Strong coupling and resonant effects in the near field of dielectric structures

Transcription

1 Strong coupling and resonant effects in the near field of dielectric structures D. Felbacq University of Montpellier, France

2 D. Felbacq University of Montpellier, France

3 Strong coupling / quantum metamaterials

4 Meta-surfaces N. Yu, F. Capasso Nat. Mat J. Lee & al. Nature 2014

5 Simplified model of a resonant meta-surface ε eff Small period along y: distributed dipoles along z Each line dipole is characterized by a scattering matrix S 0 U s p(m) = X n b p n' n (M M p ) ' n (r, ) =H n (1) (k 0 r )e in

6 Scattered field: Rayleigh expansion U s (x, y) = X n U s ne i( nx+ n y ) ε eff U s n = K X p b p n ˆn( 0 + pk) bb 0 =(1 S 0 ) 1 S 0 ba Obtained from multiple scattering theory

7 Asymptotic analysis: dipole approximation U s X (x, y) = Une s i( nx+ n y ) n Each scatterer is represented by a magnetic dipole and an electric dipole bb 0 =(1 = 0 S 0 ) 1 S A

8 Asymptotic analysis: dipole approximation y>0,u s (x, y) =R(!, 0 ) K y<0, U s (x, y) =T (!, 0 ) K X X R = K 0 P z + im apple + 0 T =1+ K n n 1 1 n 0 P z + im apple 0 n e i( nx+ n y) e i( nx ny).

9 Asymptotic analysis: transfer matrix The meta-surface is described by an impedance condition T m = T 2 R T T i 2 (R 1) 2 0 2T (R+1) 2 T 2 2i 0 T 2 R T!

10 Asymptotic analysis: extension to multiple resonators S R m = S m 1 S m G mm,m2 {1,...,N} X b 0 m = s R me i (m)+s R m b 0 kg mk G = 0 0 G G 1N G G 2N G N1 G N C A k6=m bb 0 =(1 S R G) 1 S R E i

11 Global resonances R = K 0 P z + im apple + 0 T =1+ K 0 P z + im apple 0 R and T have the same resonances as the dressed scattering matrix (coefficients b -1, b 0, b 1 ) They are given by the dispersion relation S 0 (!) (!, 0 )=1 bb 0 =(1 S 0 ) 1 S 0 ba

12 Resonance-free scatterers The dispersion relation Can be solved to obtain: S 0 (!) (!, 0 )=1 0 d 2 p 2C s 1+ r 1 16C 2 d 2! !d/ ω/d c k 0 d

13 Bloch modes for non-resonant scatterers ε eff 0 Dielectric rods : radius/d=1/10 Permittivity : 5

14 Resonant scatterers 3 S 0 (!) apple!! z!! p ω/d c 1.5 Flat dispersion bands: small group velocity k 0 d apple (!! z ) 0 (!,k)=!! p

15 Bloch modes for resonant scatterers S ε eff ω/d c!d c k d k 0 d

16 Resonant scatterers ε eff 3 2.5!d c!d/ 2 ω/d c k 0 d 0

17 Resonant scatterers ε eff 3!d c 2.5!d/ 2 ω/d c k 0 d 0

18 0 th order diffracted efficiency for resonant scatterers The resonances appears as Wood s anomalies on the efficiencies ε eff

19 Diffracted Efficiencies for resonant scatterers ε eff

20 Strong coupling (a) (b) In microcavities: open the field to Polaritonics Polariton laser, BEC of polaritons, polariton superfluidity...

21 Strong coupling : pole structure ΓBloch The scattering matrix of the resonator is dressed by the environment Bare scattering matrix :

22 Strong coupling : pole structure ΓBloch Dressed scattering matrix of the resonator :

23 Transmission : dipolar+magnetic dipole

24 weak coupling h=20 λ

25 Strong coupling h=0

26 Strong coupling h=0

27 Going complex

28 Birth of a pole-zero structure Poles are not created ex nihilo They are created/annihilated along two mechanisms : l By anti-coalescence l By bifurcation z z 0 z z p z 0 =z p 1

29 Birth of a pole-zero structure Bifurcation implies degenerescence Here the complex dynamics is dominated by anticoalescence z z 0 z z p z 0 =z p 1 Poles are associated with zeros Thm: Let f be meromorphic on a compact Riemann surface, then number of poles=number of zeroes

30 The pole-zero structure : modulus of the dressed scattering matrix of the resonator h=0 The dipole is situated between two nanorods

