1. Electromagnetic modes in nanophotonics
Philippe Lalanne
Institut d'Optique d’Aquitaine, Bordeaux – France
Photons and nanosystems
Complex nanostructures
Cold atoms, matter waves
Biophotonic
Optics & numerics (virtual reality)
Les deux dernières décennies ont vu des progrès considérables dans le domaine de la modélisation en nanophotonique, d’abord à cause de l’accroissement des capacités de calcul numérique, mais surtout parce que la nanophotonique vient de vivre une période très riche avec un foisonnement d’idées sans précédent autour des cristaux photoniques, des nanostructures plasmoniques et des métamatériaux.
La communauté a réellement appris à contrôler avec une grande précision l’interaction de la lumière à des échelles sub-micrométriques. Un bel exemple de ces progrès a été l’avènement de microcavités de grands facteurs de Q et de petits V. Et pourtant on est partie de loin.
Laboratoire Photonique, Numérique et Nanosciences (LP2N)

2. 20 years ago: the first PBG cavities
Q = 400
R = 0.96
However, it is worth mentioning that state of-the-art reflectivities for dielectric Bragg mirrors with up-to-date automated deposition techniques are still much what?
JS Foresi. et al. , Photonic-bandgap microcavities in optical waveguides, Nature 390, 143 (1997).

3. Today Q = 700,000 R = 0.99996 (?) Quantum optomechanics but also
-non-classical light sources
-nanolasers …
Aujourd’hui les microcavités à cristaux photoniques sont de petites cages de quelques µm3 de volume qui gardent la lumière pendant typiquement 1 ns.
L’enjeu n’est plus réellement d’augmenter encore le Q, il semble que l’on soit bloquer par les imperfections de fabrication et que l’on ne parvienne pas à faire des designs tolérants à ces défauts.
L’enjeu est plutôt d’utiliser ces microcavités pour étudier des effets nouveaux comme sur cet exemple d’optomécanique quantique (Kastler Brossel – LPN), ou alors de réaliser des sources brillantes de photons indiscernables par exemple, ou de nanolaseurs qui émettent des faisceaux directifs ; il s’agit non pas de réduire les pertes par diffraction hors du plan, mais bien au contraire de les contrôler pour que les seules pertes pertes hors du plan soient converties en photons effectifs (Travaux de Thalès).
Pour atteindre ce niveau de maitrise, il a fallu comprendre comment s’y prendre, trouver les paramètres physiques sur lesquels on peut jouer pour que le photon reste dans sa petite cage,
Les recherches actuelles dans ce domaine visent à refroidir le système mécanique pour l'amener dans son état quantique fondamental (mouvement de point zéro), ouvrant la voie à l'exploration du comportement d'un objet quantique de grandes dimensions et à la réalisation de nouveaux tests de mécanique quantique.
Kastler Brossel – LPN
J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, O. Painter , “Laser cooling of a nanomechanical oscillator into its quantum ground state” (arXiv: ) 

4. Gap-evanescent Bloch mode
500 nm
ridge mode Si
SiO2
Gap-evanescent Bloch mode

5. Gap-evanescent Bloch mode
500 nm
ridge mode
tapered
section
periodic
mirror Si
SiO2
Gap-evanescent Bloch mode
P. Velha et al., NJP 8, 204 (2006).

6. R=0.9996 ng=3.5 R=0.997 ng=13 R=0.993 ng=3.5 R=0.9993 ng=20
P. Velha et al., NJP 8, 204 (2006).
Y. Akahane et al., OE 13, 1202(2005)
nanoridge family
Defect-mode family
R=0.9996
ng=3.5
R=0.997
ng=13
Y. Akahane et al., Nature 425, 944 (2003)
M. Lermer et al, PRL 108, (2012)
tap.
T. Azano et al, OE. 14, 1996 (2006)
R=0.993
ng=3.5
R=0.9993
ng=20
micropillar
Heterostructure familly

