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The work conducted in this thesis examines the nature of circumstellar discs by investigating irradiance and polarization of scattered light. Two circumstellar discs are investigated. Firstly, H-band high contrast imaging data on the transitional disc of the Herbig Ae/Be star HD169142 are presented. The images were obtained through the polarimetric differential imaging (PDI) technique on the Very Large Telescope (VLT) using the adaptive optics system NACO. Our observations use longer exposure times, allowing us to examine the edges of the disc. Analysis of the observations shows distinct signs of polarization due to circumstellar material, but due to excessive saturation and adaptive optics errors further information on the disc could not be inferred. The HD169142 disc is then modelled using the 3D radiative transfer code Hyperion. Initial models were constructed using a two disc structure, however recent PDI has shown the existence of an annular gap. In addition to this, the annular gap is found not to be devoid of dust. This then led to the construction of a four-component disc structure. Estimates of the mass of dust in the gap (2:10x10⁻⁶ M⊙) are made as well as for the planet (≈1:53 x 10⁻⁵ M⊙ (0.016MJupiter)) suspected to be responsible for causing the gap. The predicted polarization was also estimated for the disc, peaking at 14 percent. The use of realistic dust grains (ballistic aggregate particles) in Monte Carlo code is also examined. The fortran code DDSCAT is used to calculate the scattering properties for aggregates which are used to replace the spherical grain models used by the radiative transfer code Hyperion. Currently, Hyperion uses four independent elements to de ne the scattering matrix, therefore the use of rotational averaging and a 50/50 percent population of grains and their enantiomers were explored to reduce the number of contributing scattering elements from DDSCAT. A python script was created to extract the scattering data from the DDSCAT output les and to apply a size distribution to the data. The second circumstellar disc investigated is the debris disc of the M dwarf star AU Mic. The disc was modelled, using the radiative transfer code Hyperion, based on F606W (HST) and JHK0-band (Keck II) scattered light observations and F606Wband polarized light observations. Initially, the disc is modelled as a two component structure using two grain types: compact silicate grains and porous dirty ice water. Both models are able to reproduce the observed SED and the F606W and H-band surface brightness pro les, but are unable to t the observed F606W degree of polarization. Therefore, a more complex/realistic grain model was examined (ballistic aggregate particles). In addition, recent millimetre observations suggest the existence of a planetesimal belt < 3 AU from the central star. This belt is included in the BAM2 model and was successful in fitting the observed SED, F606W and H-band surface brightness and F606W polarization. These results demonstrate the limitations of spherical grain models and indicate the importance of modelling more realistic dust grains.

Aims. We consider the polarization arising from scattering in an envelope illuminated by a central anisotropic source. This work extends the theory introduced in a previous paper (Al-Malki et al. 1999) in which scattering polarization from a spherically symmetric envelope illuminated by an anisotropic point source was considered. Here we generalize to account for the more realistic expectation of a non-spherical envelope shape. Methods. Spherical harmonics are used to describe both the light source anisotropy and the envelope density distribution functions of the scattering particles. This framework demonstrates how the net resultant polarization arises from a superposition of three basic “shape” functions: the distribution of source illumination, the distribution of envelope scatterers, and the phase function for dipole scattering. Results. Specific expressions for the Stokes parameters and scattered flux are derived for the case of an ellipsoidal light source inside an ellipsoidal envelope, with principal axes that are generally not aligned. Two illustrative examples are considered: (a) axisymmetric mass loss from a rapidly rotating star, such as may apply to some Luminous Blue Variables, and (b) a Roche-lobe filling star in a binary system with a circumstellar envelope. Conclusions. As a general conclusion, the combination of source anisotropy with distorted scattering envelopes leads to more complex polarimetric behavior such that the source characteristics should be carefully considered when interpreting polarimetric data

Nous avons développé une méthodologie basée sur phénomène de mélange à quatre ondre polarimétrique « Four wave Mixing FWM » et son équivalen résonant la diffusion Raman cohérente anti-Stokes (CARS, Coherent Anti-Stokes Raman Scattering) polarimétrique et réalisé des mesures sur des systèmes cristallins, simili biologiques : les membranes cellulaires connues sous le nom de « Multilamellar Vesicles MLV » et des échantillons de biologiques : la myeline, et ce, en variant les polarisations des lasers excitateurs, Pompe et Stokes. Le signal anti-Stokes émis est ensuite analysé afin d’en extraire les ordres 2 et 4 de la fonction de distribution angulaire des molécules actives constituant l’échantillon. Pour cela, plusieurs approches sont explorées telles que des algorithmes d’optimisation ou par décomposition en série de fourrier du signal polarimétrique. Ces multiples approches en traitement du signal permettent d’obtenir de manière rapide les coefficients des fonctions de distribution angulaire recherchées, et ainsi d’avoir des informations sur la symétrie des échantillons imagés, allant jusqu’à l’observation d’une symétrie d’ordre 4. La capacité de la microscopie non linéaire résolue en polarisation à sonder des ordres moléculaires est clairement démontrée et ainsi son intérêt dans l’étude de la relation entre la structure et la fonction de systèmes biologiques. / The capacity to quantify molecular orientational order in tissues is of a great interest since pathologies (skin lesion, neurodegenerative diseases, etc) can induce strong modifications in proteins’ organization. While numerous studies have been undertaken using polarization resolved second order nonlinear optical microscopy which is only specific to non-centrosymmetric organizations, higher order effects have been less explored. Four-wave mixing (FWM) microscopy and its resonant counterpart coherent anti-Stokes Raman scattering (CARS) can be of a great utility as label free diagnosis tools benefiting from less constraining symmetry rules. In this work, we implement incident polarizations tuning in FWM and CARS microscopy to probe molecular order, using a generic method to read-out symmetry information.Fourier analysis of the polarization-resolved FWM/CARS signal processed with an analytical model provides a fast and direct determination of the symmetry orders of the distribution function of the probed molecules. This method does not require a priori knowledge of the organization structure and provides quantitatively its second and fourth order symmetries. We applied this technique on different systems, from crystalline to less organized (multilamellar vesicles and proteins aggregates). We show that this new approach brings additional and more refined information on supra-molecular structures in complex media.

