- Email: [email protected]
Some aspects of image processing using foams
Accepted Manuscript Some aspects of image processing using foams A. Tufaile, M.V. Freire, A.P.B. Tufaile
PII: DOI: Reference:
S0375-9601(14)00890-1 10.1016/j.physleta.2014.09.009 PLA 22806
To appear in:
Physics Letters A
Received date: 19 July 2014 Revised date: 3 September 2014 Accepted date: 6 September 2014
Please cite this article in press as: A. Tufaile et al., Some aspects of image processing using foams, Phys. Lett. A (2014), http://dx.doi.org/10.1016/j.physleta.2014.09.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
• We have obtained the light scattering in foams using experiments. • We model the light transport in foams using a chaotic dynamics and a diffusive process. • An optical filter based on foam is proposed.
Some aspects of image processing using foams
A. Tufaile*, M. V. Freire, and A. P. B. Tufaile Soft Matter Laboratory, Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, 03828-000, São Paulo, Brazil
Abstract We have explored some concepts of chaotic dynamics and wave light transport in foams. Using some experiments, we have obtained the main features of light intensity distribution through foams. We are proposing a model for this phenomenon, based on the combination of two processes: a diffusive process and another one derived from chaotic dynamics. We have presented a short outline of the chaotic dynamics involving light scattering in foams. We also have studied the existence of caustics from scattering of light from foams, with typical patterns observed in the light diffraction in transparent films. The nonlinear geometry of the foam structure was explored in order to create optical elements, such as hyperbolic prisms and filters. Keywords: foams, hyperbolic prism, chaotic scattering, diffuse reflection, optical filter.
*
Electronic Address: [email protected]
1
I. INTRODUCTION Foams have the properties of transparency and translucency. While transparent objects let light pass trough without noticeably scattering its rays, a translucent object lets some light through, but it scatters the ray so much that whatever is on the other side cannot be seen clearly. Just like transparency, translucency depends on the geometric properties of the foam. Conceding that point, if a single soap bubble is held up to a lamp, the light will pass through, but the liquid part will scatter and absorb some light. If more and more bubbles are added to make a thicker layer, then the light will eventually disappear turning the foam an opaque object [1]. Despite the fact that the different states are naturally observed in foams such as transparency, translucency and the capacity of blocking the light, the light transport in foams is challenging because it involves multiple chaotic scattering of light through hyperbolic optical elements formed by the interface of bubbles [2, 3]. Although the laws of ray reflection and refraction are simple and related to geometrical optics, the boundary conditions for the light scattering in foams are very difficult to be determined precisely, due to the concept of sensitivity to initial conditions. In addition to this, there is the possibility of the occurrence of diffraction and other phenomena for which the ray approximation is not valid, as in the case of thin film interference of a soap bubble. On the other hand, studying the complex physical system of light in foams and other systems can be rewarding, because these systems can give valuable insights to development of new device design in optics [2, 4, 5], such as hyperbolic prisms, filters and diffusers, along with the improving of the understanding of image formation through translucent objects [6]. The problem of the transport of light in foams has been studied using different techniques, experimentally and theoretically [6, 7, 8], and since a lot of
2
excellent papers and reviews from various perspectives have been published, we have decided to examine the light transport in foams using a different approach, by making some slices of foams in transparent boxes, and injecting light in these liquid foams, mimicking a naïve tomography of light profile scattered in a liquid foam. This letter is organized as follows, the next section presents the experimental apparatus. In section III, we discuss the existence of diffusive and chaotic behaviors observed in our experiment involving the scattering of light in foams. After that, we have tried to recognize the main features of the light intensity distribution inside the foam, proposing a model for this phenomenon, with a brief discussion of the chaotic scattering of light in section IV. Furthermore, we have explored some diffraction patterns obtained from the experiment in section V, and we have used some of these features of the light transport in foams to implement an optical filter based on foams. Finally, we present the conclusions of our findings in section VI.
II. EXPERIMENTAL APPARATUS This experiment involves the light scattering in three dimensional foams. We have studied the light scattering in liquid foam confined in a setup of transparent boxes [2, 3, 9]. Each box consists of two plain parallel Plexiglas plates separated by a gap (19 x 19 x 2.0 cm3). Using different ensembles of these boxes containing this liquid foam, we could inspect different slices of the light scattered inside the foam, as it is shown in Figs. 1(a)-1(e). Initially, a laser beam is injected at the center of the face of the box containing foam. The box contains air and an amount of commercial dishwashing liquid diluted in water (V = 114 cm3). The essential surfactant is Linear Alkylbenzene Sulfonate (LAS). The surface tension is 25 dyne/cm, and the density of this detergent is
ρ = 0.95 g/cm3, with refractive indices nl = 1.333 for the liquid, and ng = 1.0 for the air.
