Bragg Fibers with Soliton-like Grating Profiles

Nonlinear dynamical system corresponding to the optical holography in a nonlocal nonlinear medium with dissipation contains stable localized spatio-temporal states, namely the grid dissipative solitons. These solitons display a non-uniform profile of the grating amplitude, which has the form of the dark soliton in the reflection geometry. The transformation of the grating amplitude gives rise many new atypical effects for the beams diffracted on such grating, and they are very suitable for the fiber Brass gratings. The damped nonlinear Schrodinger equation is derived that describes the properties of the grid dissipative soliton.


Introduction
Fiber Bragg gratings (FBG) are very popular in the contemporary applications.FBG is an extended Bragg reflection grating formed in a fiber core.FBG can be used as inline optical filters to block certain wavelengths, If the nonlinear medium possesses a nonlocal response, i.e. the matter grating is shifted relative to the light grating in a space, the localized stated are emerging in such system.They display the non-uniform profile of the dynamical grating amplitude.We have found previously that they are described by the equation similar used as inline optical filters to block certain wavelengths, or as a wavelength-specific reflector.The spectral selectivity of light beams reflected in FBG is due to periodic refractive index grating in the fiber core and beams' diffraction on this grating.FRG have narrow reflection spectrum and they are used in fiber lasers, fiber sensors and detectors, for stabilization and changes of the laser wavelength in laser diodes, and others [1].But always the amplitude of the grating is constant, and only the changes of the grating period may occur due to different external influences such as mechanical stretching or compression of the fiber, changes the temperature, etc.In our work we consider the case, what would happened if the amplitude of the grating is changed and what changes must be.We show that the answers to these questions may be found in consideration of the dynamic holography.The grid dissipative soliton can be formed on the Bragg reflection grating during the wave-interaction in a dynamic dissipative medium.These solitons are very suitable for the FBG.We show that the proposed dynamical system is an effective tool for modeling many effects of wave interaction in FBG.
The dynamic holography covers the following processes: formation of a light grating (the interference pattern) due to interaction of coherent optical waves in a nonlinear medium, creation of a refractive index grating by this light grating, and transformation of the interacting wave owing to their self-diffraction on this grating.In the dynamical regime both the light grating and the material grating mutually influence to each others and both are changed in time and space.previously that they are described by the equation similar to the complex Ginzburg-Landau equation (CGLE) [2,3].The CGLE include dissipative soliton solutions that arise in non-equilibrium systems due to the balance between gain and loss [4].The photoinduced refractive index would form the soliton-like patterns.The first experimental observation of localization effect for the photoinduced refractive index has been done in wavemixing experiments in bulk LiNbO 3 crystals [5].The same effects should be observed for the case, when the nonlinear response has time-delay regarding the impact of a light-pulse.
In present paper we derive the nonlinear Schrodinger (NLS) equation in a specific case, namely when the space shift between the light grating and the refractive index grating is equal to the quarter of the grating period and the refractive index grating is formed in the reflection geometry.The obtained damped NLS equation describes the properties of the grid dissipative soliton.

Grid dissipative soliton
The condition of the formation of the grid dissipative soliton in the reflection geometry is shown in Fig. 1.It appears, when two coherent waves interfere in an extended nonlinear medium.The straight lines show the maximums of the intensity pattern.But, the nonlinear response is nonlocal, and the dynamical grating formed by this intensity pattern is shifted in the space on quarter of the grating period, or on π/2 by phase.The maximums of the grating are shown by the dashed lines.The waves experience the self-diffraction on the dynamical phase grating, which they are created themselves.As the result, both the light grating and the refractive index grating are changing in space and time.In this case the grating becomes with non-uniform amplitude.The amplitude is described by the dark-soliton solution.Then the grating will look like it is shown in Fig. 2, where iχ (3) indicates the nonlocal Kerr-like nonlinear susceptibility, and the arrow shows the spatial direction of the nonlocal shift.arrow shows the spatial direction of the nonlocal shift.

