The introductory lecture that has been delivered at this Symposium is a condensed version of an extended course held by the author at the XII Canary Island Winter School from November 13 to November 21, 2000. The full series of lectures can be found in Landi Degl'Innocenti (2002). The original reference is organized in 20 Sections that are here itemized: 1. Introduction, 2. Description of polarized radiation, 3. Polarization and optical devices: Jones calculus and Muller matrices, 4. The Fresnel equations, 5. Dichroism and anomalous dispersion, 6. Polarization in everyday life, 7. Polarization due to radiating charges, 8. The linear antenna, 9. Thomson scattering, 10. Rayleigh scattering, 11. A digression on Mie scattering, 12. Bremsstrahlung radiation, 13. Cyclotron radiation, 14. Synchrotron radiation, 15. Polarization in spectral lines, 16. Density matrix and atomic polarization, 17. Radiative transfer and statistical equilibrium equations, 18. The amplification condition in polarized radiative transfer, and 19. Coupling radiative transfer and statistical equilibrium equations.

The introductory lecture delivered at the Symposium has covered the subjects itemized above with the exclusion of Sections 10, 11, 14, 18, and 19.

In 1811, François Arago observed the disk of the Sun with his “lunette polariscopique”. From the absence of detectable polarization compared with his laboratory observations of glowing solids, liquids, and flames he concluded that the Sun's visible surface is an incandescent gas. From this beginning, thanks to orders of magnitude technology improvements, a remarkable amount of what we know about the physics of the Sun has continued to flow from solar polarimetry. This short review compares some selected polarimetric discoveries with subsequent recent observations to illustrate the tremendous progress of solar polarimetry during the last two centuries.

The past 20 years have seen remarkable advances in spectropolarimetric instrumentation that have allowed us, for the first time, to identify some magnetic stars in most major stages of stellar evolution. We are beginning to see the broad outline of how such fields change during stellar evolution, to confront theoretical hypotheses and models of magnetic field structure and evolution with detailed data, and to understand more of the ways in which the presence of a field in turn affects stellar structure and evolution.

The quiet Sun observed in polarized light exhibits a rich and complex magnetic structuring which is still not fully resolved nor understood. The present work is intended to contribute to the debate about the origin of the quiet sun magnetic fields, in relation or not to the global solar dynamo. We present analysis of center-to-limb polarization measurements obtained with the SOT/SP spectropolarimeter onboard the Hinode satellite outside active regions, in 2007 and 2013, i.e. at a minimum and a maximum of the solar cycle, respectively. We compare the spatial fluctuation Fourier spectra of unsigned circular and linear polarization images after corrections for polarization bias and focus variations between the two data sets. The decay of active regions is clearly a source of magnetic fields in the quiet Sun. It leads to a global increase of the polarization fluctuation power spectrum in 2013 in the network. In the internetwork, we observe no variation of the polarization fluctuation power at mesogranular and granular scales, whereas it increases at sub-granular scales. We interpret these results in the following way. At the mesogranular and granular scales very efficient mechanisms of magnetic field removal are operating in the internetwork, that leads to a dissipation or a concentration of magnetic fields on smaller scales. So the cycle-invariant magnetic signal that we detect at mesogranular and granular scales must be continuously created by a dynamo mechanism which is independent of the solar cycle.

Within the literature there are at least 15 references indicating that the horizontal magnetic flux does not exactly balance vertical flux in sunspots, leading to the surprising result that div $\vec{B}$ would depart from zero. Intuitively, this has to be related to the stratification at the surface of the star, due to which horizontal and vertical typical lengths are different. This surface anisotropy results from gravity, but how does gravity influence the magnetic field? To answer this question, a scenario has been proposed in two recent publications, based on anisotropic Debye shielding. The presentation reported in this paper was devoted to investigate the possibility and causes of a non-zero div $\vec{B}$. A scaling law associated with the anisotropy is able to reestablish the nullity of div $\vec{B}$, which would lead to a renewed MHD in the solar photosphere layer. An eventual observation in the laboratory is also reported.

NOAA 11035 was a highly sheared active region that appeared in December 2009 early in the new activity cycle. The leading polarity sunspot developed a highly unusual feature in its penumbra, an opposite polarity pore with a strong magnetic field in excess of 3500 G along one edge, which persisted for several days during the evolution of the region. This region was well observed by both space- and ground-based observatories, including Hinode, FIRS, TRACE, and SOHO. These observations, which span wavelength and atmospheric regimes, provide a complete picture of this unusual feature which may constitute a force-free magnetic field in the photosphere which is produced by the reconnection of magnetic loops low in the solar atmosphere.

