Light wave driven electron dynamics in clusters



The dynamics of solid-density nanoplasmas driven by intense lasers takes place in the strongly-coupled plasma regime, where collisions play an important role. The microscopic particle-in-cell method has enabled the complete classical electromagnetic description of these processes. The theoretical foundation of the approach and its relation to existing methods are reviewed. Selected applications to laser cluster processes are presented that have been inaccessible to numerical simulation so far.

1 Introduction

When matter is exposed to intense non-relativistic laser fields, ionization takes place; the irradiated material is turned into a plasma, which absorbs energy from the laser field resulting in the heating and acceleration of charged plasma particles. Non-relativistic light-matter interaction encompasses a broad spectrum of applications. Lasers can be used for micro-machining and modification of materials, such as metals and dielectrics [1, 2]. Whereas micro-machining is of interest for high-precision industrial applications, material modification can be used to write channels into dielectrics for the realization of 3D microfluidic chips [3] or 3D integrated photonic devices for telecommunication [4]. On the other hand, the interaction of lasers and nano-objects, such as clusters and nano-layers, is of great interest for the realization of pulsed x-ray, electron, ion, and neutron sources [5]. In a laser-induced nano-plasma, a significant portion of the heated electrons leave, resulting in a positive charge up of the target and subsequent space charge acceleration of the ions. In such a hot and charged plasma, x-ray radiation is created by electron recombination. Nuclear processes can also take place during ion collisions resulting in the generation of neutrons and other nuclear particles. Finally—when irradiated at somewhat lower intensities below the damage threshold—nanostructures in general exhibit strong coherent field enhancement that is of interest for high-harmonic generation, low energy electron acceleration, and attosecond near-field microscopy [6-10]. These processes, as well as those mentioned so far, take place in the realm of strongly coupled plasma physics, where the use of traditional plasma tools—developed for weakly coupled plasmas—becomes questionable.

Modelling the interaction processes between laser light and strongly-coupled plasmas is challenging. To model strongly coupled plasmas, the classical trajectories of all electrons and ions have to be traced. On one hand, microscopic processes such as collisions have to be fully resolved, requiring a space resolution of about one atomic unit (0.529 Å). On the other hand, wave propagation phenomena need to be captured, which takes place on the order of the laser wavelength; modern experimental setups use light ranging from extreme ultraviolet (XUV) to mid-infrared (MIR) wavelengths (∼0.1 – 2.4 μm, respectively). For example, a solid density chunk of 1 μm3 holds about 1010 atoms. Thus, a microscopic description of nonlinear laser-plasma processes typically requires following the dynamics of 1010 particles with atomic scale resolution, along with light propagation (laser and scattered fields). Modelling these processes has become possible through the recent development of the microscopic particle-in-cell (MicPIC) approach.

This chapter provides an overview over the numerical method of MicPIC, its validation, and some applications. We focus on clusters exposed to intense light fields, as they present an ideal testbed for MicPIC for the following reasons. First, analytical solutions (Mie solution) exist, by which the validity of the MicPIC approach can be tested. Second, nano-plasma processes can be investigated over a wide range of sizes, changing the weight of plasma volume-to-surface processes. Third, laser-cluster interaction has important applications in the areas of nanophotonics, nonlinear optics, and strong-field laser physics [11].

The chapter is organized as follows. In Sec. 2, the validity ranges of the different theoretical approaches to classical light-matter interaction are reviewed. In Sec. 3, the formal theory behind MicPIC, its implementation, and validation are presented. In Sec. 4, the application of MicPIC to the microscopic analysis of light-matter processes in cluster nanoplasmas is explored.

2 Resolving light-matter interactions on the atomic-scale

Resolving light-matter interactions on the atomic-scale, both in terms of temporal and spatial resolution, promises an ultimate understanding of ultrafast optical phenomena. A complete microscopic analysis that would include all optical processes, from internal atomic dynamics to long-range light propagation, is impossible. In fact, rigorous modelling of atomic dynamics is a very demanding quantum mechanical problem that is limited to a few electrons. The situation changes at non-perturbative laser intensities where warm/hot plasmas are created by ionization that can be modelled with classical dynamics. In the following sections, the validity ranges of different approaches to classical light-matter interaction are reviewed. Finally, common tools used for computational classical light-matter interaction are presented and put in context with MicPIC.

2.1 Theoretical foundations of classical light-matter interaction

To understand the physical meaning of a strongly coupled plasma we briefly review the three main realms of plasma physics: weakly coupled, strongly coupled, and quantum (see Fig. 1). The relevant quantities for characterizing the nature and coupling strength of a plasma are the degeneracy parameter

display math(1)

and the Debye number

display math(2)

where math formula is the Fermi energy, math formula the Debye length, ℏ the Planck constant, math formula the electron mass, math formula the Boltzmann constant, ε0 the vacuum dielectric constant, and e the elementary electric charge. The parameters T and n are the electron temperature and number density, respectively. Physically, Θ quantifies the influence of the internal plasma pressure associated with the Pauli exclusion principle. In cold dense plasmas, where math formula, all particles tend to occupy the lowest system eigenstates. As the Pauli principle prohibits Fermions from going into an already occupied state, plasma dynamics is substantially changed by blocking effects and quantum mechanical screening, as compared to a classical plasma. In warm classical plasmas, where math formula, quantum effects of plasma electrons diminish and the relevant screening parameter is the Debye length math formula, that represents the scale over which the electrons screen the electric field, with math formula giving the average number of electrons that participate locally to the process. The Debye length and Debye number are closely related to the plasma parameter math formula that appears in the Coulomb logarithm for the evaluation of classical two-body collision integrals [12, 13]. When math formula, the plasma is hot and dilute. In this limit, plasma microfield fluctuations and particle collisions can be assumed as small perturbations and the system can be described by an effective mean-field one-body density. When math formula, the plasma is warm and dense and the system can only be described by rigorous microscopic N-body calculations.

Figure 1.

Insight into laser-plasma processes is gained when examined in terms of the corresponding electron temperature and density. The (dashed black) line math formula draws the limit between math formula regions where plasma processes are weakly and strongly coupled. Around the (solid black) line math formula and above, quantum mechanical effects come into play.

In the following we discuss the transition between the three plasma realms by the example of a metal at room temperature that is exposed to an intense laser field. At equilibrium, the lowest eigenstates of the metal are nearly fully occupied and the Pauli exclusion principle keeps electrons from going into already populated states. However, when the metal is excited by a laser pulse, the electrons absorb energy and the occupation of the lowest eigenstates becomes diluted. As a result, the Pauli exclusion principle loses importance and the dynamics can be approximated by classical equations of motions. During heating, the plasma goes first through the state of a strongly coupled plasmas (warm and dense, with math formula). This regime characterizes solid-density materials driven by relatively low-intensity lasers: during laser ablation, for example. In such a plasma many-particle collisions and the strongly fluctuating plasma micro-fields are important, which means that the classical equations of motion of all plasma particles have to be accounted for. When the plasma is heated further and expands, it goes over into a weakly coupled plasma (hot and dilute, with math formula), which can be modelled by mean-field transport equations, such as the Vlasov and Boltzman equations (see below).

An exact semi-classical analysis of light-matter interaction requires the self-consistent solution of the Quantum Liouville (von Neumann) equation coupled to the classical (not quantized) Maxwell equations. In a concise form, the Quantum Liouville equation reads [14]:

display math(3)

It describes, in the Schrödinger representation, the time evolution of the density matrix (or density operator) math formula of a quantum system characterized by the N-particle Hamiltonian math formula. Solving this fundamental set of equations is a formidable task. Therefore, different levels of approximation have been used so far to facilitate calculations. Different approaches, and how they arise from the Quantum-Liouville-Maxwell equations, are summarized in Fig. 2. Background information on statistical physics are found in Schwabl [14] and Landau-Lifshitz [12].

