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Doppler redistribution of anisotropic radiation and resonance polarization in moving scattering media. I. Theory revisited in the density matrix formalism

Sylvie Sahal-Bréchot, V. Bommier, N. Feautrier

Published Dec 1, 1998• 4 citations
OpenAlex

Abstract

Under the light of recent developments of the theory of matter-radiation interaction in the presence of magnetic field applied to non-LTE spectropolarimetry in astrophysics, we have revisited the theory of anisotropic resonance line scattering in moving media by means of the density-matrix formulation. This has led us to present a theoretical method of determination of the matter velocity field vector in the solar wind acceleration region. The example of the O Vi 103.2 nm line has been chosen for putting this theory into operation. It has been observed by the ultraviolet spectrograph \textsc{Sumer} of \textsc{Soho} in different regions of the solar wind acceleration region; it is partially formed by resonance scattering of the incident underlying transition region radiation which competes (and can predominate) with electron collisional excitation at the low densities which prevail at these high altitudes. The theory which is developed hereafter not only shows that this line is shifted and its intensity dimmed by the Doppler effect, due to the matter velocity field of the solar wind, but also predicts that it is linearly polarized, owing to the anisotropy of the incident radiation field; its two linear polarization parameters, degree and direction of polarization, are sensitive to the matter velocity field vector. Our results show that the interpretation of polarimetric data, associated to the shift and the Doppler-dimming effect, may offer a method of diagnostic of the complete velocity field vector, provided that the partial anisotropy of the incident radiation field be taken into account. In fact such a diagnostic is currently missing. Yet its interest is crucial to understand various problems in astrophysics, such as stellar winds, and especially the acceleration mechanisms of the solar wind. It is also essential for a dynamical modelling of solar structures.

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Diagnostic of the magnetic field vector using the atomic density matrix formalism.

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The authors present the work which has been done and which is being done using the atomic density matrix formalism, for the interpretation of linear polarization measurements, leading to the magnetic field diagnostic in astrophysical media. This formalism has been used for interpreting polarization data from solar prominences, leading to original results on the prominence magnetic field obtained through the Hanle effect. An attempt is made to solve the coupled problem of polarized radiative transfer and statistical equilibrium in the case of weak, intermediate and strong magnetic fields, using a method of the integral method type; the advantages of this method, with respect to other methods, of the differential method type, are briefly discussed.

•Aug 27, 1990•0 citations

Modification of a strong resonance radiation polarization in media with Doppler-broadened transitions of 0 to 1 and 1 to 0 types and in the presence of a DC electric field

Sergey V. Fomichev

For atomic media with resonant Doppler-broadened transitions of J=0 to J'=1 or J=1 to J'=0 type and taking into account the relaxation due to depolarizing collisions, a theory of non-linear optical polarization phenomena in the presence of a DC electric field is developed. The theory is valid in the case of both arbitrary resonance radiation intensity and arbitrary external DC electric field spatial orientation. The intensity-dependent contributions to variation of radiation polarization are analysed in detail in the Doppler limit. Special attention is paid to the case of linear polarization of incident radiation; in this case as well as in the presence of a DC electric field, the Doppler-enhanced non-linear optical contributions determining the emerging ellipticity and the rotation angle of polarization, respectively, are of the same order of magnitude (unlike the case of external magnetic field, in which the key effect is the rotation of polarization). It is shown, however, that in the case of the strictly transverse geometry and if the angle between the electric field direction and the initial polarization vector is equal to 45 degrees , the Doppler-enhanced non-linear optical contribution to the rotation angle becomes anomalously small. The effect of the presence of a longitudinal component of electric field on the non-linear optical polarization phenomena is investigated. In addition, although in the case of a purely longitudinal geometry for the considered transitions the electric field effect on radiation polarization modification is absent, some analytic results concerning self-rotation and deformation of a polarization ellipse in the Doppler limit for an intense initially elliptically polarized radiation are also presented.

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A density-matrix approach has been developed for the angular distribution and polarization of radiative emission during single-photon atomic transitions for a general set of steady-state excitation processes in an arbitrary arrangement of static (or quasistatic) electric and magnetic fields. Particular attention has been directed at spectroscopic observations in the intense fields of the high-power ion diodes on the Particle Beam Fusion Accelerator II (PBFA II) and SABRE devices at Sandia National Laboratories and at magnetic-field measurements in tokamak plasmas. The field-dependent atomic eigenstates are represented as expansions in a complete basis set of field-free bound and continuum eigenstates. Particular emphasis has been given to directed-electron collisional excitations, which may be produced by an anisotropic incident-electron velocity distribution. We have allowed for the possibility of the coherent excitation of the nearly degenerate field-dependent atomic substates, which can give rise to a complex spectral pattern of overlapping Stark-Zeeman components. Coherent excitations may be produced by a beam of electrons that are spin-polarized at an angle with respect to the propagation direction or by nonparallel electric and magnetic fields. Our main result is a general expression for the matrix elements of the photon-polarization density operator representing the total intensity, angular distribution, and polarization of the atomic radiative emission. For the observation of radiative emission in the direction of the magnetic field, the detection of linearly polarized emission, in addition to the usual circularly polarized radiation, can reveal the presence of a perpendicular electric field or a coherent excitation mechanism.

