Optical characterisation of metamaterials

   
 5
 
1. Introduction 
 
The word metamaterial can be split into the word meta-, that takes its origin from the 
greek term µετά which means  beyond , and the word material. These kinds of 
materials can be, theoretically, synthesized embedding some composites with specific 
shapes into a host medium forming a particular pattern. This means we are going to 
talk about something  beyond  our usual concept of material that takes its optical 
properties from the atoms it is naturally made of. We will rather consider a kind of 
medium whose primitive characteristics (as light-matter interaction) strongly depend 
on man made sub-unities (containing many atoms) which constitute its structure. The 
features of the structure involved in the fabrication, by a physical point of view, are so 
crucial that their combination endows the whole structure of new macroscopic optical 
effects which don t exist in nature. 
The growing interest in metamaterials lies in the possibility to tailor a medium optical 
response in order to obtain a variety of applications. Artificial materials with peculiar 
electromagnetic properties are not a new challenge. In 1898 Sir Jagadish Chandra 
Bose (1858   1937) first experimented microwave responses from twisted structures 
geometries today known as chiral elements1 [1]. In 1914 Lindman (1874   1952) 
created artificial chiral elements embedding randomly oriented small helices coiled 
from copper wires in a host medium [2]. Later in 1948 Kock actually was able to 
tailor the refractive index of a microwave lens, periodically arranging conducting 
elements [3].  
The capability in manipulating the interaction between matter and electromagnetic 
fields can be pushed further leading to extraordinary consequences such as negative 
refraction that, in the middle of the last century, was considered no more than an 
abstract speculation.  In 1944 Mandelshtam [4] noticed that, considering Snell s law2 
between two media, for a given incidence angle θ1 at the interface, mathematically 
two solutions are possible for the direction of the refracted ray: not only the 
                                                 
1
 Chiral materials are a subclass of bi-anisotropic media showing an intrinsic asymmetry with respect to 
the left and right. The effect of chirality on electromagnetic field propagation is a rotation of the plane 
of linearly polarized waves. 
2
 Snell Law rules light rays directions propagating through an interface between two dielectric media. 
Let n1, n2 be respectively  medium 1 and medium 2 refractive index. If  θ1 is the angle of incidence of 
the ray in medium 1, the angle of refraction θ2 in medium 2 can be obtained from the relation: 
n1sinθ1=n2sinθ2  .  
   
 6
conventional solution θ2 but also an unusual one given by π - θ2 . Let s report what he 
wrote about the latter: 
 
 Demanding [as before] that the energy in the second medium propagates from the 
boundary, we arrive to the conclusion that the phase must propagate towards the 
boundary and, consequently, the propagation direction of the refracted wave will 
make with the normal the angle π   θ 2. This derivation appears to be unusual, but 
of course there is no wonder because the phase velocity still tells us nothing about 
the direction of the energy transfer.  
 
Unfortunately he didn t study this phenomenon deeper; therefore further implications 
didn t turn out from his work. We have to wait until V.G. Veselago s seminal work in 
1967 to find the first systematic study about negative refraction and its outcomes [5]. 
Starting from the consideration that the index of refraction n of a substance can be 
calculated from: 
 
εµ=2n  with    ε : electrical permittivity µ : magnetic permeability 
 
he observed that changing both signs of the real part of ε and µ doesn t modify the 
relation above3. Even if, by that time, substances with simultaneous ε<0 and µ<0 
didn t exist, he carried on the argument showing striking consequences. Starting from 
Maxwell equations he proved that a medium with negative permittivity and 
permeability could have been considered as a material with a negative refraction 
index exhibiting a reversed Snell s law.  In particular, exploiting this phenomenon, he 
showed how a slab lens could be used to focus the radiation from a point source 
without aberration. Moreover, in such a medium, electric field, magnetic field and 
wave direction of propagation no more created a right handed triplet. This is why 
terms like Left Handed Metamaterials (LHM) and Negative Index Metamaterials 
(NIM) began to be used.  Although Veselago pointed out other interesting effects for 
NIM such as reversed Doppler effect and obtuse angle Cerenkov radiation, the lack of 
any experimental demonstration made his paper only an academic issue. This was the 
situation until 2000, when J.B Pendry published an article explaining how a 
                                                 
3
 From now on, when we speak about changing the signs of the constitutive parameters, it is implied we 
are referring to the real part of ε and µ. 
   
