DALHM D evelopment and A nalysis of L eft H anded M aterials
description
Transcript of DALHM D evelopment and A nalysis of L eft H anded M aterials
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FET - Open Domain IST-2001-35511FET - Open Domain IST-2001-35511
DALHM Development and Analysis of Left Handed
Materials
FORTH, Crete, GreeceBilkent University, Ankara, TurkeyImperial College, London, England
2nd year MeetingJuly 29-30, 2004Crete, Greece
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Computational Methods Plane wave expansion method (PWE) R. Moussa, S. Foteinopoulou & M. Kafesaki
Transfer matrix method (TMM) Th. Koschny, R. Penciu & P. Markos
Finite-difference-time-domain-method (FDTD) M. Kafesaki, R. Moussa, & S. Foteinopoulou
Effective medium theories E. N. Economou, Th. Koschny
Microwave studio T.O.Gundogdu, R. Penciu, M. Kafesaki & Lei
Zhang
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Transfer matrix method to compute scattering amplitudes
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New discretization scheme: Symmetry is preserved
This new symmetric material discretization completely eliminates the problem of the off-diagonal terms in the transfer matrix approach for sufficiently accurate computation. So we have successfully implemented a new discretization scheme that gives no off-diagonal terms.
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continuum Homogeneous Effective Medium inversion
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Generic LH related Metamaterials
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€
′ ω p
€
′ ω p
€
ωm
€
ωm
€
′ ω m
€
′ ω m
€
ωa/c
€
ωa/c
€
ωa/c
€
ε
€
μ
Resonance and anti-resonance
Typical LHM behavior
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Analytic model for the electric and magnetic response of SRRs
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Analytic model of the electric and magnetic response of LHMs
PRL (accepted, 2004)
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Electric response of LHM
Electric and magnetic response of SRR
E and M response of LHM
Electric response of wires
Electric response of cut wires
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f (GHz)
T
30 GHz FORTH structure with 600 x 500 x 500 μm3
SubstrateGaAsεb=12.3
LHM Design used by UCSD, Bilkent and ISU
LHMSRRClosed LHM
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Left-Handed MaterialsLeft-Handed Materials
SRR Parameters:
r1=2.5 mm, r2=3.6 mm,d=w=0.2 mmt=0.9 mm
Parameters:
ax=9.3 mm ay=9mm az=6.5 mmNx=15 Ny=15
w
t
dr1
r2
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Transmission data for open and closed SRRs
Bilkent & Forth
Magnetic resonancedisappears for closed SRRs
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Bilkent & Forth
Effective ωp of closed SRRs & wires is much lower than ωp of the wires.
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Best LH peak in a left-handed material
Losses: -0.3 dB/cm Bilkent & Forth
Peak at f=4 GHz=75 mm
much largerthan
size of SRR
a=3.6 mm
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Experiment Theory
Bilkent & Forth
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Bilkent & Forth
Retrieval parameters for Bilkent structure
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Transmission spectra in the low frequency region for 3 unit cells
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Transmission spectra in the higher frequency region
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Transmission S21 in the lower and higher region (1 unit cell)
10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0
0.01
0.1
1
S21
Frequency(GHz)2.0 2.5 3.0 3.5 4.0
0.01
0.1
1
S21
Frequency(GHz)
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Retrieved n in the lower frequency region
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0-6
-5
-4
-3
-2
-1
0
1
Frequency(GHz)
real n imag n
n
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10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5-10
-5
0
5
10
n
Frequency(GHz)
real n imag n
Retrieved n in the higher frequency region
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Retrieved ε, μ in the lower frequency region
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0-30
-25
-20
-15
-10
-5
0
5
10
Frequency(GHz)
ε μ
ε, μ
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Retrieved ε, μ in the higher region
10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5-10
-5
0
5
10
15
ε, μ
( )Frequency GHz
ε μ
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Closed rings
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2 4 6 8 10 12-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
n real n imag n
Frequency(GHz)
Closed rings
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Closed rings
2 4 6 8 10 12-15
-10
-5
0
5
10
15
ε μ
ε, μ
( )Frequency GHz
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Electric and Magnetic Response of SRRs and LHMs
• Electric and Magnetic Response are independent.
