### Optical properties of non-symmetrical hyperbolic media, based on

```Optical properties of asymmetrical
hyperbolic media, based on
graphene multilayers
Igor Nefedov and Leonid Melnikov
Outline
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
5. Spontaneous emission in hyperbolic media
6. Radiation of a small dipole, placed inside the
asymmetric hyperbolic medium
Hyperbolic media
Illustration of inifinite density of
modes in 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 source
D.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 emission
I.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)
through micron gaps.
Model of graphene conductivity
intraband conductivity (the Kubo formula)
interband conductivity,
G.W. Hanson, JAP, 103, 064302 (2008)
Effective permittivity
4
d=1,5 nm
2

0
-2
 c=0.8 eV
-4
 c=1. eV
-6
 c=1.2 eV
-8
-10
-12
0.5
1
,  m
1.5
2
Schematic view
z
´
x
´
Eigenwaves, non-symmetry with
respect to the Z-axis
Indefinite medium: εt =1; ε’zz =-1+iδ,
special case:
Isofrequencies. Hyperbolic dispersion
40
35
=45
30
 =1.2  m
 =1.16  m
25
kz/k

20
15
10
5
0
-1.5
=90
-1
-0.5
0
kx/k
0.5
1

1.5
Conditions for the perfect absorption
No reflection! Perfect absorption!
S.M. Hashemi, I.S. Nefedov, PRB, 86, 195411 (2012).
Normal components of wave vectors
θ=45°
80
60
Re(k(2)
)
z
(1)
Re(kz )
Im(k(2)
)
z
Re(kz)/k, Im(kz)/k
40
Re(k(2)
)
z
Im(k(2)
)
z
60
Re(kz)/k, Im(kz)/k
50
30
20
10
0
-10
Re(k(1)
)
z
40
20
0
-20
-30
1
1.05
1.1
,  m
1.15
1.2
-20
-50
0

50
z - components of wave vectors for waves propagating in
opposite directions under the fixed transverse component
kx =ksin(θ)
Absorption in graphene multilayers
1
0.9
0.8
A, |T|
2
0.7
A, =10-13
|R| 2
 c=1eV
|T| 2, =10-13
d=1.5 nm
h=80 nm
A, =10-14
|T| 2, =10-14
0.6
A, =10-12
0.5
|T| 2, =10-12
0.4
0.3
0.2
0.1
0
0.8
1
,  m
1.2
1.4
1.6
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
0.8
3 nm
d=5 nm
A, |T|
2
1.5 nm
0.6
0.4
0.2
0
1.5
2
,  m
2.5
3
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
1
A
0.9
|R| 2
0.8
|T| 2
2
A, |T| , |R|
2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-50
0

50
h=λ/10
h=λ/10
E
Thermal emission
z

Ergodic hypethesis
TM
h

z’
d
x
x’
thermal emission into a solid angle
Energy of Planck’s oscillator
Thermal emission
Hyperbolic isofrequencies
40
35
=45
30
 =1.2  m
 =1.16  m
25
kz/k

20
E
z

TM
h

z’
15
10
d
x
5
=90
x’
0
-1.5
-1
-0.5
0
kx/k
0.5
1

1.5
Far-zone thermal emission
Density of modes
Thermal emission
10
Sz/ ( ,T)
8
=48
6
4
=45

2
=90
0
-0.5
0
emission angle

0.5
A model of spontaneous emission in HM: two-level atom
- basic states
a
n

k
Equations:
b
- initial conditions
- ratio of energy stored in the field and in the atoms
Angle-averaged
in dependence on a and b
Angular dependence
of spontaneous
dipole in vacuum
dipole in hyperbolic
medium
Conclusions
•
•
•
•
•
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.
```