3
The foam is obtained by shaking the boxes. A camera was used to detect the resulting light patterns. Once the beam reaches the foam in Fig. 1(a), part is reflected or refracted, and part is transmitted. The scattering process makes the light spread out and limits the depth of light penetration. In Fig. 1(b), there is a typical image of the foam. We can observe two different cross sections of the light scattered in the foam, for example, taking the circle around the blob of light of Fig. 1(a), we can see the horizontal profile of light scattered in Fig. 1(c), and the vertical profile in Fig. 1(d). As foam ages, its structure can vary greatly, changing the profile of the light scattered, as shown in Fig. 1(e), because average bubble size increases, decreasing the number of bubbles, and consequently a larger amount of light penetrates beyond the initial region. We have used this property to find the best condition to make use of the foam as a filter. In order to explore some properties related to chaotic behavior observed in foams, we have used hyperbolic kaleidoscopes and hyperbolic prisms discussed in a previous paper [9]. The regular hyperbolic kaleidoscopes were constructed using three Christmas ball ornaments with the same size in a plane touching each other as curved mirrors, with a diameter of 5.0 cm. We also have made a hyperbolic prism as a way to obtain images of multiple reflections of light using a laser beam. The refractive index of the material of this prism is np = 1.32. The light source used when photographing the light scattered in foam boxes, the hyperbolic kaleidoscope and hyperbolic prisms was a diode LabLaser (Coherent) with wavelength of 635 nm. We have used a diffraction grating to generate the images in order to test foam as a filter. To analyze image profiles of light scattered and to simulate the light scattering in foams, we have used some routines written in commercial software (Origin).
4
III. SOME LIGHT SCATTERING PROFILES We have obtained a color map of the laser beam scattered by the foam, with a top view in Fig. 2(a) and the amplitude plot of the same color map in Fig. 2(b), in which we can see the existence of rings or halos created by the scattering. At the center of Fig. 2(a) we can observe an increase of amplitude values of light intensity due to direct incidence of the laser at this point. The colors from yellow to red represent the high intensity of light in comparison with the low intensity at the border of the halo, represented by the region of ligth intensity between the colors purple and light blue. In Fig. 2(c) we depict the cross section through the center of Fig. 2(b), paralel to the x axis. We can see two distinct regions separated by the arrows: a smooth one related to a diffusive process [7, 8] at the border, and an irregular one at the center, related to a chaotic process [2]. The difusive region is associated with a gaussian process (green curve), and the irregular region with a chaotic dynamics (red curve). We have conjectured that the diffusive behavior of light is related to wave optics while the chaotic behavior is related to the realm of geometric optics. In this way, we converted those two behaviors into a mathematical model. In this model we are proposing two main process of the transport of light in foams using the function g(x,y): a diffusive one related to Gaussian function, and the function f(x,y) representing the chaotic process, as it is shown in Fig. 2(d):
g ( x, y ) =
1 2πσ
2
e
−
x2 + y2 2σ 2
+ f ( x, y ) .
(1)
In Eq. (1), the diffusive process is present in all places, while the effects of chaotic dynamics are present mainly at the center, close to the point of entrance of the
5
light ray in the foam. The construction of a map for f(x,y) based on a deterministic
system will be discussed in the next section.
IV. CHAOTIC SCATTERING OF LIGHT
The aim of this section is to give a short outline of the chaotic dynamics involving light in foams. The chaotic process in our model can be generated using some maps based on recurrence equations [9]. The high curvature shapes of the liquid bridges of Fig. 3(a) can cause a change of the direction of the incident light ray source, scattering the trajectory of rays in many different directions through reflection, in addition to refraction. Considering the case of pure reflection, we can understand the existence of the chaotic dynamics based on the concept of optical billiards. The multiple reflections in the optical billiard are related to some properties of reflective spheres [11, 12], such as the fractal pattern of a Sierpinski gasket of Fig. 3(b) for a hyperbolic kaleidoscope formed by the scattering of light of four reflective spheres [10, 12, 13]. Even for the case of three identical reflective spheres, the scattering of light in this cavity is chaotic [3], considering the light trajectories as point particles specularly reflecting from circular scatterers of radius R arranged on a triangular lattice. The cavity acts as a hyperbolic kaleidoscope, forming a fractal pattern known as curvaceous Sierpinski triangles [3, 11]. For different values of the geometry of this system and initial conditions, we have obtained the existence of positive values of Lyapunov exponents [13] in the plot of Fig. 3(c), a way to characterize and quantify chaotic phenomena which describe the temporal evolution of small perturbations of the initial conditions. Based on this concept, the ray transfer matrix analysis can be used to generate the different mechanisms of light transport of geometrical optics [14, 15]:
6
§ x2 · § A B ·§ x1 · ¨¨ ¸¸ = ¨¨ ¸¸¨¨ ¸¸ , ©θ 2 ¹ © C D ¹©θ 2 ¹
(2)
where the input beam is represented by a vector with the components x1 and θ1, x2 and θ2 refer to the light ray after passing through the optical component, A = x2/x1 for θ1 = 0, B = x2/θ1 for x1 = 0, C = θ2/x1 for θ1 = 0 and D = θ2/θ1. For example, for the case of the reflection from a curved mirror, we have A = 1, B = 0, C = -2/R, and D = 1. Following the model of transport described by Vollmer [16] based on the Lorentz gas model, we have to consider the direction θ of the trajectory, the momentum p, R is the radius of the scatterer, and the angular momentum I of the particle, which is related to its position. The Poincaré map suggested is:
