## Abstract

An oscillating electric dipole in free space emits its energy along straight lines. We have considered the effect of a nearby interface with a material medium. Interference between the directly emitted radiation and the reflected radiation leads to intricate flow line patterns. When the interface is a plane mirror, numerous interference vortices appear, and when the distance between the dipole and the mirror is not too small, these vortices lie on four strings. At the center of each vortex is a singularity, and these singularities are due to the fact that the magnetic field vanishes at these locations. When the interface is a boundary between dielectric media, reflection leads again to interference. The pattern for the transmitted radiation depends on whether the medium is thicker or thinner than the material in which the dipole is embedded. For thicker dielectrics, the field lines bend toward the normal, reminiscent of, but not the same as, the behavior of optical rays. For thinner media, oscillation of energy across the interface appears, and above a crossing point, there is a tiny vortex. We have also considered the case of a dipole in between two parallel mirrors.

### Keywords

- vortex
- singularity
- Poynting vector
- dipole radiation
- interface
- mirror

## 1. Introduction

The common conception about the propagation of light is that the radiation travels along straight lines. Such a picture certainly seems to hold for a laser beam and for sunlight. The success of ray diagrams for the construction of images by lenses and mirrors also supports this picture. Reflection by and transmission through an interface is another example of a process that can be described by a ray picture of light. However, light is electromagnetic radiation, and ultimately any optical phenomenon must be accounted for by a solution of Maxwell’s equations for the electric and magnetic fields. In the geometrical optics limit of light propagation [1], spatial variations on the scale of a wavelength or less are neglected, and optical rays are defined as the orthogonal trajectories of the wave fronts of a propagating wave. From a different point of view, we can define the direction of light propagation as the direction of the energy flow in the radiation field. This direction is determined by the electromagnetic Poynting vector [2]. It can be shown that in the geometrical optics limit for propagation in vacuum the field lines of the Poynting vector are straight lines and are identical to the optical rays, defined with the help of the propagation of wave fronts.

The concept of optical rays breaks down when spatial variations on the scale of a wavelength or less are of concern, or when coherence in the radiation leads to macroscopic constructive and destructive interference (as for a diffraction grating). We shall consider a small particle, like an atom, molecule, or nano-particle, irradiated by a monochromatic laser beam, oscillating with angular frequency

The energy flow lines for a free (linear) dipole are straight at all distances. Any deviation from this radially outgoing pattern is due to the environment of the particle. For instance, when the particle is embedded in an absorbing medium, the imaginary part of the permittivity is responsible for a bending of the field lines in the near field toward the dipole axis [3]. Here we shall consider the case where the particle is located near an interface. Some of the emitted radiation by the dipole will be incident on the interface, and here reflection and transmission takes place. The reflected light will interfere with the incident light, and in the far field this leads to maxima and minima in the radiated power per unit solid angle. The structure of the angular power distribution of the transmitted light in the far field depends on the value of the critical angle and the distance between the particle and the surface. Usually, when a traveling plane wave is incident upon an interface, the transmitted wave is again traveling, and is bent toward the normal. When the angle of incidence approaches 90°, the transmitted wave is still traveling, and the angle of transmission is called the critical angle (for transmission). In the dipole spectrum, evanescent waves are present, and they are still transmitted as traveling waves, provided the wavelength is not too small. Therefore, above the critical transmission angle, all transmitted light comes from evanescent dipole waves, and this can lead to a large lobe in the power distribution above the critical angle [4]. Rather than considering the effects in the far field, we shall here present results for the power flow in the near field. We shall illustrate that interference gives rise to interesting flow patterns, including singularities, vortices, and strings of vortices. We also show that transmission in the near field exhibits interesting features.

