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Characterization of Fresnel images of an elongated circular grating
Optik 113, No. 7 (2002) 285–292 ª 2002 Urban & Fischer Verlag J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating http://www.urbanfischer.de/journals/optik
285
International Journal for Light and Electron Optics
Characterization of Fresnel images of an elongated circular grating J. S. Kang1, M. S. Lee1, S. Chang1, B. J. Kim2, K. C. Yuk2 1 2
Department of Physics, Hannam University, 133 Ojungdong, Taejon 300-791, Korea Department of Physics Education, Kongju National University, 182 Sinkwandong, Kongju, Korea
Abstract: We characterize theoretically and experimentally Fresnel images (or self-images) of an elongated circular (EC) grating which could be employed for the visual measurement of linear displacements in Talbot moire´ deflectometry. Based on a Fresnel diffraction integral, we estimate the shape of the Fresnel image formed at a specific distance from the EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. For a square type transmittance of the grating, we also compare the theoretical estimates with its self-images captured by an array of charge-coupled devices (CCD). We show that self-images of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two self-imaged EC gratings are found not much affected by the central aberration. Key words: Self-image elongated circular grating Talbot effect
1. Introduction In an ealier work [1], use of an elongated circular (EC) grating in moire´ deflectometry has been proposed for the fine visual determination of both the linear displacement and its direction. Moire´ patterns formed by placing two complementary EC gratings in contact (or projecting one EC grating onto the other) have been also used to measure the thermal expansion of metals (or the absolute refractive indices of optically transparent liquids). Optical moire´s of two different EC gratings have the advantage of providing ten times as good accuracy in metrology as those of conventional circular gratings [2–7]. If the EC gratings are employed to a Talbot interferometer [8–10] which requires no lens to image an object grating at the plane of a reference grating, they could be more useful in practical applications because of simplicity and compactness of the system. The performance of a Talbot interferometer is
based on the quality of Fresnel images (or self-images) of an object grating. For successful performance of a Talbot interferometer with EC gratings, therefore, it is of great importance to characterize Fresnel images of an EC grating and moire´ fringes by self-imaging an object EC grating on a reference EC grating. Using the Fresnel-Kirchhoff integral formula with a parabolic approximation of the optical path length [11], Montgomery [12] discussed the general condition for self-imaging objects of infinite aperture. The selfimaging phenomena of the cross, line, circle, spiral, and evolute gratings were well described by Cowley and Moodie [13], Winthrop and Worthington [14], Szwaykowski [15], and Mansuripur [16]. Self-images of a pinhole array were also examined for application to the pinhole-array camera by Bryngdahl [17]. However, self-images of an amplitude type EC grating which consists of concentric semicircles matched with parallel lines of finite length have not been investigated. In this paper we characterize theoretically and experimentally self-images of an amplitude type EC grating which could be employed for the visual measurement of linear displacements in Talbot deflectometry. Based on a Fresnel diffraction integral, we estimate the shape of the self-image formed at Talbot distances from the EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. For a square type transmittance of the grating, we also compare the theoretical estimates with its self-images captured by an array of charge-coupled devices (CCD). We show that self-images of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two superimposed self-imaged EC gratings are found not much affected by the central aberration.
2. Self-images through Fresnel diffraction Received 22 February 2002; accepted 6 June 2002. Correspondence to: S. Chang Fax: ++82-42-629-7444 E-mail: [email protected]
Figure 1 shows the geometry of an optical setup for observing Fresnel images (or self-images) of an elongated circular (EC) grating which is composed of the 0030-4026/02/113/07-285 $ 15.00/0
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
y
y’ x’
x
Coherent Plane wave
z’
=2p20 / O
ðq; f; zÞÞ. The origin of a rectangular (or cylindrical) coordinate system is situated at the center of the linear (or semicircular) grating as shown in fig. 2. The diffraction field A0 ðx0 ; y0 ; z0 Þ; which is observed at a distance z0 from an object grating, can be determined by putting the object field Aðx; yÞ into the Fresnel-Kirchhoff diffraction integral with a parabolic approximation of the optical path length [11, 12] 0
0
0
1 ð
0
A ðx ; y ; z Þ ¼ C
1 ð
dx 1
Object EC grating
Reference EC grating
Fig. 1. Geometry of an optical setup for observing Fresnel images (or self-images) of an elongated circular (EC) grating. The origin of a rectangular coordinate system (x; y; z) (or a circular coordinate system (q; f; z)) is on the surface of an object EC grating. Self-images of the grating are formed at Talbot distances z0 ¼ m(2p2 =l) (for m ¼ 1; 2; . . .), where po is the pitch of the grating and l is the wavelength of an incident wave.
! ðx0 xÞ2 þ ðy0 yÞ2 ; ip lz0
ð1Þ
with a factor of C ¼ exp ði2pz0 =lÞ=ðilz0 Þ: In a cylindrical coordinate system ðq; f; zÞ; the Fresnel diffraction integral (1) can also be transformed into 0
0
0
0
A ðq ; f ; z Þ ¼ C
0
1 ð
2p ð
df
pq2 dq qAðq; fÞ exp i lz0
0
2pq0 q 0 cos ðf f Þ ; exp i lz0
ð2Þ
where we have used the relations C0 ¼ C exp ipq02 =lz0 ; x ¼ q cos f ; x0 ¼ q0 cos f0 ;
y ¼ q sin f ; y0 ¼ q0 sin f0 :
Now we are to use the diffraction integrals ð1Þ and ð2Þ for examining self-images of both the parallel line part and the semicircular part in the EC grating.
