Sometimes multiple antennas are used in an antenna array to affect directivity. The top view of this arrangement is shown in Figure 6 b with a sketch of the power pattern.
When we look into a mirror,we see the effect of reflections of electromagnetic radiation. Why do waves bounce off of conductive surfaces? What is the result of these reflections on radiation? The basis for reflections is the boundary condition of the fields on the surface of a conductor. Boundary conditions for E and H fields are shown in Figure 7. Inside the conductor, charges are free to move when influenced by electric fields and current is induced by time-varying magnetic fields.
A charge nearby the conductor causes charges to migrate on the conductor surface. Any tangential component of the E field would cause the charges to move until the tangential component of E is zero. The resulting effect is equivalent to the image , or virtual charge, located below the conductor surface shown in Figure 7 c. A magnetic field that is time-varying induces a current in the perfect conductor.
The current opposes the magnetic field so that no normal component can penetrate the conductor surface. Thus the current image shown in Figure 7 c causes the resulting normal component of H to disappear at the surface.
The effect of the image is very important because antennas are often nearby conductive surfaces such as the Earth, or the sheet metal of a car or airplane, or the ground plane of a circuit board.
The fields that radiate into space are the sum of those from the antenna and those from the image. If we consider the E-field from a dipole, it is easy to see the effect. In Figure 8 a a dipole parallel to conductor is shown with its image.
When the dipole is perpendicular to the ground plane, an image of the dipole with inverted charge exists below it—as shown in Figure 8 b. In these two examples, the field at some point in space is the sum of the fields from the dipole and its image. When the field radiating from a dipole hits the conductor, as shown in Figure 8 c , the reflection can be interpreted as the wave from the image. Antennas are connected to transmitters or receivers through transmission lines. Since the antenna impedance is not a constant function of frequency, it cannot be matched to the transmission line at all frequencies.
When the antenna impedance does not match the impedance of the transmission line usually 50 W or 75 W , reflections are formed at the connection to the antenna.
Waves that come from the source are reflected back down the transmission line reducing the ability to transmit power. The VSWR , voltage standing wave ratio, is a measure of the mismatch. VSWR is the ratio of the maximum voltage to minimum voltage on the transmission line. With an impedance mismatch, the VSWR is greater than one, indicating the presence of reflections.
As the impedance at the end of the transmission line becomes higher—approaching open circuit, the VSWR approaches infinity, indicating that all the power is reflected. This situation is similar to the incidence of a light beam at an interface between two media, such as air and water, in which some light is reflected and some goes into the water. VSWR reduces the amount of power transmitted to the antenna or reduces the signal from the antenna when it is used to receive signals.
The change in VSWR and the proportion reflected are shown in Figure 9 a and 9 b , respectively, for a W system, in which the load resistance is varied. Another problem with connecting to antennas is signal unbalance caused by a ground plane.
Figure 10 a shows a dipole antenna connected to a source through a shielded cable. The shield is connected to the ground plane. Parasitic capacitance between the antenna and the ground plane causes some current to flow through the ground plane rather than through the shield. The scalar potential is most conveniently evaluated using the Lorentz gauge condition see Sect. Given the vector and scalar potentials, Eqs. The average power crossing a spherical surface whose radius is much greater than is Recall that for a resistor of resistance the average ohmic heating power is Note that the formula is only valid for.
This suggests that for most Hertzian dipole antennas: i. In an anisotropic medium the electric fields of the eigenmodes are no longer perpendicular to each other unlike in an isotropic medium. The radiation pattern of a dipole antenna can be significantly altered by placing the dipole inside an optical microcavity [ 20 ].
Interferences between reflections at the different layer interfaces cause variations in the local density of states at the location of the dipole antenna. As a result the angular emission pattern of the dipole is modified, this is also called the Purcell effect.
Such microcavities are often made by depositing a series of thin films. The lateral dimensions of the films are much larger than their thicknesses and so the microcavities are in good approximation one-dimensional. For isotropic microcavities a model for dipole radiation based on plane wave decomposition was developed by Lukosz [ 12 ] and has been applied to simulate a wide range of applications [ 13 ].
The starting point of the Lukosz method is the plane wave decomposition of the field of a dipole antenna in an infinite medium [ 21 ]. The electric field of the dipole in an infinite medium is then altered by interference from the reflections of the various interfaces of the microcavity. A schematic can be seen in Fig.
In Eq. The denominator embodies multiple beam interference: it sums all reflections between the top and bottom parts of the cavity.
For an anisotropic microcavity a similar method can be applied to calculate the radiation of a dipole in the cavity. The eigenmodes of anisotropic materials are two linearly polarized waves, the ordinary o and the extraordinary e wave instead of the TE and TM polarization.