## 2. Dipole radiation in free space

The oscillating dipole moment can be written as

where *r* be the length of

with

The dimensionless complex amplitudes

We then obtain for an electric dipole [5]

The time-averaged Poynting vector for radiation in free space is given by

For an electric dipole, we split off a factor:

so that

With the above expressions for

and here

## 3. Dipole radiation near a mirror

The simplest interface is a flat, infinite, and perfect mirror. We take the surface of the mirror as the *xy*-plane, and the dipole is located on the positive *z*-axis, at a distance *H* above the mirror. The dipole direction vector *z*-axis, and we take *yz*-plane. Therefore,

The electric field above the mirror is equal to the field of the dipole plus the electric field of an image dipole located at a distance *H* below the mirror on the *z*-axis, as illustrated in Figure 2, and the same holds for the magnetic field. The dipole moment direction of the image dipole is

The complex amplitudes of the electric and magnetic fields of the dipole are given by Eqs. (6) and (7), but with

The coordinates of the field point only come in through *q* in Eqs. (6) and (7), so we replace *q* by

*q* by *z* = 0, e.g., just above the mirror surface, the Poynting vector is along the surface.

## 4. Computation of field lines

With the above method, the Poynting vector **r**. We shall use dimensionless Cartesian coordinates *h* and **q** on the curve the vector *t* an arbitrary dummy variable. Then

In Cartesian coordinates this becomes:

and similarly for *t* does not appear explicitly, and such equations are called autonomous. The solution is determined by an initial point *t* = 0 at the initial point. The direction of the field line is the direction along the curve that follows from increasing *t*. So, the field line runs from the initial point into the direction that corresponds to the solution with

Obviously, the differential Eq. (16) will in general need to be solved numerically. An interesting exception is the case for an arbitrary (elliptical) dipole in free space for which an analytical solution can be obtained, as reported in Ref. [6]. We use *Mathematica* to solve the set and produce the field line pictures. For two-dimensional problems, the routine StreamPlot only requires the expression for *Mathematica* select these initial points. This is much faster, and works well to get an initial picture. However, in this approach *Mathematica* cuts off field lines when they get too close together, and that does not necessarily look too good.

Finally, field lines are determined by the direction of vector **q**, and not by its direction. Therefore, the vector fields **q**, have the same field lines. This can also be seen by making the change of variables

and this equation has the same solutions for the field lines as Eq. (16). It just gives a different parametrization of the curves. A popular choice is

## 5. Field lines in the symmetry plane

The dipole direction vector *yz*-plane. If we take a field point in the *yz*-plane, then also *yz*-plane, and therefore, *yz*-plane, and the same holds for the complex amplitude of the image electric field. From Eq. (7), we see that the complex amplitudes of the magnetic dipole field and the magnetic image field are along the *x*-axis if the field point is in the *yz*-plane. Therefore, the Poynting vector from Eq. (10) is in the *yz*-plane if the field point is in the *yz-*plane. Consequently, any field line through a point in the *yz*-plane stays in the *yz*-plane. This plane is the symmetry plane for the dipole near the mirror. For field lines off the symmetry plane the field lines will be 3D curves, and the flow pattern is reflection symmetric in the *yz*-plane. Field lines in 3D are difficult to visualize, so we shall only consider field lines in the symmetry plane.

Figure 3 shows the flow lines of energy for a dipole oscillating along the *z*-axis (*z*-axis. The field lines that run toward the surface smoothly bend away from the surface upon approach. At the origin of coordinates we have a singularity, since the field lines that come out of the dipole and run straight down split there. This point is marked with a white circle. Figure 4 shows the flow pattern for a dipole oscillating parallel to the surface (*z*-axis, so only the region

Figure 5 shows the field lines for a dipole oscillating under 45° with the *z*-axis (*c* is close to the dipole, and is similar to the small vortex in Figure 4. Vortices *a* and *b* have a greater extent, although they are still of subwavelength dimension. Some field lines come out of the dipole and swirl around vortex *a*, and some pass by vortex *b*. Interestingly, there are field lines that emanate from vortex *a* and end up at the center of vortex *b*. This seems to represent energy flowing from vortex *a* to vortex *b*, but that is not the case. Only the dipole is a source for the vector field of the Poynting vector. The singularity at the point labeled *S* seems like a bump in the road for the field lines that pass nearby. It can be shown analytically [7] that this singularity is a point on a singular circle in the plane of the mirror. The circle goes through the origin of coordinates and singularity *S*, and singularity *S* is located at *z*-axis, not shown in Figure 5, there seems to be no interesting structure.

