On the Transmission of Light Through Random Media
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On the Transmission of Light through Random Media PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE RIJKSUNIVERSITEIT TE LEIDEN, OP GEZAG VAN DE RECTOR MAGNIFICUS DR.W.A.WAGENAAR, HOOGLERAAR IN DE FACULTEIT DER SOCIALE WETENSCHAPPEN, VOLGENS BESLUIT VAN HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP DONDERDAG 16 OKTOBER 1997 TE KLOKKE 16.15 UUR DOOR Jeroen Clemens Jozef Paasschens GEBOREN TE VALKENSWAARD IN 1970 Promotiecommissie: Promotor: Prof. dr. C. W. J. Beenakker Referent: Prof. dr. G. W. ’t Hooft Overige leden: Prof. dr. L. J. de Jongh Prof. dr. J. M. J. van Leeuwen Prof. dr. G. Nienhuis Prof. dr. ir. W. van Saarloos Het onderzoek beschreven in dit proefschrift heeft plaatsgevonden op het Philips Natuurkundig Laboratorium te Eindhoven, als onderdeel van het Philips onderzoeksprogramma, en is tevens onderdeel van het wetenschap- pelijk programma van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The work described in this thesis has been carried out at the Philips Re- search Laboratories Eindhoven, the Netherlands, as part of the Philips research programme, and is also part of the scientific programme of the Netherlands Organization for Scientific Research (NWO). About the cover: The cover shows the image of a cat, distorted by a piece of frosted glass (an example of a random medium). The original image is regained upon reflection in a special “phase-conjugating” mirror (studied in the last chap- ter of this thesis). From Jack Feinberg, Optics Letters 7, 486 (1982), used with permission of the author. aan alle mensen met kanker . la vraye Philosophie, dans laquelle on conc¸oit la cause de tous les effets naturels par des raisons de mechanique. Ce qu’il faut fairea ` mon avis, ou bien renoncera ` tous esperance de jamais rien compren- dre dans la Physique. de ware Filosofie, waarin men mechanische re- denen veronderstelt als oorzaak van alle natuurlijke effecten. Dit moet men volgens mij wel doen, of an- ders alle hoop opgeven ooit iets van Natuurkunde te begrijpen. Christiaan Huygens, Trait´edelaLumiere. Contents 1 Introduction 1 1.1Medicalimaging......................... 1 1.2Randomlasers........................... 3 1.3Phase-conjugatingmirror.................... 5 1.4 Analogies between photons and electrons ........... 6 1.5Radiativetransfertheory..................... 9 1.6Diffusionapproximation..................... 11 1.7Scatteringtheory......................... 12 1.8Numericalsimulations...................... 14 1.9Thisthesis............................. 16 References................................ 19 2 Accuracy of the diffusion equation with extrapolated-boundary condition 23 2.1Exactcalculationofthetransmittance............. 24 2.2Comparisonwiththediffusionapproximation........ 27 References................................ 31 3 Exact solution of the time-dependent Boltzmann equation 33 3.1 Calculation of the Fourier-transformed intensity ....... 34 3.2InversionoftheFouriertransform............... 37 3.2.1 Twodimensions..................... 38 3.2.2 Fourdimensions..................... 39 3.2.3 Threedimensions..................... 40 3.2.4 Ballistic peak ....................... 43 3.3Decaylengthanddispersionrelation.............. 44 3.4Conclusion............................. 46 3.A Numerical inversion of the Fourier transform in 3 dimensions 47 3.BAnisotropicscattering...................... 48 3.B.1 Formalexactsolution.................. 49 3.B.2 Kratky-Porodtheory................... 50 3.B.3 Decaylength....................... 52 References................................ 55 CONTENTS 4 Influence of boundaries on the imaging of objects in turbid me- dia 57 4.1Perturbationtheoryforgeneralobjects............. 58 4.1.1 Bornseries......................... 60 4.1.2 Smallobjects........................ 61 4.1.3 Strengthofspecificobjects................ 64 4.1.4 Example.......................... 67 4.1.5 Amplitudemodulatedsources............. 68 4.2Optimalboundaries........................ 68 4.2.1 Sourcesandboundaries................. 68 4.2.2 Greenfunctions...................... 72 4.2.3 Reflectingboundaries.................. 75 4.2.4 Experimentalresults................... 77 4.2.5 Absorbingboundaries.................. 85 4.3Conclusions............................ 90 References................................ 93 5 Probability of reflection by a random laser 95 5.1 Calculation of the albedo ..................... 96 5.2 Probability distribution of the albedo .............. 