The holographic research is related to the coded storage and the ulterior decoded of optical information . The holographic processes involved had extended its application to almost all the optical areas; such that nowadays is not possible to conceive the development of the modern optics without this powerful tool [2-5]. The development of this area implies the generation of optical fields with suitable structure also as the research and synthesis of novel holographic materials with large refractive index values and sensible to a wide frequencies range [6-8].
In the present study, we are interested in to generate holographic rough surfaces, such that the scattered field can be amplitude self-correlated. Until our best knowledge, almost all works related to the speckle pattern are irradiance correlated. However, in order to understand the physical features of the speckle pattern is necessary to obtain the amplitude correlation function, which must be manifested in the interference features between speckle motes. This is possible because for the same illumination configurations, the roughness parameters may obey two probability density functions. Other important effect occurs during the recording process, making possible to generate regions with cusped geometry. The behavior of the electric field in the neighborhood of these cusped regions generates evanescent waves [9,10], and it is used to generate surface plasmon fields. This occurs when a metal thin film is deposited on the holographic rough surface . The surface plasmon fields appears when the power spectrum associated to the cusped region, is matched with the dispersion relation function of the surface plasmon. In the context of the angular spectrum model the cusped regions correspond to the Gibbs phenomenon .
The holographic rough surface proposed is generated by recording a set of optical fields kind cosine, the register media consists in a photo-resist film deposited on a glass substrate, using the periods of the optical field as the control parameter. This construction allows us to design the power spectrum of the holographic surface. Consequently, during the reconstruction process, the scattered field consists in a well localized speckle band.
The holographic rough surface thus generated is implemented as a beam splitter in an amplitude-correlation interferometer, which allows the interference between two speckle bands. The main feature is that the two speckle patterns coming from the same holographic rough surface obey different probability density functions; however a certain amplitude correlation function between the speckle patterns is preserved.
From the experimental results for the interference, very interesting features can be identified, one of them consists in the generation of bifurcation effects kind pitchfork [13,14]. The physical origin of this effect is explained from the boundary condition to the electromagnetic field. Since the theoretical point of view, the bifurcation effects allow to determinate the interaction between the irradiance distributions and can be interpreted as amplitude four order correlations. The geometry of the bifurcation suggests as application to be used as a kind of speckle tweezers. Another possible application consists in the alignment of nanoparticles and nanotubes inducing resonance effects close related to tunable spectroscopy [15-17]. Since the theoretical point of view, behavior such as Anderson localization can be implemented .
The structure of this chapter is as follows. In section
2. Holographic generation of rough surfaces
The holographic rough surfaces are generated by means of a superposition of cosenoidal patterns resulting of the interference between two plane waves, whose amplitude distribution is given by
The irradiance distribution takes the form
By changing the incidence angle we generate a set of cosine patterns
These irradiance distributions are recorded on the holographic material. The experimental details are as follows. The holographic plate was made by depositing a photo-resist film using a spin-coating technique on a glass substrate. The number of irradiance distributions recorded were
Assuming a lineal response of the holographic material, the mathematical representation to the resultant profile is
By considering that
The transmittance function associated to the holographic surface is obtained by normalizing the height distribution in Eq. (4), and it is given by
The statistical parameters of the surface are the following, the mean value of the rough surface profile is .
The variance of the height distributions is given by
Another important parameter is the length of correlation. This can be obtained following the classical definition of correlation, but it implies to lose the geometrical point of view, for this reason, we prefer to use an approximated relation. This can be done by noting that the maximum correlation of a cosine term with itself is
To generate the surface, we use a holographic system as it is sketched in Fig. (1). The mirrors and the holographic plate are placed on displacement mountings whose movements are controlled with a computer. This setup allows us to control the fringe period and the phase term
As a final remark of this section, during the recording process, some regions with geometrical cusped may be generated by the consonance in the maximum/minimum values of the cosine terms, as it is shown in Fig. 2.
The importance of these cusped regions appears during the reconstruction process, because they have the capacity to generate evanescent waves. We consider that the number of cusped regions follows a Poisson distribution. The amplitude value to the electric field in the neighborhood of these points can exceed in several magnitude orders the value respect other regions.
So far we have described the synthesis of holographic rough surfaces with controlled statistical parameters. To understand the physical features of the scattered field is necessary to study the amplitude distribution, this can be done by analyzing the interference effects as it is described in the following section.
3. Interference between speckle patterns
During the reconstruction process, the scattered field emerging from the rough surface, when it is illuminated with a plane wave has the structure of a speckle pattern. The speckle pattern is spatially bounded as a consequence of the recording process. This is because the recording incident angles of the plane waves take values in an established range, for the presented case the angular range is
It is a knowing fact that the amplitude function of a speckle pattern satisfies a Gaussian probability density function, and the irradiance distribution follows an exponential function . In both cases, the statistical parameters such as the variance can be controlled with the transversal size of the illumination beam. This parameter will be used to control the bifurcation effects generated during the interference process, as it is shown below.