The role of magnetic fields in late type stars, such as Asymptotic Giant Branch stars (AGBs), Post-AGBs and Planetary Nebulae (PNe), is poorly known from an observational point of view. Magnetic fields are however believed to have a non-negligible influence on the dynamics (via mass loss control, outflows shaping) and even on the chemistry (e.g. extra mixing) of these stellar objects. We are therefore presenting two different types of investigation, both based on the use of polarimetry, which aim at filling the gap between the observations on the one hand and the theoretical predictions on the other hand.

Full Stokes spectropolarimetric observations of a Mira star (χ Cyg) and a RV Tauri star (R Sct) are presented and analyzed comparatively. From their Stokes V data (circular polarization), we report the detection of a weak magnetic field at the surface of these cool and evolved radially pulsating stars. For both stars, we analyse this detection in the framework of their complex atmospheric dynamics, with the possibility that shock waves may imprint an efficient compressive effect on the surface magnetic field. We also report strong Stokes U and Stokes Q signatures associated to metallic lines (as a global trend), those linear polarimetric features appear to be time variable along the pulsating phase. More surprising, in the Stokes U and Stokes Q data, we also detect signatures associated to individual metallic lines (such as Sr i 460.7 nm, Na D2588.9 nm), that are known (from the solar case) to be easily polarizable in case of a global asymmetry at the photospheric level.

For several decades we have been cognizant of the presence of magnetic fields in early-type stars, but our understanding of their magnetic properties has recently (over the last decade) expanded due to the new generation of high-resolution spectropolarimeters (ESPaDOnS at CFHT, Narval at TBL, HARPSpol at ESO). The most detailed surface magnetic field maps of intermediate-mass stars have been obtained through Doppler imaging techniques, allowing us to probe the small-scale structure of these stars. Thanks to the effort of large programmes (e.g. the MiMeS project), we have, for the first time, addressed key issues regarding our understanding of the magnetic properties of massive (M > 8 M⊙) stars, whose magnetic fields were only first detected about fifteen years ago. In this proceedings article we review the spectropolarimetric observations and statistics derived in recent years that have formed our general understanding of stellar magnetism in early-type stars. We also discuss how these observations have furthered our understanding of the interactions between the magnetic field and stellar wind, as well as the consequences and connections of this interaction with other observed phenomena.

Observations of stable mainly dipolar magnetic fields at the surface of ~7% of single hot stars indicate that these fields are of fossil origin, i.e. they descend from the seed field in the molecular clouds from which the stars were formed. The recent results confirm this theory. First, theoretical work and numerical simulations confirm that the properties of the observed fields correspond to those expected from fossil fields. They also showed that rapid rotation does not modify the surface dipolar magnetic configurations, but hinders the stability of fossil fields. This explains the lack of correlation between the magnetic field properties and stellar properties in massive stars. It may also explain the lack of detections of magnetic fields in Be stars, which rotate close to their break-up velocity. In addition, observations by the BinaMIcS collaboration of hot stars in binary systems show that the fraction of those hosting detectable magnetic fields is much smaller than for single hot stars. This could be related to results obtained in simulations of massive star formation, which show that the stronger the magnetic field in the original molecular cloud, the more difficult it is to fragment massive cores to form several stars. Therefore, more and more arguments support the fossil field theory.

An extremely weak circularly-polarized signature was recently detected in the spectral lines of the Am star Sirius A. With a prominent positive lobe, the shape of the phase-averaged Stokes V line profile is atypical of stellar Zeeman signatures, casting doubts on its magnetic origin. We report here on ultra-deep spectropolarimetric observations of two more bright Am stars: β Uma and θ Leo. Stokes V line signatures are detected in both objects, with a shape and amplitude similar to the one observed on Sirius A. We demonstrate that the amplitude of the Stokes V line profiles depend on various line parameters (Landé factor, wavelength, depth) as expected from a Zeeman signature, confirming that extremely weak magnetic fields are likely present in a large fraction of Am stars. We suggest that the strong asymmetry of the polarized signatures, systematically observed so far in Am stars and never reported in strongly magnetic Ap stars, bears unique information about the structure and dynamics of the thin surface convective shell of Am stars.

