OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 19 — Sep. 10, 2012
  • pp: 21145–21159
« Show journal navigation

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

A. Lencina, P. Solano, J. P. Staforelli, J. M. Brito, M. Tebaldi, and N. Bolognini  »View Author Affiliations


Optics Express, Vol. 20, Issue 19, pp. 21145-21159 (2012)
http://dx.doi.org/10.1364/OE.20.021145


View Full Text Article

Acrobat PDF (5715 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

It is well known that if coherent light illuminates an optically rough surface, the scattered fields produce a non-localized interference distribution known as speckle pattern. Since their appearance, speckle fields were mainly used in optical image processing and metrological applications [1

1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

3

3. J. W. Goodman, Speckle Phenomena in Optics: theory and applications (Roberts&Company, 2007).

] and today are an active research field [4

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

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]

].

Fig. 1 Magnified images of different kinds of speckle patterns. a) Clustered speckles; b) Standard speckles. Speckle patters were obtained on air by using a frequency doubled Nd:YAG laser with λ = 532 nm, Z0 = 75 mm, ZC = 400 mm, f = 50 mm. For clustered speckles a lens pupil mask consisting of sixteen apertures of about 1 mm diameter evenly distributed on a circumference of 30 mm diameter was employed. See Sec. 2.1 for parameters details.

For several applications, i.e. optical manipulation [11

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

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]

], atom trapping [15

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

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]

], among others, where tailor made speckle distributions could be used, speckle patterns are generated at the focus of an optical system mainly because of the required sizes and field gradient properties. Furthermore, these characteristics can be improved by clustered speckle fields, issue that has not been widely studied. This fact generates additional interest to the capability of modeling clustered speckle formations around the focus region or, more generally, everywhere after the lens. In that sense, the purpose of this paper is to develop a general analytical approach to simulate three-dimensional (3D) clustered speckle fields in the semi-space after the lens. Additionally, in order to study the validity of this model, simulations are compared with experimental results. Since speckles are a statistical process the modulus of the complex coherence factor is considered [1

1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

]. Besides, its radial profile is introduced as a better tool to describe the statistical features of clustered speckles allowing to improve its description.

This paper is organized as follows: In Sec. 2 the analytical approach is developed. In Sec. 3 the experimental setup is described, whereas in Sec. 4 the simulation details are outlined. In Sec. 5 the statistical tools to analyze the clusters are established. The experimental results and its simulations are presented in Sec. 6, together with their statistical analysis. In Sec. 7 some possible applications are revised. Finally, in Sec. 8 the conclusions are established.

2. Theoretical approach

2.1. Field propagation

Let us consider a time Fourier transformed 3D (unit amplitude) scalar field E(r), where r = xî + yĵ + zk̂. 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

20. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Ch 3.

].

We consider a field that propagates a distance Z0 from the diffuser xy plane to the lens uv plane in a medium with refractive index n0 (see Fig. 2). Then, the field propagates a distance ZC towards the observation XY plane in another medium whose refractive index is nC. Mathematically, for a lens of focal distance f and characterized by a pupil function P(u, v), it is expressed as
E(x,y,z)=1Zn0ZnCE(x,y,0)P(u,v)exp{i2π(Xu)2+(Yv)22ZnC}×exp{i2π(ux)2+(vy)22Zn0}exp{i2πu2+v22f}dxdydudv,
(1)
where Zn0 = Z0/n0, ZnC = ZC/nC, and E(x, y, 0) is the field at the diffuser xy plane. Phase terms that do not contribute to the field intensity nor to the correlation are disregarded.

Fig. 2 Clustered speckle field propagation sketch. A plane wave impinges normally on a diffuser. A diffuser pupil mask with several apertures is placed behind it. After this, the speckles propagate towards a lens which also has a lens pupil mask. Then, clustered speckles are obtained in the semi-space after the lens and recorded at the observation plane. The apertures in the diffuser pupil mask are centered at the (xs, ys), whereas the apertures in the lens pupil mask are centered at (uh, vh).

