## Three-dimensional clustered speckle fields: theory, simulations and experimental verification |

Optics Express, Vol. 20, Issue 19, pp. 21145-21159 (2012)

http://dx.doi.org/10.1364/OE.20.021145

Acrobat PDF (5715 KB)

### Abstract

Clustered speckle patterns are a particular type of speckles that appear when a coherently illuminated diffuser is imaged through a multiple aperture pupil mask attached to a lens. The cluster formation is the result of the complex speckle modulations of the multiple interferences produced by the apertures. In this paper, a three-dimensional analytical approach to simulate cluster speckles everywhere after the lens is presented. This approach has the possibility of including multiple aperture masks at the lens and at the diffuser, in contrast to previous works which were also limited to the description of the patterns only at the image plane. This model contributes to the development of tailor made speckle patterns that can be used in diverse optical applications, including those lying in the focus region. The approach is validated under different conditions by comparing experimental results with simulations on a statistical basis. Some aspects of possible uses of these clusters are briefly revised, such as optical trapping, manipulation and metrology.

© 2012 OSA

## 1. Introduction

4. G. Zhang, Z. Wu, and Y. Li, “Speckle size of light scattered from 3D rough objects,” Opt. Express **20**, 4726–4737 (2012). [CrossRef] [PubMed]

5. D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. Theory and numerical investigation,” J. Opt. Soc. Am. A **28**, 1896–1903 (2011). [CrossRef]

6. M. Tebaldi, A. Lencina, and N. Bolognini, “Analysis and applications of the speckle patterns registered in a photorefractive BTO crystal,” Opt. Commun. **202**, 257–270 (2002). [CrossRef]

7. A. Lencina, P. Vaveliuk, M. Tebaldi, and N. Bolognini, “Modulated speckle simulations based on the random-walk model,” Opt. Lett. **28**, 1748–1750 (2003). [CrossRef] [PubMed]

**image plane**generated by multiple apertures attached to an imaging lens could be simulated by this analytical approach. It allowed the study of speckle fields where apertures were evenly distributed along a circumference [8

8. A. Lencina, M. Tebaldi, P. Vaveliuk, and N. Bolognini, “Dynamic behaviour of speckle cluster formation,” Waves in Random and Complex Media **17**, 29–42 (2007). [CrossRef]

*clustered speckles*in accordance with a previous Uno

*et al.*proposal [9

9. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. **114**, 203–210 (1995). [CrossRef]

10. F. Mosso, M. Tebaldi, A. Lencina, and N. Bolognini, “Cluster speckle structures through multiple apertures forming a closed curve,” Opt. Commun. **283**, 1285–1290 (2010). [CrossRef]

8. A. Lencina, M. Tebaldi, P. Vaveliuk, and N. Bolognini, “Dynamic behaviour of speckle cluster formation,” Waves in Random and Complex Media **17**, 29–42 (2007). [CrossRef]

9. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. **114**, 203–210 (1995). [CrossRef]

11. J. P. Staforelli, J. M. Brito, E. Vera, P. Solano, and A. Lencina, “A clustered speckle approach to optical trapping,” Opt. Commun. **283**, 4722–4726 (2010). [CrossRef]

**focal plane**of the objective lens. Although clustered speckle fields look similar to that obtained with apertures attached to the lens and observed at the image plane [8

8. A. Lencina, M. Tebaldi, P. Vaveliuk, and N. Bolognini, “Dynamic behaviour of speckle cluster formation,” Waves in Random and Complex Media **17**, 29–42 (2007). [CrossRef]

11. J. P. Staforelli, J. M. Brito, E. Vera, P. Solano, and A. Lencina, “A clustered speckle approach to optical trapping,” Opt. Commun. **283**, 4722–4726 (2010). [CrossRef]

14. T.M. Grzegorczyk, B.A. Kemp, and J.A. Kong, Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field, J. Opt. Soc. Am. A. **23**, 2324–2330 (2006). [CrossRef]

15. D. Boiron, C. Mennerat-Robilliard, J. M. Fournier, L. Guidoni, C. Salomon, and G. Grynberg, “Trapping and cooling cesium atoms in a speckle Field,” Eur. Phys. J. D **7**, 373–377 (1999). [CrossRef]

