Efficient method for the numerical extraction of Bergman’s spectral density function

. Effective properties of random particulate media are key to better understand their electromagnetic characteristics. For that purpose, many effective medium theories were developed. In this work, Bergman’s theory of homogenisation is explored. The core of this theory is its spectral density function (SDF), which contains the contribution of the mixture internal geometry on the effective properties. Here, we extract it with a standard but time-consuming homogenisation procedure, and a novel one, much faster. The two homogenisation methods plus the two extracted SDFs are then compared using a practical example showcasing excellent agreement between them.


Introduction
To define how light travels through a material, the knowledge of its permittivity () as a function of the wavelength () is essential.Although for homogeneous materials, such as glasses or polymers, it can be obtained from reflectivity measurements, for heterogeneous materials made of several phases only effective permittivities ( eff ) can be defined in the quasi-static regime.A number of theoretical models aim to predict these effective properties based on the knowledge of the properties of the constituent phases, let us mention the Maxwell-Garnett (MG) and Bruggeman theories.However, all these effective medium theories rely on approximation regarding the size/shape of the inclusions or the structure of the material.
On the other hand, Bergman's theory of twocomponent composites (a medium with   and inclusions with   ) does not rely on such approximations.By introducing a spectral density function (SDF) m, which accounts for the geometrical contributions, the effective permittivity is given by [1]: where  =   / (  −   ) is a complex number.The extraction of m from  eff is achieved for a specific range of ratio     ⁄ .Indeed, taking into account the inverse Stieltjes-Perron formula stating that [2]: it can be deduced that we must consider Im(    ⁄ )~0 and Re(    ⁄ )~1 − 1/(), with Re() ∈ ]0,1].Such permittivity variations potentially yield resonating electromagnetic interactions between heterogeneities [3].Accordingly, the representative volume element (RVE) required for homogenisation becomes large and thus includes a large number of heterogeneities to simulate [4].As a result, the simulations of  eff for 3D particles (spheres) is precluded and only 2D particles (circle) are considered in this work.Nonetheless, simulations are still demanding and necessitate powerful computing servers, the one we used have 1.5TB of memory and 16 cores.Therefore, in an effort to democratize Bergman's theory, we developed a novel method to simulate  eff , which relies on less tedious computation that can be deployed on an ordinary laptop computer.Finally, the SDFs resulting from the novel homogenisation method and a standard method are compared using a practical example.

System and homogenisation methods
The medium for which we want to compute the SDF is composed of circular particles with radius   = 0.1μm, randomly distributed with a filling factor   = 15%.In order to alleviate the aforementioned resonances, a minimal interparticle distance of 20nm is imposed.Particles are illuminated by a plane-wave with  = 10μm and transverse magnetic polarisation so that resonant surface modes are excited [5].
On the one hand, the standard homogenisation makes use of the coherent far-field flux of 1024 circular agglomerates with radius  = 10μm, see right inset of Fig. 1.Each agglomerate is a RVE of the random medium and is characterized by a different spatial arrangement of the ~1500 particles.By fitting the flux of a homogeneous circle to the coherent flux of the particulate system via a least-square method, the effective properties are retrieved.
On the other hand, the novel homogenisation is based on the idea that a particle with permittivity   , itself composed of particles of permittivity   and radius   (see left inset of Fig. 1), should not be detectable if embedded in the right effective medium.In other words,  eff is obtained by minimizing the coherent scattering crosssection of such composites: where 〈  〉 is the scattering coefficients of composites averaged over an ensemble.Ours is composed of 100 configurations, each made of only 10 interacting particles.

Results and discussions
The effective permittivity is extracted for 100 evenly spaced values of () in the range ]0,1], while Im(    ⁄ ) = 0.1.To compute the 100  eff values with the novel method, less than two hours were needed on a regular laptop.In comparison, the standard method took more than a month of computation on dedicated servers.Since m is a continuous function and only a finite number of  eff can be computed (100 here), extracting m from  eff is an ill-defined inverse problem.However, by applying the maximum entropy approach (MEA) [6], it is possible to obtain the least biased and most likely SDF based on the limited data available and the physical constraints (the integral of m equals   [1]).The SDF extracted from the standard and novel homogenisation are displayed in Fig. 1.There is a good agreement between them, and the approximate SDF is able to reproduce most of the features of the exact SDF.The fluctuation observed around () ∈ [0.2, 0.4] is caused by instabilities in the extraction of  eff causing overfitting of the MEA.Once the SDF of a particular structure is known, it is possible to predict the effective refractive index  eff of any two-component system by applying Eq. (1).As an example, let us consider particles made of Al 2 O 3 embedded into a SiO 2 medium.The refractive indices of each phase as a function of  (not shown) were obtained from reflectivity measurements on homogeneous material at our laboratory.The real and imaginary part of  eff in the 10 to 20μm spectral range, resulting from Eq. ( 1), is displayed with lines (solid and dashed) in Fig. 2(a) and (b), respectively.Little differences between effective properties extracted from the exact and approximate SDF are observed.To further validate the approach, the direct homogenisation of the system using either method is performed at different wavelengths; results are represented by markers (crosses and dots).The agreement between the prediction of Bergman's theory and the direct simulation is excellent, confirming that the SDF is a powerful predicting tool and that our novel method is able to accurately homogenise complex systems.As a comparison, the effective refractive index obtained via the MG formula is reported as black dotted lines.Overall, the agreement is acceptable, but in some spectral bands, it completely fails.A number of reasons explain such failure, the main one being that MG formula does not take into account the interactions between particles.

Concluding remarks
In this work, we successfully extracted the SDF of a random medium composed of nanoparticles via two different homogenisation methods giving similar results.The first one considers RVE of the medium and therefore can be considered exact, but each computation is memory and time consuming (~8h/simulation), whereas the second one relies on smaller volume, implying it is an approximation, but is much faster (~2min/simulation).A comparison of both methods is made on a system of Al 2 O 3 particles embedded into a SiO 2 medium showcasing excellent agreement.

Fig. 1 .
Fig. 1.The SDF extracted from the standard and novel homogenisation methods.Left inset: example of a composite with 10 particles inside.Right inset: agglomerate example.

Fig. 2 .
Fig. 2. The real (a) and imaginary (b) part of  eff extracted from the standard and novel homogenisation methods.Markers represent direct simulations, whereas lines are extracted from the SDF.Dotted lines correspond to MG.