§ I − pl ( sn ) sin(θ n − π sn ) · 6 ¸ § I n+1 · ¨ n ¨¨ ¸¸ = ¨ § I n+1 · ¸ , ¸¸ ¸¸ ©θ n+1 ¹ ¨¨ θ n + π + 2 arcsin¨¨ © pR ¹ ¹ ©
(3)
in which the scatterer hit in collision n+1 is sn = 0, …, 11, and l(sn) is the distance of the nth to the n+1st scatterer and the impact parameter corresponds to the modulus of I/p. This map presents a similar behavior to the phenomenon of a chaotic saddle [13], in which a particle bounces back and forth for a certain amount of time in the scattering region and leaves it through one of the several exits. This map can be modified to generate the sequence of collisions to reproduce the function f(x,y). For example, the Galton board dynamics [9] is a deterministic system in which collisions of a particle with scatterers can be modeled by the two dimensional map: x + 1, y + 1 ° x − (m − 1) / l + ( m − 1 − Δ ), y − ( n − 1) / l + ( n − 1 − Δ ) ° f ( x, y ) ≡ ® °( x − m) / r + (m + Δ ), ( y − n) / r + ( n + Δ ) °¯ x − 1, y − 1
x < 0, y < 0, m − 1 < x < (m − r ), n − 1 < y < (n − r ), (m − r ) < x < m, (n − r ) < x < n, ( M + 1) < x, ( N + 1) < y
(4)
7
where m = 0,…, M+1, n = 0,…, N+1, Δ = 1, M = N = 25, and l + r = 1, with l = ½ for unbiased collisions. We can obtain the time series from this map presenting a power spectra decay psα with α ≈ - 2. This is the case of the traditional random walk scenario, a behavior analogous to the one observed in optical billiards of Eq. (3). In light-ray propagation in foams with reflection and refraction at same time, we have seen some of those features described before. In Fig. 4(a)-(d) we can see some trajectories of light in a hyperbolic prism [2], the equivalent refractive system inspired in foams, exhibiting some of the same properties of the chaotic system of the hyperbolic kaleidoscope [3]. In this context, the chaotic scattering is a natural consequence of multiple scattering of light rays, reflecting light diffusively with high efficiency.
V. DIFFRACTION PATTERNS The ways in which the light waves are redirected in foams fall into three categories: refraction, reflection, and diffraction, analogous to the problem of quantum chaos [17]. We have observed these three ways of transmission of light through Plateau borders in foams at same time. In order to show the myriad of phenomena involved, we present some experimental results obtained. For example, what does a hexagonal image of Fig. 5(a) when light travels through the triangular Plateau border consisting of the three circular arcs of Fig. 3(a)? The propagation of light from the Plateau border to the image plane can be modeled by the wave optics approximation [14]:
U ( x, y ) ∝ ³³ u (ξ ,η )e
−
2πi ( xξ + yη ) λz
dξdη ,
(5)
where U(x,y) is the optical field in the image plane, ξ and η are the coordinates in the Plateau border plane, u(ξ,η) is the optical field in the Plateau border. Under substitutions fx= x/λz and fy= y/λz the image observed in Fig. 5(b) is a Fourier transform. In this way, the sharp edges in the input to the Fourier transform are
8
represented by bright streaks in the output, with orientation of streak being perpendicular to the edge which produced it, so the Plateau border has three sharp edges, and therefore the resulting point spread function produces a six pointed star. We also have observed the formation of caustics from Plateau borders. A caustic is a kind of envelop of light rays obtained by the combination of reflection, refraction, and diffraction. In Fig. 5(c) there is a diagram of light rays exemplifying this phenomenon, and in Fig. 5(d) we have obtained a caustic from the Plateau border. Some authors have studied some diffraction patterns similar to the ones obtained from foams of Fig. 5(e) in the context of Catastrophe theory [18]. We have obtained a phenomenological simulation of this pattern using ray tracing method along with image processing techniques in Fig. 5(f). This triangular pattern of parallelograms containing bright regions is also known as coma in classical aberration theory, and represents a family of light rays filling regions of space produced by transparent thin films.
Using the combination of these chaotic and diffusive properties, we have developed an optical filter. This filter modifies the input signal by a convolution with the function g(x,y) of Eq. (1). In Fig. 6, we are presenting the effect of filtering process using the foam. First, without the filter, the input image is shown in Fig. 6(a), and second, using our filter based on foam, the output image is presented in Fig. 6(b). The input image was obtained from a diffraction grating creating a concentric pattern of dots with a speckle noise. Comparing the pictures of Fig. 6, we can see that the level of speckle is reduced. The presence of streaks forming radial lines at the center of Fig. 6(b) is due to the occurrence of diffraction from Plateau borders, reinforcing bright dots, putting in evidence the fifth ring of dots around the center. Taking a look at the corners of both figures, the level of grey is higher after the filtering, emphasizing the Gaussian process involved in the scattering of light, and consequently blurring the speckle. To
9
sum up, in addition to diffuse reflection, this filter exhibits a mixture of chaotic behavior and diffuse refraction.
VI. CONCLUSIONS We have discussed in this letter some aspects of the scattering of light in foams confined in a transparent box. We have explored some concepts of chaotic dynamics and wave light transport in foams. We have used some optical elements, such as the hyperbolic kaleidoscope and the hyperbolic prism, to understand the dynamics behind the transport of light in foams. We are proposing the combination of two main processes of the light transport in foams: a diffusive one related to Gaussian function, and another one related to a chaotic dynamics, similar to those observed in a chaotic saddle, in which a ray of light bounces back and forth for a certain time in the scattering region, and leaves it through one of the several exits. Another topic addressed in our study was the presence of caustics obtained from the light scattered in foams, with typical patterns observed in diffraction of light by transparent thin films. Finally, we have applied these concepts in order to obtain an optical filter. This filter exhibits a mixture of chaotic behavior, diffuse reflection, and diffuse refraction.
