O.N. Kozina Saratov Branch

1. SFM 2017, 26–29September, Saratov, Russia Principal characteristics of the electromagnetic waves propagation in the asymmetrical hyperbolic medium O.N. Kozina Saratov Branch Kotel’nikov Institute of Radio-Engineering and Electronics of Russian Academy of Science Saratov, Russia kozinaolga@yandex.ru L.A. Melnikov Department of Instrumentation Engineering Yuri Gagarin State Technical University of Saratov Saratov, Russia I.S. Nefedov School of Electrical Engineering Aalto University, Aalto, Finland

2. Motivation Challenge task: Generation and controlling electromagnetic waves of the optical and THz ranges by the nanoscale devices. Challenge way: New composite materials with designated properties (metamaterials) suitable for solving of aforementioned problems have a remarkable progress. Metamaterials are artificial objects created by subwavelength structuring, are useful for engineering electromagnetic space and controlling light propagation. Such materials exhibit many unusual properties that are rarely or never observed in nature. They can be employed to realize useful functionalities in emerging metadevices based on light. One of the promising class of artificial structures in optical and THz frequencies and most unusual class is hyperbolic metamaterials (HMM). HMM display hyperbolic (or indefinite) dispersion, which originates from one of the principal components of their electric or magnetic effective tensor having the opposite sign to the other two principal components. Such anisotropic structured materials exhibit distinctive properties, including strong enhancement of spontaneous emission, diverging density of states, negative refraction and enhanced superlensing effects.

3. L.F. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, 1973 (references to E. Arbel, L.B. Felsen, 1963) Isofrequencies of Hyperbolic Medium (HM) Most popular realization of hyperbolic media are metallic nanowire arrays embedded in the dielectric host matrix and sub-wavelength metal-dielectric alternating multilayer films. εxx = εyy<0 εzz>0 εxx = εyy > 0 εzz < 0 C.R. Simovski, P.A. Belov, A.V. Atrashchenko, Y.S. Kivshar, “Wire Metamaterials: Physics and Applications,” Adv. Mater., vol. 24, pp. 4229–4248, 2012. L.A. Melnikov, O.N. Kozina, A.S.Zotkina, I.S. Nefedov “Optical characteristics of the metal-wire dielectric periodic structure: hyperbolic eigenwaves” SPIE 9031, 903117. 2014. I.V. Iorsh, I.S. Mukhin, I.V. Shadrivov, P.A. Belov, Y.S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, p. 075416, 2013. O.N. Kozina, L.A. Melnikov, I.S. Nefedov, “Dispersion characteristics of hyperbolic graphene-semiconductors multilayered structure” SPIE Vol. 9448, Article CID 9448-11 N 9448-108, 2015.

4. Fields in HM

5. Design of the AHMM Asymmetric hyperbolic metamaterial (AHMM) is a hyperbolic metamaterial with an optical axis, tilted with respect to a medium interface. Asymmetry appears as a difference in properties of waves, propagating upward and downward with respect to the metamaterial interface under a fixed transverse wave vector component. The most important feature of AHMM is the possibility to excite a very slow wave in AHMM by a plane wave, incoming from free space, while a minimal reflection may be achieved. By other words, high density of photon states, excited in AHMM, can be perfectly coupled with photons in free space.

6. Calculation model: Berreman 4x4 matrix method The algorithm for solving of the Maxwell equation based on the Berreman 4x4 matrix. This method is convenient for the investigation of the propagation of polarized light in anisotropic media. This method allows analyzing of complex optical systems where the effects of biaxiality, magnetic anisotropy, and optical activity play an important role. This is accurate and effective method which take into account direct and backward waves inside structure and correct boundary conditions. • exp(ikr-it) k=(kx,ky,kz) K =ω/c = 2π/λ The elements ij of Berreman 4x4 matrix  generally determined by main components of the dielectric tensor {,, ||}, Euler angles , , , which describes the orientation of optical axis, x-component of the incidence wave vector kx=K sin . For active medium (with losses or gain) and|| are complex. Berreman D. W., “Optics in stratified and anisotropic media: 4 x 4-matrix formulation,” Journal of the Optical Society of America, 62(4), 1157-1160 (1972). Palto S. P., “An Algorithm for Solving the Optical Problem for Stratified Anisotropic Media,” Journal of Experimental & Theoretical Physics 92(4), 552-562 (2001). 6

7. Calculation model: Berreman 4x4 matrix method Electromagnetic fields of the transmitted, incident and refracted waves on the slab with the thickness h are related by the equation: P(h) is the propagation matrix for layer with thickness h The eigenvalues of the matrix : Transmission and reflection coefficients 7

8. periodically arranged layers or wires in active host medium, titled in relation to outer boundary, with gain or losses General model of the hyperbolic medium under investigation We use the model of the hyperbolic medium which consists of periodically arranged layers in active host media, titled relatively to outer boundary. k=(kx, ky, kz)  - incidence angle h – thickness of host medium in the z direction ,  ,ψ - Euler's angles, ψ=0 N - nodes line Structure is infinite in the x and y-direction. General model correspond to AHMM (diagonal extremely anisotropic permittivity tensor ) with arbitrary value of permittivity.

