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Physics 208A Presentation
Oct. 18th, 2004
Artificial Magnetic Resonators and
Potential Applications in Nonlinear Field
Yongmin Liu
Applied Science & Technology
Outline
I.
Background of Metamaterials
II. Artificial Magnetic Resonators at THz
III. Potential Application in Nonlinearity
IV. Summary
What are Metamaterials?
Artificially fabricated structures or media that exhibit
electrodynamic properties not found in naturally occurring
materials.
* Dimension of the unit cell is less than the wavelength of
excitation EM wave, thus the effective-media theorem can be
applied.
Why Metamaterials are Interesting?
* We can design and control the properties of materials. Some
novel properties, such as negative electric permittivity,
negative magnetic permeability, negative refractive index etc.
have been explored.
Negative Permittivity
The permittivity of metal is given by
ep2
 ( )  1 
 (  i )
2
ne
2
Plasma frequency: ep 
(typically in the UV region)
 0 me
where n is the electron density, and me is the electron mass
0
Damping factor:  

where  is the electric conductivity
In the visible region, () is negative for most metals.
At lower frequencies, permittivity is imaginary.
Negative Permittivity (cont’d)
neff
r 2
0ne2
 n 2 , meff 
ln( a / r )
a
2
ep2 
neff e 2
 0 meff
6
radius of wire: r  1.0 10 m,
lattice constant: a  5.0 103 m,
n  5.675 1017 m 3
Negative  with small loss in low
frequencies can be achieved by
metallic wire lattice
ep  8.2GHz!!
ep2
 eff  1 
 (  i  0.1ep )
Pendry J.B. et al., Phys. Rev. Lett. 76, 4773 (1996)
Negative Permeability
Magnetism originates from
1) orbital motion of electrons
2) unpaired electron spins
The magnetic response of most nature
materials fades away in GHz region.
Negative  can be achieved by splitring resonator (SRR)
Artificial magnetism can be realized by
conducting, nonmagnetic split-ring
resonators. The magnetic response is able
to extend to THz, even higher frequency
with large positive or negative
permeability.
Will discuss in detail later !
Left-handed Materials (LHM) with
Negative Refractive Index (NIR)
2
n
 
Refractive index:
When  < 0 and  < 0 simultaneously, we have to choose
n   
 

  
Maxwell’s equation: k  E  H , k  H  E
n>0
( > 0 ,  > 0)
E
k
S
H
Right handed materials
(RHM)
n<0
( < 0 ,  < 0)
E
k
S
H
Left handed materials
(LHM)
LHM with NRI (cont’d)
Exotic properties of LHM:
1. Snell’s law:
RHM
sin 1
n
sin  2
k
S
LHM
Negative refraction !
2. Flat superlens
Diffraction-limit (dmin~l) free !
Fourier Expansion
of 2D object:
E ( r, t ) 

,k x ,k y
Z
(
) (
E k x , k y exp ik z z  ik x x  ik y y  it
2 2
2
2
Propagating waves: k z    c  k x  k y ,
2c 2 > k x2  k 2y ,
k z  i k x2  k 2y  2c 2 ,
2c 2 < k x2  k 2y .
Evanescent waves:
)
Veselago V.G. Sov. Phys.10, 509 (1968); Pendry J. B. PRL 85, 3966 (2000)
LHM with NRI (cont’d)
Artificially engineered metamaterials implements the concept of LHM!
Photograph of LHM
Shelby R. A. et al., Science 292, 77 (2001);
Smith D. R. et al., Science 305, 788 (2004)
Negative refraction by LHM prism
LHM with NRI (cont’d)
Imaging properties of LHM
Electric field of a point source
focused by a LHM slab
Imaging experiment in
microwave region
Simulation of subwavelength
imging by FDTD
Houck A. A. et al. PRL 90, 137401 (2003); Kolinko P. et al. Opt Exp 11, 640 (2003)
LHM with NRI (cont’d)
Metamaterials open a new field in physics, engineering
material science and optics!
Negative refraction is among the
Top 10 highlights of 2003 by
Physicsweb
http://physics.ucsd.edu/~drs/ Prof. Smith D.R. in UCSD
Outline
I.
Background of Metamaterials
II. Artificial Magnetic Resonators at THz
III. Potential Application in Nonlinearity
IV. Summary
Artificial Magnetic Resonators at THz
Concept:
1.
The magnetic-flux induced current loop to form magnetic dipole.
2.
The intrinsic conductance and inductance will cause strong paramagnetic
or diamagnetic activity around the resonance frequency.
2r
a
H
_+ _+ _+
+
_ _+ _+
Pendry J.B. et al, IEEE MTT 47, 2075 (1999)
Artificial Magnetic Resonators at THz (cont’d)
Current distribution of SRR simulated by Microwave Studio
Artificial Magnetic Resonators at THz (cont’d)
H-field of SRR simulated by Microwave Studio
Artificial Magnetic Resonators at THz
r 2
 eff  1 
a2
1
2i
3
 2
 0r   0 2Cr 3
Resonance frequency: 0 
3
 2 0Cr 3
Magnetic plasma frequency:
mp
3

