Effective Medium Modelling of Plasmonic MetaMaterials
Gennady Shvets, The University of Texas at Austin
Metamorphose School’09: “Current research and recent developments on opticalMetamorphose School 09: Current research and recent developments on optical metamaterials”, Crete, Greece, June 13, 2009
Outline of the talk
• Introduction to optical metamaterials
• Attempts to answer puzzling questions that I would have had if I was entering the field:1. Why is the optical magnetism difficult to achieve (a.k.a why are
the microwave experts looking down on us?)
2 A l i ff t d b d f ki ti l2. Are plasmonic effects good or bad for making optical metamaterials?
3. How does one analyze periodic plasmonic structures?3. How does one analyze periodic plasmonic structures?
4. Is it possible to to homogenize plasmonic nanostructures?
5 Why is everybody talking about spatial dispersion and how5. Why is everybody talking about spatial dispersion and how important is it?
• Perspectives on the optical metamaterials field: where the• Perspectives on the optical metamaterials field: where the promising applications are?
Who suggested particle clusters for ti ?magnetic response?
Shvets & Urzhumov, JOPA 7, S23 (2005);
PRL 93, 243902 (2004)
What is a MetaMaterial?
O i i t f G k d " ft /b d"Originates from a Greek word : "after/beyond"
Example: metaphysics ("beyond nature"): a branch of philosophy concerned with giving a general and fundamental account of the way the world is. (Wilkepidia)
•Metamaterials are artificially engineered materials i ti ( h i l ti l l t i l)possessing properties (e.g., mechanical, optical, electrical)
that are not encountered in naturally occurring materials.
•The emphasis of this talk in on unusual electromagnetic properties such as dielectric permittivity , magnetic permeability and refractive index npermeability , and refractive index n.
What is a MetaMaterial?
Material properties are determined by theMaterial properties are determined by the properties of the sub-units plus their spatial distribution. p
For a << effective medium theory.
F h i ffFor a ~ photonic effects.
What about meso-scale materials: bigger than atom butbigger than atom but smaller than the wavelength??g
Microscopes: the Engines of Discovery
V L h k (1676)Van Leeuwenhooke (1676): discovered bacteria, blood cells
)sin(222.1
nx
What if the imaged specimen is aspecimen is a sub-wavelengthgrating?
Difficult to resolve sub- features
L >> d/2
Small features (or large wavenumbers) of the object are lost because of the exponential j pevanescence of short-wavelength waves
Evanescence of large-k waves
•Fields in the object plane:
yikekAdkyx )( ),0(Hz
Spectral width k feature size y = k-1
•Optical system propagates A(k) through glass, vacuum
xkkiekA22
0)(
For smooth features with k < k0 = 0/c
cuum
xkkekA20
2
)(
For sharp features with k > k0 = 0/c
In v
ac
•How to overcome decay? Go near-field!
(a) get very close (L < d/2) (b) lifdecay? Go near field! (b) amplify evanescent waves
Negative Index Materials to the Rescue: = -1 = -1 n = -1 = -1, = -1 n = -1
L/2 L L/2
•Super-lens is a near-field concept amplifies evanescent waves to restore image
•Near-field superlens possible with just < 0 (Pendry, 2000)
Negative Index Materials for -waves
2/12 2 c 2F2 log(D/r)
2 ,1
Dc
pp 0 1 22
M
MF
Smith et.al., PRL'2000
•Basic Elements of a NIM: (a) Split ring resonator: just a well designed inductor ( ) p g j gresonating at << c/D gives < 0(b) Metal wires (continuous or cut): r << D to ensure that < 0 for << c/D
Sub- NIM Using Microwave Tricks•In microwave range: use “perfectly” conducting components to simulate < 0 and < 0 Smith et al (2000)components to simulate < 0 and < 0, Smith et.al., (2000)
Metal poles: = 1 – 2/2 < 0Metal poles: 1 p / < 0
Split-ring resonators, Pendry’99: “geometric” resonance at “geometric” resonance at M
0 1 22
2
MF
Mi t i k bl i i t i ti ( << /L)
22 M
•Microwave tricks enable miniaturization (p, M << c/L): (a) very thin wires still good conductors(b) split ring design high capacitance and inductance(b) split ring design high capacitance and inductance
Why is it difficult to make a ymagnetically-active metamaterial
?in the optical domain?