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Magnetic field rotation of a strong resonance radiation polarization plane in the media with Doppler-broadened transitions of 01 and 10 types

Sergey V. Fomichev

For atomic media with resonant Doppler-broadened transitions of J=0 to J'=1 or J=1 to J'=0 type and taking into account the relaxation due to depolarizing collisions a theory of nonlinear magneto-optic polarization phenomena is developed. The theory is valid in the case of both arbitrary resonance linearly-polarized radiation intensity and arbitrary external magnetic field spatial orientation. The Faraday and Voigt geometries are analysed in detail, and in the case of the Faraday geometry the effect is considered of a noncompensated magnetic field transverse component on the shape of a dependence relating the radiation polarization plane rotation angle to a magnetic field longitudinal component. In the case of the Voigt geometry an essential effect is found of the angle between the radiation initial polarization and transverse magnetic field vectors on the character of the dependence relating the radiation polarization plane rotation angle to the transverse magnetic field magnitude.

Journal of Physics B Atomic Molecular and Optical Physics•Nov 28, 1991•6 citations

The theory of optical birefringence

Laurence David Barron

The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.

•Jan 1, 1969•1 citation

Coherence structure of D1 scattering

Jan Olof Stenflo

Abstract The extensive literature on the physics of polarized scattering may give the impression that we have a solid theoretical foundation for the interpretation of spectro-polarimetric data. This theoretical framework has however not been sufficiently tested by experiments under controlled conditions. While the solar atmosphere may be viewed as a physics laboratory, the observed solar polarization depends on too many environmental factors that are beyond our control. The existence of a symmetric polarization peak at the center of the solar Na D1 line has remained an enigma for two decades, in spite of persistent efforts to explain it with available quantum theory. A decade ago a laboratory experiment was set up to determine whether this was a problem for solar physics or quantum physics. The experiment revealed a rich polarization structure of D1 scattering, although available quantum theory predicted null results. It has now finally been possible to formulate a well-defined and self-consistent extension of the theory of quantum scattering that can reproduce in great quantitative detail the main polarization structures that were found in the laboratory experiment. Here we give a brief overview of the new physical ingredients that were missing before. The extended theory reveals that multi-level atomic systems have a far richer coherence structure than previously believed.

