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 - mario.agio@phys.chem.ethz.ch

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

NMON 07.09.2007 2

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

NMON 07.09.2007 4

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

NMON 07.09.2007 7

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

NMON 07.09.2007 8

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

NMON 07.09.2007 9

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

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)

Applications

Single molecule and SNOM tip

Lifetime engineering with nanoantennae

NMON 07.09.2007 11

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

NMON 07.09.2007 12

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

NMON 07.09.2007 13

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

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

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)

NMON 07.09.2007 16

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

NMON 07.09.2007 17

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∫

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)

The issue of quenching

160

200

Cross section

80

120Cross sectionRadiativeNon-radiative

40x10

500 600 700 800 900Vacuum Wavelength [nm]

0

NMON 07.09.2007 20

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

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)

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)

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)

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)

Acknowledgments

Single molecule and SNOM tip

Nanoantennae

FDTD

NMON 07.09.2007 25