Making a sub- NIM: from -waves to plasmonics by downsizing!to plasmonics by downsizing!
Microwave SRR, =1.57cm Plasmonic SRR, =3.44m
E
k
W=0.8mm, Gap = 80m (filled with d =4 dielectric)
Downsize by about x5,000 W=160nm, Gap = 16nm (filled with d 4 dielectric)
Magnetic cutoff = 0 at = 20*Period how??
pMagnetic cutoff = 0 at = 20*Period (i) how? Plasmonics effects! (ii) Can we
Narrow gap + high- dielectric high capacitance
Plasmonics effects! (ii) Can we move to shorter wavelength by further downsizing? No!
Disappearance of magnetism for optical frequencies ( < 3m)optical frequencies ( < 3m)
Energy content of microwave structure:Electric energy
,22 BdVUEdVU
UUUUU
BE
BEBEtot
gy
8 ,
8dVUdVU BE
Magnetic energy In EE language:
22
21 ,
2LIU
CQU BE
12
LC12
Compare with the plasmonic structure:
Disappearance of magnetism for optical frequencies ( < 3m)optical frequencies ( < 3m)
Energy content of Electron kinetic energy2
1 pEnergy content of
plasmonic structure:
UUUU kinBEtot
Electron kinetic energy
ergy
21
p
,8
,8
22 BdVUEdVU BE
kinBEtot
Magnetic energylect
ric
ene
8
2
2
2 EdVU pkin
E
Energy balance at = 3.44m:
U 0 5U U 0 32U
Plasmonic parameter at= 3.44m (silver): 7.1 kin
p UUT
UE = 0.5Utot, U kin = 0.32U tot, UB = 0.18 U tot
( )m
p U
Electrostatic resonance at =-300 (or = 3 m for Ag): Tp = ∞
Y.A.Urzhumov and G.Shvets, Solid State Comm. 146, 208 (2008).
What happens when you downsize an ideal (lossless) plasmonic structure?ideal (lossless) plasmonic structure?
•Regime I: Very large unit cell (appropriate for -wave frequencies) metals behave as PEC's, miniaturization (if any) accomplished by
C/
increasing capacitance
•Regime II: Intermediate unit cell shrinking th ll th ti l th ( f /the cell causes the operating wavelength (e.g., for magnetic resonance) to shrink proportionally: = C*Period
C/
•Regime III: Small unit cell approaches the electrostatic resonance wavelength res unit
C/res
cell shrinks faster than the wavelength: > C*Period
•Regime IV: No magnetic activity for < res. Sorry…
What if > 3 m is not enough: simplify the structure!simplify the structure!
res = -330 res =3 m res = -82 res =1.5 m (or res=2.25 m for glass spacer)2.25 m for glass spacer)
E
k
silver strips
k
Strip Pair-One Film (SPOF) Lomakin, Fainman, U h G S O t Strip Pair One Film (SPOF)
res = -8.8 res = 0.5 m (or res = 0.7 m with glass spacer)
ver
film
Urzhumov, G.S., Opt. Exp. 14, 11164 (2006); Urzhumov & G.S., Solid State Comm 146 si
lvSolid State Comm. 146,208 (2008).
Effect of plasmonic losses on developing sub wavelength NIMssub-wavelength NIMs
•Why not operate very close to the electrostic plasmonic resonance (extreme minimization)? Losses destroy the NIM band!
•Total energy increases as the plasmonic number Tpi th l t t ti
Bptot UTU 12increases near the electrostatic resonance:
•Group velocity decreases as Utot increases:mg
UU
cv
p y tottotUc
•NIM propagation band is destroyed if the 1 P•NIM propagation band is destroyed if the
transit time across a single cell is longer than the decay time -1 :
1
1
pTcP
Three-dimensional example: is my fishnet better than yours?my fishnet better than yours?
Electrostatic resonances teach us about relative advantages/disadvantages of various structures!
Novel Plasmonic Double-Fishnet Structures
•Traditional double fishnet has never become a sub-negative index metamaterial
•Still room for innovation: look for
lt ti !
Planar fishnets are easy to •Too “simple” to be sub-
alternatives!
