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Speckle clustering in diffraction patterns of random objects under ring-slit illumination
1 February 1995
3 OPTICS COMMUNICATIONS Optics Communications 114 ( 1995) 203-210
El *~l-YlER
Speckle clustering in diffraction patterns of random objects under ring-slit illumination Katsuhiro Uno, Jun Uozumi, Toshimitsu Asakura Research Institute for Electronic Science, Hokkuido University, Sapporo, Hokkaido 060, Japan
Received 14 July 1994
Abstract Some optical conditions are discussed for producing speckle patterns having long intensity correlations in the diffraction geometry. A random rough surfaces under illumination of coherent light with a circular ring slit produces the speckles with a zero-order Bessel function for the intensity correlation when observed in the Fraunhofer diffraction region. This function has relatively long oscillating tails and, consequently, string or network structures appear in speckle patterns, which we may call speckle clustering. This clustering phenomenon can explain the similar appearance observed in diffraction patterns produced by random Koch fractals.
1. Introduction When a random medium is illuminated by coherent light, the waves scattered by the medium interfere each other in a random way, giving rise to a speckle pattern [ 11, which is a random distribution of bright spots, called speckle grains, resulting from constructive interference. If the random phases of the scattered waves are distributed over a range greater than 2~ and if the number of the scattering centers contributing to individual observation points is large enough, the resultant speckle pattern appears to have high contrast and is called fully developed speckles. In the standard theory of speckles [ 11, the average size of the speckle grains defines an extent of the correlation property of the speckle pattern and the two points apart beyond the speckle size are regarded to be almost uncorrelated. Recently, however, a peculiar type of speckle patterns were reported for the Fraunhofer diffraction field of random Koch curves [ 2,3]. In these patterns, speckle grains form clusters, which suggest an exElsevier Science B.V. .SSDf 0030-4018( 94100653-9
istence of the longer correlation beyond the speckle size. In the present paper, a possible mechanism for a clustering of speckle grains is proposed and discussed theoretically and experimentally from the viewpoint of the shape of apertures illuminating random objects. We first review the theoretical background concerning the intensity correlation functions of the speckle patterns. Next, we show theoretically that the illumination of an object with a ring-slit aperture causes relatively long and ringing correlation tails in the speckle intensity distribution. An experimental verification is provided for this effect. Finally, on the basis of those considerations, the origin of the speckle clustering in the diffraction patterns of random Koch fractals is discussed.
204
K. Uno et al. /Optics Communications 114 (1995) 203-210
2. Theoretical background
After Substituting Eq. (5) into Eq. (4), the complex coherence factor depends only on the difference of coordinates, Ar = rl - r2, and is reduced by
2. I. Intensity correlation of the speckle jield We first review the theoretical background for the intensity correlation of speckles. The correlation of intensities, Z(rt ) and I (rz), at two points, rl and r2, in the speckle pattern is written by Mrl,r2)
= (I(rl)I(r2)),
(1)
where the angle bracket denotes an ensemble average. Eq. ( 1) can be expressed in terms of the amplitude correlation, if the amplitude A(r) forming the intensity I(r) = IA( ’ a zero-mean circular complex IS Gaussian random variable [ 1 ] : &(rl,r2)
= (Q2[l
+ Ib4(r1,r2)1217
(2)
where we assumed that (Z(q)) = (I(r2)) = (I). In Eq. (2), ,!,&A (q , r2 ) is referred to as a complex coherence factor and is defined by pA(r,
r2)
=
(A(r (0
(3)
’
where the asterisk represents a complex conjugate. Now we assume that a random object is illuminated by coherent light with the wavelength A. When a speckle pattern is detected on the back focal plane of the lens with the focal length f located behind the object, the complex coherence factor in the speckle field is expressed, in terms of the amplitude correlation function just behind the object plane, by e’”
pA(rl,r2)
=
(I)
JJd5dS’W)a*(5’))
PACT1 7 r2)
& =-
(0
= PA(Ar)
dg(la(g)l')ex JJ (6)
This equation shows that the complex coherence tor ,%A( Ar) is a Fourier transform of the intensity tribution just behind the object.