2.1 Self-images of the parallel line part
y
( r, f)
0
L
1
exp
0
two parts: straight lines of finite length and semicircles. The field of a plane wave is incident perpendicularly to the surface of the grating. The wavelength of the incident light is denoted by l: The spacing between the two adjacent parallel lines (or semicircles) in the object EC grating is represented by po. The light fields diffracted by the grating produce an image of itself at integer multiples of distance z0 ¼ 2p2o =l. Self-images of the object EC grating are recorded by using an array of chargecoupled devices (CCD) with an image board. The object field through the linear (or semicircular) part of the EC grating is well described in a rectangular coordinate system ðx; y; zÞ (or a cylindrical coordinate system
dy Aðx; yÞ
d
x
p0 Fig. 2. Schematic diagram of an elongated circular grating in which the semicircles are matched with the parallel lines. The upper semicircles belong to the domain of 0 < q < d and 0 < f < p: The parallel lines are in the domain of jxj < d and L < y < 0: In a realistic case, d and L are large compared with a pitch po :
The parallel line part of an EC grating is just a linear grating of uniform period po and overall width 2d: The dark stripes of length L are periodically arranged parallel to the y-axis as shown in fig, 2. The transmission coefficient of the parallel line part is assumed to change periodically between 1 and a ð> 0Þ along the xaxis (fig. 3), while it remains constant along the y-axis. The ratio of the opening between the two adjacent stripes to a period is denoted by D. In a case of a square type transmission, the openning ratio D is taken as 1/2. To evaluate the diffraction integral (1), we may as well represent a rectangular type transmission coefficient tðx; yÞ of the parallel line part in a Fourier series of harmonic functions as 8 1 > < P b cos 2pn x for jxj < d and L < y < 0 ; n po tðx; yÞ ¼ n¼0 > : 0 otherwise ;
(3)
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
pffiffiffiffiffiffiffiffi0
2=lz ðdx0 nlz0 =po Þ
0
0
Wðx ; z Þ ¼ pffiffiffiffiffiffiffiffi0 2=lz
ð
ðdx0 nlz0 =p
p dt exp i t 2 2
287
ð5Þ
oÞ
If we have a screen situated at Talbot distances z0 ¼ mð2p2o =lÞ Zm for m ¼ 1; 2; . . ., then the first exponential factor in the summation of eq. (4) will always go to unity. Thus, the field (4) distributed on the screen at Talbot distances Zm can be expressed as 1 b P n 2 n¼0
2pn 0 0 Vðx ; Zm Þ exp i x po 2pn 0 þ Wðx0 ; Zm Þ exp i x ; po
A0 ðx0 ; y0 ; Zm Þ ¼ C00 Uðy0 ; Zm Þ
Fig. 3. Transmission coefficient t of an elongated circular grating which varies between 1 and a with a period of po . The portion of the transparent part in a period is denoted by D: For a square type transmission, the opening ratio D goes to 1=2:
where the coefficients are found to be of b0 ¼ a þ ð1 aÞ D for n ¼ 0 and of bn ¼ 2ð1 aÞ sin ðnpDÞ=np for n > 0: For a case where the grating is illuminated by a coherent plane wave of amplitude A0, the object field Aðx; yÞ transmitted through its parallel-line part can be described by multiplying tðx; yÞ by A0, which is a rectangular wave in the domain of jxj < d and L < y < 0: Substituting Aðx; yÞ into Eq. (1) and carrying out the integration over the whole domain, we find the diffraction field to be A0 ðx0 ; y0 ; z0 Þ ¼ CA0
ð0
ðd
dy tðx; yÞ
dx d
L
! ðx0 xÞ2 þ ðy0 yÞ2 exp ip lz0 1 P lz0 lz0 n2 Uðy0 ; z0 Þ bn exp ip 2 ¼ CA0 4 po n¼0 2pn 0 x þ Wðx0 ; z0 Þ Vðx0 ; z0 Þ exp i po 2pn 0 exp i x ; po
2=lz y
2=lz ðdx0 þnlz0 =po Þ
Vðx0 ; z0 Þ ¼ pffiffiffiffiffiffiffiffi0
ð
2=lz ðdx0 þnlz0 =po Þ
where C00 is a constant. Moreover, in a realistic case that d is large compared with both jx0 j and jlz0 =po j; the Fresnel integral functions V and W in eq. (5) become pffiffiffi equal to 2 exp ðip=4Þ [18]. Consequently, the diffraction field (6) can be written in a simple form as 1 P 2pn 0 A0 ðx0 ; y0 ; Zm Þ ¼ C000 Uðy0 ; Zm Þ bn cos i x ; po n¼0 ð7Þ pffiffiffi 00 In the above we have let C ¼ 2 exp ðip=4ÞC : The summation part on the right-hand side of eq. (7) behaves like an object field (eq. (3)) in the x direction, while the integral function U arises from Fresnel diffraction by straight lines of finite length in the y direction. Therefore, it is clear that the diffracted field (7) describes self-images of the parallel line part which include aberrations by Fresnel diffraction in the y direction. The intensity of the diffraction field (7) at the m-th Talbot distance Zm is now given by 000
Ilin ðx0 ; y0 ; Zm Þ ¼ A0 ðx0 ; y0 ; Zm Þ A0 *ðx0 ; y0 ; Zm Þ :
ð8Þ
2.2 Self-images of the semicircular part
ð4Þ
where we have defined the Fresnel integral functions pffiffiffiffiffiffiffiffi0 2=lz ðLþy0 Þ ð p dt exp i t 2 ; Uðy0 ; z0 Þ ¼ 2 pffiffiffiffiffiffiffiffi0 0 pffiffiffiffiffiffiffiffi0
ð6Þ
Next we examine the upper semicircular part of an EC grating of which the transmission coefficient varies periodically between 1 and að> 0Þ with radial variable q in the region of 0 < q < d and 0 < f < p (fig. 2). As above, we assume a transmission coefficient t of the semicircular part in a Fourier series of harmonic functions 8 1 > < P b cos 2pn q for 0 < q < d and 0 < f < p; n po tðq; fÞ ¼ n¼0 > : 0 otherwise ;
(9) p dt exp i t 2 ; 2
with the coefficients of b0 ¼ a þ ð1 aÞ D for n ¼ 0 and bn ¼ 2ð1 aÞ sin ðnpDÞ=np for n > 0: If a coherent plane wave of amplitude A0 is incident normally
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
onto the surface of the EC grating, the object field on the semicircular part may be represented by Aðq; fÞ ¼ A0 tðq; fÞ: Insertion of Aðq; fÞ into eq. (2) leads to ðd 1 P 2pn dq q bn cos q A0 ðq0 ; f0 ; z0 Þ ¼ C0 A0 po n¼0 0
ðp pq2 exp i 0 df lz 0 2pq0 q 0 cos ðf f Þ : exp i lz0
ð10Þ
The second integral in eq. (10) can be easily solved by expanding the integrand in a series of Bessel functions as follows: 2pq0 q 0 cos ðf f Þ exp i lz0 1 P 2pq0 q 2pq0 q l þ 2 cos ½lðf0 fÞ ; ¼ J0 ð i Þ J l 0 lz0 lz l¼1