• One can change the magnetic response without changing the electric response.
• GHz and THz magnetic response in artificial structures!
• The SRR has strong electric response. It’s cut-wire like.
• Effective electric response of LHM is the sum of wire and SRR.
• Effective ωp of the LHM is much lower than ωp of the wires.
• There are “phony” LH peaks when ωp < ωm PRL (accepted, 2004)
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Electric coupling to the magnetic resonance
APL 84, 2943 (2004)
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8 9 10 11-10
0
10
20
30 retrieved μ ( )for a
realμ imagμμ
( )Frequency GHz
8 9 10 11-10
0
10
20
30 retrieved ε ( )for d
ε
realε imagε
( )Frequency GHz8 9 10 11
0
1
2
3
4 retrieved μ ( )for d
μ
( )Frequency GHz
realμ imagμ
Photonics and Nanostructures (accepted, 2004)
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Magnetic response at 100 THz, almost optical frequencies
S. Linden & M. Wegener, Karlsruhe 10
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Magnetic response at 100 THz, almost optical frequencies
S. Linden & M. Wegener, Karlsruhe
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Magnetic response at 100 THz, almost optical frequencies
S. Linden & M. Wegener, Karlsruhe
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4 cases of different propagation and polarization for single ring
cell = 2.5mm
gap azimuthal = 0.3mm
ring outer side length = 2.2mm
ring width = 0.2mm
sub thickness =0.25mm
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Transmission and retrieved parameters
4 6 8 10 12 14 16 18 20-6
-4
-2
0
2
4
6
Frequency(GHz)
kparEpar kparEpen kpenEpar kpenEpen
μ
4 6 8 10 12 14 16 18 20-20
-10
0
10
20
30
40
50
60
Frequency(GHz)
kparEpar kparEpen kpenEpar kpenEpenε
4 6 8 10 12 14 16 18 200.0
0.2
0.4
0.6
0.8
1.0
kparEpar kparEpen kpenEpar kpenEpen
S21
Frequency(GHz)
k in the plane gives a negative μ region,
otherwise μ remains positive even though
a gap appears in the transmission spectra
when E field is along the ring gap.
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Opposed ring can get rid of the effect of electric coupling
2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
kparEpar kparEpen kpenEpar kpenEpen
S21
frequency(GHz)
2 4 6 8 10 12 14-4
-2
0
2
4
6
kparEpar kparEpen kpenEpar kpenEpen
frequency(GHz)
μ
2 4 6 8 10 12 14-4
0
4
8
12
16
20
24
28
32
kparEpar kparEpen kpenEpar kpenEpenε
( )frequency GHz
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Sub thickness dependence
the closer the separated opposed rings are, the weaker the electric coupling is.Here are shown transmission spectra and μ when the thickness are chosen to be
0.25mm, 0.125mm and 0.075mm
2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
t = 0.25 t = 0.25 * 0.5 t = 0.25 * 0.3
S21
frequency (GHz)2 4 6 8 10 12 14
0.6
0.8
1.0
1.2
1.4
t = 0.25mmt = 0.25mm*0.5t = 0.25mm*0.3
μ
frequency
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3D Rings & Wirescell = 2.5mm in X/Y/Zring side length = 2.2mmwire width = 0.2mmring width = 0.2mmopposed ring separation = 0.2mm
this structure is symmetric in 3D and also behaves almost the same for different polarizationssee black and red curve below.
