#### 1. Introduction

The beginning of the last century was the fruitful period that has arisen the new branches of physics concerned in the structure of the matter, for example atomic and quantum physics, condensed matter physics. The interest to very promising properties of semiconductor materials resulted in development of physics of semiconductors that in turn caused the origin of the era of electronics. It is hardly imagine our daily life without devices containing integrated circuits. Revolutionary invention of the laser in the 1957 and the following progress in the integrated optics and optical fiber fabrication gave rise to the photonics. One of its branches, namely optical fiber communication, developed considerably during more than 30 years. In the beginning of conventional fiber optics the principal requirement to the fiber media was its ability to guide light pulses without distortion for the most long distance. The other inherent characteristics such as inhomogeneity, dispersion and nonlinearity mostly were considered as side and often undesirable kinds. The modern technology of fiber fabrication successfully overcomes all the difficulties caused by the material properties of the fiber: the profile of the fiber can be made uniform, the losses are very low (

In the conventional fibers light propagates due to the total internal reflection which is implies that the refractive index of the core is high comparing with the cladding. The common hollow core fibers are leaky for the light waves, however the improvement of their waveguide properties based on the Bragg reflection of the transverse modes was proposed [1]. The idea was to create a fiber with refractive index periodically changing in the cross section along the radius of the fiber. The solid-core concentric structure made using method of modified chemical vapor deposition (MCVD) was reported [2]. A detailed review related to the state of art in the fabrication and applications of the photonic crystals is given in [3]. The interesting idea of hollow core photonic bandgap fiber fabrication was reported in [4]. The photonic crystal fiber (PCF) preform was constructed from a bunch of stacked cylinder silica capillaries. The silica canes were removed from the center of the preform to create a hollow core structure. After being fused and drawn down the fiber with perforated cross section and the hole in the center was obtained. Another technique to produce PCF from the material with low melting temperature is the extrusion. Comparing to the previously described method the extrusion can provide a huge variety of the fiber cross-section profiles. The typical materials here are the polymers [5] as well as tellurite glass or chalcogenides. The advantages of hollow core PCF are in the enhancing of the nonlinear processes at high intensities (harmonic generation, stimulated Raman and Brillouin scattering). Having intrinsic filtering properties they also might serve as spectral filtering devices. Filling the hollow core of fibers with gases, molecular aggregates, bacteria or viruses opens implementations of hollow core PCF in the biology and in chemistry. The guidance of the dielectric polystyrene particles of micron sizes under radiation pressure of argon ion laser in the hollow core photonic bandgap fiber was reported in [6]. Advantages of this non-intrusive technique are self-evident.

The presence of the gap in the permitted range of the linear wave spectra that is arises due to the structure periodicity offered an incentive in fabrication of the fiber waveguides with its refraction index periodically varying along the fiber length. Fiber Bragg gratings are produced using phase mask created by electron beam lithography. In [7] the stitching error-free 100 mm phase mask was constructed for production of the fiber Bragg gratings. The new technique for simultaneous laser writing [8] of Bragg gratings by UV laser on photosensitive germanium doped silica was reported in [9]. The review on technology of fiber Bragg grating fabrication is given in [10].

The methods for fabrication the periodic structure with nanometer-scale thin films containing metallic nanoparticles or molecules are the electron beam deposition and layer-by-layer adsorption technique [11].

Though the linear properties of light coupled with excitations of the medium inspire many researchers in the optics of nanomaterials, the consideration of nonlinear response of the medium will lead to understanding of much more complex and intriguing optical phenomena. In [12] it was considered the phenomenon of second harmonic generation in the nonlinear medium containing resonant metallic nanoparticles, and it was found the solitary wave solution for the wave of medium polarization coupled with electric field of the light pulse. In the papers [13-15] the model is developed describing the ultrashort (comparing to the relaxation time in the medium) light pulse propagating through the ideal periodic Bragg grating with resonant metallic nanoparticles. The nonlinear dielectric properties of metallic nanoparticles implanted into the thin dielectric films, which periodically placed in the optical waveguide, are described by cubically nonlinear Duffing equation. This model of nonlinear oscillator also properly describes many other nanoaggregates having dielectric properties such as ensembles of quantum dots, atoms and dipole molecules. The authors of the works [13-14] found the solitary wave solution for the electric field of the light pulse coupled with nonlinear polarization of the medium. The solution exists even for the case of inexact resonance between the frequency of dimensional quantization of nanoparticles and carrier frequency of the electromagnetic wave, as well as for inexact resonance between the wave vector of reciprocal lattice - Bragg vector and the wave vector of the light wave. The excited state of the medium polarization coupled with the light wave is named polariton [16-17]. The dispersion relation for the model considered in [13-14] corresponds to polaritonic spectrum.

Recently there was a splash of interest in the localization of light in the disordered media [18-20]. It is noteworthy that dispersion relation will give the answer concerning the possibility of light localization in the medium. For example in the one dimensional model the presence of the defect in the lattice will gives the peak in the density of states. As the group velocity is defined by the formula

In the paper [23] a model of distributed Bragg grating was considered with taking into account inhomogeneity of density of resonant atoms in nonlinear medium to investigate numerically the propagation of polaritons, coupled solitary waves of polarization and electric field, in such inhomogeneous medium. In the numerical simulation a solitary wave solution derived by the authors of papers [13-14] is used as the initial condition to study the propagation and scattering of the solitary waves at the density defects. The three types of defects: microcavity - linear medium without resonant nanoparticles, groove - the defect span with reduced density of nanoparticles, and stripe - the defect with high density of resonant atoms were discussed. The nonlinear polariton transmission and scattering at defects, trapping of solitary wave in microcavity placed in the resonantly absorbing Bragg grating are demonstrated by numerical simulations.

