Optical diffractometry
M. Taghi Tavassoly,
1,2,
*
Mohammad Amiri,
3
Ahmad Darudi,
4
Rasoul Aalipour,
1
Ahad Saber,
1
and AliReza 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 14394547, Iran
3
Physics Department, BuAli 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 including highprecision measurements of displacement, phase change, refractive index proﬁle, temperature gradient, 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 halfwave. Therefore, measurements 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 appearance 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 after 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 diffraction 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 objectﬁeldboundary.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 19thcentury 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, farﬁeld diffraction.Fraunhofer diffraction has many applications in describing 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 imposed by reﬂecting a light beam from a step or transmitting 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 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 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 coordinate system used for the intensity calculation at point
P
.For
P
0
on the left side of the step edge and given the coefﬁcients 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, respectively, at point
P
0
), and
C
0
and
S
0
represent the wellknown Fresnel cosine and sine integrals, respectively, associated with the distances between
P
0
and the source
540 J. Opt. Soc. Am. A/Vol. 26, No. 3/March 2009 Tavassoly
et al.
10847529/09/0305408/$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 coefﬁcients. However, even for
h
=0, because
r
L
r
R
, the intensity 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 immersing 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 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 bereplaced by [9]
=
kNh
n
2
−sin
2
−cos
,
5
where
n
=
N
/
N
represents the ratio of the refractive index 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 numerous 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 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, inmany cases, mirrors of a few millimeters widths are quiteadequate. Thus, the phase steps can be designed and fabricated 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 modiﬁcations. For example, to build a 1D phase step by Michelsoninterferometer one can replace the mirrors by two rectangular 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 mirror
M
1
in the beam splitter
B
.
S
. form the required phasestep.To build a phase step of desired shape by Michelson interferometer 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
immersed 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; otherwise, the scattered lights enhance the noise.In a Mach–Zhender interferometer (MZI) one can install 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 between 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 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 heightsformed by Michelson interferometer. The plots are the intensity 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 spacing. The spacing of the phase step fringes depends on thedistance of the diffractor from the light source and the observation 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
represents 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 deﬁned 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 mirrors 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 arrangement 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 associated 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 contribution 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 distribution of the corresponding fringes. A novel applicationof the phenomenon is in the measurement of the refractive index gradient that appears in many situations, suchas in a diffusion process and in media sustaining temperature 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 refractive index gradient in a biliquid diffusion process, we install 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 process 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 content at altitude
z
. As the cell is perpendicularly illuminated 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 patterns of light diffracted from the edge of a plane parallelplate installed in a rectangular cell in which sugar solution 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 aperture) leads to a uniform ﬁeld. Two parts of a 1D or 2Dphase 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 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 image 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 containing pure water over sugar solution of concentration 10% atdifferent times after the initiation of the diffusion. The established refractive index gradient has appeared as the fringes inclined 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 diffracted from a slit of 0.24 mm width and an opaque strip of thesame width as the slit. (c) The pattern and intensity proﬁle obtained 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 diffraction ﬁelds in (d) and (e) in a MZI.544 J. Opt. Soc. Am. A/Vol. 26, No. 3/March 2009 Tavassoly
et al.