Optical diffractometry - OSA Publishing

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2Department of Physics, University of Tehran, Kargar Shomally, Tehran 14394-547, Iran. 3Physics Department, Bu-Ali Sina University, Hamedan 65178, Iran.

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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 Moradi1 1

Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran 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: [email protected]

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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 the optical path difference in diffraction. This leads to many novel and interesting metrological applications including high-precision measurements of displacement, phase change, refractive index profile, temperature gradient, diffusion coefficient, and coherence parameters, to name only a few. The subject fundamentally differs from interferometry in the sense that in the latter the measurement criterion is the change in intensity or fringe location, while in the former the criterion is the change in the visibility of fringes with an already known intensity profile. The visibility can vary from zero to one as the OPD changes by a half-wave. Therefore, measurements with the accuracy of a few nanometers are quite feasible. Also, the possibility of changing the OPD in diffraction allows us to use Fresnel diffraction in Fourier spectrometry, to enhance or suppress diffracted fields, and to build phase singularities that have many novel and useful applications. © 2009 Optical Society of 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 appearance of colors in thin films back in the 17th century by Boyle and Hooks [1]. Later, numerous applications of interference in research and metrology were realized after Michelson introduced his famous interferometer in 1881 [2]. In fact, Michelson’s interferometer and many other interferometers are plates of variable thickness. The diffraction of light was discovered by Grimaldi even earlier than the interference. The more familiar diffraction phenomenon appears when the passage of a spatially coherent beam of light is partly obstructed by an opaque object. In this process the amplitude of the propagating wave experiences a sharp change at the object-field boundary. The foundation of diffraction theory was laid by Huygens in the late 17th century. It was promoted into a consistent wave theory by Fresnel and Kirchhoff in the 19th century that has been very successful in dealing with optical instruments and describing numerous optical phenomena. Based on this theory the subject of diffraction includes Fresnel diffraction (FD), Fraunhofer diffraction, and, closely related to the latter, far-field diffraction. Fraunhofer diffraction has many applications in describing optical systems and in spectrometry, but applications of FD are very limited. The limitation is imposed by the nonlinearity of FD and the inability to change the optical path difference (OPD) at will. However, a rather unfamiliar form of FD occurs as the phase of a wavefront in some region undergoes a sharp change. An abrupt change in the phase can be easily imposed by reflecting a light beam from a step or transmitting it through a transparent plate with an abrupt change 1084-7529/09/030540-8/$15.00

in thickness or refractive index. Although this kind of FD has been studied directly and indirectly by several authors [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 in FD outlined in the latter reports and use them to realize the aforementioned applications. But before doing so we briefly 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 through point S is parallel to the step edge. Using the Fresnel– Kirchhoff integral the diffracted amplitude and intensity can 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 P0, the origin of the coordinate system used for the intensity calculation at point P. For P0 on the left side of the step edge and given the coefficients of the amplitude reflection rL and rR for the left and right sides of the edge, the intensity at point P is given by [9] IL = I0rLrR关cos2共␾/2兲 + 2共C02 + S02兲sin2共␾/2兲 − 共C0 − S0兲sin ␾兴 2 兲兴 , + I0/2关共rL − rR兲2共 2 + C02 + S02兲 + 共C0 + S0兲共rL2 − rR 1

共1兲

where I0 is proportional to the illuminating intensity, ␾ = 2kh cos ␪ is the phase introduced by the step (k and ␪ stand for the wave number and incidence angle, respectively, at point P0), and C0 and S0 represent the wellknown Fresnel cosine and sine integrals, respectively, associated with the distances between P0 and the source © 2009 Optical Society of America

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z

6

␾ = kNh关冑n2 − sin2 ␪ − cos ␪兴,

P

s

Sc

T

P0

and the step edge. According to Eq. (1), the intensity at point P depends on the step height and the reflection coefficients. However, even for h = 0, because rL ⫽ rR, the intensity across a screen normal to S⬘P is not uniform and Fresnel fringes are observed. For rL = rR the normalized intensity on the left or right side of the edge, specified by ⫺ and ⫹, respectively, is expressed as [8,9] In = cos2共␾/2兲 + 2共C02 + S02兲sin2共␾/2兲 ⫿ 共C0 − S0兲sin ␾ , 共2兲 or In = A + B cos ␾ ⫿ C sin ␾ ,

共3兲

where 1

B = 2 − 共C02 − S02兲,

where n = N / N⬘ represents the ratio of the refractive index of the plate to that of the medium. Extension to 2D phase steps is straightforward [9]; however, for our objectives 1D steps are quite adequate.