7. Numerical space? PML with a complex fPML coefficient
Outre la connaissance nouvelle qu’il a fallu développer pour établir les bonnes recettes, il a été nécessaire de développer les outils numériques pour calculer des cavités à grands Q (même en prenant en compte la rugosité pour les meilleurs).
Voici un exemple de prouesses numériciennes, infructueux, mais que j’aime bien, que nous avons du développer dans mon groupe à palaiseau pour modéliser proprement une cavité L3.
Avec les progrès considérables qu’a enregistrés la modélisation numérique depuis une vingtaine d’années, il est vrai que l’on dispose d’outils extrêmement puissants et que couplés à des outils d’optimisation performant, il est possible d’obtenir des designs remarquablement pointus et de prédire des effets physiques inattendus. On pourrait être enclin à penser qu’aujourd’hui on peut tout ou presque tout modéliser, et qu’il suffit d’attraper les outils commerciaux de l’étagère et de les confier à un doctorant.
Dans la logique de l’exposé de Boris, je vais vous montrer sur deux exemples (la TEO et l’effet Purcell) que l’on ne peut pas se contenter d’utiliser des logiciels très numériques, mais que la compréhension et la conceptualisation des phénomènes physiques électromagnétiques passe parfois par des calculs numériques plus compliqués que ceux qui nécessite une modélisation directe du phénomène (c’est le cas de la TEO), ou par des calculs plus exigeants sur le plan des mathématiques.
Sans enlever aucunement les mérites des approches très numériques, je vais me poser en défenseur de l’optique électromagnétique qui s’appuie aussi sur la physique mathématique.
Il est vrai que si l’on dispose d’outils extrêmement puissants et que vous les couplés à des outils d’optimisation performant, des solutions remarquables parce que c’est plus facile de faire tourner un logiciel d’optimisation. Au lieu de se demander comment « ca marche », comment mettre en place les regles du lego, cela permet de se focaliser sur l’application, sur sa pertinence et d’envisager des problèmes réellement multiphysiques complexes.
Cela est indéniable, mais ce n’est qu’une facette du problème. Dans la logique de l’exposé de Boris, je vais me placer en défenseur de la trilogie numérique-mathématique-électromagnétisme et vous montrer
JJG: En ce qui concerne la situation de la modélisation, la situation actuelle est profondément différente de ce qu'elle était il y a une vingtaine d 'années. 
Des outils de simulation numérique sont mis à la disposition de tout un chacun. Beaucoup de choses peuvent êtres étudiées avec des outils commerciaux. On peut donc se poser la question de ce qui reste à faire en termes de développement de méthodes numériques.
e à prendre en compte la rugosité.
real coordinate transform
PML

8. outline
Deep into the mecanism of the extraordinary optical transmission
revisiting the Purcell factor: the case of metallic nano-resonator

9. The extraordinary optical transmission
l (nm)
ratio
transmission
Although the SPP Fano’s interpretation was well established, following the discovery, the role of plasmon in the EOT has been intensively debated.
To cope with this difficulty,
T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).

10. Surface plasmon assisted transmission?
Les plasmons sont générés efficacement dans le visible
C’est evidemment tentant d’attribuer l’effet au plasmon qui sont connues depuis les travaux de wood anomalie
The forced resonance explanation of Wood’s anomaly by U. Fano (1941)
A Hessel and A.A. Oliner, Appl. Opt. 4, 1275 (1965).
Polology by the French school in Marseille (Maystre, …) and US school (Peng, Tamir)
T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).

11. Main results from mode theory
Phenomenological polology
E. Popov et al., PRB 62, (2000).
Resonance-assisted tunneling
L. Martín-Moreno et al., PRL 86, 1114 (2001).
Spoof plasmon
J. Pendry et al., Science 305, 847 (2004). T
The Fano-type formula is very elegant as it well reproduce the spectral lineshape with o,nly 5 real parameters. It additionnally shows that the EOT is a resonance phenomenon. w
resonance
More insight has been provided by Martin-Moreno who showed that the resonance occurs at interfaces and that they boosts an evanescent tuneling.
resonance
"SPOOF" SPP
Pendry showed that the same resonant-assisted mechanism occurs at low frequencies, and introduced the concept of spoof plasmons. w
k//
The modes in these works are GLOBAL quantities attached to periodic ensembles; they give a good insight into the macroscopic mechanisms responsible for the transmission, but nothing is known about the individual plasmons that are launched inbetween the holes of the array.

