Optical 'Shorting Wires'

Connecting lumped circuit elements in a conventional circuit is usually accomplished by conducting wires that act as conduits for the conduction currents with negligible potential drops. More challenging, however, is to extend these concepts to optical nanocircuit elements. Here, following our recent development of optical lumped circuit elements, we show how a special class of nanowaveguides formed by a thin core with relatively large (positive or negative) permittivity surrounded by a thin concentric shell with low permittivity may provide the required analogy to 'wires' for optical nano-circuits.

The fundamental concepts for extending the classic circuit theory to nanocircuits at optical frequencies have been introduced and developed by our group in a recent letter [1]. In that work, we have shown that it may be possible to realize lumped nanoinductors, nanocapacitors and nanoresistors by utilizing sub-wavelength nanoparticles with negative-real (plasmonic), positivereal (non-plasmonic) and imaginary (lossy) permittivity, respectively. In this framework, the role of conduction current density flowing in classic circuits is replaced by the displacement current circulating in such a nanocircuit, with ω being the operating radian frequency under an i t e ω − time convention and ε the local permittivity. While for a single element it has been proven that the classic Kirchhoff laws in the circuit theory are satisfied in this nanocircuit paradigm [1] in terms of the current I defined as the flux integral of d J across the optical nanoelement and the optical potential V between its two "ends", it is more challenging to envision a complex nanocircuit containing several closely packed nanocircuit elements connected with a special pattern.
Even though we have shown that the basic parallel and series combinations of two optical nanoelements are indeed possible by suitably positioning two interconnected particles with respect to the orientation of the external field [2], and that the optical nanotransmission lines may be designed following simple circuit considerations on periodic arrangements of such elements [3]- [4], the flexibility and possibility of connecting relatively distant optical lumped nanoelements or in connecting insulated nanoparticles that are closely spaced on a nanocircuit board is not as straightforward as in a classic circuit at lower frequencies [5]. This is because at lower frequencies, at which classic circuits rely on the conduction current ( c J E σ = , with σ being the local conductivity and E the local electric field) circulating across the elements, the background material (i.e., air) or the substrate of a circuit board are naturally very poorly conducting ( 0 background materials, in some sense connecting all nanocircuit nanoelements together in an unwanted manner. In this nanocircuit paradigm, the role of material conductivity is indeed replaced by the local material permittivity ε . It follows, as discussed in [5], that the equivalent of a poorly conducting material in the framework of our optical nanocircuit theory is represented by a lowpermittivity ( ε -near-zero, ENZ) material, for which d J is close to zero despite the non-zero values of the local electric field. An ENZ material surrounding a lumped nanocircuit element would indeed "insulate" the element from the unwanted coupling with other neighboring elements, at least for what concerns the displacement current flow in it [5]. Analogously, the role of nanoconnectors may be taken, in this nano-circuit analogy, by large permittivity materials ( ε -very-large, EVL, either positive or negative large permittivity), which may "connect" relatively distant optical nanoelements with a small potential drop, since for a given level of current d J the corresponding longitudinal electric field, and the consequent voltage drop, are very low across the EVL connector.
Following these heuristic concepts [1]- [5], the representation of a complex nanocircuit system is getting closer to realization with feasible materials and technology at optical frequencies, i.e., employing naturally available plasmonic, non-plasmonic and/or polaritonic materials. In order to make the picture more complete, however, it may be desirable to envision an optical "wire" that may connect two parts of an optical nanocircuit or two or more nanocircuit elements, in the same manner as classic metallic conducting wires at lower frequencies allow flow of conduction currents with no current leakage and with negligible voltage drop. Such an optical "wire" should be able to pass the displacement current flow with essentially no phase delay, without spreading and leaking it around its path, and with a sufficiently low optical potential drop across its terminals. In the following, we investigate the possibility of providing these functionalities of an optical wire in the optical nanocircuit domain, exploiting the anomalous properties of EVL and ENZ materials.
We may heuristically envision the optical "wire" as a deeply sub-wavelength cylindrical waveguide that is formed by a core with an EVL material surrounded by a thin ENZ shell, which bounds the displacement current flow in the EVL core and does not allow current leakage across it. Here, we verify this prediction by studying this problem analytically and numerically, demonstrating how a displacement current flow with a low optical potential drop and low phase delay may be indeed obtained with such a structure. Since ENZ and EVL materials exist at infrared/optical frequencies [6] or they may be realized as layered metamaterials [7], realization of such optical nanowaveguides may be envisioned in the optical domains.
Consider the geometry of Fig. 1a: a cylindrical core of permittivity EVL ε surrounded by a shell with permittivity ENZ ε and surrounded by a background material with permittivity 0 ε . The corresponding radii are, respectively, core a and shell a . All the material permeabilities are assumed to be the same as that of free space 0 μ . An azimuthally symmetric TM mode supported by such a waveguide has the following field distributions in each region of space in Fig. 1 [8]: in the corresponding cylindrical reference system of Fig. 1, where β denotes the propagation constant of a guided mode, n z is the suitable n -th order Bessel function ( n J in the core region, a linear combination of n J and n Y in the shell region and the Hankel function ( ) 1 n H in the background region). Moreover, ε is the local permittivity in the different regions, 0 k ω εμ = is the local wave number and A a suitable coefficient (different in each region) to match the continuity of the tangential components of fields at the two interfaces. Fig. 1 shows an example of these field distributions.
The general dispersion relation for the modes guided by this structure with arbitrary values for material permittivities is well known [8]. Only for where γ is the Euler constant.
This closed-form expression is quite remarkable in its utility to find the propagation constants of guided modes supported by the waveguide of Fig. 1 in the limits of interest here, and it provides many physical insights into its anomalous guidance properties. First of all, it is evident how the values of β are proportional to the wave number in the shell region ENZ k , which tends to zero for low-epsilon shell materials. This is physically expected, due to the low variation of phase in the ENZ material, and is consistent with the findings in [9], regardless of the field excited inside the EVL core. Due to its low value, β is required to be necessarily complex in order to take into account the radiation leakage. In other words, since 0 k β < , the supported modes are leaky-modes, and thus they necessarily radiate in the background region as manifested in the intrinsic complex nature of β given in Eq. (2). (Note the analogies with the structure described in [10]).
The similarities with a standard shorting metallic conducting wire in a radio-frequency circuit become evident for the traveling modes described by (2): the phase variation along the wire is ) and the electric and magnetic field distributions around it, as depicted in Fig. 1, closely resemble those of a regular conducting wire at radio frequencies. It should be noted that a regular metallic wire also radiates some small energy when a time-varying current flows along its path, and its field distribution is distributed in the background material as described in (1). Usually the radiation losses are negligibly small in classic RF circuits, due to the very small electrical size of RF circuits and the subsequent quasi-static nature of the problem. Eq.
(2) can fully take into account the radiation associated with a displacement current propagating with 0 ENZ k β ∝ along this optical nanowire, again consistent with its lower frequency counterpart.
If we now consider the limit 0 ENZ ε = , we will note that the ENZ shell will act as a perfect magnetic conductor for this TM polarization, reducing the field to zero outside the core and imposing the stringent dispersion relation ( ) Here we consider the scenario that assumes reasonable realistic values for ENZ materials, for which which imposes some constraints on EVL ε for a fixed dimension of the inner core. (We note how, in this limit, the outer radius of the shell shell a is not crucial in ensuring the guidance properties of Eq. (2), for which condition (3) is sufficient, even though it may play a role in the amplitude of β , appearing in the denominator of Eq. (2)). The imaginary part of β , in this limit and under condition (3), is intrinsically small, since it is proportional to ENZ k . Its value, and the corresponding radiation from the wire, becomes negligible when the left-hand of Eq. (3) becomes sufficiently negative to overcome the term ( ) iπ in (2), which happens near the condition ( )  The dependence of the wave number over the geometrical parameters of the waveguide is well Inspecting the results of Fig. 2b concerning the imaginary part of the guided wave number β , it is evident how the damping of such modes due to radiation losses is relatively low, ensuring that in the sub-wavelength scale of an optical nano-circuit these nano-wires may transport the displacement current from one end to the other, with small energy loss (in Fig. 2 we have not considered the presence of material losses, which may enter into play when the nano-wire has a very subwavelength cross-section or when the imaginary part of permittivity is relatively high). The results of Fig. 2b, however, are substantially unchanged when reasonable ohmic losses for optical materials are considered, as we show in the following examples that consider realistic material losses. As an aside, the sign of