Figure 2.

Schematic view of the different levels of approximation in dealing with radiation phenomena in plasmas. Bi-directional arrows mark an equivalence while one-way arrows indicate approximations. Numerical techniques are in blue. Acronyms are given in the text. A similar series of approximations, not shown here, can be made to simplify quantum mechanical calculations.

Light-matter processes in non-degenerate plasmas (math formula) are well described by the classical Liouville equation:

display math(4)

Formally, it gives the temporal evolution of the microstate probability density math formula of a classical system characterized by the N-body Hamiltonian H. Here, math formula is a function of time as well as of the position and the momentum vectors of each particle in the system. For N particles, the problem has math formula dimensions (6N is phase space and 1 dimension in time) and

display math(5)

Following the evolution of a microstate phase-space trajectory given by the Liouville equation is equivalent to solving the 6N coupled classical equations of motion.

The collective response of a laser-driven plasma can be described by a one-body phase-space distribution function math formula, known as the lowest-order of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [15]. It is effectively obtained by integrating math formula over math formula dimensions, as shown in Fig. 2b. In this operation, the problem is reduced to 6 + 1 dimensions, but most microscopic features are lost. The Boltzmann-Maxwell (with collisions) and Vlasov-Maxwell (without collisions) equations are the two main equations obtained under these assumptions [12, 14]. Those are shown in Figs. 2c and 2d.

The probabilistic treatment of collisions in phase-space usually relies on the so-called small-angle binary-collisions approximations, which is valid for weakly coupled plasmas in which only two particles are involved in a collision and close encounters resulting in large angle deflections are rare. This is generally justified for gases but not for laser-driven solid-density plasmas that are characterized by strongly-coupled dynamics (see Sec. 2.3 for more details).

The Boltzmann-Maxwell equations can be further simplified by integrating out the momentum distribution and deriving the equations of moments with regard to math formula. The resulting two lowest order equations describe the evolution of the space-time dependent density math formula and velocity field math formula. This results in the Navier-Stokes equations (see Fig. 2e). The 3 + 1 dimensional hydrodynamic equations can be solved with computational fluid dynamics (CFD) techniques. In addition to the loss of microscopic collisions, the velocity distribution at a given point in space is replaced by a local, averaged particle velocity and temperature. Collisions can only be accounted for by introducing a phenomenological collision frequency ν.

At the coarsest level of approximation, one can assume that the fluid is homogeneous (math formula) and incompressible (math formula and math formula). Under the action of a weak electric field math formula, the assumption that all physical quantities also oscillate as math formula (this is called linearization) then leads to the following relative permittivity:

display math(6)

also shown in Fig. 2f. This particular result is known from P. Drude (1900), who similarly applied the kinetic theory to explain electrical conductivity in materials. The so-called Drude model is a cornerstone of the macroscopic electromagnetism theory. But as demonstrated, it is far from being a microscopic description of light-plasma interaction, as it involves many assumptions and approximations.

The analysis of light-plasma interaction is usually supported by numerical modelling and simulations. Figure 2 relates the various numerical methods to the various levels of approximation of plasma physics. CFD and Computational electromagnetism (CEM) are bound to modelling light propagation in materials via the use of macroscopic dielectric and magnetic constants. Microscopic features of weakly coupled plasmas are taken into account by the collisional electromagnetic particle-in-cell (PIC) method [16-18]. Electrostatic molecular dynamics (MD) [19] solves the N-body dynamics rigorously but neglects light propagation. MicPIC is effectively the only approach that bridges the microscopic and macroscopic realms of laser-matter interactions by solving rigorously and self-consistently the N-body problem corresponding to the classical Liouville equation coupled to the microscopic Maxwell equations. The three main approaches: MD, PIC, and MicPIC, and their relation to each other are described in more detail in the following sections.

2.2 Molecular dynamics

For small objects (and when the laser intensity is fairly weak), relativistic effects and the explicit treatment of the electromagnetic wave propagation can be neglected. In such cases, the classical plasma dynamics are described exactly by MD. This approach solves the equation of motion, (i.e., the Newton equation), for all the individual particles, where the force is the sum over all binary Coulomb forces (for example, see [20]). MD has been proven to work extremely well for small nanoplasmas in few-nanometer clusters, where the dipole and electrostatic approximations are justified. It has revealed the importance of microscopic processes, such as collisions and plasma microfields, in dense non-relativistic plasmas [11, 21, 22]. However, as soon as the object size is comparable to the wavelength in the medium, MD can not be used because light propagation has to be taken into account explicitly by solving Maxwell's equations. The onset of this “propagation regime” for the example of a resonantly excited cluster, i.e. the size where the electrostatic approximation breaks down, will be elucidated in detail in Sec. 4.3.

2.3 The particle-in-cell method

Electromagnetic PIC codes are the most common tools used to study wave propagation phenomena in the presence of free charges [16-18]. With this approach, particle motion is coupled to a numerical grid that samples the electromagnetic field. The temporal evolution is then calculated by solving Maxwell's equations via the finite-difference time-domain method (FDTD) along with the relativistic equations of motion for the particles [23-25]. This tool has literally revolutionized the understanding of relativistic laser-plasma interactions. Its potential is well exemplified through a major contribution to the development of compact laser-driven plasma accelerators [26, 27], to cite one success story in particular. However, PIC is also often used in cases where it is not justified and where only qualitative insight can be provided.

The general idea behind the PIC algorithm is to solve the Vlasov equation by decomposing the single-particle distribution math formula into phase-space subvolumes, where each subvolume effectively corresponds to a local collection of particles that have the same charge-to-mass ratio and move with the same velocity. When each macroparticle contains a fairly large number of physical particles, typically 104 − 106, they can be represented by continuous charge distributions. In this context, the collision between two PIC particles represents, in fact, a collision between two clouds of charged particles. This statistical continuum, or “coarse-grain”, approach eliminates the microscopic features of the laser-plasma interaction. This is justified in the highly relativistic regime and for dilute systems where collisions can be neglected. To some extent, microscopic interactions can be reintroduced in the form of Monte-Carlo binary collisions between plasma particles  [28, 29]. However, those “collisional” PIC codes remain bound to the weakly coupled regime, where microfield fluctuations are negligible and microscopic interactions are limited to small-angle binary collisions [30-33].

2.4 The microscopic particle-in-cell method

Recently, the microscopic particle-in-cell (MicPIC) method [34, 35] was introduced to overcome the limitations of the current modeling tools by connecting MD and PIC in a two-level approach. In MicPIC, only long-range electromagnetic interactions are treated on a coarse-grained PIC level. When two particles come close, the PIC field is replaced by the electrostatic one to fully resolve the microscopic (Mic) interactions. The MicPIC dynamics can be solved efficiently by using the particle-particle particle-mesh (P3M) concept introduced originally for electrostatic simulations by Eastwood and Hockney [36] and is capable of tracking 107 particles on a single CPU (∼1010 expected with parallelization). Microscopic resolution with comparable particle numbers was so far restricted to electrostatic P3M or tree schemes [37, 38], which however neglect laser propagation and magnetic fields. The key advantages of MicPIC over conventional PIC approaches are the atomistic resolution of the plasma dynamics (including the surface) as well as the capability to directly model strongly coupled plasmas. In particular, this method was used to explore lightwave-driven metal clusters and reveal the underlying electron dynamics with unprecedented detail [34, 35]. The most important aspects of MicPIC and its use for the microscopic analysis of radiation processes in plasmas is discussed in the remaining of this chapter.

