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Abstract
Optical scattering strength of fractal optical disordered media with varying fractal dimension is reported. The diffusion limited aggregation (DLA) technique is used to generate fractal samples in 2D and 3D, and fractal dimensions are calculated using the box-counting method. The degree of structural disorder of these samples is calculated using their light localization strength, via the inverse participation ratio (IPR) analyses of the optical eigenfunctions. Results show non-monotonous behavior of the disorder-induced scattering strength with the fractal dimension, attributed to competition between the increasing structural disorder due to the decrease in fractal dimension versus the decrease in scattering centers due to decreasing fractality.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Anderson postulated disorder-induced wave localization due to multiple scattering and interference effects in disordered media [1]. Since then, wave localization in disordered systems has been a subject of intense investigation, including the light wave in disordered optical media. It is also known that disordered systems with a self-similar structure can be characterized by fractal geometry. Naturally occurring porous materials, biological systems, etc., are considered as fractal in nature. A fractal system is defined by its spatial mass density correlation decay, follows as a power law in space, whose non-integer exponent is termed as the ‘fractal dimension (Df)’ of the system [2]. Fractal geometry analysis, in terms of fractal dimension, is often employed to characterize a wide range of naturally occurring optical disordered systems, such as aerosol clusters [3], biological samples [4–6], etc. The optical scattering from a refractive index (RI) fluctuating (n(r)) medium can be characterized by onsite fluctuations of n(r) and its spatial correlation decay length lc, therefore, we expect scattering properties of fractal disordered media to depend strongly on fractal dimension, i.e., spatial structural geometry/correlation of these fractal systems. There are studies to characterize scattering properties of fractal dimension for different systems, in particular biological cells/tissues for cell characterization [4,5,7,8]. In these studies, the fractal dimension is used to detect cancer. However, the relationship between the fractal dimension of a system and how its light scattering properties changes in different fractal dimensions are not well studied. In particular, a systematic comprehensive study of the nature of the variation of the degree of scattering properties of a fractal system with the fractal dimension and the strength of the refractive index fluctuations and its spatial correlation, has not been reported. When a solid system (with a starting integer dimension: 1 or 2 or 3) changes to a fractal system due to the development of the appropriate porosities in the media, the disorder properties of the system start increasing at the beginning due to the development of the refractive index contrasts in the media. However, with the further increase of the porosity, the scattering properties of these media go through a competition of two opposite factors: (i) increasing structural disorder due to the decrease in fractal dimension (i.e., increase in porosity) versus (ii) decrease (loss) in the scattering centers due to the decrease in the fractal dimension (i.e., increase in porosity). Due to this opposite competition of two factors with the increase of porosity, the degree of scattering strength due to the change in porosity (or fractal dimension) may not be a monotonous function with the change in fractal dimension, and one expects optimal scattering point(s).
The problem addressed here is important to many branches of optical condensed matter systems, where changes in fractal dimensions are used for sample characterizations by light scattering experiments. For instance, measuring the change in the fractal dimension can be used to detect the progress of cancer in biological tissues [4,9–11], where the exact position of the turning point or optimal point(s) is (are) important to exactly characterize the increase or decrease of the fractal dimension. Therefore, the exact nature of the scattering, or the degree of scattering strength, with the change in fractal dimension demands a comprehensive study to locate the optimal scattering point(s) and to characterize how the scattering properties of the system change with the change in fractal dimension.
In this work, we numerically investigate the relationship between the degree of structural disorder and fractal dimension based on their degree of light localization properties, for 2D and 3D fractal disordered optical systems. 2D and 3D fractal optical systems are generated by diffusion limited aggregation (DLA) simulation technique [12]. The degree of disorder or light-scattering properties of these systems were quantified in terms of the degree of light localization strength of these systems. The degree of light localization can be measured in terms of the mean inverse participation ratio < IPR > value of the optical eigenfunctions of the Anderson tight-binding model (TBM) Hamiltonian of the system, with closed boundary conditions. It is known that < IPR > value is proportional to the light localization properties of the system, and is a measure of the effective scattering strength of the system [13,14]. Therefore the < IPR(Df)> versus fractal dimension (Df) are generated using the light localization technique to investigate the peak scattering point(s) or optimal scattering fractal dimension(s). Furthermore, we also studied random cut (RC) lattices, a non-fractal optical disordered samples/lattice media for < IPR(Df)> versus Df variation, to characterize optimal scattering points in 2D and 3D media, and to compare the results with the fractal samples generated by DLA method.
