Feb. 9, 2011 Fourier Transforms Polarizations see Bracewell’s book: FT and Its Applications Fourier Transforms 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) f( ) F ( x ) e dx 2 ix The inverse transform is F ( x ) ) e f( 2 i x dx note change in sign Some examples (1) F.T. of box function b() B(x)e 2ix 1 for x dx 2 B (x ) 0for xand x 2 2 2 e dx 2 ix 2 e 2ix 2 2 i 2 1 2i( 2 ) 2i( 2 ) e e 2i 1 cos isin() cos isin() 2i sin() sinc() 2 “Ringing” -- sharp discontinuity ripples in spectrum When ω is large, the F.T. is narrow: first zero at other zeros at 1 1 (2) Gaussian F.T. of gaussian is a gaussian with narrower width G ( x ) 1 2 2 x / e FT g ( ) e 2 2 2 Dispersion of G(x) β Dispersion of g(σ) 1 (3) (x x ) 1 delta- function ( x x ) 0 f x or x 1 1 x x1 Note: (x)dx1 ( x x ) F ( x ) dx F ( x ) ( x x ) dx 1 F ( x ) 1 1 1 FT of thedeltafunction: f ( ) (x x1)e 2ix dx 2ix1 e (x x )dx 1 2ix1 e Amplitude of F.T. of delta function = 1 (constant with sigma) Phase = 2πxiσ linear function of sigma (4) F ( x ) ( x x ) ( x x ) 1 1 x -x1 +x1 0 2 cos( 2 x ) 2 ix 1 FT f () e e 2 ix 1 1 So, cosine with wavelength 1 x1 transforms to delta functions at +/ x1 (5) F ( x ) ( x x ) ( x x ) 1 1 -x1 x 0 x1 FT f ( ) 2 i sin 2 x ) 1 Summary of Fourier Transforms 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 POLARIZATION The solution to the wave equation we considered was i ( k r t ) E aˆ1E0e This describes a monochromatic beam which is linearly polarized – E is alwaysin thesame direction,aˆ1 More generally, consider a wave propagating in direction z The electric vector is the real part of it E xˆE1 yˆE2 e where E1 and E2 are complex numbers. They can be written in the form i1 E1 1e The real part of E is E2 2e i2 φ1 and φ2 are phases Ex 1 cos(t 1 ) E y 2 cos(t 2 ) The tip of the E vector traces an ellipse with time (1) describes an ELLIPTICALLY POLARIZED wave 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 E x ' 0 cos cost E y ' 0 sin sin t where 2 is phasedifferencebetween Ex and Ey 2 For 0 2 Ellipse traced clockwise as viewed by an observer toward whom the wave propagates Called RIGHT-HAND polarization, or negative helicity For - 2 0 Ellipse trace counter-clockwise as viewed by an observer LEFT-HAND polarization, or positive helicity Negative and Positive Helicity Show that Ex 1 cos(t 1 ) E y 2 cos(t 2 ) is generallyan ellipse Ex 1 cos(t 1 ) cos(a b) cosa cosb sin a sin b cos(t ) cos1 sin(t ) sin 1 Ey 2 cos(t ) cos2 sin(t ) sin 2 So... Ex 1 sin 2 Ey sin 1 2 cos(t ) cos1 sin 2 cos(t ) cos2 sin 1 (1) cos(t ) sin(1 2 ) since sin(a b) sin a cosb cosa sin b Ex 1 cos2 Ey sin 2 2 sin(t ) sin(1 2 ) (2) Square (1) and (2) and add 2 Ex Ex E y Ey 2 cos1 2 1 1 2 2 sin 2 1 2 2 Equation for an Ellipse E is elliptically polarized Since B is always perpendicular toE , B is also elliptically polarizedand its 0 ellipse is rotated90 with respect toE ' s ellipse. What does E look like for special cases? (1) 1 2 2 2 Ex E x E y Ey 2 cos1 2 sin 2 1 2 1 1 2 2 2 becomes 2 Ex E x E y E y 2 0 1 1 2 2 y 2 Ex E y 0 1 2 ε2 -ε1 x ε1 -ε2 Ex 1 Ey 2 LINEARLY POLARIZED Ex and Ey are in phase reach maxima together = 0 together (2) 1 2 Ex 1 2 2 E x E y Ey 2 cos1 2 sin 2 1 2 1 2 2 becomes 2 Ex E y 0 1 2 y Ex ε2 ε1 -ε1 -ε2 x 1 Ey 2 LINEARLY POLARIZED LINEAR POLARIZATION (3) 1 2 Ex 1 2 2 2 E x E y Ey 2 cos1 2 sin 2 1 2 1 2 2 becomes 2 Ex E y 1 1 2 ELLIPTICALLY POLARIZED If 1 2 thenE describes a circle y y ε2 -ε1 2 x -ε2 ε1 x CIRCULARLY POLARIZED Circular Polarization Note phase shift SUMMARY Stokes Parameters Ex 1 cos(t 1 ) and Ey 2 cos(t 2 ) can be re - written(rotatingaxes)in termsof , Ex 0 cos cos cost sin sin sin t E y 0 cos sin cost sin cos sin t where 1 cos1 o cos cos 1 sin 1 o sin sin 2 cos2 o cos sin 2 sin 2 o sin cos Given 1,1, 2 ,2 solvefor 0 , , Define STOKES PARAMETERS I 2 1 2 2 2 0 Q cos 2 cos 2 2 1 2 2 2 0 U 21 2 cos1 2 cos 2 sin 2 2 0 V 21 2 sin 1 2 sin 2 2 0 02 I V sin 2 I U t an 2 Q It is customaryto describe thepolarization of the wave in termsof the ST OKES PARAMET ERS(I,Q, V, U) insteadof 0 , , or 1 , 2 , 1 , 2 02 I V sin 2 I U t an 2 Q I: always positive proportional to flux or 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 For a monochromatic wave, you only need 3 parameters to describe it: e.g. 0 , , For pure elliptical polarization I Q U V 2 2 2 2 The Stokes parameters are not independent: you need only specify 3, then can compute the 4th 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) I 2 1 2 2 Q 2 1 2 2 U 2 1 2 cos1 2 V 2 1 2 sin 1 2 time average 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... 