parts are recorded at the first Talbot distance by using a CCD-array camera (Samsung SBC 341) with an image board (DT 3155). The CCD-array camera has the cell size of 8 mm. For use of EC gratings in Talbot deflectometry, we present the moire´ patterns of an EC grating pair made by a self-imaging method. The pitch of the object grating is taken at po ¼ 200 mm, and the reference grating (see fig. 1b) in ref. [1]) is made to be of pitch pr ¼ po ð2N 1Þ=ð2NÞ for N ¼ 12: Figure 4 shows the theoretical curves for the intensity of the image field which is evaluated at the first Talbot distance z0 ð¼ 2p2o =lÞ; where the parallel line part in an object EC grating is self-imaged; a) the image intensity of the parallel line part is plotted as a function of x0 ; while the value of y0 is fixed at L=2; and b) the intensity of the image field at x0 ¼ 0 is also plotted as a function of y0 : We have chosen the parameters as D ¼ 1=2; a ¼ 0:4; L po and 2d ¼ 23po : It is obvious in fig. 4b) that the image field distributed along the x-axis corresponds to the object field of the
ð11Þ
where Jl ðcÞ denotes the Bessel function of order l: By putting eq. (11) into eq. (10) and solving the second integral at an image plane (i.e., z0 ¼ mð2p2 =lÞ ¼ Zm Þ, one can find the image field as an integral with respect to radial variable q A0 ðq0 ; f0 ; Zm Þ ¼ C 0 A0
1 P n¼0
(
ðd dq q bn cos
2pn pq2 q exp i po lZm
0
1 ðiÞl P 2pq0 q lp 2pq0 q sin þ4 Jl l lZm 2 lZm l¼1 lðp 2f0 Þ : ð12Þ cos 2
pJ0
The intensity of the image field (12) at the m-th Talbot distance is also given by Is ðq0 ; f0 Þ ¼ A0 ðq0 ; f0 ; Zm Þ A0 *ðq0 ; f0 ; Zm Þ :
ð13Þ
The field intensity (13) has to be evaluated by means of a numerical method so as to examine the behaviour of self-images of the upper semicircular part.
3. Results and discussion Numerical calculations of the Fresnel diffraction integrals (7) and (12) are carried out to characterize selfimages of an amplitude type EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. The wavelength of the incident light is chosen to be of l ¼ 632:8 nm (He-Ne laser). For comparison of the theoretical estimates with measured results, self-images of the linear and semicircular
Fig. 4. Theoretical curves for the intensity of the image field which are evaluated at the first Talbot distance z0 (¼ 2p2o =l) from the parallel line part. We let D ¼ 1=2; a ¼ 0:4 and L po : (a) Self-images of the parallel line part are shown as a function of x0 when y ¼ L=2; and (b) the intensity of the image field is plotted as a function of y0 at x0 ¼ 0: Half shadows tail to the dark region of y0 > 0:
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
square-wave type in fig. 3. As a result of Fresnel diffraction by a straight edge, however, the y-distribution of the image intensity in fig. 4b) is oscillating in the bright region (i.e., y0 < 0Þ; and a half shadow is shown to tail to the dark region (i.e., y0 > 0Þ: In Fig. 5 we show the theoretical curves for the intensity of the image field which is evaluated at the first Talbot distance z0 ð¼ 2p2o =lÞ, where the upper semicircular part in an object EC grating is self-imaged; a) the image intensity of the upper semicircular part is plotted as a function of q0 ; while the angular position f0 remains fixed at p=2; and b) the intensities of the image field are plotted as a function of f0 at integer multiples of q0 ¼ po : Other conditions are the same as in fig. 4. The intensity of the image field in fig. 5a) is highly enhanced at the edge of each semicircular stripe, while the distribution of the object field is of the square-wave type along the q-axis. However, it should
Fig. 5. Theoretical curves for the intensity of the image field which are evaluated at the first Talbot distance z0 (¼ 2p2o =l) from the upper semicircular part. We let D ¼ 1=2 and a ¼ 0:4: a) Self-images of the upper semicircular part are shown as a function of q0 when f0 ¼ p=2; and (b) the intensities of the image field are plotted as a function of f0 when q0 ¼ mpo (for m ¼ 1 5). In cases of small q0 ; half shadows tail to the dark region of p < f < 2p.
289
be noted here that the spacing between the two self-imaged semicircles is identical with that in the object grating. Figure 5b) shows that at specific positions of q0 ’ po and 2po the speckled patterns appear in the dark region of p < f < 2p: The intensity of the speckled pattern in the dark region decreases rapidly by increasing q0 : We find in fig. 5b) that the angular distribution of the image intensity (belonging to the upper semicircular part) is substantially aberrated in the central region of small q0 but not in the off-central region. Next, to confirm the theoretical estimates, we have experimentally taken self-images of the linear and semicircular parts in an object EC grating. Figure 6
Fig. 6. Photographs of the self-images formed on the screen at the first Talbot distance z0 ¼ 2p2o =l;of which each corresponds to the semicircular part (top), the parallel line part (middle) and the elongated circular grating (bottom). The weak halfshadows are observed in the dark region.
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
shows the photographs of the self-images formed at the first Talbot distance z0 ¼ 2p2o =l;where each of them corresponds to the semicircular part (top), the parallel line part (middle) and the elongated circular grating (bottom). Both of the linear and semicircular parts are observed to be well self-imaged. The self-imaged line grating has clear-cut outlines along the x-axis but obscure boundaries at y0 ¼ 0 as predicted in fig. 4. In a case of the self-imaged semicircular grating, the speckled circles appear to the dark region of p < f < 2p and they disappear as q0 increases. This behaviour is in excellent agreement with that in fig. 5b). Figure 7 shows the microdensitometric results for a) the self-imaged parallel lines and b) the self-imaged semicircles that are presented in fig. 6. In fig. 7a) the image intensity is given as a function of x0, while y0 ¼ L=2: In fig. 7b) the curve of the image intensity is plotted as a function of q0, while f0 ¼ p=2: If the curves in fig. 7 are compared with those in fig. 4a) and fig. 5a), one can find the measured pitches in good agreement with those in the theoretical estimation.
