Random object optical field diagnostics by using carbon nanoparticles

: We propose a new approach of using carbon nanoparticles for correlation optical diagnostics of a complex scalar optical field created by scattering and diffraction of radiation off a rough surface. This surface is simulated and we generate a diffraction pattern of the amplitude and phase distribution in the far field. Carbon nanoparticles of a certain size and concentration are obtained by the bottom-up methods of hydrothermal synthesis of citric acid and urea followed by centrifugation. The optical properties of carbon nanoparticles, such as luminescence and absorption in the visible spectrum that essentially differs for different wavelengths, as well as particle size of about dozen nanometers, are the determining criteria for using these particles as probes for the optical speckle field. Luminescence made it possible to register the coordinate position of carbon nanoparticles in real time. The algorithm for reconstruction of the scalar optical field intensity distribution through the analysis of the nanoparticle positions is here displayed. The skeleton of the optical speckle field is analyzed by Hilbert transform to restore the phase. Special attention is paid to the restoration of the speckle field’s phase singularities.


Introduction
A complex optical scalar field, here a fully developed speckle field, generally formed by coherent light, because of radiation scattering on a rough surface, is an erratic distribution of maxima and minima of intensity. The basic element of this field is an optical vortex, i.e. a phase singularity, an element of the field, where the phase is undefined. At these points, the intensity takes on a zero value, and the phase gradient (local wave vector) circulates around this point. Optical vortices [1][2][3], as a feature of an optical field structure, are an integral part of the optical field and exist independently of any special excitation conditions. As a rule, optical singularities appear near the lens focus, because of edge diffraction effects, or due to the superposition of plane waves.
A number of research works have been devoted to the study of optical vortices, and, in general, the approach of singular optics is used [3,4]. Singularities are considered either in a 2D plane or in 3D space. The singularity representation differs in the way they are created as a point in a plane, or a complex dark line in 3D, which determines a certain skeleton of the wavefront. The way of creating the singularities will determine their visualization method.
The use of a reference beam is the traditional (classical) way for studying the optical field singularity for identifying its location and the skeleton (base) of the optical field, which allows us to identify the object. However, such a visualization method is not always applicable; any diffraction restriction reduces the probability of determining the localization of basic elementshere the singularity points. Here, the study of remote objects, which are the targets for searching the solutions of reverse problems, is complicated when an additional reference beam is needed for obtaining and recovering of the phase information from the recorded interference pattern. The diagnostics of remote objects is not possible in this case.
In addition, the use of the interference approach to visualize optical vortices such as interference forks is a rather difficult task, since the shift of the forks with respect to the center of the vortex is conceivable, which depends on the period of the interference pattern.
As pointed out in the review [4], both generating structured light, and its application for solving the practical problems for the singularity reproduction in the interference approach, is possible with a second beam containing an optical vortex at a nodal point. This allows one to observe rather small, specifically dark regions in the object under study having optical vortices.
To avoid the necessity of applying an additional second beam, the method of phase singularity reproduction by optical correlation technique [5,6] is used here. In general, this approach enables us to describe in statistical terms a surface with roughness, determining the transmission or reflection off the surface due to scattering of the radiation on the structure inhomogeneity. The separation of the real and imaginary parts of the complex field amplitude [6] allows one to calculate the phase of the field and to reconstruct a skeleton through the spatial distribution of the lines of phase singularities. Accordingly, the points of intersection of the lines of the real and imaginary parts of the complex field amplitude determine the position of the phase singularities. However, this empirical approach reproduces both the singularity points, and the points of the intensity minima without optical vortices, which reduces the accuracy of the performed diagnostics.
In our recent works [7][8][9][10][11], we proposed to use the two-dimensional discrete Hilbert transform to restore phase information, in compliance with the sign principle, by sequentially convolving the fixed amplitude distribution of the speckle field with the selected transformation core, facilitating an approach for the reconstruction of the phase object. The mathematical processing of the registered skeleton by the Hilbert transform results in a reconstructed phase map of the entire object.
