![Page 1: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/1.jpg)
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
![Page 2: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/2.jpg)
James Clerk Maxwell(1831–1879)
![Page 3: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/3.jpg)
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.
![Page 4: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/4.jpg)
Plasmon-phonon-polariton
Low et al., Nat. Mater. 16 (2017), 182
Diel. Permittivity,Surface plasmon-polariton
![Page 5: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/5.jpg)
Maxwell’s equations
A 2D conducting material is viewed as a boundary (hypersurface).
Σ
volume conductivity
jump eff. surface conductivity
Wavenumber of ambient space
![Page 6: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/6.jpg)
SP via classical EM reflection/transmission theory
Infinitely long Graphene sheet; conductivity σ
Reflection coefficient:
Incident field:
Reflection:
Transmission:
![Page 7: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/7.jpg)
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?
![Page 8: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/8.jpg)
The geometry can be manipulated in surprising ways….
Graphene spring[Blees et al., Nature 524 (2015) 204]
10µm
![Page 9: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/9.jpg)
Edges generate SPs[DM, Maier, Luskin, SAPM, to appear]
Prototypical problem: Scattering of wave by graphene sheet in 2D.Transverse Magnetic (TM) polarization
![Page 10: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/10.jpg)
Scattering from graphene sheet in 2D (cont.)
![Page 11: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/11.jpg)
Scattering from graphene sheet in 2D (cont.)
Analytic in lower half plane,
Analytic in upper half plane,
![Page 12: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/12.jpg)
Scattering from graphene sheet: SP unveiled
Dispersion relation
![Page 13: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/13.jpg)
Scattering from graphene sheet: Approximate formula for tangential electric field on sheet
SP contr.Incident + dir. reflected fields radiation field
![Page 14: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/14.jpg)
Numerical results by Finite Element Method
More on the numerics:M. Maier (next talk)
[DM, Maier, Luskin, SAPM, to appear]
![Page 15: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/15.jpg)
How can curvature of 2D material affect SP dispersion?
![Page 16: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/16.jpg)
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?
![Page 17: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/17.jpg)
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
![Page 18: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/18.jpg)
2D problem: Circular cylinder
From boundary conditions
Cylindrical coords.n=0
n=1e-dipole
Poisson sum.formula
![Page 19: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/19.jpg)
Dispersion relation in 2D setting
Sign controlled byconvexity/concavity SP more pronounced
on concave bend
Limitations?
Debye expansion
![Page 20: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/20.jpg)
3D setting: SphereSpherical coords.
From boundary conditions
Dispersion relation for SP:
SP more pronouncedon concave bend
![Page 21: Effects of geometry on surface plasmon-polaritons: an ... · Effects of geometry on surface plasmon-polaritons: an analytical approach. Dionisios Margetis. Department of Mathematics,](https://reader034.vdocuments.site/reader034/viewer/2022042811/5fa4d940bdc8196c913a736c/html5/thumbnails/21.jpg)
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?