Feb. 9, 2011 Fourier Transforms Polarization Friday Presentations: Schwarz (Planck), Patel (JWST), Cox (SOFIA)

1. Feb. 9, 2011Fourier TransformsPolarizationFriday Presentations:Schwarz (Planck), Patel (JWST), Cox (SOFIA)

2. Fourier Transforms see Bracewell’s book: FT and Its Applications A function’s Fourier Transform is a specification of the amplitudes and phases of sinusoidals, which, when added together, reproduce the function Given a function F(x) The Fourier Transform of F(x) is f(σ) The inverse transform is note change in sign

3. Some examples (1) F.T. of box function

4. “Ringing” -- sharp discontinuity  ripples in spectrum When ω is large, the F.T. is narrow: first zero at other zeros at

5. (2) Gaussian F.T. of gaussian is a gaussian with narrower width Dispersion of G(x)  β Dispersion of g(σ) 

6. (3) delta- function x x1 Note:

7. Amplitude of F.T. of delta function = 1 (constant with sigma) Phase = 2πxiσ linear function of sigma

8. (4) x +x1 -x1 0 So, cosine with wavelength transforms to delta functions at +/ x1

9. (5) -x1 x x1 0

10. Summary of Fourier Transforms

13. Fourier Transforms: • Sharp features in the time domain  ringing in frequency domain • Narrow feature in time domain  broad radiation spectrum • Broad feature in the time domain  narrow radiation spectrum

15. POLARIZATION

16. The solution to the wave equation we considered was This describes a monochromatic beam which is linearly polarized –

17. More generally, consider a wave propagating in direction z The electric vector is the real part of where E1and E2are complex numbers. They can be written in the form The real part of is φ1 and φ2 are phases vector traces an ellipse with time  (1) describes an ELLIPTICALLY POLARIZED wave The tip of the

18. The tips of the Ex and Ey trace out an ellipse whose major axis is tilted with respect to the x- and y- axes, by angle χ or, in the x’, y’ coordinate system

19. Ellipse traced clockwise as viewed by an observer towardwhom the wave propagates Called RIGHT-HAND polarization, or negative helicity Ellipse trace counter-clockwise as viewed by an observer  LEFT-HAND polarization, or positive helicity

20. Negative and Positive Helicity

22. So... (1) (2)

23. Square (1) and (2) and add   Equation for an Ellipse

24. What does E look like for special cases? becomes y ε2 -ε1 x ε1  LINEARLY POLARIZED -ε2 Ex and Ey are in phase reach maxima together = 0 together

25. becomes y ε2  LINEARLY POLARIZED ε1 -ε1 x -ε2

26. LINEAR POLARIZATION

27. becomes  ELLIPTICALLY POLARIZED y y ε2 -ε1 x x ε1 -ε2  CIRCULARLY POLARIZED

28. Circular Polarization Note phase shift

29. SUMMARY

31. Stokes Parameters where

32. Define STOKES PARAMETERS

34. I: always positive proportional to fluxor intensity of wave V: measures circular polarization V=0 linear polarization V>0 right hand ellipticity V<0 left hand ellipticity Q,U: measure orientation of ellipse relative to x-axis Q=U=0 for circular polarization

35. For a monochromatic wave, you only need 3 parameters to describe it: For pure elliptical polarization The Stokes parameters are not independent: you need only specify 3, then can compute the 4th

36. A more general situation will involve the superposition of many waves, each with their own wavelength and polarization. Then one defines the Stokes parameters as time averages of the ε1, ε2,χ (note – in one nanosecond, a visible wave has ~106 oscillations) time average

37. Sometimes waves are completely unpolarized: phase difference between Ex and Ey are random No prefered direction in x-y plane, so Ex and Ey don’t trace an ellipse, circle, line etc. In this case: So... The intensity will consist of a polarized part (for which I2 = U2 + V2 + Q2) and an unpolarized part. Thus,

38. Degree of Polarization Special case: V=0 no circular polarization, but can have “partial linear polarization” i.e. Some of I is unpolarized  Some of I is polarized 

39. Angle of Polarization

40. Sources of Polarization of Light in Celestial Objects (1) Refelection off solid surfaces e.g. moon; plane mirrors

41. (2) Scattering of light by molecules: Rayleigh Scattering e.g. the blue sky

42. (3) Zeeman Effect e.g. Sunspots In the presence of a magnetic field of strength B, a line will split into several components, each with different polarization e.g. classical “Normal” Zeeman effect: An oscillating charge of mass m radiates with frequency ω0 in the absence of a B field. Apply B-field of strength B  splits into 3 lines circularly polarized linearly polarized

43. (4) Scattering of light by free electrons (Thomson scattering) e.g. solar corona (5) Synchrotron emission (e.g. radio galaxies) Radiation from relativistic electrons in B-field

44. (6) Scattering by dust grains e.g. polarization of starlight by dust grains aligned in the Milky Way’s B-field --- The Davis-Greeenstein Effect The interstellar magnetic field in the Milky Way will align paramagnetic dust grains – they tend to orient their long axes perpendicular to the B-field. E-field parallel to the long axis is blocked more than E-field perpendicular to the dust grains

45. Light from stars is unpolarized, but becomes polarized as it traverses the interstellar medium (ISM). Light becomes polarized parallel to the magnetic field  map of B-field The direction of polarization is shown below as short lines superimposed on a map of the hydrogen gas distribution in Galactic latitude and longitude. Note that the hydrogen gas filaments lie mostly parallel to the polarization directions of starlight, indicating that the gas concentrations are elongated parallel to the local magnetic field. This indicates that the gas filaments cannot be strongly self-gravitating. Cleary, Heiles & Haslam 1979

46. Polarimeters Most polarimeters rely on linear polarizers, i.e. “analyzers” which sub-divide the incident light into 2 beams: one beam linearly polarized parallel to the “principal plane” of the analyzer other beam perpendicular to it. TYPES OF ANALYZERS (1) Polarizer, polarizer film invented by Land in 1928, at age 19 Absorbs the component of the electric vector which oscillates in a particular direction  usually not used in astronomy, since you hate to throw out light

48. (2) Birefringent crystal e.g. calcite Has different index of refraction for waves oscillating in x-direction vs. the y-direction

49. (3) Wollaston Prism; Nicol Prism * To get equal intensities for the parallel and perpendicular beams when the incident beam is unpolarized, you can cement 2 pieces of birefringent crystal together, with the principle planes crossed. * This configuration also results in the widest separation of the 2 beams * Nicol prism: one beam reflected at the interface.

50. (4) Pockels Cell Single crystal emersed in a controllable E field. The external E field induces bi-refringence; can be varied.