Aug 19, 2017

where β is the complex propagation constant and the transverse field dependence f(x) = Ey(x) or Hy(x) for TE and TM modes respectively. These expressions for the time-averaged z component of Poynting vector (in the direction of phase propagation parallel to the layers) are general and apply to any layer with arbitrary complex dielectric permitivity ε(x) in a multi-layer slab waveguide where Ey(x) and Hy(x) depend on the mode solution β .

The 2D plots below show all the field components and instantaneous (not time averaged) Poynting vector components at a fixed time for the TM0 mode of the lossless symmetric waveguide. The instantaneous (and time-averaged) Poynting vector component Sz is discontinuous across the interfaces as mentioned above. The time-averaged Poynting vector is the average at any given (x,z) location of the instantaneous value and corresponds to the single line above (independent of z since there is no mode attenuation). The "islands" of instantaneous Poynting vector correspond to the peaks of EM energy density in the mode propagating along the +z direction without change with Hy and Ex being in phase. The component of power flow normal to the layers, Sx, is interesting. The TIME AVERAGED value of Sx is zero at any location, but the instantaneous flux shows a forward-reverse surging effect at the interfaces in the x direction with time as clearly seen in the distribution. This is obvious when one considers that the distributions shown simply move along the z propagation direction and for a fixed z, the Sx value changes in direction with the time-averaged value being identically zero, a consequence of the waveguide being lossless (compare the lossy surface-wave mode example below where the time averaged Sx flow is NOT zero). This is the same behaviour as the cyclic energy surge in the evanescent field present for the simple plane wave reflection for a Total Internal Reflection (TIR) case (Stratton 1941 p.499). For the present symmetric waveguide, this is not surprising as the field in the outer waveguide regions are essentially evanescent TIR-like fields. Note that the Sx peaks are displaced from the Sz peaks :

The TM mode, as demonstrated above is considerably more "complex" in character depending on the layer dielectric constants. The sign of the Sz expression above for the general TM slab mode depends on the sign of Re(βε*), a consequence of the discontinuity of the normal component of electric field, Ex at the dielectric interfaces (here ε is the relative dielectric constant):

where, using the sign convention above, for lossy media, β

Note that for any medium, including media with loss, a PLANE WAVE will always have the time-averaged power flow Sz in the same direction as the propagation (phase velocity) direction, since any normal lossy medium always has

Therefore the time-averaged reverse power flow seen in one of the media of the TM surface-wave mode discussed above is a consequence of both a discontinuity in the normal component of the electric vector at the interface along with a sufficiently large negative real part of the dielectric constant in one of the two media. The negative dielectric constant is fundamentally related to the complex phase shift in the electron response in metals due to the driving optical field (Born et. al 1975).

A negative real part of a dielectric constant may not always cause a time-averaged reverse power flow in the conductor of a TM surface-wave mode. The plots below show how the time-averaged relative power flow in the two regions (integrated over the transverse direction x) of a lossless/lossy media TM0 mode configuration changes as the imaginary part of the lossy medium n2.im changes. Note that the real part of the dielectric constant for n2, epsilon2.re becomes negative for n2.im = -0.3 but the x-integrated Sz power flow in that medium Pt2 is still in the phase propagation direction +z. Pt2 only changes direction for n2.im ~ -1.102, at the condition

The 2D field plots below show the real Hy, Ex, Ez and instantaneous (not time-averaged) Sz Poynting vector components Sz and Sx at the same instant of time for a representative dielectric/metal interface exhibiting strong reverse power flow in the metal (a point at far left side of curve above). The instantaneous Poynting vector view (as compared to the more common time-averaged view) clearly demonstrates the local variation of the energy density on the wavelength scale. The field and Poynting vector patterns move unchanged in shape along the +z propagation direction (the phase velocity direction), except for an overall amplitude attenuation rate of 2βim. The relative phase of the Ex and Hy field components determines the power flow direction. In the metal, for this data point, Ex is close to 180° (163°) out of phase with the Hy field (reversed in sign), while in the dielectric, Ex and Hy are almost exactly in phase. The z component Sz of instantaneous (and also the time averaged) Poynting vector in the metal therefore is in the REVERSE direction (-z) to the phase velocity direction compared to that in the dielectric. The instantaneous Sz profile shows that in fact there is a slight surging into the forward direction (a data point at the transition

The corresponding time-averaged Poynting vector components are shown below at the left for a fixed z and demonstrate the relative values of Sx and Sz. In both cases, the distribution merely attenuates for increasing z. Note that the time-averaged Sx is everywhere negative (flux in the -x direction), showing that the energy flow is toward the metal from the dielectric and into it. The Sz component along the interface shows the flow direction change at the interface as discussed above. The phase relationship of the real values of Hy, Ex and Ez and the corresponding instantaneous Sz and Sx values are more clearly seen in temporal plots as shown at the right for a fixed x=0 location (at the interface) for the metal (0) and dielectric (1). Hy, Ez and Sx are all continuous at the interface. To facilitate easier comparison of phase relationships between the various fields and Poynting vector components on the same graph, Ex and Ez are scaled by the same fixed magnitude value of |βZ

It is easy to show using similar analysis to that above that reverse power flow also occurs in the simpler perhaps more intuitive configuration of a P-polarized (transverse magnetic field Hy) simple plane wave reflected/refracted at an interface between a lossless dielectric and a lossy medium with a negative real part of the permittivity. In this case, with an incident and reflected plane wave in the dielectric, with the same orientation as that above for the TM0 surface wave, the component of propagation constant along the interface is real and is β = k0sin(θi) where θi is the real angle of incidence. The z component of Poynting vector along the interface in the real incident medium is in the +z direction. However, again the z component of Poynting vector in the refracted negative permittivity medium is in the -z direction and this is true for ALL real angles of incidence.

**Electromagnetic Theory**, J. Stratton, 1941, McGraw Hill, pp. 517-524

**Principles of Optics**, M. Born, E. Wolf, 5th Edn. 1975, Pergamon Press. pp. 621-626

**An Introduction To Optical Waveguides**, M. J. Adams, 1981, J. Wiley & Sons, pp 66-68

**Surface Wave Character On A Slab of Metamaterial with Negative Permittivity and Permeability**, S. Mahmoud, A. Viitanen, Progress in Electromagnetics Research, #51, 2005 pp. 127-137