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Optical diffractometry
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  Optical diffractometry M. Taghi Tavassoly, 1,2, *  Mohammad Amiri, 3 Ahmad Darudi, 4 Rasoul Aalipour, 1 Ahad Saber, 1 and Ali-Reza Moradi 1 1 Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran 2 Department of Physics, University of Tehran, Kargar Shomally, Tehran 14394-547, Iran 3 Physics Department, Bu-Ali Sina University, Hamedan 65178, Iran 4 Physics Department, Zanjan University, Zanjan 45195, Iran * Corresponding author: tavasoli@iasbs.ac.ir  Received September 30, 2008; revised December 15, 2008; accepted December 16, 2008;posted January 5, 2009 (Doc. ID 101670); published February 17, 2009 Interference of light has numerous metrological applications because the optical path difference (OPD) can be varied at will between the interfering waves in the interferometers. We show how one can desirably change theoptical path difference in diffraction. This leads to many novel and interesting metrological applications in-cluding high-precision measurements of displacement, phase change, refractive index profile, temperature gra-dient, diffusion coefficient, and coherence parameters, to name only a few. The subject fundamentally differsfrom interferometry in the sense that in the latter the measurement criterion is the change in intensity orfringe location, while in the former the criterion is the change in the visibility of fringes with an already knownintensity profile. The visibility can vary from zero to one as the OPD changes by a half-wave. Therefore, mea-surements with the accuracy of a few nanometers are quite feasible. Also, the possibility of changing the OPDin diffraction allows us to use Fresnel diffraction in Fourier spectrometry, to enhance or suppress diffractedfields, and to build phase singularities that have many novel and useful applications. © 2009 Optical Societyof America OCIS codes:  260.1960, 260.6042, 120.5050, 120.6780, 120.3930, 030.1640 . 1. INTRODUCTION The interference of light by a plate was noticed as the ap-pearance of colors in thin films back in the 17th centuryby Boyle and Hooks [1]. Later, numerous applications of  interference in research and metrology were realized af-ter Michelson introduced his famous interferometer in1881 [2]. In fact, Michelson’s interferometer and manyother interferometers are plates of variable thickness.The diffraction of light was discovered by Grimaldi evenearlier than the interference. The more familiar diffrac-tion phenomenon appears when the passage of a spatiallycoherent beam of light is partly obstructed by an opaqueobject. In this process the amplitude of the propagating wave experiences a sharp change at the object-fieldboundary.The foundation of diffraction theory was laid by Huy-gens in the late 17th century. It was promoted into a con-sistent wave theory by Fresnel and Kirchhoff in the 19thcentury that has been very successful in dealing with op-tical instruments and describing numerous optical phe-nomena. Based on this theory the subject of diffraction in-cludes Fresnel diffraction (FD), Fraunhofer diffraction,and, closely related to the latter, far-field diffraction.Fraunhofer diffraction has many applications in describ-ing optical systems and in spectrometry, but applicationsof FD are very limited. The limitation is imposed by thenonlinearity of FD and the inability to change the opticalpath difference (OPD) at will.However, a rather unfamiliar form of FD occurs as thephase of a wavefront in some region undergoes a sharpchange. An abrupt change in the phase can be easily im-posed by reflecting a light beam from a step or transmit-ting it through a transparent plate with an abrupt changein thickness or refractive index. Although this kind of FDhas been studied directly and indirectly by several au-thors [3–6], systematic and detailed studies of the subject have been reported very recently [7–10]. In this paper we discuss and extend the schemes for changing the OPD inFD outlined in the latter reports and use them to realizethe aforementioned applications. But before doing so webriefly review the theoretical bases of the subject. 