The basis of phase singularities (the skeleton of the optical field) reproduces both the internal structure of the object and its spatial shape. The change of the basis in time makes it possible to follow any changes in the object in real time. The task of reproducing the phase singularity localization is formulated, both theoretically and experimentally, and there is a need to use an approach that would significantly reduce the error in identifying the position of singularities.
To restore the amplitude and phase information of the object and to expand the existing approach for optical field restoration, we use nanoscale test particles with special properties. Selection of usable luminophores is not easy. They should be quite small but larger than organic dye molecules, in order to provide the necessary diffusional pattern. In addition, they should be strongly absorbing in yellow-green wavelengths, brightly luminescing, and weakly absorbing for He-Ne laser irradiation wavelength. Essentially, their diffusional motion should be efficiently guided by the optical energy flow, and because of that they should be dipoles. Due to these reasons, for singular optical field diagnostics and the identification of singularity points' position, we chose fluorescent carbon nanoparticles.
Fluorescent carbon nanostructures, as a new class of nanostructured materials, are of considerable interest due to their unique and exciting properties, such as tunable luminescence, easy synthesis, chemical inertness to surface modification, low cost, low toxicity and biocompatibility [12]. Essentially, in our knowledge, being of spherical structure of the order of ten nanometers, they possess optical symmetry that was discovered in fluorescence polarization measurements [13] and described on the level of single nanoparticles, presenting them as collective quantum emitters [14]. As it was suggested, the origin of this unique behavior is the existence of Frenkel-type molecular excitons [15]. Such properties of the new carbon materials lead to potential applications for creating light-emitting diodes [16], solar cells (absorbing light energy), sensors [17], biosensors for bio-visualization in medical diagnostics [18]. Optical trapping, as a new tool for manipulating and moving nanostructures, has been successfully applied to carbon nano-objects [19]. The very mechanism of spatial displacement of such nanostructures is determined by the interaction of the laser field with instantly induced dipole moments of individual nanoparticles [14].
The peculiar absorption and emission properties of carbon nanoparticles, which determines the nature of optical capture, promote the use of these particles for analysis and diagnostics of complex optical fields, including speckle fields, assuming that the particles are localized not just in the phase gradient regions, but at the singularity points of the studied optical field. The intensity minima that do not contain singularities are characterized by smaller intensity gradients, the energy flows here being much weaker than in the singularity regions. Accordingly, the number and concentration of nanoparticles will increase significantly in the minimum intensity regions with singularities.
This paper demonstrates the possibility of amplitude-phase optical field reconstruction for a random scattering object over a field skeleton visualization by observing spatially redistributed carbon nanoparticles in the points of minimum intensity with singularities due to internal optical flow action. The full intensity distribution is restored from the obtained singularity points, which is subjected to Hilbert processing. The result of this action is a phase map of the object under study. Processing the field of optical radiation scattered by an object is recorded in real time. The proposed approaches here are expected to have many applications. They may be applied in the study of spatial structure of the field of a sharply focused beam [20], where the use of interference methods is impossible; in the analysis of dynamic liquid media, including the restoration of the size distribution function of micro and nanoparticles in dynamic light scattering technologies [21], in life sciences; in the study of non-stationary processes in cells; in the study of turbulent gaseous media [22], in aerodynamic applications, as well as for the solution of problems of digital holographic interference in the analysis of non-stationary objects and scenes [23,24].

Study of an optical speckle field by correlation-optical method using carbon nanoparticles
In this work, to diagnose the optical scalar field (speckle field) obtained by scattering and diffraction of radiation off a rough surface, carbon nanoparticles were applied. The reason for their choice was explained above. In our case, they were prepared using a one-step hydrothermal synthesis from citric acid and urea in autoclaves with following centrifugation at t = 190°C for 2 hours, which makes it quite easy to obtain bright carbon nanoparticles with stabilized carboxy-, amino-and other groups with a diameter from 2-3 nm to hundred nanometers [25][26][27].
