^{1}

^{*}

^{1}

^{*}

^{2}

^{*}

In the present communication, the hydrodynamic model is used to investigate the amplitude modulation as well as demodulation of an electromagnetic wave of high power helicon pump wave into another helicon wave in strain dependent dielectric material incorporating carrier heating (CH) effects. The consideration of CH in modulation and demodulation is prime importance for the adding of new dimension in analysis of amplification of acoustic helicon wave. By using the dispersion relation, threshold pump electric filed and growth rate of unstable mode from the modulation and demodulation of the high power helicon wave well above from the threshold value will be discussed in the present analysis. The numerical analysis is applied to a strain dependent dielectric material, BaTiO
_{3} at room temperature and irradiated with high power helicon wave of frequency 1.78 × 10
^{14} Hz. This material is very sensitive to the pump intensities, therefore during studies, Gaussian shape of the helicon pump wave is considered during the propagation in stain dependent dielectric material and opto-acoustic wave in the form of Gaussian profile (ω
_{0},κ
_{0}) is induced longitudinally along the crystallographic plane of BaTiO
_{3}. Its variation is caused by the available magnetic field (ω
_{c}), interaction length (z) and pulsed duration of interaction (τ). From the analysis of numerical results, the incorporation of CH effect can effectively modify the magnitude of modulation or demodulation of the amplitude of high power helicon laser wave through diffusion process. Not only the amplitude modulation and demodulation of the wave, the diffusion of the CH effectively modifies the growth rate of unstable mode of frequency in BaTiO
_{3}. The propagation of the threshold electric field shows the sinusoidal or complete Gaussian profile, whereas this profile is found to be completely lost in growth of unstable mode. It has also been seen that the growth rate is observed to be of the order of 10
^{8} - 10
^{10} s
^{-1} but from diffusion of carrier heating, and that its order is enhanced from 10
^{10} - 10
^{12} s
^{-1} with the variation of the magnetized frequency from 1 to 2.5 × 10
^{14} Hz.

A number of extensive researches have been carried out from the different researchers in worldwide to investigate the effect of modulation and demodulation of high power pump wave from nano-pulsed laser action. From this interaction of electromagnetic wave in solid-state plasmas, the numbers of application have been raised to diagnose the metals, semimetals and semiconductor [_{3}) have been used to investigate the effect of modulation and demodulation of the plane wave in nonlinear dispersive medium by using the different approximation [

Much more attention has been paid by the researcher to find out the basic problem of the frequency and amplitude modulation in the gaseous plasma. The charge carriers due to ionization of the gas, from the interaction of the high power laser wave, are responsible for arising the plasma in gaseous state. In most of study of modulation and demodulation, interaction with nonlocal effects such as diffusion of the charge exciting the change in responsible parameters such as nonlinear refractive index has been ignored during calculation. Nonlinear polarizations due to acousto-optic interaction in dielectric and semiconducting materials are playing an increasing role in optical modulation, demodulation and beam splitting [

The charge carrier is easily excited in nonlinear dispersive media when an intense laser beam passes through it, and considerable heating raises the steady state temperature which is higher than that of lattice temperature. The carrier heating provides the momentum transfer collision frequency to an electron, thus modifying the mobility and diffusion of charge carriers, as well as conductivity of the dispersive medium by adsorption of ions from the gaseous state and hence it shows the refinement effect on modulation of amplitude or frequency. It is found that increasing the diffusion through charge carrier heating makes it more difficult to reflect or transmit the light from the local equilibrium which represents the unstable and stable TE and TM linear surface wave [_{0}, k_{1} ‖ ‖ B_{s}) magnetized semiconducting plasma [

In this present article, our analysis can be employed to see the effect of hot carriers on the amplitude modulation instability of an intense helicon pump wave due to acoustic-optic interaction in diffusive strain dependent dielectric constant. The dispersion relation can be solved by considering the complex relation for diffusion dependent dispersion