12. Surface plasmon assisted transmission?
Les plasmons sont générés efficacement dans le visible
C’est evidemment tentant d’attribuer l’effet au plasmon qui sont connues depuis les travaux de wood anomalie
If one derives a model of the EOT where only SPP are assumed to carry the energy between adjacent hole chains and compares with fully-vectorial computations, then one should allow us to quantify what is really due to SPP in the EOT.

13. Microscopic SPP model r t b b a t
in-plane reflection-transmission of SPP
coupling of SPP to free-space r t b b a t
Pour distinguer ce qui est du au SPP de ce qui ne l'ai pas, on a considere un modele dans lequel le transport d'energie entre deux rangees de trous est uniquement dû aux SPP. On qualifie le modèle de "purement plasmonique".
(for periodic arrays)
H. Liu and P. Lalanne, Nature (London) 452, 448 (2008).

14. Actual SPP role in the EOT
RCWA
SPP model
a=0.68 µm
Normal incidence
0.2
Transmittance
0.1
l/a
0.95 1
1.05
1.1
1.15
H. Liu and PL, Nature (London) 452, 448 (2008).

15. q2  Transmission wavelength (nm) Normal incidence Microscopic model
0.1
q = 1 2a 1a
Microscopic model
q=2 3a
q2  Transmission
0.05 1a
q = 2
q=3
q = 4
We interpret by saying that only the SPP is involved in the EOT of the q=2, 3 … grids,
All transmission peaks for q=2, 3 … are very similar in magnitude, except the q=1 transmission peak
 Direct proof that a wave, different from the SPP, plays a role in the EOT
q = 6
750
800
850
900
wavelength (nm)
Measurement performed in Martin van Exter’s group (Leiden)

16. q2  Transmission wavelength (nm) Normal incidence
q = 1 (exp.)
q = 2 (exp.)
q = 4 (exp.)
q = 6 (exp.)
0.1
q = 1
Normal incidence
The 5 coefficients p1 (real), ab and r+t (complex) are fitted for q = 2,3 …7
q2  Transmission
0.05
q = 2
q = 4
la TEO est un exemple singulier en nanophotonique ou il y a eu un reel effort pour comprendre en profondeur le phenomene.
Le plus souvent on se contente d’une comprehension plus superficielle, on accepte le phenomene, on l’utilise et on avance, et au bout de quelques années son utilisation courante fait croire qu’on a compris.
All transmission peaks for q=2, 3 … are very similar in magnitude, except the q=1 transmission peak
 Direct proof that a wave, different from the SPP, plays a role in the EOT
q = 6
750
800
850
900
wavelength (nm)
F. van Beijnum et al., Nature 492, 411 (Dec. 2012)

17. Slow light injector w/c b Slow Fast Fast ng=4 injector Slow ng=1000
7.2
Slow 7
w/c
6.8
Fast
6.6
Once we have the tapers, we are able to design the speed bump. It is made of a waveguide working in the slow light regime, with a varying a number of periods, surrounded by the tapers and the fast waveguide.
If you shine the system from the left side for instance, the light will propagate as a high velocity, will be slow down in the taper, propagate slowly in the slow waveguide then speed up in the taper to finally go out in the fast waveguide with no backreflection at any bourdary.
To characterize the speed bump, we calculate the purcell factor, a quantity proportional to the DOS, of a source placed in the center of the structure.
0.2
0.3
0.4
0.5 b
Fast
ng=4
injector
Slow
ng=1000
Injector: 95% efficient and only two periods long?