2. THEORETICAL CONSIDERATIONS In Fig. 1 the cylindrical wavefront    strikes a 1D step of height  h . The axis of the wavefront that passes throughpoint  S  is parallel to the step edge. Using the Fresnel–Kirchhoff integral the diffracted amplitude and intensitycan be calculated at an arbitrary point  P  along   S   P ,where  S   is the mirror image of   S . The intensity at point  P  depends on the location of   P 0 , the srcin of the coordi-nate system used for the intensity calculation at point  P .For  P 0  on the left side of the step edge and given the co-efficients of the amplitude reflection  r  L  and  r  R  for the leftand right sides of the edge, the intensity at point  P  isgiven by [9]  I   L =  I  0 r  L r  R  cos 2     /2  +2  C 02 +  S 02  sin 2     /2  −  C 0 −  S 0  sin    +  I  0  /2   r  L − r  R  2  1 2  + C 02 +  S 02   +  C 0 +  S 0  r  L 2 − r  R 2   ,   1  where  I  0  is proportional to the illuminating intensity,    =2 kh  cos    is the phase introduced by the step ( k  and    stand for the wave number and incidence angle, respec-tively, at point  P 0 ), and  C 0  and  S 0  represent the well-known Fresnel cosine and sine integrals, respectively, as-sociated with the distances between  P 0  and the source 540 J. Opt. Soc. Am. A/Vol. 26, No. 3/March 2009 Tavassoly  et al. 1084-7529/09/030540-8/$15.00 © 2009 Optical Society of America  and the step edge. According to Eq. (1), the intensity at point  P  depends on the step height and the reflection co-efficients. However, even for  h =0, because  r  L  r  R , the in-tensity across a screen normal to  S   P  is not uniform andFresnel fringes are observed. For  r  L = r  R  the normalizedintensity on the left or right side of the edge, specified by  and  , respectively, is expressed as [8,9]  I  n =cos 2     /2  +2  C 02 +  S 02  sin 2     /2    C 0 −  S 0  sin   ,  2  or  I  n =  A +  B cos    C sin   ,   3  where  A =  1 2  + C 02 +  S 02 ,  B =  1 2  −  C 02 −  S 02  ,  C = C 0 −  S 0 .   4  One can form a 1D phase step in transmission by im-mersing a transparent plate in a transparent medium(liquid or gas). When a plane or cylindrical wave passesthrough the plate it experiences a sharp change in phaseat the plate edges because of an abrupt change in refrac-tive index. Intensity calculation by the Fresnel–Kirchhoff integral at a point on a screen perpendicular to the direc-tion of the transmitted light, Fig. 2, leads to equations similar to (1) and (2) except for the phase     that should bereplaced by [9]   = kNh    n 2 −sin 2   −cos    ,   5  where  n =  N   /   N    represents the ratio of the refractive in-dex of the plate to that of the medium. Extension to 2Dphase steps is straightforward [9]; however, for our objec- tives 1D steps are quite adequate. 3. STEP WITH VARIABLE HEIGHT  A phase step with variable height can be built in numer-ous ways. For example, by mounting a circular mirror andan annular mirror on the tops of two coaxial cylinders asshown in Fig. 3 one can build a circular step. The heightof the step can be varied by moving cylinder  C 1  in a ver-tical direction. To build a 1D phase step the circular mir-rors are replaced by rectangular ones. Since in FD the ef-fective parts of an aperture are the edge neighborhood, inmany cases, mirrors of a few millimeters widths are quiteadequate. Thus, the phase steps can be designed and fab-ricated in compact form. This, in turn, reduces the effectof any mechanical noise.One can also design phase steps by using Michelsonand Mach–Zhender interferometers with some modifica-tions. For example, to build a 1D phase step by Michelsoninterferometer one can replace the mirrors by two rectan-gular mirrors in such a way that each mirror reflects thealternative halves of the beam striking the beam splitter,Fig. 4(a). In this case mirror  M  2  and the image  M  1   of mir-ror  M  1  in the beam splitter  B .  S . form the required phasestep.To build a phase step of desired shape by Michelson in-terferometer one can paste two complementary masks onthe mirrors. By complementary masks we mean twomasks that are joined together so as to obstruct the entire  s  z   x  P  0  P    Sc S   h  Fig. 1. Cylindrical wave  striking a 1D phase step of height  h .The diffracted intensity at point  P  is given in the text.    N  N    Sc. Fig. 2. Profile of a transparent plate of refractive index  N   im-mersed in a liquid of refractive index  N     N  . The 1D phase stepsare formed at the edges of the plate. S  B . 1  M  2  M  1 c 2 c Fig. 3. Sketch of a circular phase step that can be built bymounting a circular mirror  M  1  and an annular mirror  M  2  on twocoaxial cylindrical stands  C 1  and  C 2 . The light reflected from thebeam splitter B.S. diffracts from the step formed by the mirrors,and the step height can be varied by displacing mirror  M  1  in a vertical direction.Tavassoly  et al.  Vol. 26, No. 3/March 2009/J. Opt. Soc. Am. A 541  beam in one of the interferometer’s arm. For instance, acircular mask and an annular mask with its inner radiusequal to that of the circular mask pasted symmetricallyon the interferometer mirrors provide a circular phasestep. The masks should be good absorbers of light; other-wise, the scattered lights enhance the noise.In a Mach–Zhender interferometer (MZI) one can in-stall the complementary masks in the interferometerarms at equal distances from the beam splitter  B .  S .2 inFig. 