31 The pole-zero structure h=0 The dipole is situated between two nanorods

32 The pole-zero structure h>>d The dipole is situated far from the nanorods

33 zooming h>>d The dipole is situated far from the nanorods

34 Time for a pole dance?

35 Introducing quantum dots in a photonic structure c

36 Band induced by Mie resonances in magnetic dipole

37 Poles of the scattering matrix Finite life-time of inner modes due to leakage Resonances at! r =! i, > 0 e i! rt = e i!t e t

38 Poles of the scattering matrix Finite life-time of inner modes due to leakage! r =! i, > 0

39 Poles of the scattering matrix Adding gain make poles shift towards the upper half of the complex plane Instability when crossing the real line: infinite lifetime

40 Poles of the scattering matrix Light Amplification in the upper sheet! r =! + i, > 0 e i! rt = e i!t e t

41 Poles of the scattering matrix Adding gain make poles shift towards the upper half of the complex plane Light Amplification in the upper sheet

42 Introducing quantum dots in the photonic structure The minus sign corresponds to the inversion regime (gainpumped QDs)

43 One scatterer with a QD: poles of the scattering matrix S(! + i )! r =! + i

44 2 scatterers with a QD: poles of the scattering matrix S(! + i )! r =! i

45 2 scatterers with a QD: poles of the scattering matrix S(! + i )! r =! + i

46 3x3 scatterers with a QD: poles of the scattering matrix S(! + i )! r =! + i

47 Transmission through the structure

48 Map of the field On resonance Off resonance

49 Validity of the model What is the meaning of a pole in the upper sheet?

50 Validity of the model Consider the temporal behavior of the field: U(x, t) = R r(x,!)!! 0 +i e i!t d!

51 Validity of the model When the pole is in the lower sheet (Γ > 0), this corresponds to an exponentially decreasing field in time. R e i!t!! 0 +i d! = (x)e i! 0t e t When the pole is in the upper sheet (Γ < 0), this corresponds to an non-causal field R e i!t!! 0 +i d! = ( x)e i! 0t e t The true behavior should be obtained by means of analytical continuation

52 Validity of the model U (x,t) (z) = R R + r(x,!)! z e i!t d!, =z <0 This expression also makes sense for Im(z) > 0. Denote eu the corresponding function. U(! 0 There is a cut line on we get: Implying: R +, from Plemelj- Sokhotski theorem, i0) = R R + r(x,!)!! 0 e i!t d! i r(x,! 0 )e i! 0t U(! 0 i0) = e U(! 0 + i0) 2i r(x,! 0 )e i! 0t

53 Validity of the model The jump is an entire function of ω 0, showing that the field can be analytically continued by posing: For z 2 C +, U(z) = e U(z) 2i r(x, z)e izt Therefore the field is exponentially growing when the pole enters the upper sheet. Physically, this means that this approach can only account for the early times, afterwards, saturation and nonlinearity cannot be neglected.

54 Towards quantum metamaterials A quantum formalism : l Set of two-level systems l Dipole coupling l Not necessarily RWA Similar to the Dicke model for super-radiance Except for the spatial variation of the field Polaritons are expected to exist as collective modes of QDs mediated by Bloch waves, inducing non-local effects (k is a good quantum number here!)

55 Conclusion Derivation of an asymptotic model for resonant metasurfaces Demonstration of the existence of confined Bloch modes with small group velocity Global effects of resonance on the diffracted efficiencies Strong coupling with a resonant dipole Approach easily extendable to: anisotropic scatterers, aperiodic structures 2D metamaterial with QDs: collective behavior near resonance lead to tunable transmitted state G centers in photonic crystals could be used to avoid broadening

56 bmission: 18of Montpellier January 2017 University RABONI, University of Bologna, Italy es Sciences Chimiques de Rennes, France ersity of South Florida, USA niversity of Bath, UK NANOMATERIALS Q Nano-engineering coatings and thin films R Nanoparticles in dielectric matrix: from synthesis to device applications for photonics, electronics, and bio sensing S ALTECH Analytical techniques for precise characterization of nano materials T Synthesis, processing and characterization of nanoscale multi functional oxide films VI U Computer modeling of thermal transport at the nanoscale V Novel multifunctional nanomaterials for energy, sensing, electronic, and detection technologies FUNCTIONAL MATERIALS W Small scale mechanical behaviour of interfaces: bridging experimental and computational modelling methods X New frontiers in laser interaction: from hard coatings to smart materials Y Paper electronics: from materials to applications Z Metamaterials: from waves to matter