3D Finite Difference Time-Domain Modeling of Acoustic Wave ...
The Finite-Difference Time-Domain Method in...
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The Finite-Difference Time-Domain Method in Nano-Optics Mario Agio Nano-Optics Group, Laboratory of Physical Chemistry, ETH Zurich ti th h i i@h h th h www.nano-optics.ethz.ch - mario.agio@phys.chem.ethz.ch NMON 07.09.2007 © ETH Zürich | Taskforce Kommunikation
Transcript of The Finite-Difference Time-Domain Method in...
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
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)
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K. S. Yee, IEEE Trans. Antennas Propag. AP-4, 302 (1966)A. Bossavit, Progress in Electromagnetic Research 32, 45 (2001)
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)
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)
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)
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)
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
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
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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)
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
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
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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
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]
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a=100nm
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).
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
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S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, Phys. Rev. Lett. 97, 017402 (2006)
Metal nanoparticle as a nanoantenna
60 nm Au spheres
Vacuum Wavelength [nm]
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B. J. Messinger, et al., Phys. Rev. B 24, 649 (1981)C. Bohren, Am. J. Phys. 51, 323 (1982)
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∫
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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)
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)
Engineering the decay rates
200
250
long axis
100
150
50
100
x100
short axis
500 600 700 800 900Vacuum Wavelength [nm]
0
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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)
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
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L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
3D antenna models
104
103
1020304050
120x38
2
10
102
900 950 1000 1050 1100Vacuum Wavelength [nm]
101
nb=1.7
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L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, Opt. Lett. 32, 1623 (2007)
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%
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L. Rogobete, et al., in preparationJ.R. Lakowicz, Anal. Biochem. 337, 171 (2005)
Acknowledgments
Single molecule and SNOM tip
Nanoantennae
FDTD
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