Recent Advances on the Effective Optical Properties of Turbid Colloids. Rubén G. Barrera Instituto de Física, UNAM Mexico

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1 Recent Advances on the Effective Optical Properties of Turbid Colloids Rubén G. Barrera Instituto de Física, UNAM Mexico

2 In In collaboration with: Augusto García Edahí Gutierrez Celia Sánchez Pérez Felipe Pérez Luis Mochán Alejandro Reyes

3 The problem index of refraction of complex fluids milk blood optical properties

4 Refractive index refractometer actually reflectance n = n + in critical-angle refractometry n blood n 2 Abbe-type refraction sinθ c = n n 2 1 non-magnetic FIT μ = μ 2 0 n = ε / ε = n + in Fresnel s relations r 2 ( n, n ; θ ) 2 2 i

5 Applications changes in structure n2 n2 +δ n2 δn 2 real time states of aggregation MODEL n 2 m n δ n

6 Index of refraction in continuum electrodynamics

7 Electrodynamics of continuous media average E = E +δe fluctuations Average spatial average E macroscopic coherent λ

8 Average power Plane waves 1 ( ) 1 S = Re E H* = Re ( E + δ E) ( H + δh) * = S + δs 2 2 S = E H + δe δh S S coh diffuse S not enough S coh S diffuse S diffuse one neglects the power carried by the diffuse field S coh

9 Average induced current AVERAGE J ind ρ ind ρ J ind ind ρ J ind ind J ind material Traditional approach P = + M t J P ( PM, ) J M ρ ind J ind + = t MATERIAL FIELDS P M polarization magnetization 0

10 εμ scheme Tradition Displacement field D = ε E + P 0 H field H B = M μ 0 Linear materials Isotropic and homogeneous on the average D = ˆ ε E H = 1 μ ˆ B 3 Dr (, ω) ε( r r ; ω) E ( r, ω) dr Frequency domain time dispersion 3 1 = Hr (, ω) = dr μ ( r r ; ω) B( r, ω) a NL non-local ε( k, ω) 1 spatial dispersion μ( k, ω) electric permittivity magnetic permeability

11 Local optics anl λ long-wavelength limit ( k 0) ε( k 0, ω) = ε( ω) μ( k 0, ω) = μ( ω) ε( ω) = ε ( ω) + i ε ( ω) dissipation μ( ω) μ 0 Index of refraction n( ω) = εω ( )/ ε = n ( ω) + in ( ω) 0 (, ε μ0) continuum

12 How about inhomogeneous materials? Can one extend the continuum approach?

13 colloids dispersed phase / homogeneous phase EXAMPLES colloidal particles / matrix ordered colloids

14 Optical properties SIZE gold colloids 2a transparency turbidity size parameter ka = 2π an λ 0 S ka 1 ka 1 diffuse S S S coh diffuse coh

15 Effective-medium approach homogenization ( ε, μ ) continuum n = ε μ small particles ka 1 always possible although it may be difficult advantage electrodynamics of continuous media UNRESTRICTED optical properties

16 Reflection & refraction unrestricted measurement critical-angle refractometry ( ε, μ ) k n n k = k0n k0 ε Fresnel s relation r ε k ε k i z 0 z p = i ε kz + ε0kz

17 Big colloidal particles ka 1 S turbidity diffuse S coh diffraction

18 Effective medium Is there an ective medium? continuum coherent diffuse incident If there is one, it should be. for the coherent beam If there is one, the theory should be incomplete

19 Energy conservation Power E = E + E 1.0 AV fluc transmitted power coherent diffuse penetration ective properties coherent beam scattering as... dissipation

20 Effective index of refraction first attempts van de Hulst dilute limit Light scattering by small particles (1957) complex n = 1 + iγ S(0) δn 3 f γ = 3 2 ( ka) 0 γ 1

21 Scattering matrix sphere s ikr inc E θ e S2( ) 0 E = s inc E θ ikr 0 S1( ) E MIE S (0) = S (0) = S(0) dc sca /dω θ TiO 2 /resin a = 0.10 μm

22 Effective index of refraction 2 Van de Hulst δ n f n S = S (0) δ n = i f 2 ( k a) π a λ δ n f n S = 2.8 Is this ective-medium theory unrestricted? compare with experiment π a λ

23 Comparison with experiment critical-angle refractometer A García-Valenzuela, RG Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, E Méndez, Optics Express, 13, 6723 (2005) ( ) R θ i

24 Results Log-normal a = nm σ = 1.23 Pure water unrestricted experiment TiO 2 / water λ 0 = 6350 n n p m = 2.73 nm =

25 Results n p = 1.47 a 0 = nm latex / water Pure water unrestricted experiment λ 0 = 6350 n n p m = 2.73 nm =

26 Why?

27 Our result IN TURBID COLLOIDAL SYSTEMS AN EFFECTIVE MEDIUM EXISTS, BUT IT IS NONLOCAL ε μ ( k, ω) ( k, ω) Fresnel s equations are not valid magnetic response Nonlocal nature of the electrodynamic response of colloidal systems Rubén G. Barrera, Alejandro Reyes-Coronado & Augusto García- Valenzuela Physical Review B 75, (2007)