3 Fundamentals of the microscopic particle-in-cell approach

In this section, MicPIC's theoretical background is presented, along with details of MicPIC's numerical implementation. The links to the electrostatic MD method and continuum models are also described. Those two traditional approaches give results that can be compared to those obtained with MicPIC in limiting cases: for objects with a size much smaller than the laser wavelength for which using the electrostatic approximation is justified and for weak-field excitation where the plasma optical response is linear, respectively.

3.1 Theoretical background

In MicPIC, each plasma particle is a single physical particle (electron or ion). To ensure numerical stability, these particles are described by a charge density, math formula, where math formula and math formula are the charge and position of the i-th particle and math formula is a normalized Gaussian shape function. The shielding parameter w0 “softens” the Coulomb forces and emulates the effective shielding of Coulomb singularities by quantum uncertainty and by the finite width of particle wavefunctions. This practice also extends to MD simulations. The effective width, w0, can be adjusted so that the minimum potential energy for an electron-ion pair is equal to a specified binding energy. It is set large enough to prevent classical recombination below the quantum energy levels.

Classically, the exact dynamics of the i-th plasma particle is governed by the force

display math(7)

obtained by averaging the electric (math formula) and magnetic (math formula) fields over the particle charge density. The self-consistent evolution of the fields is given by the microscopic Maxwell's equations:

display math(8)
display math(9)

where the current density, math formula, depends on the velocities, math formula, and effective charge densities, math formula, of plasma particles.

Departing from the above exact classical description, the particle-in-cell (PIC) approximation represents each particle by a wider Gaussian charge density, math formula, with math formula. The larger PIC particle width is key to an efficient solution of Maxwell's equations as it allows the use of a coarse numerical grid and large time steps. But as explained in the introduction, PIC underestimates and softens the fields of charged particles close to their origin. Therefore, PIC electric and magnetic fields are smoother than the actual fields, i.e., they do not contain field microfluctuations. Still, they evolve as

display math(10)
display math(11)

The PIC fields are driven by the current density math formula. With PIC, radiation fields are fully accounted for, however the microscopic nature of the particles is lost due to the large particle size. The PIC force on the i-th particle is

display math(12)

To identify and approximate the missing short-range forces, the actual force on particle i can be formally split into a microscopic portion, math formula, and a long-range PIC portion, math formula. This is done by adding/subtracting the PIC forces from/to the full expression in (7). This splitting has motivated the acronym MicPIC and yields

display math(13)

This expression is identical to the force in (7). The short-range character of the microscopic contribution, math formula, becomes evident after decomposing the PIC and the actual electric/magnetic fields into their individual particle contributions

display math(14)

This sum describes the force on the i-th particle created by the fields of all other (j) particles. Such a decomposition can be done because of the linearity of Maxwell's equations. Fields and charge densities shown with and without the “pic” superscript refer to the PIC and actual fields and charge densities, respectively. It should be stressed that the dynamics of the plasma particles are described exactly - independently of the width of the particles on the PIC level. The value of the PIC particle width wpic determines only the softness of the Coulomb force on the PIC level which, in turn, provides the radius within which the microscopic forces contribute.

For every particle j that is far enough away from the i-th particle (math formula), the actual and PIC fields produced in the region math formula are identical. Further, the variation of the (actual and PIC) fields over the PIC particle extent can be approximated by a linear Taylor expansion around math formula. Due to the even symmetry of the charge density, integration over the linear field terms gives zero. As a result only the constant field terms remain and can be pulled out of the integral. The remaining integral over the actual and PIC particle density gives zero for each index j, as their total charge is equal. This proves the short range nature of the microscopic correction.

Up to this point, everything has been derived in full generality. Now, as the only formal approximation in MicPIC, we assume that the local interactions within the microscopic correction volume are non-relativistic and electrostatic. This greatly facilitates the numerical evaluation of Eq. (14). Neglecting magnetic fields and expressing electric fields by the respective Coulomb interaction yields

display math(15)

For Gaussian distributions, the double integral can be evaluated analytically to give the difference of the particle interaction energies for actual and PIC particles. The interaction energy of two Gaussian particles with width parameter w reads

display math(16)

and allows rewriting (15) as

display math(17)
display math(18)

Here math formula is the inter-particle distance. By combining the electrostatic microscopic correction in (17) and the PIC force in (12), the final MicPIC force reads

display math(19)

The MicPIC formalism is now complete, with the plasma dynamics described and determined by the self-consistent integration of Newton's equations of motion with the force specified in (19), together with (10) and (11).

Finally, the total energy in MicPIC is evaluated as follows:

display math(20)

where the individual terms on the right-hand side describe the kinetic energy, the PIC-level electromagnetic energy, the energy resulting from microscopic correction, and the energy renormalization to remove the spurious self-energy of the particles on the PIC grid.

3.2 Numerical implementation

Numerical efficiency is the key to large-scale simulations. In the following, we demonstrate that best scaling for particle-based simulations – linear with particle number N – is achievable with MicPIC. The two conceptually different parts – the grid-based evaluation of long-range interactions and the gridless description of microscopic corrections – are linked in a two-level scheme similar to the famous P3M technique, which was originally developed for electrostatic PIC codes [36]. Update of particles position and velocity is performed using Boris' scheme that solves the Lorentz force Eq. (17). For the electromagnetic field update, we use the FDTD method [39]. Tunnel and electron impact ionization were included following [40]. MicPIC's flowchart is given in Fig. 3.

Figure 3.

Organization of time-stepping in the MicPIC algorithm.

3.2.1 The electromagnetic solver

To solve Maxwell's equations, the electric and magnetic field components are calculated using the FDTD method on a coarse, equidistant, three-dimensional mesh [39]. The time step is chosen according to the Courant criterion, math formula, with c as the vacuum speed of light, and math formula as the spatial step size. For the incident laser field, a flat beam profile (plane wave) is assumed. In order to separate the internal and induced fields (we define the sum of both as the scattered fields) from the (trivial) incident beam, we apply the total-field/scattered-field scheme [39]. Using this decomposition, it is sufficient to propagate only the scattered fields, which substantially reduces the load on the absorbing boundaries. On the surface of the simulation box, a uniaxial perfectly-matched layer (UPML) absorbs the outgoing scattered fields [39].

3.2.2 Gaussian-shape particles and microscopic force correction

The MicPIC ansatz requires a low-noise method to link the particles to the FDTD mesh. For strongly coupled plasmas – where microscopic plasma processes are important – the traditional particle-in-cell/cloud-in-cell (PIC/CIC) approach (cf. [41] for historical notes) is inappropriate because it suffers from excessive interaction screening, numerical heating, and force anisotropy. To solve these issues, MicPIC makes use of Gaussian shapes, as originally proposed by Eastwood and Hockney [36]. Low noise, negligible self-force, and weak anisotropy of the force produced by the Gaussian shapes effectively allows for the implementation of microscopic corrections with a finite cutoff radius, which is essential to simulate large numbers of particles.

In electromagnetic PIC codes, spurious numerical noise is generated when abrupt changes in the tracked quantities occur [17]. This is the case for the PIC/CIC charge-sharing scheme [18, 42], where particles are modeled by a top-hat distribution whose value and derivatives are discontinuous at the edges. As such a particle moves through the numerical grid, the current in adjacent cells is switched on and off abruptly. This creates periodic bursts of high-frequency radiation which are not well resolved by the numerical mesh. The resulting noise background hides the microscopic plasma features. The use of smooth Gaussian shape functions – along with the efficient absorbing boundaries mentioned in the previous section – solves the problem (see Fig. 4).

