Descriptions of Polarized Light Scattering Polarization...

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  • Lecture 11: Polarized Light

    Outline

    1 Fundamentals of Polarized Light2 Descriptions of Polarized Light3 Scattering Polarization4 Zeeman Effect5 Hanle Effect

  • Fundamentals of Polarized Light

    Electromagnetic Waves in Matter

    electromagnetic waves are a direct consequence of Maxwellsequations

    optics: interaction of electromagnetic waves with matter asdescribed by material equations

    polarization properties of electromagnetic waves are integral partof optics

  • Plane-Wave SolutionsPlane Vector Wave ansatz

    ~E = ~E0ei(~k ~xt)

    ~k spatially and temporally constant wave vector~k normal to surfaces of constant phase|~k | wave number~x spatial location angular frequency (2 frequency)t time

    ~E0 a (generally complex) vector independent of time and spacedamping if ~k is complexreal electric field vector given by real part of ~E

  • Polarization

    spatially, temporally constant vector ~E0 lays in planeperpendicular to propagation direction ~krepresent ~E0 in 2-D basis, unit vectors ~e1 and ~e2, bothperpendicular to ~k

    ~E0 = E1~e1 + E2~e2.

    E1, E2: arbitrary complex scalarsdamped plane-wave solution with given , ~k has 4 degrees offreedom (two complex scalars)additional property is called polarizationmany ways to represent these four quantitiesif E1 and E2 have identical phases, ~E oscillates in fixed plane

  • Description of Polarized Light

    Polarization EllipsePolarization

    ~E (t) = ~E0ei(~k ~xt)

    ~E0 = E1ei1~ex + E2ei2~eywave vector in z-direction~ex , ~ey : unit vectors in x , ydirectionsE1, E2: (real) amplitudes1,2: (real) phases

    Polarization Description2 complex scalars not the most useful descriptionat given ~x , time evolution of ~E described by polarization ellipseellipse described by axes a, b, orientation

  • Jones Formalism

    Jones Vectors

    ~E0 = Ex~ex + Ey~ey

    beam in z-direction~ex , ~ey unit vectors in x , y -directioncomplex scalars Ex ,yJones vector

    ~e =(

    ExEy

    )phase difference between Ex , Ey multiple of , electric fieldvector oscillates in a fixed plane linear polarizationphase difference 2 circular polarization

  • Summing and Measuring Jones Vectors

    ~E0 = Ex~ex + Ey~ey

    ~e =(

    ExEy

    )

    Maxwells equations linear sum of two solutions again asolutionJones vector of sum of two waves = sum of Jones vectors ofindividual waves if wave vectors ~k the sameaddition of Jones vectors: coherent superposition of waveselements of Jones vectors are not observed directlyobservables always depend on products of elements of Jonesvectors, i.e. intensity

    I = ~e ~e = exex + eyey ,

  • Jones matricesinfluence of medium on polarization described by 2 2 complexJones matrix J

    ~e = J~e =(

    J11 J12J21 J22

    )~e .

    assumes that medium not affected by polarization statedifferent media 1 to N in order of wave direction, combinedinfluence described

    J = JNJN1 J2J1

    order of matrices in product is crucialJones calculus describes coherent superposition of polarizedlight

  • Quasi-Monochromatic Lightmonochromatic light: purely theoretical conceptmonochromatic light wave always fully polarizedreal life: light includes range of wavelengthsquasi-monochromatic lightquasi-monochromatic: superposition of mutually incoherentmonochromatic light beams whose wavelengths vary in narrowrange around central wavelength 0

    1

    measurement of quasi-monochromatic light: integral overmeasurement time tmamplitude, phase (slow) functions of time for given spatiallocationslow: variations occur on time scales much longer than the meanperiod of the wave

  • Polarization of Quasi-Monochromatic Light

    electric field vector for quasi-monochromatic plane wave is sumof electric field vectors of all monochromatic beams