In order to verify, analyze, and estimate the size of nanoparticles, atomic force microscopy measurements were performed. We use water solution of prepared carbon particles. The used technological process for the production of carbon nanoparticles ensured a high stability of the formed nanostructures; their parameters did not change during the measurements. The entire technological process of manufacturing carbon nanoparticles, approaches to assessing the size and optical characteristics, which is actually necessary for studying speckle fields, are given in our paper [27]. The evaluation of the absorption and luminescence spectra of particles obtained during our experiment sets the absorption maximum at 405 nm, the luminescence maximum at a wavelength of about 450 nm, and a weak absorption at the wavelengths of He-Ne laser radiation (633 nm), which is crucial for using these particles as a speckle field skeleton probe. Absorption rate (A) of carbon the prepared nanoparticles [27] at He-Ne laser irradiation, in the field of which the speckle field under study is formed, is minimal and is in the order of 0.2 cm −1 . The transmission coefficient (τ = exp(−D)) was evaluated at the given experimental conditions, and was in the order of 80%. Determination of these optical parameters was made by estimating the optical density (D) of the used solutions.
For these carbon nanoparticles, sized from ten to hundred nanometers, a size-dependent effect [19] is not observed, which greatly simplifies the practical implementation of the experiment to detect luminescence of carbon nanoparticles followed by particle localization in the zones of an optical singularity.
Laser radiation acting on carbon nanoparticles can cause heating of the medium surrounding the particles and of the particles themselves, which force us to discuss the temperature effects. A consequence of this effect can be the redistribution of the optical flows, which again can cause a change the direction of the particle motion and eventually their capture. It is possible to quench fluorescence at temperatures above 50°, which is accompanied by increased diffusion, phonon relaxation of the shell excitation energy, consisting of active polar groups. As a result, the emission intensity decreases [28]. In order to avoid the temperature effects including destroyed changes or possible particle oscillation due to their Brownian motion leading to the essential change of particle position, the temperature regime and the exposure time are controlled. Therefore, we use laser sources with a maximum wavelength of 405 nm and a power of 5 mW for the observation of luminescence, and a He-Ne Laser for obtaining the scattered field in which the motion of carbon particles occur. Time necessary for particle redistribution and capture is about 30 seconds, which is confirmed theoretically.
The exceptional optical properties of carbon nanoparticles, specifically their interaction with external laser radiation, allow one to use them for diagnostics of optical fields, and consequently to recover the field skeleton under study, and the information retrieval of the object.
In order to study the optical fields by correlation-optical methods, we address the following problems: -simulation of an arbitrary surface with given roughness; -obtaining a diffraction pattern of the simulated object with corresponding speckle field in the far zone; -visualization of singular points due to the absorption of internal optical flows by carbon nanoparticles with their redistribution into the regions of singularity (luminescence of particles is registered); -object phase retrieval by the use of 2-D Hilbert transform without reference wave.
Thus, to study optical field carbon nanoparticles of size about λ/10 with strong absorption in the yellow-green region (405 nm), luminescence at wavelengths of 530 nm and weak absorption at the red wavelength of speckle field formation (633 nm) are used. The particles of this size are easily controlled by the speckle field by moving along the intensity gradient [29,30]. The structure of the optical speckle field is visualized and captured by a CCD camera.
Consequently, the speckle field is restored by a set of spatially redistributed carbon nanoparticles in the areas of minimum intensity with singularity, thereby reproducing the structure base (skeleton) of the optical field and the phase map in real time.

Rough surface modeling [6]
Here, it is proposed to use the simulation approach of a surface with given roughness and the results of the experiment, which were tested by the authors and previously presented [31]. There a multifunctional system for the diagnostics of rough surfaces of various structure is presented. Modeling of a rough surface was carried out according to the appropriate algorithm [31]. Simulation took place in the MATHEMATICA software package. We use the MATHEMATICA program to simulate a speckle field in the present work. A surface of the field with a square of 100 by 100 µm 2 is derived; the number of pixels in a given area is determined by a step length of λ/3. The choice of step length is determined by: i) there should be no mutual amplification or quenching of the signal from the neighboring pixels (the wavelength is divided by an odd number), ii) the obtained points should determine the pixels, in accordance with the resolution of the camera, used to record the image (1024 pixels).