19. M. Robert-de-Saint-Vincent, J. P. Brantut, B. Allard, T. Plisson, L. Pezzé, L. Sanchez-Palencia, A. Aspect, T. Bourdel, and P. Bouyer, “Anisotropic 2D Diffusive Expansion of Ultracold Atoms in a Disordered Potential,” Phys. Rev. Lett. **104**, 220602 (2010). [CrossRef] [PubMed]

## 2. Theoretical approach

### 2.1. Field propagation

*E*(

**r**), where

**r**=

*x*î +

*yj*̂ +

*zk*̂. In order to simplify the calculations the 3D space is normalized by the vacuum wavelength. All calculi are performed in the Fresnel (paraxial) or Fraunhofer approximation [20].

*Z*

_{0}from the diffuser

*x*−

*y*plane to the lens

*u*−

*v*plane in a medium with refractive index

*n*

_{0}(see Fig. 2). Then, the field propagates a distance

*Z*towards the observation

_{C}*X*−

*Y*plane in another medium whose refractive index is

*n*. Mathematically, for a lens of focal distance

_{C}*f*and characterized by a pupil function

*P*(

*u*,

*v*), it is expressed as

*Z*

_{n0}=

*Z*

_{0}/

*n*

_{0},

*Z*

_{nC}=

*Z*/

_{C}*n*, and

_{C}*E*(

*x*,

*y*, 0) is the field at the diffuser

*x*−

*y*plane. Phase terms that do not contribute to the field intensity nor to the correlation are disregarded.

7. A. Lencina, P. Vaveliuk, M. Tebaldi, and N. Bolognini, “Modulated speckle simulations based on the random-walk model,” Opt. Lett. **28**, 1748–1750 (2003). [CrossRef] [PubMed]

### 2.2. 3D model for clustered speckles

*diffuser pupil mask*(

*x*−

*y*plane in Fig. 2) formed by

*P*disjoint apertures centered at (

*x*,

_{s}*y*). Each aperture has

_{s}*M*uniformly distributed scattering points

*E*(centered at (

_{qs}*x*,

_{q}*y*)) so that for each one the Fraunhofer approximation holds. Moreover, each scattering point encloses a random phase

_{q}*ϕ*uniformly distributed on the interval [−

_{qs}*π*,

*π*]. Then,

*lens pupil mask*(

*u*−

*v*plane in Fig. 2) composed by

*N*small disjoint apertures

*a*centered at (

_{h}*u*,

_{h}*v*), i.e.

_{h}*lens law*when

*w*= 0 and defines the relationship between

*Z*

_{n0},

*Z*

_{nC}and

*f*to account for image formation. When

*w*≠ 0, the system is out of focus. To emphasize the role of scattering points and apertures, the coordinate systems are moved to their geometrical centers (

*x*,

_{q}*y*) and (

_{q}*u*,

_{h}*v*) at planes

_{h}*x*−

*y*and

*u*−

*v*, respectively. If the new variables are called

*x*

_{0}−

*y*

_{0}and

*u*

_{0}−

*v*

_{0}, the electric field is written as

*U*the shifted angular spectrum of the

_{qs}*q*-th scattering point on the

*s*-th aperture of the diffuser pupil mask.

**17**, 29–42 (2007). [CrossRef]

*w*= 0 (

**image plane**) and only a lens pupil mask with several apertures, Eq. (3) reduces to the equation for clustered speckle fields found in [8

**17**, 29–42 (2007). [CrossRef]

**17**, 29–42 (2007). [CrossRef]

**focal plane**), if a diffuser pupil mask with several apertures is considered together with a lens pupil mask with only one aperture, Eq. (3) lacks the phase factor within the summations. Even though fringe systems also appear. They originate from the integral phase contributions that depend on the diffuser aperture centers, as will be seen below. Note that this simple case is not covered by previous approaches [8

**17**, 29–42 (2007). [CrossRef]

### 2.3. Analytical approach for clustered speckles

*E*at the diffuser are considered as Dirac’s delta functions and apertures

_{qs}*a*(

_{h}*u*

_{0},

*v*

_{0}) at the lens pupil mask are rectangles of sides

*a*

_{uh}and

*a*

_{vh}. With these assumptions, Eq. (3) results

**17**, 29–42 (2007). [CrossRef]

## 3. Experimental setup

*λ*= 532

*nm*) or an He-Ne (

*λ*= 632.8

*nm*) were used. A pupil mask with one or several circular apertures is placed behind the diffuser and centered on the optical axis of the system (see Fig. 3). The resulting field is focused by an Edmund Optics semi-plan 40X DIN microscope objective with focal length

*f*= 4.39

*mm*.

