Physics 4150 – Optics Scattering, Polarization and Birefringence Light waves can scatter off water droplets, dust particles and other molecules in air Some wavelengths in white light scatter more than others (Rayleigh scattering) This is why sky is blue Polarization of light refers to direction of wave electric field Most light is unpolarized Electric field on a given ray jumps around in orientation while remaining perpendicular to ray Scattered or reflected light is polarized in particular planes Polaroid sunglasses help block polarized light from sky or ground Polaroid filters Apparent paradoxes Colors in stressed materials Birefringent materials Anisotropic (e.g., stressed) Can produce two shifted images Can create circular or elliptic polarization Can rotate plane of polarization of light wave Can display different colors

What is Rayleigh scattering? (or why is the sky blue) Occurs when resonant frequencies of molecules (e.g. O2) > light freqs. Shorter wavelengths scattered more blue scattered more than red. sky is blue and sunsets are red. Dust or smoke enhances red look of the sun by enabling more scattering Larger dust particles scatter red as well as blue and hence look white. Clouds. Think of white light from sun as a mixture of R, G and B Blue is scattered the most so sky looks blue when we look away from the sun For same reason sun looks yellow (red + green) More atmosphere allows next shortest wavelengths (green) to scatter so sunset looks red

Polarization of light

What is polarized light? Light polarized if waveform and E-fields remain in one plane E always perpendicular to ray Light ray traveling in z-direction and polarized in y-direction Light ray traveling in z-direction but polarized in x-direction Visualize polarization in x-y plane, looking at rays head-on E-vectors up and down or left and right (stacked up behind one-another). Convention for visualizing vertical and horizontal polarization Looking at ray "head-on" see E-vectors up & down y z x y z x y x

What is unpolarized light? Plane of polarization jumps around in angle in x-y plane E-vectors remain perpendicu-lar to ray (direction of travel) Amplitude of E-vector doesn't change Visualize in the x-y plane (looking into the ray) as shown at right The many crossed double sided arrows are the symbol for unpolarized light Partially polarized light Jumps direction only in a limited range of angles Electric field (force} vector x y z wave travels in z-direction y x y x

Unpolarized light can be broken up into vector sum of two in-phase orthogonally polarized components whose amplitudes jump around in time or space Feynman: x y z x y At z1 x y At z2 Unpolarized light can't have different polari- zations at same time because they would ADD Ex and Ey in phase. Retain amplitudes and overall phase until next jump

Different θ can also be interpreted as rotation of E Force, F, always equivalent to two simultaneous perpendicular forces (many ways to take components) Force, F, not pointing in x or y direc-tions always can be resolved into components: two simultaneous forces: one in x-direction, other in y-direction True for wave electric force on putative charged particle: F = qE The E-field of unpolarized light is equivalent to the vector sum of orthogonal in-phase components, Ex and Ey whose amplitudes and overall phase jump while Ex2 + Ey2 remains equal to E2 The time-average intensities are <Ex2> + <Ey2> = <E2> = E2 Since x and y are equivalent over long times, <Ex2> = <Ey2> = E2/2 y x ray coming at you x and y do NOT have to be vertical and horizontal. Can be in any two perpendicular directions. Proof: y x 90°-θ Different θ can also be interpreted as rotation of E θ

What happens when unpolarized light is passed through a polarizer? polaroid filter Suppose only the vertically polarized component of the unpolarized light comes throught the polarizer (horizontal component absorbed) It will jump around in both ampitude and phase while remaining vertically polarized It is linearly polarized but not coherent It is incoherent light The rms time-average intensity of the vetically polarized light is E2/2 x y z x y At z1 x y At z2

Linear polarization using polaroid filters

Polaroid filters use long molecules

Often use picket fence conven- tion for polarizing filters Polaroid filter is not like a picket fence although that convention is often used Often use picket fence conven- tion for polarizing filters = Video demo

Retain amplitudes and overall phase How does unpolarized light become polarized when passed through a horizontal polaroid filter? The E-field of unpolarized light is equivalent to the vector sum of orthogonal in-phase components, Ex and Ey whose amplitudes jump while Ex2 + Ey2 remains equal to E2 Polaroid filter allows only Ex or Ey to pass through Time-average, <Ex2> = <Ey2> = < E2>/2, so after passing through filter the intensity is half that of the unpolarized light Ex and Ey in phase. Retain amplitudes and overall phase until next jump

Crossed polarizers. Why does filter #3 allow light to pass? Video demo includes circular polarization

