Plasmonics aims to combine the advantages of nanometer scale electronics with the high operating frequency (terahertz and beyond) of photonics.

Control of plasmon propagation can be achieved in a two-dimensional electron gas (2DEG) by tuning the electronic properties of the 1D nanostructures it contains, which act as scatters for plasmons. Plasmonic response of these nanostructures, however, happens on a length scale much smaller than the wavelength of free space electromagnetic radiation and cannot be studied with conventional optical microscopy.

Instead, we resolve these nanoscopic phenomena using near-field optical microscopy, which has a spatial resolution of $\sim 20\,\mathrm{nm}$.

In this dissertation, we first describe the working principles of near-field optical microscopy, then analyze the plasmonic phenomena we observed around several 1D nanostructures, including a potential well in monolayer graphene, domain walls in bilayer graphene, and a low-conductivity gap in a 2DEG.

In Chapter 1, we give an overview of the basic properties of surface plasmons and graphene, followed by a brief explanation of the operating principles of near-field optical microscopy.

In Chapter 2, we study theoretically the electromagnetic interaction between a sub-wavelength particle (the `probe') and a material surface (the `sample').

The interaction is shown to be governed by a series of resonances corresponding to surface polariton modes localized near the probe.

The resonance parameters depend on the dielectric function and geometry of the probe, as well as the surface reflectivity of the material.

Calculation of such resonances is carried out for several types of axisymmetric probes: spherical, spheroidal, and pear-shaped.

For spheroids an efficient numerical method is developed, capable of handling cases of large or strongly momentum-dependent surface reflectivity.

Application of the method to highly resonant materials such as aluminum oxide (by itself or covered with graphene) reveals a rich structure of multi-peak spectra and nonmonotonic approach curves, i.e., the probe-sample distance dependence.

These features also strongly depend on the probe shape and optical constants of the model.

For less resonant materials such as silicon oxide, the dependence is weak, so that the spheroidal model is reliable.

The calculations are done within the quasistatic approximation with radiative damping included perturbatively.

In Chapter 3, we show that surface plasmons of a two-dimensional Dirac metal such as graphene can be reflected by line-like perturbations hosting one-dimensional electron states.

The reflection originates from a strong enhancement of the local optical conductivity caused by optical transitions involving these bound states.

We propose that the bound states can be systematically created, controlled, and liquidated by an ultranarrow electrostatic gate.

Using infrared nanoimaging, we obtain experimental evidence for the locally enhanced conductivity of graphene induced by a carbon nanotube gate, which supports this theoretical concept.

In Chapter 4, we show that topological bound states confined to the domain walls in bilayer graphene are the source of the wall's strong coupling to surface plasmons observed in infrared nanoimaging experiments.

These domain walls separate regions of $\mathrm{AB}$ and $\mathrm{BA}$ interlayer stacking and have attracted attention as novel examples of structural solitons, topological electronic boundaries, and nanoscale plasmonic scatterers.

The optical transitions among the topological chiral modes and the band continua enhance the local conductivity, which leads to plasmon reflection by the domain walls. The imaging reveals two kinds of plasmonic standing-wave interference patterns, which we attribute to shear and tensile domain walls.

We compute the electronic structure of both wall varieties and show that the tensile wall contains additional confined bands which produce a structure-specific contrast of the local conductivity, in agreement with the experiment. The coupling between the confined modes and the surface plasmon scattering unveiled in this work is expected to be common to other topological electronic boundaries found in van der Waals materials. This coupling provides a qualitatively new pathway toward controlling plasmons in nanostructures.

In Chapter 5, we present a comprehensive study of the reflection of normally incident plasmon waves from a low-conductivity 1D junction in a 2D conductive sheet.

Rigorous analytical results are derived in the limits of wide and narrow junctions.

Two types of phenomena determine the reflectance, the cavity resonances within the junction and the capacitive coupling between the leads.

The resonances give rise to alternating strong and weak reflection but are vulnerable to plasmonic damping.

The capacitive coupling, which is immune to damping, induces a near perfect plasmon reflection in junctions narrower than $1/10$ of the plasmon wavelength.

Our results are important for 2D plasmonic circuits utilizing slot antennas, split gates or nanowire gates.

They are also relevant for the implementation of nanoscale terahertz detectors, where optimal light absorption coincides with the maximal junction reflectance.