the finite-difference time-domain method in...
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The Finite-Difference Time-Domain Methodin Nano-Optics
Mario Agio
Nano-Optics Group, Laboratory of Physical Chemistry, ETH Zurich
ti th h i i @ h h th hwww.nano-optics.ethz.ch - [email protected]
NMON 07.09.2007© ETH Zürich | Taskforce Kommunikation
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Outline
Introduction to FDTD Applications
Typical situations in Nano-Optics
Sources
Single molecule and SNOM tip
Lifetime engineering with
Boundary conditions
Near-to-far-field transformation
nanoantennae
Metals
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The Yee algorithm
∂∂t
D ⋅ n ds∫ = H ⋅ dl∫∂
−∂∂t
B ⋅ n ds∫ = E ⋅ dl∫
f (x,y,z,t) → f (i, j,k,n)Δx,... Δt
∂∂t
f (...,t) =f (...,n +1/2) − f (...,n −1/2)
Δt= ′ f (...,n)
NMON 07.09.2007 3
K. S. Yee, IEEE Trans. Antennas Propag. AP-4, 302 (1966)A. Bossavit, Progress in Electromagnetic Research 32, 45 (2001)
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The Yee algorithm - code example
SUBROUTINE march_h! Use Yee algorithm without a Source!! Magnetic Field Components!
m=1_i1b ! For non-magnetic media: u=1.0 everywere!
DO k=1,n3-1 ; DO j=1,n2-1 ; DO i=1,n1! i l(i j k)! m=material(i,j,k)
h1(i,j,k)= h1 (i,j ,k )+ &coeff_h1(2,m)*(e_2(i,j ,k+1)-e_2(i,j,k))- &coeff_h1(3,m)*(e_3(i,j+1,k )-e_3(i,j,k))
ENDDO ENDDO ENDDOENDDO ; ENDDO ; ENDDO…RETURNEND SUBROUTINE march_h
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A. Taflove, and S. C. Hagness, Computational Electrodynamics:The Finite-Difference Time-Domain Method 3rd ed. (Artech House, Norwood, MA 2005)
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Typical situations in Nano-Optics
Sources
Boundary conditions
Near-to-far-field transformationNear to far field transformation
Staircasing and Metals
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SourcesDipole: use J in Maxwell’s equationsDipole: use J in Maxwell s equations
Plane wave: total-scattered field technique
Tightly focused beam W. A. Challener, et al.,Opt Express 23 3160 (2003) Seagate Research
Gaussian beam
Opt. Express 23, 3160 (2003) - Seagate Research
J. B. Judkins, and R. W. Ziolkowski,J. Opt. Soc. Am. A 12, 1974 (1995) - Univ. Arizona
With substrate - film
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P. B. Wong et al., IEEE Trans. Antennas Prop. 44, 504 (1996) - Stanford (radar astronomy)K. Demarest et al., IEEE Trans. Antennas Prop. 43, 1164 (1995) - Kansas (radar, remote sensing)
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Boundary conditions
λ
Convolutional Perfectly Matched Layer (CPML)
J. A. Roden, and S. D. Gedney
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Microwave Opt. Technol. Lett. 27, 334 (2000)J.-P. Bérenger, IEEE Trans. Antennas Propag. 50, 258 (2002)
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Near-to-far-field transformation
observation point
free-space Green’s function(θ,φ)
free-space Green s function
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P. B. Wong et al., IEEE Trans. Antennas Prop. 44, 504 (1996) - Stanford (radar astronomy)K. Demarest et al., IEEE Trans. Antennas Prop. 44, 1150 (1996) - Kansas (radar, remote sensing)
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Staircasing and metals
Dx (i + 12 , j,k, t) = εx (i + 1