ACKNOWLEDGMENTS This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx) and FAPESP.
References
10
[1]
D. S. Wiersma, Nature Photon. 7 (2013) 188-196.
[2]
A. Tufaile, A. P. B. Tufaile, Phys. Lett. A 375 (2011) 3693-3698.
[3]
A. P. B. Tufaile, A. Tufaile and G. Liger-Belair, J. Phys. Conf. Ser. 285 (2011) 012006.
[4]
V. G. Baryakhtar, V. V. Yanovsky, S. V. Naydenov, A. V. Kurilo, J. Exp. Theor. Phys. 103 (2006) 293-302.
[5]
C. Penfold, R. T. Collins, A. P. B. Tufaile, Y. Souche, J. Magnetism Magnetic Mat. 242-245, (2002) 964-966.
[6]
A. van der Net, L. Blondel, A. Saugey, and W. Drenckhan , Colloids Surf. A 309 (2007) 159.
[7]
M. Miri, E. Madadi, H. Stark, Phys. Rev. E 72 (2005) 031111.
[8]
S. McMurry, D. Weaire, J. Lunney, S. Hutzler, Op. Eng. 33 (1995) 3849-3852.
[9]
A. Tufaile and A. P. B. Tufaile, Phys. Lett. A 372 (2008) 6381-6385.
[10] D. Sweet, E. Ott, and J. A. Yorke , Nature 399 (1999) 315-316. [11] D. Sweet, B. W. Zeff, E. Ott, D. P. Lathrop, Physica D 154 (2001) 207-218. [12] J. Aguirre, R. L. Viana, M. A. F. Sanjuán, Rev. Mod. Phys. 81 (2009) 333-386. [13] L. Poon, J. Campos, E. Ott, C. Grebogi, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6 (1996) 251-265. [14] E. Hecht, Optics, Wesley, New York, 1997. [15] ABCD Matrices Tutorial: http://www.photonics.byu.edu/ABCD_Matrix_tut.phtml [16] J. Vollmer, Phys. Rep. 372 (2001) 131. [17] K. Schaadt, A. P. B. Tufaile, C. Ellegaard, Phys. Review E 67 (2003) 26213. [18] M. V. Berry, J. Phys. A (1975) 566-584.
Figure Captions
11
FIG. 1. (a) Diagram of the laser beam scattered by the foam inside a transparent box. The scattering process spreads the light and limits the depth of light penetration, creating a center glow located just above the interface of the box and the foam. (b) Image of the bubbles inside the foam, the bar size is 5.0 mm. (c) Top view of the light scattered by the foam in the region of the circle around the glowing point in (a). (d) Profile of the light scattered in the vertical position, with the laser beam entering at the bottom side. (e) As foams age, the bubbles coarsen, decreasing the number of bubbles, and consequently an amount of light penetrates beyond the initial region, increasing the size of the light profile.
FIG. 2. (a) Color map of the top view of the light scattering in the foam in the region of the circle around the center glow, shown in Fig. 1(a). (b) Amplitude plot of the light intensity of the previous color map (arbitrary units). In (c) there is a section of the color map for the coordinate y = 200. There are two distinct regions separated by the arrows: a smooth one and an irregular one at the center. Plot of our model of Eq. (1) in (d), with the smooth region related to a diffusive process, and the irregular points at the center, related to a chaotic dynamics.
FIG. 3. The chaotic dynamics approach: (a) the Plateau borders in foam are curved triangular sections where liquid films meet, or the interface of three bubbles. This type of geometry resembles a hyperbolic kaleidoscope in (b), in which the reflection of light presents a fractal structure. Using ray tracing techniques for three discs in a plane, we can observe the chaotic dynamics. In (c) we have obtained positive values for the Lyapunov exponents of this system, for different disc sizes and using the scattering
12
functions of the angle θ and the impact parameter. Each ray will enter the scattering region, stay there for certain time and leave toward infinite. These values of the Lyapunov exponents characterize the existence of chaotic behavior. The dashed line is a guide to the eye.
FIG. 4. The hyperbolic prism is an optical element inspired in the Plateau border, in (a) we can see the laser beam refracting inside the hyperbolic prism, while in (b) the laser beam is reflected for a different angle of incidence. In (c) there are multiple internal reflections, and in (d) the multiple reflections create a self-similar pattern.
FIG. 5. In order to explore some properties of the diffraction of light in foams, we have obtained in (a) the far field diffraction of light through a Plateau border, while in (b) a simulation of this phenomenon using Eq. (4). The diagram of an astroid base of the caustic (c) of diffracted rays in a Plateau border profile, and in (d) a caustic obtained from the experiment using the foam. A more detailed diffraction pattern is shown in (e). We have obtained a simulation of this pattern using ray tracing method in (f). FIG. 6. The input image is shown in (a) obtained from a diffraction grating creating a circular pattern of dot with a speckle noise. Using the foam as a filter, the resulting image can be viewed as a smoothed version of the original image in (b) and the speckle is filtered.