Finally, moire´ patterns of two self-imaged complementary EC gratings are investigated so as to demonstrate the feasible use of EC gratings in Talbot deflectometry. In fig. 8 we show the photographs of the moire´ patterns produced by overlapping a self-imaged object EC grating onto a reference EC grating as the object grating of pitch po ð¼ 200 mmÞ is displaced from the reference grating of pitch pr ¼ ð23=24Þ po by a) 0; b) po =4; c) po =2; d) 3po =4; e) po ; f) 5po =4; g) 3po =2; h) 7po =4; and e) 2po in the x-direction. The width of the parallel line part is chosen as 2d ¼ 23po ¼ 24pr : These moire´ fringes by means of a self-imaging method are comparable with those in fig. 3 of ref. [1] based on a contact (or lens-imaging) method. In the region of the parallel line part, one can see a vernier dark moire´ moving as a function of displacement of the object grating. The shift of the vernier dark moire´ Dx is related to the displacement of the object grating dx by Dx ¼
W dx ; po
ð14Þ
where the period of the vernier dark moire´s is given by po pr W¼ : ð15Þ po pr Substitution of po and pr into eqs. (14) and (15) leads to the formulas Dx ¼ 23 dx and W ¼ 2d: This means that if the center of the object grating shifts by a pitch po toward the positive x-direction, the vernier dark moire´ moves by 2d ð¼ 23po Þ toward the negative x-direction. In case of the semicircular part, the number of the dark moire´s is shown to increase as the center of the object grating goes away from that of the reference grating. The order m of the dark moire´s allowed in the semicircular part is determined by q jdxj q jdxj þ m ; W po W po
ð16Þ
where the bars j . . . j denote the absolute value of the argument. If we limit our interest to the region of q W; eq. (16) can be transformed into a simple form
jdxj jdxj m : po po
ð17Þ
If the object grating is displaced in the range of po dx < 2po ; the dark fringes of orders m ¼ 0; 1 and þ1 are allowed in the semicircular part as shown in figs. 8e)–h).
4. Conclusion Fig. 7. Microdensitometric curves for (a) the image intensity of the parallel lines measured at y0 ¼ L=2 as a function of x0 ; and (b) the image intensity of the semicircles measured at f0 ¼ p=2 as a function of q0 : By comparing the curves in fig. 7 with those in fig. 4a) and fig. 5a), we find the measured pitches in good agreement with those in the theoretical estimation.
In this paper, Fresnel images (or self-images) of an elongated circular (EC) grating have been examined theoretically and experimentally. Based on a Fresnel diffraction integral, we have analyzed the shape of the Fresnel image of the EC grating in which both parallel lines and semicircles are illuminated by a coherent
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
291
m=0
(a)
(b) m= -1,
(d)
0,
(c) +1
(e)
(f) m= -2, -1,
(g)
(h)
0
+1 +2
(i)
Fig. 8. Moire´ patterns produced by overlapping two self-imaged elongated circular gratings as the object grating of pitch po ð¼ 200 mmÞ is displaced from the reference grating of pitch pr ¼ (23=24) po by a) 0; b) po =4; c) po =2; d) 3po =4; e) po ; f) 5po =4; g) 3po =2; h) 7po =4; and e) 2po in the x-direction. The width of the parallel line part is chosen as 2d ¼ 23po ¼ 24pr : These moire´ fringes by means of a self-imaging method are comparable with those in fig. 3 of ref. [1] by using a lens-imaging method.
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
plane wave. For a square type transmittance of the grating, the theoretical estimation has been compared with the self-images captured by an array of chargecoupled devices (CCD). We have shown that selfimages of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two self-imaged EC gratings have been found not much affected by the central aberration. We conclude that the EC gratings can be successfully employed for the visual measurement of linear displacements in Talbot moire´ deflectometry.
References [1] Song JS, Lee YH, Jo JH, Chang S, Yuk KC: Moire´ patterns of two different elongated circular gatings for the fine visual measurement of linear displacements. Opt. Commun. 154 (1998) 100–108 [2] Theocaris PS: Moire´ Fringes in Strain Analysis. pp. 112– 123, Pergamon Press, Oxford 1969 [3] Durelli AJ, Parks VJ: Moire´ Analysis of Strain. pp. 8–11, Prentice-Hall, Englewood Cliffs 1970 [4] Kafri O, Glatt I: The Physics of Moire´ Metrology. pp. 36– 37, 102–103, John Wiley & Sons, New York 1990 [5] Patorski K, Kujawinska M: Handbook of the Moire´ Fringe Technique. pp. 17–18, 71–72, 140–143, Elsevier, Amsterdam 1993
[6] Post D, Han B, Ifju P: High Sensitivity Moire´. pp. 76–78, 108–110, Springer-Verlag, New York 1994 [7] Park YC, Kim SW: Determination of two-dimensional planar displacement by moire´ fringes of concentric circle gratings. Appl. Opt. 33 (1994) 5171–5176 [8] Talbot HF: Facts relating to optical science. Philos. Mag. 9 (1836) 403–405 [9] Rayleigh L: On copying diffraction gratings, and on some phenomenon connected therewith. Philos. Mag. 11 (1881) 196–205 [10] Yu W, Yun D, Wang L: Talbot and Fourier moire´ deflectometry and its applications in engineering evaluation. Opt. Lasers Eng. 25 (1996) 163–177 [11] Born M, Wolf E: Principles of Optics. pp. 378–383. Pergamon Press, Oxford 1980 [12] Montgomery WD: Self-imaging objects of infinite aperture. J. Opt. Soc. Am. 57 (1967) 772–778 [13] Cowley JM, Moodie AF: Fourier images: I.The point source. Proc. Phys. Soc. B70 (1957) 486–497 [14] Winthrop JT, Worthington CR: Theory of Fresnel images, I. Plane periodic objects in monochromatic light. J. Opt. Soc. Am. 55 (1965) 373–381 [15] Szwaykowski P: Self-imaging in polar coordinates. J. Opt. Soc. Am. A5 (1988) 185–191 [16] Mansuripur M: The Talbot effect. Opt. Photonics News (April 1997) 42–47 [17] Bryngdahl O: Image formation using self imaging techniques. J. Opt. Soc. Am. 63 (1973) 416–418 [18] Abramowitz M, Stegun IA (eds.): Handbook of Mathematical Functions. pp. 300–301, 322–324. NBS, Washington D.C. 1972