18. outline
Deep into the mecanism of the extraordinary optical transmission
revisiting the Purcell factor: the case of metallic nano-resonator

19. Near-field excitation
Far-field excitation
Near-field excitation
Start with the bell
We faced two difficulties in this work. The first one is linked to the nature of QNMs, which are morally radiation modes belonging to a continuum and which are not square integrable.
The second is linked to material dispersion that seems to prevent the resonance modes to form a compete basis.
What is new in the work that we have is that for plasmonic and lossy materials, we solved that problem and have derived analytical expressions for alpha.
The purpose of my talk is to discuss the consequences for the treatment (classical or semi-classical) of the interaction of light with resonance and mainly plasmonic resoanance (ie resonance that involve lossy and dispersive materials).

20. Quasi-normal modes
Befor moving to the modal expansion, let us take a look to the modes themselves.

21. Near-field excitation
QNM expansion
Far-field excitation
Near-field excitation
Completness?

22. Near-field excitation
excitation coefficient a
Only hypothesis : material is reciprocal
Far-field excitation
Near-field excitation
Very easy!
Energy of a dispersive material? yes but only when absorption is small. No energy consideration in the derivation.
Derivation based on reciprocity arguments, see C. Sauvan, J.P. Hugonin, PL, Phys. Rev. Lett. 110, (2013) & Q. Bai et al., Opt. Express 21, (2013).

23. The normalization issue
Complex coordinate transform (PML)
X = a(1+im) x
Y = a (1+im) y
Z = a (1+im) z
Analytical continuation in the complex plane with PMLs
remove the divergence problem for suitable m’s by transforming the exponentially diverging field into an exponentially damped field   

24. The normalization issue
Complex coordinate transform (PML)
X = a(1+im) x
Y = a (1+im) y
Z = a (1+im) z
is an invariant under space coordinate transforms
is invariant too and can be calculated with any PML, by computing the integral in real space and in the PML.
We will show what is the meaning of the imaginary part on the Purcell formula
First (?) time the field in the PML is explicitly considered to evaluate a physical quantity.

25. Test of the invariance with the PML
Im(R)
|E|2 R0
Real space
PML
R = (1+mi)(Re(R)-R0)
R = Re(R)
Re(R) R0
Each PML defines an analytical continuation of V in the complex plane R’+iR" for R > R0
Silver nanosphere (radius 100 nm) in air
l = i

26. R0 = 0.11 µm
V = ( i) + ( i) =
Re(V)
R0 = 1 µm
Re(V)
V =
R0 = 2 µm
Maintenant que j’ai rapidement montré les efforts en mathématique appliqué qui sont requis pour normalisé le mode résonnant, je vais pouvoir revenir a l’effet Purcell a proprement parle c’est a dire à la modification du taux d’émission d’un émetteur couplé à une cavité.
Re(V)
V = 1 2
R (µm)

27. Classical Purcell formula
Classical Lorentzian shape
Only valid for large Q (error scales as 1/Q as Q) F
Note that if Q is infinite, then there is no leakage out of the cavity and then there is no divergence problem.
Therefore, the classical expression appears as an approximate case valid for large Q’s.
Non exponential decay by Michiel de Dood

28. modal-expansion of the LDOSw
For periodic boundary and lossless systems, the system is conservative and the operator equivalent of the hamiltonien is hermitian and semidefinite under the inner product <E,E’>=integral(E*.epsilon E)
R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

29. modal-expansion of the LDOSw
- valid only when radiative and Ohmic losses are small
- the theory of lossy systems is not completed. The modes are not eigenstates but poles of the Green’s function of the system and they are called quasi-normal modes. Because they are poles, we might intuitively think that they contribute a Lorentzian peak to the LDOS, but actually they are not, strangely. Even more strangely, as we will see, the LDOS cannot be decomposed as a sum of positive quantities over the quasi-normal modes.
R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

30. modal-expansion of the LDOSw
- valid only when radiative and Ohmic losses are small
- the theory of lossy systems is not completed. The modes are not eigenstates but poles of the Green’s function of the system and they are called quasi-normal modes. Because they are poles, we might intuitively think that they contribute a Lorentzian peak to the LDOS, but actually they are not, strangely. Even more strangely, as we will see, the LDOS cannot be decomposed as a sum of positive quantities over the quasi-normal modes.
R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

31. Revisiting the Purcell formula
a detuning between the quantum emitter and cavity frequencies does not necessarily result in a Lorentzian lineshape response, as it is presently accepted by everyone. F
Derivation based on reciprocity arguments, see C. Sauvan et al., PRL 110, (2013) & Q. Bai et al., Opt. Express 21, (2013).

