Effects of geometry on surface plasmon-polaritons:

an analytical approachDionisios Margetis

Department of Mathematics, andInstitute for Physical Science and Technology (IPST), and

Center for Sci. Computation and Math. Modeling (CSCAMM),University of Maryland, College Park

Collaborators: M. Luskin (UMN), M. Maier (UMN)

IMA Hot Topics Workshop on: Mathematical Modeling of 2D MaterialsThursday, May 18, 2017

James Clerk Maxwell(1831–1879)

Perspective

• Certain 2D materials are promising for the control of light at the microscale in nano-photonics applications. Examples: graphene, black phosphorus, ….

• At the interface of such materials with air or other dielectrics: electromagnetic (EM) waves may be excited w/ unusual features at the IR range.

• Special type of surface wave: Surface plasmon-polariton (SP): Evanescent EM wave, manifestation of coupling of incident, free-space radiation with the electron plasma of material. Goal: SP wavelength << free-space wavelength.

Plasmon-phonon-polariton

Low et al., Nat. Mater. 16 (2017), 182

Diel. Permittivity,Surface plasmon-polariton

Maxwell’s equations

A 2D conducting material is viewed as a boundary (hypersurface).

Σ

volume conductivity

jump eff. surface conductivity

Wavenumber of ambient space

SP via classical EM reflection/transmission theory

Infinitely long Graphene sheet; conductivity σ

Reflection coefficient:

Incident field:

Reflection:

Transmission:

A few questions

• Should classical Maxwell's eqs. be used for SPs? Nonlinearities? Time domain analysis?

• How can one derive effective, "macroscopic" theories of EM propagation consistent with the material microstructure?

(By homogenization, coarse graining etc.)

• In the context of "macroscopic" equations, how can we develop accurate computational schemes to capture fine structure of SP? How can we test/validate such methods via analytical solutions? Insights?

The geometry can be manipulated in surprising ways….

Graphene spring[Blees et al., Nature 524 (2015) 204]

10µm

Edges generate SPs[DM, Maier, Luskin, SAPM, to appear]

Prototypical problem: Scattering of wave by graphene sheet in 2D.Transverse Magnetic (TM) polarization

Scattering from graphene sheet in 2D (cont.)

Scattering from graphene sheet in 2D (cont.)

Analytic in lower half plane,

Analytic in upper half plane,

Scattering from graphene sheet: SP unveiled

Dispersion relation

Scattering from graphene sheet: Approximate formula for tangential electric field on sheet

SP contr.Incident + dir. reflected fields radiation field

Numerical results by Finite Element Method

More on the numerics:M. Maier (next talk)

[DM, Maier, Luskin, SAPM, to appear]

How can curvature of 2D material affect SP dispersion?

Flexible plasmonics can be realized on unconventional and nonplanar substrates

[Aksu et al., Adv. Mater. 23 (2011) 4422

Schematic: Convex bend of conducting layer (towards vac)Substrate

[Smirnova et al., ACS Photonics 3 (2016) 875]How is the SP dispersion affected by a bend?

Formulation: Preliminaries[M.V. Berry, J. Phys. A: Math. Gen. 8 (1975) 1952; … Xiao et al., Photon. Res. 3 (2015) 300; Velichko, J. Opt. 18 (2016) 035008;Smirnova et al., ACS Photonics 3 (2016) 875…]

substrate substrate

Conductinglayer

Program:Formulate an exactly solvable model with circle (2D) or sphere (3D).Assume electrically large radius of curvature.Remove periodicity algebraically via Poisson summation formula* and asymptotics.

vacuum

*[T. T. Wu, Phys. Rev. B 104 (1956) 1201; H. M. Nussenzveig, J. Math. Phys. 10 (1969) 82; M. V. Berry, K. E. Mount, Rep. Prog. Phys. 35 (1972) 315]

Substrate or vacuum

Vacuum orsubstrate

.

e-dipole

2D problem: Circular cylinder

From boundary conditions

Cylindrical coords.n=0

n=1e-dipole

Poisson sum.formula

Dispersion relation in 2D setting

Sign controlled byconvexity/concavity SP more pronounced

on concave bend

Limitations?

Debye expansion

3D setting: SphereSpherical coords.

From boundary conditions

Dispersion relation for SP:

SP more pronouncedon concave bend

Conclusion-Work in progress

• We showed how edges act as induced localized sources of SPs via canonical problem.

• So far, we have studied analytically SPs propagating perp. to edge. How about the SP propagating along the edge?

• Due to the mechanical flexibility of some 2D materials, we plausibly asked: How are the dispersion relations affected by a curved substrate?

This calls for studying SP dispersion relations on manifolds in 3D. Systematic numerics?

• For relatively simple, slowly varying geometries, curvature induces BC with effective, wave number-dependent conductivity. Larger curvatures? Anisotropies?

• Generalized BCs?