4(b). The equal distance from the beam splitter as- sures that the diffracting apertures are practically thesame distance from the observation screen. In these casesthe step height can be varied by changing the OPD be-tween the interferometer’s arms. This can be done eitherby moving one of the mirrors or changing the physicalproperty of the materials occupying the arms of the inter-ferometer, say, by changing the air density.The patterns shown in Fig. 5 are typical FD patterns of light diffracted from 1D phase steps of different heightsformed by Michelson interferometer. The plots are the in-tensity profiles of the patterns (the average intensities inthe vertical direction are plotted for the FD patterns of Fig. 5). The diffraction patterns and the intensity profiles illustrated in Fig. 6 have been obtained by diffracting light from circular phase steps of different heights formedby a MZI. A fundamental difference between the fringes formedby a phase step of variable height and those formed inconventional interference is that the visibility of theformer is very sensitive to the change of OPD, while the visibility of the latter is practically insensitive to OPD.Aswe will show later, the capability of measuring a 1%change in the visibility of the step fringes provides thepossibility of measuring a change of     /400 in step height. Another remarkable difference concerns the fringe spac-ing. The spacing of the phase step fringes depends on thedistance of the diffractor from the light source and the ob-servation screen. For fixed distance and a given diffractorgeometry the intensity profile of the diffraction pattern isa known function. This provides a large volume of data onthe step height and further improves the measurementaccuracy. In addition, measurement by diffractometry isless sensitive to mechanical vibrations compared withconventional interferometry. However, the interferencefringe spacing depends on the gradient of the OPD, andthe intensity profile is not known in advance. As the patterns and the intensity profiles in Figs. 5 and6 show, the fringe visibility decreases with the distancefrom the step edge. We define the visibility for the threecentral fringes by the following expression V  = 1 2   I  maL +  I  maR  −  I  miM  1 2   I  maL +  I  maR  +  I  miM  ,   6  where  I  maL  and  I  maR  stand for the maximum intensities of the left side and right side bright fringes, while  I  miM   rep-resents the minimum intensity of the central dark fringe.Plotting Eq. (6) versus    /      =2 h cos     in the range 0–1the curve shown in Fig. 7 is obtained. According to thiscurve, as    varies in an interval of     /2 the visibility de-fined above changes from zero to one. 1  M  2  M  1  M   S  B . 2 O 1 O 2  M  2.. S  B  1.. S  B (a)(b) 1  M  h Fig. 4. (a)A1D phase step of height  h  is formed by replacing themirrors in a Michelson interferometer by two rectangular mir-rors in such a way that each mirror intersects the alternativehalves of the light beam striking the beam splitter. (b) A 1Dphase step is formed by mounting two opaque plates  O 1  and  O 2 in the arms of a MZI at equal distances from the beam splitterB.S.2 in such a way that the plates obstruct the alternativehalves of the beam reflecting from the mirrors  M  1  and  M  2 . Thestep height is varied by changing the OPD between the arms of the interferometer. −1 −0.5 0 0.5 12060100mm−1 −0.5 0 0.5 12060100140mm−1 −0.5 0 0.5 12060100140mm(a)(b)(c) Fig. 5. FD patterns of light diffracted from 1D phase steps of different heights formed in a Michelson interferometer arrange-ment and the corresponding intensity profiles over the patterns.(a)  h =   /8. (b)  h =   /4. (c)  h =3   /8.542 J. Opt. Soc. Am. A/Vol. 26, No. 3/March 2009 Tavassoly  et al.  It is interesting to recall that in the FD caused by asharp change of the amplitude the visibility of the fringesis very low. This remarkable difference can be explainedby the Cornu spiral adequately. One Cornu spiral is asso-ciated with each side of a step. The two Cornu spirals of astep differ in phase due to the step height. Only oneCornu spiral accounts for the intensity distribution atpoints far from the step edge. However, the contributionsof the two spirals should be considered at points close tothe step edge. For example, at point  P  in Fig. 1 the con-tribution of the left side Cornu is  J  1  M    in Fig. 8, while the contribution of the right side is  M    J  2   . We have    M    J  2    =   MJ  1     when the reflection coefficients on both sides arethe same. By squaring the vectorial sum    J  1  M    +  MJ      andsubstituting the corresponding coordinates in the  C -  S system, Eq. (2) is derived. For    =    the two spirals are inopposite phase and the resultant amplitude vanishes atpoints corresponding to the edge of the step. −1 −0.5 0 0.5 1−0.8−0.6−0.4−0.200.20.40.60.8 C i  S r L =r R h =  λ   /10 φ  = 2 π  /5 φ OJ 1 J’ 1 J 2 J’ 2 MJ’’M’ φ Fig. 8. Cornu spirals attributed to a 1D phase step of height  h =   /10 or    =2    /5. The bold face parts of the spirals contribute tothe amplitude at point  P  in Fig. 1 associated with points  M  and M   on the spirals.Fig. 9. Scheme of a rectangular cell and a plane parallel platethat is installed inside it to study liquid–liquid diffusion by lightdiffraction.Fig. 6. FD patterns and the corresponding intensity profiles of light diffracted from circular phase steps of different heightsformed by a MZI, (a)  h =5   /24. (b)  h =   /2. (c)  h =5   /6. 