•Planar fishnets are easy to fabricate & align (e.g. nanoimprint lithography)All d
Too simple to be sub without plasmonics: P > /2n Kafesaki et.al., PRB’07: P = 1 cm = 2cm n = 2•All to-date structures are not
sub-wavelength: P > /2n Brueck et.al., PRL’05:
P = 1 cm, = 2cm, n = 2•Novel plasmonic design sub-wavelength!P f t i d,
P = 0.8 m, = 2 m, n=1.5 •Perfect impedance matching: zero reflection!
Modified vs regular fishnet Minimum wavelength for magnetic activity from ES simulations
Modified double-fishnet: "Regular" double-fishnet:min = 1.2 m (m/d = -62) at = 1.5 m can still be sub-(close to plasmonic resonance!)
min = 0.73 m (m/d = -8.3)
To be be sub-, must operate at < 900nm, or Period < 300nm
Impedance-matched NIM at = 1550nmcontinuous silvercontinuous silver wires control eff
20 nm
impedance matching
80 nm
20 nm50 nm “cut”
silver wires
320 nm80 nm
control eff
320 nm250 nm
•This structure is sub-l h P i d /5wavelength: Period = /5
•It operates in the desirable = = -1 regime re = re = -1
•Applications: detection (SEIRA, et.)
Effect of plasmonic losses
How to chartacterize periodic plasmonic structures?plasmonic structures?
•Standard approach: compute frequency vs wavenumber k
•This is not so easy to accomplish: premittivity of metals itselfThis is not so easy to accomplish: premittivity of metals itself depends on the frequency
•There is a way to proceed: compute k vs !•There is a way to proceed: compute k vs !
•This is even better: we can find complex k’s which give us the t f ti l d i id th b d f i di t t !rate of spatial decay inside the band gap for periodic structures!
Calculating k vs omegag g
Solve using the weak formulation of the PDE:
Calculating k vs omegag g
Assume Bloch-periodic solutions:
Substitute into:
Obtain:
Now discretize and solve for k as an eigenvalueg
Array of nanorods
SPOF structure: propagation bands
Is it possible to homogenize an optical metamaterial?optical metamaterial?
•Homogenization involves introducing a set of effective parameters (epsilon, mu, bi-anisotropy coefficients) by postulating some type of field averaging procedure and/or fit to Fresnel coefficients (transmission/reflection)
•Preferably, these parameters should have a clear and intuitive physical meaning (e.g., conform with our understanding of
it i d t t )capacitance, inductance, etc.)
•The situation can be further complicated by strong non-local dependence of D and B on E and H (spatial dispersion)
Example: effective permittivityp p y
extE QQAQAQC
)()( Eeff
+Q-QL
QL
QC
)()(effeff
i i
ΔL
ffperiodic
i di
-1V +1V
(a) Impose constant voltage drop sidewalls of a single unit cell: nothing more to do for dielectric non dispersive MMs
periodic
nothing more to do for dielectric non-dispersive MMs
(b) For plasmonic MMs: Scan frequency and recover Q() t ( )compute eff()
(c) Peaks of E-field (or Q) correspond to ES resonances
Example: effective permittivity tensorp p y
A l t l fi ld
ay
Apply external field:
y
Solve Poisson’s equation:
ax
Set up boundary conditions:p y
Extracting the permittivity tensorg p y
A l t l fi ld
ay
Apply external field:
y
ax
What if you have a dispersive material with () ?with () ?
1 S th f d
ay
1. Scan the frequency and repeat the capacitance measurementy measurement
2. Find a better way: Generalized EigenvalueaxGeneralized Eigenvalue Differential Equation
Generalized Eigenvalue Equation
periodic
xE
10 ,0
0 xE00
Define
01
s 1)(
periodic
dm
s
/)(1
)(
B & S d’92 S k l ’01Bergman & Stroud’92; Stockman et.al.’01; Shvets & Urzhumov, PRL’04 & JOPA’05
Generalized Eigenvalue Equationperiodic
0
Define an eigenvalue equation:
01
periodic
)()()( 2
And use the eigenfunctions to solve:
)()()( 012
1 xsx
Examples of dipole-active electrostatic resonancesresonances
N
iFR2
1)( s() = [1 - )]-1
i i
iqs ssd 1
2 )(1)(
si resonance “frequency”
Fi oscillator strength
s = 2/ 2 = 0 14 ( = - 6) s = 2/ 2 = 0 40 ( = 1 5)
Two strongest ES hybridized dipole
s1 = /p = 0.14 ( = - 6) s2 = 2/p2 = 0.40 ( = -1.5)
y presonances.