At this point it is usual to study the quadratic terms contributions in both, the object and the image plane. If the lens pupil mask has one centered aperture and it is assumed to be an ideal imaging system, a geometrical-image approximation is appropriate to describe image formation [20, Ch. 5]. But for non ideal systems, some careful analysis has to be done [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]

]. Here, we adopt a different strategy to analyze the 3D clustered speckle field formation: the contribution from each aperture at the lens and each scattering point in the diffuser is considered.

2.2. 3D model for clustered speckles

In this theoretical approach the speckle field originates from a diffuser pupil mask (xy plane in Fig. 2) formed by P disjoint apertures centered at (xs, ys). Each aperture has M uniformly distributed scattering points Eqs (centered at (xq, yq)) so that for each one the Fraunhofer approximation holds. Moreover, each scattering point encloses a random phase ϕqs uniformly distributed on the interval [−π, π]. Then, E(x,y,0)=qsMPEqs(xxsxq,yysyq)exp(iϕqs). In addition, let us assume a lens pupil mask (uv plane in Fig. 2) composed by N small disjoint apertures ah centered at (uh, vh), i.e. P(u,v)=hNah(uuh,vvh). In this way, integrals at object and lens plane are limited to the scattering points and apertures, respectively. Under these conditions the clustered speckle field appears as
ECS=1Zn0ZnCs,q,hP,M,Nah(uuh,vvh)exp{iϕqs}Eqs(xxsxq,yysyq)×exp{i2π[(xZn0+XZnC)u+(yZn0+YZnC)v]}exp{iπw(u2+v2)}dxdydudv,
(2)
where w=Zn01+ZnC1f1 is the lens law when w = 0 and defines the relationship between Zn0, ZnC 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 (xq, yq) and (uh, vh) at planes xy and uv, respectively. If the new variables are called x0y0 and u0v0, the electric field is written as
ECS=1Zn0ZnCs,q,hP,M,Nexp{i2π[(xs+xqZn0+XZnC)uh+(ys+yqZn0+YZnC)vhw(uh2+vh2)/2]+iϕqs}×exp{i2π[(xs+xqZn0+XZnCwuh)u0+(ys+yqZn0+YZnCwvh)v0]}×Uqs(u0+uhZn0,v0+vhZn0)ah(u0,v0)exp{iπw(u02+v02)}du0dv0.
(3)
being Uqs the shifted angular spectrum of the q-th scattering point on the s-th aperture of the diffuser pupil mask.

From Eq. (3), note that the apertures in the lens plane act as band pass filters of the shifted angular spectrum for each scattering point in the diffuser. If the angular spectrum is larger than the width of the apertures, then apertures dominate the field correlations [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]

].

2.3. Analytical approach for clustered speckles

3. Experimental setup

The approach detailed above allows us to simulate speckle fields in very general situations, in particular for clustered speckles appearing at the focal plane. An experimental setup capable of producing such a clustered speckles is depicted in Fig. 3. The clustered speckle field is obtained by employing an inverted microscope where an expanded laser beam passes through the diffuser. According to the case, a frequency doubled Nd:YAG (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.

Fig. 3 Experimental setup employed to study the 3D clustered speckle features. An inverted microscope is modified to obtain and to record clustered speckles at different planes. Computer display: reconstruction of the longitudinal intensity profile of the laser beam around the focal plane employed to set the focal plane position.

On the other hand, the field between the image plane and the objective focal plane is imaged by an Edmund Optics 60X DIN microscope objective. Images are captured by using a monochrome CMOS uEye camera (1.3Mpixel). In some cases, the air gap between objectives is filled with cedar oil from Riedel-de Haën, with nominal refractive index 1.515 – 1.520.

To achieve a complete comparison with simulated results, images are spatially calibrated. This is performed by means of a 1 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.

In order to define the planes where images should be taken, the diffuser and the pupil are previously removed and the focus of the system is found (see inset on Fig. 3).

4. Simulation details

In this section the diffuser parameters used to simulate the clustered speckles are detailed. Scattering points on the diffuser pupil mask are obtained by generating random points uniformly distributed on each aperture. Each point encloses also a random phase uniformly distributed on the interval [−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 3MP 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 ≈ 102. 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]

]: since each scattering point fills an area at the image plane, a minimum number of points to cover a given area exist. By taking a higher number, it is ensured that the probability density function of the intensity will have an exponentially decreasing behavior, which indicates the Gaussian regime.