This thesis is devoted to improve our knowledge on the theory of polarized line formation in a magneto-turbulent medium, and in a scattering dominated magnetized medium, where partial redistribution (PRD) effects become important. Thus the thesis consists of two parts. In the first part we carry out a detailed investigation on the effect of random magnetic fields on Zeeman line radiative transfer. In the second part we develop the theory of polarized line formation in the presence of arbitrary magnetic fields and with PRD. We present numerical methods of solution of the relevant transfer equation in both part-I and II. In Chapter I we give a general introduction, that describes the basic physical concepts required in both parts of the thesis. Chapters 2-6 deal with the part-I, namely stochastic polarized Zeeman line formation. Chapters 7-10 deal with part –II, namely the theory and numerics of polarized line formation in scattering media. Chapter II is devoted to the future outlook on the problems described in part-I and II of the thesis. Appendices are devoted to additional mathematical details. Part-I of the Thesis: Stochastic polarized line formation in magneto-turbulent media Magneto-convection on the Sun has a size spectrum that spans several orders of magnitudes and hence develops turbulent elements or eddies the sizes of which are much smaller than the spatial resolution of current spectro-polarimeters (about 0.2 arcsec or 150km at the photospheric level). We were thus strongly motivated to consider the Zeeman effect in a medium where the magnetic field is random with characteristic scales of variation comparable to the radiative transfer characteristic scales. In Chapter 2, we consider the micro-turbulent limit and study the mean zeeman absorption matrix in detail. The micro-turbulent limit refers to the case when the scales of fluctuations of the random field are much smaller than the photon mean free paths associated to the line formation. The ‘mean’ absorption and anomalous dispersion coefficients are calculated for random fields with a non-Zero mean value - isotropic or anisotropic Gaussian distributions that are azimuthally invariant about the direction of the mean field. The averaging method is described in detail, and fairly explicit expressions for the mean coefficients are established. A detailed numerical investigation of the mean coefficients illustrates two simple effects of the magnetic field fluctuations: (i) broadening of the components by fluctuations of the field strength, leaving the π-components unchanged, and (ii) averaging over the angular dependence of the π and components. Angular averaging can modify the frequency profiles of the mean coefficients quite drastically, namely, the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in the direction of the line-of-sight. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions, which are related to the derivatives of the Voigt and Faraday-Voigt functions. In chapter 3, we study these functions in detail. Simple recurrence relations are established and used for the calculation of the functions themselves and of their partial derivatives. Asymptotic expansions are also derived. In Chapter 4, we consider the Zeeman effect from a magnetic field which has a finite correlation length(meso-turbulence) that can be varied from zero to infinity and thus made comparable to the photon mean free-path. The random vector magnetic field B is modeled by a Kubo-Anderson process – a piecewise constant Markov process characterized by a correlation length and a probability distribution function(PDF) for the random values of the magnetic field. The micro- and macro-turbulent limits are recovered when the correlation length goes to zero or infinity respectively. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tools to determine the scale of unresolved features in the solar atmosphere. In Chapter 5, using statistical approach, we analyze the effects of random magnetic fields on Stokes line profiles. We consider the micro and macro-turbulent regimes, which provide bounds for more general random fields with finite scales of variations. The mean Stokes parameters are obtained in the micro-turbulent regime, by first averaging the Zeeman absorption matrix Φ over the PDF P(B) of the magnetic field and then solving the concerned radiative transfer equation. In the macro-turbulent regime, the mean solution is obtained by averaging the emergent solution over P(B). In this chapter, we consider the same Gaussian PDFs that are used to construct (Φ) in chapter 2. Numerical simulations of magneto-convection and analysis of solar magnetograms provide the empirical PDF for the magnetic field line-of-sight component on the solar atmosphere. In Chapter 6, we explore the effects of different kinds of PDFs on Zeeman line formation. We again consider the limits of micro and macro-turbulence. The types of PDFs considered are: (a) Voigt function and stretched exponential type PDFs for fields with fixed direction but fluctuating strength. (b) Cylindrically symmetrical power law for the angular distribution of magnetic fields with given field strength. (c) Composite PDFs accounting for randomness in both strength and direction obtained by combining a Voigt function