 2 0Cr 3 (1  r 2 / a 2 )
Typical value: r  2.0 103 m
a  5.0 103 m, d  1.0 104 m
f 0  2.94GHz, f mp  4.17GHz
Pendry J.B. et al, IEEE MTT 47, 2075 (1999)
Dispersion of eff with frequency
Artificial Magnetic Resonators at THz (cont’d)
Sample
50um
Cu
S: space
W: width
L :length
G: gap
Au/Ti
quartz
L:26m, S:10m, W:4m
d=L+S, G: 2m, :1.5x103W
Ye T.J. et al., Science 303, 1494 (2004)
Artificial Magnetic Resonators at THz (cont’d)
IR
I0
=30o
Die
Simulation
(THz)
Experiment
(THz)
D1
1.22
1.27±0.07
D2
0.88
0.96±0.05
D3
0.91
0.85±0.15
Experimentally and theoretically
ellipsometric results
Artificial Magnetic Resonators at THz (cont’d)
3.0
2.5
Re(Mue)
2.0
1.5
1.0
0.5
0.0
LSR300
LSR350
LSR400
-0.5
-1.0
-1.5
0
10000
20000
30000
40000
50000
60000
Frequency (GHz)
k (cm-1)
f (THz)
λ/a
LSR400
1100
33.00
2.52525
LSR350
1282
38.46
2.47629
LSR300
1490
44.70
2.48571
Near-infrared (45THz) magnetic
resonance is achieved by novel
design. Final goal is visible region.
Outline
I.
Background of Metamaterials
II. Artificial Magnetic Resonators at THz
III. Potential Application in Nonlinearity
IV. Summary
Potential Application in Nonlinearity
Extremely high intensity is the key to nonlinear phenomena!
Brabec T et al., Rev. Mod. Phys. 72, 545 (2000)
Potential Application in Nonlinearity (cont’d)
When resonance takes place, the energy is strongly localized
inside the small resonators. Local fields can be many orders
higher than that in free space.
For a capacitor of 1nm 1nm 1nm , one single photon can
create an electric field about 108V/cm.
Localized E-filed with 103 times larger than the external field.
Potential Application in Nonlinearity (cont’d)
Embed the magnetic resonator into dielectric matrix whose
permittivity is intensity-dependent.
Two aspects of the nonlinear response:
1) The strong localized field changes the dielectric
permittivity, since D  D (|E|2)
2) Nonlinear eigenfrequency adjusts correspondingly
due to the change of capacitance.
External H fieldIntensity of the local E fieldValue of
permittivity Capacitance  Eigenfrequency
Potential Application in Nonlinearity (cont’d)
Effect nonlinear permittivity:
 eff (| E |2 )   D (| E |2 ) 
 p2
 (1  i  )
Effective permeability:
(a 2 / d 2 ) 2
eff ( H )  1  2
  02NL  i
where 
2
0 NL
dg
c 2
(H )  ( )
a h D (| E g ( H ) |2 )
2
2
2
Consider Kerr nonlinearity:  D (| E | )   D 0   | E | / Ec
Ec is a characteristic electric field, and   1corresponds
to focusing or defocusing nonlinearity respectively.
Zharov A.A. et al., PRL 91, 037401 (2003)
Potential Application in Nonlinearity (cont’d)
(1  X 2 )[( X 2  W 2 ) 2  W 2 2 ]
| H |  A E
X6
W   / 0 , X  0 NL / 0 , 0 is the eigenfrequency in linear limit
2
Transition of eff
from – to +
2
2
c
Jump of eff due
to external H field
Outline
I.
Background of Metamaterials
II. Artificial Magnetic Resonators at THz
III. Potential Application in Nonlinearity
IV. Summary
Summary
 The unprecedented properties associated with metamaterials, such
as negative refraction, superlensing etc. are reviewed.
 The principle of achieving negative permeability, which is critical
in realizing LHM is interpreted. Magnetic resonators with resonance
frequency above THz is successfully demonstrated.
 The strong localized field inside the resonator can cause nonlinear
effect. As one example, the hysteresis-type dependence of the
magnetic permeability on the field intensity is theoretically studied.
 It is the right time to start the new topic--nonlinear effects in
metamaterials. The engineering of nonlinear composite materials will
open a number of applications such as swithers, frequnecy
multipliers etc.