3. STEP WITH VARIABLE HEIGHT

Sc

1

共5兲

x

h

Fig. 1. Cylindrical wave ⌺ striking a 1D phase step of height h. The diffracted intensity at point P is given in the text.

A = 2 + C02 + S02,

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C = C0 − S0 .

共4兲

A phase step with variable height can be built in numerous ways. For example, by mounting a circular mirror and an annular mirror on the tops of two coaxial cylinders as shown in Fig. 3 one can build a circular step. The height of the step can be varied by moving cylinder C1 in a vertical direction. To build a 1D phase step the circular mirrors are replaced by rectangular ones. Since in FD the effective parts of an aperture are the edge neighborhood, in many cases, mirrors of a few millimeters widths are quite adequate. Thus, the phase steps can be designed and fabricated in compact form. This, in turn, reduces the effect of any mechanical noise. One can also design phase steps by using Michelson and Mach–Zhender interferometers with some modifications. For example, to build a 1D phase step by Michelson interferometer one can replace the mirrors by two rectangular mirrors in such a way that each mirror reflects the alternative halves of the beam striking the beam splitter, Fig. 4(a). In this case mirror M2 and the image M1⬘ of mirror M1 in the beam splitter B.S. form the required phase step. To build a phase step of desired shape by Michelson interferometer one can paste two complementary masks on the mirrors. By complementary masks we mean two masks that are joined together so as to obstruct the entire

One can form a 1D phase step in transmission by immersing a transparent plate in a transparent medium (liquid or gas). When a plane or cylindrical wave passes through the plate it experiences a sharp change in phase at the plate edges because of an abrupt change in refractive index. Intensity calculation by the Fresnel–Kirchhoff integral at a point on a screen perpendicular to the direction of the transmitted light, Fig. 2, leads to equations similar to (1) and (2) except for the phase ␾ that should be replaced by [9]

M1

c1

B.S M2

Nc

T

c2 N

Sc. Fig. 2. Profile of a transparent plate of refractive index N immersed in a liquid of refractive index N⬘ ⫽ N. The 1D phase steps are formed at the edges of the plate.

Fig. 3. Sketch of a circular phase step that can be built by mounting a circular mirror M1 and an annular mirror M2 on two coaxial cylindrical stands C1 and C2. The light reflected from the beam splitter B.S. diffracts from the step formed by the mirrors, and the step height can be varied by displacing mirror M1 in a vertical direction.

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Fig. 4. (a) A 1D phase step of height h is formed by replacing the mirrors in a Michelson interferometer by two rectangular mirrors in such a way that each mirror intersects the alternative halves of the light beam striking the beam splitter. (b) A 1D phase step is formed by mounting two opaque plates O1 and O2 in the arms of a MZI at equal distances from the beam splitter B.S.2 in such a way that the plates obstruct the alternative halves of the beam reflecting from the mirrors M1 and M2. The step height is varied by changing the OPD between the arms of the interferometer.

beam in one of the interferometer’s arm. For instance, a circular mask and an annular mask with its inner radius equal to that of the circular mask pasted symmetrically on the interferometer mirrors provide a circular phase step. The masks should be good absorbers of light; otherwise, the scattered lights enhance the noise. In a Mach–Zhender interferometer (MZI) one can install the complementary masks in the interferometer arms at equal distances from the beam splitter B.S.2 in Fig. 4(b). The equal distance from the beam splitter assures that the diffracting apertures are practically the same distance from the observation screen. In these cases the step height can be varied by changing the OPD between the interferometer’s arms. This can be done either by moving one of the mirrors or changing the physical property of the materials occupying the arms of the interferometer, 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 heights formed by Michelson interferometer. The plots are the intensity profiles of the patterns (the average intensities in the 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 formed by a MZI.