facdis-
2.2. Circular and ring apertures 2.2.1. Circular aperture Use of a circular aperture is the most typical illumination condition. When an illuminated region on the rough surface is circular, the intensity just behind the object is uniform within this region. This illumination produces the intensity correlation of the speckle field expressed, from Eqs. (2) and (6)) by
&(Ap)
= (Z)2 ( 1 + /2 “$$/2),
(7)
where pa is the radius of the aperture and the argument has been replaced by Ap = 2n-]Arl/Af for convenience. It is noted again that the intensity correlation is a function of only the difference of coordinates. 2.2.2. Ring slit aperture Next, we consider an unusual illumination using a ring-slit aperture. A ring slit is mathematically expressed by a subtraction of two concentric circular apertures. When their radii are represented by a and b where a < b, the intensity correlation of the speckles generated by this slit illumination is given by [4]
,
i$(rl.C-r2.f) >
’
(b2 - a2)2
2bJlCbAp)
AP
where a(& is the complex amplitude at the point c just behind the object plane and @ is a quadratic phase factor determined by two positions t-1 and r2. If the correlation length of the object amplitude a( 6) is much smaller than the size of the illumination area, the amplitude correlation function can be approximated
In the limiting case that the inner radius a approaches the outer radius b, the above equation is approximated
by
by
@(&a*(&‘))
= (la(S)12)&g
- 5’).
(5)
- 2aJ1 (aAp) AP
&(AP)
. 11 2
= (Z)2]1 +Jif(aAp)l.
(8) (9)
205
K. Unoet al. /Optics Communications114 (1995) 203-210 Object
He-Ne
I_zy Computer
Fig. 2. Experimental setup for detecting Fraunhofer diffraction patterns.
aAp Fig. 1. Intensity correlations of the speckles produced with random rough surfaces masked with (a) a circular aperture and (b) a ring-slit aperture with the negligible slit width.
It is noted that, in another limiting case of a + 0, Eiq. (9) reduces to Fq. (7) with pa = b. The correlation functions of Qs. (7) and (9) are shown in Figs. la and b, respectively. As seen in these figures, the correlation function of the speckle intensity due to the ring slit has long tails consisting of ringing side lobes, and the peak width is smaller than that derived from the circular aperture. If we define the peak width w by the first point where the intensity correlation function R,( Ap)/(Q2 becomes 1, we have 3.831 WN
PO
(10)
,
2.401 a for the circular aperture and the ring-slit aperture with a + b, respectively. It is noted that the peak width represents the average size of speckle grains.
3. Experiments
and discussion
3.1. Circular and ring slit apertures The experimental
setup is shown in Fig. 2. The ob-
ject consisting of a ground glass and an aperture is illuminated by collimated coherent light. The Fraunhofer diffraction pattern is formed at the back focal plane of the lens and observed by the CCD camera through the magnifying lens. Values of the intensity of the pattern are stored in the frame memory of a computer as discrete data with 256 levels. Three different apertures are prepared: a circular aperture with pc 2: 1 mm and two ring-slit apertures having different widths of a/b = 0.95 and 0.69. The roughness of the ground glass is greater than the wavelength of light and the correlation area is much smaller than the aperture size. These conditions allows us to treat the diffraction field as a zero-mean circular complex Gaussian random process. When the circular aperture is used, we observe a speckle pattern shown in Fig. 3a. On the other hand, the observed speckle patterns for the ring-slit apertures are shown in Figs. 3b and c for the cases of a/b = 0.95 and 0.69, respectively. It is noted in Fig. 3b that, in case of the narrow slit aperture of a/b = 0.95, the speckle grains form clusters which also appear to have a network structure. The speckle clustering is not observed appreciably in Fig. 3a, while it is recognized only slightly in Fig. 3c. The intensity correlation functions evaluated numerically from Figs. 3a-c are shown in Figs. 4a-c, respectively. These correlation functions were obtained by applying the FFI’ operation twice for patterns in Fig. 3 on the basis of the Wiener-Khintchine theorem. As seen in Fig. 4b, long correlation tails clearly exist as side lobes of the main peak for the case of the narrow slit aperture. Note that the condition of a/b = 0.95 is close to the limiting case of a/b = 1 and that, hence, Fig. 4b is close to the theoretical curve of Fig. lb. Appreciable side lobes are also recognized in the
206
K. Uno et al. /Optics Communications 114 (1995) 203-210
Fig. 3. Speckle patterns in a Fraunhofer diffraction plane produced by the ground glass masked with (a) the circular aperture, ring-slit apertures with (b) a/b Y 0.95 and (c) a/b N 0.69. Three apertures have the same outer radius of b Y 1 mm.