We present an exceptional data set acquired with the Vacuum Tower Telescope (Tenerife, Spain) covering the pre-flare, flare, and post-flare stages of an M3.2 flare. The full Stokes spectropolarimetric observations were recorded with the Tenerife Infrared Polarimeter in the He i 1083.0 nm spectral region. The object under study was active region NOAA 11748 on 2013 May 17. During the flare the chomospheric He i 1083.0 nm intensity goes strongly into emission. However, the nearby photospheric Si i 1082.7 nm spectral line profile only gets shallower and stays in absorption. Linear polarization (Stokes Q and U) is detected in all lines of the He i triplet during the flare. Moreover, the circular polarization (Stokes V) is dominant during the flare, being the blue component of the He i triplet much stronger than the red component, and both are stronger than the Si i Stokes V profile. The Si i inversions reveal enormous changes of the photospheric magnetic field during the flare. Before the flare magnetic field concentrations of up to ~1500 G are inferred. During the flare the magnetic field strength globally decreases and in some cases it is even absent. After the flare the magnetic field recovers its strength and initial configuration.

Flux emergence phenomena are relevant at different temporal and spatial scales. We have studied a flux emergence region underneath a filament. This filament elevated itself smoothly, and the associated CME reached the Earth. In this study we investigate the size and the amount of flux in the emergence event. The flux emergence site appeared just beneath a filament. The emergence acquired a size of 24 Mm in half a day. The unsigned magnetic flux density from LOS-magnetograms was around 1 kG at its maximum. The transverse field as well as the filament eruption were also analysed.

On the agenda of modern astrophysics is the exploration of not only disk-integrated stellar magnetic fields but surface mapping of them. However, it is hardly possible to expect that spatial resolution better than some dozens or hundreds pixels over stellar disk will be achieved for this goal in the foreseeable future. Among other reasons this fact makes very important observations of the average and large-scale magnetic fields of the Sun, which can be naturally used for testing polarimetric measurements on other stars, especially on solar-type stars. In this study we explore different aspects of observations of solar magnetic fields (SMF) with low spatial resolution, including Sun-as-a-star observations, which are characterized by extremely low magnetic flux densities. Comparison of disk-integrated and spatially resolved Stokes observations of the Sun allow us to demonstrate how Stokes V profiles depend on the distribution of large-scale magnetic fields in the disk center. It is shown that center-to-limb variations of magnetic strength ratios (MSR) and area asymetries, most likely could be interpreted as the manifestation of kG magnetic flux tubes. We have made cross-calibration of the full-disk magnetograms obtained by space-borned SDO/HMI and by the ground-based STOP telescope, and pretty good agreement is found. Finally, the absence of significant systematic time variations of MSRs with solar cycle is demonstrated.

The near-infrared absorption line Fe I 15648 Å, which has a Landé g-factor of 3, shows a particularly large Zeeman splitting. We regularly take full-disk polarization maps of the Sun in the Fe I 15648 Å line (as well as the He I 10830 Å line) with an infrared spectropolarimeter installed at the Solar Flare Telescope of the National Astronomical Observatory of Japan (NAOJ). It is known that weak, mostly horizontal magnetic fields are ubiquitously distributed in the quiet regions of the Sun, while the strong magnetic fields are concentrated in active regions and network boundaries. The weak horizontal field has not been sufficiently investigated due to the difficulty of such observations. The polarization maps in Fe I 15648 Å show the magnetic field strength at each pixel, regardless of the filling factor, so we can easily isolate the weak horizontal field signals from strong magnetic field ones using the Stokes V profiles of the Fe I 15648 Å line. Here we present instrumental aspects and observational results of solar near-infrared full-disk polarimetry. We highlight the weak horizontal field inferred from Fe I 15648 Å.

The Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) provides continuous monitoring of the Sun's vector magnetic field through full-disk photospheric data with both high cadence and high spatial resolution. Here we investigate the evolution of AR 11249 from March 6 to March 7, 2012. We make use of HMI Stokes imaging, SDO/SHARPs, the HMI magnetic field line-of-sight (LOS) maps and the transverse components of the magnetic field as well as LOS velocity maps in order to detect regions with significant flux emergence and/or cancellation. In addition, we apply the Local Correlation Tracking (LCT) technique to the total and signed magnetic flux data and derive maps of horizontal velocity. From this analysis, we were able to pinpoint localized shear regions (and a shear channel) where penumbrae and pore formation areas, with strong linear polarization signals, are stretched and squeezed, showing also important downflows and upflows. We have also utilized Hinode/SP data and compared them to the HMI-SHARPs and the HMI-Stokes spectrograms. The aforementioned shear channel seems to correspond well with the X-class flare main channel of March 7 2012, as observed in AIA/SDO 171, 304 and 1600 Å.