Figure 4.

Comparison between the inter-particle force produced by the top-hat [18, 42] and the Gaussian distribution functions (force normalized to the value for Coulomb at unit cell distance math formula). The Gaussian function minimizes the fluctuations in the long-range force, down to the limit of the absorbing boundaries (for details on the uniaxial perfectly-matched layer (UPML), see [39]). In the periodic cases (without UPML), forces reache a constant value due to the self-action of the particle's periodic images.

In MicPIC, the Gaussian-shape particle method is implemented with the following support function

display math(21)

with the following normalization:

display math(22)

In three dimensions: math formula. The usual way to define the mesh weights/currents in cell-based PIC codes is to calculate, respectively, the amount of charge within and the amount of charge that travels into/out the cells touched by the particle. The Gaussian weighting is motivated differently, that is, by a local sampling of the charge and current densities produced by the i-th particle at a grid point rg via

display math(23)

Errors which result from sampling the Gaussian functions with a finite number of points can be limited (intuitively) by using enough sampling points and an appropriate ratio of the PIC particle size wpic, in relation to the grid spacing math formula (for details, see supporting material of [34]). For math formula, which is used for simulations later on, the evaluation of weights only for grid points closer than math formula from the center of the particle conserves 99.998% of the total particle charge.

The main barrier to the implementation of microscopic corrections in MicPIC is the inability of a square mesh to efficiently represent a spherically symmetric particle. This issue is usually referred to as anisotropy and manifests itself as an inter-particle force that depends not only on the inter-particle distance but also on the angle that makes the inter-particle distance vector with respect to the numerical PIC mesh [36]. Using wide Gaussian PIC particles allows to reduce the anisotropy to the level where the inter-particle force in all directions can be fit with a unique two-body potential (see Fig. 5a). This allows for the removal of the PIC-level force which is then replaced by the correct (analytic) microscopic force via local MD (see Fig. 5b). As a result of this operation, MicPIC particles appear to their neighbors as nearly symmetric and effectively much smaller than the actual mesh size. Through this step, the microscopic plasma features are recovered on the particle level.

Figure 5.

Force fit and correction for two 7-point Gaussian particles with math formula as a function of the inter-particle distance. The force is rescaled to the Coulomb force at a distance of math formula. (a) The force when the particle separation is along a cell edge and a cell diagonal is fit using math formula, where math formula is the two-body potential defined at Eq. (16). (b) The fitting two-body force allows for the correction of short-range interactions within a finite cutoff radius.

In MicPIC, the cutoff radius must be made large enough so that the transition between the Mic and PIC forces is smooth enough to avoid numerical heating and ensure global energy conservation. It was found that a cutoff radius of math formula is sufficient for most applications. Finally, the physical particle size w0 is adjusted so that the minimum potential energy for an electron-ion pair is equal to a specified binding energy, but set large enough to prevent classical recombination below the quantum energy levels. For an electron interacting with Ar1 + or Xe1 + , this typically means an effective particle size of math formulaÅ.

3.2.3 Linear scaling with MicPIC

In MicPIC, the microscopic correction, (i.e., the particle-particle portion of the P3M decomposition), is conveniently implemented as local MD, on relatively small subsets of the total particle ensemble. This is possible because the microscopic correction forces are short-ranged and binary forces need to be evaluated only within a finite sphere of radius rcut. To evaluate the microscopic correction within the respective cutoff spheres, particles within the cutoff radius, rcut, have to be identified. To do so efficiently, the cell index method was applied [43], which results in linear scaling with particle number N per step. Note that a direct treatment (comparison of all particle pairs) would result in an math formula scaling and would destroy the linear scaling of MicPIC.

Like in PIC, mesh-particle size and grid spacing must be fine enough to resolve the length scale of field propagation processes. Apart from that constraint, math formula and wpic can be chosen freely, without any limitation on the number of PIC particles per grid cell. This freedom can be utilized for load balancing between the Mic and PIC routines. By exploiting the fact that the cutoff radius, PIC particle width, and grid spacing are chosen in a fixed relation to each other, the total numerical effort for the propagation of a fixed time interval can be expressed as [34, 35]

display math(24)

where the parameters a1, a2, and a3 are constant prefactors for the microscopic correction, the current injection/force evaluation (on the PIC level), and the solution of Maxwell's equation, respectively. For very small/large values of rcut, load is distributed dominantly on the PIC or Mic parts. Hence, the cutoff radius (or equivalently, the PIC particle width) determines the numerical balancing between the microscopic and PIC parts. A value for the PIC particle width was selected to provide the fastest code execution. This typically leads to wpic values smaller than those needed to resolve the physical processes.

3.2.4 Typical numerical parameters

The results presented in Sec. 4 were obtained with the following parameters. We used a 3003-cell PIC mesh with mesh size math formulaÅ, a UPML boundary condition [39], a PIC particle width math formula, and a cutoff radius math formula. This results in almost equal numerical load between the Mic and PIC portions of the code, for the cases where the particle number is the largest. Note that the numerical effort is independent of actual particle widths, which is of the order of math formula Å  for the electrons and ions in our study. Other particular values are given in the discussion of the considered scenarios.

3.3 Link to molecular dynamics

For small objects, MicPIC can be compared to MD. This permits the identification of propagation-effect signatures and defines regimes where the electrostatic approximation does hold. For this chapter, the same MD code as in [40] was used. Also, to ensure a fair comparison between the two approaches, the same representation for the plasma particles was used for both MD and MicPIC: Physical particles are modeled by Gaussian charge distributions with width w0. It is important to remember that MD takes into account the electrostatic interactions between all particles, resulting in an N2 scaling. It also includes the external laser field in the dipole approximation. Accordingly, the MD force on the i-th plasma particle is given by

display math(25)

and is evaluated using massively parallel computation techniques. In MD, the energy absorption is given by the total energy difference before and after laser excitation. It should be stressed that MD is only valid in the limit of weak light scattering – defined as light scattering where source depletion and energy loss due to the re-emission of light can be neglected.

3.4 Link to continuum models

To study the light which has been scattered by an ionized cluster in the linear excitation regime, it is convenient to represent the nanoplasma as an homogeneous metallic sphere of radius R. This problem can be modeled with continuum theories with the help of the following Drude-like relative dielectric constant:

display math(26)

with χ0 a real-valued background susceptibility, ν the collision frequency, and math formula the plasma frequency.

To include both electron-ion (math formula) and electron-surface (math formula) collisions in the cluster response, we use a cluster-radius-dependent collision frequency, math formula. Here ν0 is the regular bulk collision frequency (math formula) and math formula describe the additional surface collisions (math formula) [44], where ν1 is an effective electron velocity.

Such a macroscopic dielectric function takes into account microscopic effects via an effective collision frequency and linear susceptibility. This approach allows for their inclusion in macroscopic theories (such as Mie and nanoplasma [45, 46]), for the modeling of laser absorption and scattering by a cluster.

In Mie theory, the frequency-dependent absorption and scattering cross sections are defined as [45, 46]

display math(27)


display math(28)

The Mie coefficients, math formula and math formula, and the details of their calculation can be found in supplementary literature (cf., [45, 46]). In the small-sphere limit, the Mie theory reduces to the quasi-electrostatic Rayleigh theory [46]. In this limit where propagation effects are neglected (indicated below by superscript “stat”), the single-frequency absorption cross section becomes

display math(29)

Note that this result is equivalent to the heating rate used in the well-known nanoplasma model [47]; therefore we will denote the small-sphere electrostatic limit as nanoplasma theory from here on. The corresponding small-sphere expression for the scattering cross section is

display math(30)

We emphasized that the pole in Eqs. (29) and (30) is connected to the famous Mie plasmon resonance, describing a resonant collective dipole oscillation of the conduction electrons in a sphere. In fact, for math formula, the absorption and scattering cross sections diverge due to the plasmon resonance. Both cross sections, however, remain finite for finite collision frequency. For small background susceptibility (math formula), the absorption cross section is peaked at the classical Mie plasmon frequency defined as math formula, irrespective of the value of the collision frequency.