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0 0.5 1 mm Fig. 5. FD patterns of light diffracted from 1D phase steps of different heights formed in a Michelson interferometer arrangement and the corresponding intensity profiles over the patterns. (a) h = ␭ / 8. (b) h = ␭ / 4. (c) h = 3␭ / 8. (c)

−1 −0.5

A fundamental difference between the fringes formed by a phase step of variable height and those formed in conventional interference is that the visibility of the former is very sensitive to the change of OPD, while the visibility of the latter is practically insensitive to OPD. As we will show later, the capability of measuring a 1% change in the visibility of the step fringes provides the possibility of measuring a change of ␭ / 400 in step height. Another remarkable difference concerns the fringe spacing. The spacing of the phase step fringes depends on the distance of the diffractor from the light source and the observation screen. For fixed distance and a given diffractor geometry the intensity profile of the diffraction pattern is a known function. This provides a large volume of data on the step height and further improves the measurement accuracy. In addition, measurement by diffractometry is less sensitive to mechanical vibrations compared with conventional interferometry. However, the interference fringe spacing depends on the gradient of the OPD, and the intensity profile is not known in advance. As the patterns and the intensity profiles in Figs. 5 and 6 show, the fringe visibility decreases with the distance from the step edge. We define the visibility for the three central fringes by the following expression

V=

1 共I + ImaR兲 2 maL 1 共I + ImaR兲 2 maL

− ImiM + ImiM

,

共6兲

where ImaL and ImaR stand for the maximum intensities of the left side and right side bright fringes, while ImiM represents the minimum intensity of the central dark fringe. Plotting Eq. (6) versus ⌬ / ␭ 共⌬ = 2h cos ␪兲 in the range 0–1 the curve shown in Fig. 7 is obtained. According to this curve, as ⌬ varies in an interval of ␭ / 2 the visibility defined above changes from zero to one.

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iS

0.8

r =r L R h = λ/10

0.6

φ = 2π/5

M’

J’

2

J’1

0.4 φ

0.2

J’’

φ M

0

O

C

−0.2 −0.4 J

1

−0.6

J

2

−0.8 −1

−0.5

0

0.5

1

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

Fig. 6. FD patterns and the corresponding intensity profiles of light diffracted from circular phase steps of different heights formed by a MZI, (a) h = 5␭ / 24. (b) h = ␭ / 2. (c) h = 5␭ / 6.

ciated with each side of a step. The two Cornu spirals of a step differ in phase due to the step height. Only one Cornu spiral accounts for the intensity distribution at points far from the step edge. However, the contributions of the two spirals should be considered at points close to the step edge. For example, at point P in Fig. 1 the conជ tribution of the left side Cornu is J 1M in Fig. 8, while the ជ ជ contribution of the right side is M⬘J2⬘ . We have 兩M ⬘J2⬘ 兩 ជ = 兩MJ1⬘ 兩 when the reflection coefficients on both sides are ជ ជ the same. By squaring the vectorial sum 共J 1M + MJ⬘兲 and substituting the corresponding coordinates in the C-S system, Eq. (2) is derived. For ␾ = ␲ the two spirals are in opposite phase and the resultant amplitude vanishes at points corresponding to the edge of the step.

It is interesting to recall that in the FD caused by a sharp change of the amplitude the visibility of the fringes is very low. This remarkable difference can be explained by the Cornu spiral adequately. One Cornu spiral is asso1

Visibility

Calculation

0.5

0 0

0.25

0.5 ∆/λ

0.75

1

Fig. 7. Calculated visibility versus the optical path difference divided by wavelength ⌬ / ␭ for three central fringes in FD from a 1D phase step.

Fig. 9. Scheme of a rectangular cell and a plane parallel plate that is installed inside it to study liquid–liquid diffusion by light diffraction.

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150 100 50 0 −1

(a)

150 100 50 0 −1 (b) 150 100 50 0 (c) −1 5min.

15min.

30min.

60min.

120min.

350min.

Fig. 10. Diffraction patterns of the light diffracted from the edge of a plane parallel plate immersed in a rectangular cell containing pure water over sugar solution of concentration 10% at different times after the initiation of the diffusion. The established refractive index gradient has appeared as the fringes inclined with respect to the plate edge.

