Reevaluation of radiation reaction and consequences for light-matter interactions at the nanoscale

In the context of electromagnetism and nonlinear optical interactions damping is generally introduced as a phenomenological, viscous term that dissipates energy, proportional to the temporal derivative of the polarization. Here, we follow the radiation reaction method presented in [G. W. Ford and R. F. O'Connell, Phys. Lett. A, 157, 217 (1991)], which applies to non-relativistic electrons of finite size, to introduce an explicit reaction force in the Newtonian equation of motion, and derive a hydrodynamic equation that offers new insight on the influence of damping in generic plasmas, metal-based and/or dielectric structures. In these settings, we find new damping-dependent linear and nonlinear source terms that suggest the damping coefficient is proportional to the local charge density, and nonlocal contributions that stem from the spatial derivative of the magnetic field and discuss the conditions that could modify both linear and nonlinear electromagnetic responses.

velocity in the absence of an externally applied force. A comprehensive review of the subject for charged particles in harmonically varying fields in classical, quantum, relativistic and nonrelativistic domains may be found in [4] and references therein. An additional controversy over radiation emission when the charged particle is placed in a constant, uniform accelerating field was resolved after some debate [5]. That notwithstanding, the question has still not been fully resolved, and an equation of motion that accurately describes the motion of a radiating charge is still not available.
From a classical standpoint the electron is generally described as a point particle with rest mass m0. The essence of the problem begins with the original derivation summarized in the Abraham-Lorentz equation of motion, which takes the form [4][5][6][7] where the dots indicate full time derivatives, and (1) may be found in references [4][5][6][7]. A method alternative to Eq.(1) that avoids unphysical, runaway solutions may be obtained by assuming the electron has uniform charge density over its volume, and thus finite radius, by defining the reaction force in terms of the applied force as follows [9,10]: where 13 0~1 0 c cm τ − is the classical electron radius. Although Eq.(2) had originally been presented as a Taylor series expansion [9], the authors in [10] showed that Eq.(2) is exact and valid in the quantum domain, provided a proper interpretations of the variables is made. Our present goal is to use Eq.(2) as the starting point to derive a new hydrodynamic equation of motion that includes linear and nonlinear damping contributions, and we contrast the result that emerges from our development with the hydrodynamic equation that follows from the usual introduction of a phenomenological damping coefficient [11,12], independent of the type of externally applied force. In addition, we will explore the consequences of using this approach to the dynamics that ensues in media whose physical description requires multiple polarization components.
We first develop the equation of motion of a charged particle and derive the electron polarization component for a nonrelativistic electron gas. Assuming the particle is acted upon by an external electromagnetic field that induces a Lorentz force, and that it is subject to internal electron gas pressure, Eq.2 becomes: where p n ∇ − is the pressure term, and we have used the total derivative identity ( ) on both sides of Eq. (2). An immediate observation that can be made in examining Eq.(3) is that the number of nonlinear source terms has more than tripled, a result that may be consequential at ultra-high intensities and/or in a relativistic context. After we identify the current density as where n=n(r,t), and its partial derivative with respect to time as  n is the charge density in the absence of applied fields, and is presently taken to be uniform throughout the volume. Assuming either an ideal or a quantum electron gas leads to the same lowest order, linear pressure contributions [12,13], which may be written for a quantum gas as follows: The last pressure term on the left hand side is purely nonlinear and its contribution is neglected in the development below. Retaining lowest order terms on both sides of the Eq.(4) we obtain: The surviving terms are the only linear terms proportional to 0 τ in Eq.3. Finally, using the Ampere-Maxwell's equation: to substitute for t ∂ ∂ E , we can rewrite Eq.(6) in a manner that allows us to clearly identify linear and nonlinear damping terms: For silver or gold, for instance, the data suggests for purely free electrons, with a characteristic length that approaches the size of the atom. We will return to discuss this issue below.
We now take a closer look at Eq. (8) where k is the spring constant, and the subscript b helps us differentiate between free and bound effective masses and characteristic times. Neglecting all nonlinear terms in both Eqs. (8) and (10) we determine the lowest order, linear contributions of damping and any interplay that may ensue between free and bound electrons. Multiplying Eq.(10) by b n e , where b n is now assumed to be constant for bound charges, and keeping only linear terms we obtain: 1 where the total polarization is the vector sum of the two components, Total b f = + P P P . We have adopted the subscript f to describe the free electron polarization. Combining Eqs. (8), (11) and (12) leads to the following coupled, linear equations of motion: coupling the two polarizations of the system. Clearly this is not the case in any traditional analysis, where one simply injects a phenomenological damping coefficient into each of Eqs.(13) [13][14][15], a procedure that conceals coupling and magnetic effects. We also note that, similarly to what occurs in the free electron portion of the material, damping introduces a nonlocal contribution in the dynamics of bound charges as well, in the form of a spatial derivative of the magnetic field.
An examination of the data of noble metals like silver and gold in the visible and UV ranges suggests that bound oscillators (d-shell electrons) may be damped at a rate Eliminating the free electron polarization and solving for the bound and free electron polarizations, respectively, yields: The bound electron polarization is explicitly influenced by the dynamics of free electrons, and vice versa. In this regard we note that at high frequencies metals are typically characterized by many bound electron resonances [16]. Also at high frequencies the number of (d-shell) electrons each atom contributes to the dielectric constant increases [17], boosting the effective bound electron as is usually done [13][14][15]. In summary, we have applied a new formalism starting with the equation of motion with radiation reaction/damping to derive a modified hydrodynamic equation where the damping term is a function of local charge density. The formalism also introduces new linear, nonlinear, and nonlocal source terms that depend on the spatial derivative of the magnetic field, which under the right conditions can influence both linear and nonlinear dynamics, e.g. circular dichroism and harmonic generation. We also showed that in a multi-component medium, well-represented by a typical metal at near-IR wavelengths and below, or by systems where both free electrons and ionic species are the result of ionization, as may occur at high intensities or in a relativistic regime [18], the new, more general formulation generates interference between the polarizations, triggering shifts in the position of the resonances and changes in the amplitude of the dielectric constant, a fact that requires a reassessment of the oscillator parameters that are used to fit any measured data.
Of some interest will be the development of Eq.(2) in the relativistic regime [19].