Use and abuse of the effective-refractive-index concept in turbid colloidal systems

Transcription

1 Use and abuse of the ective-refractive-index concept in turbid colloidal systems Rubén G. Barrera* Instituto de Física, Benemérita Universidad Autónoma de Puebla *Permanent address: Instituto de Física, UNAM

2 In In collaboration with: Alejandro Reyes Augusto García

3 and with Felipe Pérez Edahí Gutierrez

4 Also,I acknowledge very interesting discussions with: Eugenio Méndez Luis Mochán Peter Halevi

5 Motivation internal-reflection configuration critical angle sinθ c = n n 2 1 n milk n 2 δn 2 Real time states of aggregation

6 Question What is the index of refraction of milk? it is white and turbid

7 Colloid Inhomogeneous phase dispersed within a homogeneous one homogeneous phase colloidal particles DISORDER

8 continuous phase disperse phase name examples liquid solid sol milk, paints, blood, liquid liquid emulsion oil/water, water/benzene liquid gas foam foam, whipped cream solid solid solid sol composites, policrystals, rubys solid liquid solid emulsion milky quartz, solid gas solid foam porous media, opals gas solid solid aereosol smoke, powder gas liquid liquid aereosol fog Photonic crystals and metamaterials: ordered colloids?

9 Measurements reflectance around the critical angle.inconsistencies

10 Turbidity light scattering coherent coherent (average) diffuse (fluctuating) OPTICAL SPECTRUM 400 λ 800 nm turbid

11 ensemble Total field E = E +δe probability 3 dr V T on the average homogeneous and isotropic

12 Small particles nano size parameter ka 2π a = λ 1 a λ 2π 0.1 µ m the diffuse field can be neglected average i.e. macroscopic electrodynamics

13 Effective medium ective medium Effective-medium theories Homogenization theories n ε µ [{ },{ }] n optical structural unrestricted ective properties Continuum Electrodynamics

14 Extended ective medium BIG λ a 0.1 µ m 2π colloid ective medium ε µ average n INCOMPLETE!!? fluctuations Continuum Electrodynamics

15 Energy conservation Power E = E + δe ka ~1 1.0 transmitted power coherent diffuse penetration

16 Attempts MODEL: Random system of identical spheres vacuum a µ = µ 0 ε = ε ( ω) n p = p ε ( ω)/ ε p i ε p( ω) ε0 + σp( ω) ω 0 identical nonmagnetic local k 0 = ω c

17 van de Hulst Light scattering by small particles (1957) f 1 dilute scattering matrix E e S 0 E s ikr inc 2 = s inc E ikr 0 S1 E n = 1 + iγs(0) 3 f γ = 3 2 ( ka) 0 δn complex volume filling fraction sphere S (0) = S (0) = S(0) 1 2 dc sca /dω θ TiO 2 /resin a = 0.10 µm

18 van de Hulst scattering 2.5 δ n f n S = 2.8 n S = π a λ

19 Craig Bohren J. Atmos Sci. 43, 468 (85) transmission reflection n n = 1 + iγs(0) = 1 + iγs ( π ) 1 Proposition r = µ ε µ + ε ε = 1 + iγ S(0) + S1( π) µ = 1 + iγ S(0) S1( π) MAGNETIC?

20 RG Barrera & A García-Valenzuela JOSA A 20, 296 (2003) COHERENT SCATTERING MODEL TE µ ( θ ) = 1+ i iγs cos (1) 2 ( θi ) θ ( (1) (1) 2 ) + TE ε ( θ ) = 1+ iγ 2 S ( θ ) S ( θ )tan θ i MAGNETIC i i i i (1) 1 S+ ( θi) = S(0) + S1( π 2 θi) 2 (1) S ( θi) = S(0) S1( π 2 θi) Small particles S(0) = S ( π 2 θ i ) 1 µ = 1 NON-MAGNETIC TE Normal incidence BOHREN ε = 1 + iγ (0) 1( ) S + S π µ = 1 + iγ S(0) S1( π) Comment: a very unconfortable result