*μm*, 2

*μm*, 3

*μm*and 4

*μm*calibrated polystyrene particles set from Kisker-Biotech. For each particle size, its diameter in pixels is measured when they are focused on the camera. Then, from a linear regression applied to a particle size versus pixel plot, the calibration appears as 0.135

*μm/pixel*.

## 4. Simulation details

*π*,

*π*] to ensure a fully developed speckle, which is usually the experimental case. All random numbers are generated by a linear congruential generator. This is enough for the simulations presented here, because only 3

*MP*random numbers are necessary. In our case, the number three accounts for the position and phase of the scattering points,

*P*is the number of apertures in the diffuser pupil mask, and

*M*≈ 10

^{2}. The last value is estimated by considering the number of scattering points needed to obtain a Gaussian speckle pattern at the image plane [7

7. A. Lencina, P. Vaveliuk, M. Tebaldi, and N. Bolognini, “Modulated speckle simulations based on the random-walk model,” Opt. Lett. **28**, 1748–1750 (2003). [CrossRef] [PubMed]

*mm*in the lens pupil mask. This fact, as will be shown later, does not play an important role in the final results. The aperture is centered with the optical axis of the system (

*N*= 1 with

*u*

_{1}=

*v*

_{1}= 0).

*Z*

_{n0}and

*Z*

_{nC}are measured relative to the first and second principal plane, respectively. In all cases

*Z*

_{n0}= 138

*mm*.

## 5. Statistical tools

*μ*(

_{A}*x*

_{1},

*x*

_{2};

*y*

_{1},

*y*

_{2})| is usually evaluated and considered here [1,3]. Moreover, according to Reed’s moment theorem [22

22. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory **IT-8**, 194–195 (1962). [CrossRef]

*μ*(

_{A}*x*

_{1},

*x*

_{2};

*y*

_{1},

*y*

_{2})| with the transversal intensity autocorrelation by being

*R*(

_{I}*x*

_{1},

*x*

_{2};

*y*

_{1},

*y*

_{2}) ≡ 〈

*I*(

*x*

_{1},

*y*

_{1})

*I*(

*x*

_{2},

*y*

_{2})〉, and 〈.〉 stands for statistical average over the ensemble.

23. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A **7**, 827–832 (1990). [CrossRef]

*x*and

*y*, it can be calculated by correlating only one intensity pattern

*I*(

*x*,

*y*,

*z*) = |

*E*(

_{CS}*x*,

*y*,

*z*)|

^{2}(avoiding using several images to compute the ensemble average). Then, by considering the convolution theorem from Fourier theory, it has where Δ

*x*=

*x*

_{2}−

*x*

_{1}, Δ

*y*=

*y*

_{2}−

*y*

_{1}and

*ℱ*{.} is the Fourier transform operator. Then, from a single measured or simulated intensity pattern of clustered speckles, its spatial statistic can be retrieved by employing FFT algorithms.

*μ*(Δ

_{A}*x*, Δ

*y*)|. This is defined as where

*r*= (Δ

*x*

^{2}+ Δ

*y*

^{2})

^{1/2}and

*θ*= arctan(Δ

*y*/Δ

*x*).

## 6. Results and discussion

### 6.1. Clustered speckles at the objective focal plane

*mm*diameter. The aperture diameters are 0.46

*mm*and 0.90

*mm*approximately. A doubled Nd:YAG laser (

*λ*= 532

*nm*) is used and the gap between the objectives is filled with oil. From Fig. 4 it is apparent that the theoretical approach allows simulating quite well the speckles obtained experimentally. The |

*μ*| are very coincident in all cases. The one aperture case shows only one broad peak. On the other hand, Fig. 4 reveals that for the cases of six and ten apertures, the width of the main correlation peak is reduced considerably. Moreover, secondary spots at given positions also appears. This is a signature of clustered speckle formation. Unlike the case of ring apertures [9

_{A}9. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. **114**, 203–210 (1995). [CrossRef]

*mm*diameter. Here, it seems the net effect of the apertures is only to reduce drastically the speckle size. But, as it is mentioned in Sec.5 a better tool to analyze clustered speckles is the radial profile of the modulus of the transversal complex coherence factor and it is analyzed below.

