Surface electromagnetic waves at gradual interfaces between lossy media
SSurface Electromagnetic Waves at Gradual Interfaces Between Lossy Media
Igor I. Smolyaninov
Abstract — A low loss propagating electromagnetic wave is shown to exist at a gradual interface between two lossy conductive media. Such a surface wave may be guided by a seafloor-seawater interface and it may be used in radio communication and imaging underwater. Similar surface waves may also be guided by various tissue boundaries inside a human body. For example, such surface wave solutions may exist at planar interfaces between skull bones and grey matter inside a human head at 6 GHz.
Index Terms — Surface electromagnetic wave, underwater communication, bioelectromagnetics
I. I
NTRODUCTION
Abrupt step-like planar interfaces between media having different electromagnetic properties are known to sometimes support surface electromagnetic waves (SEW). However, long-range low-loss propagation of such SEWs is known to occur only in some limited circumstances. The most well-known cases of such low-loss SEWs include surface plasmons (SP), which propagate along metal-dielectric interfaces [1], and Zenneck waves [2], which may propagate along interfaces between highly lossy conductive media and low-loss dielectrics. In both cases the wave vector k of such SEWs along the interface may be determined as: += ck (1) where and are complex dielectric permittivities of the adjacent media, resulting in Im( k )< HEORETICAL C ONSIDERATION Let us solve macroscopic Maxwell equations in a non- magnetic ( =1) medium, in which the dielectric permittivity I. I. Smolyaninov is with the Saltenna LLC, 1751 Pinnacle Drive, Suite 600 McLean VA 22102-4903 USA (phone: 443-474-1676; e-mail: [email protected]) and with the University of Maryland (e-mail: [email protected] ). Fig. 1. Geometry of the problems. The dielectric permittivity of the medium depends only on z coordinate, which is illustrated by halftones. The gradual transition layer thickness between the two media equals . is continuous, and it depends on z coordinate only: = (z), as shown in Fig.1. For such a geometry let us search for an electromagnetic wave propagating in the x direction, so that its field is proportional to )( tkxi e − . The macroscopic Maxwell equations may be written as = D , = B , BiE = , and DiH −= (2) which leads to a wave equation ( ) EcE = (3) After straightforward transformations this wave equation may be re-written as EczEE z )( =−− (4) Depending on the polarization state (TE or TM) of the electromagnetic wave solution, Eq.(4) may be re-written in the form of the following effective Schrodinger equations: yyy EkEczzE 22 222 )( −=−− (5) for the TE polarized wave (in which E z =0), and zzzz EkEzczzzEzE ln)(ln −= +−−− (6) for the TM polarized wave (in which z E ). Eq.(6) may be further transformed by introducing the effective wave function as / = z E , which leads to ( ) /4321)( kVzzzczz −=+−= +−−+− (7) for the TM polarized wave. In both cases (Eq.(5) for the TE wave and Eq.(7) for the TM wave) -k plays the role of effective energy in the respective Schrodinger equations. However, the effective potential energies in the TE and the TM cases are different. In the TE case the effective potential energy is )()( czzV −= , so that the only solutions of Eq.(5) having propagating character (Im( k )< Let us confirm the analytical theoretical consideration above by detailed simulations in two situations of practical importance. First, let us analyze propagation of the newly found SEWs along the seawater-seabed interface and demonstrate that they may be used in long-distance underwater radio communication. The sandy seabed conductivity of =1 S/m has been measured in [5], while the average conductivity of seawater is typically assumed to be =4 S/m [3]. In the ELF-VHF radio frequency ranges the dielectric permittivities of seawater and sandy seabed are defined by their conductivities: '" = ii (12) where is the dielectric permittivity of vacuum. Therefore, in these two media /2 and it is safe to assume that remains approximately constant within the gradual transition layer between the seabed and the seawater. We will also assume that the large-scale roughness of the sandy seabed defines the width of the ( z) transition layer. Under these assumptions Eq.(11) may be re-written as ( ) /4321 zzciV +−−= (13) The effective potential energy defined by Eq.