285
International Journal for Light and Electron Optics
Characterization of Fresnel images of an elongated circular grating J. S. Kang1, M. S. Lee1, S. Chang1, B. J. Kim2, K. C. Yuk2 1 2
Department of Physics, Hannam University, 133 Ojungdong, Taejon 300-791, Korea Department of Physics Education, Kongju National University, 182 Sinkwandong, Kongju, Korea
Abstract: We characterize theoretically and experimentally Fresnel images (or self-images) of an elongated circular (EC) grating which could be employed for the visual measurement of linear displacements in Talbot moire´ deflectometry. Based on a Fresnel diffraction integral, we estimate the shape of the Fresnel image formed at a specific distance from the EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. For a square type transmittance of the grating, we also compare the theoretical estimates with its self-images captured by an array of charge-coupled devices (CCD). We show that self-images of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two self-imaged EC gratings are found not much affected by the central aberration. Key words: Self-image elongated circular grating Talbot effect
1. Introduction In an ealier work [1], use of an elongated circular (EC) grating in moire´ deflectometry has been proposed for the fine visual determination of both the linear displacement and its direction. Moire´ patterns formed by placing two complementary EC gratings in contact (or projecting one EC grating onto the other) have been also used to measure the thermal expansion of metals (or the absolute refractive indices of optically transparent liquids). Optical moire´s of two different EC gratings have the advantage of providing ten times as good accuracy in metrology as those of conventional circular gratings [2–7]. If the EC gratings are employed to a Talbot interferometer [8–10] which requires no lens to image an object grating at the plane of a reference grating, they could be more useful in practical applications because of simplicity and compactness of the system. The performance of a Talbot interferometer is
based on the quality of Fresnel images (or self-images) of an object grating. For successful performance of a Talbot interferometer with EC gratings, therefore, it is of great importance to characterize Fresnel images of an EC grating and moire´ fringes by self-imaging an object EC grating on a reference EC grating. Using the Fresnel-Kirchhoff integral formula with a parabolic approximation of the optical path length [11], Montgomery [12] discussed the general condition for self-imaging objects of infinite aperture. The selfimaging phenomena of the cross, line, circle, spiral, and evolute gratings were well described by Cowley and Moodie [13], Winthrop and Worthington [14], Szwaykowski [15], and Mansuripur [16]. Self-images of a pinhole array were also examined for application to the pinhole-array camera by Bryngdahl [17]. However, self-images of an amplitude type EC grating which consists of concentric semicircles matched with parallel lines of finite length have not been investigated. In this paper we characterize theoretically and experimentally self-images of an amplitude type EC grating which could be employed for the visual measurement of linear displacements in Talbot deflectometry. Based on a Fresnel diffraction integral, we estimate the shape of the self-image formed at Talbot distances from the EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. For a square type transmittance of the grating, we also compare the theoretical estimates with its self-images captured by an array of charge-coupled devices (CCD). We show that self-images of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two superimposed self-imaged EC gratings are found not much affected by the central aberration.
2. Self-images through Fresnel diffraction Received 22 February 2002; accepted 6 June 2002. Correspondence to: S. Chang Fax: ++82-42-629-7444 E-mail: [email protected]
Figure 1 shows the geometry of an optical setup for observing Fresnel images (or self-images) of an elongated circular (EC) grating which is composed of the 0030-4026/02/113/07-285 $ 15.00/0
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
y
y’ x’
x
Coherent Plane wave
z’
=2p20 / O
ðq; f; zÞÞ. The origin of a rectangular (or cylindrical) coordinate system is situated at the center of the linear (or semicircular) grating as shown in fig. 2. The diffraction field A0 ðx0 ; y0 ; z0 Þ; which is observed at a distance z0 from an object grating, can be determined by putting the object field Aðx; yÞ into the Fresnel-Kirchhoff diffraction integral with a parabolic approximation of the optical path length [11, 12] 0
0
0
1 ð
0
A ðx ; y ; z Þ ¼ C
1 ð
dx 1
Object EC grating
Reference EC grating
Fig. 1. Geometry of an optical setup for observing Fresnel images (or self-images) of an elongated circular (EC) grating. The origin of a rectangular coordinate system (x; y; z) (or a circular coordinate system (q; f; z)) is on the surface of an object EC grating. Self-images of the grating are formed at Talbot distances z0 ¼ m(2p2 =l) (for m ¼ 1; 2; . . .), where po is the pitch of the grating and l is the wavelength of an incident wave.
! ðx0 xÞ2 þ ðy0 yÞ2 ; ip lz0
ð1Þ
with a factor of C ¼ exp ði2pz0 =lÞ=ðilz0 Þ: In a cylindrical coordinate system ðq; f; zÞ; the Fresnel diffraction integral (1) can also be transformed into 0
0
0
0
A ðq ; f ; z Þ ¼ C
0
1 ð
2p ð
df
pq2 dq qAðq; fÞ exp i lz0
0
2pq0 q 0 cos ðf f Þ ; exp i lz0
ð2Þ
where we have used the relations C0 ¼ C exp ipq02 =lz0 ; x ¼ q cos f ; x0 ¼ q0 cos f0 ;
y ¼ q sin f ; y0 ¼ q0 sin f0 :
Now we are to use the diffraction integrals ð1Þ and ð2Þ for examining self-images of both the parallel line part and the semicircular part in the EC grating.
