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Electromagnetic modes in nanophotonics Philippe Lalanne Institut d'Optique d’Aquitaine, Bordeaux – France Laboratoire Photonique, Numérique et Nanosciences (LP2N) Photons and nanosystems Complex nanostructures Cold atoms, matter waves Biophotonic Optics & numerics (virtual reality)

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JS Foresi. et al., Photonic-bandgap microcavities in optical waveguides, Nature 390, 143 (1997). Q = 400 R = years ago: the first PBG cavities

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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: ) Today Q = 700,000 R = (?) Quantum optomechanics but also -non-classical light sources -nanolasers …

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ridge mode Gap-evanescent Bloch mode 500 nm SiO 2 Si

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500 nm SiO 2 tapered section Si periodic mirror P. Velha et al., NJP 8, 204 (2006). Gap-evanescent Bloch mode ridge mode

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

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PML PMLPML PMLPML Numerical space? PML with a complex f PML coefficient real coordinate transform

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

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The extraordinary optical transmission T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998). (nm) ratio transmission

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ee ee ee ee ee ee ee ee ee ee Surface plasmon assisted transmission? T. W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Nature 391, 667 (1998).

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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). 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. 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. T "SPOOF" SPP k // resonance More insight has been provided by Martin-Moreno who showed that the resonance occurs at interfaces and that they boosts an evanescent tuneling. Pendry showed that the same resonant-assisted mechanism occurs at low frequencies, and introduced the concept of spoof plasmons.

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Surface plasmon assisted transmission? 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.

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Microscopic SPP model t in-plane reflection- transmission of SPP coupling of SPP to free-space H. Liu and P. Lalanne, Nature (London) 452, 448 (2008). (for periodic arrays)

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Actual SPP role in the EOT H. Liu and PL, Nature (London) 452, 448 (2008) a=0.68 µm Transmittance /a Normal incidence RCWA SPP model

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Microscopic model q 2 Transmission wavelength (nm) 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 Normal incidence Measurement performed in Martin van Exter’s group (Leiden) q=2 2a2a 1a1a 3a3a q=3 1a1a q = 1 q = 4 q = 2 q = 6

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q 2 Transmission wavelength (nm) q = 1 (exp.) q = 2 (exp.) q = 4 (exp.) q = 6 (exp.) 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 F. van Beijnum et al., Nature 492, 411 (Dec. 2012) Normal incidence q = 1 q = 4 q = 2 q = 6 The 5 coefficients p 1 (real), and + (complex) are fitted for q = 2,3 …7

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Fast n g =4 Slow n g =1000 injector Slow light injector /c/c Fast Slow Injector: 95% efficient and only two periods long?

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

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Far-field excitation Near-field excitation

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Quasi-normal modes

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

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excitation coefficient Far-field excitation Near-field excitation 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). Energy of a dispersive material? yes but only when absorption is small. No energy consideration in the derivation. Very easy! Only hypothesis : material is reciprocal

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Complex coordinate transform (PML) X = (1+im) x Y = (1+im) y Z = (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 0 The normalization issue

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Complex coordinate transform (PML) X = (1+im) x Y = (1+im) y Z = (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. First (?) time the field in the PML is explicitly considered to evaluate a physical quantity. The normalization issue

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Silver nanosphere (radius 100 nm) in air = i |E| 2 Im(R) Re(R) R0R0 R = Re(R) R0R0 Each PML defines an analytical continuation of V in the complex plane R’+iR" for R > R 0 Real space PML Test of the invariance with the PML

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R 0 = 0.11 µm R 0 = 1 µm R 0 = 2 µm Re(V) 120 R (µm) V = ( i) + ( i) = V =

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Classical Lorentzian shape F F Classical Purcell formula Only valid for large Q (error scales as 1/Q as Q )

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DOS modal-expansion of the LDOS R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996).

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DOS R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996). modal-expansion of the LDOS

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DOS R.K. Chang and A.J. Campillo, Optical processes in microcavities, (World Scientific, 1996). modal-expansion of the LDOS

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F Revisiting the Purcell formula F Derivation based on reciprocity arguments, see C. Sauvan et al., PRL 110, (2013) & Q. Bai et al., Opt. Express 21, (2013).

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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). Non-Lorentzian response with metallic resonance

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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. Sauvan et al., Phys. Rev. Lett. 110, (2013). Multi-resonance case 85 nm 145 nm 45 nm Au 80 nm 20 nm

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Plasmon induced hot carriers tomorrow Problème multiphysique? Avec des retombées importantes (?) dans d’autres domaines des sciences (chimie? Bio?) M.W. Knight et al., Science 332,702 (2011). -photodetectors with spectral responses circumventing band gap limitations -chemical catalysis close to nanostructured metal surfaces

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

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

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

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Fast n g =4 Slow n g =1000 injector Slow light injector /c/c Fast Slow Injector: 95% efficient and only two periods long?