and the
K. Unoet al. /Optics Communications114 (1995) 203-210
Fig. 3.
case of Fig. 4c. It is clear that these side lobes in the correlation functions reflect a clustering appearance in speckle patterns. On the other hand, as seen in Fig. 4a, the side lobes for the case of the circular aperture are very weak, reflecting non-appreciable speckle clustering in Fig. 3a. Another noticeable point is the difference in the correlation peak width W. By comparing Figs. 3a-c, we recognize that the average speckle size is the largest for Fig. 3a, while it is the smallest for Fig. 3b. This tendency is reflected in the peak width w of the intensity correlation: When the ring slit becomes broader and approaches the circular aperture, the peak width w becomes broader. Experimental values of w obtained from Figs. 4a-c are 3.79, 2.07, and 2.80, respectively. These behaviors of intensity correlations agree well with the theoretical dependence of Figs. la and b.
207
Cont.
3.2. The speckle produced by a random Koch fractal In the previous studies [2,3], the clustering phenomenon was also reported for the speckles produced by randomized Koch curves. In this subsection, this phenomenon is discussed in comparison with the speckle clustering due to the ring-slit illumination. Random Koch curves are fractals which are selfsimilar in a statistical sense. They are generated by introducing a certain kind of randomness into the generation process of regular Koch curves [ 31. A random Koch curve employed in the experiment is shown in Fig. 5. In this object, the recursive replacement was performed five times with the randomness of u = 15 and the fractal dimension of D = log4/ log3 [2,3]. The object was photographed on a negative film, which was placed at the object plane without the aperture in Fig. 2. The observed speckle pattern is shown in Fig.
P oApx
(b)
I
O-10
I
I
0
I
1
l(
b Apx
b Apx Fig. 4. Intensity correlation functions of the speckle patterns produced by (a) the circular aperture, and the ring-slit apertures of (b) a/b N 0.95 and (c) a/b ry 0.69, respectively. The cross-sections of these functions along the horizontal axis px are also shown at the bonom.
K. Uno et al. /Optics Communications 114 (1995) 203-210
(a)
(b)
/
Fig. 5. (a) Random Koch curve with the fractal dimension D = log4/log3. It is constructed by recursive substitution with (b) a generator and (c) an initiator by introducing randomness in the generator [ 21.
209
6, where the power-law decay of the average intensity is compensated as described in the previous work [ 31. In this pattern, speckle clustering can be clearly seen. The intensity correlation function (above pattern) calculated from this speckle pattern is shown in Fig. 7 together with its cross-section (lower figure). Fig. 7 shows an appreciable side lobes as a consequence of speckle clustering. An existence of the long correlation tail in the speckles from regular and random Koch curves was also demonstrated in a separate paper [ 51 by means of the lacunarity of the speckle patterns. It is noted that the random Koch fractal has approximately a ring-slit shape. From this feature, it can be concluded that the cause of speckle clustering is a ring-like structure of the object. Strictly speaking, the random Koch fractal is not a circular ring, but is considered as a ring made of a connection of many arcs with different scales. As a result, the side lobes of the corresponding intensity correlation is not so large as in the case of the narrow ring slit and looks rather smoothed out slightly.
Fig. 6. Speckle pattern in a Fraunhofer diffraction plane produced by the random Koch fractal shown in Fig. 5a. The average intensity which decays with a power function of q --D is compensated in this pattern [ 31.