On the other hand, although experimentally clustered speckle fields are passed through a microscope objective with circular boundaries, theoretical simulations are carried out with a square aperture of side 4 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 u1 = v1 = 0).

It should be highlighted that the microscope objective is considered as a thin lens by using the concept of principal planes (see Appendix). They are calculated with the help of the manufacturer specifications. Then Zn0 and ZnC are measured relative to the first and second principal plane, respectively. In all cases Zn0 = 138 mm.

5. Statistical tools

Due to the statistical nature of the speckles, only averages have sense for quantitative analysis. To identify its spatial structure the modulus of the transversal complex coherence factor |μA(x1, x2; y1, y2)| is usually evaluated and considered here [1

1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

,3

3. J. W. Goodman, Speckle Phenomena in Optics: theory and applications (Roberts&Company, 2007).

]. 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]

] and assuming the electric field possesses circular Gaussian statistics, it is possible to relate |μA(x1, x2; y1, y2)| with the transversal intensity autocorrelation by
|μA(x1,x2;y1,y2)|=RI(x1,x2;y1,y2)RI(x1,x1;y1,y1)1,
(5)
being RI(x1, x2; y1, y2) ≡ 〈I(x1, y1)I(x2, y2)〉, and 〈.〉 stands for statistical average over the ensemble.

Taking into account that the transversal intensity autocorrelation is stationary [23

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]

], i.e. it is independent of x and y, it can be calculated by correlating only one intensity pattern I(x,y,z) = |ECS(x, y, z)|2 (avoiding using several images to compute the ensemble average). Then, by considering the convolution theorem from Fourier theory, it has
RI(Δx,Δy)=1{|{I}|2}
(6)
where Δx = x2x1, Δy = y2y1 and {.} 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.

The modulus of the transversal complex coherence factor is a proper tool to characterize the spatial structure of standard speckles, in particular, to define an average speckle size. However, as will be seen below, for clustered speckles the situation is quite different because the appearance of secondary loci of not null correlation. Then, a useful tool will be the averaged radial profile of |μAx, Δy)|. This is defined as
RPμ(r)=12π02π|μA(r,θ)|dθ,
(7)
where r = (Δx2 + Δy2)1/2 and θ = arctan(Δyx).

6. Results and discussion

Now the approach for clustered speckles here developed is tested by comparing the experimental results with the simulated ones by using Eq. (4). The statistical analysis is performed by means of Eqs. (5) and (7). Three main points are addressed: the skill to reproduce (statistically) the features of the clustered speckle at the focal plane; the possibility to simulate 3D clustered speckles around the focal plane; and the interpretation of the clustered speckle formation as intra-speckle modulations coming from the coherent addition of the speckle field generated by each aperture.

6.1. Clustered speckles at the objective focal plane

The analysis begins with clustered speckles at the focal plane obtained by different diffuser pupil masks (see Fig. 4). One, six and ten apertures are employed. For the case of one aperture, this is centered on the optical axis of the system. For the other cases, apertures are evenly distributed on a circumference of 4 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 |μA| 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

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]

] the cluster appearance is indicated by the presence of spots at geometrical loci given by the number of apertures and the circumference diameter where they are distributed. Besides, it is observed that the cluster is modified if the apertures size increase as it is apparent from Fig. 4 for the case of ten apertures of 0.90 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.

Fig. 4 Simulated and experimental speckle images for the cases of diffuser pupil masks with one, six and ten apertures of 0.46 mm and 0.90 mm diameter. In the case of one aperture, this is centered on the optical axis of the system. For the cases of six and ten apertures, they are evenly distributed on a circumference of 4 mm diameter. For all cases the |μA| is calculated by means of Eq. (5). All images display a square region of 12 μm × 12 μm.

Fig. 5 Radial profiles of the modulus of the complex coherence factor calculated from Fig. 4 by using Eq. (7).

6.2. 3D clustered speckles around the objective focal plane

In this subsection the volume plots of clustered speckles around the focal plane obtained experimentally are compared with those simulated by using Eq. (4).