2.1 Self-images of the parallel line part
y
( r, f)
0
L
1
exp
0
two parts: straight lines of finite length and semicircles. The field of a plane wave is incident perpendicularly to the surface of the grating. The wavelength of the incident light is denoted by l: The spacing between the two adjacent parallel lines (or semicircles) in the object EC grating is represented by po. The light fields diffracted by the grating produce an image of itself at integer multiples of distance z0 ¼ 2p2o =l. Self-images of the object EC grating are recorded by using an array of chargecoupled devices (CCD) with an image board. The object field through the linear (or semicircular) part of the EC grating is well described in a rectangular coordinate system ðx; y; zÞ (or a cylindrical coordinate system
dy Aðx; yÞ
d
x
p0 Fig. 2. Schematic diagram of an elongated circular grating in which the semicircles are matched with the parallel lines. The upper semicircles belong to the domain of 0 < q < d and 0 < f < p: The parallel lines are in the domain of jxj < d and L < y < 0: In a realistic case, d and L are large compared with a pitch po :
The parallel line part of an EC grating is just a linear grating of uniform period po and overall width 2d: The dark stripes of length L are periodically arranged parallel to the y-axis as shown in fig, 2. The transmission coefficient of the parallel line part is assumed to change periodically between 1 and a ð> 0Þ along the xaxis (fig. 3), while it remains constant along the y-axis. The ratio of the opening between the two adjacent stripes to a period is denoted by D. In a case of a square type transmission, the openning ratio D is taken as 1/2. To evaluate the diffraction integral (1), we may as well represent a rectangular type transmission coefficient tðx; yÞ of the parallel line part in a Fourier series of harmonic functions as 8 1 > < P b cos 2pn x for jxj < d and L < y < 0 ; n po tðx; yÞ ¼ n¼0 > : 0 otherwise ;
(3)
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
pffiffiffiffiffiffiffiffi0
2=lz ðdx0 nlz0 =po Þ
0
0
Wðx ; z Þ ¼ pffiffiffiffiffiffiffiffi0 2=lz
ð
ðdx0 nlz0 =p
p dt exp i t 2 2
287
ð5Þ
oÞ
If we have a screen situated at Talbot distances z0 ¼ mð2p2o =lÞ Zm for m ¼ 1; 2; . . ., then the first exponential factor in the summation of eq. (4) will always go to unity. Thus, the field (4) distributed on the screen at Talbot distances Zm can be expressed as 1 b P n 2 n¼0
2pn 0 0 Vðx ; Zm Þ exp i x po 2pn 0 þ Wðx0 ; Zm Þ exp i x ; po
A0 ðx0 ; y0 ; Zm Þ ¼ C00 Uðy0 ; Zm Þ
Fig. 3. Transmission coefficient t of an elongated circular grating which varies between 1 and a with a period of po . The portion of the transparent part in a period is denoted by D: For a square type transmission, the opening ratio D goes to 1=2:
where the coefficients are found to be of b0 ¼ a þ ð1 aÞ D for n ¼ 0 and of bn ¼ 2ð1 aÞ sin ðnpDÞ=np for n > 0: For a case where the grating is illuminated by a coherent plane wave of amplitude A0, the object field Aðx; yÞ transmitted through its parallel-line part can be described by multiplying tðx; yÞ by A0, which is a rectangular wave in the domain of jxj < d and L < y < 0: Substituting Aðx; yÞ into Eq. (1) and carrying out the integration over the whole domain, we find the diffraction field to be A0 ðx0 ; y0 ; z0 Þ ¼ CA0
ð0
ðd
dy tðx; yÞ
dx d
L
! ðx0 xÞ2 þ ðy0 yÞ2 exp ip lz0 1 P lz0 lz0 n2 Uðy0 ; z0 Þ bn exp ip 2 ¼ CA0 4 po n¼0 2pn 0 x þ Wðx0 ; z0 Þ Vðx0 ; z0 Þ exp i po 2pn 0 exp i x ; po
2=lz y
2=lz ðdx0 þnlz0 =po Þ
Vðx0 ; z0 Þ ¼ pffiffiffiffiffiffiffiffi0
ð
2=lz ðdx0 þnlz0 =po Þ
where C00 is a constant. Moreover, in a realistic case that d is large compared with both jx0 j and jlz0 =po j; the Fresnel integral functions V and W in eq. (5) become pffiffiffi equal to 2 exp ðip=4Þ [18]. Consequently, the diffraction field (6) can be written in a simple form as 1 P 2pn 0 A0 ðx0 ; y0 ; Zm Þ ¼ C000 Uðy0 ; Zm Þ bn cos i x ; po n¼0 ð7Þ pffiffiffi 00 In the above we have let C ¼ 2 exp ðip=4ÞC : The summation part on the right-hand side of eq. (7) behaves like an object field (eq. (3)) in the x direction, while the integral function U arises from Fresnel diffraction by straight lines of finite length in the y direction. Therefore, it is clear that the diffracted field (7) describes self-images of the parallel line part which include aberrations by Fresnel diffraction in the y direction. The intensity of the diffraction field (7) at the m-th Talbot distance Zm is now given by 000
Ilin ðx0 ; y0 ; Zm Þ ¼ A0 ðx0 ; y0 ; Zm Þ A0 *ðx0 ; y0 ; Zm Þ :
ð8Þ
2.2 Self-images of the semicircular part
ð4Þ
where we have defined the Fresnel integral functions pffiffiffiffiffiffiffiffi0 2=lz ðLþy0 Þ ð p dt exp i t 2 ; Uðy0 ; z0 Þ ¼ 2 pffiffiffiffiffiffiffiffi0 0 pffiffiffiffiffiffiffiffi0
ð6Þ
Next we examine the upper semicircular part of an EC grating of which the transmission coefficient varies periodically between 1 and að> 0Þ with radial variable q in the region of 0 < q < d and 0 < f < p (fig. 2). As above, we assume a transmission coefficient t of the semicircular part in a Fourier series of harmonic functions 8 1 > < P b cos 2pn q for 0 < q < d and 0 < f < p; n po tðq; fÞ ¼ n¼0 > : 0 otherwise ;
(9) p dt exp i t 2 ; 2
with the coefficients of b0 ¼ a þ ð1 aÞ D for n ¼ 0 and bn ¼ 2ð1 aÞ sin ðnpDÞ=np for n > 0: If a coherent plane wave of amplitude A0 is incident normally
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
onto the surface of the EC grating, the object field on the semicircular part may be represented by Aðq; fÞ ¼ A0 tðq; fÞ: Insertion of Aðq; fÞ into eq. (2) leads to ðd 1 P 2pn dq q bn cos q A0 ðq0 ; f0 ; z0 Þ ¼ C0 A0 po n¼0 0
ðp pq2 exp i 0 df lz 0 2pq0 q 0 cos ðf f Þ : exp i lz0
ð10Þ
The second integral in eq. (10) can be easily solved by expanding the integrand in a series of Bessel functions as follows: 2pq0 q 0 cos ðf f Þ exp i lz0 1 P 2pq0 q 2pq0 q l þ 2 cos ½lðf0 fÞ ; ¼ J0 ð i Þ J l 0 lz0 lz l¼1