When the dimensionless distance *h* between the dipole and the surface increases, so does the number of vortices. In Figures 3–5, this distance was taken as *z*-axis. Figure 7 shows an enlargement of one of these vortices, and it follows from the pattern that there must be another singularity nearby. The vortices on the left have a clockwise rotation and Figure 8 shows an enlargement. For the vortices on the left the field lines come out of the center of the vortex, and for the vortices on the right the field lines end at the center of the vortex, as in Figures 7 and 8, respectively. In Figure 6, the field lines that start or end at a vortex are drawn in bold. Many other field lines are present. They run from the dipole to infinity, either without coming in the neighborhood of the vortices (the four field lines on top of the picture) or they cross the array of vortices once or twice.

## 6. Location of the vortices

At a singularity, the Poynting vector vanishes. This can be due to *xy*-plane, the complex amplitude of the magnetic field is along the *x*-axis. For

and similarly

The solutions of Eq. (19) are curves in the *yz*-plane, and Eq. (20) also represents a set of curves in the *yz*-plane. At intersections between these sets of curves the magnetic field is zero, and this corresponds to the center of a vortex. Figure 9 shows the curves for the same parameters as shown in Figure 5. The solid lines are the solutions of Eq. (19) and the dashed lines are the solutions of Eq. (20). The three intersections *a, b*, and *c* are the centers of the three vortices in Figure 5. Interestingly, at the center of the bump on the mirror, labeled *S*, the magnetic field also vanishes. This point is located at

and it is easy to check that this is indeed a solution of both Eqs. (19) and (20). Here, the magnetic field is zero, but there is no vortex. For the case of the parameters for Figure 6, the solutions of Eqs. (19) and (20) are shown in Figure 10.

## 7. Vortex strings

In Figure 10, the intersections between the solid and dashed curves are not always precisely to determine and this gets worse with increasing *h*. By manipulating Eqs. (19) and (20), a different set of equations can be obtained. We find [8]

The solid curve in Figure 11 is the solution of Eq. (22) and the dashed curves are the solutions of Eq. (23). Vortices appear at the intersections. The parameters are the same as for Figure 10. Interestingly, Eq. (23) is independent of the orientation angle *S* on the mirror, as can easily be checked from Eq. (21).

The vortices in Figure 11 appear to lie on two ‘strings’. The left string starts at the dipole and runs to point *S* on the mirror. The second string is in between the left string and the *z*-axis. It can be seen from Figure 6 that all the vortices on the left string have a counterclockwise rotation, as in Figure 7. On the right string the vortices have a clockwise rotation, as in Figure 8. Figure 12 shows a larger view of the same graph as in Figure 11. The solid line passes the dipole and continues in the upper right part of the graph. It appears that far away there are also intersections between the solid curve and the dashed curves, and these correspond also to the location of vortices. The vortices appear to lie on a third string. It can be shown that these vortices have a counterclockwise rotation.

The left string ends at point S on the mirror. The location of this point depends on the dipole angle *z*-axis, and so is the string pattern. We see that a fourth string of vortices appears, and from symmetry it follows that these have a clockwise rotation. When we look again at Figure 12, the solid curve must intersect the dashed curves outside the picture in the upper right corner. Therefore, there is a fourth string very far away, and outside the picture. Apparently, there are always four vortex strings in electric dipole radiation near a mirror.

## 8. Dielectric interface

An interesting generalization of the free dipole near the mirror is the case of a dipole embedded in a dielectric medium, and near an interface with another dielectric material. For this problem, we reverse the *z*-axis, as compared to Figure 2, and we place the dipole on the negative *z*-axis, at a distance *H* below the interface. This is illustrated in Figure 14. The dielectric constant of the embedding medium is *z*-axis, e.g., *z*-component of a wave vector can change. Set *z*-components of the wave vectors can be expressed in terms of the functions

The wave vector of an incident partial wave of the source field has a *z*-component of *z*-components *p*-polarized plane wave with parameter

For the dipole oscillating along the *z*-axis, all partial waves are *p* polarized.

