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540 J. Opt. Soc. Am. A/Vol. 26, No. 3 /March 2009 Tavassoly et al.

Optical diffractometry

M. Taghi Tavassoly,1,2,* Mohammad Amiri,3 Ahmad Darudi,4 Rasoul Aalipour,1 Ahad Saber,1 and Ali-Reza Moradi1

1Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran2Department of Physics, University of Tehran, Kargar Shomally, Tehran 14394-547, Iran

3Physics Department, Bu-Ali Sina University, Hamedan 65178, Iran4Physics 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 bevaried 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.

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. INTRODUCTIONhe interference of light by a plate was noticed as the ap-earance of colors in thin films back in the 17th centuryy Boyle and Hooks [1]. Later, numerous applications ofnterference in research and metrology were realized af-er Michelson introduced his famous interferometer in881 [2]. In fact, Michelson’s interferometer and manyther interferometers are plates of variable thickness.he diffraction of light was discovered by Grimaldi evenarlier than the interference. The more familiar diffrac-ion phenomenon appears when the passage of a spatiallyoherent beam of light is partly obstructed by an opaquebject. In this process the amplitude of the propagatingave experiences a sharp change at the object-fieldoundary.The foundation of diffraction theory was laid by Huy-

ens in the late 17th century. It was promoted into a con-istent wave theory by Fresnel and Kirchhoff in the 19thentury that has been very successful in dealing with op-ical instruments and describing numerous optical phe-omena. Based on this theory the subject of diffraction in-ludes Fresnel diffraction (FD), Fraunhofer diffraction,nd, closely related to the latter, far-field diffraction.raunhofer diffraction has many applications in describ-

ng optical systems and in spectrometry, but applicationsf FD are very limited. The limitation is imposed by theonlinearity of FD and the inability to change the opticalath difference (OPD) at will.However, a rather unfamiliar form of FD occurs as the

hase of a wavefront in some region undergoes a sharphange. An abrupt change in the phase can be easily im-osed by reflecting a light beam from a step or transmit-ing it through a transparent plate with an abrupt change

1084-7529/09/030540-8/$15.00 © 2

n thickness or refractive index. Although this kind of FDas been studied directly and indirectly by several au-hors [3–6], systematic and detailed studies of the subjectave been reported very recently [7–10]. In this paper weiscuss and extend the schemes for changing the OPD inD outlined in the latter reports and use them to realize

he aforementioned applications. But before doing so weriefly review the theoretical bases of the subject.

. THEORETICAL CONSIDERATIONSn Fig. 1 the cylindrical wavefront � strikes a 1D step ofeight h. The axis of the wavefront that passes throughoint S is parallel to the step edge. Using the Fresnel–irchhoff integral the diffracted amplitude and intensity

an be calculated at an arbitrary point P along S�P,here S� is the mirror image of S. The intensity at pointdepends on the location of P0, the origin of the coordi-

ate system used for the intensity calculation at point P.or P0 on the left side of the step edge and given the co-fficients of the amplitude reflection rL and rR for the leftnd right sides of the edge, the intensity at point P isiven by [9]

IL = I0rLrR�cos2��/2� + 2�C02 + S0

2�sin2��/2� − �C0 − S0�sin ��

+ I0/2��rL − rR�2� 12 + C0

2 + S02� + �C0 + S0��rL

2 − rR2 �� , �1�

here I0 is proportional to the illuminating intensity, �2kh cos � is the phase introduced by the step (k and �tand for the wave number and incidence angle, respec-ively, at point P0), and C0 and S0 represent the well-nown Fresnel cosine and sine integrals, respectively, as-ociated with the distances between P and the source

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Tavassoly et al. Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. A 541

nd the step edge. According to Eq. (1), the intensity atoint P depends on the step height and the reflection co-fficients. However, even for h=0, because rL�rR, the in-ensity across a screen normal to S�P is not uniform andresnel fringes are observed. For rL=rR the normalized

ntensity on the left or right side of the edge, specified byand �, respectively, is expressed as [8,9]

In = cos2��/2� + 2�C02 + S0

2�sin2��/2� � �C0 − S0�sin �,

�2�

r

In = A + B cos � � C sin �, �3�

here

A = 12 + C0

2 + S02, B = 1

2 − �C02 − S0

2�, C = C0 − S0. �4�

One can form a 1D phase step in transmission by im-ersing a transparent plate in a transparent medium

liquid or gas). When a plane or cylindrical wave passeshrough the plate it experiences a sharp change in phaset the plate edges because of an abrupt change in refrac-ive index. Intensity calculation by the Fresnel–Kirchhoffntegral at a point on a screen perpendicular to the direc-ion of the transmitted light, Fig. 2, leads to equationsimilar to (1) and (2) except for the phase � that should beeplaced by [9]

s

z

x

P

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h

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ig. 1. Cylindrical wave � striking a 1D phase step of height h.he diffracted intensity at point P is given in the text.

