We study propagation of electromagnetic waves in a nano-coaxial waveguide for frequencies around and below the surface plasmon frequency. We show, that for frequencies sufficiently lower than the surface plasmon frequency, the waveguide supports a plasmon polariton mode that resembles, and indeed reduces to the conventional TEM mode of the conventional coaxial transmission line, known in the radiotechnology.
©2008 Optical Society of America
Recently, transmission of visible light through nanoscopic, coaxial transmission lines based on carbon nanotubes, has been demonstrated . The light propagates, without any frequency cut-off, along narrow channels much smaller than the light wavelengths λ, and over large distances much larger than λ. This resembles transmission of radio-waves through the conventional coaxial cable  (known since 1884 ) via its basic mode, the transverse electromagnetic (TEM) wave, that does not possess any cut-off frequency (i.e. it is gapless). In this paper we demonstrate theoretically, that a metallic nanocoax indeed supports a gapless polaritonic mode, the 0th order transverse magnetic (TM0) mode, which for frequencies sufficiently lower than the surface plasmon frequency of the metal, resembles, and for much smaller frequencies, reduces to the text-book TEM coax mode. This allows for a “radio-engineering” treatment of the light propagation through nanocoaxial transmission lines, involving, for example, an impedance analysis for the line matching and coupling . While the dispersion of the gapless polaritonic mode has been calculated by various authors [4–9], its detailed character in the low frequency limit has not been identified.
2. Theoretical study
2.1 Analytical approach
We begin with a standard analytical approach, based on matching solutions of the Maxwell equations (in cylindrical coordinates) across the nanocoax interfaces. This approach has been commonly used in the theory of fiber optics transmission lines. In the case of the nanocoax, the difference is the presence of metallic regions, in which the dielectric function is a strong function of frequency, and below the plasma frequency becomes negative. Optics of such systems, with “negative dielectrics”, was investigated in general by Takahara et al , and a “gapless” mode was referred to as a “one dimensional (1D) optical wave”. We call it a “TEM-like” mode.
The geometry of the coaxial waveguide is shown schematically in Fig. 1. The metal core is a solid cylinder with radius a. The outer conductor is a hollow metallic cylinder with the inner radius b, and the outer radius c. Since we are interested here only in internally guided modes, i.e. modes with fields localized between the core and the outer conductor, the outer conductor can be assumed to be infinitely thick (i.e. c→∞). This is a good approximation for any coaxial waveguide with outer conductor much thicker than the field penetration depth of these internally guided modes. We assume that both conductors are made of the same, non-magnetic metal described by the Drude dielectric function
where ω is the frequency, ωp is the metal’s plasma frequency, γ is the damping parameter, and εb is the contribution from the “bound” electrons. A dielectric material, with dielectric constant ε 2, fills the space between the conductors. All fields are assumed to be transverse, and ∞ expi(kzz-ωt), where kz is the z-component of the wave vector. Each material interface is assumed to be abrupt, with standard boundary conditions used to match the electromagnetic fields.
Employing this procedure leads to a set of equations, which have a solution if
where , m=1,2,3, and In,Kn are the nth-order modified Bessel functions of the first and the second kind, respectively.
Equation (2) is the desired dispersion relation for the “TEM-like” mode. In the limit of b→∞ (a single electrode transmission line) it reduces to the formula for the “negative dielectric pin” of Ref. , as expected
To evaluate the solutions of Eq. (2) we chose a nanocoax made of silver, a low loss metal, which also is known to have negligible, non-local, quantum mechanical corrections . The Drude parameters of this metal are ħωp=9.9 eV, ħγ=0.04eV, and εb=6.8 . Results for other metals are similar if properly scaled in plasmon units (i.e. by expressing the frequency as ω/ωp, and the damping parameter as γ/ωp). For simplicity, we also assume that ε 2=1. It is clear from Eq. (2), that only ratios of dielectric functions are relevant, and thus changing ε 2 amounts to a simple simultaneous scaling of kz, ω, 1/a and 1/b. We chose the following nanocoax dimensions: a=50 nm and b=150 nm.
