Numerical study of the degree of light scattering strength versus fractal dimension in optical fractal disordered media: Applications in strong to weak disordered media

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 are calculated using their light localization strength, using 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 the competition between increasing structural disorder due to decrease in fractality versus decrease in scattering centers due to decreasing fractality.

localization, or light localization strength of the optical disorder systems. The degree of structural disorder or disorder strength are measured in terms of the inverse participation ratio (defines later, termed as I 2 ) values of the optical eigenfunctions of Anderson-tight binding model (TBM) Hamiltonian of the systems, constructed under closed boundary conditions. While the fractal systems are represented as closed boundary lattice systems with fluctuating indexes of refraction, the I 2 calculation method is particularly useful as an entire system's localization strength can be characterized by a single positive real number, as the ensemble averaged I 2 ,, <I 2 > , is proportional to the degree of structural disorder of these systems. By comparing <I 2 > values to its average fractal dimension, a direct comparison between optical scattering based on scattering centers' density and fractality can be demonstrated.
In this paper, we show by numerical simulations and analysis, that the relationship between fractality and the degree of optical scattering is non-monotonous for both 2D and 3D fractal matrixes. We treat our fractal matrix as an optical lattice system, with its lattice points as the scattering centers, by appointing refractive index for each lattice points. Different fractal geometries result in different distributions of the scattering centers with long-range or power law correlations, which consequently leads to different degrees of disorder or disorder strengths.
Once we generated different fractal lattices, we use the Anderson-disorder TBM with nearest neighbor interactions, under closed boundary conditions, a well-studied method [8,9] to construct the Hamiltonian, H, for the system as shown in Eq.1.
In the Eq. (1), ) is defined as the optical potential of the site and t is the inter-lattice site hopping strength, which is restricted to only the nearest neighbors. |i> and |j> are the optical wave functions at the i and j lattice sites, respectively, and <ij> indicates the nearest neighbors. The eigenvalues and eigenfunctions (Ψ s) of the system are obtained after diagonalizing the Hamiltonian H. The I 2 is defined over a sample length L as: where Ψ i is the i th eigenfunction of the Hamiltonian of the sample size L. Ensemble averaged I 2 , <I 2 >, is proportional to the degree of disorder, the higher value, the higher the disorder in the system is [7].
The fractal dimension of the systems follows the scaling power law N ∝ r Df , with D f being the fractal dimension, and was taken by the well know 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 N-cubes [N(r)] with side length r, then the N-cube counting fractal dimension is defined as [2,10]: Counting the number of N-cube of different size (r), the fractal dimension of equation (4) can be estimated by the slope of the linear fit of ln(1/r) versus ln(N(r)) points. The N-cube sizes were simulated by sample size r×r, and sample length r = 1, (2) 1 , (2) 2 … 2 W , where W is an integer. This method proved accuracy within ±0.05 of the fractal dimension of deterministic fractals with known fractal dimensions, such as such as Sierpinski's Carpet and Sierpinski's Triangle.

II. METHOD
Generation of fractal samples through simulations: Below, we describe optical lattice generation by both the DLA ( fractal) and RC (randon cut/pseudo fractal) simulation methods.
(i) Diffusion limited algorithm method (DLA): Fractal structure systems are constructed by numerical simulations using a diffusion limited aggregation (DLA) algorithm, method is a well-known process found in a wide array of fields such as biology, chemistry, material sciences, etc., whereby particles undergoing a random walk cluster together to form aggregates [11][12][13]. The rules for DLA fractal growths are as follow: the simulation 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 ) [11]. Lower sticky probability values allow the particles to 'fall' into gaps inside the lattice, as shown in Fig.1.(a) and (b). If particle wonders a certain radius (kill radius) from the seed, the particle is destroyed and the process restarts.
(ii) Random Cut (RC) model: We also perform a simple random cut method starting from net filled matrix space for disorder system, that does not have exact fractal correlation, for a 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 point randomly. Two different stages are shown in Fig. 2 (c) and (d), with lesser number of cuts and a greater number of cuts, respectively. This method does not generate a true fractal but rather a random disorder but may not have the long-range spatial correlation decay length, volume occupied ratio which is a simple measurement of how "full" the system is, rather than the calculated fractal dimension.
Once we generated the 2D/3D, matrices, TBM Hamiltonians are formed. We kept the onsite