�

N

N �

Sc.ig. 2. Profile of a transparent plate of refractive index N im-ersed in a liquid of refractive index N��N. The 1D phase steps

re formed at the edges of the plate.

� = kNh��n2 − sin2 � − cos ��, �5�

here n=N /N� represents the ratio of the refractive in-ex of the plate to that of the medium. Extension to 2Dhase steps is straightforward [9]; however, for our objec-ives 1D steps are quite adequate.

. STEP WITH VARIABLE HEIGHTphase step with variable height can be built in numer-

us ways. For example, by mounting a circular mirror andn annular mirror on the tops of two coaxial cylinders ashown in Fig. 3 one can build a circular step. The heightf the step can be varied by moving cylinder C1 in a ver-ical direction. To build a 1D phase step the circular mir-ors are replaced by rectangular ones. Since in FD the ef-ective parts of an aperture are the edge neighborhood, inany cases, mirrors of a few millimeters widths are quite

dequate. Thus, the phase steps can be designed and fab-icated in compact form. This, in turn, reduces the effectf any mechanical noise.

One can also design phase steps by using Michelsonnd Mach–Zhender interferometers with some modifica-ions. For example, to build a 1D phase step by Michelsonnterferometer one can replace the mirrors by two rectan-ular mirrors in such a way that each mirror reflects thelternative halves of the beam striking the beam splitter,ig. 4(a). In this case mirror M2 and the image M1� of mir-or M1 in the beam splitter B.S. form the required phasetep.

To build a phase step of desired shape by Michelson in-erferometer one can paste two complementary masks onhe mirrors. By complementary masks we mean twoasks that are joined together so as to obstruct the entire

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ig. 3. Sketch of a circular phase step that can be built byounting a circular mirror M1 and an annular mirror M2 on two

oaxial cylindrical stands C1 and C2. The light reflected from theeam splitter B.S. diffracts from the step formed by the mirrors,nd the step height can be varied by displacing mirror M1 in aertical direction.

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542 J. Opt. Soc. Am. A/Vol. 26, No. 3 /March 2009 Tavassoly et al.

eam in one of the interferometer’s arm. For instance, aircular mask and an annular mask with its inner radiusqual to that of the circular mask pasted symmetricallyn the interferometer mirrors provide a circular phasetep. The masks should be good absorbers of light; other-ise, the scattered lights enhance the noise.In a Mach–Zhender interferometer (MZI) one can in-

tall the complementary masks in the interferometerrms at equal distances from the beam splitter B.S.2 inig. 4(b). The equal distance from the beam splitter as-ures that the diffracting apertures are practically theame distance from the observation screen. In these caseshe step height can be varied by changing the OPD be-ween the interferometer’s arms. This can be done eithery moving one of the mirrors or changing the physicalroperty of the materials occupying the arms of the inter-erometer, say, by changing the air density.

The patterns shown in Fig. 5 are typical FD patterns ofight diffracted from 1D phase steps of different heightsormed by Michelson interferometer. The plots are the in-ensity profiles of the patterns (the average intensities inhe vertical direction are plotted for the FD patterns ofig. 5). The diffraction patterns and the intensity profiles

llustrated in Fig. 6 have been obtained by diffractingight from circular phase steps of different heights formedy a MZI.

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ig. 4. (a) A 1D phase step of height h is formed by replacing theirrors in a Michelson interferometer by two rectangular mir-

ors in such a way that each mirror intersects the alternativealves of the light beam striking the beam splitter. (b) A 1Dhase step is formed by mounting two opaque plates O1 and O2n the arms of a MZI at equal distances from the beam splitter.S.2 in such a way that the plates obstruct the alternativealves of the beam reflecting from the mirrors M1 and M2. Thetep height is varied by changing the OPD between the arms ofhe interferometer.

A fundamental difference between the fringes formedy a phase step of variable height and those formed inonventional interference is that the visibility of theormer is very sensitive to the change of OPD, while theisibility of the latter is practically insensitive to OPD. Ase will show later, the capability of measuring a 1%

hange in the visibility of the step fringes provides theossibility of measuring a change of � /400 in step height.nother remarkable difference concerns the fringe spac-

ng. The spacing of the phase step fringes depends on theistance of the diffractor from the light source and the ob-ervation screen. For fixed distance and a given diffractoreometry the intensity profile of the diffraction pattern isknown function. This provides a large volume of data on

he step height and further improves the measurementccuracy. In addition, measurement by diffractometry isess sensitive to mechanical vibrations compared withonventional interferometry. However, the interferenceringe spacing depends on the gradient of the OPD, andhe intensity profile is not known in advance.