The Daniel K. Inouye Solar Telescope (DKIST), formerly Advanced Technology Solar Telescope when it begins operation in 2019 will be by a significant margin Earth's largest solar research telescope. Science priorities dictate an initial suite of instruments that includes four spectro-polarimeters. Accurate polarization calibration of the individual instruments and of the telescope optics shared by those instruments is of critical importance. The telescope and instruments have been examined end-to-end for sources of polarization calibration error, allowable contributions from each of the sources quantified, and techniques identified for calibrating each of the contributors. Efficient use of telescope observing time leads to a requirement of sharing polarization calibrations of common path telescope components among the spectro-polarimeters and for those calibrations to be repeated only as often as dictated by degradation of optical coatings and instrument reconfigurations. As a consequence the polarization calibration of the DKIST is a facility function that requires facility wide techniques.

The Solar Orbiter is the next solar physics mission of the European Space Agency, ESA, in collaboration with NASA, with a launch planned in 2018. The spacecraft is designed to approach the Sun to within 0.28 AU at perihelion of a highly eccentric orbit. The proximity with the Sun will also allow its observation at uniformly high resolution at EUV and visible wavelengths. Such observations are central for learning more about the magnetic coupling of the solar atmosphere. At a later phase in the mission the spacecraft will leave the ecliptic and study the enigmatic poles of the Sun from a heliographic latitude of up to 33○.

A central instrument of Solar Orbiter} is the Polarimetric and Helioseismic Imager, SO/PHI. It will do full Stokes imaging in the Landé g = 2.5 Fe I 617.3 nm line. It is composed of two telescopes, a full-disk telescope and a high-resolution telescope, that will allow observations at a resolution as high as 200 km on the solar surface. SO/PHI will also be the first solar polarimeter to leave the Sun-Earth line, opening up new possibilities, such as stereoscopic polarimetry (besides stereoscopic imaging of the photosphere and stereoscopic helioseismology). Finally, SO/PHI will have a unique view of the solar poles, allowing not just more precise and exact measurements of the polar field than possible so far, but also enabling us to follow the dynamics of individual magnetic features at high latitudes and to determine solar surface and sub-surface flows right up to the poles.

In this paper an introduction to the science goals and the capabilities of SO/PHI will be given, as well as a brief overview of the instrument and of the current status of its development.

We present the innovative soft X-ray spectro-polarimeter, SolpeX. This instrument consists of three functionally independent blocks. They are to be included into the Russian instrument KORTES, to be mounted onboard the ISS. The three SolpeX units are: a simple pin-hole X-ray spectral imager, a polarimeter, and a fast-rotating drum multiple-flat-crystal Bragg spectrometer. Such a combination of measuring blocks will offer a new opportunity to reliably measure possible X-ray polarization and spectra of solar flares, in particular during the impulsive phase. Polarized Bremsstrahlung and line emission due to the presence of directed particle beams will be detected, and measurements of the velocities of evaporated hot plasma will be made. In this paper we discuss the details of the construction of the SolpeX units. The delivery of KORTES with SolpeX to the ISS is expected to happen in 2017/2018.

Magnetic fields in the solar atmosphere provide the energy for most varieties of solar activity, including high-energy electromagnetic radiation, solar energetic particles, flares, and coronal mass ejections, as well as powering the solar wind. Despite the fundamental role of magnetic fields in solar and heliospheric physics, there exist only very limited measurements of the field above the base of the corona. What is needed are direct measurements of not only the strength and orientation of the magnetic field but also the signatures of wave motions in order to better understand coronal structure, solar activity, and the role of MHD waves in heating and accelerating the solar wind. Fortunately, the remote sensing instrumentation used to make magnetic field measurements is also well suited to measure the Doppler signature of waves in the solar structures. We present here a mission concept for the Waves And Magnetism In the Solar Atmosphere (WAMIS) experiment which is proposed for a NASA long-duration balloon flight.