Neither MD analysis nor the nanoplasma model account for propagation effects, thus they apply only to the small cluster limit (math formula). However, as MD and the nanoplasma model are based on the same approximations, they describe the same physical scenario and must fit, even outside their range of validity. On the other hand, both the Mie theory and MicPIC model include propagation and are valid for all cluster radii. However, the Mie theory is valid in the linear response regime only. Finally, the microscopic parameters of the dielectric function in both continuum theories (such as the collision frequency) had to be determined and are used as fitting parameters. It should be noted that the dielectric function for both continuum approaches must be the same. Finally, to compare Mie/nanoplasma model with MicPIC/MD numerical data for excitation by short laser pulses, the cross sections must be averaged over the pulse spectrum. For a Gaussian pulse math formula with carrier frequency ω0 and duration τ, the normalized intensity spectrum is math formula. The effective cross sections then follows from

display math(31)

Normalizing the cross sections to the geometrical cross section of the sphere yields the single frequency

display math(32)

and spectrally averaged efficiency factors

display math(33)

for absorption and scattering.

4 Microscopic analysis of laser-driven nanoclusters

Laser light interaction with nanometer-sized atomic and molecular clusters offers various opportunities to explore and control ultrafast many-particle dynamics. The size of clusters can range from that of molecules to small droplets of condensed matter. This scalability makes them a valuable testing ground in nanoplasma science [11] and ultrafast nanoplasmonics [48], where laser-driven phenomena emerge from the interplay between microscopic and macroscopic processes. Although the pivotal role of resonant collective electron excitation in clusters is well accepted [49-51], little is known about how collective enhancement proceeds in large clusters, where electromagnetic propagation effects can not be neglected.

For the first time, MicPIC allows a fully microscopic analysis of ultrafast strongly-coupled light-matter processes. In particular, for the investigation of the linear, nonlinear, and non-perturbative excitation regimes of rare-gas clusters driven by few-cycle infrared laser fields. The main model systems considered in this book chapter are pre-ionized clusters that are resonant with near-infrared laser radiation (math formula nm). Such clusters have to be created in a pump-probe experiment by laser excitation followed by cluster expansion. Metallic clusters provide high valence electron densities already at low temperature where quantum effects – that are not yet included in MicPIC – are essential. This important question is being investigated.

Due to the different timescales involved – sub-femtosecond for ionization to tens or hundreds of picoseconds for cluster explosion – it is still demanding to model real pump-probe experiments in a single simulation. Instead, the pump and probe scenarios are modeled separately here to highlight the respective main features resulting from propagation effects. For the pump phase (Sec. 4.1), a small Ar cluster was simulated (math formula), starting from its neutral ground state configuration, to demonstrate the capability of MicPIC to model overdense, strongly-coupled plasmas including atomic-scale ionization processes. It shows the applicability of the method to typical dynamical aspects of laser-cluster interactions, such as microfield effects in the ionization dynamics, hydrodynamic nanoplasma expansion, and Coulomb explosion. For the probe phase (Secs. 4.24.5), which was studied in both the linear and nonlinear excitation regimes, singly ionized spherical nanoplasmas with a density chosen such that the Mie plasmon is in resonance with the driving (800 nm) laser field were considered.

In the linear response regime (Secs. 4.2 and 4.3), a systematic analysis of radius-dependent resonant absorption and scattering of laser energy is provided. The comparison between MicPIC and MD simulations shows that electrostatic MD loses its validity when math formula, where R is the cluster radius. It is here that the propagation regime is entered, and consequently effects like radiation damping, field attenuation (skin effect), as well as polaritonic plasmon red-shifts begin to substantially influence the optical response. Peak absorption cross sections of up to five times the size of the geometric cross section indicate the occurrence of a pronounced absorption paradox in resonant clusters with sizes of some tens of nanometers. By fitting the MicPIC and MD data to the corresponding continuum approaches – the Mie theory and the nanoplasma model, respectively – it is possible to determine, quantitatively, the contributions of both the bulk and surface collisions to the overall collision frequency, as well as their respective influence on the nanoplasmonic response (Sec. 4.4). In fact, it is observed that the contribution from surface collisions remains important even when the effects associated with light propagation come into play. Based on the results, the maximal plasmon lifetime can be related, again quantitatively, to the competition of surface collisions and radiation damping. These results are general and true for all types of clusters – assuming they were pre-ionized and properly expanded before being resonantly pumped.

In resonant nanoplasmas excited by high laser intensities (Sec. 4.5) – where the nanoplasma response is no longer linear – electrons can undergo extreme collective motions [34, 35]. In this nonlinear regime, insight into the internal dynamics can be gained by looking at the scattering spectrum. The MicPIC analysis allows to reveal the origin of the different harmonics created at even and odd multiples of the laser's fundamental wavelength. The particular case of the second harmonic is studied in detail and it is shown that the interplay between electromagnetic and collisional processes plays an important role in obtaining quantitative predictions of the nonlinear susceptibility of clusters.

4.1 Nanoplasma formation in a small rare-gas cluster

In this section, focus is on describing the full plasma dynamics (including the laser-induced metallization) of an initially neutral rare-gas cluster exposed to an intense laser field. The main goal is to demonstrate that MicPIC is capable of describing strongly-coupled plasma dynamics, including ionization. For this benchmark scenario, a spherical math formula nm Argon cluster (math formula) was used. This had an face-centered cubic (FCC) structure and atomic Wigner-Seitz radius math formula Å. A width parameter of math formulaÅ is used to prevent classical recombination below the quantum mechanical energy levels. For cluster excitation a 25 fs laser pulse (800 nm) with peak intensity math formula is considered. Atomic ionization dynamics, including tunnel and electron impact ionization, is implemented according to [40]. This takes into account the effects of ionization threshold lowering due to local plasma fields. It should be noted that ionization is described with full atomic scale resolution, (i.e., each atom/ion is tested for ionization separately). Since propagation effects are negligible for this cluster size, the results can be compared to MD to validate MicPIC. Figure 6 shows the cluster dynamics calculated with both methods and demonstrates excellent agreement.

Figure 6.

Time evolution of ArN math formula under a 25 fs laser pulse (800 nm) at intensity math formula including ionization and starting from the neutral cluster ground state as calculated with MicPIC (solid) and MD (dashed); (a) kinetic and absorbed energies, (b) cluster radius extracted from the ionic rms radius and charge density in units of the critical density, (c) inner ionization with contributions from tunnel and impact ionization, (d) Debye Number. Note that MicPIC and MD results show excellent agreement with deviations of the order of the linewidth. In panel (a) Etot, Ekin, tot, Ekin, ion and Ekin, el denote the total absorption of the cluster, the total kinetic energy, the ion kinetic energy, and the electron kinetic energy, respectively. Reproduced from [35]. ©IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Published under a CC BY-NC-SA licence.

