nstrments Vol 8 pp hang Yanpeng h Hanng ho Wenlng and L Hefe Applcaton of the Forer transform n electronc speckle photography" EXP Mechancs Vol 4 No pp APPENDICES Fgre : The Constrcton of the Speckle Pattern Fgre : Interferometrc Technqe for Measrng Thckness of a Transparent Plate

9 Actve Speckle Photography Method sng Forer Transform for Measrng the Thckness of a Transparent Plate 69 Fgre 3: The Calbraton Crve for Measrng the Frnge Spacng; n the -As Drecton Fgre 4: The Calbraton Crve for Measrng the Separaton between a Par of Identcal Speckle Pattern Fgre 5: Image of Frnges n the Spectral Feld: The Thckness of the Transparent Plate was 58 mm and the Angle of ts Rotaton was 05 o

10 70 Nasser A Mostafa Fgre 6: a b Images of Frnges n the Spectral Feld: The Thckness of the Transparent Plates was 6 mm and 406 mm Respectvely the Plates Tlted by an Angle 05 o Fgre 7: The Dependence of the Rato on the Angle of Incdence Fgre 8: ab Images of Frnges n the Spectral Feld: The Thckness of the Transparent Plates was 6 mm and 406 mm Respectvely the Plates Tlted by an Angle o

11 Actve Speckle Photography Method sng Forer Transform for Measrng the Thckness of a Transparent Plate 7 Fgre 9: a b Images of Frnges n the Spectral Feld: The Wavelength of sed Lght was and m Respectvely The Thckness of the Transparent Plates was 6 mm and the Plate Tlted by an Angle 05 o Fgre 0: Dependence of ncertanty of Measrements on the Changng of Wavelengths for Dfferent Thcknesses

12 7 Nasser A Mostafa Table : ncertanty Components and ther Relatve Standard ncertantes Sorce of ncertanty Frnge spacng n -drecton λ Separaton between a par of speckle Δζ Refracton angle γ Nmber of the transparent plates n Wavelengthλ Probablty Dstrbton Rectanglar Type B Rectanglar Type B Rectanglar Type B Rectanglar Type B Rectanglar Type B Standard ncertanty µm Senstvty Coeffcent ncertanty Contrbton µm µm µm µm µm µm Combned Standard ncertanty = 568 µm Epanded ncertanty k = 74 µm