parts are recorded at the first Talbot distance by using a CCD-array camera (Samsung SBC 341) with an image board (DT 3155). The CCD-array camera has the cell size of 8 mm. For use of EC gratings in Talbot deflectometry, we present the moire´ patterns of an EC grating pair made by a self-imaging method. The pitch of the object grating is taken at po ¼ 200 mm, and the reference grating (see fig. 1b) in ref. [1]) is made to be of pitch pr ¼ po ð2N 1Þ=ð2NÞ for N ¼ 12: Figure 4 shows the theoretical curves for the intensity of the image field which is evaluated at the first Talbot distance z0 ð¼ 2p2o =lÞ; where the parallel line part in an object EC grating is self-imaged; a) the image intensity of the parallel line part is plotted as a function of x0 ; while the value of y0 is fixed at L=2; and b) the intensity of the image field at x0 ¼ 0 is also plotted as a function of y0 : We have chosen the parameters as D ¼ 1=2; a ¼ 0:4; L po and 2d ¼ 23po : It is obvious in fig. 4b) that the image field distributed along the x-axis corresponds to the object field of the
ð11Þ
where Jl ðcÞ denotes the Bessel function of order l: By putting eq. (11) into eq. (10) and solving the second integral at an image plane (i.e., z0 ¼ mð2p2 =lÞ ¼ Zm Þ, one can find the image field as an integral with respect to radial variable q A0 ðq0 ; f0 ; Zm Þ ¼ C 0 A0
1 P n¼0
(
ðd dq q bn cos
2pn pq2 q exp i po lZm
0
1 ðiÞl P 2pq0 q lp 2pq0 q sin þ4 Jl l lZm 2 lZm l¼1 lðp 2f0 Þ : ð12Þ cos 2
pJ0
The intensity of the image field (12) at the m-th Talbot distance is also given by Is ðq0 ; f0 Þ ¼ A0 ðq0 ; f0 ; Zm Þ A0 *ðq0 ; f0 ; Zm Þ :
ð13Þ
The field intensity (13) has to be evaluated by means of a numerical method so as to examine the behaviour of self-images of the upper semicircular part.
3. Results and discussion Numerical calculations of the Fresnel diffraction integrals (7) and (12) are carried out to characterize selfimages of an amplitude type EC grating in which both parallel lines and semicircles are illuminated by a coherent plane wave. The wavelength of the incident light is chosen to be of l ¼ 632:8 nm (He-Ne laser). For comparison of the theoretical estimates with measured results, self-images of the linear and semicircular
Fig. 4. Theoretical curves for the intensity of the image field which are evaluated at the first Talbot distance z0 (¼ 2p2o =l) from the parallel line part. We let D ¼ 1=2; a ¼ 0:4 and L po : (a) Self-images of the parallel line part are shown as a function of x0 when y ¼ L=2; and (b) the intensity of the image field is plotted as a function of y0 at x0 ¼ 0: Half shadows tail to the dark region of y0 > 0:
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
square-wave type in fig. 3. As a result of Fresnel diffraction by a straight edge, however, the y-distribution of the image intensity in fig. 4b) is oscillating in the bright region (i.e., y0 < 0Þ; and a half shadow is shown to tail to the dark region (i.e., y0 > 0Þ: In Fig. 5 we show the theoretical curves for the intensity of the image field which is evaluated at the first Talbot distance z0 ð¼ 2p2o =lÞ, where the upper semicircular part in an object EC grating is self-imaged; a) the image intensity of the upper semicircular part is plotted as a function of q0 ; while the angular position f0 remains fixed at p=2; and b) the intensities of the image field are plotted as a function of f0 at integer multiples of q0 ¼ po : Other conditions are the same as in fig. 4. The intensity of the image field in fig. 5a) is highly enhanced at the edge of each semicircular stripe, while the distribution of the object field is of the square-wave type along the q-axis. However, it should
Fig. 5. Theoretical curves for the intensity of the image field which are evaluated at the first Talbot distance z0 (¼ 2p2o =l) from the upper semicircular part. We let D ¼ 1=2 and a ¼ 0:4: a) Self-images of the upper semicircular part are shown as a function of q0 when f0 ¼ p=2; and (b) the intensities of the image field are plotted as a function of f0 when q0 ¼ mpo (for m ¼ 1 5). In cases of small q0 ; half shadows tail to the dark region of p < f < 2p.
289
be noted here that the spacing between the two self-imaged semicircles is identical with that in the object grating. Figure 5b) shows that at specific positions of q0 ’ po and 2po the speckled patterns appear in the dark region of p < f < 2p: The intensity of the speckled pattern in the dark region decreases rapidly by increasing q0 : We find in fig. 5b) that the angular distribution of the image intensity (belonging to the upper semicircular part) is substantially aberrated in the central region of small q0 but not in the off-central region. Next, to confirm the theoretical estimates, we have experimentally taken self-images of the linear and semicircular parts in an object EC grating. Figure 6
Fig. 6. Photographs of the self-images formed on the screen at the first Talbot distance z0 ¼ 2p2o =l;of which each corresponds to the semicircular part (top), the parallel line part (middle) and the elongated circular grating (bottom). The weak halfshadows are observed in the dark region.
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J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
shows the photographs of the self-images formed at the first Talbot distance z0 ¼ 2p2o =l;where each of them corresponds to the semicircular part (top), the parallel line part (middle) and the elongated circular grating (bottom). Both of the linear and semicircular parts are observed to be well self-imaged. The self-imaged line grating has clear-cut outlines along the x-axis but obscure boundaries at y0 ¼ 0 as predicted in fig. 4. In a case of the self-imaged semicircular grating, the speckled circles appear to the dark region of p < f < 2p and they disappear as q0 increases. This behaviour is in excellent agreement with that in fig. 5b). Figure 7 shows the microdensitometric results for a) the self-imaged parallel lines and b) the self-imaged semicircles that are presented in fig. 6. In fig. 7a) the image intensity is given as a function of x0, while y0 ¼ L=2: In fig. 7b) the curve of the image intensity is plotted as a function of q0, while f0 ¼ p=2: If the curves in fig. 7 are compared with those in fig. 4a) and fig. 5a), one can find the measured pitches in good agreement with those in the theoretical estimation.