21 Our new result PRB 75, (2007) IN TURBID COLLOIDAL SYSTEMS THE EFFECTIVE MEDIUM EXISTS BUT ITS ELECTROMAGNETIC RESPONSE IS NONLOCAL Electromagnetic response TOTAL GENERALIZED EFFECTIVE CONDUCTIVITY J ind = σˆ E LINEAR OPERATOR Nonlocal 3 J ( r; ω) σ ( r, r ; ω) E ( r ; ω) d r ind =

22 local vs nonlocal E ext E S E I ISOLATED SPHERE σ ( r; ω) S = σ ( ω) LOCAL J ( r; ω) = σ ( r; ω) E ( r; ω) ind S I = NL σ S ( r, r ; ω) E ( r ; ω) d r NONLOCAL S 0 r r V V 3 ext S S

23 Generalized NL conductivity σ NL S ( rr, ; ω) = + NL 3 Ur ( ; ω) δ( r r) I G0 ( rr, ; ω) σs ( r, r; ω) dr iωµ 0 σs ( r; ω) NL iωµσ Trrω (, ; ) 0 (, S r r; ω) NL iωµσ 0 (, p p ;) ω Tpp (, ;) S ω T matrix

24 Total induced current J ( r; ω) = J ( r; ω) ind i EXCITING FIELD i ind, i = NL σ S r ri,, 1, 2,... i 1, i+ 1,... N NONLOCAL Er (, ω) 3 r r; ω) E ( r ; r r r r r ; ω) d r i exc i E ext + i

25 Effective-Field Approximation valid in the dilute regime NL J ( r; ω) = σ (, r r; ω) E( r, ω) ind i S 3 r ri d r i NL J ( r ; ω) r r ; ) ind = σ (, (, ) i S 3 r r i ω E r ω d r i σ ( r r ; ω) GENERALIZED NONLOCAL CONDUCTIVITY GENERALIZED NONLOCAL OHM S LAW

26 Momentum representation NONLOCAL ind J ( p, ω) = σ ( p, ω) E ( p, ω) 3 dr i... V T NL σ ( p, ω) = n σ ( p, p = p ; ω) 0 S n 0 = N V σ NL S ( rr, ; ω) FT σ NL S ( p, p ; ω) NL σ S ( p, p = p; ω) 3 X 3 = 9 INTEGRAL EQUATION

27 LT scheme homogeneous and isotropic on the average L T σ ( p; ω) = σ ( p, ω) pp ˆˆ+ σ ( p, ω)(1 pp ˆˆ) 2 generalized ective nonlocal dielectric function i ε ( p; ω) = 1 ε0 + σ ( p; ω) ω tradition L T ε ( p, ω) ε ( p, ω) pa 0

28 Small pa LT ( ) [0] LT ( )[2] ε (, ω) = ε ( ω) + ε ( ω) ( ) p pa +... ε ε ε p 0 LOCAL LIMIT 2 NONLOCAL DEPENDENCE 0 Calculation procedure Phys. Rev. B, 75, (2007)

29 Results ε T ( p, ω ) = 1+ f ε T ( p, ω) 1 = f

30 T Re[ ε ( p, ω)] 1 5 f for Ag ( radius = 0.1 µ m) λ µm 0.62 µm 0.45 µm 0.30 µm 0.22 µm pa

31 T Re[ ε (ε T ( p, ω)] 1 for Ag ( radius = 0.1 µ m) (p,ω)/ f ε 0-1)/f for Ag (radius=0.1µm) pa λ 0 [µm]

32 Electromagnetic modes dispersion relation p = p + ip longitudinal transverse L ε ( p, ω ) = 0 T p = k 0 ε ( p, ω) L p ( ω) T p ( ω) ective index of refraction T p ( ω) k n ( ω) = 0 GENEALOGY T p = k 0 ε ( p, ω) T p = k 0 ε ( p 0; ω) nonlocal local

