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

Ag TiO2 ~ 10 nm

Periodic subwavelength metal-dielectric multilayer structure: uniaxial extremely anisotropic medium

x z

𝜁 = 𝜁∥ 𝜁∥ 𝜁⊥

Anisotropic medium

Ag TiO2 ~ 10 nm

Periodic subwavelength metal-dielectric multilayer structure: uniaxial extremely anisotropic medium

x z

𝜁 = 𝜁∥ 𝜁∥ 𝜁⊥ Bruggeman’s effective medium theory: Maxwell’s equation with plane waves 𝑙∥

2

𝜁⊥ + 𝑙⊥

2

𝜁∥ = 𝑙0

2

Extraordinary or TM wave dispersion

𝜁∥ = 𝑔𝜁𝑛 + (1 − 𝑔)𝜁𝑒 𝜁⊥ = 𝜁𝑛𝜁𝑒 𝜁𝑛(1 − 𝑔) + 𝜁𝑒𝑔

Fill fraction of metal

𝜁∥ 𝜁⊥

Wavelength (µm) Effective permittivity

For example: f = 1/3

Effective permittivity

𝑙∥

2

𝜁⊥ + 𝑙⊥

2

𝜁∥ = 𝜕2 𝑑2 Hyperbolic isofrequency contour !

𝜁∥ 𝜁⊥

𝜁∥. 𝜁⊥ < 0 possible

Wavelength (µm) Effective permittivity

For example: f = 1/3

Effective permittivity

𝑙∥

2

𝜁⊥ + 𝑙⊥

2

𝜁∥ = 𝜕2 𝑑2 Hyperbolic isofrequency contour !

𝜁∥ 𝜁⊥

𝜁∥. 𝜁⊥ < 0 possible

Wavelength (µm) Effective permittivity

For example: f = 1/3

λ = 500 nm Elliptic λ = 700 nm Hyperbolic

Effective permittivity

Types of hyperbolic metamaterials

kx kz kz kx kz ky

𝜁∥ < 0 ; 𝜁⊥ > 0 𝜁∥ > 0 ; 𝜁⊥ < 0 Type I Type II

Applications

Enhanced spontaneous emission: Extremely high Photonic Density Of States (PDOS) High-resolution subwavelength imaging, Hyperlens : no diffraction limit

And many others: extremely confined waveguide, cavities, negative refraction,…

Galfsky, T. et al., Optica, vol. 2, 62-65. (2015) Liu, Z. et al., Science, vol. 315, 1686. (2007)

Limits of effective medium theory

𝑙∥/𝑙0

EMT

𝑙∥/𝑙0

EMT

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

Origin of hyperbolic properties: plasmonic  Field extremely confined  Strong variation of the field on the scale of a single layer

𝑙∥/𝑙0

EMT D = 27 nm Brillouin zone:

𝜌 𝐸

Limits of effective medium theory

Origin of hyperbolic properties: plasmonic  Field extremely confined  Strong variation of the field on the scale of a single layer

𝑙∥/𝑙0

EMT D = 27 nm D = 9 nm Brillouin zone:

𝜌 𝐸

Limits of effective medium theory

Dispersion

𝑙∥/𝑙0 𝑔

0 (Hz)

EMT

𝑙∥/𝑙0

EMT Exact

𝑔

0 (Hz)

Dispersion

𝑙∥/𝑙0

EMT Exact Typical plasmon saturation

𝑔

0 (Hz)

Dispersion

𝑙∥/𝑙0

EMT Exact Interesting from the point of view of isofrequency

𝑔

0 (Hz)

Dispersion

Typical plasmon saturation

Hyperbolic mode

𝑙∥/𝑙0 𝑙∥/𝑙0 𝑔

0 (Hz)

Hyperbolic mode

𝑙∥/𝑙0 𝑙∥/𝑙0 𝑔

0 (Hz)

Hyperbolic mode always exists!

Origin of hyperbolic dispersion

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

Origin of hyperbolic dispersion: coupling of elementary excitation

𝑔

0 (Hz)

𝑙∥ (𝑛−1)

Origin of hyperbolic dispersion: coupling of elementary excitation

Origin of hyperbolic dispersion: coupling of elementary excitation

Holes in multilayer structure

Rectangular nanorod array, still hyperbolic?

x y

Single nanorod mode

𝑙∥/𝑙0 𝑔

0 (Hz)

𝑙∥/𝑙0 𝑔

0 (Hz)

Single nanorod mode

Nanorod array

𝑔

0 (Hz)

𝑙∥/𝑙0

𝑔

0 (Hz)

𝑙∥/𝑙0

Hyperbolic mode always exist too

Nanorod array

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.