KINETIC THEORY OF ABSORPTION OF ULTRASHORT LASER PULSES BY ENSEMBLES OF METALLIC NANOPARTICLES UNDER CONDITIONS OF SURFACE PLASMON RESONANCE

This paper presents a theory that allows one to calculate the energy absorbed by spheroidal metal nanoparticles when irradiated by ultrashort laser pulses of different duration in the region of surface plasmon resonance. Simple analytical expressions are obtained to calculate the absorption energy dependent on the frequency of carrier laser wave, on the pulse duration, and on the ratio between the axes of the ellipsoids. It is shown that at the frequency of the carrier (laser) wave, which coincides with that of the surface plasmon, the maximum absorption is observed for spherical nanoparticles. As the carrier frequency deviates from the surface plasmon one, two maxima appear in the absorption spectrum, dependent on the ratio of spheroidal axes: one corresponds to the elongated particles and the other to the flattened ones.


INTRODUCTION
Metal nanoparticles are interesting objects from both the point of view of condensed matter physics and the practical point of view. An increase in local electric fields near nanoparticles, due to surface plasmon resonance [1], caused by laser radiation, makes them useful for use in solar energy [2], biology [3], and medicine [4]. Metal nanoparticles serve as a basis for a new direction in the development of electronicsnanoplasmonics [5]. Since the beginning of the XXI century intensive research has begun on the creation of an element base for integrated circuits on plasmons, the use of plasmons in energetics, and the creation of surface plasmon amplification (SPASERsurface plasmon amplification by stimulated emission of radiation) analogs of a laser in which plasmons are used instead of photons [6]. Even more interesting is the behavior of metal nanoparticles in the field of ultrashort laser radiation. First, the short duration of such pulses (of the order of 10 -15 s) makes it possible to study the dynamics of electronic processes in metal nanoparticles, as well as all kinds of nonlinear optical phenomena. Second, the ultrashort laser pulse (ULP) contains almost all harmonics, including those that coincide with plasmon resonances, which make a major contribution to the absorption of light by metal nanoparticles. If the nanoparticle is spherically symmetric, it will be characterized by one plasmon resonance, and if the particle is ellipsoidal in shape, then there will be three such resonances. This feature of the absorption of ULP by metal nanoparticles will be studied in this paper. Another feature that effects the absorption is associated with the shape of nanoparticles. Earlier, it was shown that in the case of asymmetric metal nanoparticles, the optical conductivity, which is determined by the electrical absorption and the width of the plasmon resonance line, becomes a tensor value [8].
This paper considers the peculiarities of the absorption of ULP by metal nanoparticles with sizes much smaller than the free path of electrons in them. In particular, the energy absorbed by metal nanoparticles in the process of plasmon resonance was found, when the carrier frequency of ULI coincides with that of plasmon resonance, and when this frequency deviates from the resonant one.

MATERIALS AND METHODS
Let us consider the case when an ensemble of metal nanoparticles is irradiated with a laser pulse, the electric field of which is given by the following expression where ω0is carrier frequency of the electromagnetic wave (laser pulse), In addition to the electrical component, the laser pulse also has a magnetic component, which can be found from the corresponding Maxwell equation: , .

H r t E r t ct
, .

H r t E r t ct
The simplest relationship between the Fourier components of these quantities looks like: where 00 / m k k  unitary vector. Now we find the Fourier component of the electric field of the laser pulse (1), which can be written as follows: In the simplest case, when 0 0  , from (4) we will obtain: The electric field of the laser pulse induces inside the metal nanoparticle a potential electric field   , in E r t , and a magnetic vortex electric field   , vr E r t . If the characteristic size of the nanoparticle R satisfies the inequality 0 1 kR , then the Fourier coordinate dependence of the components of the electric and magnetic fields of the laser wave can be neglected [7]. For an ellipsoidal nanoparticle, this allows us to obtain the following expressions for internal potential is the complex dielectric constant of the metal nanoparticle, Ljgeometric factors (depolarization coefficients), , , , , , , vr j r t . Therefore, the total energy absorbed by the metal nanoparticle will be equal to [9] where Vis the volume of the nanoparticle, w(t)is the power absorbed by the nanoparticle.
In the general case, the current   can be found as a solution of the corresponding linearized Boltzmann kinetic equation which will be present in the following form: in vr Here γ is the collisions frequency of electrons in the particle bulk.
Equation (12) must be supplemented by appropriate boundary conditions. As such, we choose the condition of diffuse reflection of electrons from the inner surfer of the nanoparticle In (13), n v is the normal to the surface S component of the electron velocity. The substantiation of such boundary conditions can be found, in particular, in [11]. Equation (12) Dependent on the frequency of the laser wave, its polarization and the shape of the nanoparticles, the dominant mechanism of energy absorption in a nanoparticle can be both electrical and magnetic absorption [11]. On the other hand, dependent on how much the frequency of the incident laser wave differs from that of the plasmon resonance, individual or collective absorption mechanisms may be prevalent. In this paper, we consider the absorption of ultrashort laser pulses in the region of plasmon resonance. Thus, we will take into account the collective (plasmon) absorption mechanism, which is part of the electrical absorption. Therefore, magnetic absorption can be neglected by placing   ,0 vr Er   . In this case, the solution of equation (12) can be found as follows: Within the zero approximation for a small Now let us consider the energy absorption of laser radiation by metal nanoparticles under conditions of plasmon resonance. In this case, the predominant mechanism of energy absorption will be electrical absorption, because under conditions of plasmon resonance there is a collective mechanism of energy absorption. Thus, substituting (10) and (14) in (9), we can find the following expression for the total energy If the plasmon attenuation is small (18)

RESULTS AND DISCUSSION
To illustrate the effect of particle shape on the nature of the absorption of pulsed laser radiation, we limit ourselves to considering nanoparticles in the form of ellipsoids of rotation (spheroids). The geometric factors L included in (18) for nanoparticles having the shape of an elongated ( RR   ) or flattened ( RR   ) spheroid ( R  and Rhalf-axis along and across the axis of rotation of the spheroid) have the form [13]   We confine ourselves to considering metal nanoparticles of spheroidal shape and choose for convenience such a polarization of the electric field of the incident laser wave at which it would be possible to excite simultaneously plasmon oscillations both along and across the axis of rotation of the spheroid. If component 0 E is considered to be directed along and 0 E is transverse to the axis of rotation of the spheroid, then in the general case such polarization will be where is the angle between the axis of rotation of the spheroid and the vector of the electric field strength of the incident laser wave 0 E . Substituting (22) into (21), we obtain:

CONCLUSION
The kinetic theory of absorption of ULP by metal nanoparticles of ellipsoidal shape is considered. Simple analytical expressions are obtain that allow one calculating the absorption energy as dependent on the frequency of the carrier laser wave, the pulse duration and the ratio between the axes of ellipsoids (particle shape).
At the carrier wave frequency, which coincides with the surface plasmon frequency, the maximum absorption is observed for spherical nanoparticles. As the carrier frequency deviates from the surface plasmon frequency, two maxima appear in the absorption spectrum, which depend on the ratio of the spheroidal axes: one corresponds to the elongated particles and the other to the flattened ones. As the frequency deviates from the resonant one, the peak of absorbed energy first decreases in absolute value, then splits into two, and finally stabilizes for both elongated and flattened spheroidal nanoparticles.