Emitted fluorescence as a measure of the penetration depth 94

In order to give an estimate of the penetration depth in our setup, we dissolved fluorescent dye in water and by varying the angle of incidence we varied also the penetration depth. In the interpretation of the measured data we are following the analysis of Edwards et al. [73].

The evanescent wave intensity at the interface z = 0 with polarization per-pendicular to the plane of incidence for unit incoming intensity is [74]

I_⊥(α) = 4 cos²α

1−(n₂/n₁)². (6.11)

The emitted intensityI_F of the fluorophores excited with the evanescent wave can then be written with equations 6.9, 6.10 and 6.11 to be

IF ∝AI⊥(α) Z _∞

0

c(z)e^−ξ(α)zdz. (6.12) Here we have cast the quantum yield of the fluorophores, the efficiency of the collecting optics and the quantum efficiency of the detector into the pre-factor A.

The fluorophore concentration is assumed to be constant over the whole sample cell (c(z) =c). From the equation (6.12) we obtain

I_F ∝AI_⊥(α)ξ⁻¹(α)c. (6.13)

The emitted fluorescence is then proportional to the penetration depth ξ⁻¹, rest of the terms being practically constant within the measurement range (see 6.4.1).

6.2.3 Evanescent-wave dynamic light scattering

Principles of dynamic light scattering

In order to understand the evanescent wave dynamic light scattering (EWDLS), we review shortly the principles of dynamic light scattering (DLS) and after that we discuss the differences between these two scattering techniques.

The principle of the DLS can be seen readily in Fig. 6.2 where laser light (plane wave) impinges a system of scatterers with a wavevectorki (incident) and the scattered light is collected far away from the sample in well defined direction with a wavevector k_f (final). The phase shift of the scattered light of the jth particle (B) with the particle sitting at the origin (O) is

φj = 2π

λ (AB+BC) = (ki−kf)·rj =q·rj (6.14) where

|q|= 4πn λ₀ sinθ

2. (6.15)

λ0 is the wavelength in vacuum andn is the index of refraction of the medium where the scatterers are in. The total scattered electric field from all the particles is then

E ∝X

j

e^−iq·rj (6.16)

wherej is the index of the scatterers in the scattering volume. The measured pre-detection signal J(t) is then

J(t)∝X

jk

e^{−iq·∆rjk}, where ∆r_jk =r_k−r_j. (6.17) From the equation (6.17) are derived the well known Siegert relation hJ(0)J(τ)i = hJ(0)i² +|hE^∗(0)E(τ)i|². For more detailed discussion see Berne and Pecora [75].

The differences of the EWDLS to the DLS can be readily understood by looking at the Fig. 6.2 (b): in EWDLS we have an evanescent wave as an

r

Figure 6.2: a) The principle of DLS: Coherent light (k_i) impinges a group of scatterers. The scattered light with wavevector k_f is collected in one well de-fined direction. The collected scattered light interferes at the detector forming a fluctuating signal, as the scatterers move, which then again can be analyzed with a correlator. b) In EWDLS the evanescent wave is used as incident light source. As it decays exponentially from the surface the scatterers are not homo-geneously illuminated. Furthermore the strongly reduced size of the scattering volume causes a spread in the scattering vector q [76]. The components of the scattering vector along (q_k) and perpendicular (q_⊥) to the surface are also shown.

incident light source and therefore the incident electric field is no longer spatially homogeneous but decays exponentially with distance from surface. This fact has to be considered when the theoretical model for the field autocorrelation function is derived (see Lan et al. [69]). Furthermore the size of the scattering volume is comparable to the inverse of scattering vectorq. This leads to the fact thatqis no longer well defined. The spread in the scattering vector is given approximately by ∆q ≈ 2πcos(θ/2)/2r, where r is the radius of the scattering volume [76].

However as contributions from negative and positive ∆q are equally probable the error tends to be averaged out as long as enough measurements are recorded.

Furthermore the use of evanescent wave as a incident light source leads always to a static scattering from the surface, in addition to the fluctuating scattered field reflecting the dynamics of the sample. As static and fluctuating fields add up coherently the measured intensity correlation function is related to the field autocorrelation function by a modified Siegert relation representing partial het-erodyning

g⁽²⁾(τ) = 1 + 2j_lj_sg⁽¹⁾(τ) +j_s²[g⁽¹⁾(τ)]² (6.18) where

j_s = hJ_si

J_l+J_b+hJ_si, j_l = J_l

J_l+J_b+hJ_si and

j_b = 1−j_l−j_s. (6.19)

The termhJ_si is the averaged pre-detection signal of the dynamic part of the scattering. Jl is the so called local oscillator and it originates from the surface scattering and finally J_b represent the scattering form the solvent. Furthermore g⁽¹⁾(τ) is the normalized autocorrelation function

g⁽¹⁾(τ) = hE_s(0)E_s^∗(τ)i

hJ_si (6.20)

of the the scattered field E_s from the scatterers under study. For detailed discussion see Flammer et al. [77].

Non-linearity effects of the PMT to the correlation function

The ideal detector [77] has the following statistical properties: firstly there is linear dependence between the expected photocounts (n) collected during one sampling time ts and the pre-averaged pre-detection signal Jt (α being here the quantum efficiency of the detector andJ(t⁰) is the pre-detection signal):

n=αtsJt Jt= 1 t_s

Z _t+ts

t

J(t⁰)dt⁰. (6.21) Secondly the expected photocounts at different time intervals are statistically independent:

ntnt+τ =ntnt+τ. (6.22) The real photodetector exhibits memory effects and therefore it is not ideal.

The non-linearity effect of the detector to the expected photocounts including the fluctuations of the pre-averaged pre-detection signal is given by

hni=αt_shJ_ti where ²=αhJ_tiθ. The θ and φ are called deadtime and updating parameter, respectively (for physical meaning see Fig. D.1 (b)). Also the autocorrelation function can be written in terms of pre-averaged pre-detection signal

hn₀n_τi = (αt_shJ_ti)²hhJ₀J_τi

The autocorrelation function in equation (6.24) can then be written in terms of equations (6.19) and (6.20) in a lengthy form

hn₀n_τi

hni² = 1 + X4

i=1

f_i(², φ, j_l, j_b)[g⁽¹⁾]ⁱ (6.25) and a similar form can be also derived for cross-correlation function (see Ap-pendix C for details) The remaining problem is to find out the parameters ², θ and φ. ² can be written in terms of measured count rate r and deadtime θ resulting in ² = rθ+r²θ²(1 +_M¹) +O(θ³) where M is the effective number of modes transmitted by the fiber. Now by performing a homodyne experiment, where j_l=j_b = 0, the final form of the equation (6.25) in terms of r, M,θ and φ is as the lag time τ → 0. This can be understood so that as count rate raises, the intercept of the measured intensity correlation function decreases and from this behavior can the parameters M, θ and φ be solved (for more details see Flammer et al.[77] and appendix D).

6.3 Materials and methods