    ~E (t) = ~E0 (t) ei(~k ~xt) .

    can write this way because 0measured intensity of quasi-monochromatic beam~Ex ~Ex

    +~Ey ~Ey

    = lim

    tm>

    1tm

    tm/2tm/2

    ~Ex (t)~Ex (t) + ~Ey (t)~Ey (t)dt ,

    : averaging over measurement time tm.measured intensity independent of timeQuasi-monochromatic: frequency-dependent material properties(e.g. index of refraction) are constant within

  • Stokes and Mueller Formalisms

    Stokes Vectorneed formalism to describe polarization of quasi-monochromaticlightdirectly related to measurable intensitiesStokes vector fulfills these requirements

    ~I =

    IQUV

    =

    ExEx + EyEyExEx EyEyExEy + EyEx

    i(ExEy EyEx

    ) =

    E21 + E

    22

    E21 E222E1E2 cos 2E1E2 sin

    Jones vector elements Ex ,y , real amplitudes E1,2, phasedifference = 2 1

    I2 Q2 + U2 + V 2 .

  • Stokes Vector Interpretation

    ~I =

    IQUV

    =

    intensitylinear 0 linear 90

    linear 45 linear 135circular left right

    degree of polarization

    P =

    Q2 + U2 + V 2

    I

    1 for fully polarized light, 0 for unpolarized lightsumming of Stokes vectors = incoherent adding ofquasi-monochromatic light waves

  • Mueller Matrices4 4 real Mueller matrices describe (linear) transformationbetween Stokes vectors when passing through or reflecting frommedia

    ~I = M~I ,

    M =

    M11 M12 M13 M14M21 M22 M23 M24M31 M32 M33 M34M41 M42 M43 M44

    N optical elements, combined Mueller matrix is

    M = MNMN1 M2M1

  • Polarized Light in Solar Physics

    Magnetic Field Maps from Longitudinal Zeeman Effect

  • Second Solar Spectrum from Scattering Polarization

  • Scattering Polarization

    Single Particle Scatteringlight is absorbed and re-emittedif light has low enough energy, no energy transfered to electron,but photon changes direction elastic scatteringfor high enough energy, photon transfers energy onto electroninelastic (Compton) scatteringThomson scattering on free electronsRayleigh scattering on bound electronsbased on very basic physics, scattered light is linearly polarized

  • Polarization as a Function of Scattering Angle

    same variation of polarization with scattering angle applies toThomson and Rayleigh scatteringscattering angle projection of amplitudes:

    1 for polarization direction perpendicular to scattering planecos for linear polarization in scattering plane

    intensities = amplitudes squaredratio of +Q to Q is cos2 (to 1)total scattered intensity (unpolarized = averaged over allpolarization states) proportional to 12

    (1 + cos2

    )

  • Solar Continuum Scattering Polarization

    (from Stenflo 2005)

    due to anisotropy of the radiation fieldanisotropy due to limb darkeninglimb darkening due to decreasing temperature with heightlast scattering approximation without radiative transfer

    http://www.astro.phys.ethz.ch/papers/stenflo/pdf/AA429_713_2005.pdf

  • Solar Spectral Line Scattering Polarization

    resonance lines exhibit large scattering polarization signals

  • Jupiter and Saturn

    (courtesy H.M.Schmid and D.Gisler)

    Planetary Scattered Light

    Jupiter, Saturn show scatteringpolarizationmultiple scattering changespolarization as compared to singlescatteringmuch depends on cloud heightequivalent effect to studyextrasolar planetary systemsExPo (Extreme Polarimeter)development here in Utrecht

  • Zeeman Effect

    photos.aip.org/

    Splitting/Polarization of Spectral Linesdiscovered in 1896 by Dutch physicistPieter Zeemandifferent spectral lines show differentsplitting patternssplitting proportional to magnetic fieldsplit components are polarizednormal Zeeman effect with 3components explained by H.A.Lorentzusing classical physicssplitting of sodium D doublet could notbe explained by classical physics(anomalous Zeeman effect)quantum theory and electrons intrinsicspin led to satisfactory explanation

    http://photos.aip.org/

  • Quantum-Mechanical Hamiltionianclassical interaction of magnetic dipol moment ~ and magneticfield given by magnetic potential energy