2 , j,k)Ex (i + 12 , j,k,t)
Dy (i, j + 12 ,k, t) = εy (i, j + 1
2 ,k)Ey (i, j + 12 ,k,t)
D (i j k + 1 t) ε (i j k + 1 )E (i j k + 1 t)
Dx (i + 12 , j,k, t) = εx (i + 1
2 , j,k,t − ′ t )Ex (i + 12 , j,k, ′ t )dt∫
Dz(i, j,k + 12 ,t) = εz (i, j,k + 1
2)Ez (i, j,k + 12 ,t)
x ( 2 j ) x ( 2 j ) x ( 2 j )∫
-10
0
2.5
3.0
Gold:Drude+Lorentz
-40
-30
-20
1.0
1.5
2.0 Drude+Lorentz
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500 600 700 800 900 1000 1100Wavelength [nm]
-50500 600 700 800 900 1000 1100
Wavelength [nm]
0.5
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Staircasing and metals100
101 7
10-4
10-3
10-2
10-1
100
r=15nm
r=20nmr=25nm
r=30nm
3
4
5
6Δ=1nmΔ=0.5nm
500 600 700 800 900 1000Wavelength [nm]
10-7
10-6
10-5
10
r=5nm
r=10nm
500 600 700 800 900 1000Wavelength [nm]
0
1
2
F. Kaminski, V. Sandoghdar, and M. Agio, J. Comput. Theor. Nanosci. 4, 635 (2007)
Ey
d1.0
1.2
Hz
ExΔ
0.2
0.4
0.6
0.8
analyticalCP f=0.75Δstaircase f=0.75 Δstaircase f=0.25 Δ
NMON 07.09.2007 10
Δf 0.0 1.0 2.0 3.0 4.0
Wavevector [k/ks]
0.0
A. Mohammadi, and M. Agio, Opt. Express 14, 11330 (2006)
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Applications
Single molecule and SNOM tip
Lifetime engineering with nanoantennae
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Single molecule and SNOM tip
TipxFar-field Detector
Source
Molecule
(θ,φ)
z
Near-field Detector
Molecule
Far-field detection
Tip parameters: core (SiO2), cladding (Al)cladding thickness 200nm, aperture radius 50nm.Molecule parameters: oriented along x resonant
600nm
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Molecule parameters: oriented along x, resonantat λ=615nm, distance tip-molecule d=40-600nm
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The tip near field
200
300
100
6.0
4.0
200
300
100
0.3200
300
100
3.0
2.0
0
-200
300
-1002.0
0
-200
300
-100
0.2
0.1
0
-200
300
-100 1.0x
|Ex| |Ey| |Ez|
-300
0-200-300 100 300nm
200-1000.0
-300
0-200-300 100 300nm
200-1000.0
-300
0-200-300 100 300nm
200-1000.0y
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The moleculeε (ω) = ε +
Δεωo2
εx (ω) = ε∞ +ωo
2 −ω 2 − 2iγω
α x ≈εx (ω) −ε∞
ε (ω) + 2ε=
Δεωo2
Δε⎛ ⎞ ⎡ ⎤ =Δε
3ε + Δε′ ω o2
′ ω 2 −ω 2 − 2iγωεx (ω) + 2ε∞ 3ε∞ ωo2 1+
Δε3ε∞
⎛
⎝ ⎜
⎞
⎠ ⎟ −ω 2 − 2iγω
⎡
⎣ ⎢
⎤
⎦ ⎥
3ε∞ + Δε ω o ω 2iγω
γ << ′ ω ⇒ α ≈ − π Δε ′ ω o⎛ ⎜
⎞ ⎟ (Δ − iγ)L(ω) L(ω) = 1 γ Δ = ω − ′ ω γ << ω o ⇒ α x ≈ −
2γ 3ε∞ + Δε⎝ ⎜
⎠ ⎟ (Δ − iγ)L(ω), L(ω) =
π Δ2 + γ 2 , Δ = ω − ω o
Etip(r → ∞) = Etip(rm ) ⋅ um( )gud = Bgud, Esc(r → ∞) = −A2
(Δ − iγ)L(ω)Bfud2
Etot = Etip + Esc = B g − f A2
(Δ − iγ)L(ω)⎡ ⎣ ⎢
⎤ ⎦ ⎥ ud
E 2 f
NMON 07.09.2007 14
S =Etot
2
Etip
2 =1+γ
4πV 2L(ω) −VL(ω)(Δ cosψ + γ sinψ), V (θ,φ) = A
fg
, ψ =ψ f −ψg
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Fitting the FDTD results with theory
0 161 0 162 0 163 0 1640 161 0 162 0 163 0 164
(without film)
1.1
1.2
0.161 0.162 0.163 0.164
1.1
1.20.98
1.00
0.161 0.162 0.163 0.164
0.98
1.00
0.8
0.9
1.0
0.8
0.9
1.0
0.92
0.94
0.96
0.92
0.94
0.96
d=40nm - θ=0 - φ=0
0.161 0.162 0.163 0.164Frequency [a/λ]
d=280nm - θ=40 - φ=90
0.161 0.162 0.163 0.164Frequency �[a/λ]
NMON 07.09.2007 15
a=100nm
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Angle ScanDistance Scan 1.4
1.6
aperture=100nm
Changing the Tip
Changing the parameters - collective result
1.3
1.4
1.5
1.6