210
K. Uno et al. / Optics Communications I14 (1995) 203-210
is shown theoretically and experimentally that, in a statistical term, this apparent phenomenon is elucidated by the long and ringing tails of the intensity correlation function of the speckle patterns. Namely, an increase of the correlation tail manifests itself as stronger speckle clustering, which, in the present study, is generated by the narrow slit aperture. On the other hand, the average size of speckle grains is reduced with a decrease in the slit width of the ring aperture. However, it remains an unsettled question why the speckle clustering produced by a narrow annular aperture takes the characteristic network structure having a snake-like appearance. There is a room for further investigation at this problem. The clustering phenomenon which is visible in t!re diffraction patterns produced by random Koch fractals was also discussed and attributed to the ring-like shape of the object. It is noted that the present study does not exclude the possibility that other illuminating geometries than ring-slit apertures may cause long correlation tails in the speckle intensity distributions. It is also worth noting that the shape of the illuminating aperture can control the intensity correlation property of the resulting speckles. This fact is particu!arly interesting, since the intensity correlation properties are important factors in various methods in speckle metrology such as displacement and velocity measurements by means of the speckle correlation.
Fig. 7. Intensity correlation function of the speckle pattern shown in Fig. 6. The cross-section of this function along the horizontal axis pX is also shown at the bottom.
4. Conclusion We discussed the relation between the object shape and the clustering of the resulting speckles. It has been shown that the illumination of a random object with a ring-slit aperture gives rise to the speckle clustering. It
References 1l] J. W. Goodman,
in: Laser speckle and related phenomena (Springer, Berlin, 1975). [2] J. Uozumi, H. Kimura and T. Asakura, Waves in Random Media 1 (1991) 73. [3] K. Uno, J. Uozumi and T. Asakura, Appl. Optics 32 ( 1993) 2722. [4] M. Born and E. Wolf, in: Principles of optics, 6th Ed. (Pergamon, New York. 1980). [ 51 K. Uno, J. Uozumi and T. Asakura, Waves in Random Media, submitted for publication.
3 OPTICS COMMUNICATIONS Optics Communications 114 ( 1995) 203-210
El *~l-YlER
Speckle clustering in diffraction patterns of random objects under ring-slit illumination Katsuhiro Uno, Jun Uozumi, Toshimitsu Asakura Research Institute for Electronic Science, Hokkuido University, Sapporo, Hokkaido 060, Japan
Received 14 July 1994
Abstract Some optical conditions are discussed for producing speckle patterns having long intensity correlations in the diffraction geometry. A random rough surfaces under illumination of coherent light with a circular ring slit produces the speckles with a zero-order Bessel function for the intensity correlation when observed in the Fraunhofer diffraction region. This function has relatively long oscillating tails and, consequently, string or network structures appear in speckle patterns, which we may call speckle clustering. This clustering phenomenon can explain the similar appearance observed in diffraction patterns produced by random Koch fractals.
1. Introduction When a random medium is illuminated by coherent light, the waves scattered by the medium interfere each other in a random way, giving rise to a speckle pattern [ 11, which is a random distribution of bright spots, called speckle grains, resulting from constructive interference. If the random phases of the scattered waves are distributed over a range greater than 2~ and if the number of the scattering centers contributing to individual observation points is large enough, the resultant speckle pattern appears to have high contrast and is called fully developed speckles. In the standard theory of speckles [ 11, the average size of the speckle grains defines an extent of the correlation property of the speckle pattern and the two points apart beyond the speckle size are regarded to be almost uncorrelated. Recently, however, a peculiar type of speckle patterns were reported for the Fraunhofer diffraction field of random Koch curves [ 2,3]. In these patterns, speckle grains form clusters, which suggest an exElsevier Science B.V. .SSDf 0030-4018( 94100653-9
istence of the longer correlation beyond the speckle size. In the present paper, a possible mechanism for a clustering of speckle grains is proposed and discussed theoretically and experimentally from the viewpoint of the shape of apertures illuminating random objects. We first review the theoretical background concerning the intensity correlation functions of the speckle patterns. Next, we show theoretically that the illumination of an object with a ring-slit aperture causes relatively long and ringing correlation tails in the speckle intensity distribution. An experimental verification is provided for this effect. Finally, on the basis of those considerations, the origin of the speckle clustering in the diffraction patterns of random Koch fractals is discussed.