Derivation of damped NLS equation for grid dissipative soliton
To derive the damped NLS equation, which describes the The basic test of the creation of such a grating is the amplification of one a wave, or, to say by other words, the energy transfer between waves.Since the dynamical grating is shifted in a determined direction relative to the intensity pattern, the two interacting waves become not equivalent.The amplitude of the grating is determined by the intensity ratio of two interacting waves.Such wave is To derive the damped NLS equation, which describes the grid soliton, we consider the dynamical problem (see Fig. 4).It includes the interaction of two coherent plane waves from one laser source in an extended nonlinear medium.amplified, which has the wave-vector directed to the grating shift.So, in our pictures, Figs.1,2, the wave 1 is a signal, and the wave 2 is the pump.The best case is when the pump is much higher than the signal.Then the pump will transfer its energy to the signal, which will be amplified.In the case of equal input intensities, or when the signal is higher than pump, the grating amplitude is decreased and the profile is moved, as the result there is almost no energy transfer.
The grid dissipative soliton has steady state solution in a form of the dark soliton in reflection geometry: Where Q is the amplitude of the refractive index grating; γ is the coefficient of the nonlocal nonlinear gain; τ is Wave interaction in dynamical dissipative medium.
The physical mechanism of the nonlinearity is characterized by a physical value Q, for which we can write the evolution equation.The evolution equation for the amplitude of the nonlinearity in the simplest form, which includes the nonlinear gain proportional to the light intensity and the relaxation of the nonlinearity with the time constant τ, is the following: Also there are coupled-wave equations in the nonlinear medium, where the coupling coefficient is described by the function Q(t,z): (3) γ N is the coefficient of the nonlocal nonlinear gain; τ is the time constant; p is the constant, which depends on the input intensity ratio.
Interesting to note, to reach the optimal condition for the energy transfer, the medium should not have a strong nonlinear response.The calculations in Fig. 3 show that.These exists the optimal value of the nonlinear response, which is not very strong, when the created dynamical grating reaches its maximum.In the system (2)-( 3 The coupling coefficient may be function, which describes changes of dielectric permittivity.Also the system (2)-( 3) implies that there is the spatial shift between the matter grating and the light grating on the quarter of the period in the direction of z-axis [3].
The system (2)-( 3) of three equations can be rewrote in the form of one nonlinear equation of the second order for the function Q: (5) Further procedure permits to transform this nonlinear equation to the NLS equation.It would be convenient to use the presentation of the function Q in the following form: Q=(Q+Q * )/2, where we introduce an auxiliary function Q which is a complex-valued function, and Q * is its complex conjugate.Then we apply the reductive perturbation technique, in which we expand the function in the curved coordinates on a small parameter ε [6]: We consider the nonlinear equations that appears at different orders of ε.Note, the reductive perturbation technique is applied to transform the sine-Gordon equation into the Nonlinear Schrodinger equation [6].
For the first order of ε we obtain the wave equation for function F 0 , for which we can write solution in a form We obtain the NLS equation with exponential damping on the time in the nonlinear term: This equation represents the interest to further investigation in dependence on their parameters, as well as its connection with the experimental conditions.

Conclusion
We have shown that the nonlinear problem of the dynamic holography, which consists of the selfdiffraction of waves in a dynamical nonlinear medium, essentially includes dissipative soliton solution, which we called the grid dissipative soliton.This soliton describes a localized profile of the grating amplitude.The distribution of the light intensity in the interference pattern will have the same profile.In this case the deal is with the interference pattern, which can be either standing or running.The same are the matter wave in a form of a sinusoidal phase grating.
We have derived the damped NLS equation for the case of the self-diffraction of coherent waves in the reflection geometry in a dynamic nonlinear medium with time relaxation.The obtained NLS equation has the steady state solution, which represents a dark soliton.
The grid dissipative soliton is defined by the intensity ratio of input waves, and in its own turn for function F 0 , for which we can write solution in a form of a plane wave relative to the coordinates T 0 and Z 0 , but with the amplitude A depending on all others coordinates: where L is the operator, and q is the wave-vector of the grating.For the second order of ε we obtain the following equation with two operators L and M: from which we can assume the function F 1 is equaled to zero (F 1 =0) to avoid the appearance of secular terms, and for the function of A we obtain the following PDE: Finally, for the case of the third order of ε, we obtain the equation with two new operators N and R: from which we can assume the function F 2 is equaled to be zero (F 2 =0) for the same reason as we did for the function F 1 : to avoid the secular terms.We can apply new variables ζ and η determined by the following way: intensity ratio of input waves, and in its own turn determines the intensities of output waves.The dynamical case predicts many effects, which may be implemented in Bragg fibers in the condition when to crate the grating with non-homogeneous amplitude, but with a certain determined profile calculated from the dynamical task.Among such effects are a giant amplification of a seed pulse, amplification of individual pulses in a series, and the pulse retardation.We show that for modelling and prediction of new effects the considered dynamical problem is an effective tool.There are many direction to expand the dynamical task, in particularly, to consider the dynamic holography with not coherent waves.

Fig. 1 .
Fig. 1.The self-diffraction of two coherent waves in a dynamical dissipative medium in the reflection geometry.The curve illustrates the grating amplitude distribution.