The holographic rough surface is implemented as a beam splitter to perform a four arm interferometer as it is sketched in Fig.3.
The interference effects, on the arm
The expression for the interference is described as follows. The amplitude on an arbitrary point
where the amplitude terms for reflected and transmitted fields can be approximated as
An important parameter is the number of trajectories
The irradiance associated to the scattered field, takes the form
and the mean irradiance is given by
A remarkable feature is that the interference area is controlled by means of the size of the illumination beam. This means that the
The fact that the holographic rough surface exhibits two probability density functions is easily understood because the roughness profile is depending on the illumination wavelength. When the illuminating light is coming from free space, the probability density function for the amplitude field is
The average size of the speckle pattern is greater for the scattered field generated by reflection, this is because the refractive index, expressed as quotation between the wavelength implies that
The sub-index in the square brackets in Eq. (11) means that the mean value for the interference term must be calculated using the joint probability density function, which is an unknown function. However an approximate expression can be proposed by noting that one of the amplitude values can be scaled, such that the mean interference term may be calculate by using a Gaussian probability density function whose arguments depends on the relative difference between two arbitrary points
As a finally remark, by a visual analysis of the interference pattern it is easily to detect the existence of regions where the interference fringes are “branched”, which corresponds to the bifurcation effects under study. The experimental shown in Fig. 4 are for normal incidence illumination and for angle of
4. Description of bifurcation effects
The bifurcation effects are generated when one or more parameters that characterize the system change continually, such that, when they acquire some critical value, modify and generate new physical properties of the system. For the present study, this parameter is the size of the transversal section of the illumination beam, which allows controlling the size of the speckle motes and then generates in a controlled way a transversal interaction between two or more speckle motes. The influence of this parameter is implicit in the number of trajectories
The transversal interaction between the interference fringes bounded by the speckle motes is shown in Fig. 6.c). The interaction between two interference fringe is the responsible of the generation of the bifurcation effects. The physical origin of this effect is in the boundary condition of the electromagnetic field which are dependent on the phase value in the contact point, in general, exists a discontinuity in the phase value when the interference fringes becomes closer. In order to satisfy the continuity of the tangential components of the electric field and the normal components of the electric displacement, a balance in energy and phase between the interference fringes occurs. This means that, when the two motes are far away no interaction occurs, when they become closer part of the energy must be transferred between the fringes, i.e. the energy flows through the fringes acquiring a final equilibrium value, modifying the interference fringes geometry. In general the fringes associated to each mote has different values in the phase function, which means that the fringes are shifted one respect to other. When a jump in the phase value of
To describe the transversal interaction between interference fringes, we assume that two speckle neighborhood motes has irradiance I1 and I2 respectively, the energy transfer satisfies
The nature of the solutions for the Helmholtz equation, depends on the difference between the irradiance values on a contact point, also as the sign of the dot product between the coupling vectors. The simplest case occurs when
When the dot product is negative,
When the dot product is positive,
The study presented offers the possibility to generate local optical vortex by controlling in an alternating way the size of the illuminating beam, i.e. we pass from regions with bifurcation to region without bifurcation, controlling locally the geometry of the interference fringes. An important application of this effect is the possibility to transfer angular moment to particles placed in the neighborhood of the contact point of two speckle motes.
5. Final remarks and conclusions
In order to have a complete description of the scattered field generated by the holographic rough surface is necessary to describe the optical field emerging from the neighborhood of the cusped points. We approximate the effects by noting that the cusped point can be consider as a quasi-point source, its representation, using the angular spectrum model, can be obtained using the Weyl representation given by 
It must be noted that the representation consists in two kinds of waves. One of them occurs when and it corresponds to homogeneous plane waves. Another case occurs when , the spatial frequency
In general the surface plasmon waves propagate short distances, typically 100
and for a metal thin film of thickness
From last representation, we have that to generate the coupling of illumination light-surface plasmon fields-scattered light, the mean distance between two cusped points
As a conclusion, in this chapter, we described the generation of a holographic rough surface using an incoherent convergence of holograms kind cosine. The surface was implemented to perform an interferometer that allows correlating two speckle patterns. The structure of the interference fringes shows the generation of bifurcation effects kind pitchfork. The splitting of the interference fringes is related with the size of the speckle motes and it is a consequence of the boundary conditions of the electromagnetic field. The superposition of cosine patterns generates cusped points randomly distributed and during the reconstruction process this cusped points generates evanescent waves. When the holographic rough surface is covered with a metal film, the cusped points generate surface plasmon fields.
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