6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
kxEz kxEy
S21
Frequency (GHz)
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Retrieved Z, n
9 10 11 12 13 14 150
2
4
6
8 kxEz
Frequency (GHz)
Z
9 10 11 12 13 14 15-6
-5
-4
-3
-2
-1
0
Frequency (GHz)
n
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n and ε μ
6 8 10 12 14-50
-40
-30
-20
-10
0
10
20
Frequency(GHz)
real ε realμ
ε μ
there are multiple negative index regions from the retrieval code
6 8 10 12 14-10
-8
-6
-4
-2
0
real n imag n
n
Frequency(GHz)
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Going to multi-gap structures (1)
(a) better than (b) (wider SRR dip); (c) better than (d) (stronger dip); (e) like the conventional SRR but weaker dip (for large separation)
Problem: Increase of ωm (ωm close to ω0 )
Gaps act like capacitors in series: ωm2(n gaps)
~ n ωm2(1 gap)
Reason: requirement for higher symmetry, for use in 3D LH structures
a) b) c) d) e)
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Going to multi-gap structures (2)
Solution: Make the gaps smaller or change the design
Improvements?
Up to a point
Only the left one
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Promising multi-gap structures from 1D study
a) b)
(a): Detailed study on progress (in 1D)
(b): Not studied in detail yet
(c): Good LH T
3D structures
a) b) c) Best combination: (b)+(c)
c)
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Two-sided SRR Structures: No coupling to Electric Field
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2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
t = 0.25 t = 0.25 * 0.5 t = 0.25 * 0.3
S21
frequency (GHz)
2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
kparEpar kparEpen kpenEpar kpenEpen
S21
frequency(GHz)
Two-sided SRRs do not have coupling to electric field
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Multi-Azimuthal SRR
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2-gaps SRR
• adding another gap at the opposite side of ring helps to inhibit the electric coupling
• Magnetic properties remain the same
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testing with different k and polarization
16 18 20 22 24 26 28 30-1
0
1
2
3
Frequency(GHz)
retrieved shows that there is no magneticresonance when K is out of the SRR plane
K||e|_ K||e|| K|_e|_ K|_e||μ
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4-gaps SRR
introducing 4 cuts at each side of the ring respectively may help to build a near-isotropic structure
unit cell = 25mm
ring side = 22mm
ring width = 2mm
azimuthal = 50um
sub thickness = 2.5mm
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20 25 30 35 40 45 50-0.5
0.0
0.5
1.0
1.5
2.0
realmu immu
Frequency (GHz)
μ
20 25 30 35 40 45 50-1.0
-0.5
0.0
0.5
1.0
ren imn
n
Frequency (GHz)20 25 30 35 40 45 50
-4
-3
-2
-1
0
1
2
3
4
Frequency (GHz)
rez imz
Z
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Comparison of 1/2/4-gaps SRRall SRR have a total azimuthal length 50μm
While keeping the sum of the gap widths the same, the 1, 2 or 4 cuts SRR have different resonance frequencies. 4-cuts SRR has a much higher resonance region than single cut ones. To build a LHM with 4cuts SRR, wires need to be more compact to enhance plasma frequency.
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LHM composed of 4-cuts SRR and Wires
• Unit cell dimension = 2.5mm
• Ring side length = 2.2mm
• Ring width = 0.2mm
• Wire width = 0.6mm
• Ring thickness = 17micron
• Wire thickness = 50micron
• Sub thickness = 0.25mm
• Sub permittivity = 2.2
• Deposit permittivity = 9.61
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18 20 22 24 26 28-6
-5
-4
-3
-2
-1
0
1
retrivied n
Frequency
real(n) imag(n)
n
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18 20 22 24 26 28-6
-4
-2
0
2
4
6
8retrivied ε, μ
Frequency
(realε) (realμ)
ε,μ
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T and R of a Metamaterial
€
ts =exp(−ikd)
cos nkd( )−1
2z +
1
z ⎛ ⎝
⎞ ⎠sin nkd( )
€
z =με
€
n = με
€
μ ω( )=1−ωmp