(13) is plotted in Fig.2 for two different cases of seabed surface roughness and two different operating frequencies (note that in both cases Im( V )< Fig. 4. Loss angle of various human tissues at 6 GHz (based on the data assembled in [7]). Various tissues present in the head are indicated by arrows. The green dashed lines indicate difference of the loss angle between the skull bones and the grey matter. complications, let us consider an idealized geometry in which a planar skull bone is separated from the grey matter tissue by a 0.5 mm thick transition layer, in which the loss angle remains approximately constant at 6 GHz, as illustrated in Fig.5(a). The thickness of the transition region was chosen based on the lattice-like appearance of the bones of the cranial vault with a typical scale of the order of 500 m - see for example Fig.3 from [8] (note also that the bone-tendon junctions also have similar transition layer thicknesses – see Fig.5 from [9]). A SEW solution having Im( k )< Fig. 7 Simulations of a SEW cavity excitation by a 6 GHz point source positioned at different distances from the interface. The distance between an excitation source and the interface increases progressively from (a) to (d). The SEW electric field images are 25 mm x 25 mm. The SEW cavity is formed by two 0.4 mm diameter cylindrical metal scatterers placed within the transition layer at half SEW wavelength (2mm) from each other. Under such circumstances the SEW cavity is excited rather efficiently even if the distance between the cavity and the point source reaches centimeter-scale distances. placed in the immediate vicinity of the transition layer. The SEW is not excited when the point source is separated by more than ~ 1 mm from the transition layer. While this result is encouraging from the EM radiation safety prospective, simulations depicted in Fig. 7 look much less reassuring. In these simulations a SEW cavity is formed within the transition layer by placing two 0.4 mm diameter cylindrical metal scatterers inside the layer, which are separated by 2mm, which approximately equals one half of the SEW wavelength at 6 GHz (note that the SEW wavelength is much smaller than the free space wavelength at 6 GHz). As can be seen from Fig.7, under such circumstances the SEW cavity is excited rather efficiently even if the distance between the cavity and the point source reaches centimeter-scale distances. IV. C ONCLUSION We have demonstrated that a low loss propagating surface electromagnetic wave may exist at a gradual interface between two lossy conductive media. Such a surface wave may be guided by a seafloor-seawater interface and it may be used in radio communication and imaging underwater. Similar surface waves may also be guided by various tissue boundaries inside a human body. For example, such surface wave solutions may exist at planar interfaces between skull bones and grey matter inside a human head at 6 GHz. A possibility of deeply sub- wavelength SEW cavities (or “hot spots”) has been revealed in numerical simulations of SEW -related effects in human tissues. Since 6 GHz signals are widely used in the currently deployed 5G networks and novel wi-fi interfaces, it would be prudent to re-examine EM radiation safety issues associated with the potential excitation and scattering of the newly discovered surface electromagnetic waves inside a human body. R EFERENCES [1] A. V. Zayats, I. I. Smolyaninov, and A. Maradudin, "Nano-optics of surface plasmon-polaritons", Physics Reports , Vol. 408, pp. 131-314, 2005. [2] K. A. Michalski and J. R. Mosig , “The Sommerfeld half -space problem revisited: from radio frequencies and Zenneck waves to visible light and Fano modes”, Journal of Electromagnetic Waves and Applications , Vol. 30, pp. 1-42, 2016. [3] I. I. Smolyaninov, Q. Balzano, C. C. Davis, D. Young, “Surface wave based underwater radio communication”, IEEE Antennas and Wireless Propagation Letters Vol. 17, pp. 2503-2507, 2018. [4] L. D. Landau, E. M. Lifshitz. Quantum Mechanics: Non-relativistic theory (Vol. 3, § 45) (Elsevier, 2013). [5] H. Müller, T. von Dobeneck, C. Hilgenfeldt, B. SanFilipo, D. Rey, B. Rubio, “Mapping the mag netic susceptibility and electric conductivity of marine surficial sediments by benthic EM profiling”, Geophysics, Vol. 77, pp. 1JF-Z19, 2012. [6] B. V. Numerov, "A method of extrapolation of perturbations",