2 1 2 2 Q U V 0 Q2 U 2 V 2 0 The intensity will consist of a polarized part (for which I2 = U2 + V2 + Q2) and an unpolarized part. Thus, I Q U V 2 2 2 2 Degree of Polarization intensityof polarizedpart of wave totalintensityof wave Q2 U 2 V 2 I 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 I unpolarize d I Q 2 U 2 I polarized Q 2 U 2 Angle of Polarization angle of maximumpolarization U tan2 Q Sources of Polarization of Light in Celestial Objects (1) Refelection off solid surfaces e.g. moon; plane mirrors (2) Scattering of light by molecules: Rayleigh Scattering e.g. the blue sky (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 eB 0 2mc 0 circularly polarized linearly polarized (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 (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 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 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 (2) Birefringent crystal e.g. calcite Has different index of refraction for waves oscillating in x-direction vs. the y-direction (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. (4) Pockels Cell Single crystal emersed in a controllable E field. The external E field induces bi-refringence; can be varied. (5) Wire grid analyzers and Dipole Antennas * Grid of parallel wires * Dipole antenna -- radio receivers Most sensitive to radiation which is linearly polarized with E parallel to the dipole for optimum efficiency, need to build “dual-polarization” receivers (6) Retarders e.g. sheet of mica A material produces a phase difference between the beams polarized parallel and perpendicular to the principle plane of the crystal – called retardance, τ Transforms a beam with Stokes Parameters I,Q,U,V to one with different Stokes Parameters I’, Q’, U’, V’ Quarter Wave Plate 2 If incident beam is circularly polarized: I=V, then the output beam is linearly polarized I=sqrt(Q2 + U2) Half Wave Plate Changes right-handed circular polarization to left-handed, or Changes linear polarization at angle θ to linear pol. at angle -θ Single channel polarimeters combine these elements to measure either circular or linear polarization: Some Practical Considerations for Polarimeters (1) Birefringent materials are scarce, and difficult to obtain in large sizes limits sizes of instruments, resolution of spectrographs (2) Refractive indices of material are a function of λ waveplates have retardance τ which is a function of λ need to achromatize (3) The night sky is polarized. Moonlight is VERY polarized. sky polarization must be monitored. (4) Spectropolarimetry * The wavelength dependence of the % polarization origin of polarization (e.g. dust vs. electron scattering vs. synchrotron) * Diffraction gratings, however, polarize light very strongly, like mirrors. design spectropolarimetry so that the polarizer is before diffraction grating in light path, not after (5) Telescope induced polarization * light is polarized by reflection * turns out, the induced linear polarization by each reflection is exactly canceled on axis for Cassegrain focus reflectors Prime focus reflectors refractors * TERRIBLE at Newtonian, Coude, Naysmyth foci \ because of the plane mirror reflection Other Applications of Polarization Rainbows are polarized David Lien, PSI EPOD 03/02/06 These two images of a late afternoon rainbow in Tucson, Arizona were taken within 20 seconds of each other (taken in July of 2005). The difference? Each picture was taken through a polarizing filter which was rotated 90 degrees between the two photographs. The light waves that are ultimately redirected to create the rainbow are reflected at the back of the raindrop, and the angle of redirection is very close to Brewster's angle for a water-air interface. Light reflecting off of a surface at Brewster's angle is 100% polarized. Since the sunlight resulting in a rainbow reflects off the back of a drop over a small range of angles, rainbows are not 100% polarized (~95% polarization is typical). Note that these images also show faint anticrepuscular rays created by storm clouds along the western horizon. Anti-glare windows LCD Displays Bees can see polarized light polarization of blue sky enables them to navigate Humans: Haidinger’s Brush Vikings: Iceland Spar? Many invertebrates can see polarization, e.g. the Octopus Not to navigate (they don’t go far) Perhaps they can see transparent jellyfish better? unpolarized polarized Polarization and Stress Tests In a transparent object, each wavelength of light is polarized by a different angle. Passing unpolarized light through a polarizer, then the object, then another polarizer results in a colorful pattern which changes as one of the polarizers is turned. CD cover seen in polarized light from monitor 3D movies Polarization is also used in the entertainment industry to produce and show 3-D movies. Three-dimensional movies are actually two movies being shown at the same time through two projectors. The two movies are filmed from two slightly different camera locations. Each individual movie is then projected from different sides of the audience onto a metal screen. The movies are projected through a polarizing filter. The polarizing filter used for the projector on the left may have its polarization axis aligned horizontally while the polarizing filter used for the projector on the right would have its polarization axis aligned vertically. Consequently, there are two slightly different movies being projected onto a screen. Each movie is cast by light which is polarized with an orientation perpendicular to the other movie. The audience then wears glasses which have two Polaroid filters. Each filter has a different polarization axis - one is horizontal and the other is vertical. The result of this arrangement of projectors and filters is that the left eye sees the movie which is projected from the right projector while the right eye sees the movie which is projected from the left projector. This gives the viewer a perception of depth.