Explanation of three polaroid filter sandwich With only two filters (picket fences) E vertically-polarized after passing through 1st filter. No horizontal component. Nothing passes through 2nd filter With 3rd filter Yellow vector is completely equivalent to sum of two green vectors Only one of the green vectors gets through filter 3 Green vector is completely equivalent to sum of two red vectors Only one of red vectors gets through filter 2 Revisit case of only two filters Breakup of yellow vector into green and red vectors still makes sense 1st filter E after 1st filter E = 0 after 2nd filter 2nd filter 1st filter E after 1st filter E after 3rd filter E after 2rd filter 2nd filter 3rd filter = =

Polarization is sometimes used to see 3D color movies However the two orthogonal polarizations used in modern 3D movies are left and right circularly polarized rather than vertically and horizontally polarized

Linear polarization by reflection

When unpolarized light reflects off horizontal surface (water or beach) reflected light is horizontally polarized Brewster's angle Special angle of incidence for which refracted ray ⊥ reflected ray Decompose E of incident unpolarized light into two ⊥ components horizontal polarization: Eh parallel to reflecting horizontal surface, ⊥ ray E2 ⊥ to both ray and Eh E2 cannot be sustained in reflected ray because it would be parallel to ray and hence not a transverse field Only Eh survives in reflected ray Similar effect when look up at blue sky near right angle to sun Explanation coming

Polaroid sunglasses (and camera filters) block bright reflected light from snow, beach, road, water, sky, etc. Polaroid sunglasses block light polarized in plane of eyes Horizontal plane (for vertical head) Reflected light from highway or scattered light from sun are mostly polarized in horizontal plane Polaroid filter works because of long molecules lined up parallel to each other like hairs Components of light with polarization parallel to molecules are absorbed Components of light polarized perpendicular to molecules are transmitted (go through) Crossed polarizers Third polarizer at in-between angle lets light through again Stressed plastic between polarizers shows colors which reveal stress patterns (birefringence)

Polarizing Lenses: Example Without Polarizing lenses With Polarizing lenses

Linear polarization by Scattering

Linear polarization by scattering We see blue sky because sunlight is scattered by atmosphere Light from points in sky 90°from the sun will be linearly polarized When sun is overhead, the sky near the horizon will be most polarized

Polarization by Scattering When sun is at horizon, point of maximum polarization will be straight up in the sky. The part of the sky directly opposite the sun will be unpolarized, as will areas of the sky very near the sun Other areas of the sky will be partially polarized

Polaroid camera filters and the sky

Questions If Earth's atmosphere were much higher so that more Raleigh scattering of sunlight could occur what color would the sky appear to be? Red Green Blue Cyan Magenta If you look directly at a lightbulb wearing polaroid sunglasses and your head vertical, what do you see? [ light from an ordinary lightbulb is unpolar-ized] Complete blocking of the light Blocking of some of the horizontally polarized light and most of the other (perpendicular) polarization Blocking of most of the horizontally polarized light and very little of the other (perpendicular) polarization Blocking of most of the vertically polarized light No blocking of any of the light

Circularly and elliptically polarized waves When there is a phase difference between orthogonal polarization vectors in a light wave their vector sum E-field will be ellipically or circularly polarized The orthogonal vectors must be 90° out of phase and have equal amplitudes for the vector sum to be circularly polarized! Other phase differences or relative amplitudes produce ellliptic polarization, of which linear polarization is a limiting case E is vector sum of Ex and Ey. Tip of E- vector will trace out helix in z at given time or circle in x-y plane at a given z

Conventions for circularly polarized light Two different conventions Summary of conventions Online textbook ⇒ Antiparallel convention in optics Both conventions in physics (Jackson, EM Theory) ⇒ Parallel convention is con- sistent with positive helicity

More explanation of circular polarization k parallel to thumb Left/right handedness determined by poin-ting left or right thumb in wave propag-ation direction, and matching curl of one's fingers to direction of rotation of field at one z. Convention of non-optics physics com-munity (Plasma physics, high energy, etc) For k into paper, clockwise E(t) at one z = RT CIRC Polarized x E k y k antiparallel to thumb Alternative convention, — left/right handednes determined by pointing left/right thumb toward −k, and matching curl of fingers to rotation of E Many optics textbooks use this second convention E k E at different times at one pt For k out of paper, clockwise = RT CIRC Polarized E at different pts. along ray at one time E k E k Looking from other side, k points out of paper, thumb still || to k, but now counter-clockwise STILL RT CIRC Polarized.

