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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 Optical Society of America
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 [1–3] and today are an active research field [4, 5].
As applications have evolved, different kinds of speckle patterns arose. In particular, modulated speckles appeared in photorefractive applications [6]. However, the lack of a proper theoretical model for modulated speckles pushed research to the development of an approach which explains the observed modulations. That approach came with the aid of the random walk model and an adequate treatment of quadratic phase terms [7]. Then, modulated speckles at the 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]. These fields presented a clustered structure leading to the denomination of clustered speckles in accordance with a previous Uno et al. proposal [9]. See Fig. 1 for a comparison of clustered and standard speckles. In a few words, the cluster results from intra speckle modulations generated by the interference patterns produced by paired apertures. These interference patterns generate a cluster that resembles a symmetrized version of the curve where apertures are distributed as it is shown in Ref. [10]. In that work it is established that: cluster size is inversely proportional to the circumference diameter; the number of spots that form the cluster is proportional to the apertures number; and the cluster structure does not depend on the aperture radius. Moreover, as it was previously observed, the emergence of long correlation spots at geometrical loci is considered as a signature of cluster appearance [8, 9].
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.
Recently, an approach to optical trapping with clustered speckles was presented in Ref. [11]. There, clustered speckles were generated by attaching a multiple aperture pupil mask to the diffuser. In contrast to previous works, in this case clusters appear at the 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], they come from different setups and lie at different planes. In particular, note that if the apertures are attached to the diffuser, clustered speckles appear at the focal plane but not at the image plane. This is not described by current analytical approaches, revealing the lack of a general model.
For several applications, i.e. optical manipulation [11–14], atom trapping [15–19], 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]. 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].
We consider a field that propagates a distance Z0 from the diffuser x − y plane to the lens u − v plane in a medium with refractive index n0 (see Fig. 2). Then, the field propagates a distance ZC towards the observation X − Y 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
where Zn0 = Z0/n0, ZnC = ZC/nC, and 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.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]. 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 (x − y 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, . In addition, let us assume a lens pupil mask (u − v plane in Fig. 2) composed by N small disjoint apertures ah centered at (uh, vh), i.e. . 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
where 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 x − y and u − v, respectively. If the new variables are called x0 − y0 and u0 − v0, the electric field is written as 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].
For 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]. In this case, it has a phase factor within the summations that can not be disregarded because it contributes to the fringes system formation on the intensity. Each aperture pair contributes to a fringe system whose frequency depends on the apertures center separation and its direction coincides with the line that joins the aperture centers. Furthermore, if the intensity of one centered scattering point is considered, then it is proportional to the transversal intensity correlation [8]. On the other hand, for (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]. In a general case, where apertures are present both at the diffuser and at the lens plane, a complex fringes system exists. This is an additional degree of freedom that allows the generation of a variety of speckle fields such as clustered speckles both at the focal plane and at the image plane.
2.3. Analytical approach for clustered speckles
In this section an analytical expression for the clustered speckles everywhere after the lens is calculated. To achieve this, two assumptions are made: scattering points Eqs at the diffuser are considered as Dirac’s delta functions and apertures ah(u0, v0) at the lens pupil mask are rectangles of sides auh and avh. With these assumptions, Eq. (3) results
being erf{·} the error function [21]. Equation (4) summarizes the 3D analytical approach sought. In Sec. V it is validated by comparing experimental results with their corresponding simulations. Attention is concentrated on clustered speckles resulting from considering only a diffuser pupil mask. We study the clustered speckle fields at the focal plane and its surroundings, and the contribution of the speckles generated by each aperture to the formation of clustered speckles. The case of only a lens pupil mask was analyzed in previous works and it is not considered here. The general case, i.e. with diffuser and lens pupil masks, can be inferred from the results presented here and from previous results [8].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]: 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 [1,3]. Moreover, according to Reed’s moment theorem [22] 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
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], 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
where Δx = x2 − x1, Δy = y2 − y1 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.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 |μA(Δx, Δy)|. This is defined as
where r = (Δx2 + Δy2)1/2 and θ = arctan(Δy/Δx).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] 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.
Figure 5 shows the radial profiles obtained from Fig. 4 by using Eq. (7). Experimental and simulated profiles are superimposed. Now, the correspondence between experimental results and simulations is apparent. It is observed that for the one aperture case, an increase in the diffuser apertures diameter does not modify the speckle size, defined by the FWHM of the correlation peak (blue dashed line). In accordance with the theory of speckle [3] their sizes should be the same if the lens pupil mask dominates the field correlation. On the other hand, for the pupils with six apertures, the main correlation peak width is drastically reduced but a secondary peak appears. As it is mentioned in the previous paragraphs, this peak indicates the appearance of clusters. This peak does not change its position by changing the aperture size (green dotted line in Fig. 5). This is in accordance with previous observations [8] where it is stated that peak position depends on the number of apertures and the diameter of the circumference where they are located. In that work is observed that by increasing the apertures number the secondary peaks tend to spread from the central peak. At the same time, third order peaks close to the main peak appear. This fact can be observed in the case of ten apertures. In particular, note that for apertures of of 0.90 mm diameter the secondary peak disappears and another peak close to the main peak appears. In Ref. [8] is also established that by increasing the aperture size the clustered speckle 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, 9]. This feature is observed in Fig. 5 in the case of ten apertures of 0.90 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).