32. Non-Lorentzian response with metallic resonance
Circle: Green-tensor calculation (decay in all modes)
Blue line: revised Purcell formula (with a single mode)
Sauvan et al., Phys. Rev. Lett. 110, (2013).

33. Multi-resonance case
the contribution of a quasi-normal mode to the total power radiated by a source may be detrimental (it may reduce the decay rate), even when the frequencies of the source and the mode are matched.
20 nm
80 nm
45 nm
L’image que donne ce formalisme est quand meme singulièrement differente de l’interpretation classique qui attribue à chaque mode une reponse lorentzienne et qui fait l’hypothese que plus il y a de modes, plus l’effet purcell est grand.
145 nm Au
Sauvan et al., Phys. Rev. Lett. 110, (2013).
85 nm

34. tomorrow Plasmon induced hot carriers Problème multiphysique?
-photodetectors with spectral responses circumventing band gap limitations
-chemical catalysis close to nanostructured metal surfaces
Demain sera fait de quoi?
Je crois que les problèmes d’envergure qui donneront toutes leurs lettres de noblesse à l’optique électromagnétique seront des problèmes problèmes à l’interfacage de différentes discipline de la physique, possiblement avec des retombées dans des domaines autres que la physiques. L’éloboration des théories à l’interface et des methodes numériques associés seront sans doute fantastiquement excitantes.
Théorie de la fonctionnelle de la densité
en essayant de lui donner toutes ses lettres de noblesse.Je ne sais
These hot carriers are capable of inducing chemical reactions in molecules in the vicinity of the surfaces of plasmonic nanostructures, which would otherwise be energetically very demanding.
Plasmon induced hot carriers also provide an efficient mechanism to convert light into electric current that can be used for developing alternative solar-energy harvesting devices,
or to design efficient photodetectors with spectral responses circumventing band gap limitations.
Although direct excitation of hot carriers on metal surfaces is possible and has since long been exploited in the field of surface femtochemistry, the utilization of surface plasmon decay to increase the efficiency of the hot carrier generation process is relatively novel.
M.W. Knight et al., Science 332,702 (2011).
Problème multiphysique?
Avec des retombées importantes (?) dans d’autres domaines des sciences (chimie? Bio?)

35. Remerciements Haitao Liu Christophe Sauvan Jean Paul Hugonin
(Institut d’Optique)
Haitao Liu
(Nankai Univ)
Jean Paul Hugonin
(Institut d’Optique)

36. Hybrid techniques a-FMM only a-FMM + FE FE FMM
J. P. Hugonin, M. Besbes and PL, Opt. Lett. 33, 1590 (2008). 36

37. Left-handed metamaterials
(l = 1.5 µm)
(l = 60 mm)
D. Smith et al., "Composite medium with simultaneously negative permeability and permittivity", PRL 84, 4184 (2000).
Fast
ng=4
Refractive index
l (µm)
J. Valentine et al., Nature (London) 455, 376 (2008)

38. Slow light injector w/c b Slow Fast Fast ng=4 injector Slow ng=1000
7.2
Slow 7
w/c
6.8
Fast
6.6
Once we have the tapers, we are able to design the speed bump. It is made of a waveguide working in the slow light regime, with a varying a number of periods, surrounded by the tapers and the fast waveguide.
If you shine the system from the left side for instance, the light will propagate as a high velocity, will be slow down in the taper, propagate slowly in the slow waveguide then speed up in the taper to finally go out in the fast waveguide with no backreflection at any bourdary.
To characterize the speed bump, we calculate the purcell factor, a quantity proportional to the DOS, of a source placed in the center of the structure.
0.2
0.3
0.4
0.5 b
Fast
ng=4
injector
Slow
ng=1000
Injector: 95% efficient and only two periods long?

