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 formalism 1) make use of the completeness theorem for the normal modes of waveguides H y = E z = 2) Use orthogonality of normal modes PL, J.P. Hugonin and J.C. Rodier, PRL 95, (2005) x Tableau: *Preciser la decomposition modale avec a(x) qui inclut le exp(ikx). *Dire qu'on est dans la convention Hy,-Ez pour le backward-propagating mode de Hy,Ez *dire que l'orthogonalite n'est pas definie au sens de Poynting *dire que c'est les champs transverses Hy et Ez qui sont concernes. Tableau: *Preciser la decomposition modale avec a(x) qui inclut le exp(ikx). *Dire qu'on est dans la convention Hy,-Ez pour le backward-propagating mode de Hy,Ez *dire que l'orthogonalite n'est pas definie au sens de Poynting *dire que c'est les champs transverses Hy et Ez qui sont concernes.

General theoretical formalism x exp[-Im(k sp x)]

x x/ax/a e SP 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 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

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 Important case for plasmonics analytical expressions Important case for plasmonics analytical expressions

SPP generation n2n2 n1n1

 + (x)  - (x) n2n2 n1n1 SPP generation

 + (x)  - (x) n2n2 n1n1 SPP generation

n2n2 n1n1  + (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). Valid only at x =  w/2 only!

total SP excitation probability results obtained for gold Total SP excitation efficiency solid curves (model) marks (calculation) 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, (2008) & S. Ravets et al., "Surface plasmons in the Young slit doublet experiment", JOSA B 26, B28 (special issue plasmonics 2009). n2n2 n1n1  + (x)  - (x) Analytical model 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

n2n2 n1n1  + (x)  - (x) -Immersing the sample in a dielectric enhances the SP excitation (  n 2 /n 1 ) -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 describe material properties Analytical model PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

 + (x)  - (x)  + (x)  - (x)

Grooves SS S =  + [t 0  exp(2ik 0 n eff h)] / [1-r 0 exp(2ik 0 n eff h)] PL, J.P. Hugonin and J.C. Rodier, JOSAA 23, 1608 (2006).

groove SS S =  + [t 0  exp(2ik 0 n eff h)] / [1-r 0 exp(2ik 0 n eff h)] h/ S Can be much larger than the geometric aperture! =0.8 µm =1.5 µm gold

S =  + [t 0  exp(2ik 0 n eff h)] / [1-r 0 exp(2ik 0 n eff h)] SS r r = r  +  2 exp(2ik 0 n eff h)] / [1-r 0 exp(2ik 0 n eff h)] Grooves Mention the alpha squared dependence. Mode-to-mode reciprocity has been used for that (dessine au tableau). Mention that is is a weak process in general, since alpha is always smaller than 1. Mention the alpha squared dependence. Mode-to-mode reciprocity has been used for that (dessine au tableau). Mention that is is a weak process in general, since alpha is always smaller than 1. 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G If  =  ', the scattered field is unchanged : E'(r')=E(r'/G) ' H inc H' inc = H inc r' r'/G '' - All dimensions are scaled by a factor G - The incident field is unchanged H inc ( )=H inc ( ') The difficulty comes from the dispersion :  '    '/  = G 2 (for Drude metals)

A perturbative analysis fails Unperturbed system  E' = j  µ 0 H'  H' = -j  (r)E' Perturbed system  E = j  µ 0 H  H = -j  (r)E -j  E E s =E-E' & H s =H-H'  E s = j  µ 0 H s  H s = -j  (r)E s –j  E Indentation acts like a volume current source : J= -j  E  (E' H') (E H)

c1c1 c1c1 One may find how c 1 or c  1 scale.  E   (r-r 0 ) A perturbative analysis fails

"Easy" task with the reciprocity c1c1 c1c1 Source-mode reciprocity c 1 =  E SP (  1) (r 0 )  J c  1 =  E SP (1) (r 0 )  J provided that the SPP pseudo-Poynting flux is normalized to 1 ½  dz [E SP (1)  H SP (1) )]  x = 1 E SP = N 1/2 exp(ik SP x)exp(i  SP z), with N  |  | 1/2 /(4  0 ) J   (r-r 0 )

c1c1 c1c1 One may find how c 1 or c  1 scale. Then, one may apply the first-order Born approximation (E = E'  E inc /2). Since E inc ~ 1, one obtains that dielectric and metallic indentations scale differently.  E   (r-r 0 ) A perturbative analysis fails What is wrong with that. Simply that the E- field on the particle is not the incident field. There is a local field effect which is very strong for large  's. This is difficuly in general, and a solution can be provided only case by case. What is wrong with that. Simply that the E- field on the particle is not the incident field. There is a local field effect which is very strong for large  's. This is difficuly in general, and a solution can be provided only case by case.   ~ 1 for dielectric ridges  ~  for dielectric grooves  ~  for metallic ridges

(results obtained for gold) H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue) Numerical verification  H SP |  -1/2 (µm)

(results obtained for gold) Numerical verification  H SP |  -1/2 (µm) H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

(results obtained for gold) Numerical verification  H SP |  -1/2 (µm) H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

(results obtained for gold) Numerical verification (µm)  H SP |  -1/2 H. Liu et al., IEEE JSTQE 14, 1522 (2009 special issue)

Local field corrections: small sphere (R<< ) hh bb incident wave E 0 E = E 0 E = E 0 for small  only ! 3b3b  + 3  b E J.D. Jackson, Classical Electrodynamics  h -  b Just plug  in perturbation formulas fails for large .

GG ' H inc 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 = H inc

GG ' H inc 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 = H inc

GG ' H inc 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 = H inc

GG ' H inc 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 = H inc

GG ' H inc '' The initial launching of the SPP changes : H SP  |  -1/2 H' inc = H inc    ') 1/2