Finally, moire´ patterns of two self-imaged complementary EC gratings are investigated so as to demonstrate the feasible use of EC gratings in Talbot deflectometry. In fig. 8 we show the photographs of the moire´ patterns produced by overlapping a self-imaged object EC grating onto a reference EC grating as the object grating of pitch po ð¼ 200 mmÞ is displaced from the reference grating of pitch pr ¼ ð23=24Þ po by a) 0; b) po =4; c) po =2; d) 3po =4; e) po ; f) 5po =4; g) 3po =2; h) 7po =4; and e) 2po in the x-direction. The width of the parallel line part is chosen as 2d ¼ 23po ¼ 24pr : These moire´ fringes by means of a self-imaging method are comparable with those in fig. 3 of ref. [1] based on a contact (or lens-imaging) method. In the region of the parallel line part, one can see a vernier dark moire´ moving as a function of displacement of the object grating. The shift of the vernier dark moire´ Dx is related to the displacement of the object grating dx by Dx ¼
W dx ; po
ð14Þ
where the period of the vernier dark moire´s is given by po pr W¼ : ð15Þ po pr Substitution of po and pr into eqs. (14) and (15) leads to the formulas Dx ¼ 23 dx and W ¼ 2d: This means that if the center of the object grating shifts by a pitch po toward the positive x-direction, the vernier dark moire´ moves by 2d ð¼ 23po Þ toward the negative x-direction. In case of the semicircular part, the number of the dark moire´s is shown to increase as the center of the object grating goes away from that of the reference grating. The order m of the dark moire´s allowed in the semicircular part is determined by q jdxj q jdxj þ m ; W po W po
ð16Þ
where the bars j . . . j denote the absolute value of the argument. If we limit our interest to the region of q W; eq. (16) can be transformed into a simple form
jdxj jdxj m : po po
ð17Þ
If the object grating is displaced in the range of po dx < 2po ; the dark fringes of orders m ¼ 0; 1 and þ1 are allowed in the semicircular part as shown in figs. 8e)–h).
4. Conclusion Fig. 7. Microdensitometric curves for (a) the image intensity of the parallel lines measured at y0 ¼ L=2 as a function of x0 ; and (b) the image intensity of the semicircles measured at f0 ¼ p=2 as a function of q0 : By comparing the curves in fig. 7 with those in fig. 4a) and fig. 5a), we find the measured pitches in good agreement with those in the theoretical estimation.
In this paper, Fresnel images (or self-images) of an elongated circular (EC) grating have been examined theoretically and experimentally. Based on a Fresnel diffraction integral, we have analyzed the shape of the Fresnel image of the EC grating in which both parallel lines and semicircles are illuminated by a coherent
J. S. Kang et al., Characterization of Fresnel images of an elongated circular grating
291
m=0
(a)
(b) m= -1,
(d)
0,
(c) +1
(e)
(f) m= -2, -1,
(g)
(h)
0
+1 +2
(i)
Fig. 8. Moire´ patterns produced by overlapping two self-imaged elongated circular gratings as the object grating of pitch po ð¼ 200 mmÞ is displaced from the reference grating of pitch pr ¼ (23=24) po by a) 0; b) po =4; c) po =2; d) 3po =4; e) po ; f) 5po =4; g) 3po =2; h) 7po =4; and e) 2po in the x-direction. The width of the parallel line part is chosen as 2d ¼ 23po ¼ 24pr : These moire´ fringes by means of a self-imaging method are comparable with those in fig. 3 of ref. [1] by using a lens-imaging method.
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plane wave. For a square type transmittance of the grating, the theoretical estimation has been compared with the self-images captured by an array of chargecoupled devices (CCD). We have shown that selfimages of the EC grating are substantially aberrated near the center of the interface between the linear and semicircular parts but not in its off-central region. Moire´ patterns by two self-imaged EC gratings have been found not much affected by the central aberration. We conclude that the EC gratings can be successfully employed for the visual measurement of linear displacements in Talbot moire´ deflectometry.
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[6] Post D, Han B, Ifju P: High Sensitivity Moire´. pp. 76–78, 108–110, Springer-Verlag, New York 1994 [7] Park YC, Kim SW: Determination of two-dimensional planar displacement by moire´ fringes of concentric circle gratings. Appl. Opt. 33 (1994) 5171–5176 [8] Talbot HF: Facts relating to optical science. Philos. Mag. 9 (1836) 403–405 [9] Rayleigh L: On copying diffraction gratings, and on some phenomenon connected therewith. Philos. Mag. 11 (1881) 196–205 [10] Yu W, Yun D, Wang L: Talbot and Fourier moire´ deflectometry and its applications in engineering evaluation. Opt. Lasers Eng. 25 (1996) 163–177 [11] Born M, Wolf E: Principles of Optics. pp. 378–383. Pergamon Press, Oxford 1980 [12] Montgomery WD: Self-imaging objects of infinite aperture. J. Opt. Soc. Am. 57 (1967) 772–778 [13] Cowley JM, Moodie AF: Fourier images: I.The point source. Proc. Phys. Soc. B70 (1957) 486–497 [14] Winthrop JT, Worthington CR: Theory of Fresnel images, I. Plane periodic objects in monochromatic light. J. Opt. Soc. Am. 55 (1965) 373–381 [15] Szwaykowski P: Self-imaging in polar coordinates. J. Opt. Soc. Am. A5 (1988) 185–191 [16] Mansuripur M: The Talbot effect. Opt. Photonics News (April 1997) 42–47 [17] Bryngdahl O: Image formation using self imaging techniques. J. Opt. Soc. Am. 63 (1973) 416–418 [18] Abramowitz M, Stegun IA (eds.): Handbook of Mathematical Functions. pp. 300–301, 322–324. NBS, Washington D.C. 1972