1.The emblematic example of the EOT-extraordinary optical transmission (EOT)-limitation of classical "macroscopic" grating theories-a microscopic pure-SPP model of the EOT

2.SPP generation by 1D sub- indentation-rigorous calculation (orthogonality relationship)-the important example of slit-scaling law with the wavelength

3.The quasi-cylindrical wave-definition & properties-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs-definition of scattering coefficients for the quasi-CW

How to know how much SPP is generated?

General theoretical formalismGeneral theoretical formalism

1) make use of the completeness theorem for the normal modes of waveguides

Hy= 

Ez= 

2) Use orthogonality of normal modes

(z)(x)Ha(z)H(x)α(x)α (rad)σσ σSP

(z)(x)Ea(z)E(x)α(x)α (rad)σσ σSP

(x)α(x)α2(z)Ez)(x,Hdz SPy

(x)α(x)α2(z)Hz)(x,Edz SPz

PL, J.P. Hugonin and J.C. Rodier, PRL 95, 263902 (2005)

x

General theoretical formalismGeneral theoretical formalism

xexp[-Im(kspx)]

x

x/a

SPe

SPee S

P

General theoretical formalism applied to

gratings

General theoretical formalism applied to grooves

ensembles

• 11-groove optimized SPP coupler with 70% efficiency.• The incident gaussian beam has been removed.

General theoretical formalism

0

0.1

0.2

0.3

0.4

• 11-groove optimized SPP coupler with 70% efficiency.• The incident gaussian beam has been removed.

General theoretical formalism

• 11-groove optimized SPP coupler with 70% efficiency.• The incident gaussian beam has been removed.

0

0.1

0.2

0.3

0.4

1.Why a microscopic analysis?-extraodinary optical transmission (EOT)-limitation of classical "macroscopic" grating theories-a microscopic pure-SPP model of the EOT

2.SPP generation by sub- indentation-rigorous calculation (orthogonality relationship)-the important slit example-scaling law with the wavelength

3.The quasi-cylindrical wave-definition & properties-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs-definition of scattering coefficients for the quasi-CW

SPP generation

n2

n1

+(x) -(x)

n2

n1

SPP generation

+(x) -(x)

n2

n1

SPP generation

n2

n1

+(x) -(x)

1) assumption : the near-field distribution in the immediate vicinity of the slit is weakly dependent on the dielectric properties

2) Calculate this field for the PC case

3) Use orthogonality of normal modes

Analytical model

PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

012

1

1/2

221

2

Iw'nn1

Iw

nεε

nn

π4αα

(x)α(x)α2(z)Ez)(x,Hdz SPy

(x)α(x)α2(z)Hz)(x,Edz SPz

Valid only at x = w/2 only!

tota

l SP

exc

itat

ion

pro

bab

ility

results obtained for gold

Tot

al S

P e

xcit

atio

n e

ffic

ien

cy

solid curves (model)

marks (calculation)

Question: how is it possible that with a perfect metallic case, ones gets accurate results?

Question: how is it possible that with a perfect metallic case, ones gets accurate results?

describe geometrical properties-the SPP excitation peaks at a value w=0.23 and for visible frequency, ||2 can reach 0.5, which means that of the power coupled out of the slit half goes into heat

EXP. VERIFICATION : H.W. Kihm et al., "Control of surface plasmon efficiency by slit-width tuning", APL 92, 051115 (2008) & S. Ravets et al., "Surface plasmons in the Young slit doublet experiment", JOSA B 26, B28 (special issue plasmonics 2009).

n2

n1

+(x) -(x)

Analytical model1) assumption : the near-field distribution

in the immediate vicinity of the slit is weakly dependent on the dielectric properties

2) Calculate this field for the PC case

3) Use orthogonality of normal modes

012

1

1/2

221

2

Iw'nn1

Iw

nεε

nn

π4αα

n2

n1

+(x) -(x)

-Immersing the sample in a dielectric enhances the SP excitation ( n2/n1)

-The SP excitation probability ||2 scales as ||-1/2

1) assumption : the near-field distribution in the immediate vicinity of the slit is weakly dependent on the dielectric properties

2) Calculate this field for the PC case

3) Use orthogonality of normal modes

012

1

1/2

221

2

Iw'nn1

Iw

nεε

nn

π4αα

describe material properties

Analytical model

PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

+(x) -(x)

+(x) -(x)

Grooves

S S

S =  + [t0 exp(2ik0neffh)] / [1-r0 exp(2ik0neffh)]

PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

groove

S S

S =  + [t0 exp(2ik0neffh)] / [1-r0 exp(2ik0neffh)]

0 0.2 0.4 0.6 0.80

1

2

3

h/

S

Can be much larger than the geometric aperture!