The importance and significance of the problem reported here are therefore results of a full numerical characterization of the degree of scattering strength with the change in the fractal dimension and the strength of the refractive index fluctuations, importantly, identifying the optimal scattering points. The degree of scattering strength is expressed in one parameter, the < IPR > or (Ld) value, and this provides the opportunity to characterize the system simpler way through the important statistical parameters of the disordered samples.
2. Method
2.1. Theoretical framework of light scattering in closed systems
The Schrodinger equation and Maxwell’s wave equations can be projected to each other in all dimensions, as both the equations can be projected to Helmholtz equation with certain conditions, therefore, several well-developed methods of electronic systems of scattering can be used for optical systems [15]. We treat numerically generated fractal optical samples using DLA and RC methods as optical lattice systems, with its lattice points as the scattering centers, by appointing a refractive index value for each lattice point, two state values: lower (0) and higher (1). Different fractal geometries result in different distributions of the scattering centers with long-range or power-law spatial correlations, which consequently leads to the different degrees of disorder or disorder strengths. Once samples are generated by DLA or RC simulations with different fractal dimensions, we use the Anderson-disorder tight binding model (TBM) with nearest neighbor interactions, under closed boundary conditions, a well-studied method in electronic system [16,17]. The Hamiltonian, H, for the system can be written as [17]:
In Eq. (1), ${\varepsilon _i} = (d{n_i}/{n_0})$ is defined as the optical potential of ${i^{th}}$ site with dni is the fluctuation above the mean refractive index n0 [18], t is the inter-lattice site hopping strength restricted to only the nearest neighbors. |i > and |j > are the optical wave functions at the ith and jth lattice sites, respectively, and <ij > indicates the nearest neighbors. The eigenvalues and eigenfunctions (or Ψ values) of the system are obtained by diagonalizing the Hamiltonian H. Further, assuming the sample length L and the lattice size a, we can define the mean IPR value (<IPR >) over all the eigenvalues of the sample as:
Likewise, we can generalize Eq. (2) for 3D, where the volume element will be dxdydz and N=(L/a)3, dx = dy = dz = a.
2.2. Calculation of the structural disorder or disorder strength (Ld)
The fractal dimensions of the system, follow the scaling power law Nr∝ r-Df, with Df being the fractal dimension, and was taken by the well-known Minkowski–Bouligand dimension, or box-counting method. In this method, if a dimensional space containing a specific fractal structure is partitioned with a number of Nr-boxes[Nr(r)] (box: area r2 in 2D and volume r3 in 3D) with side length r, then the Nr-boxes counting fractal dimension is defined as [2,19]:
Counting the number of Nr-boxes of different sizes or values, the fractal dimension using Eq. (4) can be estimated by the slope of the linear fit of ln(1/r) versus ln(Nr(r)) points. This method has proven accuracy of the fractal dimension calculation within ±0.05 for different deterministic fractals with known analytical fractal dimensions, such as Sierpinski's carpet and Sierpinski's triangle.
3. Numerical simulation
3.1. Theoretical framework of light scattering in closed systems
In general, there are two types of scattering probes to characterize a sample: open and closed scattering probes. In an open scattering probe, the probing signal and scattering signals are measured outside of the sample such as reflection and transmission intensities. In a closed scattering experiment, the sample is probed by closed scattering parameters, where the eigenfunctions and eigenvalues of the system are measured. However, in both methods, the same physical scattering parameters of the sample can be probed. For example, for disordered systems, the main parameters of the sample are the STD of the local refractive index fluctuations dn and its spatial correlation length lc.
Generation of fractal samples by DLA and RC simulations: In the following, we describe optical lattice generation by both the DLA (fractal) and RC (random cut, non-fractal) simulation methods.