204
K. Uno et al. /Optics Communications 114 (1995) 203-210
2. Theoretical background
After Substituting Eq. (5) into Eq. (4), the complex coherence factor depends only on the difference of coordinates, Ar = rl - r2, and is reduced by
2. I. Intensity correlation of the speckle jield We first review the theoretical background for the intensity correlation of speckles. The correlation of intensities, Z(rt ) and I (rz), at two points, rl and r2, in the speckle pattern is written by Mrl,r2)
= (I(rl)I(r2)),
(1)
where the angle bracket denotes an ensemble average. Eq. ( 1) can be expressed in terms of the amplitude correlation, if the amplitude A(r) forming the intensity I(r) = IA( ’ a zero-mean circular complex IS Gaussian random variable [ 1 ] : &(rl,r2)
= (Q2[l
+ Ib4(r1,r2)1217
(2)
where we assumed that (Z(q)) = (I(r2)) = (I). In Eq. (2), ,!,&A (q , r2 ) is referred to as a complex coherence factor and is defined by pA(r,
r2)
=
(A(r (0
(3)
’
where the asterisk represents a complex conjugate. Now we assume that a random object is illuminated by coherent light with the wavelength A. When a speckle pattern is detected on the back focal plane of the lens with the focal length f located behind the object, the complex coherence factor in the speckle field is expressed, in terms of the amplitude correlation function just behind the object plane, by e’”
pA(rl,r2)
=
(I)
JJd5dS’W)a*(5’))
PACT1 7 r2)
& =-
(0
= PA(Ar)
dg(la(g)l')ex JJ (6)
This equation shows that the complex coherence tor ,%A( Ar) is a Fourier transform of the intensity tribution just behind the object.
facdis-
2.2. Circular and ring apertures 2.2.1. Circular aperture Use of a circular aperture is the most typical illumination condition. When an illuminated region on the rough surface is circular, the intensity just behind the object is uniform within this region. This illumination produces the intensity correlation of the speckle field expressed, from Eqs. (2) and (6)) by
&(Ap)
= (Z)2 ( 1 + /2 “$$/2),
(7)
where pa is the radius of the aperture and the argument has been replaced by Ap = 2n-]Arl/Af for convenience. It is noted again that the intensity correlation is a function of only the difference of coordinates. 2.2.2. Ring slit aperture Next, we consider an unusual illumination using a ring-slit aperture. A ring slit is mathematically expressed by a subtraction of two concentric circular apertures. When their radii are represented by a and b where a < b, the intensity correlation of the speckles generated by this slit illumination is given by [4]
,
i$(rl.C-r2.f) >
’
(b2 - a2)2
2bJlCbAp)
AP
where a(& is the complex amplitude at the point c just behind the object plane and @ is a quadratic phase factor determined by two positions t-1 and r2. If the correlation length of the object amplitude a( 6) is much smaller than the size of the illumination area, the amplitude correlation function can be approximated
In the limiting case that the inner radius a approaches the outer radius b, the above equation is approximated
by
by
@(&a*(&‘))
= (la(S)12)&g
- 5’).
(5)
- 2aJ1 (aAp) AP
&(AP)
. 11 2
= (Z)2]1 +Jif(aAp)l.
(8) (9)
205
K. Unoet al. /Optics Communications114 (1995) 203-210 Object
He-Ne
I_zy Computer
Fig. 2. Experimental setup for detecting Fraunhofer diffraction patterns.
aAp Fig. 1. Intensity correlations of the speckles produced with random rough surfaces masked with (a) a circular aperture and (b) a ring-slit aperture with the negligible slit width.
