Coherent Backscattering from Multiple Scattering Systems - KOPS ...

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5 Experiments 0.35 0.3 0.25 linear 2nd order polynomial 3rd order polynomial 4th order polynomial laser power [a.u.] 0.2 0.15 0.1 0.05 0 −0.05 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 diode signal x 10 4 Figure 5.1: Calibration. The graph shows an example calibration function of a single photodiode. The errors of both the power and the diode signals are hardly recognizable in the graph and are therefore not drawn explicitly. Overall, 3rd order polynomials seem to be most appropriate to describe the gain characteristics of the photodiodes and their electronics. knowledge of the parameter kl ∗ is needed to characterize the phase transition from diffusive transport to a localizing state [7, 48]. Our recent progress in this field [48] and the closely related experiments on extremely wide backscattering cones, where conservation of energy demands a sizeable correction to the present-day theory, prompted us to attentively review our backscattering experiments and the evaluation of the backscattering data. An improved theory was contributed by Akkermans and Montambaux, who managed to eliminate certain improper assumptions from the calculations. 5.1.1 The power scale of the wide angle setup One difficulty of the coherent backscattering experiments is to distinguish between the inevitable background of stray light and electronic dark count, the incoherent background (which might be hidden not only by the backscattering cone, but also by the compensating intensity cutback), and the enhancement of the backscattering cone (which does not necessarily reach a factor of 2 due to incomplete blocking of single scattering). In other words, the experiments require an independent intensity scale. We figured that for experiments with our wide angle setup (see sec. 3.2) this independent scale could be provided by a reference sample with a very high kl ∗ , as it is already used to calibrate 44
5.1 Conservation of energy in coherent backscattering the photodiodes. For such samples, the angular distribution of the backscattered intensity is known to be proportional to the diffuson α d (θ) given in eqn. 2.14, as any deviations from the incoherent background (i.e. the backscattering cone and the energy cutback) are too narrow or too small to be detected by the setup. The backscattering of the reference is also proportional to the incoming laser power P and the total fraction of the backscattered intensity, the albedo A. The latter is reduced by energy losses in the sample, mostly due to absorption of photonic energy, but also because of photon leakage at the side and the rear surfaces. For the calibration of the photodiodes only a relative power scale is needed, as one is merely interested in compensating different gains of the diodes and their electronics. Therefore the exact value of the albedo is not important, as long as it is independent of the inserted light power. If however the albedos are known for sample and reference, one can derive an absolute power scale for the diode signals. Herein lies the misapprehension of [22, 23, 47], where the reference sample was also used to determine the intensity level of the diffuson. As the different albedos of sample and reference were neglected, this approach confuses relative and absolute power scale and therefore yields slightly wrong results. The photodiodes are calibrated by a series of measurements of the diode signals d θ at different incoming light powers P, so that a relation between the diode signals and the light power scattered in a certain direction θ can be derived. Previously used were relations like d θ ∝ α d (θ) · P [19], d θ ∝ cos θ · P, or even simply d θ ∝ P [22, 23, 47]. The latter however distort both diffuson and cooperon for angles θ > 0, so that they can be used only if the backscattering cone is comparatively narrow. The effect of the different albedos of sample and reference can be incorporated in the calibration function by the albedo mismatch b ℵ = A reference /A sample : d θ ∝ α d (θ) · ℵ · P (5.2) The albedo mismatch rescales the emitted intensity of the reference, so that the backscattering of reference and sample is directly comparable. The precision of the experimental result is determined by the precision of the reference measurements. The order of the fitted polynomial must therefore be chosen such that the errors are not artificially reduced. Based on the observations on fig. 5.1 we use 3rd order polynomials to fit the data and obtain the calibration functions for the photodiodes. 5.1.2 Calculating the albedo The albedo A of a multiple scattering sample is defined as the ratio of the (back-)scattered light power P out and the inserted power P in . The albedo of a sample without any losses would [b] ℵ = hebr. aleph 45
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 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: 5 Experiments 5.1 Conservation of e
Page 55 and 56: 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
Page 79 and 80: Bibliography [1] http://www.schneid
Page 81 and 82: Bibliography [34] E. Larose, L. Mar
Page 83 and 84: Figures and Tables Figures Backscat
Page 85 and 86: Figures and Tables Tables 4.1 Collo
Page 87 and 88: MATLAB codes Angular intensity dist
Page 89 and 90: Evaluation of the wide angle data E
Page 91 and 92: Evaluation of the wide angle data %
Page 93 and 94: Evaluation of the wide angle data f
Page 95 and 96: Evaluation of the small angle data
Page 97 and 98: Evaluation of the small angle data
Page 99: Evaluation of the small angle data

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