Proceedings of the International Astronomical Union•Dec 1, 2014•0 citations
Under the light of recent developments of the theory of matter-radiation interaction in the presence of magnetic field applied to non-LTE spectropolarimetry in astrophysics, we have revisited the theory of anisotropic resonance line scattering in moving media by means of the density-matrix formulation. This has led us to present a theoretical method of determination of the matter velocity field vector in the solar wind acceleration region. The example of the O Vi 103.2 nm line has been chosen for putting this theory into operation. It has been observed by the ultraviolet spectrograph \textsc{Sumer} of \textsc{Soho} in different regions of the solar wind acceleration region; it is partially formed by resonance scattering of the incident underlying transition region radiation which competes (and can predominate) with electron collisional excitation at the low densities which prevail at these high altitudes. The theory which is developed hereafter not only shows that this line is shifted and its intensity dimmed by the Doppler effect, due to the matter velocity field of the solar wind, but also predicts that it is linearly polarized, owing to the anisotropy of the incident radiation field; its two linear polarization parameters, degree and direction of polarization, are sensitive to the matter velocity field vector. Our results show that the interpretation of polarimetric data, associated to the shift and the Doppler-dimming effect, may offer a method of diagnostic of the complete velocity field vector, provided that the partial anisotropy of the incident radiation field be taken into account. In fact such a diagnostic is currently missing. Yet its interest is crucial to understand various problems in astrophysics, such as stellar winds, and especially the acceleration mechanisms of the solar wind. It is also essential for a dynamical modelling of solar structures.
The authors present the work which has been done and which is being done using the atomic density matrix formalism, for the interpretation of linear polarization measurements, leading to the magnetic field diagnostic in astrophysical media. This formalism has been used for interpreting polarization data from solar prominences, leading to original results on the prominence magnetic field obtained through the Hanle effect. An attempt is made to solve the coupled problem of polarized radiative transfer and statistical equilibrium in the case of weak, intermediate and strong magnetic fields, using a method of the integral method type; the advantages of this method, with respect to other methods, of the differential method type, are briefly discussed.
For atomic media with resonant Doppler-broadened transitions of J=0 to J'=1 or J=1 to J'=0 type and taking into account the relaxation due to depolarizing collisions, a theory of non-linear optical polarization phenomena in the presence of a DC electric field is developed. The theory is valid in the case of both arbitrary resonance radiation intensity and arbitrary external DC electric field spatial orientation. The intensity-dependent contributions to variation of radiation polarization are analysed in detail in the Doppler limit. Special attention is paid to the case of linear polarization of incident radiation; in this case as well as in the presence of a DC electric field, the Doppler-enhanced non-linear optical contributions determining the emerging ellipticity and the rotation angle of polarization, respectively, are of the same order of magnitude (unlike the case of external magnetic field, in which the key effect is the rotation of polarization). It is shown, however, that in the case of the strictly transverse geometry and if the angle between the electric field direction and the initial polarization vector is equal to 45 degrees , the Doppler-enhanced non-linear optical contribution to the rotation angle becomes anomalously small. The effect of the presence of a longitudinal component of electric field on the non-linear optical polarization phenomena is investigated. In addition, although in the case of a purely longitudinal geometry for the considered transitions the electric field effect on radiation polarization modification is absent, some analytic results concerning self-rotation and deformation of a polarization ellipse in the Doppler limit for an intense initially elliptically polarized radiation are also presented.
A density-matrix approach has been developed for the angular distribution and polarization of radiative emission during single-photon atomic transitions for a general set of steady-state excitation processes in an arbitrary arrangement of static (or quasistatic) electric and magnetic fields. Particular attention has been directed at spectroscopic observations in the intense fields of the high-power ion diodes on the Particle Beam Fusion Accelerator II (PBFA II) and SABRE devices at Sandia National Laboratories and at magnetic-field measurements in tokamak plasmas. The field-dependent atomic eigenstates are represented as expansions in a complete basis set of field-free bound and continuum eigenstates. Particular emphasis has been given to directed-electron collisional excitations, which may be produced by an anisotropic incident-electron velocity distribution. We have allowed for the possibility of the coherent excitation of the nearly degenerate field-dependent atomic substates, which can give rise to a complex spectral pattern of overlapping Stark-Zeeman components. Coherent excitations may be produced by a beam of electrons that are spin-polarized at an angle with respect to the propagation direction or by nonparallel electric and magnetic fields. Our main result is a general expression for the matrix elements of the photon-polarization density operator representing the total intensity, angular distribution, and polarization of the atomic radiative emission. For the observation of radiative emission in the direction of the magnetic field, the detection of linearly polarized emission, in addition to the usual circularly polarized radiation, can reveal the presence of a perpendicular electric field or a coherent excitation mechanism.
For atomic media with resonant Doppler-broadened transitions of J=0 to J'=1 or J=1 to J'=0 type and taking into account the relaxation due to depolarizing collisions a theory of nonlinear magneto-optic polarization phenomena is developed. The theory is valid in the case of both arbitrary resonance linearly-polarized radiation intensity and arbitrary external magnetic field spatial orientation. The Faraday and Voigt geometries are analysed in detail, and in the case of the Faraday geometry the effect is considered of a noncompensated magnetic field transverse component on the shape of a dependence relating the radiation polarization plane rotation angle to a magnetic field longitudinal component. In the case of the Voigt geometry an essential effect is found of the angle between the radiation initial polarization and transverse magnetic field vectors on the character of the dependence relating the radiation polarization plane rotation angle to the transverse magnetic field magnitude.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.
Abstract The extensive literature on the physics of polarized scattering may give the impression that we have a solid theoretical foundation for the interpretation of spectro-polarimetric data. This theoretical framework has however not been sufficiently tested by experiments under controlled conditions. While the solar atmosphere may be viewed as a physics laboratory, the observed solar polarization depends on too many environmental factors that are beyond our control. The existence of a symmetric polarization peak at the center of the solar Na D1 line has remained an enigma for two decades, in spite of persistent efforts to explain it with available quantum theory. A decade ago a laboratory experiment was set up to determine whether this was a problem for solar physics or quantum physics. The experiment revealed a rich polarization structure of D1 scattering, although available quantum theory predicted null results. It has now finally been possible to formulate a well-defined and self-consistent extension of the theory of quantum scattering that can reproduce in great quantitative detail the main polarization structures that were found in the laboratory experiment. Here we give a brief overview of the new physical ingredients that were missing before. The extended theory reveals that multi-level atomic systems have a far richer coherence structure than previously believed.