3.2. Diffusion limited aggregation method (DLA): fractal samples generation
Fractal samples are generated by numerical simulations using a diffusion-limited aggregation (DLA) algorithm, a well-known process found in a wide array of fields such as biology, chemistry, material sciences, etc. In this technique, particles undergoing a random walk cluster together to form aggregates [12,20,21]. The rules for DLA fractal growths, in brief, are as follows: the simulation first creates a stationary ‘seed’ particle where the aggregation is then built by adding a second particle at some radius r away from the seed particle. The second particle undergoes a random walk until it encounters the seed particle, at which point it sticks with a certain probability, varying from 1 (full stick) to 0 (no stick) [12]. Lower sticky probability values allow the particles to ‘fall’ into gaps inside the lattice. If particle wonders a certain radius (kill radius) from the seed, the particle is destroyed and the process restarts. Typical DLA generated samples are shown in Figs. 1(a) and (b).
Fig. 1. Fractal structures generated by diffusion limited aggregation (DLA) simulation method in 2D with increasing stickiness parameter from left (a) for Df = 1.68 to right (b) for Df = 1.89. Numerically generated disorder matrix by random cut (RC) method simulation in 2D with randomly cutting lattice points or randomly decreasing lattice points from left (c) for Df = 1.87 to right (d) for Df = 1.33.
3.3. Random cut (RC) model: nonfractal samples generation
We also perform a simple random cut (RC) of lattice method starting from the net filled sample of filled lattice/matrix for disorder system, that does not have exact fractal correlation, for comparison with DLA method. In the RC method, 2D/3D filled matrices are generated by taking a lattice filled with points/particles, or scattering centers at every lattice point, and removing, or cutting the lattice points randomly. Two different stages of random cut are shown for 2D in Fig. 1(c)-(d), with lesser number of cuts (empty points) and a greater number of cuts from a filled lattice/matrix, respectively. This method does not generate a true fractal but rather a random disorder sample without any long-range spatial correlation. The volume occupied ratio which is a simple measurement of how “full” the system is, rather than the calculated fractal dimension.
4. Results
4.1. Fractal dimension versus scattering strength variation in 2D
Once we generated the 2D/3D matrices, TBM Hamiltonians are formed for these matrices. We kept the onsite refractive index potential ɛ constant at occupied position and 0 at non-occupied position throughout the lattice and the hopping parameter t as constant equal to 1. Simulations are performed in both 2D and 3D systems, using both DLA and RC techniques. 2D square lattices of length L = 8, 16, 32, and 64 corresponding to TBM Hamiltonian matrix size (N = L2), N = 64, corresponding to TBM Hamiltonian matrix size of (N = L3), N = 64, 512, and 4096, respectively, are constructed and simulated. In each complete iteration of a DLA/RC method, generated optical lattices are analyzed for its fractal dimension Df and average < IPR(Df)> value. In this method, we systematically scan the fractal dimension Df by the box counting method and corresponding structural disorder strength by calculating < IPR(Df) > . For DLA simulation, the arbitrary number of particles are simulated until the aggregation is built, and each aggregation is analyzed for its Df value and corresponding < IPR(Df)> value using Eqs. (1)–(4). For an expected system size L, the particles were added at a radius r of twice L with a kill radius of twice r, i.e. 2r.
In the Hamiltonian, keeping the relative interaction strength ratio ɛ/t in Eq. (1) as constant, the < IPR(Df)> vs Df curves are plotted for each system. The fractal dimension is adjusted directly for the RC method by controlling the number of particles cutting/removing from the lattice. The fractal dimension adjusted for the fractal system generated using DLA method is by controlling the stickiness parameter value. Since the largest stickiness value of 1 creates DLA fractals with larger values of the fractal dimension. The DLA and RC methods develop both 2D and 3D fractal systems that are simulated for eigenfunctions and then IPR value of a system.
As can be seen in Fig. 2(a)-(b), <IPR(Df)> vs Df curves for different energy ratio constant ɛ/t (0.625-10) values are shown for: (a) DLA generated fractal systems and (b) RC generated non-fractal disorder systems. These demonstrate a recognizable optimum/maximum, suggesting that there is indeed a fractal dimension to which optimal scattering occurs. This particular fractal dimension was dubbed the turning point (or optimal scattering point), or Dft.
Fig. 2. 2D and lower fractal dimensions < IPR(Df)> vs Df plots. Averaged < IPR(Df)> against Df plots for: (a) DLA and (b) RC systems simulations shown with L=64, for different ɛ/t values. Energy ratio ɛ/t vary between 0.625 and 10 with intervals of 0.625, and the large ratios a correspond to larger maximum < IPR > values.