0 0.25 0.5 0.75 100.51 ∆  /   λ        V      i     s      i      b      i      l      i      t     y Calculation Fig. 7. Calculated visibility versus the optical path differencedivided by wavelength    /    for three central fringes in FD from a1D phase step.Tavassoly  et al.  Vol. 26, No. 3/March 2009/J. Opt. Soc. Am. A 543  4. METROLOGICAL APPLICATIONS Some straightforward applications of the effect are in themeasurements of displacement, film thickness, refractiveindex, and dispersion of a transparent film or plate thatcan be realized with high accuracy by fitting Eq. (1) or (2) on the experimentally obtained normalized intensity dis-tribution of the corresponding fringes. A novel applicationof the phenomenon is in the measurement of the refrac-tive index gradient that appears in many situations, suchas in a diffusion process and in media sustaining tem-perature gradients. There are optical methods based oninterferometry, holography, and moiré deflectometry formeasuring the refractive index gradient [11–13]. How- ever, the method we describe here is remarkably simpleand highly accurate. For example, to measure the refrac-tive index gradient in a biliquid diffusion process, we in-stall a transparent plane parallel plate of thickness  e  andrefractive index  N   inside a rectangular transparent cell of width  W  , as shown in Fig. 9. Then, the cell is filled with the given liquids in the proper way. As the diffusion pro-cess proceeds, the refractive index along the vertical edgeof the plate varies, and a step of height  h =  e   N  − n   z   isformed, where  n   z   is the refractive index of the cell con-tent at altitude  z . As the cell is perpendicularly illumi-nated by a coherent beam of wavelength   , the visibilityof the step fringes repeats along the plate edge as  e n   z  changes by   . The plot of fringe visibility versus  z  pro- vides the index very accurately in the neighborhood of theplate edge. The patterns in Fig. 10 are the diffraction pat-terns of light diffracted from the edge of a plane parallelplate installed in a rectangular cell in which sugar solu-tion was diffusing into water, at different times after thebeginning of diffusion. The inclinations and the spacingsof the oblique fringes show very clearly the states of thediffusion process. 5. EXPERIMENTAL REALIZATION OFBABINET’S PRINCIPLE  According to Babinet’s principle, superposition of thefields diffracted from two complementary apertures (twoapertures that are connected together form an infinite ap-erture) leads to a uniform field. Two parts of a 1D or 2Dphase step for the case of zero step height are complemen-tary apertures. The diffraction patterns and intensity pro-files shown in Figs. 11(a) and 11(b) are obtained by dif- fracting light from a slit and an opaque strip of the samewidth as the slit in similar conditions. However, when theobjects are installed in a MZI in such a way that the im-age of one object in the second  B .  S . is superimposed onthe other object, illumination of both objects leads to thediffraction pattern and the intensity profile shown in Fig.11(c) that confirms Babinet’s principle experimentally.The patterns and the plots in Figs. 11(d)–11(f) illustrate 5min. 15min. 30min. 60min. 120min. 350min. Fig. 10. Diffraction patterns of the light diffracted from theedge of a plane parallel plate immersed in a rectangular cell con-taining pure water over sugar solution of concentration 10% atdifferent times after the initiation of the diffusion. The estab-lished refractive index gradient has appeared as the fringes in-clined with respect to the plate edge. −1 −0.5 0 0.5 1050100150mm−1 −0.5 0 0.5 1050100150mm−1 −0.5 0 0.5 1050100150mm−1 −0.5 0 0.5 1050100150mm−1 −0.5 0 0.5 1050100150mm−1 −0.5 0 0.5 1050100150mm(e)(d)(c)(b)(a)(f) Fig. 11. Experimental realization of Babinet’s principle. (a), (b)The diffraction patterns and intensity profiles of the light dif-fracted from a slit of 0.24 mm width and an opaque strip of thesame width as the slit. (c) The pattern and intensity profile ob-tained by superimposing the diffracted fields in (a) and (b) in aMZI. (d), (e) The diffraction patterns and intensity profiles of thelight diffracted from two complementary straight edges. (f) Thepattern and intensity profile obtained by superimposing the dif-fraction fields in (d) and (e) in a MZI.544 J. Opt. Soc. Am. A/Vol. 26, No. 3/March 2009 Tavassoly  et al.
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