Surface contribution to the electric dipole moment near an interface, and its effect on power emission.

When a small particle is located near an interface, its electric dipole moment can be induced by laser irradiation. Since the laser light reflects at the interface, this leads to an interference pattern, and the dipole moment is determined by this total field. In addition, the dipole radiation emitted by the particle reflects at the interface, and this field adds to the external field. In this fashion, the dipole moment is altered by the field it emits, and as such the emitted radiation modifies its own source. We have derived a simple expression to take this back action into account. We introduce two resolvent functions, Υ‖(h) and Υ⊥(h), which depend on the dimensionless distance h between the particle and the interface. These functions exhibit resonance features due to the underlying back-action mechanism. It is shown that two, one, or no resonance peaks appear in the induced dipole moment. Whether these peaks are present depends on the parameters under consideration. The power emitted by the particle depends on h due to interference between the source radiation and the reflected radiation. With the surface induced contribution to the dipole moment included, an additional h dependence appears. This dependence shows the resonance peaks, which may be amenable to experimental observation.

Propagation of magnetic dipole radiation through a medium.

An oscillating magnetic dipole moment emits radiation. We assume that the dipole is embedded in a medium with relative permittivity ϵr and relative permeability µr, and we have studied the effects of the surrounding material on the flow lines of the emitted energy. For a linear dipole moment in free space the flow lines of energy are straight lines, coming out of the dipole. When located in a medium, these field lines curve toward the dipole axis, due to the imaginary part of µr. Some field lines end on the dipole axis, giving a nonradiating contribution to the energy flow. For a rotating dipole moment in free space, each field line of energy flow lies on a cone around the axis perpendicular to the plane of rotation of the dipole moment. The field line pattern is an optical vortex. When embedded in a material, the cone shape of the vortex becomes a funnel shape, and the windings are much less dense than for the pattern in free space. This is again due to the imaginary part of µr. When the real part of µr is negative, the field lines of the vortex swirl around the dipole axis opposite to the rotation direction of the dipole moment. For a near-single-negative medium, the spatial extent of the vortex becomes huge. We compare the results for the magnetic dipole to the case of an embedded electric dipole.

Photon statistics of resonance fluorescence in the limit of separated spectral lines.

We have studied the statistics of fluorescent photons emitted by a two-state atom in a laser beam in the limit where either the detuning or the Rabi frequency is large. For this case, the spectrum of resonance fluorescence has three separated lines. We have obtained closed form expressions for the conditional probability density for the emission of the nth photon and for the probability for emission of n photons in a time interval [0,T]. Our solutions are complementary to the known solutions for the case of perfect resonance.

Theory of field attenuation in photon detection, with an application to resonance fluorescence.

Detection of photons from electromagnetic radiation can be considered as the appearance of random events on the time axis. When an attenuator is placed in front of the detector, which attenuates the intensity by a factor of α, the statistical properties of the detected photons are altered. We show that simple relations exist between the statistical functions of the photons detected from the attenuated field and the same functions for the photons that would be detected from the unattenuated field. We also derive several recurrence relations for the statistical functions involving their dependence on the parameter α. For photon detection from resonance fluorescence, the parameter α appears naturally as the probability that an emitted photon is detected. In this case, there is no attenuator, but the parameter α appears in the same way. We show that the probability for the emission (α=1) of n photons in a given time interval can easily be computed, and with the general theory we can then obtain the result for the detection of n photons (α<1).

Damping of the dipole vortex.

When a circular electric dipole moment, rotating in the x-y plane, is embedded in a material with relative permittivity ε(r) and relative permeability µ(r), the field lines of energy flow of the emitted radiation are dramatically influenced by the surrounding material. For emission in free space, the field lines swirl around the z axis and lie on a cone. The direction of rotation of the field lines around the z axis is the same as the direction of rotation of the dipole moment. We found that when the real part of ε(r) is negative, the rotation of the field lines changes direction, and hence the energy counter-rotates the dipole moment. When there is damping in the material, due to an imaginary part of ε(r), the cone turns into a funnel, and the density of the field lines diminishes near the location of the source. In addition, all radiation is emitted along the z axis and the x-y plane, whereas for emission in free space, the radiation is emitted in all directions. It is also shown that the displacement of the dipole image in the far field depends on the material parameters and that the shift can be much larger than the shift of the image in free space.