Fig. 6 Clustered speckle around the focal plane of Fig.3. Simulated and experimental intensity contours at half-intensity are displayed. A doubled Nd:YAG laser, λ = 532 nm is employed and the gap between the objectives is filled with oil. The diffuser pupil mask has six apertures of approximately 0.46mm diameter distributed in a circumference of 4 mm diameter.

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

Fig. 7 Comparison between experimental and simulated images taken from pupil image formation to the lens focal plane. All ZC-distances are measured respect to the second principal plane of the objective in the approximation of a thin lens (see Appendix). The gap between objectives is filled with and without immersion oil which constitutes the two cases to be compared. A He-Ne laser with λ = 632.8 nm is used. The diffuser pupil mask is the same as in Fig. 6. All images display a square region of 160 μm × 160 μm.
Fig. 8 Experimental and simulated clustered speckle pattern formation at the focal plane. Images are a X10 magnified region extracted from Fig. 7 displaying an area of 16 μm × 16 μm.

Additionally, the case with immersion oil exhibits also a good agreement between experimental images and simulations that can be interpreted in the same way as the case without oil. However, the pupil image is formed at a longer distance (ZC = 6.80 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.

7. Possible applications of clustered speckle

In this section some aspects of the envisaged uses of clustered speckles are reviewed. Attention is placed on the potential advantage of using clustered speckles instead of standard speckles.

7.1. Optical manipulation

As analyzed in [11

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]

], clustered speckles reduce light volumes several times in relation to standard speckles. For instance, by using the same set-up of Fig. 3 with 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

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]

]. In this line, by including multiple aperture pupils masks, different kinds of clustered speckles could be obtained. It would enable to trapp particles of different sizes with different spatial distribution and density, thus allowing studies on optical binding of particles [14

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]

] with random potentials.

7.2. Atoms trapping

7.3. Photorefractive register

Several applications of speckles in photorefractive materials have been shown [6

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

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]

]. The speckle size introduces an additional parameter to control register efficiency. The volume nature permits control of grating thickness by considering the speckle-speckle superposition in a registering setup where two speckle fields interfere in a photorefractive crystal. However, it is unclear what the effective thickness is when multiple speckle fields interfere at the crystal. In this case, a clustered speckle is registered. Then by simulating this field and analyzing its intensity correlations, the effective thickness of the register could be inferred allowing optimization of the optical system parameters to obtain a higher diffraction efficiency in the read-out process.

7.4. Speckle metrology

One of the first uses of speckles was in metrological applications [26

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

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]

]. Modulated speckle patterns where used to dislocate the relevant information away from the zero order focus [28

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]

]. Metrologycal performance was improved by using a lens pupil mask with multiple apertures [29

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

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]

]. If a diffuser pupil mask with multiple apertures is also employed, not only the overall displacement of the diffuser could be measured but also the relative displacements among the apertures could be obtained. This procedure should increase the information obtained in the measurement process.

7.5. Vortex metrology

In recent years a technique that employs vortex to perform displacement measurements was developed [31

31. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006). [CrossRef] [PubMed]

,32

32. G. H. Sendra, H. J. Rabal, R. Arizaga, and M. Trivi, “Vortex analysis in dynamic speckle images,” J. Opt. Soc. Am. A 26, 2634–2639(2009). [CrossRef]

]. Vortices appearing in the pseudophase of an analytical signal obtained from an intensity pattern are correlated to determine the measure. Clustered speckles should increase the vortex density in the pseudophase, allowing increasing the precision of the displacement determination. Finally, correlation of vortex in clustered speckle fields is an open issue that should be addressed in future researches.

8. Conclusions

Appendix: Objective principal planes calculation

The simplicity of a thin lens can be retained for a thick lens (or even a system of lenses) by appealing to the concept of principal planes [20, pp. 449–451 ]. In Fig. 9, the relevant information to find these planes in a DIN objective is detailed. This objective has an object-to-image distance OI¯=195mm, whereas the distance from the rear shoulder of the objective to the I-plane is SI¯=45mm. The focal length of the Edmund Optics 40x semi-plan objective is f = 4.39 mm and its working distance is EI¯=0.6mm. This last parameter sets the length of the objective at SE¯=44.4mm. These are all the quantities needed to find the principal planes.