Screw sense of spatial helix to parallel k when use antiparallel convention for finding rotation in time k parallel to thumb ⇒ t-dependence For k into paper, clockwise = RIGHT CIRC Polarized x E k E at different times at one pt k antiparallel to thumb ⇒ t-dependence ψ = kz − ωt ParametricPlot3D[{Cos[ψ], Sin[ψ], ψ/2π}, {θ,0,4π}] k k is in direction of helix advancement with correct RH rule kz E k E at different times at one pt For k out of paper, clockwise = RIGHT CIRC Polarized

Elliptically polarized light Two ways to make elliptically polarized light Let Ex and Ey have different amplitudes Let phase difference between equal-amplitude Ex and Ey be different from 90° (next slide)

Phase difference between two orthogonal equal-amplitude vector components of E-field determine kind of polarization

Circularly polarized light Animated GIF

Summary of linear, circular and elliptic polarization Can understand all of these polarizations as well as unpolarized light by thinking of the electric field vector as the sum of two orthogonal component vectors. Type of polarization depends on relative amplitude and phase of components

Birefringence Wikipedia

Polarization in birefringent crystals (half and quarter wave plates) Different indices of refraction depending on direction of wave electric field vector with respect to crystal axis Birefringent materials Anisotropic (e.g., stressed) Can produce two shifted images Can create circular or elliptic polarization Can rotate plane of polarization of light wave Can display different colors

Anisotropic media have different properties in different direction Calcite Free electrons in a DC magnetic field, B0 Plasma in B-field Thin metal film in B-film Structured but isotropic Anisotropic Unstructured and isotropic B0

Birefringent anisotropic media have a special direction called the optic axis y z E k Optical axis along z Propagation along z Ex and Ey waves see same index of refraction Normally incident monochromatic polarized light with k || optic axis has same index of refraction, n, for both components of E Normally incident light with k ⊥ optic axis has different n for wave with component of E || optic axis (E|| = Eextraordinary) and wave with component ⊥ optic axis (E⊥ = Eordinary) next = ckext/ω ≠ nord = ckord/ω Different wavenumbers, kext ≠ kord Different wavelengths, λext ≠ λord Different phase velocities, ω/kext ≠ ω/kord Slow versus fast wave x y z E k Optical axis along z Propagation along x Ez ( = E||) waves and Ey ( = E⊥) waves see different indices of refraction Optical axis

Double image of object viewed through birefringent crystal Two images Wolfram Demo Wikipedia Theory extraordinary wave (E polarized in x-z plane) ordinary wave (E polarized along y) images z x optic axis unpolarized ray from dot

Theory of EM waves in an anisotropic birefractive media

Quarter Wave Plate Uses Birefringence To Create Circularly Polarized Light Since they have different phase velocities, the component of E || to optic axis and the component of E ⊥ to axis can be phase-shifted relative to each other to rotate plane of polarization. For normal incidence and plate thickness to produce 90° phase difference between 2 components of E, circularly-polarized light will exit the plate. For other thicknesses, the phase differences lead to the following polarizations of the emerging light 

Polarized monochromatic light incident normally (k || x) on birefringent crystal. E in y-z plane at angle to optic axis (|| z) Polarization and quarter and half-wave plates video Polarized white light with E at 45° to optic axis enters crystal normally (k || x) E components: Ez and Ey || and ⊥ to optic axis Indices of refraction: ny and nz for Ey, Ez ky = (ω/c)ny and kz = (ω/c)nz Ey and Ez waves have same amplitude. Ey, Ez waves exit crystal with phase difference, Δψ = dΔk = ωdΔn/c Different polarization for different Δψ Quarter-wave plate: Δψ = ±π/2 +2πj gives circular polarization Half-wave plate: Δψ = ±π +2πj rotates plane of polarization of entering wave Elliptic polarization for other Δψ Back (x=d) E k, x Front (x=0) d E z, optic axis y Front E Ez Ey

Birefringent material between crossed polarizers Inside the birefringent plate the two components of the wave E-field have different wavenumbers: k1 = (ω/c)n1 and k2 = (ω/c)n2 so the phase difference between the two components upon exiting the plate = (ω/c)Δn·(plate thickness)

Analyzing stress in plastics between crossed linear polaroid filter Stress causes plastics to become birefringent which causes the plane of polarization to rotate E-fields corresponding to different wavelength rays are rotated by different amounts depending on stress and thickness, leading to pattern of colors after second crossed linear polarizer.

π/2 phase difference but could be any phase difference Why do we see different colors from incident white light when a birefringent material is put between crossed linear polarizers? White light has spectrum of wavelengths (colors) which do not interfere with each other. After 1st polarizer each wavelength is linearly polarized After 2nd polarizer each wavelength is again linearly polarized with contributions from ordinary and extraordinary waves. Birefringent crystals have different indices of refraction for different ω in white light spectrum because Δψ = dΔk = ωdΔn/c Ordinary and extraordinary waves at each wavelength (color) interfere constructively, destructively or in between. Different colors experience different degrees of constructive or destructive interference making them brighter or dimmer π/2 phase difference but could be any phase difference

Double image of object viewed through birefringent crystal Two images Wolfram Demo extraordinary-wave bends away from normal ordinary wave bends more away from normal optic axis x-wave bends towards normal Unpolarized o-wave bends more towards normal Unpolarized light ray from dot enters crystal