The hydro-dynamical model is considered for the numerical calculation for the propagation of high power helicon pump wave in one-component homogeneous n-type piezoelectric semiconductor plasma of infinite extent with electrons as major charge carriers, along the direction z of externally applied static magnetic field,

where,

where

electron plasma frequency. By using analysis of Fourier analysis, the modified Maxwell’s equations for unbounded

plasma is given by

tion is the direction of the wave vector k and whose magnitude is the index of refraction. The basic equation involve in the analysis are the zeroth and first-order momentum transfer, Maxwell equation, continuity equation and the equation of motion of the lattice of a piezoelectric semiconductor. The classical equation of motion for carriers of charges e and effective mass m is

If we assume a constant drift velocity

Equation (3) yields

Since the applied magnetic field B parallel to the

In general, when high intensity wave interacts with semiconductor which contained high mobility charge, they gain momentum and energy as a result from electron collision. This collision develops the heat from momentum transfer (MTCF) through the relation

The power absorbed per electron from the pump electric field becomes

where ^{*} denotes the complex conjugate of the quantity and re denotes the real part. This power is dissipated in collision of electron from the acoustic phonon in the Brillouin active medium. Following Conwell [

where

temperature of the medium.

to the ionic self-diffusion coefficient D (in cm^{2}∙s^{−1})

(7) and (8), we obtain the expression for electron temperature and modified diffusion coefficient as

The equation of motion for an element of volume dxdydz and density ρ is

The propagation of the acoustic wave (AW) in a crystal with SDDC is just possible by only longitudinal EKWs due to the piezoelectric effect, which is slightly induced by the transverse electric field of the helicon. By using the Poisson equation

The piezo-electrically excited longitudinal plasma oscillation can be obtained from Equation (13) by considering (3) and (4)

where_{0} being the equilibrium electron concentration and n_{1} the electron concentration perturba-

tion. Assuming that

where

If the AW frequency

frequency. The following procedure adopted by Ghosh and Agarwal [

in which

netic wave velocity in the crystal with lattice dielectric constant

As the density perturbation in the plasma has been assumed to vary as

in which

dispersive electron plasma wave and

cient. Equations (9) and (4) one may write

and the side band waves vary as

where

In the slow wave limit, it is the quasi-static approx

From Equation (15) in absence of piezoelectric coupling coefficient

We assume

and

where

Since

In the presence of laser field one can obtain the threshold value of the electric field necessary for the onset instability by putting

So the threshold value of the electric field is obtain by [

Above equation shows that for instability the condition _{3} high dielectric constant material. For a value of

In this section, the numerical results of the possibility of modulation instability and the amplification of the acousto-helicon wave which arising from interaction of the pump helicon wave with acousto-helicon wave have been analyzed through Equations (28) and (29) for strain dependent dielectric material. The carrier heating on acousto-helicon interaction modifies the dependent parameter such as electron momentum transfer collision frequency (MTCF) (Equation (7)) and diffusion of charge carrier at the different temperature (Equation (10)) and hence consequently modifies the threshold electric field and modulation of a high power helicon wave effectively. The modulation instability and the amplification of acousto-helicon waves from Equations (26) and (27) was solved numerically as results from the transfer of modulation helicon wave to acousto-helicon wave for the different values of the semiconductor plasma parameters such as_{3} cubic crystals at 300K, with following typical constant are taken:

_{3} is that its crystals structure is easily modified from the addition of impurities or creating the defect inside the lattice structure. Its easy tendency is due to formation of the solid solution with foreign atoms of the same size or fitted in the octahedral hole of the lattice arrangement. The electric field amplitude which is considered in the present investigation can be directly calculated from the pump intensity I_{0}, expressed by the following expres-

sion:

