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. 2018 Jan 30;8(1):1921.
doi: 10.1038/s41598-018-20239-x.

Limits of Kirchhoff's Laws in Plasmonics

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Free PMC article

Limits of Kirchhoff's Laws in Plasmonics

Gary Razinskas et al. Sci Rep. .
Free PMC article

Abstract

The validity of Kirchhoff's laws in plasmonic nanocircuitry is investigated by studying a junction of plasmonic two-wire transmission lines. We find that Kirchhoff's laws are valid for sufficiently small values of a phenomenological parameter κ relating the geometrical parameters of the transmission line with the effective wavelength of the guided mode. Beyond such regime, for large values of the phenomenological parameter, increasing deviations occur and the equivalent impedance description (Kirchhoff's laws) can only provide rough, but nevertheless useful, guidelines for the design of more complex plasmonic circuitry. As an example we investigate a system composed of a two-wire transmission line and a nanoantenna as the load. By addition of a parallel stub designed according to Kirchhoff's laws we achieve maximum signal transfer to the nanoantenna.

Conflict of interest statement

The authors declare that they have no competing interests.

Figures

Figure 1
Figure 1
Classical vs. plasmonic transmission lines. (a) Two classical TWTLs of identical characteristic impedance Z0 connected in parallel resulting in an equal splitting of electrical currents at the point junction. (b) Equivalent junction of two plasmonic TWTLs, an idealized building block of optical nanocircuits. (inset) The guided antisymmetric mode (electric field intensity |E|2 at free-space wavelength λ = 830 nm) is directly launched from the left and propagates along the nano-sized TWTL, where the mode intensity is split at the parallel junction. The mode symmetry is defined by the symmetry of the longitudinal field component Ex with respect to the TWTL mid-plane.
Figure 2
Figure 2
Parallel junction of plasmonic TWTLs. (a) Optical near-field intensity distribution in a cut through the gap centered between small-sized nanowires (width = height = 30 nm, gap = 10 nm) of an unconnected TWTL. Inset: modal profile of the guided antisymmetric mode with most of its intensity localized in the nanowire gap. The dashed white line indicates the plane used to record the near-field intensity cut. (b) Same as in (a) but for a parallel junction of TWTLs. (c) Intensity difference plot obtained from the maps with (b) and without (a) the upward pointing semi-infinite TWTL. (d) Standing wave pattern forming along the TWTL due to partial reflection of the antisymmetric mode at the TWTL junction used to extract the reflection coefficient. The theoretically expected behavior (red dashed line) corresponding to |Γ| = 1/3 and θΓ = π is added as a guide for the eye. The intensity distribution (black solid line) is recorded mid-height in the gap center of the horizontal TWTL. (eh) The same as (ad), but for a parallel junction of wider TWTLs (width = height = 120 nm, gap = 10 nm).
Figure 3
Figure 3
Variation of plasmonic TWTL dimensions. (a) Power splitting ratio, (b) reflection amplitude (black curve) and phase (red curve), and (c) λeff (black curve) and κ (red curve) as a function of nanowire width. The gap is kept constant at 10 nm. (d) Power splitting ratio, (e) reflection amplitude (black curve) and phase (red curve), and (f) λeff (black curve) and κ (red curve) as a function of the gap size. The nanowire width and the height are kept constant at 30 nm.
Figure 4
Figure 4
Finite stub reflectivity tuning. (a) Sketch of the investigated system featuring an infinite TWTL connected in parallel with a finite stub of length L. The shown antisymmetric mode is directly launched from the left and propagates along the nano-sized TWTL. (b) The mode’s standing wave pattern along a cut at midheight through the TWTL for high impedance stub (L = 120 nm, top) and low impedance stub (L = 180 nm, bottom). (c) Reflection amplitude (top) and phase (bottom) for systems of varying stub length L. The red dots are obtained by fitting of FDTD simulation data with the model described in Eq. 2, while the blue solid lines are obtained by the analytical model (Supporting Information).
Figure 5
Figure 5
Tuning the reflectivity in a nanoantenna-terminated TWTL by a parallel stub. (a) Sketch of the investigated system featuring a finite TWTL terminated by an optical antenna of length lant connected in parallel with a finite stub of length L at a distance d from the antenna. The shown antisymmetric mode is directly launched from the left and propagates along the nano-sized TWTL. Inset: Equivalent circuit representation of the system. (b) Reflection amplitude (top) and phase (bottom) for varying stub length L in a system with open end termination (lant = 70 nm) and d = 200 nm. (b) Reflection amplitude (top) and phase (bottom) for varying stub length L in a system with resonant antenna termination (lant = 230 nm) and d = 200 nm. The red dots are obtained by fitting of FDTD simulation data with the model described in Eq. 2, while the blue solid lines are obtained by the analytical model (Supporting Information).

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