I.V. Iorsh, I.V. Shadrivov, P.A. Belov,
and Yu.S. Kivshar
Benasque, 03.03-08.0.3, 2013
Isotropic media:
D  E
Anisotropic media:
D E
  xx 0
   0  yy
 0
0

0

0
 zz 
Disp. equation:
2

kx2  k y 2  kz 2  2
c
Isofrequency
Disp. equation :
2
surface:
kx 2 k y kz 2  2


 2
 xx  yy  zz c
Isofrequency surface:
Hyperbolic medium:
D E
 xx   yy   ||
 zz   
  ||  0
Disp. equation :
k 2
 ||

k2

  2 / c2
Isofrequency surface :
P(t ) e
t /
Transition rate (Fermi Golden Rule):

1


eg
3
| e | d | g |2  (r , eg , e)
For atom in vacuum:
 0 ( ) 
1
0

 d 2
nd 2 3
0 
3 c3
LDOS
Note: Fermi Golden Rule is not an
exact result, but rather a first
approximation solution of the
integro-differential equation
obtained from time-dependant
perturbation theory
E.M. Purcell
(1912-1997)
f  3Q / 4 V
3
2
Purcell worked with RF range and small metallic
20
Cavities: enhancement of the order of 10
1
ds

  k E (k )
(2 ) 2 isofrequency
surface
isofrequency surface unbound 
 =
Narimanov et al, Appl. Phys. B: 100, 215–218 (2010)
Wire medium
Magnetized plasma (for RF)
Graphite (for UV)
J. Sun et al. Appl. Phys. Lett. 98, 101901 (2011)

Within the effective media approximation the layered metal dielectric
nanostructure can be described as a hyperbolic media


  0
0

 Me
0

0
 d   DdD
0    Me Me
d Me  d D

0 ;
  (d  d D )
      Me D Me
 d  d
 p2
  
 (  i )
Me
D
D
|| ,  
 Me   D
Me

•Extremum is observed at the
bulk plasmon frequency .

R  
D
R  Im G(0,0,  )
3

k||3dk||
3
Im G (0, 0,  )  Re 
(rTM ) 
4 0 k03 k02  k||2
k
3

k||3dk||
3 0 k|| dk||
3
 
Re(rTM )  
Im(rTM )
3
2
2
3
2
2
4 0 k0 k0  k||
4 k0 k0 k||  k0
T. Tumkur, G. Zhu, P. Black, Yu. A. Barnakov,
C. E. Bonner, and M. A. Noginov,
APL 99, 151115, (2011)
O. Kidway, S.V. Zhukovsky, J.E. Sipe,
OL, 36,13,(2011)
 RAD  100 s
 NR  1ns

Efficiency is very low
But what if to utilize Purcell effect?
 RAD   RAD / R
From the other hand, THz frequency range lies well below the characteristic bulk
plasmon frequencies in the conventional metal-dielectric multilayers, which
significantly limits the achievable values of the Purcell factors.
1.Hyperbolic isofrequency contours
in metal-dielectric nanostructures
arise due to near field Bloch waves
2.Near field Bloch waves – essentially are
The coupled surface plasmon polaritons
3. Graphene sheet supports surface plasmon modes
which can be coupled if we organise an array
of graphene sheets.
Multilayer graphene structure should behave
As a hyperbolic metamaterial
TE : cos( KD)  cos(k z D) 
2i k0
sin(k z D)
kz
TM : cos( KD)  cos(k z D) 
2i k z
sin(k z D)
 k0
R  108 !

R 
D
 10  m
D  0.005 m
3
  1 ps
Phys. Rev. B 87, 075416 (2013)
4
ck0 D
4
ck0 D
TM

1: R
1: R‖
TM
3


 ck0 d 
3 
c

;

 exp  

2  2‖ Im( ) | 
 2‖ Im( ) | 
2

3 
c



8(k0 d )  2 | Im( ) | 
Largest Purcell factors correspond to:
4
1
ck0 D
2
local approach:  (, k|| )   ( )
works only for: k||

 dk
||
0
kF   / vF
kF
  dk||
0
2 | Im( ) | 
coth(  d / (2vF )) 
vF
Vogel, Welsch, “Quantum optics”:
2
d s c
3

2
Im( ( s ,  ))Gik (r , s ,  )G (r , s , )  ImG(r , r , )
*
jk
 Im G(0,0,  )
To separate the far field and near field:
 RAD   d 3 s
2
c2
  d 3s

Im( ( s ,  ))Gik (0, s ,  )G *jk (0, s ,  ) 
2
c2
Im( ( s ,  ))Gik (0, s ,  )G *jk (0, s ,  )
Perpendicular magnetic field couples
the TE and TM polarized Bloch waves:
A B
(A  B ) 2  2 H2 sin 2 (k z d )
cos( K1,2 d ) 


,
2
4

2i k0
A  cos(k z d ) 
sin( k z d ),
c kz
B  cos(k z d ) 
2i k z
sin(k z d ).
c k 0
Coupling term

Multilayered graphene structures could be
used as a new realization of hyperbolic
metamaterials for THz range to boost the
terahertz transitions in semiconductor
devices.
  xx

 0
 0

0
ˆnloc
 yy 0 
0  
1
 xx    2
kz 2
1  2 2 f (k x , k z )
k0
 yy
0
1
   2
1  2 f (k x , k z )
2 Im( ) / (ck0 d )
k02 d sin(k z d ) / k z
k02
f (k x , k z ) 
 2
2(cos( k z d )  cos( k z d )) k z  k z2
k z2   k02  k x2