The Bragg gratings are usually fabricated by laser writing on the photoresist with subsequent etching of the unexposed regions, or by the chemical layer deposition methods. In the fabrication process the inaccuracy of measurement of the distance between layers and inexactitude of the layer deposition is always exists. There are two types of the errors can occur: the former is when the error is determined only by measurement of the whole sample (laser writing). In this case during the fabrication process the error in the distance between resonant films is corrected in such a way that the error at the length of the sample is comparable to the measurement error. In the second case the error accumulates all deviations of lattice period if consequent deposition of the layers continues without control of the length of the waveguide, for example in layer-by-layer adsorption using method of modified chemical vapor deposition. If the layers are deposited one by one and the width of each dielectric layer is checked, the error in the total length will grow as a square root of the length of the sample. In this technological process the width of one layer is not correlated with the neighbors.

The purpose of the numerical simulation discussed in this work is to investigate the evolution of the ultrashort electromagnetic pulse propagating through the Bragg grating formed by the layers of resonant nanoparticles almost periodically placed in the host dielectric waveguide. As the method of the grating fabrication by the laser is more precise, here it is considered the case when the layers of resonant nanoparticles are arranged so that the distance between layers can be slightly small or large comparing to the average value of the lattice spacing, but the total error is not exceeds the error of measurement. The influence of such slight deviations is described by random multiplicative noise in the phases of the forward and backward components of the electric field of the electromagnetic wave.

#### 2. Methods

It is performed numerical simulation of ultrashort electromagnetic pulse propagation in the waveguide medium consisting of almost periodically placed in the host dielectric waveguide thin dielectric films with metallic nanoparticles. Periodicity of the waveguide results in strong scattering of the incident electromagnetic wave propagating in the Bragg grating. In the model equations both forward and backward components of electric field of electromagnetic wave are taken into account. The length of the ultrashort electromagnetic pulse with picosecond duration is equal to hundreds of double periods of the grating. The wavelength of the carrier wave of the light pulse is comparable with a period of the grating, which is defined by the distance between the adjacent thin films with resonant nanoparticles. Thus the condition of Bragg resonance is almost fulfils. The model equations describing polaritonic wave propagating in the periodic Bragg grating in the approximation of the slowly varying amplitudes were obtained in [13]:

Dimensionless variables

The solitary wave solution found in [13] reads as

Initial phases are set in a such way that

The model which describes ultrashort electromagnetic pulse propagation in perfectly periodic structure with spatially localized in the grating layer with inhomogeneity of density of resonant atoms in nonlinear medium was considered in the paper [23]. In the numerical simulation an exact solitary wave solution defined by equations (2) and (3) was used as an initial condition to study the propagation and scattering of the solitary waves at the density defects. The model equations were the following:

Distribution of nanoparticles density in different types of defects was defined by the parameter

It was shown in [23] that the slow solitary wave propagating through nonlinear RABG can be captured in a wide defect - microcavity placed in the Bragg grating. Inside the microcavity the light pulse propagates by reflecting from the boundaries of the defect, where its radiation partly scatters.

A thorough derivation of the continuous equations for the disordered media is a distinct and difficult problem to solve. Here is used the phenomenological model which accounts for the phase variations of the slowly varying amplitudes of forward and backward electric components of ultrashort electromagnetic pulse. These random phases assumed to appear due to randomness of deviations of the lattice constant. The equations of the model describing the ultrashort electromagnetic pulse propagation in the imperfect Bragg grating with resonant absorption are the following:

Wave vector detuning

The noise in the phase is considered as Gaussian and delta-correlated, correlation between the phases of wave at the points

In numerical simulations the condition of the exact resonance is assumed:

#### 3. Results

In the numerical statistical experiment the parameters (the amplitude, width – pulse duration at the half of the maximum, and position of the pulse maximum) of solitary polaritonic wave propagating in imperfect Bragg grating were measured. It was performed a set of 370 of realizations of the random grating characterized with the standard deviation

One can see that the scattering of solitary wave occurs on numerous fluctuations of the grating, while the polariton propagates in the random Bragg grating. To investigate the evolution of the nonlinear polariton in the random Bragg grating, the two moments (the average value and the standard deviation) of following parameters of solitary polaritonic wave were measured: amplitude and the width (defined as the pulse duration at the half of the amplitude maximum), position of the pulse maximum for the forward and backward electric field components

The amplitude and the width of the pulse propagating through the random Bragg grating were calculated and processed for all the realizations. It was found that the averaged values of the pulse amplitude and the width are oscillating with decay to certain value. In the set of pictures of Figs. 2-4 are represented the plots showing the evolution over time

The amplitude and width of the pulse are oscillating with the decay to certain value. The next plots at the Fig. 5 are representing the time evolution of the ratio

The ratio of the averaged over realizations amplitude and the pulse width tends to the initial ratio (corresponding to solitary wave in the model of perfect periodic Bragg grating) for the forward wave

At the Fig. 6 is illustrated the behavior of the second moments - standard deviations of amplitude and width of the pulse from their average values (first moments) calculated previously.

From the plots shown at the Fig. 6 one can see that for standard deviation in the phase

The probability density function (PDF) does not significantly depend on the number of the realizations for neither

#### 4. Conclusion

In this paper are presented the results of numerical modeling of polaritonic solitary wave propagation in the Bragg grating with random lattice spacing. It is shown that random fluctuations in the phases of electromagnetic waves resulting from variation of lattice spacing lead to periodic oscillations (with the period