**17**, 29–42 (2007). [CrossRef]

*mm*diameter the secondary peak disappears and another peak close to the main peak appears. In Ref. [8

**17**, 29–42 (2007). [CrossRef]

*range*is reduced. That is, the intensity of the second or third order peaks diminishes as far as the aperture size increases. This is apparent for the case of ten apertures of 0.90

*mm*diameter in comparison with the 0.46

*mm*ones. This effect is traduced in a effective reduction of the speckle size which mainly depends on the circumference diameter where apertures are distributed [8

**17**, 29–42 (2007). [CrossRef]

**114**, 203–210 (1995). [CrossRef]

*mm*diameter where the clustered speckle size is reduced by approximately one order of magnitude (blue dotted line) compared to the one aperture case (standard speckles).

### 6.2. 3D clustered speckles around the objective focal plane

*mm*diameter is considered. Successive images are taken along the optical axis (

*z*–direction) ranging from −25

*μm*to +25

*μm*(measured from the focus) with steps of 1

*μm*. Small step-to-step lateral errors coming from both, the translational stage and minor misalignments, are corrected by using subpixel image registration techniques [24

24. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. **33**(2), 156–158 (2008). [CrossRef] [PubMed]

### 6.3. Clustered speckle formation: from pupil image to focal plane

**image plane**is interpreted as intra speckle modulations produced by the interference of the patterns belonging to each aperture on the lens pupil mask [8

**17**, 29–42 (2007). [CrossRef]

**focal plane**is studied. To perform this task the same setup as in previous sections is employed. Experimental results from the image plane of the system towards the focal plane are compared with simulations obtained through Eq. (4). As a first step, wide-view images are analyzed and two indices of refraction are considered (see Fig. 7). In this case, the clustered speckle details are not observed due to the wide-view, but the process that led to the fringe formation can be inferred. In a second step, a comparison of close-view clustered speckles at the focal plane is done (see Fig. 8). Then, agreement between experimental results and theoretical simulations is fully established.

*λ*= 632.8

*nm*is used in this case. Two different refraction index conditions at the objectives gap are considered in order to validate the 3D simulated field propagation. The two bottom rows compare the cases without immersion oil (air only). On the other hand, the two top rows represent the case with oil. Images every 50

*μm*are taken, until 250

*μm*are reached. All distances are indicated by

*Z*relative to the second principal plane of the objective lens. For the case shown at the bottom of Fig. 7 (without oil), the simulation exhibits a good general agreement with the experiment disregarding the differences coming from the square apertures considered in the simulations. At

_{C}*Z*= 4.54

_{C}*mm*pupil image formation takes place. Gradually, as far as images are taken closer to the focus, the speckle fields belonging to each aperture tend to overlap. Because of the coherent nature of the process these fields interfere, even though they are statistically independent. This interference takes place inside of each speckle and leads to the cluster formation. Both experimental and simulated images start to exhibit higher modulations as they approach the complete formation of clustered patterns at the focal plane

*Z*= 4.39

_{C}*mm*and beyond. Although these comparisons are mainly qualitative, since the corresponding |

*μ*| can not be well calculated from the wide-view of Fig. 7, they are in good agreement with the approach here shown.

_{A}*Z*= 6.80

_{C}*mm*) compared to the case without oil. The difference is not unexpected and can be explained by the change in the refraction index which enlarges the effective optical path length. Moreover, the experimental images exhibit noise in the area between apertures which is not observed without oil. We think that this effect can be mainly due to scattering and optical aberrations when the immersion oil is overstretched. On the other hand, the immersion oil flows perfectly when the distance between objectives is shorter and the noise tends to be minimized, as is shown in Fig. 8.