 7
Veselago s slab lens would be able to carry near-field information, restoring both the 
propagating and evanescent field in the image plane, resulting in a lens with 
theoretically infinite resolution [6]. Pendry s letter and new advanced fabrication 
techniques developed during decades, brought Veselago s work in the limelight and 
made metamaterials a cutting-edge field of research as proved by the exponential 
growth in the number of publications in this area.  
Indeed, during the last nine years, nanofabrication techniques made scientists capable 
to produce structures which, theoretically, can exhibit simultaneously negative 
constitutive parameters. Therefore a lot of efforts have to be put in the 
characterisation process: the possibility to experimentally analyse the behaviour of a 
metamaterial is very important both to confirm previously predicted phenomena and 
to provide new hints for further studies. That s why the realisation of a reliable 
experimental set-up allowing to obtain trustable results is essential for metamaterials 
investigation. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 8
2. Theoretical Background 
 
Electromagnetic metamaterials are artificial homogeneous structures showing 
characteristics not available in nature. The word homogeneous means that the average 
feature size l of these media is small enough in order not to be resolved by the 
electromagnetic wave they are interacting with. Let λ be the wavelength of the 
electromagnetic field involved, then we must have l<<λ. Usually the minimum 
requirement for a medium to be considered homogeneous is set as: l=λ/4. This 
condition is very important since it allows us to leave out scattering and diffraction, 
regarding at refraction as the dominant phenomenon. In this way we can analyze such 
a material as an electromagnetically uniform structure which can be described with its 
constitutive parameters: electric permittivity ε  and magnetic permeability µ . As 
mentioned before, the signs of ε   and µ  play a fundamental role in the medium 
behaviour. 
Beginning from Maxwell equations we can understand where a negative refractive 
index comes from. Maxwell s differential laws in a medium can be written as:  
eD ρ=⋅∇
r
  Gauss  Law (electric)    
t
B
E
∂
∂
−=×∇
r
r
  Faraday s Law       
 
0=⋅∇ B
r
  Gauss  Law (magnetic)      
s
J
t
DH
r
r
r
+
∂
∂
=×∇   Ampere s Law   
 
 
With: 
E
r
: electric field intensity vector [V/m]; H
r
: magnetic field intensity vector [A/m] 
eρ : free charge density [C/m3];     sJ
r
: free current density [A/m2] 
D
r
: electric displacement field vector [C/m2] 
B
r
: magnetic induction vector [T]  
 
Vectors D
r
 and B
r
are related with E
r
 and H
r
 through: 
 
 
 
(2.1) 
   
 9
 
 
EEEPED
re
rrrrrr
εεεχεε ==+=+= 000 )1(        
EP e
rr
χε 0=
4
 , er χε += 1  and rεεε 0=                         
 
( ) HHHMHB
rm
rrrrrr
µµµχµµ ==+=+= 000 )1(  
 HM m
rr
χ= , mr χµ += 1  and rµµµ 0=  
 
With: 
P
r
: polarization vector  M
r
: magnetization vector 
0ε : vacuum electric permittivity                    :0µ vacuum magnetic permeability 
r
ε : relative electric permittivity                     
r
µ : relative magnetic permeability 
ε : electric permittivity  µ : magnetic permeability 
eχ : electric susceptibility  mχ : magnetic susceptibility 
 
Applying curl operator to the Faraday s and AmpŁre s laws we can obtain: 
 
022 =+∇ EE
rr
εµω      
022 =+∇ HH
rr
εµω  
 
 
If we now consider the electromagnetic field with harmonic behaviour both in space 
and time, we can eventually write: 
 
HEk
rrr
ωµ=×  where:         ωεµ≡= kk
r
 is the wave vector.  
EHk
rrr
ωε−=×  
 
                                                 
4
 This is true in the hypothesis of a polarization linear response respect to the applied electric field. If 
the latter reaches too high values we should expand the response to higher orders: 
...
3)3(2)2()1( +++∝ EEEP eee χχχ    
Anyway we won t discuss non-linear metamaterials in this treatise. 
(2.2) 
(2.3) 
(2.4) 
   
 10
Consider now an electromagnetic wave propagating in the zr  direction. According to 
the above relations, for a  standard  material ( i.e. with positive ε  and µ ), E
r
, H
r
and 
k
r
 form a right-handed triplet: 
E
H
k
z
 
Fig. 2-1: right-handed triplet for an electromagnetic wave in a standard medium 
 
Let s switch now both the signs of ε  and µ . According to the previous relations, now 
we are dealing with a left-handed triplet as shown in the next figure: 
E
H
k
z
 
Fig. 2-2: left-handed triplet for an electromagnetic wave in metamaterials 
 
In this way all the relations above are satisfied and wave vector has an opposite sign 
respect to the direction of propagation. Actually if we consider all the signs  
combinations between ε  and µ , we come up with four possibilities: 
 
ε>0
µ>0
Propagating waves
Dieletric Right-handed 
materials
Evanescent waves
Plasmas (metals below 
plasma frequency)
ε 0
µ
<
>0
Propagating waves
Left-handed 
Electromagnetic 
Metamaterials
ε 0
µ
<
<0
Evanescent waves
Gyrotropic materials in 
certain frequency ranges
ε 0
µ
>
<0
µ
ε
 