As the patterns and the intensity profiles in Figs. 5 andshow, the fringe visibility decreases with the distance

rom the step edge. We define the visibility for the threeentral fringes by the following expression

V =12 �ImaL + ImaR� − ImiM

12 �ImaL + ImaR� + ImiM

, �6�

here ImaL and ImaR stand for the maximum intensities ofhe left side and right side bright fringes, while ImiM rep-esents the minimum intensity of the central dark fringe.lotting Eq. (6) versus � /� ��=2h cos �� in the range 0–1he curve shown in Fig. 7 is obtained. According to thisurve, as � varies in an interval of � /2 the visibility de-ned above changes from zero to one.

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ig. 5. FD patterns of light diffracted from 1D phase steps ofifferent heights formed in a Michelson interferometer arrange-ent and the corresponding intensity profiles over the patterns.

a) h=� /8. (b) h=� /4. (c) h=3� /8.

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Tavassoly et al. Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. A 543

It is interesting to recall that in the FD caused by aharp change of the amplitude the visibility of the fringess very low. This remarkable difference can be explainedy the Cornu spiral adequately. One Cornu spiral is asso-

ig. 6. FD patterns and the corresponding intensity profiles ofight diffracted from circular phase steps of different heightsormed by a MZI, (a) h=5� /24. (b) h=� /2. (c) h=5� /6.

0 0.25 0.5 0.75 10

0.5

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ig. 7. Calculated visibility versus the optical path differenceivided by wavelength � /� for three central fringes in FD from aD phase step.

iated with each side of a step. The two Cornu spirals of atep differ in phase due to the step height. Only oneornu spiral accounts for the intensity distribution atoints far from the step edge. However, the contributionsf the two spirals should be considered at points close tohe step edge. For example, at point P in Fig. 1 the con-ribution of the left side Cornu is J1M� in Fig. 8, while theontribution of the right side is M�J2�

� . We have �M�J2�� �

�MJ1�� � when the reflection coefficients on both sides are

he same. By squaring the vectorial sum �J1M� +MJ��� andubstituting the corresponding coordinates in the C-System, Eq. (2) is derived. For �= the two spirals are inpposite phase and the resultant amplitude vanishes atoints corresponding to the edge of the step.

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ig. 8. Cornu spirals attributed to a 1D phase step of height h� /10 or �=2 /5. The bold face parts of the spirals contribute to

he amplitude at point P in Fig. 1 associated with points M and� on the spirals.

ig. 9. Scheme of a rectangular cell and a plane parallel platehat is installed inside it to study liquid–liquid diffusion by lightiffraction.

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544 J. Opt. Soc. Am. A/Vol. 26, No. 3 /March 2009 Tavassoly et al.

. METROLOGICAL APPLICATIONSome straightforward applications of the effect are in theeasurements of displacement, film thickness, refractive

ndex, and dispersion of a transparent film or plate thatan be realized with high accuracy by fitting Eq. (1) or (2)n the experimentally obtained normalized intensity dis-ribution of the corresponding fringes. A novel applicationf the phenomenon is in the measurement of the refrac-ive index gradient that appears in many situations, suchs in a diffusion process and in media sustaining tem-erature gradients. There are optical methods based onnterferometry, holography, and moiré deflectometry for

easuring the refractive index gradient [11–13]. How-ver, the method we describe here is remarkably simplend highly accurate. For example, to measure the refrac-ive index gradient in a biliquid diffusion process, we in-tall a transparent plane parallel plate of thickness e andefractive index N inside a rectangular transparent cell ofidth W, as shown in Fig. 9. Then, the cell is filled with

he given liquids in the proper way. As the diffusion pro-ess proceeds, the refractive index along the vertical edgef the plate varies, and a step of height h=e�N−n�z�� isormed, where n�z� is the refractive index of the cell con-ent at altitude z. As the cell is perpendicularly illumi-ated by a coherent beam of wavelength �, the visibilityf the step fringes repeats along the plate edge as en�z�hanges by �. The plot of fringe visibility versus z pro-ides the index very accurately in the neighborhood of thelate edge. The patterns in Fig. 10 are the diffraction pat-erns of light diffracted from the edge of a plane parallellate installed in a rectangular cell in which sugar solu-ion was diffusing into water, at different times after theeginning of diffusion. The inclinations and the spacingsf the oblique fringes show very clearly the states of theiffusion process.