33 Comparisons Exact T p = k 0 ε ( p, ω) n ( ω ) = T p ( ω) k 0 Long wavelength approximation local = = ε [0] ( ω) p k ε ω [0] 0 ( ) n

34 Comparisons Quadratic approximation [0] T [2] 2 p = k ε ω + 0 ( ) ε ( ω)( pa) +... nonlocal [0] ε ( ω) n = 2 T [2] 1 ( ka) ε ( ω) 0 Light-cone approximation 2 2 T p = k ε ( p = k, ω) n = + γ i S(0) nonlocal van de Hulst

35 Re[ n ] for Ag ( radius = 0.1 µ m & f = 0.02) LWA QA LCA Exact λ 0 [µm]

36 nim[ nfor ] for Ag (radius=0.1µm Ag ( = 0.1 & µ mf =0.02) & f = 0.02) LWA QA LCA Exact λ 0 [µm]

37 Reflection problem nonlocal nature translational invariance ε ε T L ( p, ω ) ( p, ω ) n ( ω) ε ( rr, ) ε ( r r ) n ( ω) cannot be used in local CE (Fresnel s relations)

38 Abuse θ i nonlocal nature T ε ( p = k, ω ) vdh n 0 ( ω) LCA R Fresnel ( θ vdh ; n ) i IEMM Isotropic ective-medium model

39 Internal Reflection configuration milk? Internal reflection configuration great sensitivity IEMM R Fresnel vdh ( θ ; n ) i TiO 2 / water A García-Valenzuela, RG Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, E Méndez, Optics Express, 13, 6723 (2005) ( ) R θ i

40 Comparison TiO 2 / water a0 = 112 nm σ = 1.33 Pure water IEMM CSM

41 How to tomeasure n? Latex spheres / water Use refraction (propagation) A. Reyes-Coronado, A García-Valenzuela, C. Sánchez-Pérez, RG Barrera New Journal of Physics 7 (2005) 89 [1-22]

42 Comparison van de Hulst Fit with n NEXT STEP CBS

43 emscheme J = J + J ind P M L ε ( p, ω) = ε ( p, ω) 2 µ ( p, ω) = 1 k0 1 2 p (, ) (, ) ( T L ε ω ε ω ) p p magnetic response!

44 Results µ ( p, ω) = 1 2 k0 1 (, ) (, ) 2 p ( T L ε ) p ω ε p ω 2 1 k0 1 (, ) (, ) 2 µ ( p, ω) = p ε ω ε ω ( T L ) p p

45 1 1 Re 1 for Ag ( radius = 0.1 µ m) f µ ( p, ω) λ µm 0.62 µm 0.45 µm 0.30 µm 0.22 µm pa OPTICAL MAGNETISM

46 Conclusions We have developed an ective-medium approach to describe the optical properties of turbid colloids in the bulk, that is useful and complimentary to the multiple-scattering approach In turbid colloidal systems the ective index of refraction, due to its nonlocal character, is able to describe the propagation of light, but it cannot describe its reflection This is important because the naïve use of the ective index of refraction in the calculation of reflection amplitudes has been done many times without too much (intellectual) reflection There is a nonlocal magnetic response in turbid colloidal systems even when its components are non magnetic (optical magnetism)

47 Perspectives BULK LONGITUDINAL MODES ε L ( p, ω ) = 0 PROPAGATING MODES DO THEY EXIST? ENERGY TRANSFER 1 S = E H = E µ ˆ B p( ω) E = k n ( ) p B ˆ 0 ω S pˆ =? LEFTHANDED?

48 Coherent-beam specroscopy n ( λ ) 0 nonlocal ective-medium approach particle-size distribution optical properties of the colloidal particles

49 Reflection Coherent beam spectroscopy J ext