Note: red-shifting of strongestof strongest resonance due to particle-particle interactioninteraction
Solutions of the driven equationq
*),(, yxdxdywhere
Now use a simple definition of the permittivity tensor:
k )(where jk
kj EE 0)(
0
Effective Permittivity Tensor!y
where
Note: f0 does not vanish only if the plasmonic inclusion forms a continuous phase!continuous phase!
Reminder:dm
s
/)(1
1)(
Consequence: Drude conductivity for small frequencies!
Summary: effective permittivity of plasmonic MMsplasmonic MMs
•Assume a fully symmetric (C4v) structure: ssu e u y sy e c (C4v) s uc u e:scalar dielectric permittivity:
N
iqs
FF0
)()(1)(
i iq sss 1 )()(
•This term is only present when the metal crosses the cell’s boundary continuous
d i i f l f i
s 1)(
conductivity for low frequencies
1)(2
p
dm
s
/)(1
)( )(
1)(
i
pm
Example: strip pair-single film (SPOF)extE +Q-Q
z
hsd
L
effx
2w2wh
fd
sd zL
ΔL
xL
(a) Electrostatic simulation: scan frequency (m) obtain qs()magnetic
(b) Compare quasi-static (intuitive!)
qs with the electromagnetically
gresonance, band edge
extracted eff
(c) Sub- (/7) NIM structure qs ≈eff
Does this result in a true homogenization?
Multi-scale expansion theory for equations with oscillating periodic coefficients
X: large scale ll l
Expand potential: )(),(),()( 210 OyXyXx
X: large scale y: small scale
Derive effective permittivity for
y
yXX ),()( 0
0)(
X
XX j
ijeff
i
j y )()(
yi
j
ijijeff y
yyy
)()()()(
1Generalization of Bergman-Stockman-Stroud permittivity:
0))(()( )(1 j
j yyy
Small-scale potential functions:
Physical meaning: Match dipole moment with that of homogeneous medium
complicated field profile in metamaterial
uniformly polarizedhomogeneous medium
Generalized quasi-static permittivity tensor for nanostructures with a continuous plasmonic phase:
n n
ijn
ij
ijijeff ss
fs
f 00
dVPd
0Ed eff
Generalized sum rule:Permittivity determined by strength of dipole-active resonances:
2/1;n
nn
jn
in
ijn
V
rdddf
ij
p
n
ijn
ij
VV
ff 0
0
p
1)(2 p
Note: ES permittivity of every structure with a continuous metal phase (e.g. fishnet) has a ...1)( 2
pcontinuous metal phase (e.g. fishnet) has a
metal-like permittivity pole at =0
Unexpected connection: effective pepsilon and Extraordinary Optical Transmission
Experiment vs Theory: Transmission, Absorption, Reflection through perforated filmsReflection through perforated films
•Reflection drops, 1.0
n Reflection: p ,transmission and absorption spike in the vicinity of SPP(0,1)
0.8
abso
rptio FTIR
Theory
vicinity of SPP(0,1) resonance
•Resonant effects 0.4
0.6
nsm
issi
on,
Transmission: FTIR Theory
Fano spike
observed for both S and P-polarizations, for all sample orientations0.2
0.4
ctio
n, t
ran Theory
p
•Resonances sensitive to polarization and sample
i i10 10.5 11 11.5 12 12.5
0 Ref
lec
Absorption
orientation10 10.5 11 11.5 12 12.5
Wavelength (microns)
Radius: 1 m P i d 7 Experiments and simulationsPeriod: 7 m Wavelength: 11.8m Incidence angle: 21deg
Experiments and simulations agree: but what is the physics??
Quasi-static resonances of a periodically perforated filmperforated film
N
iFF01)(s() = [1 - )]-1
si resonance “frequency”
i i
qs sss 1 )()(1)(
si resonance frequency
Fi oscillator strength
Bergman & Stroud’92; Stockman et.al.’01; P t l f ti fil Shvets & Urzhumov, PRL’04 & JOPA’05•Present only for continuous films
•Responsible for enhanced transmission
LSP 0 55 SPP(1 0) 7LSP at = -0.55 period independent
SPP(1,0) at = -7 depends on period Note:
•This technique is reminiscent of the Fano resonances approach
•No parameter fitting to EM simulations is required
How does the effective explain extra transmission and absorbance?