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Spatial coherence in complex systems p2p2 p1p1 Phys. Rev. A. 89, (2014). applications of the formalism Quantum plasmonic Application to sensing Driving external field ( L ) J. Yang et al. (submitted)

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Cross Section Wavelength (nm) Application to sensing J. Yang et al., (in preparation)

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Plasmon induced hot carriers tomorrow Problème multiphysique? Avec des retombées importantes (?) dans d’autres domaines des sciences (chimie? Bio?) M.W. Knight et al., Science 332,702 (2011). -photodetectors with spectral responses circumventing band gap limitations -chemical catalysis close to metal surfaces D=10 nm Phys. Rev. Lett. 110, (2013) Théorie quantique de la fonctionnelle de la densité Modèle hydrodynamique Modèle de drude Nonlocal effects

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JS Foresi. et al., Photonic-bandgap microcavities in optical waveguides, Nature 390, 143 (1997). Q = 400 R = 0.96 The beginning…

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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: ) Today Q = 700,000 R = (?)

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45 PML PMLPML PMLPML REAL coordinate transform Non-regular sampling = modification of the expansion basis PML with a complex f PML coefficient real coordinate transform

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

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a/ /a PhC Bloch mode (CB) Re(k z ) gap PhC Bloch mode (VB) Im(k) 0 air semi- conductor

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a/ /a PhC Bloch mode Re(k z ) gap valence band: the field in inside the high-index material conduction band: the field in inside the low-index material (the holes)

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a/ /a PhC Bloch mode Re(k z ) gap Im(k) gap Bloch mode radiation Bloch modes gap Bloch mode Evanescent (gap) mode engineering

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gap-evanescent Bloch mode ridge mode

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500 nm SiO 2 tapered section Si periodic mirror P. Velha et al., NJP 8, 204 (2006). Gap-evanescent Bloch mode ridge mode

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500 nm periodic mirror tapered section SiO 2 tapered section Si periodic mirror P. Velha et al., NJP 8, 204 (2006).

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Loss (µm) 400 Periodic mirror tapered mirror Computational results VB

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

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Microscopic SPP model t in-plane reflection- transmission of SPP coupling of SPP to free-space H. Liu and P. Lalanne, Nature (London) 452, 448 (2008).

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SPP-CW coupled-mode equations H. Liu and P. Lalanne, Nature (London) 452, 448 (2008). (for periodic arrays)

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Dual wave picture x xx SPP Quasi- cylindrical wave xx

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a=0.68 µm a=1 µm Transmittance /a Influence of the metal conductivity H. Liu & PL, Nature 452, 448 (2008). a=3 µm Normal incidence RCWA SPP model

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H. Liu & PL, J. Opt. Soc. Am. A 27, 2542 (2010). Perfect metal RCWA a=a= /a SPP model Transmittance

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Mode 2Mode 1 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.

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excitation coefficient Far-field excitation Near-field excitation 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). Energy of a dispersive material? yes but only when absorption is small. No energy consideration in the derivation. Very easy!

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(µ, A ) Driving external field ( L ) Fano coefficient (complex number) Analytical treatment of quantum plasmonic systems J. Yang et al., (submitted 2014)

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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

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a = 8 mm = 60 mm) k E H D. Smith et al., "Composite medium with simultaneously negative permeability and permittivity", PRL 84, 4184 (2000). Left-handed materials at = 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. 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. µ transparent n > 0 opaque n imaginary (magnetic metal) opaque n imaginary (metal) SRR only transparent n < 0

66 k E H D. Smith et al., "Composite medium with simultaneously negative permeability and permittivity", PRL 84, 4184 (2000). Left-handed materials at = 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. 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. µ transparent n > 0 transparent n < 0 opaque n imaginary (magnetic metal) opaque n imaginary (metal) a = 8 mm = 60 mm) SRR +rods

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hole evanescent mode TE 10 H-symmetric gap SPP Tracking light-flow: « fluid dynamics » silver dielectric (n=1.39)

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α α t sp r sp TE 01 gap-SPP ρ α α τ (resonant term) Hole-mode scatteringGap-mode scattering Tracking light-flow: « fluid dynamics »

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

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71 t SP r SP horizontal gap plasmon vertical evanescent mode Fishnet (microscopic model) fundamental Bloch mode Q = 35g = 0 J. Yang et al., Phys. Rev. Lett. 107, (2011). Quantitative interpretation n<0 (resonant term)

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72 g = 0 (Q = 35) g = 0.01 (Q = 60) g = 0.02 (Q = 800) S. Xiao et al., Nature (London) 466, 735 (2010) dye transparency threshold 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 n eff. n – ig (resonant term) r SP t SP Quantitative interpretation n<0

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

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Antenna-nanogap plasmonic devices Negative index materials Photonic crystal devices

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