Radiative Transfer with Polarization

Transcription

1 The Radiative Transfer Equation with Polarization Han Uitenbroek National Solar Observatory/Sacramento Peak Sunspot, USA Hale COLLAGE, Boulder, Feb 16, 2016

2 Today s Lecture Equation of transfer with polarization Solving the transfer equation in a multi-dimensional environment Formation height, contribution function and response function

3 Basic Radiative Transfer: Transport Equation Transport along a ray: di λ ds = η λ χ λ I λ (1) di λ χ λ ds = di λ dτ λ = S λ I λ ; S λ η λ χ λ Integral form, the formal solution: I (τ) = I (0)e τ + τ 0 S(τ )e (τ τ ) dτ ; dτ = χds

4 A plane electromagnetic wave B E E( r, t) = (A sin(kz ωt), 0, 0) B( r, t) = (0, A sin(kz ωt), 0)

5 Partially polarized light Most sources of electromagnetic radiation contain a large number of atoms or molecules that emit light. The orientation of the electric fields produced by these emitters may not be correlated, in which case the light is said to be unpolarized If there is partial correlation between the emitters, the light is partially polarized Partially polarized light can be described as the superposition of a completely unpolarized component, and a completely polarized one

6 General description of polarized light E( r, t) = (E 1 cos(kz ωt), E 2 cos(kz ωt + φ), 0) x y z Linear Polarization: E 1 = E 2 φ = 0

7 General description of polarized light E( r, t) = (E 1 cos(kz ωt), E 2 cos(kz ωt + φ), 0) x z y Circular Polarization: E 1 = E 2 φ = 90

8 The Polarization Ellipse, General Description of Polariztion George Gabriel Stokes, 1852 S 0 I I S 1 S 2 = Q U pi cos(2ψ) cos(2χ) pi sin(2ψ) cos(2χ) S 3 V pi sin(2χ)

9 Stokes parameters Q V U I Q U = V p E E 2 2 E 2 1 E 2 2 2E 1 E 2 cos(φ) 2E 1 E 2 sin(φ) Q 2 + U 2 + V 2 I 2

10 Polarization altering materials Dichroic: Media in which the amplitude of waves propagating in one of the modes is reduced. Example: polarizer Birefringent: Media in which the two modes accrue a differential propagation delay. Example: wave plate

11 Linear polarizer and quarter-wave plate Left Handed Circularly Polarized Light Linearly Polarized Light Linearly Polarized Light Quarter Wave Plate Linear Polarizer

12 Stokes parameters give full description of polarized radiation field. Müller matrix describes interactions with materials The 4-element Stokes vector S = (I, Q, U, V ) give a full description of the intensity and polarization state of the radiation field Interactions of the polarized radiation field with material can be described through 4 4 matrices, the so-called Müller matrices M. I M 11 M 12 M 13 M 14 I Q Q S out = U V out = M S in = M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 Based on constraints that follow from the process it describes the Müller matrix M has 16 elements, of which only 7 are independent U V in

13 Radiative Transfer Equation ds dω I da I + di l Source function for radiative transport along a ray: di λ ds = j λ k λ I λ S λ j λ /k λ

14 Equation of Polarized Radiative Transfer ds I dω da I + di l Transfer Equation: di ds = KI + j I = (I, Q, U, V ), (Stokes vector) K = α c 1 + α c Φ, (Absorption matrix) j = (j c + j l Φ)e 0, e 0 = (1, 0, 0, 0)

15 Line Absorption Matrix z B γ x x y φ I φ Q φ U φ V Φ = φ Q φ I ψ V ψ U φ U ψ V φ I ψ Q φ V ψ U ψ Q φ I φ I = φ sin 2 γ (φ + + φ ), φ = 1 [ 2 φ0 1 2 (φ + + φ ) ] φ Q = φ sin 2 γ cos 2χ φ U = φ sin 2 γ sin 2χ φ V = 1 2 (φ + φ ) cos γ

16 Zeeman splitting and Doppler broadening φ 0 = H(a, v + v los ); φ ± = H(a, v ± v B + v los ) H(a, v) = a π exp ( y 2 ) (v y) 2 + a 2 dy v n v los = λ ; c λ D eλ 2 B v B = g L 4πmc λ D λ D = v broadλ ; v broad = 2kT /m c