39. Driving external field (wL)
applications of the formalism
Application to sensing
J. Yang et al. (submitted) p2 p1
Spatial coherence in complex systems
Phys. Rev. A. 89, (2014).
Driving external field (wL)
Quantum plasmonic
J. Yang et al. (submitted)

40. Application to sensing5 4 3
Cross Section 2 1
700
750
800
850
900
950
Wavelength (nm)
J. Yang et al., (in preparation)

41. tomorrow Plasmon induced hot carriers Nonlocal effects
-photodetectors with spectral responses circumventing band gap limitations
-chemical catalysis close to metal surfaces
Nonlocal effects
Théorie quantique de la fonctionnelle de la densité
Modèle hydrodynamique
Modèle de drude
D=10 nm
Pour conclure je me suis volontairement posé en défenseur de l’optique électromagnétique, avec un grand O et un grand E.
Comme je l’ai dit nous avons vécu une belle décade, et les plus anciens diraient probablement, un beau demi-siècle.
Demain sera fait de quoi?
Je crois que le grand O et grand E va devenir encore plus grand en s’attaquant à des problèmes à l’interfacage de differentes discipline.
Théorie de la fonctionnelle de la densité
en essayant de lui donner toutes ses lettres de noblesse.Je ne sais
These hot carriers are capable of inducing chemical reactions in molecules in the vicinity of the surfaces of plasmonic nanostructures, which would otherwise be energetically very demanding.
Plasmon induced hot carriers also provide an efficient mechanism to convert light into electric current that can be used for developing alternative solar-energy harvesting devices,
or to design efficient photodetectors with spectral responses circumventing band gap limitations.
Although direct excitation of hot carriers on metal surfaces is possible and has since long been exploited in the field of surface femtochemistry, the utilization of surface plasmon decay to increase the efficiency of the hot carrier generation process is relatively novel.
M.W. Knight et al., Science 332,702 (2011).
Phys. Rev. Lett. 110, (2013)
Problème multiphysique?
Avec des retombées importantes (?) dans d’autres domaines des sciences (chimie? Bio?)

43. The beginning…
Q = 400
R = 0.96
However, it is worth mentioning that state of-the-art reflectivities for dielectric Bragg mirrors with up-to-date automated deposition techniques are still much what?
JS Foresi. et al. , Photonic-bandgap microcavities in optical waveguides, Nature 390, 143 (1997).

44. Today
Q = 700,000
R = (?)
Optomécanique quantique
Les recherches actuelles dans ce domaine visent à refroidir le système mécanique pour l'amener dans son état quantique fondamental (mouvement de point zéro), ouvrant la voie à l'exploration du comportement d'un objet quantique de grandes dimensions et à la réalisation de nouveaux tests de mécanique quantique.
J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groeblacher, M. Aspelmeyer, O. Painter , “Laser cooling of a nanomechanical oscillator into its quantum ground state” (arXiv: ) 

45. REAL coordinate transform
PML with a complex fPML coefficient
PML P M L P M L
real coordinate transform
PML
Non-regular sampling = modification of the expansion basis

46. Hybrid techniques a-FMM only a-FMM + FE FE FMM
J. P. Hugonin, M. Besbes and PL, Opt. Lett. 33, 1590 (2008). 46

47. a/l Re(kz) p/a gap Im(k) air semi-conductor PhC Bloch mode (CB)
Im(k)
PhC Bloch mode (VB)
Re(kz)
p/a

48. conduction band: the field in inside the low-index material (the holes)
PhC Bloch mode
gap
valence band: the field in inside the high-index material
Re(kz)
p/a

49. Evanescent (gap) mode engineering
a/l
radiation Bloch modes
PhC Bloch mode
gap Bloch mode
gap Bloch mode
radiation Bloch modes
gap
Im(k)
Re(kz)
p/a

50. ridge mode
gap-evanescent Bloch mode

51. Si SiO2 tapered periodic section mirror 500 nm
ridge mode
tapered
section
periodic
mirror Si
SiO2
Gap-evanescent Bloch mode
P. Velha et al., NJP 8, 204 (2006).