13
(b)
(a)
(c)
(d)
(e)
PII: DOI: Reference:
S0375-9601(14)00890-1 10.1016/j.physleta.2014.09.009 PLA 22806
To appear in:
Physics Letters A
Received date: 19 July 2014 Revised date: 3 September 2014 Accepted date: 6 September 2014
Please cite this article in press as: A. Tufaile et al., Some aspects of image processing using foams, Phys. Lett. A (2014), http://dx.doi.org/10.1016/j.physleta.2014.09.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
• We have obtained the light scattering in foams using experiments. • We model the light transport in foams using a chaotic dynamics and a diffusive process. • An optical filter based on foam is proposed.
Some aspects of image processing using foams
A. Tufaile*, M. V. Freire, and A. P. B. Tufaile Soft Matter Laboratory, Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, 03828-000, São Paulo, Brazil
Abstract We have explored some concepts of chaotic dynamics and wave light transport in foams. Using some experiments, we have obtained the main features of light intensity distribution through foams. We are proposing a model for this phenomenon, based on the combination of two processes: a diffusive process and another one derived from chaotic dynamics. We have presented a short outline of the chaotic dynamics involving light scattering in foams. We also have studied the existence of caustics from scattering of light from foams, with typical patterns observed in the light diffraction in transparent films. The nonlinear geometry of the foam structure was explored in order to create optical elements, such as hyperbolic prisms and filters. Keywords: foams, hyperbolic prism, chaotic scattering, diffuse reflection, optical filter.
*
Electronic Address: [email protected]
1
I. INTRODUCTION Foams have the properties of transparency and translucency. While transparent objects let light pass trough without noticeably scattering its rays, a translucent object lets some light through, but it scatters the ray so much that whatever is on the other side cannot be seen clearly. Just like transparency, translucency depends on the geometric properties of the foam. Conceding that point, if a single soap bubble is held up to a lamp, the light will pass through, but the liquid part will scatter and absorb some light. If more and more bubbles are added to make a thicker layer, then the light will eventually disappear turning the foam an opaque object [1]. Despite the fact that the different states are naturally observed in foams such as transparency, translucency and the capacity of blocking the light, the light transport in foams is challenging because it involves multiple chaotic scattering of light through hyperbolic optical elements formed by the interface of bubbles [2, 3]. Although the laws of ray reflection and refraction are simple and related to geometrical optics, the boundary conditions for the light scattering in foams are very difficult to be determined precisely, due to the concept of sensitivity to initial conditions. In addition to this, there is the possibility of the occurrence of diffraction and other phenomena for which the ray approximation is not valid, as in the case of thin film interference of a soap bubble. On the other hand, studying the complex physical system of light in foams and other systems can be rewarding, because these systems can give valuable insights to development of new device design in optics [2, 4, 5], such as hyperbolic prisms, filters and diffusers, along with the improving of the understanding of image formation through translucent objects [6]. The problem of the transport of light in foams has been studied using different techniques, experimentally and theoretically [6, 7, 8], and since a lot of
2
excellent papers and reviews from various perspectives have been published, we have decided to examine the light transport in foams using a different approach, by making some slices of foams in transparent boxes, and injecting light in these liquid foams, mimicking a naïve tomography of light profile scattered in a liquid foam. This letter is organized as follows, the next section presents the experimental apparatus. In section III, we discuss the existence of diffusive and chaotic behaviors observed in our experiment involving the scattering of light in foams. After that, we have tried to recognize the main features of the light intensity distribution inside the foam, proposing a model for this phenomenon, with a brief discussion of the chaotic scattering of light in section IV. Furthermore, we have explored some diffraction patterns obtained from the experiment in section V, and we have used some of these features of the light transport in foams to implement an optical filter based on foams. Finally, we present the conclusions of our findings in section VI.
II. EXPERIMENTAL APPARATUS This experiment involves the light scattering in three dimensional foams. We have studied the light scattering in liquid foam confined in a setup of transparent boxes [2, 3, 9]. Each box consists of two plain parallel Plexiglas plates separated by a gap (19 x 19 x 2.0 cm3). Using different ensembles of these boxes containing this liquid foam, we could inspect different slices of the light scattered inside the foam, as it is shown in Figs. 1(a)-1(e). Initially, a laser beam is injected at the center of the face of the box containing foam. The box contains air and an amount of commercial dishwashing liquid diluted in water (V = 114 cm3). The essential surfactant is Linear Alkylbenzene Sulfonate (LAS). The surface tension is 25 dyne/cm, and the density of this detergent is
ρ = 0.95 g/cm3, with refractive indices nl = 1.333 for the liquid, and ng = 1.0 for the air.