About the physics of this benchmark scenario, the following conclusions can be drawn (see Fig. 6). First, plasma formation starts with tunnel ionization in the leading edge of the laser pulse (see Fig. 6c). The threshold behavior reflects the high nonlinearity of the tunnel ionization rate as function of intensity. The ignition due to tunnel ionization produces only a few electrons which are then efficiently heated to launch an impact-ionization avalanche, which rapidly produces a highly overdense nanoplasma (about 60 times overcritical) (see Figs. 6b and 6c). Note that electron impact ionization is by far the dominant ionization channel, contributing about 90% of the ionization events. Because of the overcritical nature (the cluster efficiently screens the external laser field), heating due to inverse bremsstrahlung is expected to proceed mainly at the cluster surface (surface heating). During the pulse, the nanoplasma electrons are heated moderately. This leads to an electron temperature of about 20 eV at the end of the pulse. The high electron density combined with the moderate temperature results in a strongly-coupled nanoplasma. The evolution of plasma coupling is indicated in Fig. 6d in terms of the Debye number (see Eq. (2)). The calculated evolution demonstrates that a strongly coupled plasma is formed shortly after initial ionization.

The fact that strong coupling persists for the rest of the dynamical evolution requires a fully correlated treatment of the microscopic collisions to describe the dynamics correctly. Note that the concept of Debye screening as a linear screening theory breaks down for strong coupling and is used here for the identification of the coupling regime only.

Focusing on ion dynamics, cluster expansion begins near the peak of the pulse. This expansion leads to a conversion of the electron's thermal energy into ionic kinetic energy via adiabatic expansion cooling (hydrodynamic expansion), see Figs. 6(a) and 6(b). However, the increase of total kinetic energy (for both the electrons and ions) shows multiple forces contributing to expansion. Coulomb repulsion occurs due to outer-cluster ionization (Coulomb explosion), and charge spill out at the surface contributes to the expansion. The cluster expansion efficiently lowers the nanoplasma density, which is about 20 times overcritical at the end of the simulation. After substantial further expansion, the cluster is expected to reach the conditions for Mie plasmon resonance – defined as 3ncrit.

The excellent agreement with MD results demonstrates that MicPIC correctly accounts for microscopic processes (ionization and collisions) and is capable to efficiently model strongly coupled nanoplasmas. A comparison of the necessary computation times for both methods (1 day on 512 cores for MD; 14 days on a single core for MicPIC) highlights the striking numerical performance of MicPIC over MD already for this moderate system size. For larger systems, which will become accessible after parallelization of the MicPIC code, the numerical advantage will be even more pronounced. Most importantly, because of the inclusion of field propagation effects, only MicPIC will be applicable.

4.2 Cluster dynamics in the linear response regime

The linear response regime of the resonant nanoplasma system is studied with a laser intensity of math formula. The time evolution of selected simulation observables is displayed in Fig. 7 for a small (R = 10 nm) and a larger (R = 30 nm) resonant cluster. Both the small and large cluster results show the excitation of pronounced dipole oscillations of the cluster electrons. The resonant nature of the interaction is reflected by an increase in the dipole amplitude even after the pulse peak. The decay of the oscillations after the laser pulse reflects damping of the collective motion via electron-ion and electron-surface collisions, as well as radiation damping. While collisional damping converts energy from the collective motion into thermal nanoplasma energy, part of the collective excitation energy is removed from the system via the emission of light. Signatures of both effects can be found in Fig. 7.

Figure 7.

Time evolutions of dipole moments (a-b) and energies (c-d) for resonant excitation of 10 nm (left column) and 30 nm (right column) clusters in the linear response regime (math formula) as calculated with MicPIC (solid lines) and MD (dashed lines). In the bottom panels Ekin, Epot, and Etot denote the total kinetic energy, the potential energy, and the total energy of the cluster, respectively. Ekin, col shows the kinetic energy of the center- of-mass motion for the electron cloud. For better comparability, all data are normalized to the number of electrons N. The given plasmon energies and lifetimes are results from fitting a damped harmonic oscillator to the decaying dipole moments (fit regions as indicated). Reproduced from [35]. ©IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Published under a CC BY-NC-SA licence.

First, the conversion of collective excitation energy into heat can be directly extracted. The collective center-of-mass motion of the electron cloud periodically converts potential into kinetic energy, and vice versa. As this process decays, thermalization occurs, and once completed the energy stored in the collective motion has been fully transformed into heat, see Figs. 7c and 7d.

Second, the importance of radiation effects as a function of cluster size is revealed by the dipole signals and energies calculated by MD and MicPIC. For the 10 nm cluster, both approaches are in almost perfect agreement; however, they differ substantially for the 30 nm system. The total cluster energy after the laser pulse is higher for MD than for MicPIC, as MicPIC provides additional accuracy by accounting for the attenuation of the laser field in the cluster due to the skin effect. In fact, in the case of the 30-nm cluster, the cluster diameter is comparable to the skin depth. Further, the MicPIC dipole signal amplitude reveals a faster decay for the larger cluster than for the smaller one. Both differences are due to energy lost from the cluster by radiation. Finally, the frequency of the dipole oscillation after the pulse is slightly lowered in the MicPIC data, which can be attributed to the polaritonic red-shift of the Mie plasmon via field retardation [44]. The difference between MicPIC and MD final total energies is of the order of the total absorption. This demonstrates the large error incurred by a pure electrostatic treatment of the 30 nm system. It is thus estimated that the neglect of propagation effects limits the MD analysis to small clusters where math formula.

4.3 Linear absorption and scattering of light

For a systematic analysis of propagation and surface effects, a radius-dependent comparison of MicPIC and MD results against the predictions of Mie theory and the nanoplasma model was performed. In order to make this comparison possible, the parameters of the dielectric function presented for the continuum models [eq.  (26)], had to be determined from the absorption and scattering efficiencies obtained from MicPIC and MD simulations. The total scattered radiation energy is measured using the Larmor formula [45]

display math(34)

where math formula is the dipole moment of the cluster. The absorbed energy is given by the difference in the total energy before and after the laser excitation. It should be stressed that the determination of absorption and scattering cross sections with MD is only valid in the limit of weak light scattering, as the energy loss due to the re-emission of light is not taken into account.

The absorption and scattering efficiencies as obtained from MD, the nanoplasma model, MicPIC and the Mie theory are plotted in Fig. 8. MD data points are given only up to 30 nm, as this was the maximum size which could be simulated given the available computation resources (1 week on 1024 processors).

Figure 8.

Absorption (Qabs) and scattering (Qsca) efficiency of metallic clusters excited at resonance (800 nm) by 7 fs laser pulses with peak intensity math formula. The efficiency is defined as the absorbed and scattered energy normalized by the pulse fluence and geometrical cluster cross section. Reproduced from [34]. © (2012) by the American Physical Society.

As the excitation takes place in the linear response regime, the results provided by MicPIC for any cluster size can be directly compared with the predictions of the Mie theory [45], which fully accounts for propagation effects – including skin effect, field attenuation, and radiation damping. In the small-cluster limit (math formula), propagation effects are negligible, and MD describes the classical cluster dynamics exactly. Similarly, in the same limit, the Mie theory simplifies to the nanoplasma model. The nanoplasma model can be compared to MD, as it uses the same approximations. Matching the nanoplasma model to the results from MD provides the coefficients for the real-valued background susceptibility, math formula, the bulk collision frequency, math formula, and the effective electron velocity, math formula, as a measure of the contribution of surface collisions to the absorption. Using these parameters in the Mie model yields an excellent agreement with MicPIC over the whole 0 to 40-nm size range. The agreement of the microscopic simulations with the respective continuum model – moreover based on the same dielectric function – proves the consistency and the validity of the MicPIC approach.