=0.8 µm

=1.5 µm

gold

S =  + [t0 exp(2ik0neffh)] / [1-r0 exp(2ik0neffh)]

S S

r

r = r + 2 exp(2ik0neffh)] / [1-r0 exp(2ik0neffh)]

Grooves

PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

1.Why a microscopic analysis?-extraodinary optical transmission (EOT)-limitation of classical "macroscopic" grating theories-a microscopic pure-SPP model of the EOT

2.SPP generation by sub- indentation-rigorous calculation (orthogonality relationship)-the important slit example-scaling law with the wavelength

3.The quasi-cylindrical wave-definition & properties-scaling law with the wavelength

4.Multiple Scattering of SPPs & quasi-CWs-definition of scattering coefficients for the quasi-CW

G

If =', the scattered field is unchanged : E'(r')=E(r'/G)

'

Hinc

H'inc= Hinc

r'

r'/G

'

- All dimensions are scaled by a factor G- The incident field is unchanged Hinc()=Hinc(')

The difficulty comes from the dispersion : ' '/ = G2 (for Drude metals)

A perturbative analysis fails

Unperturbed system

E' = jµ0H' H' = -j(r)E'

Perturbed system

E = jµ0H H = -j(r)E -jE

Es=E-E' & Hs=H-H'Es = jµ0Hs Hs = -j(r)Es –jE

Indentation acts like a volume current source : J= -jE

(E' H')

(E H)

c1c1

One may find how c1 or c1 scale.

E(r-r0)

A perturbative analysis fails

"Easy" task with the reciprocity

c1c1

Source-mode reciprocityc1 = ESP

(1)(r0)J

c1 = ESP(1)(r0)J

provided that the SPP pseudo-Poynting flux is normalized to 1

½ dz [ESP(1) HSP

(1))]x = 1

ESP = N1/2 exp(ikSPx)exp(iSPz), with N||1/2/(40)

J(r-r0)

c1c1

One may find how c1 or c1 scale.Then, one may apply the first-order Born approximation (E = E'Einc

/2).Since Einc ~ 1, one obtains that dielectric and metallic indentations scale differently.

E(r-r0)

A perturbative analysis fails

~ 1 for dielectric ridges

~ for dielectric grooves

~ for metallic ridges

100

101

10-1

100

(results obtained for gold)H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

Numerical verificationNumerical verification

HS

P|

-1/2

(µm)

100

101

10-1

100

(results obtained for gold)

Numerical verificationNumerical verification

HS

P|

-1/2

(µm)

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

100

101

10-1

100

(results obtained for gold)

Numerical verificationNumerical verification

HS

P|

-1/2

(µm)

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

100

101

10-1

100

(results obtained for gold)

Numerical verificationNumerical verification

(µm)

HS

P|

-1/2

H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

Local field corrections: small sphere (R<<)

h

b

incident wave E0

E = E0 E = E0 for small only !3b

+3b

++

+

++

--

-

--

E

J.D. Jackson, Classical Electrodynamicsh-b

Just plug in perturbation formulas fails for large .

G

'

Hinc

assumptions : the field distribution in the plane in the immediate vicinity of the slit becomes independent of the metallic permittivity

(as ||). (the particle dimensions are larger than the skin depth)

'

H'inc= Hinc

G

'

Hinc

assumptions : the field distribution in the plane in the immediate vicinity of the slit becomes independent of the metallic permittivity

(as ||). (the particle dimensions are larger than the skin depth)

'

H'inc= Hinc

G

'

Hinc

assumptions : the field distribution in the plane in the immediate vicinity of the slit becomes independent of the metallic permittivity

(as ||). (the particle dimensions are larger than the skin depth)

'

H'inc= Hinc

G

'

Hinc

assumptions : the field distribution in the plane in the immediate vicinity of the slit becomes independent of the metallic permittivity

(as ||). (the particle dimensions are larger than the skin depth)

'

H'inc= Hinc

G

'

Hinc

'

The initial launching of the SPP changes : HSP |-1/2

H'inc= Hinc

')1/2