The normalized fields of the eigenmodes are given by Eq. The polarization state of the light is determined by the complex amplitudes of the ordinary and extra-ordinary waves and their difference in phase. One must be careful to simulate all changes in polarization that occur during propagation and reflection or transmission in an anisotropic cavity.
In anisotropic media e- and o-waves are coupled when reflection or transmission at an interface takes place, therefore the reflection coefficients have to be replaced by reflection matrices. Equation 27 and 28 should be replaced by:. To calculate the reflection matrix we employ the scattering matrix method introduced by Ko [ 22 ].
Other methods, such as the Berreman 4x4 matrix method [ 23 ], can also be applied but we have chosen the scattering matrix method because it is numerically more stable when dealing with evanescent waves and total internal reflection. A sketch of the input and output waves in the scattering matrix method is shown in Fig.
The fields emitted to the outside layer N or 0 E o u t can be calculated from E c a v :. In summary see Fig. In this section we demonstrate the abilities of the method described in section 2 by applying it to three problems: the radiation of a dipole in an infinite medium, the emission inside a multilayer structure containing an anisotropic layer and one-dimensional waveguides in liquid crystal.
For these calculations the method is implemented as a computer program. We calculate F by evaluating the integral in Eq. F is then calculated as a function of d a schematic is shown in Fig. The dipole moment is chosen so that this dipole radiates a power of 1 Watt in vacuum. In Fig. Solid lines: anisotropic plane wave expansion in a homogeneous medium.
Dashed lines: isotropic plane wave expansion in a layer as a function of the layer thickness. Figure 6 shows F in an anisotropic medium for dipoles p x , p y and p z. The solid lines represent F calculated with the anisotropic plane wave decomposition. The dashed lines show F for a dipole in an isotropic layer of different thicknesses.
The results of the simulation for a dipole in a homogeneous anisotropic medium are the same as the results of the analytical formula Eq. The reflections at the layer interfaces lead to interference effects, which are a function of the thickness d of the layer and of the projections of the wave vector k x and k y.
The constructive and destructive interferences in the emission pattern, lead, after integration, to an oscillating behavior of the emitted power F. For very thick isotropic layers, the anisotropic material no longer influences F. As an example of an anisotropic multilayered structure we calculate the radiation pattern of a dipole into a film of anisotropic material deposited on a thick glass substrate and covered with a metal mirror. The configuration depicted in Fig.
The thicknesses of the layers are 5nm , nm and nm for the emitting layer, 5CB and Al respectively. The dipole is situated in the middle of the emitting layer. The radiation of a dipole is investigated by considering the power density K k x , k y per interval d k x d k y.
The unit of K is W. Again we have chosen the dipole so that 1 Watt would be emitted by the dipole in vacuum. These plane waves radiate energy to the outside of the device, and are sometimes called leaky modes. In a completely lossless structure waveguided modes have zero width. In our simulation the modes have a finite width because of absorption in the aluminum layer.
However extra-ordinary polarized waves are not evanescent, so extra-ordinary polarized modes are seen. Figure 10 and Fig. Figure 10a and Fig. Figure 10b and Fig. For e-polarized waves the behavior of waveguided modes depends on the azimuth angle of propagation. In an anisotropic multilayered structure the o- and e-waves are coupled by reflection and transmission at the interfaces in the cavity.
This coupling influences the polarization and direction in which a dipole emits. We have simulated the emission from a dipole in the following layer structure see Fig. The optical axes of the anisotropic layers are parallel to the xy -plane. The refractive indices of the materials and the wavelength are the same as in the previous example.
A small absorption term 0. Because of the different orientations of the c -axis coupling occurs between the ordinary and extra-ordinary waves. We can distinguish 3 regions in the emission pattern. A net flux of radiation of the ordinary waves towards the emitter is possible because of coupling between e- and o-waves. Waveguides in liquid crystals are another example of anisotropic microcavities.
We consider radiation emitted by an electric dipole, embedded in a medium with permittivity and permeability. For a linear dipole in free space, the field lines of energy flow are straight, but when the imaginary part of is finite, the field lines in the material become curves in the near field of the dipole. Therefore, the energy flow is redistributed due to the damping in the material. For a circular dipole in free space, the field lines of energy flow wind around the axis perpendicular to the plane of rotation of the dipole moment.
When has an imaginary part, this flow pattern is altered drastically. Furthermore, when the real part of is negative, the direction of rotation of the flow lines reverses.