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0 mm

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4. METROLOGICAL APPLICATIONS Some straightforward applications of the effect are in the measurements of displacement, film thickness, refractive index, and dispersion of a transparent film or plate that can be realized with high accuracy by fitting Eq. (1) or (2) on the experimentally obtained normalized intensity distribution of the corresponding fringes. A novel application of the phenomenon is in the measurement of the refractive index gradient that appears in many situations, such as in a diffusion process and in media sustaining temperature gradients. There are optical methods based on interferometry, holography, and moiré deflectometry for measuring the refractive index gradient [11–13]. However, the method we describe here is remarkably simple and highly accurate. For example, to measure the refractive index gradient in a biliquid diffusion process, we install a transparent plane parallel plate of thickness e and refractive 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 process proceeds, the refractive index along the vertical edge of the plate varies, and a step of height h = e关N − n共z兲兴 is formed, where n共z兲 is the refractive index of the cell content at altitude z. As the cell is perpendicularly illuminated by a coherent beam of wavelength ␭, the visibility of the step fringes repeats along the plate edge as en共z兲 changes by ␭. The plot of fringe visibility versus z provides the index very accurately in the neighborhood of the plate edge. The patterns in Fig. 10 are the diffraction patterns of light diffracted from the edge of a plane parallel plate installed in a rectangular cell in which sugar solution was diffusing into water, at different times after the beginning of diffusion. The inclinations and the spacings of the oblique fringes show very clearly the states of the diffusion process.

0 (e)

0.5

150 100 50 0

0 0.5 1 mm Fig. 11. Experimental realization of Babinet’s principle. (a), (b) The diffraction patterns and intensity profiles of the light diffracted from a slit of 0.24 mm width and an opaque strip of the same width as the slit. (c) The pattern and intensity profile obtained by superimposing the diffracted fields in (a) and (b) in a MZI. (d), (e) The diffraction patterns and intensity profiles of the light diffracted from two complementary straight edges. (f) The pattern and intensity profile obtained by superimposing the diffraction fields in (d) and (e) in a MZI. (f)

−1 −0.5

5. EXPERIMENTAL REALIZATION OF BABINET’S PRINCIPLE According to Babinet’s principle, superposition of the fields diffracted from two complementary apertures (two apertures that are connected together form an infinite aperture) leads to a uniform field. Two parts of a 1D or 2D phase step for the case of zero step height are complementary apertures. The diffraction patterns and intensity profiles shown in Figs. 11(a) and 11(b) are obtained by diffracting light from a slit and an opaque strip of the same width as the slit in similar conditions. However, when the objects are installed in a MZI in such a way that the image of one object in the second B.S. is superimposed on the other object, illumination of both objects leads to the diffraction 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

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Fig. 12. Enhancing and suppressing light diffraction. (a) FD pattern of light diffracted from a slit. (b), (c) The patterns obtained by superimposing constructively (enhanced mode) and destructively (suppressed mode) the light diffracted from two similar slits installed in a MZI’s arms. (d) The profiles of the intensity distribution of the corresponding diffraction patterns.

experimental realization of Babinet’s principle due to superposition of the fields diffracted from two complementary straight edges mounted in the arms of a Michelson interferometer.

6. ENHANCING AND SUPPRESSING DIFFRACTED FIELDS: SPATIAL COHERENCE WIDTH MEASUREMENT The superimposing of the diffracted fields of two objects raises the possibility of enhancing or suppressing diffraction. For example, by installing two similar slits in the arms of a MZI in such a way that one slit is superimposed on the image of the other in the second beam splitter, one can enhance or suppress the diffracted field by introducing a phase difference equal to an even or odd number of ␲ between the arms of the interferometer. The diffraction patterns in Figs. 12(b) and 12(c) are the enhanced and suppressed versions of the diffraction pattern of a single slit that is shown in Fig. 12(a). The plots in Fig. 12(d) are the intensity profiles of the corresponding diffraction patterns. For the enhanced case the intensities at a majority of the points are four times more than the intensities at the corresponding points for the single slit. That means the recording CCD has responded linearly at these intensities. Comparing the plots (b) and (c) in Fig. 12(d) we note that for a phase change of ␲ the intensity varies significantly, and this provides more precise phase change measurement. Suppression of diffraction is very useful in the studies of minute inhomogeneities and anisotropies in transpar-

d

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b

80 a

60 40 −1

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0 mm (e)

