1.The emblematic example of the EOT -extraordinary optical transmission (EOT) -limitation of...

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-the importance of the quasi-CW-definition & properties-scaling law with the wavelength

4.Microscopic theory of sub- surfaces -definition of scattering coefficients for the quasi-CW-dual wave picture microscopic model

=0°

0.95 1 1.05 1.1 1.15

0

0.1

0.2

a=0.68 µm

a=0.94 µm

a=2.92 µm

Tra

nsm

itta

nce

/a

RCWASPP model

What is due to SPP in the EOT

Dual wave picture

x-m exp(ik0x)

=exp(ikSPx)

= 940 nm

P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

Dual wave picture

=exp(ikSPx)

= 940 nm

P. Lalanne et J.P. Hugonin, Nature Phys. 2, 556 (2006).

quasi-

Why calling it a quasi-CW?

Why calling it a quasi-CW?

Young’s slit experiment

laser

F L1 L2PBS S CCDλ/2

titanium

glass

Au

N. Kuzmin et al., Opt. Lett. 32, 445 (2007).

Young’s slit experiment

0 10 20 30 40

0 10 20 30 40

(°) in air

=850 nm

N. Kuzmin et al., Opt. Lett. 32, 445 (2007).S. Ravets et al., JOSA B 26, B28 (2009).

(°)

=0.6 µm

=1 µm

=3 µm

=10 µm

PC

S. Ravets et al., JOSA B 26, B28 (2009).

Computational resultsF

ar-

fiel

d in

ten

sity

(a.

u.)

SPP mainly

Quasi-CW only

Quasi-CW & SPP

Quasi-CW mainly

Hinc

Es, Hs= scattered fieldE actual fieldEs = jµ0Hs Hs = -j(r)Es –jE

1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy. When is it reliable?

Polarizability tensorpx=(xxEx,inc + xzEz,inc)x

pz=(zxEx,inc+zzEz,inc)z

Hinc


1/Hypothesis : the sub- indentation can be replaced by an effective dipole p=pxx+pyy.

2/The effective dipoles px and py are unknown. They probably depend on many parameters especially for sub- indentation that are not much smaller than (such as resonant grooves)

Hinc



2/The effective dipoles px and py are unknown.

3/ We solve Maxwell's equation for both dipole sourceE = jµ0HH = -j(r)E +(Es,xx+ Es,yy) (x,y)

The bad scenario

=800 nm, gold

x/ x/

H (

a.u

.)The actual scenario

The two dipole sources approximately generate the same field

Hinc



2/The effective dipoles px and py are unknown.

3/ The field scattered (in the vicinity of the surface for a given frequency) has always the same shape :

(x,y)= [SPEinc] SP(x,y) + [CWEinc] CW(x,y)

Cauchy theorem

kSP /k0 kSP /k0

m2

md2

d

0

i

d)xikexp()0z,x(H

H = HSP + HCW

)xikexp(k

kSP

dm

md20

2SP

HSP =

HCW= Integral over a single real variable

A single pole singularity : SPP contributionTwo Branch-cut singularities : m and d

Dominated by the branch-point singularity

Analytical expression for the quasi-CW

Maths have been initially developped for the transmission theory in wireless telegraphy with a Hertzian dipole radiating over the earth

I. Zenneck, Propagation of plane electromagnetic waves along a plane conducting surface and its bearing on the theory of transmission in wireless telegraphy, Ann. Phys (1907) 23, 846866.R.W.P. King and M.F. Brown, Lateral electromagnetic waves along plane boundaries: a summarizing approach, Proc. IEEE (1984) 72, 595611.R. E. Collin, Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies, IEEE Antennas Propag. Mag. (2004) 46, 6479.

