Vector Poynting

If we assume that there is a linear relation (k, o) = e0x(k, co)S(k, o), the convolution theorem yields 

a susceptibility that depends on frequency and wave vector implies that the relation between P(x, t) and E(x, t) is nonlocal in time and space. Such spatially dispersive media lie outside our considerations. However, spatial dispersion can be important when the wavelength is comparable to some characteristic length in the medium (e.g., mean free path), and it is well at least to be aware of its existence it can have an effect on absorption and scattering by small particles (Yildiz, 1963 Foley and Pattanayak, 1974 Ruppin, 1975, 1981). 

Consider an electromagnetic field (E, H), which is not necessarily time harmonic. The Poynting vector S = E X H specifies the magnitude and direction of the rate of transfer of electromagnetic energy at all points of space it is 

The net rate W at which electromagnetic energy crosses the boundary of a closed surface A which encloses a volume V is 

The instantaneous Poynting vector (2.38) is a rapidly varying function of time for frequencies that are usually of interest. Most instruments are not capable of following the rapid oscillations of the instantaneous Poynting vector, but respond to some time average (S)  

Considering a volume V with a surface A, the following energy conservation theorem holds [1]  

The power furnished to the external current source (left-hand side) in the volume V equals the power radiated out from the surface A (first term at the right-hand side) plus the power dissipated in the volume V by the Joule effect (second term at the right-hand side). The Joule effect dissipation can also be rewritten as  

The Poynting vector for an uniform plane-wave reads  

---

There is a close similarity with planar electromagnetic cavities (H.-J. Stockmann, 1999). The basic equations take the same form and, in particular, the Poynting vector is the analog of the quantum mechanical current. It is therefore possible to experimentally observe currents, nodal points and streamlines in microwave billiards (M. Barth et.al., 2002 Y.-H. Kim et.al., 2003). The microwave measurements have confirmed many of the predictions of the random Gaussian wave fields described above. For example wave function statistics, current flow and... 

The average energy flux in the evanescent wave is given by the real part of the Poynting vector S = (c/47t)ExH. However, the probability of absorption of energy per unit time from the evanescent wave by an electric dipole-allowed transition of moment pa in a fluorophore is proportional to lnfl - El2. Note that Re S and pa E 2 are not proportional to each other they have a different dependence on 0. 

Figure 2. The FDTD simulation results for the electric field components Ex (a) and (c) and the Poynting vector component (h) and Sx (d). The waveguide is along z axis, the dashed lines contouring the non-etched 0.6 pm thick region of the core.

Figure 17. z-component of time-averaged Poynting vector in the focal plane. 

The intensity transmitted to the waveguide can be related to the Poynting vector of the incident ffee-space radiation and that of the PhC-mode a few microns away from the interface, which confirms the 80% coupling efficieny. 

It is more complex to express the attenuation along the direction of the Poynting vector, so that we may see the absorption of rays in the material, but it is interesting to quote one result, calculated by Batterman and Cole, for 1 mm thick... 

Figure 4.15 The energy flow through the crystal and the geometry of diffracted and forward-diffracted beams, for the case shown in Figure 4.14. P and P g are the Poynting vectors associated with the tie-points A and B...

When it is clear from the context that it is the time-averaged Poynting vector with which we are dealing, the brackets enclosing S will be omitted. 

Once we have obtained the electromagnetic fields inside and scattered by the particle, we can determine the Poynting vector at any point. However, we are usually interested only in the Poynting vector at points outside the particle. The time-averaged Poynting vector S at any point in the medium surrounding the particle can be written as the sum of three terms ... 

S, the Poynting vector associated with the incident wave, is independent of position if the medium is nonabsorbing Ss is the Poynting vector of the scattered field and we may interpret Sext as the term that arises because of interaction between the incident and scattered waves. 

Up to this point we have considered only extinction by a single particle. However, the vast majority of extinction measurements involve collections of very many particles. Let us now consider such a collection, which is confined to a finite volume, the scattering volume. The total Poynting vector is... 

The Poynting vector, therefore, is in the direction es. When the incident beam is normal to the cylinder axis ( = 90°), the cone reduces to a cylinder. 

We showed in Section 3.3 that the total Poynting vector S in the region surrounding an arbitrary particle can be written as the sum of three terms ... 

At energies on either side of 8.8 eV a small aluminum sphere presents a much smaller target to incident photons. At 5 eV, for example, the absorption efficiency of a sphere with x = 0.3 is about 0.1 as far as absorption is concerned, the sphere is much smaller than its geometrical cross-sectional area. The field lines of the Poynting vector, shown in Fig. 12.46, are what are to be... 

Figure 12.4 Field lines of the total Poynting vector (excluding that scattered) around a small aluminum sphere illuminated by light of energy 8.8 eV (a) and 5 eV (b). The dashed vertical line in (a) indicates the effective radius of the sphere for absorption of light.

The imaginary part of the dielectric function of SiC at its Frohlich frequency in the infrared (about 932 cm- ) is close to that of aluminum at 8.8 eV. So Fig. 2Aa also shows the field lines of the Poynting vector around a small SiC sphere illuminated by light of frequency 932 cm-1. At nearby frequencies, 900... 

---