During the simulation, the points with a step of about of 5 µm are considered and the height of the inhomogeneity is randomly generated within the interval 0 -h max (h max = 2 µm). The height of the surface inhomogeneity is obtained using spline interpolation of the generated points. Intermediate values of heights were obtained by the interpolation of periodically located random heights. At this stage of modeling, the standard MATHEMATICA function -ListInterpolation was used in the context of the 1 st order linear B-spline interpolation. To obtain a complete picture of the simulated surface roughness, a Gaussian filter with a radius of 3 µm was used, with smoothing by 30 points, giving the possibility to reproduce the relief of the rough surface (Fig. 1). The choice of parameters used in the simulation is carried out through a systematic analysis of their values, followed by matching the simulation results with an experimentally recorded rough surface profile [1,6,31]. The resulting smoothed distribution of heights was the object of our detailed analysis.  Figure 1 shows the two-dimensional and three-dimensional coordinate distributions of the surface inhomogeneities obtained in the modeling process.

Obtaining a diffraction image of a simulated surface
To simulate the diffraction image of a surface with a given roughness, the diffraction integral in the Rayleigh-Sommerfeld approximation [6] is used, and in the discrete approximation, the double integral turns into a double sum: where: F(x, y)-is the aperture function, which determines the amplitude transmission of a rough surface; h(x, y) describes the relief of the analyzed surface, and determines the phase delay of the beams diffracted on the surface φ(x, y) = 2kh(x, y), k = 2π/λ is the wave number; λ -wavelength.
R(x, y, z, ξ, ζ) = √︂ z 2 + (x − ξ) 2 + (y − ζ) 2 is the distance between a point at the object and a point in the observation plane; z is the distance between the plane where the object is localized and the observation plane; x, y, ξ, ζ -rectangular coordinates in the plane of the object and in the plane of observation, respectively. In our case z = 1000 µm = 10a, where a -is a linear size of the object. Here X, Y are the number of points of the analyzed object selected in x and y directions, respectively, with a step size of λ/3.
Determining the real Re[U(ξ, ζ)] and imaginary Im[U(ξ, ζ)] parts of the calculated field makes it possible to evaluate the field amplitude modulus, intensity, and phase. In the far field, a simulated diffraction pattern of 240 by 240 µm 2 (400 × 400 pixels) and the corresponding phase map are shown in Fig. 2. The field is based on the radiation from a He-Ne laser (λ=633 nm). The area of the diffraction pattern of 30 × 30 µm 2 (60 × 60 µm 2 ) marked by white squares in Fig. 2 was used to demonstrate the effectiveness of the given model to reconstruct the optical field intensity distribution by a coordinate-wise analysis of the tracked motion of carbon nanoparticles until they are eventually captured by regions of intensity minima having singularities. For a better assessment, the simulation results obtained for the areas of different spatial scales (Fig. 2,  squares 1,2). Marked area of about 30 × 30 µm 2 is later used for phase information retrieval using Hilbert analysis.

Reconstructing of the intensity distribution (computer simulation)
The next step is the search for ways to recover the intensity distribution through processing of the recorded tracks of the carbon nanoparticles' motion in the complex optical field, which is obtained by the light scattered off the rough surface, e.g. the diffraction pattern described above (Fig. 2, selected parts)).
A cuvette with a water solution of carbon nanoparticles of size λ/10, with known properties is placed in the optical field. Randomly suspended particles move in this field because of the internal optical flows, and concentrate in regions of minimum intensity with singularities. It is then possible to reconstruct the tracks of nanoparticle motion and the intensity distribution. Luminescence reveals the motion of particles.
The measured value of the absorption rate (A = 0.2 cm −1 ) at λ = 633 nm, allows one, using the Fresnel theory, to calculate the imaginary part of the refractive index (n), i.e. the absorption coefficient, as k = (A · λ) / 4π [32,33].