5min. 15min. 30min. 60min. 120min. 350min.

ig. 10. Diffraction patterns of the light diffracted from thedge of a plane parallel plate immersed in a rectangular cell con-aining pure water over sugar solution of concentration 10% atifferent times after the initiation of the diffusion. The estab-ished refractive index gradient has appeared as the fringes in-lined with respect to the plate edge.

. EXPERIMENTAL REALIZATION OFABINET’S PRINCIPLEccording to Babinet’s principle, superposition of theelds diffracted from two complementary apertures (twopertures that are connected together form an infinite ap-rture) leads to a uniform field. Two parts of a 1D or 2Dhase step for the case of zero step height are complemen-ary apertures. The diffraction patterns and intensity pro-les shown in Figs. 11(a) and 11(b) are obtained by dif-racting light from a slit and an opaque strip of the sameidth as the slit in similar conditions. However, when thebjects are installed in a MZI in such a way that the im-ge of one object in the second B.S. is superimposed onhe other object, illumination of both objects leads to theiffraction pattern and the intensity profile shown in Fig.1(c) that confirms Babinet’s principle experimentally.he patterns and the plots in Figs. 11(d)–11(f) illustrate

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(e)

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(c)

(b)

(a)

(f)

ig. 11. Experimental realization of Babinet’s principle. (a), (b)he diffraction patterns and intensity profiles of the light dif-

racted from a slit of 0.24 mm width and an opaque strip of theame width as the slit. (c) The pattern and intensity profile ob-ained by superimposing the diffracted fields in (a) and (b) in aZI. (d), (e) The diffraction patterns and intensity profiles of the

ight diffracted from two complementary straight edges. (f) Theattern and intensity profile obtained by superimposing the dif-raction fields in (d) and (e) in a MZI.

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Tavassoly et al. Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. A 545

xperimental realization of Babinet’s principle due to su-erposition of the fields diffracted from two complemen-ary straight edges mounted in the arms of a Michelsonnterferometer.

. ENHANCING AND SUPPRESSINGIFFRACTED FIELDS: SPATIAL COHERENCEIDTH MEASUREMENT

he superimposing of the diffracted fields of two objectsaises the possibility of enhancing or suppressing diffrac-ion. For example, by installing two similar slits in therms of a MZI in such a way that one slit is superimposedn the image of the other in the second beam splitter, onean enhance or suppress the diffracted field by introduc-ng a phase difference equal to an even or odd number of

between the arms of the interferometer. The diffractionatterns in Figs. 12(b) and 12(c) are the enhanced anduppressed versions of the diffraction pattern of a singlelit that is shown in Fig. 12(a). The plots in Fig. 12(d) arehe intensity profiles of the corresponding diffraction pat-erns.

For the enhanced case the intensities at a majority ofhe points are four times more than the intensities at theorresponding points for the single slit. That means theecording CCD has responded linearly at these intensi-ies. Comparing the plots (b) and (c) in Fig. 12(d) we notehat for a phase change of the intensity varies signifi-antly, and this provides more precise phase changeeasurement.Suppression of diffraction is very useful in the studies

f minute inhomogeneities and anisotropies in transpar-

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(c)(b)

(a)

b

a

c

(d)

ig. 12. Enhancing and suppressing light diffraction. (a) FDattern of light diffracted from a slit. (b), (c) The patterns ob-ained by superimposing constructively (enhanced mode) and de-tructively (suppressed mode) the light diffracted from two simi-ar slits installed in a MZI’s arms. (d) The profiles of the intensityistribution of the corresponding diffraction patterns.

nt media and objects such as optical fibers and lightaveguides, where diffraction from the geometry of thebject leads to very noisy data. To illustrate this pointore clearly we put two similar copper rods (wires) in the

rms of a MZI and adjusted the OPD to get full suppres-ion. Then, by passing different electrical currentshrough one of the rods we built up different temperatureradients around it and recorded the diffraction patternshown in Figs. 13(a)–13(c). The intensity distributions ofhe latter patterns and of the diffraction pattern of thenheated rods, but in enhanced condition, Fig. 13(d), arelotted in Fig. 13(e). The intensity changes provide infor-ation about the phase changes, and the slopes of the

urves indicate the presence of temperature gradients.his experiment suggests that by replacing the mirrors inMichelson interferometer with two thin rods (wires) one

ould construct an optical diffractometer to measuremall forces applied to one of the rods.