It is noted that, in another limiting case of a + 0, Eiq. (9) reduces to Fq. (7) with pa = b. The correlation functions of Qs. (7) and (9) are shown in Figs. la and b, respectively. As seen in these figures, the correlation function of the speckle intensity due to the ring slit has long tails consisting of ringing side lobes, and the peak width is smaller than that derived from the circular aperture. If we define the peak width w by the first point where the intensity correlation function R,( Ap)/(Q2 becomes 1, we have 3.831 WN
PO
(10)
,
2.401 a for the circular aperture and the ring-slit aperture with a + b, respectively. It is noted that the peak width represents the average size of speckle grains.
3. Experiments
and discussion
3.1. Circular and ring slit apertures The experimental
setup is shown in Fig. 2. The ob-
ject consisting of a ground glass and an aperture is illuminated by collimated coherent light. The Fraunhofer diffraction pattern is formed at the back focal plane of the lens and observed by the CCD camera through the magnifying lens. Values of the intensity of the pattern are stored in the frame memory of a computer as discrete data with 256 levels. Three different apertures are prepared: a circular aperture with pc 2: 1 mm and two ring-slit apertures having different widths of a/b = 0.95 and 0.69. The roughness of the ground glass is greater than the wavelength of light and the correlation area is much smaller than the aperture size. These conditions allows us to treat the diffraction field as a zero-mean circular complex Gaussian random process. When the circular aperture is used, we observe a speckle pattern shown in Fig. 3a. On the other hand, the observed speckle patterns for the ring-slit apertures are shown in Figs. 3b and c for the cases of a/b = 0.95 and 0.69, respectively. It is noted in Fig. 3b that, in case of the narrow slit aperture of a/b = 0.95, the speckle grains form clusters which also appear to have a network structure. The speckle clustering is not observed appreciably in Fig. 3a, while it is recognized only slightly in Fig. 3c. The intensity correlation functions evaluated numerically from Figs. 3a-c are shown in Figs. 4a-c, respectively. These correlation functions were obtained by applying the FFI’ operation twice for patterns in Fig. 3 on the basis of the Wiener-Khintchine theorem. As seen in Fig. 4b, long correlation tails clearly exist as side lobes of the main peak for the case of the narrow slit aperture. Note that the condition of a/b = 0.95 is close to the limiting case of a/b = 1 and that, hence, Fig. 4b is close to the theoretical curve of Fig. lb. Appreciable side lobes are also recognized in the
206
K. Uno et al. /Optics Communications 114 (1995) 203-210
Fig. 3. Speckle patterns in a Fraunhofer diffraction plane produced by the ground glass masked with (a) the circular aperture, ring-slit apertures with (b) a/b Y 0.95 and (c) a/b N 0.69. Three apertures have the same outer radius of b Y 1 mm.
and the
K. Unoet al. /Optics Communications114 (1995) 203-210
Fig. 3.
case of Fig. 4c. It is clear that these side lobes in the correlation functions reflect a clustering appearance in speckle patterns. On the other hand, as seen in Fig. 4a, the side lobes for the case of the circular aperture are very weak, reflecting non-appreciable speckle clustering in Fig. 3a. Another noticeable point is the difference in the correlation peak width W. By comparing Figs. 3a-c, we recognize that the average speckle size is the largest for Fig. 3a, while it is the smallest for Fig. 3b. This tendency is reflected in the peak width w of the intensity correlation: When the ring slit becomes broader and approaches the circular aperture, the peak width w becomes broader. Experimental values of w obtained from Figs. 4a-c are 3.79, 2.07, and 2.80, respectively. These behaviors of intensity correlations agree well with the theoretical dependence of Figs. la and b.
207
Cont.
3.2. The speckle produced by a random Koch fractal In the previous studies [2,3], the clustering phenomenon was also reported for the speckles produced by randomized Koch curves. In this subsection, this phenomenon is discussed in comparison with the speckle clustering due to the ring-slit illumination. Random Koch curves are fractals which are selfsimilar in a statistical sense. They are generated by introducing a certain kind of randomness into the generation process of regular Koch curves [ 31. A random Koch curve employed in the experiment is shown in Fig. 5. In this object, the recursive replacement was performed five times with the randomness of u = 15 and the fractal dimension of D = log4/ log3 [2,3]. The object was photographed on a negative film, which was placed at the object plane without the aperture in Fig. 2. The observed speckle pattern is shown in Fig.
