On the Transmission Line Analogy for Modeling Plasmonic Nanowire Circuits

Plasmonics https://doi.org/10.1007/s11468-024-02542-8 RESEARCH On the Transmission Line Analogy for Modeling Plasmonic Nanowire Circuits D. Dragoman1,2 Received: 5 August 2024 / Accepted: 9 September 2024 © The Author(s) 2024 Abstract Modeling slot waveguides using the analogy with transmission lines in microwaves proved itself to be an accurate and simple method for characterizing plasmonic field propagation. Here, the possibility of generalizing the applicability of this method to plasmonic circuits consisting of nanowires is analyzed. It is found that it can be applied as long as the circuit can be divided in regions with known transverse field distributions and propagation constants, the total matrix characterizing plasmon propagation being composed of propagation and interface matrices, as in slot waveguides, the elements of the latter being, however, defined in terms of butt-coupling transmission coefficients at the interface and not using the simple characteristic impedance expression used for slot waveguides. Keywords Plasmonic nanowire circuit · Transmission line analogy · Plasmon propagation modeling Introduction Plasmonic circuits [1, 2] have attracted recently a large amount of interest due to the possibility of implementing on-chip integrated photonic devices with various functionalities. To fully exploit the advantages of plasmonic circuits, both theoretical and experimental studies must be pursued further in order to resolve or at least alleviate problems such as those related to the decay length of plasmons and/or suitable materials and structures for on-chip integration [3–5], as well as to enhance the efficiency of plasmonic circuits’ modeling. The last issue is not a trivial one. Different simulation methods have been used to adequately estimate as fast and with as low computing resources as possible the performances of several plasmonic circuits (some examples can be found in [6–14]). Among these methods, the one relying on the analogy with transmission lines in microwaves proved itself suitable enough to model the plasmonic field propagation through slot waveguides and circuits [15–19]. The advantage of this method is its simplicity and ease of * D. Dragoman 1 Faculty of Physics, University of Bucharest, P.O. Box MG‑11, 077125 Bucharest, Romania 2 Academy of Romanian Scientists, 3 Ilfov Street, 050044 Bucharest, Romania application even in circuits with multiple ports, as those used for instance to implement plasmonic logic gates [20]. Whether slot waveguides cover a significant number of potential applications of plasmonic circuits, it would be of considerable interest to generalize the use of the transmission line method to simulate the outcome of circuits consisting of plasmonic nanowires. The aim of this manuscript is to analyze the possibility to extend the transmission line method to compute the output of plasmonic circuits made up from metallic nanowires placed in a dielectric environment. Unlike slot waveguides, i.e., the so-called metal–insulator-metal (MIM) waveguides, nanowires, or the so-called insulator–metal-insulator (IMI) waveguides have much larger propagation/decay lengths and spatial field extensions for narrow thicknesses of the inner layer [21]. Whether the first behavior is an advantage in waveguide circuits, the last one is problematic and stands at the origin of not using the transmission line analogy method for nanowires and their circuits. In this work, we ponder on this latter problem and argue that the transmission line analogy method could be used also for plasmonic nanowire circuits provided that the impedance mismatch, i.e., the ratio of characteristic impedances, at any interface between regions with different spatial distributions of the transverse electric field is not calculated with the simple formula valid for MIM circuits but is expressed in terms of the transmission coefficient determined by the overlap of Vol.:(0123456789) Plasmonics the electric fields at the interface. The last parameter is easily computed in a similar way as for butt coupling. Transmission Line Analogy Method for Nanowire Circuits To test whether the transmission line analogy method is suitable for modeling circuits consisting of nanowires, we refer first in detail to an illustrative example. It is one of the simplest nanowire plasmonic circuits, consisting of a succession of aligned metallic nanowires of different dimensions placed in air. The schematic diagram of this circuit is illustrated in Fig. 1a, left, wi and li, i = 1,2,3 denoting, respectively, the widths and lengths of the nanowires made from a metal with relative electric permittivity εm (silver, the dielectric constants of which were taken from [22]), placed in a dielectric environment (in our case, air, with dielectric constant εd = 1). In Fig. 1a, right, the spatial distribution of the electric field along the circuit (more precisely, the transverse modes in each � � Pi li = � nanowire) is illustrated; as can be seen, the electric field has maximum values at both metal/dielectric interfaces and propagates along the nanowires. Throughout this paper, the imaginary part of the dielectric constant in the metal will be neglected; the interest is in the propagation characteristics rather than in the decay length. The latter can be introduced in the formalism, as discussed at the end of this study, through an overall attenuation constant of the calculated transmittance. For the complementary MIM circuit, shown in Fig. 1b, the transmittance of such a structure would be simply obtained as T = 1∕|M11 |2 (1) M = P1 (l1 )I12 P2 (l2 )I23 P3 (l3 ) (2) where M11 is the (1,1) element of the total matrix where the propagation matrix in each region i with constant thickness and length li and the matrix at the interface between regions j and k are given, respectively, by [16] � � � � ⎛ Z ∕Z + Z ∕Z Z ∕Z − Zj ∕Zk ⎞ k j j k k j exp(−iki li ) 0 ⎟ � � � , Ijk = 12 ⎜� 0 exp(iki li ) ⎜ Z ∕Z − Z ∕Z ⎟ Z Z ∕Z + ∕Z k j j k k j j k ⎝ ⎠ � (3) (a) (b) Fig. 1  a The IMI circuit analyzed in the example (left) and the field distribution along it (right), and b its complementary MIM structure Plasmonics Here ki is the propagation constant in region i and Zi ≅ ki wi 𝜔𝜀0 𝜀d (4) is the characteristic impedance associated with the ith region of the MIM structure, through which the electromagnetic field is guided by the dielectric layer with width wi and relativity permittivity εd. This formalism is valid whenever the transverse dimension of the MIM waveguide is much smaller than the light wavelength and approximates very well the numerical simulations obtained using FDTD methods. In the actual IMI structure illustrated in Fig. 1a, however, two issues appear: (i) the guiding medium is a metal, rather than a dielectric, such that, at first sight, the complex-valued metal permittivity ε m should be used instead of ε d in Eq. (4) and, more importantly, (ii) given the ratio of the skin (...truncated)