0.500.751.001.251.50
0.4
0.6
0.8
1.0
1.2aperture=100nmaperture=200nmaperture=100nm - small
0.15
0.02
1.2
1.3
0 20.3
0.40.5
0.000.25
0 3
0.4
0.50.0
0.2
0.4
0 10 20 30 40 50 60
0.05
0.10
0.000 100 200 300 400 500 600Distance [nm]
�0.2
�0.10.00.1
0.2
-0.1
0.0
0.1
0.2
0.3
d=40nm - φ=0
Angle θ [deg]
θ=0 - φ=0
Distance [nm]
0 100 200 300 400 500 600Distance [nm]
-0.2
I. Gerhardt, G. Wrigge, P. Bushev, G. Zumofen, M. Agio, R. Pfab, and V. Sandoghdar,Ph R L tt 98 033601 (2007)
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Phys. Rev. Lett. 98, 033601 (2007)I. Gerhardt, G. Wrigge, M. Agio, P. Bushev, G. Zumofen, and V. Sandoghdar,Opt. Lett. 32, 1420 (2007).
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Lifetime engineering with nanoantennae Fluorescence signalFluorescence signal
2 rt r nr, ,
oo o o
o o o o oS γη η γ γ γγ
∝ ⋅ = = +d Etγ
2 SS Kηη∝ ⋅ =d E
2d E
o o
S KS
ηη
∝ =d E
r2
t
,o
K γηγ
⋅= =
⋅
d Ed E
NMON 07.09.2007 17
S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006)
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Metal nanoparticle as a nanoantenna
60 nm Au spheres
Vacuum Wavelength [nm]
NMON 07.09.2007 18
B. J. Messinger, et al., Phys. Rev. B 24, 649 (1981)C. Bohren, Am. J. Phys. 51, 323 (1982)
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Calculation of decay rates
Quantum-classical analogyPPo
=γγ o
Poynting theorem
n
Pt = Pr + Pnr
1 { }∫S
JV
Pt = −12
Re J∗(r,ω) ⋅ E(r,ω){ }drV∫
Pr = S(r,ω) ⋅ ndaS∫
NMON 07.09.2007 19
L. Rogobete, and C. Henkel, Phys. Rev. A 70, 063815 (2004)F. Kaminski, V. Sandoghdar, and M. Agio, J. Comput. Theor. Nanosci. 4, 635 (2007)
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The issue of quenching
160
200
Cross section
80
120Cross sectionRadiativeNon-radiative
40x10
500 600 700 800 900Vacuum Wavelength [nm]
0
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R. Ruppin, J. Chem. Phys. 76, 1681 (1982)L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
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Engineering the decay rates
200
250
long axis
100
150
50
100
x100
short axis
500 600 700 800 900Vacuum Wavelength [nm]
0
NMON 07.09.2007 21
J . Gersten, and A. Nitzan, J. Chem. Phys. 75, 1139 (1981)L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
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2D antenna models
1.0 3.0
0.8
2 0
2.5
0.4
0.6
1.5
2.0
X4
0.20.5
1.0
0.0500 600 700 800 900 1000 1100
Vacuum Wavelength [nm]
0.0
NMON 07.09.2007 22
L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
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3D antenna models
104
103
1020304050
120x38
2
10
102
900 950 1000 1050 1100Vacuum Wavelength [nm]
101
nb=1.7
NMON 07.09.2007 23
L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
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Application:improving the quantum efficiencyp g q y
rr r t t nr nr r a, ,o o o γγ γ γ γ γ γ γ η→ → = + + =
1η=
r r t t nr nr r anr r
, ,γ γ γ γ γ γ γ ηγ γ+
( ) ( )r r a(1 ) / /oo o oη η γ γ η η
=− +
γ ηηo =1%, γ r
γ ro =103, ηa = 80% →
ηηo
= 74, η = 74%
NMON 07.09.2007 24
L. Rogobete, et al., in preparationJ.R. Lakowicz, Anal. Biochem. 337, 171 (2005)
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Acknowledgments
Single molecule and SNOM tip
Nanoantennae
FDTD
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