*X*10) region extracted from Fig. 7 is shown. Their corresponding |

*μ*| and radial profile are calculated for each one. Both results agree well with each other at the focal plane where clustered speckle formation is observed. Note that clustered speckles are similar to that obtained in [8

_{A}**17**, 29–42 (2007). [CrossRef]

**image plane**. However, recall that the clustered speckles in Fig. 8 are obtained at the

**focal plane**. To summarize, this analytical approach, which encompasses previous ones, broadens the range of cases to be considered.

## 7. Possible applications of clustered speckle

### 7.1. Optical manipulation

11. J. P. Staforelli, J. M. Brito, E. Vera, P. Solano, and A. Lencina, “A clustered speckle approach to optical trapping,” Opt. Commun. **283**, 4722–4726 (2010). [CrossRef]

*λ*=532 nm and a diffuser pupil mask consisting of ten apertures of about 0.9

*mm*evenly distributed on a circumference of 4

*mm*diameter, the transversal dimension of the spots that constitute the cluster is on the order of the wavelength or less, whereas for a single aperture pupil mask the speckle is approximately 4

*μm*. By considering water as a surrounding medium, an analysis for particles in the Rayleigh regime showed that for the case of ten apertures, particles smaller than 50

*nm*could be trapped. But if the apertures are reduced to 0.46

*mm*the system could trap particles of at most 40

*nm*. On the other hand, in a gaseous environment with low refractive index, the conditions change, and the particles are pushed away from the beam. Last years, standard speckles have been used as sieve selector by particle size, by inducing photophoretic forces in carbon agglomerated nanoparticles [12

12. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Selective trapping of multiple particles by volume speckle field,” Opt. Express **18**, 3137–3142 (2010). [CrossRef] [PubMed]

13. V. G. Shvedov, A. V. Rode, Ya. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Speckle field as a multiple particle trap,” Proc. SPIE **7715**, 77150K (2010). [CrossRef]

14. T.M. Grzegorczyk, B.A. Kemp, and J.A. Kong, Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field, J. Opt. Soc. Am. A. **23**, 2324–2330 (2006). [CrossRef]

### 7.2. Atoms trapping

### 7.3. Photorefractive register

6. M. Tebaldi, A. Lencina, and N. Bolognini, “Analysis and applications of the speckle patterns registered in a photorefractive BTO crystal,” Opt. Commun. **202**, 257–270 (2002). [CrossRef]

25. M. Tebaldi, L. Ángel Toro, M. Trivi, and N. Bolognini, “Optical processing by fringed speckles registered in a BSO crystal,” Opt. Eng. **39**, 3232–3238 (2000). [CrossRef]

### 7.4. Speckle metrology

26. J. N. Butters and J. A. Leendertz, “A double exposure technique for speckle pattern interferometry,” J. Phys. E **4**, 277–279 (1971). [CrossRef]

27. R. P. Khetan and F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry. 1: Single aperture method,” Appl. Opt. **15**, 2205–2215 (1976). [CrossRef] [PubMed]

28. F.P. Chian and R.P. Khetan, “Strain analysis by one-beam laser speckle interferometry. 2: Multiaperture method,” Appl. Opt. **18**, 2175–2186 (1979). [CrossRef]

29. M. Tebaldi, L. Ángel, M. Trivi, and N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. **182**, 95–105 (2000). [CrossRef]

30. L. Ángel, M. Tebaldi, and N. Bolognini, “Multiple rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. **46**, 2676–2682 (2007). [CrossRef] [PubMed]

### 7.5. Vortex metrology

## 8. Conclusions

## Appendix: Objective principal planes calculation

*I*-plane is

*f*= 4.39

*mm*and its

*working distance*is

*Z*be the image distance and

_{C}*Z*

_{0}the object one. They satisfy

*Z*

_{0}/

*Z*= 40 due to the objective magnification. By putting this into the lens law

_{C}*Z*, then

_{C}*Z*

_{0}can then be obtained. Thus the positions of the principal planes can be calculated by simple substraction.