The turning point occurs for the RC (non-fractal) method (Fig. 2(b)) roughly when a maximum contrast between empty and occupied sites happens for RC systems. That is, at least, for RC disordered systems, the turning points (or optimal/maximum points) should occur when roughly half of the lattice sites are filled and half are empty. We found that for both the methods, the turning points show dependency on systems length size L, as well as on the energy ratio ɛ/t. Increasing L both shifts the turning point Dft towards a larger fractal dimension and stretches the curve in the vertical direction or to a higher localization value. The averaged turning (or optimal) fractal dimension point for the 2D RC method with an energy ratio of 1 at a system length L of 8, 16, 32, and 64 are 1.69, 1.79, 1.85 and 1.88 ± 0.05, respectively. The vertical stretching of Fig. 2(a)-(b) in < IPR > for higher energy ratio values is due to extremely high disorder-induced eigenfunction localization. The change of the energy ratio proved the most interesting in relation to the turning point. For 2D cases, both methods increasing ɛ ratio correspond, expectantly, to an increase in the maximum < IPR(Df)> value as shown in Fig. 2(a) for DLA case, and in Fig. 2(b) for the RC case. The maximum localization strength relative to the minimum can be described by the increase in onsite potential relative to the hopping parameter. As such, the maximum < IPR(Df)> values at low energy ratio ɛ/t are not relatively strong compared to their minimal, and show instability below a ratio of 1, as the order of kinetic energy overwhelms the potential.
4.2. Fractal dimension versus scattering strength variation in 3D
In Fig. 3(a)-(b), for 3D < IPR(Df) > vs Df curves are plotted for DLA and RC. The 3D systems present somewhat similar findings to 2D as shown for DLA [Fig. 3(a)] and RC [Fig. 3(b)]. The value of IPR is lower in 3D, suggesting weaker localization in 3D. There are two turning points, however, remains indefinitely regardless of how large the energy ratio becomes, for both DLA fractal samples and RC random samples. Figure 3(a), for DLA systems, show curves are similar to its 2D counterpart, a local maximum < IPR > value manifests at fractal dimension Dft = 2.24 for all ɛ/t >3 and the second turning point Dft at 2.95 ± 0.05. In Fig. 3(b), curved for RC method is presented. The saturating dimensions for two turning points are Dft = 2.24 and 2.95 ± 0.05, and is again symmetrical to the number of particles present in the system. The relative < IPR > strength of the double turning points, i.e. the optimal points/peaks, is much larger than the valley compared to its 2D counterpart as seen in Fig. 2(b). ɛ/t value below 1 display extreme instability turning points for 3D systems compared to their 2D counterparts, as the system is less localized.
Fig. 3. 3D and lower fractal dimension < IPR(Df)> plots. Averaged < IPR(Df)> against Df plots for (a) DLA and (b) RC systems simulation shown for L=16 and different ɛ/t values. Energy ratio ɛ/t varies between 0.625 and 10 with interval of 0.625; large ratios correspond to larger maximum < IPR > values.
4.3. Scattering strength versus energy ratio ɛ/t for 2D and 3D
In Figs. 4(a)-(b), we plot the Dft(ɛ/t) vs ɛ/t curves for: (a) DLA and RC samples in 2D and (b) DLA and RC samples in 3D for constant system sizes. As shown in Fig. 4(a), for 2D RC lattice, a bifurcation occurs for Dft(ɛ/t) vs ɛ/t plot ($\color{blue}{\square}$) for ɛ/t value between 2 and 10, resulting in double turning points; the split in turning points is rather abrupt and temporary as one increases the energy ratio, before returning to its original turning point of Dft=1.88 ± 0.05, which corresponds to approximately half of the lattice sites filled. The fractals generated by DLA method ($\color{red}{\textrm{X}}$) only ever singular turning point rises and then saturates after a ratio of around 3, to Dft = 1.94 ± 0.05 that, unlike its RC counterpart, persists to a larger value of the ratio. This distinction between the two turning points of the RC method versus the singular for the DLA arises from the fact that, in terms of the < IPR(Dft)>, a random distribution will behave the same as the inverse of said distribution and as such, the turning points position of the RC method is symmetrical around the number of particles present and absent. For example, double turning points for 80% filling are corresponding to 20% and 80% of the lattice filled and these two peaks merge at 50%.