[ ]
Im β in Fig. 2 also ensures that the guided modes in that specific example are all forward, independent of the sign of ε EVL . It may be proven, more in general, that the sign of ENZ ε is the one to affect directly the direction of propagation of this anomalous mode: backward propagation is obtained when ENZ ε has a negative real part (i.e., before the plasma frequency of the material).
This may be seen more in detail in Fig. 3, where the full dispersion of the guided modes for a nano- has been plotted considering the necessary dispersion of the involved plasmonic materials. In this example, the ENZ shell has indeed been assumed to be plasmonic in nature with a Drude-like dispersion    a metallic wire can still operate as intended, even when it is bent, here we show that this optical "wire" still functions with its interesting properties, when it is bent. Both figures reveal interesting features of such an optical "wire", consistent with the findings described above. In Fig. 4a it is evident how the displacement current is all concentrated in the inner core of the nano-wire, similar to a classic shorting conducting metallic wire at lower frequencies. This is also confirmed by the field distributions reported in Fig. 1, which shows a small longitudinal electric field component in the core, and which resemble those of a classic wire carrying a uniform current. The ENZ shell in this geometry indeed allows confining the displacement current inside the core with a very low optical potential drop (We note that despite the displacement current being high in the core region, the corresponding longitudinal electric field, which contributes to the potential drop along the wire, is very small due to the high value of permittivity of the core material). Fig. 4b shows the full-wave simulation of a bent nanowire, demonstrating the striking guidance properties of such an optical "wire". It can be seen how the mode is tightly guided around the subwavelength waveguide with a very low phase variation along the line. Despite the electrical length of the bent nano-wire shown in Fig. 4b, which is much longer than what would be required in nanocircuit applications, the phase delay along the wire is still negligible and the phase follows the nanowire despite its bend. Also the calculated potential drop along the nanowire is substantially lower than the one expected along a similar electrical length in free-space or along a regular waveguide, since the distribution of longitudinal electric field in the inner region is substantially lower than the transverse electric field in the surrounding free-space (consistent with Fig. 1a).
To conclude, we have reported how the anomalous guiding properties of an EVL thin cylindrical nanowire surrounded by a thin ENZ plasmonic shell may act as an optical "wire" with interesting potentials in the framework of optical nanocircuit theory and nanoelectronics. The analysis presented here has also derived a closed-form solution for the complex guided wave numbers of this optical wire. Various methods for realization of this optical wire may be forecasted: one way may consist of the use of sub-wavelength stacks of plasmonic and non-plasmonic materials, which, depending on the orientation of the field, may act as effective ENZ or EVL materials (see, e.g., [7]).
Another possibility may be by utilizing a relatively high permittivity substrate: since the present analysis has been developed relative to a generic background permittivity 0 ε , it is expected that the use of a relatively high-permittivity background may reasonably relax the requirements of a low permittivity for the ENZ shell.
This work is supported in part by the U.S. Office of Naval Research (ONR) grant number N 00014 -07-1-0622