Optical properties of asymmetrical hyperbolic media, based on

graphene multilayers

Igor Nefedov and Leonid Melnikov

1. Hyperbolic dispersion of electromagnetic waves in graphene multilayers

2. Properties of asymmetric hyperbolic media

3. Total absorption in asymmetric graphene multilayers

4. Thermal emission from asymmetric hyperbolic metamaterial, made of graphene multilayers

5. Spontaneous emission in hyperbolic media

6. Radiation of a small dipole, placed inside the asymmetric hyperbolic medium

Outline

Hyperbolic media

L.F. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, 1973 (references to E. Arbel, L.B. Felsen, 1963)

infinite power, radiated by a point-like sourceD.R. Smith, D. Schurig, PRL 90 2003 Term indefinite medium, negative refraction, near-field focusing

M. A. Noginov, et al. Optics Letters 35, 1863 (2010)Control of spontaneous emissionI.S. Nefedov, PRB, 82, 155423 (2010) Hyperbolic dispersion in 2D periodic arrays of metallic carbon nanotubes.

I.S. Nefedov, C.R. Simovski, PRB, 84, 195459 (2011) Giant radiation thermal heat transfer through micron gaps.

Illustration of inifinite density of modes in hyperbolic media

Model of graphene conductivity

intraband conductivity (the Kubo formula)

interband conductivity,

G.W. Hanson, JAP, 103, 064302 (2008)

Effective permittivity

0.5 1 1.5 2-12

-10

-8

-6

-4

-2

0

2

4

, m

c=0.8 eV

c=1. eV

c=1.2 eV

d=1,5 nm

Schematic view

z´

x´

Indefinite medium: εt =1; ε’zz =-1+iδ,

Eigenwaves, non-symmetry with respect to the Z-axis

special case:

Isofrequencies. Hyperbolic dispersion

-1.5 -1 -0.5 0 0.5 1 1.50

5

10

15

20

25

30

35

40

kx/k

k z/k

=45

=90

=1.2 m

=1.16 m

Conditions for the perfect absorption

No reflection! Perfect absorption!

S.M. Hashemi, I.S. Nefedov, PRB, 86, 195411 (2012).

Normal components of wave vectors

z - components of wave vectors for waves propagating in opposite directions under the fixed transverse component kx =ksin(θ)

1 1.05 1.1 1.15 1.2-30

-20

-10

0

10

20

30

40

50

60

, m

Re(

k z)/k,

Im(k z)/

k

Re(kz(2))

Re(kz(1))

Im(kz(2))

-50 0 50-20

0

20

40

60

80

R

e(k z)/

k, Im

(k z)/k

Re(kz(2))

Im(kz(2))

Re(kz(1))

θ=45°

Absorption in graphene multilayers

0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

, m

A, |

T|2

A, =10-13

|R|2

|T|2, =10-13

A, =10-14

|T|2, =10-14

A, =10-12

|T|2, =10-12

c=1eV

d=1.5 nmh=80 nm

Absorption (black) and transmission (red) versus wavelength, calculated for different relaxation times τ. Green line shows absorption in the same thickness multilayer with horizontally arranged graphene sheets

Different interlayer distances

1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

, m

A, |

T|2

d=5 nm1.5 nm

3 nm

Absorption (black) and transmission (red) versus wavelength, calculated for different distance between graphene sheets d. Chemical potential μc =0.5 eV.Number of graphene sheets Ng =100.

Absorption, dependence on the incidence angle

-50 0 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A, |

T|2 , |

R|2

A

|R|2

|T|2

h=λ/10

h=λ/10

Thermal emissionErgodic hypethesis

Energy of Planck’s oscillator

z

x

z ’

x ’

E

TM

h d

thermal emission into a solid angle

Thermal emissionHyperbolic isofrequencies

-1.5 -1 -0.5 0 0.5 1 1.50

5

10

15

20

25

30

35

40

kx/k

k z/k

=45

=90

=1.2 m

=1.16 m

z

x

z ’

x ’

E

TM

h d

Far-zone thermal emissionDensity of modes

Thermal emission

-0.5 0 0.50

2

4

6

8

10

Sz/

(,T

)

emission angle

=48

=45

=90

A model of spontaneous emission in HM: two-level atom

- basic states

Equations:

- initial conditions

- ratio of energy stored in the field and in the atoms

a

b

n k

Angular dependence of spontaneous radiation rate

Angle-averaged spontaneous radiation rate in dependence on a and b

Electric dipole radiation, HFSS simulation

dipole in vacuum

dipole in hyperbolic medium

• Graphene multilayers can exhibit properties of hyperbolic media in the near-infrared and visible ranges

• Perfect absorption of TM-polarized waves in a considerably wide wavelength range can be achieved in optically ultra-thin graphene multilayer structures with tilted anisotropy axes

• The perfect absorption is provided by the perfect matching with free space and a very large attenuation constant.

• High-directive thermal emission can be obtained from asymmetric graphene multilayer structures. This effect is caused by enhanced level of spontaneous emission inside hyperbolic media and ability of modes with a very high density to be emitted from ASHM without total internal reflection.

• A small source, placed incide a slab of asymmetric hyperbolic medium, can produce a high-directive radiation in far zone.

Conclusions