When a monochromatic electromagnetic plane-wave arrives at the flat interface between its transparent host (i.e., the incidence medium) and an amplifying (or gainy) second medium, the incident beam splits into a reflected wave and a transmitted wave.
In general, there is a sign ambiguity in connection with the k-vector of the transmitted beam, which requires at the outset that one decide whether the transmitted beam should grow or decay as it recedes from the interface. The question has been posed and addressed most prominently in the context of incidence at large angles from a dielectric medium of high refractive index onto a gain medium of lower refractive index. Here, the relevant sign of the transmitted k-vector determines whether the evanescent-like waves within the gain medium exponentially grow or decay away from the interface. We examine this and related problems in a more general setting, where the incident beam is taken to be a finite-duration wavepacket whose footprint in the interfacial plane has a finite width. Cases of reflection from and transmission through a gainy slab of finite-thickness as well as those associated with a semi-infinite gain medium will be considered. The broadness of the spatio-temporal spectrum of our incident wavepacket demands that we develop a general strategy for deciding the signs of all the k-vectors that enter the gain medium. Such a strategy emerges from a consideration of the causality constraint that is naturally imposed on both the reflected and transmitted wavepackets.
A uniformly-charged spherical shell of radius $R$, mass $m$, and total electrical charge $q$, having an oscillatory angular velocity $Omega(t)$ around a fixed axis, is a model for a magnetic dipole that radiates an electromagnetic field into its surr
ounding free space at a fixed oscillation frequency $omega$. An exact solution of the Maxwell-Lorentz equations of classical electrodynamics yields the self-torque of radiation resistance acting on the spherical shell as a function of $R$, $q$, and $omega$. Invoking the Newtonian equation of motion for the shell, we relate its angular velocity $Omega(t)$ to an externally applied torque, and proceed to examine the response of the magnetic dipole to an impulsive torque applied at a given instant of time, say, $t=0$. The impulse response of the dipole is found to be causal down to extremely small values of $R$ (i.e., as $R to 0$) so long as the exact expression of the self-torque is used in the dynamical equation of motion of the spherical shell.
For an oscillating electric dipole in the shape of a small, solid, uniformly-polarized, spherical particle, we compute the self-field as well as the radiated electromagnetic field in the surrounding free space. The assumed geometry enables us to obta
in the exact solution of Maxwells equations as a function of the dipole moment, the sphere radius, and the oscillation frequency. The self field, which is responsible for the radiation resistance, does not introduce acausal or otherwise anomalous behavior into the dynamics of the bound electrical charges that comprise the dipole. Departure from causality, a well-known feature of the dynamical response of a charged particle to an externally applied force, is shown to arise when the charge is examined in isolation, namely in the absence of the restraining force of an equal but opposite charge that is inevitably present in a dipole radiator. Even in this case, the acausal behavior of the (free) charged particle appears to be rooted in the approximations used to arrive at an estimate of the self-force. When the exact expression of the self-force is used, our numerical analysis indicates that the impulse-response of the particle should remain causal.
We present a brief overview of Sommerfelds forerunner signal, which occurs when a monochromatic plane-wave (frequency {omega}={omega}_s) suddenly arrives, at time t=0 and at normal incidence, at the surface of a dispersive dielectric medium of refrac
tive index n({omega}). Deep inside the dielectric host at a distance z0 from the surface, no signal arrives until t=z0/c, where c is the speed of light in vacuum. Immediately after this point, a weak but extremely high frequency signal is observed at z=z0. This so-called Sommerfeld forerunner (or precursor) is highly chirped, meaning that its frequency, which is much greater than {omega}_s immediately after t=z0/c, declines rapidly with the passage of time. The incident light with its characteristic frequency {omega}_s eventually arrives at t~z0/vg , where vg is the group velocity of the incident light inside the host medium. Brillouin has identified a second forerunner that occupies the interval between the end of the Sommerfeld forerunner at t~n(0)z0/c and the beginning of the steady signal. This second forerunner is commonly referred to as the Brillouin forerunner (or precursor). Given that the incident wave has a sudden start at t=0, its frequency spectrum spans the entire range of frequencies from minus infinity to infinity. Consequently, the high-frequency first forerunner cannot be considered a superoscillation, nor can the low-frequency second forerunner be regarded as a suboscillation. The goal of the present paper is to extend the Sommerfeld-Brillouin theory of precursors to bandlimited incident signals, in an effort to determine the conditions under which these precursors would continue to exist, and to answer the question as to whether or not such precursors, upon arising from a bandlimited incident signal, constitute super- or sub-oscillations.
A recipe is presented for constructing band-limited superoscillating functions that exhibit arbitrarily high frequencies over arbitrarily long intervals.
Using the Finite-Difference-Time-Domain (FDTD) method, we compute the electromagnetic field distribution in and around dielectric media of various shapes and optical properties. With the aid of the constitutive relations, we proceed to compute the bo
und charge and bound current densities, then employ the Lorentz law of force to determine the distribution of force-density within the regions of interest. For a few simple cases where analytical solutions exist, these solutions are found to be in complete agreement with our numerical results. We also analyze the distribution of fields and forces in more complex systems, and discuss the relevance of our findings to experimental observations. In particular, we demonstrate the single-beam trapping of a dielectric micro-sphere immersed in a liquid under conditions that are typical of optical tweezers.
The force of electromagnetic radiation on a dielectric medium may be derived by a direct application of the Lorentz law of classical electrodynamics. While the lights electric field acts upon the (induced) bound charges in the medium, its magnetic fi
eld exerts a force on the bound currents. We use the example of a wedge-shaped solid dielectric, immersed in a transparent liquid and illuminated at Brewsters angle, to demonstrate that the linear momentum of the electromagnetic field within dielectrics has neither the Minkowski nor the Abraham form; rather, the correct expression for momentum density has equal contributions from both. The time rate of change of the incident momentum thus expressed is equal to the force exerted on the wedge plus that experienced by the surrounding liquid.
Two formulations of the Lorentz law of force in classical electrodynamics yield identical results for the total force (and total torque) of radiation on a solid object. The object may be surrounded by the free space or immersed in a transparent diele
ctric medium such as a liquid. We discuss the relation between these two formulations and extend the proof of their equivalence to the case of solid objects immersed in a transparent medium.