52. Si SiO2 periodic tapered tapered mirror section periodic section
500 nm
periodic
mirror
tapered
section
tapered
section
periodic
mirror Si
SiO2
P. Velha et al., NJP 8, 204 (2006).

53. Computational results
Periodic
mirror
tapered
mirror
400
Loss VB
l (µm)

54. Heterostructure familly
Y. Akahane et al., Nature 425, 944 (2003)
P. Velha et al., NJP 8, 204 (2006).
nanoridge family
Defect-mode family
R=0.9996
ng=4
R=0.997
ng=13
tap.
R=0.993
ng=4
R=0.9993
ng=20
micropillar family
Heterostructure familly
T. Azano et al, OE. 14, 1996 (2006)
M. Lermer et al, PRL 108, (2012)
P. Lalanne et al., Laser Photonics Rev. 2, (2008).

56. Microscopic SPP model r t b b a t
in-plane reflection-transmission of SPP
coupling of SPP to free-space r t b b a t
Pour distinguer ce qui est du au SPP de ce qui ne l'ai pas, on a considere un modele dans lequel le transport d'energie entre deux rangees de trous est uniquement dû aux SPP. On qualifie le modèle de "purement plasmonique".
H. Liu and P. Lalanne, Nature (London) 452, 448 (2008).

57. SPP-CW coupled-mode equations
Coupling only with the nearest neighbors
Tight-binding approach
Numerical solution consists in solving a linear system with N unknowns (N is the number of hole rows)
If periodic, then it is analytical
(for periodic arrays)
H. Liu and P. Lalanne, Nature (London) 452, 448 (2008).

58. Dual wave picture SPP x/l x Quasi- cylindrical wave x/l
Actually we know that SPPThat we have interpreted by saying that the field launched on the surface is a superposition of two waves, the SPP mode, and another field which is not a mode and which is decaying much faster.
x/l

59. Influence of the metal conductivity
RCWA
SPP model
a=0.68 µm
Normal incidence
a=1 µm
0.2
Transmittance
0.1
a=3 µm
Increase again the period, move to the far infrared, the EOT remains, but the SPP model completely fails.
TRANSITION : It is even worse for microwaves
l/a
0.95 1
1.05
1.1
1.15
H. Liu & PL, Nature 452, 448 (2008).

60. Perfect metal Transmittance l/a a=
0.2
0.4
0.6
0.8 1
RCWA
SPP model
Transmittance
l/a
Increase again the period, move to the far infrared, the EOT remains, but the SPP model completely fails.
I am back to the comment by Fano explaining that the SPP is completely expelled from the metal and becomes a regular plane wave in vacuum grazing the surface of the metal
0.999 1
1.001
1.002
H. Liu & PL, J. Opt. Soc. Am. A 27, 2542 (2010).

61. the contribution of a quasi-normal mode to the total power radiated by a source may be detrimental (it may reduce the decay rate), even when the frequencies of the source and the mode are matched.
Mode 1
Mode 2
The agreement between the red curve and the exact calculation is remarkable, especially if one consider that the total SE rate is evaluated only with two modes.
Personnaly I consider this result as a strong support to our formalism that evidence that V is complex and that the frequency response of the coupling is in general not Lorentzian.

62. Near-field excitation
excitation coefficient a
Far-field excitation
Near-field excitation
Very easy!
Energy of a dispersive material? yes but only when absorption is small. No energy consideration in the derivation.
Derivation based on reciprocity arguments, see C. Sauvan, J.P. Hugonin, PL, Phys. Rev. Lett. 110, (2013) & Q. Bai et al., Opt. Express 21, (2013).