Redistribution of energy flow in a material due to damping.

The field lines of energy flow of the radiation emitted by a linear dipole in free space are straight lines, running radially outward from the source. When the dipole is embedded in a medium, the field lines are curves when the imaginary part of the relative permittivity is finite. It is shown that due to the damping in the material all radiation is emitted in directions perpendicular to the dipole axis, whereas for a dipole in free space the radiation is emitted in all directions except along the dipole axis. It is also shown that some field lines in the near field form semiloops. Energy flowing along these semiloops is absorbed by the material and does not contribute to the radiative power in the far field.

Optical vortices and singularities due to interference in atomic radiation near a mirror.

We consider radiation emitted by an electric dipole close to a mirror. We have studied the field lines of the Poynting vector, representing the flow lines of the electromagnetic energy, and we show that numerous singularities and subwavelength optical vortices appear in this energy flow pattern. We also show that the field line pattern in the plane of the mirror contains a singular circle across which the field lines change direction.

Nanoscale shift of the intensity distribution of dipole radiation.

The energy flow lines (field lines of the Poynting vector) for radiation emitted by a dipole are in general curves, rather than straight lines. For a linear dipole the field lines are straight, but when the dipole moment of a source rotates, the field lines wind numerous times around an axis, which is perpendicular to the plane of rotation, before asymptotically approaching a straight line. We consider an elliptical dipole moment, representing the most general state of oscillation, and this includes the linear dipole as a special case. Due to the spiraling near the source, for the case of a rotating dipole moment, the field lines in the far field are displaced with respect to the outward radial direction, and this leads to a shift of the intensity distribution of the radiation in the far field. This shift is shown to be independent of the distance to the source and, although of nanoscale dimension, should be experimentally observable.

Far-field detection of the dipole vortex.

The energy flow lines (field lines of the Poynting vector) of electric dipole radiation exhibit a vortex structure in the near field when the dipole moment of the source is in circular rotation. The spatial extend of this vortex is smaller than a wavelength and may not be observable by a measurement in the near field. We show that the rotation of the field lines close to the source affects the image of the dipole in the far field, and this opens the possibility for observation of this vortex by a measurement in the far field.

Subwavelength displacement of the far-field image of a radiating dipole.

The field lines of the Poynting vector for light emitted by a dipole with a rotating dipole moment show a vortex pattern near the location of the dipole. In the far field, each field line approaches a straight line, but this line does not appear to come exactly from the location of the dipole. As a result, the image of the dipole in its plane of rotation seems displaced. Secondly, the image in the far field is displaced as compared with the image of a source for which the field lines run radially outward. It turns out that both image displacements are the same. The displacements are of subwavelength scale, and they depend on the angles of observation. The maximum displacement occurs for observation in the plane of rotation and equals lambda/pi, where lambda is the wavelength of the light.

Integral equation formulation for reflection by a mirror.

When light is incident on a mirror, it induces a current density on its surface. This surface current density emits radiation, which is the observed reflected field. We consider a monochromatic incident field with an arbitrary spatial dependence, and we derive an integral equation for the Fourier-transformed surface current density. This equation contains the incident electric field at the surface as an inhomogeneous term. The incident field, emitted by a source current density in front of the mirror, is then represented by an angular spectrum, and this leads to a solution of the integral equation. From this result we derive a relation between the surface current density and the current density of the source. It is shown with examples that this approach provides a simple method for obtaining the surface current density. It is also shown that with the solution of the integral equation, an image source can be constructed for any current source, and as illustration we construct the images of electric and magnetic dipoles and the mirror image of an electric quadrupole. By applying the general solution for the surface current density, we derive an expression for the reflected field as an integral over the source current distribution, and this may serve as an alternative to the method of images.

Current density in a perfect mirror.