III. NUMERICAL RESULTS
For both 2D and 3D systems generated by DLA and RC methods are simulated for eigenfunctions. As can be seen in the localization.. The change of the energy ratio proved the most interesting in relationship to the turning point. For 2D cases, both methods increasing the ratio correspond, expectantly, to an increase in the maximum <I 2 > value as shown in Fig. 3 (a) for LDA cases and Fig. 3 (b) for 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 <I 2 (D f )> values at low ε/t energy ratio are not relatively strong compared to their minimal, and show instability below a ratio of 1, as the kinetic energy overwhelms the potential. For 2D lattice generated with RC methods, a bifurcation occurs at energy ratio between 2 and 8, resulting in two 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 D t =1.88±0.05, which corresponds to approximately half of the lattice sites filled. The DLA methods' only ever singular turning point rises and saturates after a ratio of 3 to D t =1.94±0.05 that, unlike its RC counterpart, persists indefinitely regardless of how large the ratio becomes.
This distinction between the two turning points of the RC method verses the singular for the DLA arises from the fact that, in terms of the I 2 (D f ), a random distribution will behave the same as the inverse of said distribution and as such, the bifurcation of the RC method is symmetrical around the number of particles present, i.e. the two peaks, the turning points, correspond to 20% and 80% of the lattice filled, with the trough corresponding to 50%. A phase diagram was constructed by plotting the energy ratio ε/t against the turning point D t for both 2D DLA and RC methods as shown in Fig. 5(a)..
The 3D systems present somewhat similar findings as shown in Fig.4  becomes (simulated to an energy ratio of 10 8 ), saturating to D t = 2.24 and 2.95±0.05, and is again symmetrical to the number of particles present in the system. The relative I 2 strength of the two turning points, the peaks, is much larger than the valley compared to its 2D counterpart as seen in Fig. 4 (a), however, the value of I 2 is lower in 3D, suggesting waeker localization in 3D. While 3D DLA, similar to its 2D counterpart, only ever displayed one turning point per energy ratio, a local maximum I 2 value manifests at fractal dimension 2.24 at ratios larger than 3, corresponding to the second turning point D t of the 3D RC method as seen in Fig. 4(d). Energy ratios below 1 display extreme instability turning points for 3D systems compared to their 2D counterparts. Fig. 5(b) shows the phase diagram of energy ratio against the turning point for the 3D systems.

IV. CONCLUSIONS
Overall, the Inverse Participation Ratio (IPR) coupled with fractal analysis provides remarkable analysis techniques for fractal disordered systems. Our preliminary numerical simulations suggest that system's fractal dimension can influence large amounts of optical localization in a system, and interesting optimal localization strength occur at certain constant hopping potential ratios in random scattering lattices. This is to be expected because of the large separation of energy states that are symmetrical for random constant potentials of complimentary particle lattice percentage occupation. We have extended our results from weak to strong disorder to probe the optimal localization, by this parametric localization.
Important of turning point, or optimal scattering fractal dimension (D t ) in biological systems for cancer detection: Most biological systems are weakly fractal disordered optical media, therefore, for biological system ε/t<1 values are more relevant for light scattering from these systems. 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 same D t for different ε/t values. It is known that tissues are optical fractal disordered media. The light scattering properties of tissues are important to understand the abnormalities in these tissues. The structural abnormalities or alterations progress with the progress of diseases such as carcinogenesis or brain abnormalities. There is no clear study to indicate which is the optimal dimension in 2D or 3D in tissues.
This means, due to the non-monotonous nature of the fractal dimension, the scattering properties can increase or decrease with the increase of the fractal dimension, depending on the side of the peak or optimal scattering dimension (D t ). Therefore, it depends on the initial tissue structure or its fractal dimension and the direction of the change of the fractal dimension.