63. Analytical treatment of quantum plasmonic systems
Driving external field (wL)
(µ,wA)
We faced two difficulties in this work. The first one is linked to the nature of QNMs.
The second is linked to material dispersion that prevents orthogonality.
What is new in the work that we have is that for plasmonic and lossy materials, we solved that problem and have derived analytical expressions for alpha.
The purpose of my talk is to discuss the consequences for the treatment (classical or semi-classical) of the interaction of light with resonance and mainly plasmonic resoanance (ie resonance that involve lossy and dispersive materials).
Fano coefficient (complex number)
J. Yang et al., (submitted 2014)

64. En appui sur la physique mathématique et la puissance accrue des calculateurs, l’optique électromagnétique a pris son essor en même temps que les nano-technologies, et a ainsi favorisé la conception d’objets photoniques aux dimensions sub-longueurs d’onde. Cette discipline a connu des progrès et un impact considérables au cours des deux dernières décades: compression temporelle et confinement spatial, super-résolution, exaltation géante et lumière lente, indices négatifs, cristaux photoniques et méta-matériaux, fibres micro-structurées, plasmonique et nano-antennes, laser aléatoire, imagerie en milieu désordonné, polarisation de speckle… Les applications
couvrent de nombreux secteurs : télécommunications, éclairage, spatial, santé, énergie et environnement, défense…
Claude Amra, Marseille

65. Left-handed materials at l = 60 mm
The composite LHM employed by Smith et al. The medium consists of pairs of split resonators, created lithographically on a circuit board, and an array of metallic posts.
When the incident wave has a magnetic field parallel to the ring axis, the magnetic resonance can be excited.
SRR only E H k
opaque
n imaginary
(metal)
transparent
n > 0
a = 8 mm
(l = 60 mm) e
opaque
n imaginary
(magnetic metal)
transparent
n < 0
D. Smith et al., "Composite medium with simultaneously negative permeability and permittivity", PRL 84, 4184 (2000).

66. Left-handed materials at l = 60 mm
The composite LHM employed by Smith et al. The medium consists of pairs of split resonators, created lithographically on a circuit board, and an array of metallic posts.
When the incident wave has a magnetic field parallel to the ring axis, the magnetic resonance can be excited.
SRR +rods E H k
opaque
n imaginary
(metal)
transparent
n > 0
a = 8 mm
(l = 60 mm) e
opaque
n imaginary
(magnetic metal)
transparent
n < 0
D. Smith et al., "Composite medium with simultaneously negative permeability and permittivity", PRL 84, 4184 (2000).

67. Optical negative refraction
p = 860 nm ~ l/2
Ag/ MgF2
from measurements n1sin(i1)=n2sin(i2)
Refractive index
from calculation of the fundamental
Bloch mode
l (µm)
J. Valentine et al., Nature (London) 455, 376 (2008)

68. Tracking light-flow: « fluid dynamics » hole evanescent mode TE10
H-symmetric gap SPP
silver
dielectric (n=1.39)

69. Tracking light-flow: « fluid dynamics »
Hole-mode scattering
Gap-mode scattering ρ
TE01 α α α
tsp
rsp α
gap-SPP
gap-SPP τ
(resonant term)

70. Tracking light-flow: « fluid dynamics »
horizontal gap plasmon
vertical evanescent mode
Fishnet (microscopic model)
fundamental Bloch mode
J. Yang et al., Phys. Rev. Lett. 107, (2011).

71. Quantitative interpretation n<0
(resonant term)
tSP
rSP
horizontal gap plasmon
vertical evanescent mode
Fishnet (microscopic model)
fundamental Bloch mode
Quantitative picture of the magnetic resonance !!
The horizontal resonance is delocalised over 4 periods.
Q = 35
g = 0
J. Yang et al., Phys. Rev. Lett. 107, (2011).

72. Quantitative interpretation n<0 transparency threshold
(resonant term)
n – ig
dye
tSP
However, at the transparency threshold, the transversal SPP resonance becomes delocalized over 25 unit cells: fishnets with gain are not 3D metamaterials but rather as complex 1D layered systems with negative neff.
rSP
S. Xiao et al., Nature (London) 466, 735 (2010)
transparency threshold
g = 0.02
(Q = 800)
The scattering coefficients only weakly depend on gain
g = 0.01
(Q = 60)
g = 0
(Q = 35)

73. 3/ NIM: modes with negative effective indices
2/ EOT: surface plasmon modes
4/ Antenna: quasi-normal modes
1/ PhC : engineering evanescent Bloch modes

74. Negative index materials
plasmonic devices
Antenna-nanogap
Photonic crystal devices