Electromagnetic radiation incident upon a perfect mirror induces a current density on the surface of the conducting material of the mirror. It is shown that this surface current density can be expressed directly in terms of the source current density, which generates the incident field, without evaluating the electric and magnetic fields first.

Boundary conditions in an integral approach to scattering.

Scattering of electromagnetic radiation by an object of arbitrary shape or a structured surface, infinite in extent, is considered. When radiation is incident on an interface separating vacuum from a material medium, a current density is induced in the bulk and a surface current density may appear on the boundary surface. The electromagnetic field is then the sum of the incident field and the field generated by the current densities. This concept leads to expressions for the electric and magnetic fields that can easily be shown to be exact integrals of Maxwell's equations both in the vacuum and in the medium. At the boundary surface, the electric and magnetic fields must be discontinuous, with the discontinuity determined by the surface charge and current densities. This is usually referred to as boundary conditions for Maxwell's equations. We show that the integrals for the electric and magnetic fields automatically satisfy these boundary conditions, no matter the origin of the current densities.

Reflection and refraction of multipole radiation by an interface.

Reflection and refraction of electromagnetic multipole radiation by an interface is studied. The multipole can be electric or magnetic and is of arbitrary order (dipole, quadrupole). From the angular spectrum representation of the radiation emitted by the multipole, I have obtained the angular spectrum representations of the reflected and transmitted fields, which involve the Fresnel reflection and transmission coefficients. The intensity distribution in the far field is evaluated with the method of stationary phase. The result is very simple in appearance and can be expressed in terms of two auxiliary functions of a complex variable. By exchanging the Fresnel coefficients for s and p polarization, the result for an electric multipole can be obtained from the result for a magnetic multipole.

Transmission of dipole radiation through interfaces and the phenomenon of anti-critical angles.

Radiation emitted by an electric dipole consists of traveling and evanescent plane waves. Usually, only the traveling waves are observable by a measurement in the far field, since the evanescent waves die out over a length of approximately a wavelength from the source. We show that when the radiation is passed through an interface with a medium with an index of refraction larger than the index of refraction of the embedding medium of the dipole, a portion of the evanescent waves are converted into traveling waves, and they become observable in the far field. The same conclusion holds when the waves pass through a layer of finite thickness. Waves that are transmitted under an angle larger than the so-called anti-critical angle theta (1) ac are shown to originate in evanescent dipole waves. In this fashion, part of the evanescent spectrum of the radiation becomes amenable to observation in the far field. We also show that in many situations the power in the far field coming from evanescent waves greatly exceeds the power originating in traveling waves.

Spatial separation of the traveling and evanescent parts of dipole radiation.

Electric dipole radiation consists of traveling and evanescent plane waves. When radiation is detected in the far field, only the traveling waves will contribute to the intensity distribution, as the evanescent waves decay exponentially. We propose a method to spatially separate the traveling and evanescent waves before detection. It is shown that when the radiation passes through an interface, evanescent waves can be converted into traveling waves and can subsequently be observed in the far field. Let the radiation be observed under angle theta(t) with the normal. Then there exists an angle theta(ac) such that for 0 < or = theta(t) < theta(ac) all intensity originates in traveling waves, whereas for theta(ac) < theta(t) < pi/2 only evanescent waves contribute. It is shown that with this technique and under the appropriate conditions almost all far-field power can be provided by evanescent waves.

Traveling and evanescent parts of the electromagnetic Green's tensor.

The angular spectrum representation of the electromagnetic Green's tensor has a part that is a superposition of exponentially decaying waves in the +z and -z directions (evanescent part) and a part that is a superposition of traveling waves, both of which are defined by integral representations. We have derived an asymptotic expansion for the z dependence of the evanescent part of the Green's tensor and obtained a closed-form solution in terms of the Lommel functions, which holds in all space. We have shown that the traveling part can be extracted from the Green's tensor by means of a filter operation on the tensor, without regard to the angular spectrum integral representation of this part. We also show that the so-called self-field part of the tensor is properly included in the integral representation, and we were able to identify this part explicitly.