According to the law of energy conservation, the reflection coefficient ρ = 1 − k − τ, made it possible to determine the real part of the refractive index (n) from the relation ρ = (n−1) 2 +k 2 (n+1) 2 +k 2 , which is sufficient to use for strongly absorbing elements, carbon nanoparticles, and in order to simplify the calculated ratios. Using values for n and k, the real and imaginary parts of the relative permittivity are calculated as, ε r = n 2 − k 2 and ε i = 2n k. Particle polarizability α (in the Gauss system) is: We use the obtained optical parameters of the applied carbon nanoparticles for the calculation of the optical force that determines the action due to the internal energy flows.
The presented nanoparticles are described by the Rayleigh light scattering mechanism [34][35][36][37], which is combined with Mie theory, giving each component of the resulting optical force acting on the particle -gradient component ⃗ F grad , absorption ⃗ F abs and scattering ⃗ F scatt components [27][28][29][30]: where C scatt = k 4 |α | 2 4π , C abs = kα ′′ are the scattering and absorption cross section (in a Gaussian system of units), k = 2π λ is the wavenumber. Here (2) Figure 3 shows the ratio of the components of the optical force for different size of carbon nanoparticles at an observation wavelength for nanoparticles in a speckle field. The distribution of the gradient force is analyzed in the x(y) directions, i.e. in the cross section of the beam. The scattering and absorbing components exert their effect in the direction of the action of the Pointing vector (longitudinal z-direction). Therefore, when analyzing the distribution and influence of these forces in the transverse plane, the action of these components does not appear. An analysis of the magnitude of the active component of the gradient force shows that the minimum regions with singularities determine a force value that can change the particles' position in time.
Naturally, in the case of three-dimensional visualization of the complex dynamics of particle motion, all three components need to be taken into account. An analysis of the components of the optical force demonstrates that the value of the gradient component of the optical field far exceeds the absorbing component, i.e. temperature effects due to absorption of radiation in the medium will be absent.
At the same time, the value of the scattering component is such that it reveals the displacement of the nanoparticle in the longitudinal direction. The nanoparticle motion will be observed until they are captured by the intensity minimum. Preliminary modeling showed that the influence of optical energy flows on nanoparticles is best manifested for particles with a size of λ/10 which allows us to use them to restore the object's phase information.
The equation of motion of the i-particle (i=1..N, where N is the total number of nanoparticles analyzed in the speckle field) under the action of optical force becomes: where ⃗ F opt i = ⃗ F grad i + ⃗ F abs i + ⃗ F scatt i -the resulting optical force, the components of which are set according to (1), (2): • m i = 4 3 πr i 3 ρZ -mass of i-carbon nanoparticle, here r ii-particle radius, • ρ = 2.26 g/sm 3 Z -water density [38], • η = 8, 9 · 10 −4 Pa · secZ -dynamic viscosity of the medium (water) at T=25°C [39].
In the scalar case, taking into account the Cartesian x(y) components, the equation of motion has the form m dv x,y i dt = F opt x,y i − 6πrηv x,y i . Accordingly, the acceleration that carbon particles acquire during their motion is a x,y i = dv x,y i dt , which allows us to determine the coordinates of the particle at a given time t. The initial moment of observation is taken as the zero value.
A computer simulation is performed (Fig. 4), where time t -are running values for each nanoparticle, given by the observation interval, which is determined by the time of capture of the nanoparticle by the intensity minimum with singularities in the area of 30 × 30 µm 2 (Fig. 5). Figure 4 demonstrates the changes in carbon nanoparticles position in time: a) the particles are randomly localized in a scalar complex optical field, b) based on the calculated coordinates we determine the track length (∆s) that the nanoparticle passes over a selected period before its capture, c) the calculated final position of the carbon nanoparticles.   )). In 30 second (Fig. 5(c)) all carbon nanoparticles redistribute into the minimum area with singularities moving along the gradient lines of intensity (green tracks). Figure 6 demonstrates the position of the particles in the intensity area of 60 × 60 µm 2 at 0.375 sec. The video presents the motion of particles during the period of observation (Visualization 1) until their capture in the intensity minima. The observation time here is 5 second.