Now let us consider the important case where the mechanistic approaches of spatial lattice formation is due to the diffusion process of photo-excited electrons or holes from the surface of semiconducting material by irradiation of the pump wave. The intensity distribution from the diffusion process between the threshold electric field can be expressed

where

fastest imaginable diffusion process from the charge carriers may be due to free flight of the particles between the different lattice site of stain dependent material with the upper limit D, which can be expressed by the diffu-

sion coefficient of an ideal gas

ly. The thermo-dynamical equilibrium actually controlled the diffusion of the charge carriers by the jumping from one lattice position to other place by the interaction of acousto-helicon wave, then the residence time τ of

the particle on its site is given by

neighboring present sites. The hot carriers easily diffuse in entire lattice of BaTiO_{3}. The random walk theory in a 3d can be considered for the self diffusion process and it is calculated from the following expression

The entire diffusion process changes the modulation and amplification of the acousto-helicon wave which results from transfer of momentum and energy through pump wave. The numerical results are plotted in _{+} are plotted in

is solved by using Equation (28) after assuming that growth rate is directly proportional to

^{14} Hz. These modulations tend to be minimum at

The growth rate having the same profile like threshold electric field but their variation is not uniform above and below the propagating plane. They show the slight variation in magnetite below and above the plane at_{+}, is expressed by sum of the different phase based on the diffusion, as shown in

Pulse duration is important parameter to interact with the pump wave with semiconducting material. Not only change the interaction parameter, but also change the diffusion process of carrier heating by transferring the maximum amount of energy and momentum. This interaction changes the surface profile of modulation and amplification of the pump wave. The pulse duration modifies the carrier-lattice interaction by means of a collision time approximation. This effect is replaced

Similarly, Equation (29) is again solved for calculating the surface profile of threshold electric field for a BaTiO_{3} structure; in this case the profile is plotted from 10^{14} Hz to 2.5 × 10^{14} Hz.

by introducing the diffusion related parameter

cycle from electron-electron interaction, which changed the nature beyond the pump frequency_{3} with and without carrier heating (CH), Equation (29) are numerically used against the cyclotron frequency ^{14} Hz.

Similarly, Equation (29) is again presented graphically in ^{14} Hz to 2.5 × 10^{14} Hz. During investigation, the results are found that the growth rate is non-uniform and non-sinusoidal with cyclotron frequency. ^{14} Hz to 2.5 × 10^{14} Hz, whereas

Below the pump frequency, the mass dependence threshold electric field controls the growth rate and the amplitude modulation. This figure illustrate the variation of growth rate with cyclotron frequency, which shows that growth rate with positive and negative are always in phase but it does not follows the sinusoidal nature: hence it shows the frequency and amplitude modulation from diffusion of hot carriers.

The results of this paper suggest that the modulation and growth rate of an electromagnetic wave can be easily achieved in many ferroelectric materials such as BaTiO_{3}. The acousto-helicon is excited by the modulation of high power helicon pump wave in a longitudinally magnetized strain dependent dielectric material. The CH effect always increases the magnitude of the modulation and growth rate either interacting with change in interaction length (z) or oscillating the applied electric field by changing the pulse duration. The diffusion of carrier heating just modifies only the magnitude by retaining the shape of profile either in threshold electric field or growth rate. Hence, hot carriers are always much more favorable to modulate the pump wave via the interaction of acousto-helicon wave. This interaction parameter of the plasma media is very limited between the carriers heating and the lattice via diffusion process and is applied over a wide range of parametric wave number k or k_{+}. The sinusoidal nature is found only in threshold electric field; meanwhile this nature is completely lost in growth rate of frequency and amplitude modulation. Non-uniform nature of growth rate is observed during the change in cyclotron frequency from 10^{14} Hz to 2.5 × 10^{14} Hz.

The authors are very much thankful to Principal Govt. M.V.M College for encouragement.

ShivaniSaxena,SanjayDixit,SanjaySrivastava, (2015) Effect of Hot Carrier on Amplitude Modulation and Demodulation of Gaussian High Power Helicon Wave in Homogeneous Longitudinally Magnetized Strain Dependent Dielectric Material. Open Journal of Acoustics,05,139-152. doi: 10.4236/oja.2015.54012