0/H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays", Opt. Exp. 12, 3629-41 (2004).1/G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model", Nature Phys. 2, 262-267 (2006).2/PL and J.P. Hugonin, Nature Phys. 2, 556 (2006).3/B. Ung, Y.L. Sheng, "Optical surface waves over metallo-dielectric nanostructures", Opt. Express (2008) 16, 90739086. 4/Y. Ravel, Y.L. Sheng, "Rigorous formalism for the transient surface Plasmon polariton launched by subwavelength slit scattering", Opt. Express (2008) 16, 2190321913.5/W. Dai & C. Soukoulis, "Theoretical analysis of the surface wave along a metal-dielectric interface", PRB accepted for publication (private communication).6/L. Martin Moreno, F. Garcia-Vidal, SPP4 proceedings 2009.7/PL, J.P. Hugonin, H. Liu and B. Wang, "A microscopic view of the electromagnetic properties of sub- metallic surfaces", Surf. Sci. Rep. (review article under proof corrections, see ArXiv too)

Analytical expression for the quasi-CW

Maxwell's equationsE = jµ0HH = -j(r)E +(Es,xx+ Es,yy) (x,y)

Analytical solution Hz,CW

[m/mEs,x + nd/SEs,y] Ex,CW

Ey,CW

S is either d or m, whether the Dirac source is located in the dielectric material or in the metallic mediumUnder the Hypothesis that

•|m| >> d

•z < •x > /2

Hy

Ex

Ez

x/λ0 10 20 30

z

x

quasi-CW for gold at =940 nm

dominant

dominant

Intrinsic properties of quasi-CW

Hy,CW Hy,0

Ex,CW = F(x) Ex,0

Ez,CW Ez,0

• F(x) is a slowly-varying envelop • [Hy,0, Ex,0, Ez,0] is the normalized field associated to the limit case of the reflection of a plane-wave at grazing incidence

Hypothesis•z < •x > /2

z

x

Hy,CW

PL et al., Surf. Sci. Rep. (review article under production, 2009)

•linear z-dependence for z < •Main fields are almost null for z ( /2 m

•nearly an exp(ik0x) x-dependence for x >>

Hy,CW Hy,0

Ex,CW = F(x) Ex,0

Ez,CW Ez,0

Grazing plane-wave field

z

x

Hy,CW


102100

10-2

100

10-4

10-6

|F(x

)| (

a.u

.)

x/

1/x3

1/x

silver @ =1 µm

Closed-form expression for F(x)


Closed-form expression for F(x)

Highly accurate form for any xF(x) = exp(ik0x) W[2nSP-ndx/] (x/)3/2 with W(t) an Erflike function

Highly accurate for x < 10

F(x) = exp(ik0x) (x/)-m

m varies from 0.9 in the visible to 0.5 in the far IR

m=0.83 for silver @ 852 nm

Scaling law

(result for silver)

10-1

10-2

100

10-3

101 102100 101 102100

=0.633 µm =1 µm

=9 µm=3 µm

x/ x/

|H| (

a.u.

)|H

| (a.

u.)

|H| (

a.u.

)|H

| (a.

u.)

x/ x/

10-1

10-2

100

10-3

HSP

HCW

(x/)-1/2 (PC)

PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)

Scaling law

(result for silver)

10-1

10-2

100

10-3

101 102100 101 102100

=0.633 µm =1 µm

=9 µm=3 µm

x/ x/

|H| (

a.u.

)|H

| (a.

u.)

|H| (

a.u.

)|H

| (a.

u.)

x/ x/

10-1

10-2

100

10-3

HSP

HCW

(x/)-1/2 (PC)

PL and J.P. Hugonin, Nature Phys. 2, 556 (2006)

m2nd

3



3.The quasi-cylindrical wave-the importance of the quasi-CW-definition & properties-scaling law with the wavelength-experimental evidence


d

|S|2

|S0|2

|S/S

0|2

d (µm)

Slit-Groove experiment

G. Gay et al. Nature Phys. 2, 262 (2006)

d Fall off for d < 5

frequency = 1.05 k0

kSP=k0 [1-1/(2Ag)] 1.01k0

silver

=852 nm

promote an other model than SPP & quasi-CW (CDEW model)

