Coherent Backscattering from Multiple Scattering Systems - KOPS ...

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5 Experiments 220 200 180 l* [µm] 160 140 120 100 2 4 6 8 10 12 14 16 lens position [mm] Figure 5.12: Transport mean free path in dependence of the lens position. Between 7 and 10 mm the transport mean free path of teflon has a slight maximum of approximately 180 µm, which can be clearly derived from the trend of the data. (The orange line was drawn as a guide to the eye.) with the refractive index of teflon, n = 1.35, and the diffusion coefficient is D = 16500 m2 /s, as suggested by the time of flight experiments (see sec. 4.2.2). With eqn. 2.16 one obtains a FWHM of about 0.01 ◦ , which corresponds to 12 pixels on the CCD chip. [47, 49] c report experimentally determined cone widths of about 0.03 ◦ , which would correspond to 35 pixels in our setup. This however is not that much wider than the intensity profile of the focus spot (see sec. 3.3.4), which therefore will have a considerable influence on the shape of the intensity enhancement. Consequently, the measured backscattering cone of teflon can be expected to be significantly widened, and the convolution of the fitted theory with the focus spot profile will become essential to yield correct results. The accuracy of the setup is for the most part determined by the divergence of the incoming laser beam (which should be as low as possible) and the correct position of the focussing lens in front of the camera. The former seems to be comparatively uncritical, as the beam can easily be adjusted to have a divergence lower than ±0.01 ◦ , which is the range where no significant change in the cone shape seems to occur. The appropriate distance between lens and camera is more difficult to determine. Fig. 5.12 shows that the fitted values for the transport mean free path can fluctuate strongly, even if the lens–camera distance is set correctly. Hence it is reasonable to measure a series of backscattering cones where the setting is deliberately varied slightly for each take. The correct [c] In [49] there is obviously a typing error at this point. 60
5.2 The coherent backscattering cone in high resolution 1 0.8 l* = 175µm l* = 180µm l* = 185µm l* = 220µm teflon data intensity [a.u.] 0.6 0.4 0.2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 scattering angle [deg] Figure 5.13: Coherent backscattering cone of teflon. The backscattering data can be fitted with a cone (i.e. eqn. 2.18) with l ∗ around 180 µm. From eqn. 2.7 one would expect l ∗ = 220 µm, which is only slightly narrower. value of l ∗ can then be read from the trend of the data, even if these contain some outliers that could otherwise be quite confusing. The reason for this tendency to produce unreliable data is that the intensity profile of the mirror reflex is rendered by extremely few data points. The convolution of mirror profile and theoretical cone shape, which is used to fit the data, is therefore prone to slight variations in the slope of the backscattering cone, which result in apparent variations of l ∗ . Although one instinctively tries to make the focus spot as small as possible to enhance the angular resolution, it is actually more preferable to go for larger focus spots, which are rendered more accurately. If the information about the intensity profile of the focus spot is incorporated in the data evaluation afterwards, this actually does not reduce the angular resolution of the experiment. For the teflon sample the data suggest l ∗ ≈ 180 µm, which is consistent with the results obtained with an older version of the setup [19, 47]. It is also not much lower than the 220 µm that one calculates from eqn. 2.7 using the diffusion coefficient D = 16500 m2 /s. An estimate with D = 27500 m2 /s on the other hand would give a transport mean free path of about 370 µm, which agrees neither with the old nor the new experimental results. From the measured transport mean free path one can derive a diffusion coefficient D = 13300 m2 /s, which is probably the most correct of the three values, although it was not possible to confirm it in time of flight experiments for technical reasons. 61
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 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 67: 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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