Fig. 2-3: media grouping with respect to ε  and µ  signs 
   
 11
 
Obviously we are mostly interested at the third square where the conditions to obtain 
metamaterials are shown. Nevertheless, later we will talk about plasmas too, since 
plasmonic resonances play a fundamental role in metamaterials as we will see. 
Gyrotropic materials are anisotropic substances showing negative magnetic 
permeability only under particular conditions; we won t discuss them deeper in this 
thesis.  
A negative ε  is nothing new since it is exhibited by all metals below their plasma 
frequency. Nevertheless, because of the lack of magnetic charges, a negative 
permeability is more difficult to reach.  
A simple way to describe a material in interaction with an electromagnetic field, is to 
express its electrons  motion through a forced, damped harmonic oscillator [7]. We 
can describe the system considering the displacement of an electric charge due to  the 
force impressed by the incoming electric field: 
Eqrm
dt
rdm
dt
rdm eee
rr
rr
−=+Γ+ 202
2
ω  where tjeEE ω0
rr
= 5 
This is known as the Lorentz model and it takes in account all the terms of a harmonic 
oscillator. 
Regarding at the left side, the first term expresses the acceleration of the charges 
induced by the electric field, the second describes the damping factor due to viscous 
friction where Γ  is the damping coefficient and the third term accounts for the 
restoring forces with a characteristic frequency 0ω , typical of a harmonic oscillator. 
At the right side we have the driving term i.e. the force acting on each electron with 
charge 0<− q . 
Since the electric field has an oscillatory dependence tje ω∝  we can assume the 
displacement rr  will follow the same behaviour:  
tj
err
ω
0
rr
=  
 We can now easily solve the differential equation switching to the frequency domain, 
obtaining: 
                                                 
5
 Within this thesis we will interchange expressions like )( kztjeF −∝ ω
r
 and  
)( tkzi
eF ω−∝
r
 to describe 
the same harmonic behaviour.  Thus, we are always allowed to set ij −= without changing the 
meaning of a formula. 
(2.5) 
(2.6) 
   
 12
( )20
2
0
0 ωωω +Γ+−
−
=
jm
Eq
r
e
r
r
               (2.7) 
 
It s now straightforward to find the electric susceptibility and thus the permittivity. 
Considering  (2.2), we can find6: 
 
2
0
2
2
,
1
ωωω
ω
ε
+Γ+−
+=
j
p
Lorentzr , 
0
2
ε
ω
e
e
p m
qN
=  is the plasma frequency. 
 
When the acceleration term is small compared to the other terms we can neglect it and 
obtain the so called Debye model: 
2
0
2
,
1
ωω
ω
ε
+Γ
+=
L
p
Debyer
j
 
 
If it is the restoring term to be much smaller than the others, we are in the so called 
Drude model: 
Γ+−
+=
ωω
ω
ε
j
p
Druder 2
2
,
1  
 
Since in the Drude model we neglected the restoring forces it is actually the proper 
one if we want to describe a metal. In this case the damping factor is:  
l
vF=Γ  
 
Where l  is the mean free path of an electron in a metal and Fv  is its Fermi velocity. 
This term represents the frequency of the collisions between an electron and the ions 
of the lattice within which it s travelling7. This is why this factor is well known as 
collision frequency.  
                                                 
6
 In fact we can set: ( )EErqNP ee
rrrr
100 −==−= εεχε  
7
 If we think about the physics of the phenomenon it s pretty clear that this term has to be linked with 
the conductivity of the metal σ . We expect that the more the collision frequency is high, the more the 
conductivity goes down because of the interruption in one electron path. In fact we have: 
Γ
=
e
e
m
qN 2
σ . 
 
(2.8) 
(2.9) 
(2.10) 
(2.11) 
   
 13
The use of the Drude model can lead us to a negative permittivity if 22 Γ−< pωω . 
Splitting the permittivity into its real and imaginary part we have:  
 
                                         ( )EErqNP ee
rrrr
100 −==−= εεχε                                  (2.12) 
( )22
2
22
2
,
1
Γ+
Γ
−
Γ+
−=
ωω
ω
ω
ω
ε
pp
Druder j                                  (2.13) 
 
Since it is the imaginary part that accounts for losses, we notice that it increases if 
frequency decreases. To have an idea of the values involved let s have a look at the 
plot below: 
 
 
Fig. 2-4: real and imaginary part of gold permittivity  
 
The diagram shows, as example, the permittivity of bulk gold obtained numerically 
applying the Drude model. It s pretty clear that real and imaginary part not only have 
different trends but also very different absolute values. According to the plot, above 
the plasma wavelength, the real part becomes negative as expected and goes down. To 
reach a negative permittivity, we can consider, for instance, a set of thin parallel