The impact of light propagation can be extracted directly through the comparison between the electrostatic and electromagnetic descriptions. For example, for clusters in the size regime math formulanm, MicPIC absorption is substantially reduced when compared to the prediction of MD. This reflects the onset of field attenuation and radiation damping as clear signatures of electromagnetic propagation effects. Values for Qabs beyond unity for math formulanm imply absorption cross sections higher than the geometric cross section of the sphere. In analogy to the extinction paradox  [52], we refer to this phenomenon as to the absorption paradox. Maximum absorption that occurs at math formulanm indicates an optimal cluster size for efficient resonance-enhanced absorption for the investigated pulse parameters. For substantially larger clusters, the strong increase of field attenuation and radiation damping leads to a rapid decrease of the absorption efficiency.

The main results obtained from the comparison in Fig. 8 are the macroscopic material parameters in the dielectric function. These parameters quantify the effect of microscopic processes in continuum models. Without knowledge of their magnitude, continuum models are confined to qualitative predictions. So far, experimental measurements or theoretical determination of optical material parameters has been proven difficult. Here, MicPIC opens up new capabilities. In particular, when two or more different processes contribute to a parameter (such as bulk and surface collisions in a cluster or nanoparticle) it is very difficult to unravel the various processes and determine their relative importance. For example, the quantitative description of surface versus bulk collision effects in metallic nanoparticles and nanoplasmas (based on first principles and including the full electromagnetic response) has not, as of the writing of this paper, been possible. The new insights which can be gained from direct determination of the optical parameters will be demonstrated in terms of the plasmon lifetime below.

4.4 Competition of bulk and surface effects with radiation damping in resonant clusters

Using the parameters provided for the dielectric function determined in the previous section, the relative importance of surface and bulk effects can be studied quantitatively as function of the cluster radius. In particular, this can be done by comparing the absorption cross sections provided by the Mie theory with and without the surface collision term math formula introduced in the dielectric function defined in Sec. 3.4 (with numerical values given in Sec. 4.3). The case math formula corresponds to the bulk limit.

Figure 9 shows the radius-dependent spectral profiles of the absorption efficiencies. Also indicated is the intensity spectrum of the laser pulse used in the above analysis for the calculation of spectrally averaged efficiencies. Inspection of the absorption efficiencies in Fig. 9a,b shows that the peak efficiency (solid line) exhibits a pronounced polaritonic redshift with increasing particle size, irrespective of the surface damping term. However, the radius dependence of the peak value and width of the absorption efficiency profiles show a clear sensitivity to surface effects up to (about) a radius of 80 nm. In the absence of the surface term, the efficiency profiles increase more rapidly with radius for small particles, are narrower, and reach a substantially higher maximum peak value near 40nm when compared to the prediction including the surface contribution, compare Figs. 9a and 9b. This obvious deviation of the absorption properties highlights the striking influence of surface collisions.

Figure 9.

Spectral profiles of absorption efficiencies from Mie theory with full (a) and bulk-only (b) collision frequency as function of the particle radius (for parameters see Sec. 4.3). Panel (c) shows the corresponding size-dependent plasmon lifetimes extracted from a Lorentzian fit of the absorption profiles. Reproduced from [35]. ©IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Published under a CC BY-NC-SA licence.

The width of Mie plasmon resonance is an experimentally relevant and accessible parameter. To analyze the effect of cluster size on the linewidth, the absorption profiles were fit with a Lorenzian curve. The resulting plasmon lifetime (proportional to inverse linewidth) is displayed in Fig. 9c for both versions of the dielectric function. The following conclusions can be extracted from the comparison.

Including the bulk collision term only, i.e., if math formula, the lifetime is maximal for small particles and decreases monotonically with increasing cluster radius, due to radiation damping. In contrast, with the inclusion of the surface terms, i.e., if math formula, the lifetime exhibits a maximum for a finite, non-zero, cluster size. The decrease in lifetime for small particles directly reflects surface damping, which is increasingly important here because of the high surface-to-volume ratio. For large particles, the full result converges asymptotically to the bulk behavior. The maximum lifetime thus results from the competition of the surface and bulk collisions with radiation damping.

Maximal lifetimes of the Mie plasmon in the few tens-of-nanometers range have been observed experimentally for spherical noble-metal nanoparticles [53-55]. Both the measured optimal particle size and peak lifetime are of the same order of magnitude as the theoretical result. Deviations in the range of 50% are attributed to the model system. A more detailed analysis requires modeling of the particles at solid-density, investigations are under way.

4.5 Microscopic analysis of nonlinear light scattering

In the previous section, the resonant interaction of light with nano-clusters was discussed in the linear limit of weak laser intensities, in the 1011 W/cm2 range. It was shown that MicPIC determines the linear response in a quantitative manner.

In this section, the same model systems are studied in situations where they are pumped by higher laser intensities – in the 1013 W/cm2 range – where the cluster response becomes nonlinear. As measurement of cluster nonlinearities is difficult due to experimental uncertainties such as cluster size and density, not much quantitative information exists so far. Here, MicPIC is used to examine the magnitude of the nonlinear susceptibility of clusters. In the weak intensity limit, the Mie resonance strongly enhances the linear susceptibility (see Fig. 8). It is of particular interest now to examine the impact of the Mie resonance on the nonlinear susceptibility.

The harmonic radiation created during the resonant laser-cluster interaction is calculated in the following way. First, the total cluster dipole moment is determined by summing over the individual electronic trajectories,

display math(35)

where math formula and math formula are the positions of the center of the cluster and of the i-th electron at time t, respectively. Then, the harmonic power spectrum generated by the nonlinear motion of the electrons is obtained by inserting the Fourier transform of the total dipole moment into the Larmor radiation formula [45]

display math(36)

Figure 10 shows the harmonic power spectrum of a 30-nm cluster resonantly driven at 800 nm by a 7 fs pulse with peak intensity of 6 × 1013 W/cm2. We find two contributions to the harmonic spectrum: electronic motion in the polarization plane (math formula) leads to odd harmonics while motion along the pulse propagation (math formula, where math formula is the Poynting vector) leads to even harmonics.

Figure 10.

Power spectrum of the light scattered by a resonant 30-nm cluster from an ultrashort infrared (math formula nm) laser pulse (math formula W/cm2). Spectrum was obtained using equation (36), with the electronic polarization math formula defined in Eq. (35). (a) The collective electronic motion along the electric field polarization axis (math formula) exhibits a series of odd harmonics extending up to the ultraviolet range (math formula). (b) The asymmetric electron motion along the propagation vector (math formula) leads to even harmonics. Comparative runs where the magnetic field was neglected (labeled “no B”) or simulation conditions similar to MD were used (MD) indicate that breaking of symmetry is in good part due to propagation effects that are not accounted for by MD. In both (a) and (b), spectra are normalized by the maximum emitted power at math formula (the linear scattering peak).

The upper plot of Fig. 10 shows odd harmonics that are emitted parallel to the incident laser pulse (math formula). The nonlinearity of this process arises from the fact that the laser displaces the electron cloud relative to the ions. The electrons that are driven beyond the cluster surface feel a nonlinear restoring force, resulting in the creation of odd harmonic radiation [56, 57]. For the cluster sizes investigated here, propagation effects do not play a major role in odd harmonic generation, as they are well described by electrostatic MD (see Fig. 10a). It is therefore more interesting to focus on the even harmonics, which are not predicted by MD (see Fig. 10b).