9. Real AHMM: values of parameters are specified. Asymmetrical hyperbolic graphene-semiconductors multilayered structure. Graphene multilayer slab formed of periodically arranged graphene sheets with period d, embedded into a SiC host matrix. Conductivity of graphene The Kubo model of conductivity () for graphene sheet was used [Hanson G. W., J. Appl. Phys. 103 064302 (2008)] Permittivity of SiC H. Mutschke, A.C. Andersen, D. Cl.ement, Th. Henning, G. Peiter, “Infrared properties of SiC particles”, Astron. Astrophys. 345, 187–202 (1999).

10. Results. Based characteristics of asymmetrical hyperbolic graphene-semiconductors multilayered structure (real AHMM). Real (solid lines) and imaginary (dashed lines) part of transverse permittivity versus frequency, calculated at different values of structure’s period, EF= 28 meV,=10-12c, T=300ºK. Real part of the dynamic conductivity, calculated for different values of Fermi energy EF. Real part of the dynamic graphene conductivity changes sign at terahertz frequencies and graphene becomes as the active material for different value of quasi-Fermi energy. The real part of the conductivity is negative at frequencies above 2.7 THz and reaches a minimum of Re(σgr) = −0.16 mS at f≈ 6.5 THz, which corresponds to a maximum of amplification. Structure posses hyperbolic properties when Re()<0. At the same time, Im() characterizes the gain properties of HMM: amplification exist when Im()<0. It is easy to see existence of the frequency regions where given structures possess hyperbolic properties and amplification properties simultaneously for every observed period of structure. Frequency region suitable for THz amplification in hyperbolic material is increased when period of structure decreases.

11. Results. Based characteristics of the general model of AHMM Comparison of results given by the (rigorous) model which takes into account periodicity (Floquet–Bloch, solid lines) and the (approximate) effective medium theory (Maxwell–Garnett, dashed lines). Dispersion of a uniaxial medium (effective medium theory ) Dispersion equation of periodical structure (Floquet–Bloch theorem) c=0.2eV k0d=0.0026 h=2 at the frequency f =63 THz

12. Results. Dispersion characteristics of asymmetrical hyperbolic graphene-semiconductors multilayered structure (real AHMM). d=4nm W=400nm c=0.2 eV =45º =78º T=300ºK Transmission (red), reflection (blue) and absorption(black) versus the wavelength for the TM waves calculated foe different rralaxation time. Solid line:  =10-12c Dashed line:  =10-13c Transmission (red), reflection (blue) and absorption(black) versus the wavelength for the TM waves. =10-13c

13. ||= - 7 + i 0.0001 (losses) = 6 - i 0.0015 (gain) K h=100 Eigenvalues i,k of the matrix  vs incidence angle  for different orientation of the optical axis: =0 and =/2 3,4 - extraordinary waves at =0 1,2 -ordinary waves =0, =/2  =/5  = 0 3,4 - extraordinary waves at  = /2 Rei,k - solid curves Im i,k - dashed curves

14. = 57/128 ||= - 7+i 0.0001 (losses) = 6-i 0.0015 (gain) K h=100 The transmission (blue curves) and reflection (red curves) coefficients vs incidence angle   = 0  = /2  =/5  = 0

15. Conclusion • Effective medium homogenization model is valid for a broad range of values normalized wave number and adopted for analyzing the periodic graphene-SiC lattice. • Graphene multilayers exhibit properties of hyperbolic medium from the visible to the infrared ranges under appropriate choice of the chemical potential and period of the multilayer lattice. • The convenient method of calculation of light propagation in HMM slabs based on Berreman 4x4 matrix was used to investigate numerically transmission, reflection, and field distribution in the HMM slab under arbitrary angles of incidence of plane wave on the boundary surface of the slab. • This method was used for calculation of the threshold condition for the medium with embedded active atoms or ions. This condition demands the zero value of Berreman matrix determinant and gives the angles of radiation depending on the direction of optical axis of HMM. • Total absorption can be obtained in optically ultra-thin slabs of asymmetric graphene-SiC metamaterials. • The chemical potential of graphene must be nonzero and the larger it is selected, the shorter the absorption wavelengths obtained. • The absorption band can be shifted to shorter wavelengths by decreasing the graphene interlayer period. • Transmission and reflection from the slab of materials considered together with intrinsic upward and downward waves are presented in dependence on wavelength, showing the wavelength regions suitable for light control. • Fabrication of the proposed graphene asymmetric hyperbolic metamaterial is realistic in the modern level of technologies.