To measure the spatial coherence width of the light il-uminating a MZI we install two similar slits or pinholesn the arms of the interferometer in such way that one ob-ect is superimposed on the image of the other with zeroPD. Then, by displacing one of the slits (or pinholes) in

ts plane we can obtain a double slit of desired separationhat is useful for spatial coherence studies.

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(a) (b)

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ig. 13. Diffraction patterns of the light diffracted from a cop-er wire of thickness 0.4 mm carrying different electric currentsfter its original diffracted field had been suppressed by the fieldiffracted from another similar wire installed in the other arm ofMZI. (a) I=0.12 A. (b) I=0.25 A. (c) I=0.41 A. (d) The pattern

btained by superimposing constructively the diffracted fieldsrom the two wires with no electric current. (e) The profiles of thentensity distributions of the corresponding diffraction patterns.

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546 J. Opt. Soc. Am. A/Vol. 26, No. 3 /March 2009 Tavassoly et al.

. APPLICATION TO THE STUDY OFPTICAL FIELDS IN THE NEIGHBORHOODF PHASE SINGULARITIES

n recent decades the behavior of optical fields in theeighborhood of phase singularities—points of indetermi-ate phases and zero amplitudes—has attracted many re-earchers [14,15]. Also, more recently a number of works

ig. 14. Diffraction patterns of white light diffracted from 1Dhase steps of slightly different step heights around h=140 nm.

ave been published in which the behavior of polychro-atic light around phase singularities has been studied

16–20]. Spectral modification, red-shifting, blue-shifting,nd anomaly in the appearance of colors have been pre-icted and observed in experiments [21–24]. In many re-orted cases the field amplitudes in the neighborhoods ofhe singularities were small and this restricted the ex-erimental studies of the subject severely. However, asas been reported in [8], the amplitudes in the neighbor-ood of the singularities produced by the FD from phaseteps are as large as the amplitudes at other points, andhis eases the experimental studies. Additionally, in theseases the depth of the singularity can be usefully varied.ery recently the FD—the phase step approach—haseen applied to the study of the phase singularity atrewster’s angle [25].One can design phase singularities for the study of

pectral modification in the following ways. One way is tonstall two complementary apertures in the arms of a MZIo get a uniform optical field implied by Babinet’s prin-iple. Then, by changing the OPD between the two arms

0.4 0.45 0.5 0.55 0.60

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0.2

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ig. 15. Spectrum of a light beam incident on a 1D phase step ofeight �0 /4. (b) The normalized spectra of the diffracted lights atwo points symmetrical with respect to the step edge ��0560 nm�.

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Tavassoly et al. Vol. 26, No. 3 /March 2009/J. Opt. Soc. Am. A 547

f the interferometer by an odd number of � /2 of theavelength used, the field amplitude for that wavelengthecomes zero at the points associated with the edge of thetep. The patterns in Fig. 14 are the diffraction patternsf white light diffracted from 1D steps of slightly differenteights. The nonsymmetrical distribution of the colors is

nteresting. The curve in Fig. 15(a) represents the spec-rum of the light incident on a 1D phase step, while theurves denoted left and right in Fig. 15(b) are the spectraf the diffracted lights at two points symmetrical with re-pect to the edge. The spectral modifications are drastic.

Another design for a phase singularity is to install twoimilar apertures, namely, two similar slits, in the arms of

MZI so that the image of one slit coincides with thether. Then, by changing the OPD by an odd multiple of/2 of the given wavelength, one obtains a singularity ofhe shape of the slit.

. SUMMARYhis report shows that light diffraction from phase stepsith variable height is a rich subject with many novel ap-lications, and the ideas may be applicable to particlesther than photons.

CKNOWLEDGMENThe corresponding author sincerely acknowledges theupport provided by the Center of Excellence of theinistry of Higher Education and Technology (Iran).

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5. R. C. Saust, “Fresnel diffraction at a transparent lamina,”Proc. Phys. Soc. London 64, 105–113 (1950).

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7. M. T. Tavassoly, H. Sahloll-bai, M. Salehi, and H. R.Khalesifard, “Fresnel diffraction from step in reflection and

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8. M. T. Tavassoly, M. Amiri, E. Karimi, and H. R.Khalesifard, “Spectral modification by line singularity inFresnel diffraction from 1D phase step,” Opt. Commun.255, 23–34 (2005).

9. M. Amiri and M. T. Tavassoly, “Fresnel diffraction from 1Dand 2D phase steps in reflection and transmission mode,”Opt. Commun. 272, 349–361 (2007).

0. A. Sabatyan and M. T. Tavassoly, “Application of Fresneldiffraction to nondestructive measurement of the refractiveindex of optical fibers,” Opt. Eng. (Bellingham) 46,128001–7 (2007).

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