99 5.3Numericalsimulationsandconclusions............101 References................................103 6 Duality between absorption and amplification 105 6.1 Proof of duality ..........................106 6.2Singlemodeexample.......................108 6.3Numericalsimulationsandconclusions............111 References................................113 7 Localization in a non-conservative disordered multi-mode wave- guide 115 7.1Formulationofthescatteringproblem.............117 7.2 Fokker-Planck equation .....................118 7.3 Localization length . .....................123 7.3.1 Single-modewaveguide.................123 7.3.2 Multi-modewaveguide.................125 7.4Numericalresultsandconclusions...............127 References................................129 CONTENTS 8 Brightness of a phase-conjugating mirror behind a random me- dium 131 8.1Formulationoftheproblem...................132 8.2Phase-incoherentsolution....................135 8.2.1 Radiativetransfertheory................135 8.2.2 Neglect of angular correlations .............136 8.2.3 ExactsolutionoftheBoltzmannequation.......137 8.3Phase-coherentsolution.....................140 8.3.1 Scatteringmatrices....................140 8.3.2 Random-matrix theory ..................143 8.3.3 Numericalsimulations..................149 8.4ComparisonwithAndreevreflection..............150 8.A Calculations in the coherent regime . .............152 References................................159 Samenvatting 161 List of Publications 165 Curriculum Vitæ 167 1 Introduction In optics, a random medium is a system where the dielectric constant ε(r) varies randomly as function of the position r. Light propagating through such a medium is scattered by the fluctuations in ε. Furthermore, a non- zero imaginary part of ε leads to absorption or amplification of the radi- ation. In this thesis we study the transmission of light through various types of random media. The initial motivation of this research was a prob- lem in medical imaging. 1.1 Medical imaging The human body is a random medium, at least on a certain length scale. Due to the structure of cells and their components the dielectric constant of human tissue fluctuates randomly from point to point. The goal of med- ical imaging is to study the inside of a body, by transmitting radiation through it. X-rays have been used for that purpose for over a century, but they damage the tissue. Therefore it is advantageous to use longer wave- lengths in the near infrared or in the visible parts of the spectrum. In this frequency region the scattering is greatly enhanced with respect to that of x-rays. It is a difficult problem to reconstruct an image from light which has been multiply scattered. An additional complication is absorption, which should be as low as possible. The absorption by tissue is dominated by the absorption in water. In the range between small (ionizing) wavelengths and wavelengths too large for imaging purposes, the absorption minimum of water is around 500 nm. Near that wavelength, however, blood (haemoglobin) and pig- ments (melanin) have their absorption peak. The effective absorption by tissue is smallest at approximately 800 nm. Experiments are therefore done in the near-infrared regime. There are two major applications of medical imaging. First of all, one can study the skin. The small thickness of the skin means that the amount of scattering can be rather low. The second application is the detection of tumors. This is possible either inside the head or in the female breast (mammography). Both regions of the body are void of bones and hence relatively homogeneous. Our research was motivated by the problem of 2 INTRODUCTION Figure 1.1: Refraction of a photon-density wave. Shown are constant ◦ phase contours (in 20 intervals) for the propagation of a diffusive photon- density wave through a planar boundary that separates two regions with different mean free path. From Ref. [1]. optical mammography. The goal of mammography is the detection of an object, the tumor, inside a tissue, which is rather homogenous. This is done by detecting at various points around the breast the amount of light transmitted, coming from a single source. By repeating the measurement for different sources one gathers a large amount of data, from which in principle the interior of the breast can be reconstructed. To be able to do a reconstruction, one needs to understand first the optical properties of the tissue when no tumor or other inhomogeneity is present. These are determined by two length scales, the mean free path for scattering and the absorption length. The angle-dependence of scatter- ing is also of importance. (In general, tissue scatters predominantly in the forward direction.) The next step is to calculate the influence of such an in- homogeneity on the detected signals. The sensitivity of the measurement depends on the geometry and boundary conditions, and on the positions of the sources and detectors. The source of light need not be a continuous wave, but could be a short pulse. The time-dependence of the detected signal can be used to gain extra