Coherent Backscattering from Multiple Scattering Systems - KOPS ...

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2 Theory can be separated into three independent differential equations using the ansatz Ψ(r, θ, φ) = R(r) · Θ(θ) · Φ(φ): 1 R ( d dr sin θ Θ r 2 dR ) + k 2 r 2 = α dr ( sin θ dΘ ) = β − α sin 2 θ dθ d dθ 1 d 2 Φ Φ dφ 2 = −β Setting β = m 2 and α = n(n + 1) where m = 0, 1, 2, . . . and n = m, m + 1, . . . we obtain the complete solution Ψ for the wave equation, and <strong>from</strong> this the vector harmonics ⃗ M = ⃗∇ × (⃗rΨ) and ⃗ N = ⃗∇× ⃗ M k . The solution of the radial part of the wave equation R(r) is given by a linear combination of the spherical Bessel functions j n (ρ) = √ π 2ρ J n+1/2(ρ) and y n = √ π 2ρ Y n+1/2(ρ) where J ν (ρ) and Y ν (ρ) are Bessel functions of first and second kind, and ρ = kr. For the outgoing scattered wave the appropriate linear combination is given by one of the spherical Bessel functions of the third kind or spherical Hankel functions, h (1) n (ρ) = j n (ρ) + iy n (ρ). The zenith part of the wave equation Θ(θ) is solved by associated Legendre functions Pn m (cos θ). For the following calculations it is convenient to define the angle-dependent functions π n = Pn 1 sin θ and τ n = dP1 n dθ . It can be shown that the solution for the scattered electric field is given by ⃗ Escat = ∞ ∑ i n 2n + 1 ) E 0 (ia nNn ⃗ − b nMn ⃗ n(n + 1) n=1 where the applicable vector harmonics are given by ⃗M n = cos φ π n (cos θ) h (1) n (ρ) ê θ − sin φ τ n (cos θ) h (1) n (ρ) ê φ ⃗N n = cos φ n(n + 1) sin θ π n (cos θ) h(1) n (ρ) ρ ê r + cos φ τ n (cos θ) [ρh(1) n (ρ)] ′ ρ ê θ − − sin φ π n (cos θ) [ρh(1) n (ρ)] ′ ρ ê φ and the coefficients a n and b n are a n = m ψ n(mx) ψ ′ n(x) − ψ n (x) ψ ′ n(mx) m ψ n (mx) ξ ′ n(x) − ξ n (x) ψ ′ n(mx) ; b n = ψ n(mx) ψ ′ n(x) − m ψ n (x) ψ ′ n(mx) ψ n (mx) ξ ′ n(x) − m ξ n (x) ψ ′ n(mx) 6
2.2 Single scattering – Mie theory Figure 2.2: Vibration ellipse. The tip of the electric field vector of a light wave traces out an ellipse with semimajor axis r 1 , semiminor axis r 2 , azimuth γ, and ellipticity η = arctan(r 2 /r 1 ). with the size parameter x = ka, the relative refractive index m = k scat k , and the Riccati-Bessel functions ψ n (z) = z j n (z) and ξ n (z) = z h (1) n (z). The prime indicates differentiation with respect to the argument in parentheses. ⃗M has no radial component, and in the far-field limit with large ρ = kr the radial component of N ⃗ is negligible compared to the transverse component, as the spherical Hankel function is asymptotically given by h (1) n (kr) ≈ − (−i)n e ikr ikr . The scattered electromagnetic field is therefore basically transverse, and E θ ≈ E 0 · E φ ≈ −E 0 · e ikr −ikr · cos φ · S 2(cos θ) with S 2 = e ikr −ikr · sin φ · S 1(cos θ) with S 1 = n c ∑ n=1 n c ∑ n=1 2n + 1 n(n + 1) (a nτ n + b n π n ) 2n + 1 n(n + 1) (a nπ n + b n τ n ) Both sums converge, so that the series can be terminated after n c steps without causing major inaccuracies. A good assumption is n c ≈ x. We now focus on a certain scattering plane, which is defined by the wave vectors of the incoming and emerging waves. Intensity and polarization of the scattered light are then merely a function of the scattering angle θ and the polarization of the incoming light with respect to the scattering plane. In a non-absorbing medium, both incoming and scattered electromagnetic waves can be characterized by a Stokes vector [30] ⎛ ⎞ ⎛ ⎞ I 1 ⃗p = ⎜Q ⎟ ⎝U⎠ = ⎜cos(2η) cos(2γ) ⎟ ⎝cos(2η) sin(2γ) ⎠ V sin(2η) 7
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: 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 and 28: 2.6 On polarization and interferenc
Page 29 and 30: 2.7 The theory of coherent backscat
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
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5.2 The coherent backscattering con
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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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