3
The foam is obtained by shaking the boxes. A camera was used to detect the resulting light patterns. Once the beam reaches the foam in Fig. 1(a), part is reflected or refracted, and part is transmitted. The scattering process makes the light spread out and limits the depth of light penetration. In Fig. 1(b), there is a typical image of the foam. We can observe two different cross sections of the light scattered in the foam, for example, taking the circle around the blob of light of Fig. 1(a), we can see the horizontal profile of light scattered in Fig. 1(c), and the vertical profile in Fig. 1(d). As foam ages, its structure can vary greatly, changing the profile of the light scattered, as shown in Fig. 1(e), because average bubble size increases, decreasing the number of bubbles, and consequently a larger amount of light penetrates beyond the initial region. We have used this property to find the best condition to make use of the foam as a filter. In order to explore some properties related to chaotic behavior observed in foams, we have used hyperbolic kaleidoscopes and hyperbolic prisms discussed in a previous paper [9]. The regular hyperbolic kaleidoscopes were constructed using three Christmas ball ornaments with the same size in a plane touching each other as curved mirrors, with a diameter of 5.0 cm. We also have made a hyperbolic prism as a way to obtain images of multiple reflections of light using a laser beam. The refractive index of the material of this prism is np = 1.32. The light source used when photographing the light scattered in foam boxes, the hyperbolic kaleidoscope and hyperbolic prisms was a diode LabLaser (Coherent) with wavelength of 635 nm. We have used a diffraction grating to generate the images in order to test foam as a filter. To analyze image profiles of light scattered and to simulate the light scattering in foams, we have used some routines written in commercial software (Origin).
4
III. SOME LIGHT SCATTERING PROFILES We have obtained a color map of the laser beam scattered by the foam, with a top view in Fig. 2(a) and the amplitude plot of the same color map in Fig. 2(b), in which we can see the existence of rings or halos created by the scattering. At the center of Fig. 2(a) we can observe an increase of amplitude values of light intensity due to direct incidence of the laser at this point. The colors from yellow to red represent the high intensity of light in comparison with the low intensity at the border of the halo, represented by the region of ligth intensity between the colors purple and light blue. In Fig. 2(c) we depict the cross section through the center of Fig. 2(b), paralel to the x axis. We can see two distinct regions separated by the arrows: a smooth one related to a diffusive process [7, 8] at the border, and an irregular one at the center, related to a chaotic process [2]. The difusive region is associated with a gaussian process (green curve), and the irregular region with a chaotic dynamics (red curve). We have conjectured that the diffusive behavior of light is related to wave optics while the chaotic behavior is related to the realm of geometric optics. In this way, we converted those two behaviors into a mathematical model. In this model we are proposing two main process of the transport of light in foams using the function g(x,y): a diffusive one related to Gaussian function, and the function f(x,y) representing the chaotic process, as it is shown in Fig. 2(d):
g ( x, y ) =
1 2πσ
2
e
−
x2 + y2 2σ 2
+ f ( x, y ) .
(1)
In Eq. (1), the diffusive process is present in all places, while the effects of chaotic dynamics are present mainly at the center, close to the point of entrance of the
5
light ray in the foam. The construction of a map for f(x,y) based on a deterministic
system will be discussed in the next section.
IV. CHAOTIC SCATTERING OF LIGHT
The aim of this section is to give a short outline of the chaotic dynamics involving light in foams. The chaotic process in our model can be generated using some maps based on recurrence equations [9]. The high curvature shapes of the liquid bridges of Fig. 3(a) can cause a change of the direction of the incident light ray source, scattering the trajectory of rays in many different directions through reflection, in addition to refraction. Considering the case of pure reflection, we can understand the existence of the chaotic dynamics based on the concept of optical billiards. The multiple reflections in the optical billiard are related to some properties of reflective spheres [11, 12], such as the fractal pattern of a Sierpinski gasket of Fig. 3(b) for a hyperbolic kaleidoscope formed by the scattering of light of four reflective spheres [10, 12, 13]. Even for the case of three identical reflective spheres, the scattering of light in this cavity is chaotic [3], considering the light trajectories as point particles specularly reflecting from circular scatterers of radius R arranged on a triangular lattice. The cavity acts as a hyperbolic kaleidoscope, forming a fractal pattern known as curvaceous Sierpinski triangles [3, 11]. For different values of the geometry of this system and initial conditions, we have obtained the existence of positive values of Lyapunov exponents [13] in the plot of Fig. 3(c), a way to characterize and quantify chaotic phenomena which describe the temporal evolution of small perturbations of the initial conditions. Based on this concept, the ray transfer matrix analysis can be used to generate the different mechanisms of light transport of geometrical optics [14, 15]:
6
§ x2 · § A B ·§ x1 · ¨¨ ¸¸ = ¨¨ ¸¸¨¨ ¸¸ , ©θ 2 ¹ © C D ¹©θ 2 ¹
(2)
where the input beam is represented by a vector with the components x1 and θ1, x2 and θ2 refer to the light ray after passing through the optical component, A = x2/x1 for θ1 = 0, B = x2/θ1 for x1 = 0, C = θ2/x1 for θ1 = 0 and D = θ2/θ1. For example, for the case of the reflection from a curved mirror, we have A = 1, B = 0, C = -2/R, and D = 1. Following the model of transport described by Vollmer [16] based on the Lorentz gas model, we have to consider the direction θ of the trajectory, the momentum p, R is the radius of the scatterer, and the angular momentum I of the particle, which is related to its position. The Poincaré map suggested is:
§ I − pl ( sn ) sin(θ n − π sn ) · 6 ¸ § I n+1 · ¨ n ¨¨ ¸¸ = ¨ § I n+1 · ¸ , ¸¸ ¸¸ ©θ n+1 ¹ ¨¨ θ n + π + 2 arcsin¨¨ © pR ¹ ¹ ©