Fig. 4. Turning points Dft against energy ratio ɛ/t in 2D and 3D. (a) 2D system with size L2=(64)2. (b) 3D system with size L3=163. Blue squares ($\color{blue}{\square}$) represent RC method while red Xs ($\color{red}{\textrm{X}}$) represent DLA.
Figure 4(b) shows the turning point vs energy ratio plot, that is Dft(ɛ/t) vs ɛ/t of energy ratio against the turning point for the 3D systems. For RC method, the double turning point values are shown ($\color{blue}{\square}$), the saturation values are 2.24 and 2.95 ± 0.05. The double turning points for the DLA is 2.24 (∼constant value) and 2.95 ± 0.05, are shown by symbol ($\color{red}{\textrm{X}}$).
5. Discussions and conclusions
We have studied the degree of light scattering properties of fractal optical disordered media, for varying fractal dimension and strength of the disorder, for starting fractal dimensions 2D and 3D. This provides the scattering from a vast range of samples in the parameter space of fractal dimension and energy ratio. In particular, weak to strong disordered systems in 2D or less and 3D or less fractal dimensions. As the results show, the inverse participation ratio (IPR) coupled with fractal analysis provides remarkable analysis and characterization technique for optical fractal disordered systems. The method provides the degree of scattering properties of fractal disordered system in a single parameter, the < IPR > value. <IPR(Df)> vs Df curves for constant ɛ/t ratio values are non-monotonous for both 2D and 3D fractal samples, as well as random cut disordered samples. The maximum/optimal scattering strengths occur at certain fractal dimension(s), we have called as optimal/maximum fractal dimension, or turning fractal dimension Dft. This is because of the large separation of energy states that are symmetrical for random constant potentials of complementary particle lattice percentage occupation (for 80% filling, it is 80% and its complementary 20%). We have extended our results from weak to strong disorder to probe the optimal localization points.
Strong to weak disordered samples parameters regimes and their examples: The weak and strong scattering samples are parametrized by potential and hopping factor ratio, in some sense potential energy and kinetic energy ratio or the value: ɛ/t. If the ɛ/t <1 is small, we can think about the weakly scattering case, while ɛ/t >1 is the strongly scattering case. As ɛ/t is ratio of two independent parameters ɛ and t, therefore, the value of the ratio can be adjusted by relative values. For example, for strong scattering regime, we can take either larger ɛ, or smaller t, or both. Biological samples are weakly disordered media while certain higher dielectric media composite with metal particles can be a strongly optical disordered media.
Importance of a turning point or optimal scattering fractal dimension (Dft) in biological systems for cancer detection: Most biological systems are fractal, weakly disordered optical media, therefore, for biological systems ɛ/t ≤ 1 values are more relevant for light scattering from these systems, from the whole samples parameters and probing light. It can be seen from the simulation results that for a fractal disordered optical media, the localization properties of the systems are quite similar related to the turning points, or optimal points. That is, approximately the same Dft for different ɛ/t values, for all the whole parameter space in 2D or 3D. It is known that tissues/cells are fractal disordered optical media. The light scattering properties of tissues are important to understand the abnormalities in these tissues, such as progress of cancer. The structural abnormalities or alterations progress with the progress of diseases such as carcinogenesis or brain abnormalities [5,6]. There is no clear study to indicate which are the optimal scattering fractal dimensions in 2D or 3D tissues. This means, due to the non-monotonous nature of the fractal dimension in scattering, the degree of scattering can increase or decrease with the change of the fractal dimension, depending on the position of the fractal turning point Dft values. Therefore, it depends on the initial tissue structure or its fractal dimension and the direction of the change of the fractal dimension.
Recently, spectroscopic methods are used for tissues fractal dimension calculations. For examples, using light scattering method, low-coherence backscattering, and Mie scattering method, fractal dimensions are calculated and used for detection of the progression of cancer [22–24]. It has been shown in these works that the fractal dimension changed systematically in tissue samples with the progress of cancer. With the new light from our calculations, we may able to exactly calculate the double value of the fractal dimension for same scattering strength, and another set of fractal dimension values, and to address the correct values of the fractal dimensions, if they exist other side of the turning point.
Funding
National Institutes of Health (R01EB003682, R01EB016983).
Acknowledgements
Partial support by National Institutes of Health (NIH) grants (R01EB003682 and R01EB016983) and Mississippi State University to Dr. Pradhan are acknowledged.
Disclosures
The authors declare no conflicts of interest.
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