or a stretched exponential with an angular power law. The composite PDF proposed has an angular distribution peaked about the vertical direction for strong fields and is nearly isotropically distributed for weak fields, which could mimic solar surface random fields. We also describe how the averaging technique for a normal Zeeman triplet may be generalized to the more common case of anomalous Zeeman splitting patterns. Part-II of the Thesis: Polarized line formation in scattering media-Theory and numerical methods Many of the strongest and most conspicuous lines in the Second Solar Spectrum are strong lines that are formed rather high, often in the chromosphere above the temperature minimum. From the standard, unpolarized and non-magnetic line-formation theory such lines are known to be formed under the conditions that are very far from local thermodynamic equilibrium. They are characterized by broad damping wings surrounding the line core. Doppler shifts in combination with collisions cause photons that are absorbed at a given frequency to be redistributed in frequency across the line profile in a complex way during the scattering process. Two idealized, limiting cases to describe this redistribution are “frequency coherence” and “complete redistribution” (CRD), but the general theory that properly combines these two limiting cases goes under the name “partial frequency redistribution” (PRD). Resonance lines which are usually strong can be properly modeled only when PRD is taken into account. To use these strong lines for magnetic field diagnostics we need a line scattering theory of PRD in the presence of magnetic fields of arbitrary strength. In the second part of the thesis we develop such a theory and derive the polarized PRD matrices. These matrices are then used in the polarized line transfer equation to compute the emergent Stokes parameters. Polarized scattering in spectral lines is governed by a 4 x 4 matrix that describes how the Stokes vector is scattered in all directions and redistributed in frequency within the line. In Chapter 7, using a classical approach we develop the theory for this redistribution matrix in the presence of magnetic fields of arbitrary strength and direction, and for a J = 0 → 1 → 0 transition. This case of arbitrary magnetic fields is called the Hanle-Zeeman regime, since it covers both the partially overlapping weak and strong-field regimes, in which the Hanle and Zeeman effects respectively dominate the scattering polarization. In this general regime the angle-frequency correlations that describe the so-called PRD are intimately coupled to the polarization properties. We also show how the classical theory can be extended to treat atomic and molecular scattering transitions for any combinations of J quantum numbers. In chapter 8 , we show explicitly that for a J = 0 → 1 → 0 scattering transition there exists an equivalence between the Hanle-Zeeman redistribution matrix that is derived through quantum electrodynamics(Bommier 1997b) and the one derived in Chapter 7 starting from the classical, time-dependent oscillator theory of Bommier & Stenflo (1999). This equivalence holds for all strengths and directions of the magnetic field. Several aspects of the Hanle-Zeeman redistribution matrix are illustrated, and explicit algebraic expressions are given, which are of practical use for the polarized line transfer computations. In chapter 9, we solve the polarized radiative transfer equation numerically, taking into account both the Zeeman absorption matrix and the Hanle-Zeeman redistribution matrix. We compute the line profiles for arbitrary field strengths, and scattering dominated line transitions. We use a perturbation method (see eg. Nagendra et al. 2002) to solve the Hanle-Zeeman line transfer problem. The limiting cases of weak field Hanle scattering and strong field Zeeman true absorption are retrieved. The ilntermediate regime, where both Zeeman absorption and scattering effects are important, is studied in some detail. Numerical method used to solve the Hanle-Zeeman line transfer problem in Chapter 9 is computationally expensive. Hence it is necessary to develop fast iterative methods like PALI (Polarized Approximate Lambda Iteration). As a first step in this direction we develop such a method in Chapter 10 to solve the transfer problem with weak field Hanle scattering. We use a ‘redistribution matrix’ with coupling between frequency redistribution and polarization and no domain decomposition. Such a matrix is constructed by angle-averaging the frequency dependent terms in the exact weak field Hanle redistribution matrix for a two-level atom with unpolarized ground level (that can be obtained by taking the weak field limit of the Hanle-Zeeman redistribution matrix). In the past, the PALI technique has been applied to redistribution matrices in which frequency redistribution is ‘decoupled’ from scattering polarization, the decoupling being achieved by an adequate decomposition of the frequency space into several domains. In this chapter, we examine the consequences of frequency space decomposition, and the resulting decoupling between the frequency redistribution and polarization, on the solution of the polarized transfer equation for the Stokes parameters.