P oApx
(b)
I
O-10
I
I
0
I
1
l(
b Apx
b Apx Fig. 4. Intensity correlation functions of the speckle patterns produced by (a) the circular aperture, and the ring-slit apertures of (b) a/b N 0.95 and (c) a/b ry 0.69, respectively. The cross-sections of these functions along the horizontal axis px are also shown at the bonom.
K. Uno et al. /Optics Communications 114 (1995) 203-210
(a)
(b)
/
Fig. 5. (a) Random Koch curve with the fractal dimension D = log4/log3. It is constructed by recursive substitution with (b) a generator and (c) an initiator by introducing randomness in the generator [ 21.
209
6, where the power-law decay of the average intensity is compensated as described in the previous work [ 31. In this pattern, speckle clustering can be clearly seen. The intensity correlation function (above pattern) calculated from this speckle pattern is shown in Fig. 7 together with its cross-section (lower figure). Fig. 7 shows an appreciable side lobes as a consequence of speckle clustering. An existence of the long correlation tail in the speckles from regular and random Koch curves was also demonstrated in a separate paper [ 51 by means of the lacunarity of the speckle patterns. It is noted that the random Koch fractal has approximately a ring-slit shape. From this feature, it can be concluded that the cause of speckle clustering is a ring-like structure of the object. Strictly speaking, the random Koch fractal is not a circular ring, but is considered as a ring made of a connection of many arcs with different scales. As a result, the side lobes of the corresponding intensity correlation is not so large as in the case of the narrow ring slit and looks rather smoothed out slightly.
Fig. 6. Speckle pattern in a Fraunhofer diffraction plane produced by the random Koch fractal shown in Fig. 5a. The average intensity which decays with a power function of q --D is compensated in this pattern [ 31.
210
K. Uno et al. / Optics Communications I14 (1995) 203-210
is shown theoretically and experimentally that, in a statistical term, this apparent phenomenon is elucidated by the long and ringing tails of the intensity correlation function of the speckle patterns. Namely, an increase of the correlation tail manifests itself as stronger speckle clustering, which, in the present study, is generated by the narrow slit aperture. On the other hand, the average size of speckle grains is reduced with a decrease in the slit width of the ring aperture. However, it remains an unsettled question why the speckle clustering produced by a narrow annular aperture takes the characteristic network structure having a snake-like appearance. There is a room for further investigation at this problem. The clustering phenomenon which is visible in t!re diffraction patterns produced by random Koch fractals was also discussed and attributed to the ring-like shape of the object. It is noted that the present study does not exclude the possibility that other illuminating geometries than ring-slit apertures may cause long correlation tails in the speckle intensity distributions. It is also worth noting that the shape of the illuminating aperture can control the intensity correlation property of the resulting speckles. This fact is particu!arly interesting, since the intensity correlation properties are important factors in various methods in speckle metrology such as displacement and velocity measurements by means of the speckle correlation.
Fig. 7. Intensity correlation function of the speckle pattern shown in Fig. 6. The cross-section of this function along the horizontal axis pX is also shown at the bottom.
4. Conclusion We discussed the relation between the object shape and the clustering of the resulting speckles. It has been shown that the illumination of a random object with a ring-slit aperture gives rise to the speckle clustering. It
References 1l] J. W. Goodman,
in: Laser speckle and related phenomena (Springer, Berlin, 1975). [2] J. Uozumi, H. Kimura and T. Asakura, Waves in Random Media 1 (1991) 73. [3] K. Uno, J. Uozumi and T. Asakura, Appl. Optics 32 ( 1993) 2722. [4] M. Born and E. Wolf, in: Principles of optics, 6th Ed. (Pergamon, New York. 1980). [ 51 K. Uno, J. Uozumi and T. Asakura, Waves in Random Media, submitted for publication.