28 1 1 Re 1 μ ω ( k, ) f spheres Ag / vacuum a = 0.1 μm ka

29 1 1 Re 1 μ ω ( k, ) f spheres TiO 2 / vacuum a = 0.1 μm ka

30 Electromagnetic modes ε μ ( k, ω) ( k, ω) dispersion relation NON-LOCAL ε L T ε ( k, ω) ( k, ω) generalized k = k + ik longitudinal L ε ( k, ω ) = 0 k L ( ω) transverse T k = k 0 ε ( k, ω) T k ( ω) ective index of refraction T k ω = k0n ( ) ( ω)

31 Approximations Local (long wavelength) k = k ε ( k 0, ω) = k ε ( ω) T 0 0 LOCAL OR NONLOCAL n ( ω) = εω ( ) Light-cone approximation (LCA) T k ( ω) = k ε ( k, ω) T 0 0 T n ( ω) = ε ( k, ω) 1 + iγs(0) LCA is close to the exact 0 n ( ω ) = T k ( ω ) k van de Hulst has a non-local ancestry Thus it is restricted 0

32 Howtomeasuretheectiverefractiveindex? Critical-angle refractometry Reflectance from a medium with a non-local response

33 Reflectance Reflectance of a half-space with an ective non-local response translation-invariance ε μ ( k, ω) ( k, ω) ε ( r r ; ω) μ ( r r ; ω) bulk like Surface region 2a ε μ ( rr, ; ω) ( rr, ; ω) ANISOTROPIC Although there are attempts there is still not a reliable solution to the reflectance problem but there will be one soon

34 Practical problems Surface sensitive Probability interface pr ( ) j z = 0 3 drj if zj > 0 = V 0 if zj < 0. surface region

35 An alternative experimental set up to measure the ective refractive index

36 Non-local refraction 2a probability interface k Probability interfaces Generalized Snell s Law for complex refractive indices θ i bulk z surface regions i k = k sin θ = n ( ω)sinθ i t local or non-local ancestry absorption generalized Snell s Law n N ( θ, n, n ) i surface region weak dependence

37 Experimental setup for refraction and extinction measurements A. Reyes-Coronado, A García-Valenzuela, C. Sánchez-Pérez, RG Barrera New Journal of Physics 7 (2005) 89 [1-22]

38 1.2 ml H 2 O incial Latex particles in solution f = 1.2% Diameter: 0.31 μm

39 1.2 ml of water ml in solution ml in solution ml in solution ml in solution ml in solution

40 Particle sizing

41

42 Effective-medium & multiple-scattering ective medium scattering theory iωμ0 σ ( k, k ) T( k, k ) coherent-scattering model dilute regime generalized conductivity 3 Jind( k, ω) = σ ( k, k ; ω) E ( k, ω) d k r = 2 γksm( π 2 θi) ( i i 2 z + z ) z γ (0) ik k k ks pol. s m = 1 pol. p m 2 3 f γ = 3 = 2 ( ka) 0 TOTAL BULK σ ( kk, ; ω) J. Opt. Soc. Am. A/Vol. 20, No. 2/February 2003 Journal of Quantitative Spectroscopy & Radiative Transfer (2003)

43 Results Log-normal a = nm σ = 1.23 Pure water unrestricted experiment CSM TiO 2 / water λ 0 = 6350 n n p m = 2.73 nm =

44 CONCLUSIONS We have provided arguments for the use of refraction as a secure way for the experimental determination of the ective index of refraction in turbid colloids.

45

46 n for Ag (radius=0.1μm & f =0.02) LWA QA LCA Im[n ] 0.02 Exact 0.01 LCA is close to the exact van de Hulst has a non-local ancestry Thus it is restricted λ 0 [μm]

47 Craig Bohren J. Atmos Sci. 43, 468 (85) multiple-scattering theory Effective-medium approach transmission reflection n n = 1 + iγs(0) = 1 + iγs ( π ) 1 Proposition ε = 1 + iγ S(0) + S1( π) μ = 1 + iγ S(0) S1( π) MAGNETIC? r = μ μ + ε ε

48 Half-space multiple-scattering theory r k k EFA dilute E exc p E coherent-scattering model r = 2 γksm( π 2 θi) ( i i 2 z + z ) z γ (0) ik k k ks i k pol. s m = 1 pol. p m = 2 3 f γ = 3 2 ( ka) 0 Z = 0 J. Opt. Soc. Am. A/Vol. 20, No. 2/February 2003 Journal of Quantitative Spectroscopy & Radiative Transfer (2003)

49 Effective medium assume to be unrestricted assume to be unrestricted n = 1 + iγ S(0) μ = ε = 1 n f = 0.1 Fresnel s relations

50 Comparison with experiment critical-angle refractometer A García-Valenzuela, RG Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, E Méndez, Optics Express, 13, 6723 (2005) ( ) R θ i

51 Comparison with experiment critical-angle refractometer A García-Valenzuela, RG Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, E Méndez, Optics Express, 13, 6723 (2005)

52 Comparison with experiment Internal reflection configuration great sensitivity R assume to beunrestricted Fresnel vdh ( θ ; n, μ = 1) i Coherent scattering model 2 γksm( π 2 θi) ( i i 2 z + z ) z γ (0) R = ik k k ks 2 TiO 2 / water A García-Valenzuela, RG Barrera, C. Sánchez-Pérez, A. Reyes-Coronado, E Méndez, Optics Express, 13, 6723 (2005) ( ) R θ i

53 probability interfaces 2a (a) d bulk 2a Probability interfaces (b) Generalized Snell s Law for complex refractive indices. z surface regions surface regions probability interface 2a bulk k θt z θ i bulk like surface region i k