6
8Real(ε
eff)
Imag(εeff
) Absorption on
Korobkin et. al., Appl.Phys.A 88, 605 (2007)
0
2
4
itti
vity
Imag(εeff
)Real(ε
SiC)
Imag(εSiC
)
Absorption maximum
rans
mis
sio
−4
−2
0
lect
ric
per
mit
t
Transmission maximum D = 2 m P = 7 m
rptio
ntr
−8
−6
−4
die
lec
Transmission minimum red: absorption
blue: transmission
Ext
raab
sor
•Enhanced transmission is due to correction to Real( ff )
10.4 10.6 10.8 11 11.2 11.4−10
λ, microns
EEnhanced transmission is due to correction to Real(eff )
•Enhanced absorption is due to enhanced Imag(eff) predicted & measured 40% absorbance!
•In metamaterials with continuous plasmonic phase resonances result in enhanced transmission!
How did the effective medium description do?
0.35
0.4Absorption (EM)Transmission (EM)
Quasi-Electrostatic (ES) R d hif
0.2
0.25
0.3
0.35 Transmission (EM)Absorption (ES)Transmission (ES)
Q ( )approach:
Compute electric dipole ( t th Fso
rptio
n
Red shift: ~(P/)2,
(D/)2
D = 2 m P = 7 m
0.1
0.15
0.2 resonances (strengths Fiand “frequencies” si)
C t ( )mis
sion
/abs
−0.05
0
0.05 Compute eff()
Fresnel formulas transmission/absorptiontr
a tr
ansm
10 10.5 11 11.5 12 12.5
−0.1transmission/absorption
N F
Ext
Wavelength (microns)
N
i i
ieff ss
FF1
0 )(1)(1)(
( ) [1 )] 1
F0 = 0.88, s0 = 0 ( = -∞)F1 = 0.04, s1 = 0.12 ( = -7)
s() = [1 - )]-1
si resonance “frequency” Fi oscillator strength
1 1F2 = 0.005, s2 = 0.2 ( = -4)
Subtle effects: spatial dispersionp p
Spatial dispersion in “artificial plasma”
P d t l PRL ’96 •Pendry et.al., PRL ’96 thin-wire arrays have effective = 1 – 2/2effective 1 p /
•Early work: W. Rothman: “Plasma simulation byPlasma simulation by artificial dielectrics”, ’62
)8/log(16
20
22
22p rdd
c Plasma frequency is detemined
by wire spacing d and radius r00)g( 0 y p g 00
Is this plasma-like really correct? important for the i f f d b li i !existence of surface waves and sub- applications!
If it supports plasma waves, it must be plasma!
Wire width w = d/5z
Wire width w d/5
Cutoff pd/2c = 0.42
Wavenumber k d = x
Wavenumber kzd =
Freq.: d/2c = 0.45y
1
(a)
1
(b)
“Photon”: E k0
0.5
z 0
0.5
z
Bulk Plasmon: E ||k
d/2c = 0.56
−1 0 1
−1
−0.5
−1 0 1
−1
−0.5d/2c = 0.45
−1 0 1y
−1 0 1y
Shvets et.al., SPIE Proc. v. 5218, 156 (2003)
Bulk and Surface Modes in 3-D Wire Mesh
wave wave
d=0.58cm w=0.1cm
wwwww|| wave
ddddddsurface wave
Surface waves exist but their dispersion differs from that of plasmonic materials manifestation of spatial dispersion
Shapiro, Shvets, Sirigiri, Temkin., Opt. Lett. 31, 156 (2006)
What is the spatial dispersion: start with 2D
Propagation of TM waves in periodic 2D structure of thin rods: origin of the cutoff
Simulate/analyze Ez in single cell: (d2/dx2 + d2/dy2) Ez = (kz
2 - 2/c2) Ez
Boundary conditions: Ez = 0 at wires & E ( d/2 ) E ( d/2 ) ik dEz(x=d/2,y) = Ez(x=-d/2,y) eik
xd
Ez(x,y=d/2) = Ez(x,y=-d/2) eiky
d
Electric/magnetic fields in 2D
•Non-vanishing fields: Ez B Er (only forEz, B, Er (only for finite kz)
C l E C•Color: Ez, Contours: magnetic field lines
•Plotted: cutoff point
•EM waves run along gthe wires for finite kz
)22/log(8
2
22
rddc
p Numerical tool: COMSOL )22/log( 0rddNumerical tool: COMSOL
Numerical simulations and effective
Parameters: d = 2, r0/d = 0.05, p = 1
• Thin-wire structure is isotropic
• Dispersion relation is
2 = p2 + kz
2c2 + 2(kx2 + ky
2)c2p y
where = 0.953
Fields can be described using cell-averaged quantities: <E> and <B> related by diagonal dielectric tensor ij: xx = yy =1
0000
222
22
222
2
|| 09.01ck
ckck
p
||00
ckck zz
Spatial dispersion of dielectric tensor
•Consider simple example of kz2 > 0 and kx = ky = 0
<Ex> = <Ey> = 0 but <Ez> = 1 what is Poynting flux <Pz > ?