    U = ~ ~B

    ~ the magnetic moment and ~B the magnetic field vectormagnetic moment of electron due to orbit and spinHamiltonian for quantum mechanics

    H = H0 + H1 = H0 +e

    2mc

    (~L + 2~S

    )~B

    H0 Hamiltonian of atom without magnetic fieldH1 Hamiltonian component due to magnetic field

    e charge of electronm electron rest mass~L the orbital angular momentum operator~S the spin operator

  • Energy States in a Magnetic Field

    energy state ENLSJ | characterized bymain quantum number N of energy stateL(L + 1), the eigenvalue of ~L2

    S(S + 1), the eigenvalue of ~S2

    J(J + 1), the eigenvalue of ~J2,~J = ~L + ~S being the total angular momentumM, the eigenvalue of Jz in the state NLSJM|

    for the magnetic field in the z-direction, the change in energy isgiven by

    ENLSJ(M) = NLSJM|H1|NLSJM

  • The Land g Factor

    based on pure mathematics (group theory, Wiegner Eckarttheorem), one obtains

    ENLSJ(M) = 0gLBM

    with 0 = e~2m the Bohr magneton, and gL the Land g-factor

    in LS coupling where B sufficiently small compared to spin-orbitsplitting field

    gL = 1 +J(J + 1) + L(L + 1) S(S + 1)

    2J(J + 1)

  • hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html

    Spectral Lines -Transitions betweenEnergy States

    spectral lines aredue to transitionsbetween energystates:

    lower level with 2Jl + 1sublevels Ml

    upper level with 2Ju + 1sublevels Mu

    not all transitionsoccur

    http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html

  • Selection rulenot all transitions between two levels are allowedassuming dipole radiation, quantum mechanics gives us theselection rules:

    Lu Ll = L = 1Mu Ml = M = 0,1Mu = 0 to Ml = 0 is forbidden for Ju Jl = 0

    total angular momentum conservation: photon always carriesJphoton = 1normal Zeeman effect: line splits into three components because

    Land g-factors of upper and lower levels are identicalJu = 1 to Jl = 0 transition

    anomalous Zeeman effect in all other cases

  • Effective Land Factor and Polarized Componentseach component can be assigned an effective Land g-factor,corresponding to how much the component shifts in wavelengthfor a given field strengthcomponents are also grouped according to the linear polarizationdirection for a magnetic field perpendicular to the line of sight components are polarized parallel to the magnetic field (pi for

    parallel) components are polarized perpendicular to the magnetic field

    (sigma for German senkrecht)

    for a field parallel to the line of sight, the -components are notvisible, and the components are circularly polarized

  • Bernasconi et al. 1998

    Zeeman Effect in Solar Physicsdiscovered in sunspots byG.E.Hale in 1908splitting small except for insunspotsmuch of intensity profile dueto non-magnetic areafilling factora lot of strong fields outside ofsunspotsfull Stokes polarizationmeasurements are key todetermine solar magneticfields180 degree ambiguity

    http://aa.springer.de/bibs/8329002/2300704/small.htm

  • Fully Split Titanium Lines at 2.2m

    Redi et al. 1998

    http://aa.springer.de/papers/8338003/2301089/small.htm

  • Hanle Effect

    Bianda et al. 1998

    Depolarization and Rotationscattering polarizationmodified by magnetic fieldprecession around magneticfield depolarizes and rotatespolarizationsensitive 103 times smallerfield strengths that Zeemaneffectmeasureable effects even forisotropic field vectororientations

    http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1999sopo.conf...31B&db_key=AST&data_type=HTML&format=&high=43fb7cfc5930508