(3)
in which the scatterer hit in collision n+1 is sn = 0, …, 11, and l(sn) is the distance of the nth to the n+1st scatterer and the impact parameter corresponds to the modulus of I/p. This map presents a similar behavior to the phenomenon of a chaotic saddle [13], in which a particle bounces back and forth for a certain amount of time in the scattering region and leaves it through one of the several exits. This map can be modified to generate the sequence of collisions to reproduce the function f(x,y). For example, the Galton board dynamics [9] is a deterministic system in which collisions of a particle with scatterers can be modeled by the two dimensional map: x + 1, y + 1 ° x − (m − 1) / l + ( m − 1 − Δ ), y − ( n − 1) / l + ( n − 1 − Δ ) ° f ( x, y ) ≡ ® °( x − m) / r + (m + Δ ), ( y − n) / r + ( n + Δ ) °¯ x − 1, y − 1
x < 0, y < 0, m − 1 < x < (m − r ), n − 1 < y < (n − r ), (m − r ) < x < m, (n − r ) < x < n, ( M + 1) < x, ( N + 1) < y
(4)
7
where m = 0,…, M+1, n = 0,…, N+1, Δ = 1, M = N = 25, and l + r = 1, with l = ½ for unbiased collisions. We can obtain the time series from this map presenting a power spectra decay psα with α ≈ - 2. This is the case of the traditional random walk scenario, a behavior analogous to the one observed in optical billiards of Eq. (3). In light-ray propagation in foams with reflection and refraction at same time, we have seen some of those features described before. In Fig. 4(a)-(d) we can see some trajectories of light in a hyperbolic prism [2], the equivalent refractive system inspired in foams, exhibiting some of the same properties of the chaotic system of the hyperbolic kaleidoscope [3]. In this context, the chaotic scattering is a natural consequence of multiple scattering of light rays, reflecting light diffusively with high efficiency.
V. DIFFRACTION PATTERNS The ways in which the light waves are redirected in foams fall into three categories: refraction, reflection, and diffraction, analogous to the problem of quantum chaos [17]. We have observed these three ways of transmission of light through Plateau borders in foams at same time. In order to show the myriad of phenomena involved, we present some experimental results obtained. For example, what does a hexagonal image of Fig. 5(a) when light travels through the triangular Plateau border consisting of the three circular arcs of Fig. 3(a)? The propagation of light from the Plateau border to the image plane can be modeled by the wave optics approximation [14]:
U ( x, y ) ∝ ³³ u (ξ ,η )e
−
2πi ( xξ + yη ) λz
dξdη ,
(5)
where U(x,y) is the optical field in the image plane, ξ and η are the coordinates in the Plateau border plane, u(ξ,η) is the optical field in the Plateau border. Under substitutions fx= x/λz and fy= y/λz the image observed in Fig. 5(b) is a Fourier transform. In this way, the sharp edges in the input to the Fourier transform are
8
represented by bright streaks in the output, with orientation of streak being perpendicular to the edge which produced it, so the Plateau border has three sharp edges, and therefore the resulting point spread function produces a six pointed star. We also have observed the formation of caustics from Plateau borders. A caustic is a kind of envelop of light rays obtained by the combination of reflection, refraction, and diffraction. In Fig. 5(c) there is a diagram of light rays exemplifying this phenomenon, and in Fig. 5(d) we have obtained a caustic from the Plateau border. Some authors have studied some diffraction patterns similar to the ones obtained from foams of Fig. 5(e) in the context of Catastrophe theory [18]. We have obtained a phenomenological simulation of this pattern using ray tracing method along with image processing techniques in Fig. 5(f). This triangular pattern of parallelograms containing bright regions is also known as coma in classical aberration theory, and represents a family of light rays filling regions of space produced by transparent thin films.
Using the combination of these chaotic and diffusive properties, we have developed an optical filter. This filter modifies the input signal by a convolution with the function g(x,y) of Eq. (1). In Fig. 6, we are presenting the effect of filtering process using the foam. First, without the filter, the input image is shown in Fig. 6(a), and second, using our filter based on foam, the output image is presented in Fig. 6(b). The input image was obtained from a diffraction grating creating a concentric pattern of dots with a speckle noise. Comparing the pictures of Fig. 6, we can see that the level of speckle is reduced. The presence of streaks forming radial lines at the center of Fig. 6(b) is due to the occurrence of diffraction from Plateau borders, reinforcing bright dots, putting in evidence the fifth ring of dots around the center. Taking a look at the corners of both figures, the level of grey is higher after the filtering, emphasizing the Gaussian process involved in the scattering of light, and consequently blurring the speckle. To
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sum up, in addition to diffuse reflection, this filter exhibits a mixture of chaotic behavior and diffuse refraction.
VI. CONCLUSIONS We have discussed in this letter some aspects of the scattering of light in foams confined in a transparent box. We have explored some concepts of chaotic dynamics and wave light transport in foams. We have used some optical elements, such as the hyperbolic kaleidoscope and the hyperbolic prism, to understand the dynamics behind the transport of light in foams. We are proposing the combination of two main processes of the light transport in foams: a diffusive one related to Gaussian function, and another one related to a chaotic dynamics, similar to those observed in a chaotic saddle, in which a ray of light bounces back and forth for a certain time in the scattering region, and leaves it through one of the several exits. Another topic addressed in our study was the presence of caustics obtained from the light scattered in foams, with typical patterns observed in diffraction of light by transparent thin films. Finally, we have applied these concepts in order to obtain an optical filter. This filter exhibits a mixture of chaotic behavior, diffuse reflection, and diffuse refraction.