BEecP zz
4
does not vanish after cell-averaging
ccEEk
BEcP jiij .
84
Landau & Lifshits, “Electrodynamics of C ti M di ”k84 Continuous Media”
vanishes does not vanish
2
222
2
|| 1),(ck
kz
pz
Strong dependence on kz
G. Shvets, AIP Conf. Proc., v. 647, 371 (2002); P. Belov et. al., Phys. Rev. B 67, 113103 (2003).
Even for 3-D meshes the spatial dispersion is still very strong!is still very strong!
2
)()()( mlkkck 20 )()(),(
ijlmijij k
(from Agranovich, Ginzburg, “Crystal Optics with Spatial Dispersion”)(from Agranovich, Ginzburg, Crystal Optics with Spatial Dispersion )
2
2
0 1)(
pFor cubic symmetry the tensor is characterized by three independents
Example of a wave with Hz, Ex, Ey field components (TM):
20 )(
y pconstants only:
Example of a wave with Hz, Ex, Ey field components (TM):
222 20)( kkkk
2
22
13
32
22
12
2
0
0
22
)(00)(
),(xyyx
yxyxij kkkk
kkkkck
yy
How do we extract these parameters?
For cubic symmetry the dispresion of waves near Gamma-point is:
d
w
d
w
d
w
d
w
dd
w
(from Agranovich, Ginzburg, “Crystal Optics with Spatial Dispersion”)
Dependence of the spatial anisotropy ffi i t ( ’ ) th /d ticoefficients (’s) on the w/d ratio
•Only 2 decays as a power law
0.2
α2er
0 .2
α2er a power law (2 = 1.3 w/d)
•Other s decay
00 0.02 0.04 0.06 0.08 0.1
ram
ete
00 0.02 0.04 0.06 0.08 0.1
ram
ete
•Other -s decay, at best, logarithmically)
-0.2
α
α3
αpa
r
-0 .2
α
α3
αpa
r
g y)-0.4
w/d
α1-0.4
w/d
α1
Note: in many cases, only 2 matters!
Spatial dispersion does not always show
Simplest geometry: wave normally incident on a vacuum/mesh interface
xzy ekkeHHeEE 000 ,,
vacuum/mesh interface
Dispersion relation:
22w 2
221
220 1
ck p
dw3.12
2
22
cp
from Smith, Schultz, Markos, Soukoulis, PRE’02
c
…but sometimes it does, with vengeance!
•Vacuum/mesh interface at z=0: BEcP
wwwww assume a surface wave with Hy, Ex, Ez components
BEcP4
dddddd
•Require the continuity of energy flux:
ccEEk jiij .
8
Outline of the talk
• Introduction to optical metamaterials
• Attempts to answer puzzling questions that I would have had if I was entering the field:1. Why is the optical magnetism difficult to achieve (a.k.a why are
the microwave experts looking down on us?)
2 A l i ff t d b d f ki ti l2. Are plasmonic effects good or bad for making optical metamaterials?
3. How does one analyze periodic plasmonic structures?3. How does one analyze periodic plasmonic structures?
4. Is it possible to to homogenize plasmonic nanostructures?
5 Why is everybody talking about spatial dispersion and how5. Why is everybody talking about spatial dispersion and how important is it?
• Perspectives on the optical metamaterials field: where the• Perspectives on the optical metamaterials field: where the promising applications are?