Coherent Backscattering from Multiple Scattering Systems - KOPS ...

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2 Theory waves enter and leave the loop. As the light waves are always in phase at this point, constructive interference leads to an enhanced photon density compared to the normal diffusive behavior. In strongly scattering media the probability of photons running on closed loops is enhanced, so that a macroscopically reduced diffusion can be observed. Eventually this leads to a complete breakdown of photon transport and a transition to a localizing state, which is called Anderson localization [13]. The critical amount of disorder for this phase transition can be expressed by the criterion proposed by Ioffe and Regel [27], namely that kl ∗ ≈ 1. The width of the steady-state coherent backscattering cone, which is inversely proportional to this quantity, is therefore an important experimental measure for the disorder in the sample [12]. 2.7 The theory of coherent backscattering To develop a theory for the steady-state coherent backscattering cone, the easiest case to consider is a uniform plane wave with infinite spatio-temporal coherence impinging perpendicularly on the surface of a non-absorbing multiple scattering medium. In this case the photon flux distribution has the simple form given in eqn. 2.11, and furthermore is translationally invariant, i.e. j back (⃗r ⊥,A ,⃗r ⊥,B , θ) becomes j back (⃗r ⊥ , θ) with⃗r ⊥ =⃗r ⊥,B −⃗r ⊥,A . We define the cooperon α c (θ) as the coherent and the diffuson α d (θ) as the incoherent addition of the flux emerging <strong>from</strong> the time-inverted paths in the sample: ∫ α d (θ) = ∫ j back (⃗r ⊥ , θ) d⃗r ⊥ j back (⃗r ⊥ , θ = 0) d⃗r ⊥ and α c (θ) = ∫ j back (⃗r ⊥ , θ) · e i ⃗ k⃗r⊥ d⃗r ⊥ ∫ (2.13) j back (⃗r ⊥ , θ = 0) d⃗r ⊥ where⃗ k is the wave vector of the emitted light wave. The diffuson is sometimes also referred to as the incoherent background of the backscattering cone. If single and low order scattering can be blocked completely, the backscattered intensity measured in an experiment is the sum of these two contributions. Otherwise, the height of the cooperon compared to the diffuson is reduced, as single scattering contributes to the incoherent, but not to the coherent addition of the photon flux [38]. Evaluating the above equations yields [10] ( ) α d (θ) = µ z0 l + µ ∗ µ+1 z 0 l + 1 ∗ 2 (2.14) and α c (θ) = 2 ( z 0 l ∗ + 1 2 1−e −2qz 0 ql ∗ + 2µ µ+1 ) ( ql ∗ + µ+1 2µ ) 2 (2.15) 20
2.7 The theory of coherent backscattering diffuson / cooperon 2 1.8 1.6 1.4 1.2 1 0.8 0.6 diffuson α d (θ) cooperon α c (θ) α d (θ) + α c (θ) 0.4 0.2 0 −90 −60 −30 0 30 60 90 scattering angle [deg] Figure 2.12: Diffuson and cooperon. The graph shows diffuson α d (θ) and cooperon α c (θ) calculated with eqns. 2.14 and 2.15 for λ = 590 nm, kl ∗ = 5, and non-reflective sample surface. with µ = cos θ and q = k| sin θ|. In exact backscattering direction θ = 0, diffuson and cooperon are both equal to one, so that the backscattered intensity is enhanced by a factor of two (fig. 2.12). For larger angles the cooperon drops rapidly to zero, and the backscattered intensity approaches the value of the diffuson. The initial slope of the coherent backscattering cone close to θ = 0 and therefore also the width of the cone is strongly affected by internal reflections. For non-reflective sample surfaces the inverse angular width is approximately 4 3 kl∗ [58]. Internal reflections – described by the reflectivity R of the surface – narrow the cone to f FWHM −1 = ( 1 + 1 + R ) 2 1 − R 3 kl∗ (2.16) The numerator of the cooperon in eqn. 2.13 is a Fourier transform of the flux distribution [38]: ∫∫ ∫ j back (⃗r ⊥ , t) · e iqr ⊥ dt d⃗r ⊥ = [ ] ∫ FT j back (⃗r ⊥ , t) dt = j(t) · e −q2 Dt dt (2.17) The cooperon is therefore not only the superposition of interference patterns as a function of the distance of first and last scatterer, but also a superposition of Gaussian distributions as a [f] FWHM = full width at half maximum 21
Page 1 and 2: Dissertation Coherent Backscatterin
Page 3 and 4: Ein kurzer Überblick Streuung ist
Page 5 and 6: Ein kurzer Überblick portweglänge
Page 7 and 8: Contents Ein kurzer Überblick i Da
Page 9 and 10: 1 Introduction There have been many
Page 11 and 12: 2 Theory In scattering theory, the
Page 13 and 14: 2.2 Single scattering - Mie theory
Page 15 and 16: 2.2 Single scattering - Mie theory
Page 17 and 18: 2.3 Random walk and diffusion scatt
Page 19 and 20: 2.3 Random walk and diffusion of pr
Page 21 and 22: 2.4 The influence of boundaries Fig
Page 23 and 24: 2.5 Photon flux from a surface The
Page 25 and 26: 2.6 On polarization and interferenc
Page 27: 2.6 On polarization and interferenc
Page 31 and 32: 2.7 The theory of coherent backscat
Page 33 and 34: 3 Setups 3.1 Laser System The key p
Page 35 and 36: 3.2 Wide Angle Setup 1 .0 0 .8 h e
Page 37 and 38: 3.3 Small Angle Setup 1 0.998 0.996
Page 39 and 40: 3.3 Small Angle Setup Figure 3.6: T
Page 41 and 42: 3.4 Time Of Flight Setup Figure 3.8
Page 43 and 44: 4 Samples 4.1 Sample characterizati
Page 45 and 46: 4.1 Sample characterization techniq
Page 47 and 48: 4.2 The samples sample particle siz
Page 49 and 50: 4.2 The samples Figure 4.5: Fluidiz
Page 51 and 52: 5 Experiments 5.1 Conservation of e
Page 53 and 54: 5.1 Conservation of energy in coher
Page 65 and 66: 5.2 The coherent backscattering con
Page 77 and 78: 6 Summary The focus of the work pre
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Bibliography [1] http://www.schneid
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Bibliography [34] E. Larose, L. Mar
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Figures and Tables Figures Backscat
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Figures and Tables Tables 4.1 Collo
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MATLAB codes Angular intensity dist
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Evaluation of the wide angle data E
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Evaluation of the wide angle data %
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Evaluation of the wide angle data f
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Evaluation of the small angle data
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