Numerical solutions of Eq. (2) are shown in Fig. 2 (crosses). The mode is gapless, with a linear (acoustic) dispersion for small frequencies (see Fig. 2(a)). In fact, it is a plasmon polariton, a hybrid of photons and plasmon excitations, exactly like its slot waveguide “cousin” . The imaginary part of k z, inverse of which is the TEM-like mode propagation length L, is shown in Fig. 2(b). There is a plateau for frequencies above γ, and well below the renormalized plasma frequency ( ) , where Im(k z) ranges from 0.00005 nm-1 to 0.0001 nm-1, which corresponds to L=1/Im(k z) ranging from 20 µm to 10 µm. This is in agreement with the experimental results of Ref. . For frequencies approaching p, Im(kx) rapidly increases (and therefore L decreases). This is a characteristic of the plasmonic domain, and reflects increasing losses due to the EM energy entering the metal as a plasma wave. Inset in Fig. 2(b) shows the very low frequency domain, for frequencies below γ. The solid line is for the TEM-like mode, and the dashed curve is for the TEM mode of a classic, macroscopic coaxial cable, given by the textbook formula 
where σ is the static conductivity of the metal, and δ is the penetration depth.
From Fig. 2(b) it is clear, that for ħω<0.03 eV the curve for the “TEM-like” mode is indistinguishable from that for the classic TEM mode (given by Eq. 4). Simultaneously, in this frequency range the dispersion of the “TEM-like” is purely “acoustic” (see Fig. 2(a)). There is a ~ 36% reduction of the slope of this “acoustic” section (i.e. reduced velocity), as compared with the light propagating in free space (velocity equal to speed of light). This renormalization of the light-line dispersion is well known [7,9,10]. This effect is negligible for gap dimensions much larger than the penetration depth δ of light into metal (conventional coax), but is significant in the nanocoax. Apart from this renormalization of the mode velocity, the “TEM-like” mode dispersion relation (both for real and imaginary wave vector) in the acoustic section is identical to that for the classical TEM mode of the conventional coax.
2. 2 Numerical approach
To further develop this point, we have performed computer simulations of the fields inside the nanocoax using the finite-difference time-domain (FDTD) method, described in detail in our earlier work . This method allows calculating the mode structure of all modes for real k z. Our results are consistent with the results obtained in Ref. . The circles in Fig. 2(a) show the dispersion of the “TEM-like” mode obtained in our simulation, in a perfect agreement with the solution of Eq. (2). In addition, a branch of a higher, surface plasmon guided (gapped) mode  is shown as solid circles in the same figure.
Figure 3 shows color maps representing radial components of the electric field (E ρ) inside the nanocoax for the two modes of Fig. 2(a). The left map is for the “TEM-like” mode, and the right for the gapped mode. In each plot the color intensity is proportional to the field amplitude, and color change represents the sign (field direction) change. Both maps are obtained for k z=0.005 nm-1, so that according Fig. 2(a) the “TEM-like” mode has frequency 0.98 eV/ħ, and the gapped mode 3.4 eV/ħ. The gapped mode is well in the plasmonic domain, which results in a large penetration of fields into metal, clearly visible in Fig. 3. In contrast, the fields of the low frequency “TEM-like” mode have a marginal penetration into metal. The inset in Fig. 3 shows E ρ (solid line) and ρE ρ (dashed line) plotted vs ρ. Since ρE ρ is essentially constant, it must be that essentially E ρ~1/ρ, which is precisely the form for the pure TEM macroscopic mode .
In conclusion, we have investigated propagation of electromagnetic waves in a coaxial nano-waveguide made of silver. We show, that for frequencies sufficiently lower than the surface plasmon frequency, but still in the infrared or visible ranges, the waveguide supports a plasmon polariton mode, the “TEM-like” mode, that resembles and indeed reduces to the conventional TEM mode of the conventional, macroscopic coaxial transmission line, known in the radiotechnology. This gapless mode is capable of transmitting electromagnetic waves with wavelength far exceeding the coax diameter, and therefore allows for truly nanoscopic dimensions of the nanocoax transmitting light waves. Existence of this mode was demonstrated in Ref. . The nanocoax operating with the “TEM-like” mode will enable numerous potential applications in the nano optics .
References and links
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