L. AIGOUYESPCI, Paris

2 µm

2 µm

Near field validation

TM

slit slit

G. JULIE, V. MATHETIEF, Orsay

gold experiment

computation

=975 nm

E//E

Field recorded on the surface

=974 nm

gold

Ez = ASP sin(kSPx) + Ac [ik0-m/(x+d)] - Ac [ik0+m/(x-d)]

m0

d)(xx)exp(ik

m0

d)(xx)exp(-ik

standing SPP right-travelingcylindrical wave

left-travelingcylindrical wave

fitted parameterASP (real)Ac (complex)(m=0.5)

slit slit

L. Aigouy et al., PRL 98, 153902 (2007)

total field

cylindrical waveSPP

x

z

A direct observation?

If one controls the two beam intensity and phase (or the separation distance) so that there is no SPP generated on the right side, do I observe a pure quasi-CW on the right side of the doublet?



3.The quasi-cylindrical wave-the importance of the quasi-CW-definition & properties-scaling law with the wavelength-experimental evidence


SPPCW

Defining scattering coefficients for CWs

elastic scattering coefficients like the SPP transmission may be easily defined.

You may also define inelastic scattering coefficients with other modes like the radiated plane waves

You may use mode orthogonality and reciprocity relationships.

?

The SPP is a normal mode

CW-to-SPP cross-conversion

CW SPP

tc

CW

SPP

rc

Other scattering coefficients

CW SPP

CW = SP

CW = SP

CW

CW

SP

SP

X. Yang et al., Phys. Rev. Lett. 102, 153903 (2009)

Cross-conversion scattering coefficients

CW SPP

Ansatz : THE SCATTERED FIELDS ARE IDENTICAL.(if the two waves are normalized so that they have identical amplitudes at the slit)


Scaling law for rc and tc

-0.4

-0.2

0

0.2

0 5 10 5 100

-0.4

-0.2

0

0.2

100

10110

-4

10-2

10-1

(µm)

|m|1

Im(rc)

Re(rc)

Im(tc)

Re(tc)

tc

(µm) (µm)

|rc|2 & |tc|2

|tc|2

|rc|2

rc

Other scattering coefficients

CW SPP

CW = SP

CW = SP

CW

CW

SP

SP


Mixed SPP-CW model for the extraordinary optical

transmission

pure SPP model

mixed SPP-CW model

tA = t +2

(P-1+1) (+) rA =

2

(P-1+1) (+)

P = 1/(u-1 - 1) +n=1, HCW(na)

xk( )

SPP

CW

H. Liu et al. (submitted)

tA = t +2

u-1 (+)rA =

22

u-1 (+)

=0°

0.95 1 1.05 1.1 1.15

0

0.1

0.2

a=0.68 µm

a=0.94 µm

a=2.92 µm

Tra

nsm

itta

nce

/a

RCWASPP model

Pure SPP-model prediction of the EOT

H. Liu & P. Lalanne, Nature 452, 448 (2008).

=0°

0.95 1 1.05 1.1 1.15

0

0.1

0.2

0.3

a=0.68 µm

a=0.94 µm

a=2.92 µm

Tra

nsm

itta

nce

/a

RCWASPP modelSPP-CW model

Mixed SPP-CW model for the extraordinary optical

transmission

RCWA CW-SPP model

xk( )

SPP

CW

a=900 nm

SPP+quasi-CW coupled-mode equations

AnBn

An-1Bn-1

It is not necessary to be periodicIt is not necessary to deal with the same indentations

Conclusion

Two different microscopic waves, the SPP mode and the quasi-CW, are at the essence of the rich physics of metallic sub- surfaces

Their relative weights strongly vary with the metal permittivity

They echange their energy through a cross-conversion process, whose efficiency scales as |m|1

The local SPP elastic or inelastic scattering coefficients are essential to understand the optical properties, since they apply to both the SPP and the quasi-CW

gold @ =800 nm

Quasi-SWCylindrical SPP

Quasi-Spherical Wave

Cylindrical SPPy/

E (

nor

mal

)

SPP, quasi-CW, C-SPP, quasi-SP

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