ACKNOWLEDGMENTS This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Instituto Nacional de Ciência e Tecnologia de Fluidos Complexos (INCT-FCx) and FAPESP.
References
10
[1]
D. S. Wiersma, Nature Photon. 7 (2013) 188-196.
[2]
A. Tufaile, A. P. B. Tufaile, Phys. Lett. A 375 (2011) 3693-3698.
[3]
A. P. B. Tufaile, A. Tufaile and G. Liger-Belair, J. Phys. Conf. Ser. 285 (2011) 012006.
[4]
V. G. Baryakhtar, V. V. Yanovsky, S. V. Naydenov, A. V. Kurilo, J. Exp. Theor. Phys. 103 (2006) 293-302.
[5]
C. Penfold, R. T. Collins, A. P. B. Tufaile, Y. Souche, J. Magnetism Magnetic Mat. 242-245, (2002) 964-966.
[6]
A. van der Net, L. Blondel, A. Saugey, and W. Drenckhan , Colloids Surf. A 309 (2007) 159.
[7]
M. Miri, E. Madadi, H. Stark, Phys. Rev. E 72 (2005) 031111.
[8]
S. McMurry, D. Weaire, J. Lunney, S. Hutzler, Op. Eng. 33 (1995) 3849-3852.
[9]
A. Tufaile and A. P. B. Tufaile, Phys. Lett. A 372 (2008) 6381-6385.
[10] D. Sweet, E. Ott, and J. A. Yorke , Nature 399 (1999) 315-316. [11] D. Sweet, B. W. Zeff, E. Ott, D. P. Lathrop, Physica D 154 (2001) 207-218. [12] J. Aguirre, R. L. Viana, M. A. F. Sanjuán, Rev. Mod. Phys. 81 (2009) 333-386. [13] L. Poon, J. Campos, E. Ott, C. Grebogi, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6 (1996) 251-265. [14] E. Hecht, Optics, Wesley, New York, 1997. [15] ABCD Matrices Tutorial: http://www.photonics.byu.edu/ABCD_Matrix_tut.phtml [16] J. Vollmer, Phys. Rep. 372 (2001) 131. [17] K. Schaadt, A. P. B. Tufaile, C. Ellegaard, Phys. Review E 67 (2003) 26213. [18] M. V. Berry, J. Phys. A (1975) 566-584.
Figure Captions
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FIG. 1. (a) Diagram of the laser beam scattered by the foam inside a transparent box. The scattering process spreads the light and limits the depth of light penetration, creating a center glow located just above the interface of the box and the foam. (b) Image of the bubbles inside the foam, the bar size is 5.0 mm. (c) Top view of the light scattered by the foam in the region of the circle around the glowing point in (a). (d) Profile of the light scattered in the vertical position, with the laser beam entering at the bottom side. (e) As foams age, the bubbles coarsen, decreasing the number of bubbles, and consequently an amount of light penetrates beyond the initial region, increasing the size of the light profile.
FIG. 2. (a) Color map of the top view of the light scattering in the foam in the region of the circle around the center glow, shown in Fig. 1(a). (b) Amplitude plot of the light intensity of the previous color map (arbitrary units). In (c) there is a section of the color map for the coordinate y = 200. There are two distinct regions separated by the arrows: a smooth one and an irregular one at the center. Plot of our model of Eq. (1) in (d), with the smooth region related to a diffusive process, and the irregular points at the center, related to a chaotic dynamics.
FIG. 3. The chaotic dynamics approach: (a) the Plateau borders in foam are curved triangular sections where liquid films meet, or the interface of three bubbles. This type of geometry resembles a hyperbolic kaleidoscope in (b), in which the reflection of light presents a fractal structure. Using ray tracing techniques for three discs in a plane, we can observe the chaotic dynamics. In (c) we have obtained positive values for the Lyapunov exponents of this system, for different disc sizes and using the scattering
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functions of the angle θ and the impact parameter. Each ray will enter the scattering region, stay there for certain time and leave toward infinite. These values of the Lyapunov exponents characterize the existence of chaotic behavior. The dashed line is a guide to the eye.
FIG. 4. The hyperbolic prism is an optical element inspired in the Plateau border, in (a) we can see the laser beam refracting inside the hyperbolic prism, while in (b) the laser beam is reflected for a different angle of incidence. In (c) there are multiple internal reflections, and in (d) the multiple reflections create a self-similar pattern.
FIG. 5. In order to explore some properties of the diffraction of light in foams, we have obtained in (a) the far field diffraction of light through a Plateau border, while in (b) a simulation of this phenomenon using Eq. (4). The diagram of an astroid base of the caustic (c) of diffracted rays in a Plateau border profile, and in (d) a caustic obtained from the experiment using the foam. A more detailed diffraction pattern is shown in (e). We have obtained a simulation of this pattern using ray tracing method in (f). FIG. 6. The input image is shown in (a) obtained from a diffraction grating creating a circular pattern of dot with a speckle noise. Using the foam as a filter, the resulting image can be viewed as a smoothed version of the original image in (b) and the speckle is filtered.
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(b)
(a)
(c)
(d)
(e)