## Acknowledgments

## References and links

1. | J. C. Dainty, |

2. | M. Françon, |

3. | J. W. Goodman, |

4. | G. Zhang, Z. Wu, and Y. Li, “Speckle size of light scattered from 3D rough objects,” Opt. Express |

5. | D. Li, D. P. Kelly, and J. T. Sheridan, “Three-dimensional static speckle fields. Part I. Theory and numerical investigation,” J. Opt. Soc. Am. A |

6. | M. Tebaldi, A. Lencina, and N. Bolognini, “Analysis and applications of the speckle patterns registered in a photorefractive BTO crystal,” Opt. Commun. |

7. | A. Lencina, P. Vaveliuk, M. Tebaldi, and N. Bolognini, “Modulated speckle simulations based on the random-walk model,” Opt. Lett. |

8. | A. Lencina, M. Tebaldi, P. Vaveliuk, and N. Bolognini, “Dynamic behaviour of speckle cluster formation,” Waves in Random and Complex Media |

9. | K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. |

10. | F. Mosso, M. Tebaldi, A. Lencina, and N. Bolognini, “Cluster speckle structures through multiple apertures forming a closed curve,” Opt. Commun. |

11. | J. P. Staforelli, J. M. Brito, E. Vera, P. Solano, and A. Lencina, “A clustered speckle approach to optical trapping,” Opt. Commun. |

12. | V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Selective trapping of multiple particles by volume speckle field,” Opt. Express |

13. | V. G. Shvedov, A. V. Rode, Ya. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Speckle field as a multiple particle trap,” Proc. SPIE |

14. | T.M. Grzegorczyk, B.A. Kemp, and J.A. Kong, Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field, J. Opt. Soc. Am. A. |

15. | D. Boiron, C. Mennerat-Robilliard, J. M. Fournier, L. Guidoni, C. Salomon, and G. Grynberg, “Trapping and cooling cesium atoms in a speckle Field,” Eur. Phys. J. D |

16. | G. Grynberg, P. Horak, and C. Mennerat-Robilliard, “Spatial diffusion of atoms cooled in a speckle field,” Europhys. Lett. |

17. | R. C. Kuhn, O. Sigwarth, C. Miniatura, D. Delande, and C. A. Müller, “Coherent matter wave transport in speckle potentials,” New J. Phy. |

18. | M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, “Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond,” Adv. Phys. |

19. | M. Robert-de-Saint-Vincent, J. P. Brantut, B. Allard, T. Plisson, L. Pezzé, L. Sanchez-Palencia, A. Aspect, T. Bourdel, and P. Bouyer, “Anisotropic 2D Diffusive Expansion of Ultracold Atoms in a Disordered Potential,” Phys. Rev. Lett. |

20. | J. W. Goodman, |

21. | M. Abramowitz and I. A. Stegun, |

22. | I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory |

23. | L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A |

24. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. |

25. | M. Tebaldi, L. Ángel Toro, M. Trivi, and N. Bolognini, “Optical processing by fringed speckles registered in a BSO crystal,” Opt. Eng. |

26. | J. N. Butters and J. A. Leendertz, “A double exposure technique for speckle pattern interferometry,” J. Phys. E |

27. | R. P. Khetan and F. P. Chiang, “Strain analysis by one-beam laser speckle interferometry. 1: Single aperture method,” Appl. Opt. |

28. | F.P. Chian and R.P. Khetan, “Strain analysis by one-beam laser speckle interferometry. 2: Multiaperture method,” Appl. Opt. |

29. | M. Tebaldi, L. Ángel, M. Trivi, and N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. |

30. | L. Ángel, M. Tebaldi, and N. Bolognini, “Multiple rotation assessment through isothetic fringes in speckle photography,” Appl. Opt. |

31. | W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express |

32. | G. H. Sendra, H. J. Rabal, R. Arizaga, and M. Trivi, “Vortex analysis in dynamic speckle images,” J. Opt. Soc. Am. A |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(050.1220) Diffraction and gratings : Apertures

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 13, 2012

Revised Manuscript: July 27, 2012

Manuscript Accepted: July 30, 2012

Published: August 31, 2012

**Citation**

A. Lencina, P. Solano, J. P. Staforelli, J. M. Brito, M. Tebaldi, and N. Bolognini, "Three-dimensional clustered speckle fields: theory, simulations and experimental verification," Opt. Express **20**, 21145-21159 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21145

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