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).
Figure 6 shows contour images of simulated and experimental 3D clustered speckles around the focal plane of the objective lens. Insets reveal the full-view of the clustered speckles at the focal plane, and the regions considered for volume analysis are highlighted. The same setup as in the previous section is employed. The case of six apertures of approximately 0.46 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]. Experimental contour plots are built by processing images with a standard 3D discrete contour plot routine. Simulations are obtained by sampling Eq. (4) with an appropriate resolution and employing the same routine. From Fig. 6 it is apparent that the heretofore developed analytical approach reproduces the main features of 3D clustered speckle fields. By employing the statistical tools described in Sec. 5, the average speckle diameter, length and transversal light fraction are calculated (see results on Fig. 6). Note that, all features are well reproduced by comparing simulations with experimental results. As expected, simulated contours are “cleaner” than the experimental ones, because different sources of noise present in the experiment are not accounted for in the simulation: edge effects from apertures on the diffuser pupil mask, spurious reflections from small misalignments or diffraction from edges due to the finite size of the optical elements, among other noise sources. Despite this, the 3D approach for clustered speckles allows for the simulation of clustered speckle fields in the neighborhood of the focal plane, thus facilitating the visualization of the volume nature and features of these type of fields.
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
The clustered speckle formation at the 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]. Unlike previous works, the cluster speckle formation at the 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.
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.
Figure 7 exhibits a comparison between experimental and simulated images taken from the image plane, where the image of the diffuser pupil mask takes place, towards the objective focal plane, where clustered speckle patterns appear. The same diffuser pupil mask of the previous section is employed. A He-Ne laser with 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 ZC 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 ZC = 4.54 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 ZC = 4.39 mm and beyond. Although these comparisons are mainly qualitative, since the corresponding |μA| can not be well calculated from the wide-view of Fig. 7, they are in good agreement with the approach here shown.
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.
Figure 8 shows a close-up comparison of clustered speckle patterns taken from experimental and simulated images. A magnified (X10) region extracted from Fig. 7 is shown. Their corresponding |μA| 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] where the cluster is formed at the 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
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], 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, 13]. 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] with random potentials.
7.2. Atoms trapping
Several years ago, the intensity profile of a standard speckles was mapped onto an atomic ensemble in a far-detuned regime [15, 16]. That experiment proved the feasibility to trap atoms with standard speckle fields. At the same time, but in a nearly resonant regime, speckle fields were used to cool atoms. As a result different longitudinal and transversal temperatures and diffusion coefficients were measured and related to the characteristic lengths of the speckle. Since that time, there were several works related to atoms trapped in disordered potentials which study Anderson localization and other issues, but in all cases standard speckles were employed [17–19]. By including clustered speckle potentials, anisotropic temperatures and diffusion coefficients could be studied. Moreover, by controlling the clustered speckles it would be possible to control the properties of the trapped atoms. In particular, the regular clusters could be employed to study the effects of its regularity on the atoms density.
7.3. Photorefractive register
Several applications of speckles in photorefractive materials have been shown [6, 25]. 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, 27]. Modulated speckle patterns where used to dislocate the relevant information away from the zero order focus [28]. Metrologycal performance was improved by using a lens pupil mask with multiple apertures [29, 30]. 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 [31,32]. 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
An analytical approach for 3D clustered speckles is proposed and successfully tested. Two different refraction index conditions, two wavelengths and different diffuser pupil mask are studied to validate the model. The analysis is performed on an statistical basis by calculating the modulus of the transversal complex coherence factor on experimental and simulated patterns. In addition, its radial profile is introduced as a general tool for these kind of fields. Results exhibit high agreement between experimental and simulated images. For apertures located at the diffuser, clustered speckle patterns appear at the focal plane. On the other hand, as analyzed in previous works, when apertures are attached to the lens, clustered speckles are obtained at the image plane. In both cases, it is observed that clusters appear when multiple aperture pupil masks are employed. The cluster formation can be interpreted by the interference of the speckle fields produced by paired apertures. Secondary spots at geometrical loci on the modulus of the complex coherence factor are the signature of cluster appearance. Their position depend on the number of apertures and the diameter of the circumference where they are arranged. This last parameter mainly determines the speckle size itself for clustered speckles. Moreover, the aperture size determines the range of the cluster. The theoretical approach developed in this paper allows the simulation and interpretation of cluster formation in several cases. It should be emphasized that this theoretical approach permits broadening the range of cases to be described that cannot be considered with previous ones. This point opens a very actual possibility to begin studies with tailored clustered speckle fields for a wide range of applications such as described in this work, among others.
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 , whereas the distance from the rear shoulder of the objective to the I-plane is . The focal length of the Edmund Optics 40x semi-plan objective is f = 4.39 mm and its working distance is . This last parameter sets the length of the objective at . These are all the quantities needed to find the principal planes.
Fig. 9 Scheme used for calculation of principal planes. , and . 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 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.
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