17 Fe i nm polarization for different field strengths Stokes I / I cont Wavelength [nm] Stokes Q / I Wavelength [nm] Stokes U / I Wavelength [nm] Stokes V / I Wavelength [nm]

18 The dynamic and Inhomogeneous Solar Atmosphere Courtesy Yukio Katsukawa, NAOJ, Japan

19 Hydrostatic Model FAL C (average quiet Sun)

20 Formal solution in 1-D Solution to transfer equation: I ν (τ ν ) = 0 S ν (t)e t dt

21 A vertical cross section through a 3-D Simulation 8 6 x [arcsec] x [arcsec] z [km] T [10 3 K] x [arcsec]

22 Formal solution in 2- and 3-D If each ray is interpolated, the size of the transfer problem is of order N 2 N!

23 Short-Characteristics in Multi-Dimensional Geometry Kunasz & Auer (1988), J. Quant. Spectrosc. Radiat. Transfer, 39, 67 C B A τb I B = I A e τ AB + S(τ)e (τ τab) d τ τ A

24 Formation height of a spectral feature How do we determine the formation height or region of influence of a spectral feauture? What is the formation height of line X? First look at the transfer equation and formal solution:

25 Formation height of a spectral feature How do we determine the formation height or region of influence of a spectral feauture? What is the formation height of line X? First look at the transfer equation and formal solution: Equation of radiative transfer: di ds = χi + η = χ(i S); S = η/χ

26 Formation height of a spectral feature How do we determine the formation height or region of influence of a spectral feauture? What is the formation height of line X? First look at the transfer equation and formal solution: Equation of radiative transfer: di ds = χi + η = χ(i S); S = η/χ Integral form, the formal solution: I (τ) = I (0)e τ + τ 0 S(τ )e (τ τ ) dτ ; dτ = χds

27 Simplest case: Eddington Barbier relation Eddington Barbier relation: S = a + bτ I = a + b = S λ (τ = 1) [nm] z [km] λ[nm] Source Function [J m 2 s 1 Hz 1 sr 1 ] x [km]

28 Contribution function I (λ) = h0 ( ) S(λ, h)e τ(λ,h) dτ(λ, h) dh. dh

29 Contribution function I (λ) = h0 ( ) S(λ, h)e τ(λ,h) dτ(λ, h) dh. dh Height [km] Contribution function [J m 2 s 1 Hz 1 sr 1 km 1 ] Wavelength [nm]

30 Contribution function Fe i nm line (LTE) Height [km] Contribution function [J m 2 s 1 Hz 1 sr 1 km 1 ] Wavelength [nm]

31 Response function Definition: I (λ) I (λ) = h0 h0 R I,X (λ, h) X (h)dh R I,X (λ, h) X (h)dh

32 Response function: Numerical Solution Numerical derivation: I (λ) = h0 R I,X (λ, h) X (h)dh

33 Response function: Numerical Solution Numerical derivation: I (λ) = h0 R I,X (λ, h) X (h)dh Using delta function X (h ) = x(h )δ(h h): I h (λ) = R I,X (λ, h) = h R I,X (λ, h ) x(h )dh 1 x(h) I h (λ)

34 Example of Response Function: the Ca i nm Line Height [Mm] R T,I Wavelength [nm] 0

35 Response function Ca i Stokes V to B Height [Mm] R B,V Wavelength [nm]

36 Contribution function Na i D 2 line Height [km] Contribution function [J m 2 s 1 Hz 1 sr 1 km 1 ] Wavelength [nm]

37 Response function Na i D 2 Stokes V to B Height [Mm] Response function Wavelength [nm]

38 Conclusions Radiative transfer of the 4 Stokes parameters describes propagation of polarized radiation throgh a medium and its interaction with the medium. Polarization measurements provide additional diagnostics of astrophysical bodies through the encoding of physical properties in the polarization. Eddington Barbier and contribution and response functions can be used to estimate the formation height of a spectral feature, with increasing fidelity.