In that case, the energy in the field rotates opposite to the direction of rotation of the dipole moment. It is indicated that in metamaterials with a negative index of refraction this may lead to an observable effect in the far field. When optical radiation from a localized source is observed at a large distance, it appears as if the light travels along straight lines. Similarly, light scattered or reflected by an object seems to travel from the object to an observer along straight lines, known as optical rays.
These lines are the flow lines of the energy in the radiation field. In close vicinity of the source, however, these flow lines are in general curves, and intricate field line patterns may appear. Such structures can be found when the flow of radiation is resolved on a scale smaller than a wavelength.
Particularly interesting is the possible presence of singularities and optical vortices. The first prediction of the existence of an optical vortex was made by Braunbek and Laukien in [ 1 ]. They considered the diffraction of a plane wave around the edge of a conducting half plane, and they found that a vortex should appear at the illuminated side of the plane, and close to the edge.
Another mechanism that may lead to singularities and vortices in the energy flow is interference. We have shown recently [ 2 , 3 ] that when a point source is located near a reflecting surface, numerous vortices are present in the energy-flow pattern when the source is about a wavelength away from the surface. A different type of vortex in the energy-flow pattern results from a rotation inside the source.
We shall show in the next section that radiation emitted by an electric dipole may have such a vortex structure [ 4 ], and such vortices appear in multipole radiation of any order [ 5 ]. A much more subtle effect is the redirection of energy flow when the radiation passes through a material medium.
A material will in general absorb radiation along its path of propagation, but we shall show that the presence of material will in general also lead to a redistribution of the power flow.
It will be shown that in media the field lines curve due to the presence of the media, and that vortices which are present due to a rotation in the source are altered dramatically by the embedding medium. The most important source of electromagnetic radiation is arguably the oscillating electric dipole. When the current density in a localized source oscillates harmonically with angular frequency , it has an electric dipole moment of the form where is the complex amplitude.
When the dimension of the source is small compared to the wavelength of the light, then electric dipole radiation gives the dominant contribution to the emitted radiation compared to higher order multipoles. For example, when a small particle, like an atom, a molecule or a nanoparticle, is irradiated by a laser of angular frequency , then the emitted, or scattered, radiation will be electric dipole radiation, at least to a very good approximation.
The setup is illustrated in Figure 1. The emitted electric field will have the form with the complex amplitude, and the magnetic field can be represented in a similar form.
The Poynting vector is defined as This is the time-averaged Poynting vector, in which terms that oscillate at twice the optical frequency have been dropped, since these average to zero on a time scale of an optical cycle. The complex amplitudes of the electric and magnetic fields of an oscillating dipole are well known [ 6 ], and this Poynting vector can readily be constructed.
The field lines of energy flow are the field lines of the vector field. When we indicate by a point on a field line, then the curve can be parametrized as , where is a dummy variable. At any point along the field line, the vector must be on the tangent line, and therefore the field lines are solutions of the autonomous differential equation Here, can be any positive function of since a field line pattern only depends on the directions of the vectors of the vector field, and not on their magnitude.
A common choice for the function is , so that the right-hand side of 4 becomes a unit vector, and the parameter is equal to the arc length measured along a field line. Each numerical step size is then a step along the field line, and this more or less guarantees an equal spacing of numerical data along each field line.
Through each point in space we have a field line, and 4 can be integrated numerically, given this starting point. Let us first consider a linear dipole moment for which with. Then the dipole moment oscillates harmonically along the axis, and the Poynting vector is found to be with the spherical coordinates with respect to the axis.
The constant is defined as with , and this equals the total power emitted by the dipole. We see from 5 that is proportional to the radial unit vector for all field points, and therefore the field lines of are straight lines, emanating from the location of the dipole. The field line pattern is shown in Figure 2. A linear dipole moment is induced when a particle is illuminated by a linearly polarized laser, as in the setup in Figure 1 , and the oscillation direction is along the direction of the polarization of the beam.
When this laser is left-circular polarized then the electric field vector of the beam rotates counterclockwise when looking into the oncoming beam. If we take the propagation direction as the axis, then this electric field rotates counterclockwise in the plane when viewed down the positive axis.
Then the complex amplitude of the induced dipole moment is and the dipole moment itself is This dipole moment rotates counterclockwise in the plane, so it has the same direction of rotation as the incident beam. The Poynting vector is found to be where we have set for the dimensionless distance between the dipole and the field point.
In units of , a distance of corresponds to one optical wavelength. Interestingly, 4 for the field lines can be solved in closed form for this simple system [ 4 , 7 ]. The term proportional to in 9 corresponds to power flowing into the radially outward direction, but now a term proportional to appears. This contribution gives rise to a rotation of the field lines around the axis. A typical field line is shown in Figure 3. The variables on the axes are , and so forth.
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