The resulting coordinate set of intensity values, which is generally random, and defines an unstructured data set, is converted into a regular continuous array by using a linear B-spline interpolation of the 1 st order.
Further, by mathematical processing, from a continuous distribution, a discrete periodic distribution of intensity points is extracted, and using the higher order interpolation, the intensity distribution is refined (Fig. 7(b)).

Phase restoring by using a Hilbert transform
The final step of our investigation is to extract the phase information from the reconstructed intensity distribution of the field (area of about 30 × 30 µm 2 was analyzed). MATHEMATICA software package allows us to restore the phase by applying a Hilbert filter [40,41] to the intensity distribution of the speckle field under study. The corresponding phase map is presented in Fig. 8(a). As it is shown by the simulation results, singularity information in Hilbert transform action is lost. Here, the points of phase singularities, marked in the picture by red points, are determined by the final position of carbon nanoparticles in the optical field ( Fig. 8(b)). The same points are obtained through the intersection of real and imaginary parts of the complex field amplitude (Fig. 8(c), green and blue lines). Red points in white squares determine the location of intensity minimum without singularities. The error in the results of reconstructing the phase using traditional mathematical methods of phase recalculating through the values of the real and imaginary parts of the complex field amplitude, described in 2.2, and using a Hilbert transform, proposed in MATHEMATICA, is about 15%. The estimation of this error was carried out pixel by pixel, by determining the deviation of the phase on the phase map reconstructed using the Hilbert transform from the values obtained using traditional mathematical methods.
An increase of the probability of the retrieval of amplitude zeros (phase singularities) is possible by the approach proposed in [42][43][44][45][46]. Here, the zeros of U(ξ, η) are transformed into branch points of log[U(ξ, η)] by applying the function log[U(ξ, η)] = log |U(ξ, η)| + iφ(ξ, η). Such an approximation allows one to use the two-dimensional logarithmic Hilbert transform that involves extracting the zero information in the complex lower half-plane, in which a real variable is changed into a complex one. At this stage of data processing, information about zeros is also lost. Further, applying of exponential filters exp[−2π y c ξ], exp[−2π y c η], y c > 0 (y cinterpolation constant) in the object plane (ξ, η) makes it possible to leave the zero information in the complex lower half-plane. This revises the obtained Hilbert phase on each line parallel to the x 1 , x 2 axis in the Fourier transform plane, as described in detail in [42]. In this case, the accuracy of reproducing the phase map with singularities of the optical field increases to 92%.
As a conclusion, it can be noted that the approach used to reconstruct the amplitude and phase information of an optical field based on stationary scattering object opens up additional possibilities both for non-stationary scattering object diagnostics and macro-analysis of the shape of the remote object for solving the aerodynamics and space problems as well as for micro-diagnostics of the image in non-stationary processes in cells in the life sciences.

Conclusion
A new approach of using carbon nanoparticle to study the optical field obtained because of diffraction and scattering on a stationary object in real time is demonstrated. Sarbon nanoparticles of size about λ/10, characterized by strong absorption and luminescence at yellow-green wavelengths and weak absorption at He-Ne wavelength were used for the first time for probing a speckle field. Visualization of singularity points by these nanoparticles, and their motion in the optical field due to the action of internal optical flows, makes it possible to reconstruct the intensity distribution of an optical field in real time. Applying Hilbert transform for phase retrieval of the optical field, describing the random light scattering object, provides the information about the phase map of the optical field. The error in reconstructing of the phase map using both the Hilbert transform and traditional approaches for estimating the components of the complex amplitude does not exceed 15%. Obtained results open up new possibilities in the diagnostics of non-stationary objects in liquid and turbulent media, as well as for study the non-stationary processes in cells in the life sciences.