2
ω 2 −ω m02 + iΓm0
€
ε ω( ) =1−ωep
2
ω2 −ω e02 + iΓe 0
UCSD and ISU, PRB, 65, 195103 (2002)
€
rs = − ts exp(+ikd)i(z −1 / z)sin(nkd) / 2
d
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z, n
Inversion of S-parameters
d
UCSD and ISU, PRB, 65, 195103 (2002)
€
e ik
€
teik
€
re− ik
€
ε =nz
€
μ =nz
€
n =1kd
cos−1 12 ′ t
1− r 2 − ′ t 2( )[ ]
⎛ ⎝
⎞ ⎠ +
2πmkd
€
z = ±1 + r( )
2− ′ t 2
1− r( )2 − ′ t 2
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Refractive index n Permittivity ε Permeability μ
Im n > 0Re n > 0
Im ε < 0 ??? Re ε > 0
Im μ > 0Re μ < 0
Energy Losses Q in a passive medium are always positive in spite of the fact that Im ε < 0
€
Q(ω) = ′ ′ ε | E |2 + ′ ′ μ | H |2
€
Q(ω) = 2ω | H |2 ′ ′ n (ω) ′ z (ω)
Q(ω) > 0, provided that Im n(ω) > 0 and Re z(ω) > 0
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1d single-ring SRR: retrieved Re n() via cHEM inversionfor different length of the unit cell: 6x10x9 ... 6x10x14
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TMM simulated 1d single-ring SRR:retrieved Re n() via cHEM inversion
for different resonance frequencies
Emulate small SRR gap: we fill the gap with dielectic, eg.eps=300Vacuum case as before
π/(Nz)
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TMM simulated 1d single-ring SRR: retrieved eps() and mu() via cHEM inversion
for different resonance frequencies
Emulate small srr gap:we fill the gap with dielectic, eg. eps=300
Vacuum case as before
No negative Im εω and Im μω are observed !
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TMM simulated 1d single-ring off-plane LHM: retrieved Re n() and Im n() via cHEM inversion
Re n()
Im n()
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TMM simulated 1d single-ring off-plane LHM: retrieved ε() and μ() via cHEM
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Intermediate summary:continuum homogeneous effective material (cHEM)
● cHEM inversion basically works, we find length-independent(!) effective material behavior
Re n(ω) seems to be cut-off at Brillouing zone. Discrepancy between n(ω) and z(ω): where is the resonance? Resonance/anti-resonance coupling. Negative imaginary parts in εω or μω Deformed resonances, i.e. unexpected shallow negative μω What is all this structure at higher frequencies?
but problems:
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Model: Effective periodic material (PEM)
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PEM analytic SRR model: retrieved n(), z() and eps(), mu() via cHEM
n() z()
eps() mu()
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PEM analytic LHM model: retrieved n(ω), z() and eps(), mu() via cHEM
n() z()
eps() mu()
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TMM simulated 1d single-ring SRR:retrieved (cHEM) + calculated (PEM-to-cHEM) n(), z()
Re n() Im n()
Re z() Im z()
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TMM simulated 1d single-ring SRR:retrieved core+avrg eps(), mu() via lattice PEM inversion
eps,muSRR
eps,muLHM
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3D Rings & Wires
cell = 2.5mm in X/Y/Z
ring side length = 2.2mm
wire width = 0.2mm
ring width = 0.2mm
opposed ring separation = 0.2mm
this structure is symmetric in 3D and also behaves almost same for different polarizationsee black and red curve below. different size for gap azimuthal is chosen, they are 0.3mm, 0.2mm and 0.1mm.
6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
kxEz gap=.3 kxEy gap=.3 kxEz gap=.2 kxEz gap=.1S21
Frequency (GHz)
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Retrieved Z, n
9 10 11 12 13 14 150
2
4
6
8
10
kxEz gap=.3 kxEy gap=.3 kxEz gap=.2 kxEz gap=.1
Frequency (GHz)
Z
9 10 11 12 13 14 15-6
-5
-4
-3
-2
-1
0
Frequency (GHz)
n
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Retrieved n and ε μ
6 8 10 12 14-6
-5
-4
-3
-2
-1
0
real n imag n
n
Frequency(GHz)6 8 10 12 14
-50
-40
-30
-20
-10
0
10
20
Frequency(GHz)
real ε realμ
ε μ
there are multiple negative index regions from retrieval code