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Fig. 13. Diffraction patterns of the light diffracted from a copper wire of thickness 0.4 mm carrying different electric currents after its original diffracted field had been suppressed by the field diffracted from another similar wire installed in the other arm of a MZI. (a) I = 0.12 A. (b) I = 0.25 A. (c) I = 0.41 A. (d) The pattern obtained by superimposing constructively the diffracted fields from the two wires with no electric current. (e) The profiles of the intensity distributions of the corresponding diffraction patterns.

ent media and objects such as optical fibers and light waveguides, where diffraction from the geometry of the object leads to very noisy data. To illustrate this point more clearly we put two similar copper rods (wires) in the arms of a MZI and adjusted the OPD to get full suppression. Then, by passing different electrical currents through one of the rods we built up different temperature gradients around it and recorded the diffraction patterns shown in Figs. 13(a)–13(c). The intensity distributions of the latter patterns and of the diffraction pattern of the unheated rods, but in enhanced condition, Fig. 13(d), are plotted in Fig. 13(e). The intensity changes provide information about the phase changes, and the slopes of the curves indicate the presence of temperature gradients. This experiment suggests that by replacing the mirrors in a Michelson interferometer with two thin rods (wires) one could construct an optical diffractometer to measure small forces applied to one of the rods. To measure the spatial coherence width of the light illuminating a MZI we install two similar slits or pinholes in the arms of the interferometer in such way that one object is superimposed on the image of the other with zero OPD. Then, by displacing one of the slits (or pinholes) in its plane we can obtain a double slit of desired separation that is useful for spatial coherence studies.

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1 0.8 0.6 0.4 0.2 0 0.4

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0.5 λ (µm)

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(a)

×10−5 1

Left Right

0.8 0.6 0.4 0.2 0 0.4

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0.5 λ (µm)

0.55

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(b)

Fig. 15. Spectrum of a light beam incident on a 1D phase step of height ␭0 / 4. (b) The normalized spectra of the diffracted lights at two points symmetrical with respect to the step edge 共␭0 = 560 nm兲.

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

7. APPLICATION TO THE STUDY OF OPTICAL FIELDS IN THE NEIGHBORHOOD OF PHASE SINGULARITIES In recent decades the behavior of optical fields in the neighborhood of phase singularities—points of indeterminate phases and zero amplitudes—has attracted many researchers [14,15]. Also, more recently a number of works

have been published in which the behavior of polychromatic light around phase singularities has been studied [16–20]. Spectral modification, red-shifting, blue-shifting, and anomaly in the appearance of colors have been predicted and observed in experiments [21–24]. In many reported cases the field amplitudes in the neighborhoods of the singularities were small and this restricted the experimental studies of the subject severely. However, as has been reported in [8], the amplitudes in the neighborhood of the singularities produced by the FD from phase steps are as large as the amplitudes at other points, and this eases the experimental studies. Additionally, in these cases the depth of the singularity can be usefully varied. Very recently the FD—the phase step approach—has been applied to the study of the phase singularity at Brewster’s angle [25]. One can design phase singularities for the study of spectral modification in the following ways. One way is to install two complementary apertures in the arms of a MZI to get a uniform optical field implied by Babinet’s principle. Then, by changing the OPD between the two arms

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of the interferometer by an odd number of ␭ / 2 of the wavelength used, the field amplitude for that wavelength becomes zero at the points associated with the edge of the step. The patterns in Fig. 14 are the diffraction patterns of white light diffracted from 1D steps of slightly different heights. The nonsymmetrical distribution of the colors is interesting. The curve in Fig. 15(a) represents the spectrum of the light incident on a 1D phase step, while the curves denoted left and right in Fig. 15(b) are the spectra of the diffracted lights at two points symmetrical with respect to the edge. The spectral modifications are drastic. Another design for a phase singularity is to install two similar apertures, namely, two similar slits, in the arms of a MZI so that the image of one slit coincides with the other. Then, by changing the OPD by an odd multiple of ␭ / 2 of the given wavelength, one obtains a singularity of the shape of the slit.

8. SUMMARY This report shows that light diffraction from phase steps with variable height is a rich subject with many novel applications, and the ideas may be applicable to particles other than photons.

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9. 10.

11. 12.

13.

14. 15. 16. 17.

ACKNOWLEDGMENT The corresponding author sincerely acknowledges the support provided by the Center of Excellence of the Ministry of Higher Education and Technology (Iran).

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7.

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