The lower plot in Fig. 10 reveals a series of even harmonics that are emitted perpendicular to the propagation direction of the incident laser beam (math formula). The even harmonic radiation is created from two contributions: the Lorentz force (math formula) and the force due to the quadrupole electric field contribution [58, 59]. Both terms are not accounted for in electrostatic MD codes, as can be seen from the green line in Fig. 10b. Whereas the Lorentz term could be included into the electrostatic treatment, the quadrupole moment is a genuine propagation effect that is omitted in an electrostatic analysis. Furthermore, in the previous sections we have seen that a microscopic resolution of the surface is important to describe electron collisions and the Mie resonance correctly. Therefore, regular PIC codes can not be used to model harmonic generation as collisions are neglected. Currently, MicPIC is the only numerical tool that can account for all processes necessary to describe classical even harmonic generation in clusters correctly.

Until implementation of MicPIC, the magnitude of the two processes contributing to even harmonic generation discussed above was unknown. However, using this tool their respective weights can be determined by toggling the Lorentz math formula force term from on (red line) to off (blue line) for particle propagation. The difference of one order of magnitude in the 2nd harmonic reveals the dominance of the Lorentz contribution. For the fourth harmonic the situation is reversed; propagation effects dominate and the Lorentz contribution is negligible. To fully explore this phenomena would require a lesson on the underlying physics of two effects (necessary to determine the magnitude of each), and would go beyond the scope of this text. Nevertheless, MicPIC's potential as a tool for use in the identification of new physics as well as the generation of quantitative predictions has been demonstrated.

Along the same lines, further experimental runs were completed which used the same laser intensity but different cluster sizes. It was found that the extracted quantitative values for the second-order susceptibility were different. Details follow.

From the total cluster dipole moment p defined at Eq. (35), the effective dipole-polarization density of a single cluster math formula is defined, where V is the cluster volume. The rapid decay of the amplitude of the harmonic spectrum shown in Fig. 10 suggests the following series expansion

display math(37)

where math formula is the incident laser field. The expansion coefficients math formula define the linear (math formula) and nonlinear (math formula) susceptibilities. For background information on nonlinear optics and this common procedure, see [60].

To determine math formula, the power of the scattered radiation is calculated by inserting math formula – obtained via Eq. (37), with math formula defined by the proper analytical form of the driving laser pulse – into Eq. (36). Values for any of the different susceptibilities are then obtained by matching the harmonic peaks of that reference spectrum with the spectrum obtained from the MicPIC data.

The effective math formula of a single cluster is plotted versus cluster radius in Fig. 11. It is observed that there is a strong dependence of math formula on the cluster radius that follows a trend similar to that of the plasmon lifetime (see Fig. 9). This suggests that collision processes play an important role in determining the nonlinear susceptibility of clusters. Values for the effective second-order susceptibility math formula are on the order of math formula, which is of the same order of magnitude than typical bulk second-harmonic materials [60]. However, phase matching is not straightforward because the second harmonic signal is emitted perpendicularly to the driving laser field. Nevertheless, the scattered light associated with the second-order dynamics in clusters is an interesting avenue for diagnostics, but unlikely for efficient second harmonic generation (SHG).

Figure 11.

Second-order susceptibility of a preionized cluster driven at resonance by an ultrashort infrared (math formula nm) laser pulse (math formula W/cm2) as a function of the cluster radius. MicPIC predicts a strong dependence on cluster size for the effective single-cluster susceptibility. Values are on the same order as those of typical nonlinear crystals [60]. The analytical fit [math formula, where math formula with R being the cluster radius in nm] recalls the shape of the plasmon lifetime in Fig. 9.

5 Conclusions

In summary, the potential of MicPIC to quantitatively analyze optical phenomena in which both propagation and microscopic effects play a role was demonstrated. This new approach holds the potential to explore heretofore inaccessible regimes of laser-matter interaction – such as the microscopic analysis of laser machining and modification of solids/droplets with infrared (IR) to extreme ultraviolet (XUV) radiation [1, 2, 61, 62] – as well as to study optical processes in nano-photonics and nano-plasmonic structures [63], which can have a thickness of a few atomic layers only. Due to MicPIC's atomic-scale resolution, atomic-level processes are fully accessible, which allows for the quantitative determination of material susceptibilities in terms of microscopic parameters, such as the bulk and surface collision rates. As a particular example, MicPIC was used here to analyze the linear and (perturbative) nonlinear interaction regimes of clusters with near-infrared laser light, at resonance.

In the linear regime, it was shown that MicPIC can determine material parameters for systems that are strongly influenced by boundary effects, whose detailed account is key to modeling the optical properties of nanostructures. In particular, results revealed a pronounced maximum lifetime of the Mie plasmon on the order of 10 fs for cluster radii of a few tens of nanometers, which is in reasonable agreement with experiments for spherical noble-metal nanoparticles. Most importantly, it was found that the dependence of the plasmon lifetime on the cluster size is due to a competition between the bulk and surface collisions, in a range of sizes where the electromagnetic propagation can not be neglected.

In the nonlinear regime, MicPIC was applied to analyze harmonic generation in clusters during the pump stage of pump-probe experiments. It was shown that MicPIC can be used to identify the different physical processes contributing to the even and odd harmonic series. In particular, the magnitude of χ(2) of the model cluster systems was obtained as a function of the cluster size. Interestingly enough, results suggest that second-order scattering is influenced by both the bulk and surface collisions, again in a range of sizes where the electromagnetic propagation can not be neglected. The standard model for SHG in metallic nanoparticles at solid-density – which assumes that only the surface contributes to second-order scattering – is insufficient to explain SHG in warm nanoplasmas driven at the Mie resonance.

MicPIC's implementation – and how it is effectively capable of simulating light propagation phenomena with atomic-scale resolution – was outlined. MicPIC's parallelized version will be able to bridge the 4 orders of magnitude that separate macroscopic and microscopic processes of light-matter interaction by tracking up to 1010 classical particles in addition to the evolution of the electromagnetic field – this is roughly the number of atoms contained in a μm3-sized chunk of a solid-density material.

As shown, MicPIC is currently appropriate to model intense laser-matter interaction, where the laser turns the medium early into a strongly-coupled plasma whose dynamics can be described classically. Tunnel and electron impact ionization are included by rate equations. Other quantum effects – such as the presence of discrete energy levels and the influence of the Pauli exclusion principle – start to play a role at moderate intensities and cannot be neglected in general. In future works, it will have to be clarified to which extent, and how, quantum effects can be added to MicPIC to approach an exact description of nanoplasmonic and nonlinear optical processes. Once this is done, MicPIC will immediately find a wealth of new applications in various areas of ultrafast optics, making it the de facto tool to analyze ultrafast light-matter processes.


  • Image of creator

    Charles Varin received his PhD in physics from Laval University in 2006, funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). From 2006 to 2008, he pursued postdoctoral studies at the University of Ottawa, funded by the Fonds québécois de la recherche sur la nature et les technologies (FQRNT). Since then, he has been working as a postdoctoral research assistant with professor Thomas Brabec.

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    Christian Peltz has been studying physics at the University of Rostock and received his master degree in 2009. Since then he is working on his PhD project in the group of professor Thomas Fennel in Rostock. His main research interests are computational many-body physics, strong-field cluster physics, laser plasmas, and attosecond science.

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    Thomas Brabec has received his PhD and Habilitation from the Vienna University of Technology in 1992 and in 1998, respectively. Since 2002 he has been professor and Canada Research Chair at the physics department of the University of Ottawa. His areas of research are: nonlinear optics, strong field laser physics, attosecond science, and many-body classical and quantum dynamics.

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    Thomas Fennel has received his PhD degree in physics from the University of Rostock in 2005. He pursued postdoctoral studies at the University of Ottawa (2006-2007) and within SFB652 at the University of Rostock, where became assistant professor for theoretical physics in 2010. His research focuses on computational quantum and classical many-body physics, intense laser-cluster interactions, ultrafast x-ray nanophotonics, and attosecond science.