The setup is rotation symmetric around the *z*-axis, so we only need to consider the solution in the *yz*-plane, with *yz*-plane, the magnetic field is along the *x*-axis, and the Poynting vector is in the *yz*-plane. We obtain for the complex amplitudes of the dimensionless electric and magnetic reflected fields [9]

and for the transmitted fields we find

Here, *z*-axis we have *z*-axis. The magnetic source field, Eq. (7), also vanishes on the *z*-axis. Therefore, the Poynting vector is zero on the *z*-axis, and so the *z*-axis is a singular line of the flow lines pattern.

For the case of Figure 15, we have *z*-axis the field lines leave the interface under approximately

More interesting is the case for transmission into a thinner medium, as illustrated in Figure 16. The indices of refraction are

## 9. Dipole in between mirrors

An interesting variation of the mirror problem is the case for a dipole in between parallel mirrors, as depicted in Figure 18. A second mirror is placed at a distance *D* above the first mirror. Now, the dipole has a mirror image in both mirrors. In order to satisfy the boundary conditions at both mirrors, a mirror image must have again an image in the other mirror, and so on. This leads to an infinite sequence of images. Let us label the images with *m*. We then find that the images are located at

with *m* integer. For *m* = 0, this is the actual dipole in Figure 18. The image with *m* = −1 is the image dipole from Figure 2. Images with *m* even have a dipole orientation vector *m* odd have

This can be combined as

for the orientation of image *m*. The dimensionless distance between the mirrors is denoted by

Figure 19 shows the energy flow pattern for a horizontal dipole midway between the mirrors. The distance between the mirrors is

## 10. Conclusions

An oscillating electric dipole in free space emits its energy along straight lines. Most radiation is emitted perpendicular to the dipole axis, and none comes out along the dipole axis. We have studied the effect of a nearby interface on this flow pattern. Reflection of radiation at the interface leads to interference between the directly emitted radiation and the reflected radiation. A mirror is impenetrable for radiation, and so all radiation bounces back at the interface. This also implies that the field lines of energy flow must be parallel to the mirror at the mirror surface. This effect is shown in Figure 3 for a dipole oscillating perpendicular to the surface, and one wavelength away from the surface. The radiation comes out of the dipole, more or less as for emission in free space, but at the mirror surface the field lines bend, and the energy flows away along the mirror surface. For a dipole oscillating parallel to the surface, a typical flow pattern is shown in Figure 4. Again, at the mirror surface the field lines run away parallel to the surface, but in between the surface and the dipole several singularities appear, and there is also a vortex very close to the dipole. For the case shown in Figure 5, the dipole oscillates under 45° with the normal to the surface, and we see that two large vortices appear and one very small one. The rotation direction of the energy flow in the two large vortices is in opposite directions, and some energy flows from one vortex to the other. When the distance between the dipole and the surface is much larger than a wavelength, numerous tiny (subwavelength) vortices appear, and we found that the vortices are located on a set of four strings. This is shown most clearly in Figure 13.

When the surface is an interface between two dielectrics, we also need to consider the radiation transmitted into the substrate. Here, we only consider the simplest case of a dipole oscillating perpendicular to the interface. This can be generalized to arbitrary oscillation directions, and also to the case where the surface is an interface with a layer of material, and this layer is located on a substrate of yet another kind of material [10]. Figure 15 illustrates a typical case of transmission into a thicker medium. The field lines bend toward the normal, just like optical rays would. However, the refraction angle for the flow lines does not follow Snell’s law for optical rays. Figure 16 shows field lines for transmission into a thinner medium. Now the field lines bend away from the normal, but some field lines bend so much that they return to the other side of the interface. There is oscillation of energy back and forth through the interface. An enlargement is shown in Figure 17, and we observe that a vortex appears just above the location where the energy goes back and forth through the interface.

We have also considered the case where the oscillating dipole is located in between two mirrors. For a horizontal dipole, the emitted radiation bends near the surfaces of the two mirrors, and then flows away horizontally, as shown in Figure 19. For the case in Figure 20, the dipole oscillates under 45° with the normal. Two vortices appear. Some of the radiation that is emitted to the bottom-right of the picture originally flows to the right along the surface of the lower mirror, but then turns around, swings by the dipole, and then continues to the left, along the surface of the top mirror. When the dipole is not located midway between the mirrors, as in Figure 21, numerous vortices appear, and the pattern repeats indefinitely to the left and right in the range outside the picture.