Fig. 9 Scheme used for calculation of principal planes. OI¯=195mm, and SI¯=45mm. F indicates the focal plane, PP1 and PP2 refer to the first and second principal planes, respectively. ZC and Z0 are the image and object distances, respectively. For the Edmund Optics 40X semi-plan objective, the working distance is EI¯=0.6mm and f = 4.39 mm is the objective focal distance. Note that the scheme is not to scale.

Let ZC be the image distance and Z0 the object one. They satisfy Z0/ZC = 40 due to the objective magnification. By putting this into the lens law ZC, then Z0 can then be obtained. Thus the positions of the principal planes can be calculated by simple substraction.

Acknowledgments

We are especially thankful to Dr. Carlos Saavedra for his valuable commentaries and discussions throughout the whole process of this work, and for revising this manuscript. We also thank Dr. Esteban Vera, for helping us to correct the images by subpixel image registration techniques. This work was supported by Grants Milenio ICM P06-067F and CONICYT-PFB08024 from Chile and PIP0863, PICT1167, PICT1343 and UNLP 11/I125 from Argentina. A. Lencina acknowledges support from the CONICYT-Chile through PBCT red 21. Juan Pablo Staforelli acknowledges support from FONDECYT 11110145. J.M. Brito is a BecasChile fellow.

References and links

1.

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

2.

M. Françon, Laser Speckle and Applications in Optics (Academic Press, 1979).

3.

J. W. Goodman, Speckle Phenomena in Optics: theory and applications (Roberts&Company, 2007).

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]

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]

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]

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]

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]

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]

16.

G. Grynberg, P. Horak, and C. Mennerat-Robilliard, “Spatial diffusion of atoms cooled in a speckle field,” Europhys. Lett. 49, 424–430 (2000). [CrossRef]

17.

R. C. Kuhn, O. Sigwarth, C. Miniatura, D. Delande, and C. A. Müller, “Coherent matter wave transport in speckle potentials,” New J. Phy. 9, 1–39 (2007).

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. 56, 243–379 (2007). [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]

20.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Ch 3.

21.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, (Dover Publications, 1965) p. 297.

22.

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

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]

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]

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]

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]

31.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006). [CrossRef] [PubMed]

32.

G. H. Sendra, H. J. Rabal, R. Arizaga, and M. Trivi, “Vortex analysis in dynamic speckle images,” J. Opt. Soc. Am. A 26, 2634–2639(2009). [CrossRef]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).
  2. M. Françon, Laser Speckle and Applications in Optics (Academic Press, 1979).
  3. J. W. Goodman, Speckle Phenomena in Optics: theory and applications (Roberts&Company, 2007).
  4. G. Zhang, Z. Wu, and Y. Li, “Speckle size of light scattered from 3D rough objects,” Opt. Express20, 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. A28, 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]
  8. A. Lencina, M. Tebaldi, P. Vaveliuk, and N. Bolognini, “Dynamic behaviour of speckle cluster formation,” Waves in Random and Complex Media17, 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]
  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]
  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]
  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. Express18, 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. SPIE7715, 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]
  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. D7, 373–377 (1999). [CrossRef]
  16. G. Grynberg, P. Horak, and C. Mennerat-Robilliard, “Spatial diffusion of atoms cooled in a speckle field,” Europhys. Lett.49, 424–430 (2000). [CrossRef]
  17. R. C. Kuhn, O. Sigwarth, C. Miniatura, D. Delande, and C. A. Müller, “Coherent matter wave transport in speckle potentials,” New J. Phy.9, 1–39 (2007).
  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.56, 243–379 (2007). [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]
  20. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), Ch 3.
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, (Dover Publications, 1965) p. 297.
  22. I. S. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. TheoryIT-8, 194–195 (1962). [CrossRef]
  23. L. Leushacke and M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A7, 827–832 (1990). [CrossRef]
  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]
  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]
  26. J. N. Butters and J. A. Leendertz, “A double exposure technique for speckle pattern interferometry,” J. Phys. E4, 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]
  31. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, and M. Takeda, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express14, 120–127 (2006). [CrossRef] [PubMed]
  32. G. H. Sendra, H. J. Rabal, R. Arizaga, and M. Trivi, “Vortex analysis in dynamic speckle images,” J. Opt. Soc. Am. A26, 2634–2639(2009). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited