July 17, 2017

It is well known (Stratton 1941, Collin 1991) that the interface between 2 semi-infinite regions with at least one region having finite loss associated with either a finite conductivity or equivalently an imaginary component of the dielectric constant supports an interface guided wave or Zenneck wave. The wave is a TM0 transverse magnetic mode and is characterized by the field amplitude decaying exponentially into both media away from the interface. The greater the medium loss, the less the field penetrates into the lossy medium. This surface wave mode has no cutoff wavelength and continues to propogate at arbitrarily low frequency. It is similar to the fundamental "principal" axially symmetric wire wave mode for an isolated cylindrical conductor. A general 2 dimensional interface between 2 semi-infinite media is shown below. In the general case, both media are lossy which can be described by either a finite conductivity and real refractive index or equivalentally using media complex refractive indices n1, n2 (or complex dielectric constants). In the geometry shown below, the mode propogation direction is along the z axis and the x axis is normal to the region interface and the magnetic field vector Hy is in the y direction. The electric field vector has components in both x and z directions:

there the values for the constants vacuum permeability μ

where the field dependence normal to the interface in the x direction is:

and

and the roots are chosen so that both p1 and p2 are positive for a wave solution f(x) that attenuates in both +/- x directions away from the interface. The characteristic attenuation distances of the surface wave in the two media are:

For this TM0 mode, the electric field components in the two regions are:

and the axial z component of the power flow Poynting vector is given by:

Integrating over x on either side of the interface provides the total power per unit width of y carried by the surface wave in each region:

The power flow across the interface at x=0, which is continuous (Hy and Ez are both continuous) is given by:

For an ideal

The time-averaged Poynting vector flux through the bounding rectangular surface S must be zero (since the time-averaged energy density doesn't change in steady state, and there is no conversion to heat in region 2 since σ=0 in region 2) :

This flux condition leads immediately to the power attenuation rate 2β

which, because k2 is real, is exactly equivalent to:

Note that in the general Poynting theorem expression above, loss in a medium can be described as either a real conductivity σ or alternately as a complex medium refractive index or equivalently a complex dielectric constant ε = n^2. Note that a complex dielectric constant means that the time-averaging of the energy density term εE(t)^2 for a steady-state harmonic time dependence of the fields will lead to a non-zero value for the rate of change of energy density which represents loss. This is exactly equivalent to the result using a real conductivity value to describe the loss. Therefore either approach may be used to specify media absorption loss. The equivalency is:

The corresponding TM0 surface wave "mode index", which is symmetric in interchange of n1 and n2 is:

and the power attenuation of the surface wave in the z propogation direction is:

(With the sign convention of Hy(x,z) above, β

For

where ε

With medium 1 having a real dielectric constant

and the phase velocity of the TM0 mode along the z direction is approximately equal to the velocity of light in dielectric medium 1. Properties for this ideal interface are calculated below :

- Using input data from first row: vacuum wavelength λ
_{0}(m) and complex media refractive indices n1, n2 - Using input data from second row: frequency f and complex media "inner" dielectric constants ε1, ε2 (these are ε/ε
_{0}) and conductivities σ1, σ2 (mhos/m)

The expressions above for β and n

λ = 0.633 µm n1 = 0.068 -j4.0 n2 = 1.52 +j0.0

The TM0 surface wave mode index is:

nand the surface wave attenuation is:_{m}= 1.643127 - j0.004711

α = 935.2 cm-1 Loss = 4062 dB/cmNote that the real part of the TM0 mode index, 1.643, is greater than the real dielectric mode index,1.520, so the phase velocity of the TM0 surface wave is lower than the phase velocity of a plane wave in that dielectric medium.

The magnetic field amplitude for this example is shown below:

The second example uses parameters in the short-wave radio frequency range. A 30 MHz surface wave propogates over an ideal flat salt-water body. In this case the appropriate material parameters are the low frequency real dielectric constant ~ 81 and typical salt-water conductivity ~ 4 mhos/m. These "low frequency" parameters are fairly constant in the frequency range from DC to ~ 1 GHz. Using the calculator above with the second method, the TMO surface wave mode index is:

nand the surface wave travels along the air/water interface at very nearly the speed of light in air. The surface wave attenuation is:_{m}= 0.999993 -j0.000208

α = .262 km-1 Loss = 1.14 dB/kmThe TM0 radio wave exponential penetration depth into the water is ~ 4.7 cm and above the water into air is ~ 112 m. At this frequency, salt water is a good conductor since σ/(εω) ~ 30.

The graph below shows the TM0 wave field penetration into air and typical salt water, and the propogation loss versus frequency over a range where the dielectric constant and conductivity are approximately constant. For salt water the "good conductor" condition σ/(ωε) >>1 is valid below ~ 300 MHz. In this region, the surface wave loss increases as the square of the frequency:

The "principal" solution is the solution with β close to k2 and for or any reasonable wire radius, ha<<1 leading to a simple iterative method (Sommerfeld 1952) for solving the above equation for

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

**Field Theory of Guided Waves**, R. E. Collin, 1991, IEEE Press, pp.697-700

**Fields and Waves in Communication Electronics**, S. Ramo, J. Whinnery, T. Van Duzer, 1984, J. Wiley & Sons. pp. 279-283

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

**Electrodynamics**, A. Sommerfeld, Lectures on Theoretical Physics, 1952, VIII Academic Press, pp 188-185