and formalism to describe the dis ispersion Fabio Vaianella*, Bjorn - PowerPoint PPT Presentation

Hyperbolic metamaterials: : basic ic properties and formalism to describe the dis ispersion Fabio Vaianella*, Bjorn Maes 18th Annual Workshop of the IEEE Photonics Benelux Chapter University of Mons Micro- and Nanophotonic Materials Group * Fabio.Vaianella@umons.ac.be

Summary • Hyperbolic properties • Elementary excitations • Nanorods

Anisotropic medium Periodic subwavelength metal-dielectric multilayer structure: uniaxial extremely anisotropic medium Ag ~ 10 nm 𝜁 ∥ 0 0 z 0 𝜁 ∥ 0 𝜁 = TiO2 0 0 𝜁 ⊥ x

Anisotropic medium Periodic subwavelength metal-dielectric multilayer structure: uniaxial extremely anisotropic medium Ag ~ 10 nm 𝜁 ∥ 0 0 z 0 𝜁 ∥ 0 𝜁 = TiO2 0 0 𝜁 ⊥ x Bruggeman’s effective medium theory: Maxwell’s equation with plane waves 𝜁 ∥ = 𝑔𝜁 𝑛 + (1 − 𝑔)𝜁 𝑒 2 2 𝑙 ∥ + 𝑙 ⊥ 2 = 𝑙 0 𝜁 𝑛 𝜁 𝑒 𝜁 ⊥ 𝜁 ∥ 𝜁 ⊥ = 𝜁 𝑛 (1 − 𝑔) + 𝜁 𝑒 𝑔 Extraordinary or TM wave dispersion Fill fraction of metal

Effective permittivity Effective permittivity 𝜁 ⊥ For example: f = 1/3 𝜁 ∥ Wavelength (µm)

Effective permittivity 𝜁 ∥ . 𝜁 ⊥ < 0 possible Effective permittivity 𝜁 ⊥ 2 2 = 𝜕 2 𝑙 ∥ + 𝑙 ⊥ For example: 𝑑 2 𝜁 ⊥ 𝜁 ∥ f = 1/3 𝜁 ∥ Hyperbolic isofrequency contour ! Wavelength (µm)

Effective permittivity 𝜁 ∥ . 𝜁 ⊥ < 0 possible Effective permittivity 𝜁 ⊥ 2 2 = 𝜕 2 𝑙 ∥ + 𝑙 ⊥ For example: 𝑑 2 𝜁 ⊥ 𝜁 ∥ f = 1/3 𝜁 ∥ Hyperbolic isofrequency contour ! Wavelength (µm) λ = 500 nm λ = 700 nm Elliptic Hyperbolic

Types of hyperbolic metamaterials k z k z k x k x k z k y 𝜁 ∥ > 0 ; 𝜁 ⊥ < 0 𝜁 ∥ < 0 ; 𝜁 ⊥ > 0 Type I Type II

Applications Enhanced spontaneous emission: High-resolution subwavelength imaging, Extremely high Photonic Density Of States (PDOS) Hyperlens : no diffraction limit Galfsky, T. et al., Optica, vol. 2, 62-65. (2015) Liu, Z. et al., Science, vol. 315, 1686. (2007) And many others: extremely confined waveguide, cavities, negative refraction ,…

Limits of effective medium theory EMT 𝑙 ∥ /𝑙 0

Limits of effective medium theory EMT 𝑙 ∥ /𝑙 0 Origin of hyperbolic properties: plasmonic  Field extremely confined  Strong variation of the field on the scale of a single layer

Limits of effective medium theory EMT 𝑙 ∥ /𝑙 0 D = 27 nm Brillouin zone: 𝜌 𝐸 Origin of hyperbolic properties: plasmonic  Field extremely confined  Strong variation of the field on the scale of a single layer

Limits of effective medium theory EMT D = 9 nm 𝑙 ∥ /𝑙 0 D = 27 nm Brillouin zone: 𝜌 𝐸 Origin of hyperbolic properties: plasmonic  Field extremely confined  Strong variation of the field on the scale of a single layer

Dispersion EMT 0 (Hz) 𝑔 𝑙 ∥ /𝑙 0

Dispersion EMT 0 (Hz) Exact 𝑔 𝑙 ∥ /𝑙 0

Dispersion EMT 0 (Hz) Exact Typical plasmon saturation 𝑔 𝑙 ∥ /𝑙 0

Dispersion EMT 0 (Hz) Exact Typical plasmon saturation 𝑔 Interesting from the point of view of isofrequency 𝑙 ∥ /𝑙 0

Hyperbolic mode 0 (Hz) 𝑔 𝑙 ∥ /𝑙 0 𝑙 ∥ /𝑙 0

Hyperbolic mode 0 (Hz) Hyperbolic mode always exists! 𝑔 𝑙 ∥ /𝑙 0 𝑙 ∥ /𝑙 0

Origin of hyperbolic dispersion Ag TiO2 Coupling between Coupling between gap plasmon mode? slab plasmon mode? More coupling of SPPs through metal or dielectric?

Origin of hyperbolic dispersion: coupling of elementary excitation 0 (Hz) 𝑔 𝑙 ∥ (𝑛 −1 )

Origin of hyperbolic dispersion: coupling of elementary excitation

Origin of hyperbolic dispersion: coupling of elementary excitation

Holes in multilayer structure y x Rectangular nanorod array, still hyperbolic?

Single nanorod mode 0 (Hz) 𝑔 𝑙 ∥ /𝑙 0

Single nanorod mode 0 (Hz) 𝑔 𝑙 ∥ /𝑙 0

Nanorod array 0 (Hz) 𝑔 𝑙 ∥ /𝑙 0

Nanorod array 0 (Hz) Hyperbolic mode always exist too 𝑔 𝑙 ∥ /𝑙 0

Conclusions  New opportunities for photonics : Extremely high index, high PDOS, …  Hyperbolic properties originate from coupling of elementary excitations  Extremely confined  Lot of losses. Still much work to do

Thank you for your attention This work is financially supported by the F.R.I.A.-F.N.R.S.