PHYS 450 Spring semester Lecture 13: Polarized Light. Ron Reifenberger Birck Nanotechnology Center Purdue University. Historical Timeline

Transcription

1 PHYS 450 Spring semester 2017 Lecture 13: Polarized Light Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 13 1 Historical Timeline 1669 Bartholinus describes image doubling properties of calcite Start of a 150 year odyssey: To explain light, you must explain birefringent properties of calcite! 1808 Malus discovers that calcite can modulate brightness of light passing through it. Studied polarization of light reflected from various surfaces (1809) First to describes light as being polarized Malus Law for crossed polarizers: I=I o cos 2 () 1817 Young realizes light must have a perpendicular component 1821 Fresnel claims light must be 100% transverse 1823 Fresnel derives equations for reflection of light (consistent with Maxwell s equations in 1860s) 1828 Nicol cuts thin slab of calcite makes first transmissive polarizer, produces plane polarized light 1928 Land invents a sheet-type dichroic linear polarizer (while an undergrad at Harvard University) 2 1

2 Polarization and the Plane of the E-field For light (electromagnetic radiation), polarization refers to the orientation of the electric (E-field) vector E. Polarization direction is by convention defined by the direction of the E-field. No fields at all! No fields at all! Snap shot at fixed time E o E=E y (z,t) B o z B=B z (z,t) zt, coskzt E coskzt xˆ E E o 2 k o 2 f f remains constant, v and can change Wave traveling in +z direction (cos could be replaced by sin). Phase between E, B radiative wave 3 y Essential Mathematics of Polarization x' y' z θ E o cos(θ) E o x Short hand notation: Linear polarized: Unpolarized: or -E o sin(θ) E E LP zt, E coskzt o or zte kztxˆ E kzt, cos cos ' sin cos yˆ' xˆ LP o o 4 2

3 INPUT: Unpolarized light Action of an IDEAL Polarizer All planes of E-field are possible Polarizing direction OUTPUT: Polarized light Polarizer Not all polarizers are ideal! 5 Reflection from an interface depends on incident polarization Incoming Ray θ i Reflected Ray n 1 Incoming Ray θ i Reflected Ray n 1 n 2 n 2 p-polarization (E-field lies in plane of incidence) s-polarization (E-field perpendicular to plane of incidence) 6 3

4 plane of incidence k i B i E i θ i θ r B r k r Er Fresnel Equations (deduced by Fresnel in 1823) y n 1 plane of incidence k i E i k r E r B i θ i θ r B r y n 1 Interface θ t E t z x n 2 Interface θ t E t z x n 2 p-polarization B t k t s-polarization B t k t r p E n cos or E n cos n t n cosi cos 1 2 oi 1 t 2 i r s E n cos or E n cos n i n cost cos 1 2 oi 1 i 2 t R p r 2 p R s r 2 s 7 Theory: Reflectivity vs. Incidence Angle ~ 4% Reflectivity s-polarized p-polarized Grazing incidence n2 tani tanb n1 Normal incidence x Incident Angle, θ i (in degrees) 8 4

5 Typical data for =543.5 nm Reflectivity s-polarized p-polarized p-pol data s-pol data Reflectivity s-polarized p-polarized p-pol data s-pol data Incident Angle, θ i (in degrees) Incident Angle, θ i (in degrees) 9 Measure Brewster s Angle for Prism Alignment! collimator fixed 1. Align face of prism with axis of prism table s rotation 2. Continuously adjust polarizer to block s and pass p 3. Continuously adjust prism to minimize p 4. When intensity is minimum, measure 5. Infer i =(-)= B 6. Repeat, take average value 10 5

6 Equipment: Establishing =0 Beware: internal reflections occur inside prism 11 Birefringent properties of calcite The understanding of polarization is closely related to study of crystalline CaCO 3, the most abundant mineral on earth after SiO 2 Unit Cell Legend: Green=Ca Black=C Red=O nm two CO 3 molecular ions twisted by 60 o 12 6

7 Optical Properties of Calcite: Calcite Crystal Structure a) E app CO 3 plane E ind Birefringence means the refractive index depends on the polarization direction of an EM wave + E app C + + O O b) E app // CO 3 plane a) Fast direction v // b) Slow direction E ind opposes E app v E app + O + C O E ind + E ind reinforces E app Crystal structure: a) Reduced polarizability lower ε lower n higher v fast b) Enhanced polarizability higher ε higher n lower v slow 13 Calcite Rhomb Birefringence in Calcite a) b) Slide over A Ordinary ray A What you see when viewed from the top Extraordinary ray Optical Axis Optical Axis

8 a) Sample A: Incident ray parallel to optical axis What s Going On? Cut three samples from a block of calcite b) Sample B: Incident ray perpendicular to optical axis c) Sample C: Incident ray at oblique angle to optical axis (naturally occurring calcite) Crystal Optical Axis Calcite Calcite Calcite Incident ray is linearly polarized 15 Birefringence in Calcite O ray passes through crystal E ray does not obey Snell s law. The ratio undeviated, suffering no refraction as sin i it should. Obeys Snell s law for ALL sin r = n 2 angles of incidence. depends on i velocity of E-ray depends on direction of incident ray. Snell s law n 1 sin i = n 2 sin r r Optical Axis n o Principal section (including all parallel planes) 71 o n 1 =1.00 Unpolarized light, angle of incidence i=0 o 16 8

9 Birefringence when viewed through a polarizer Ordinary ray A Polarizer Extraordinary ray A Polarizer Optical Axis Optical Axis 17 Wave Plates A waveplate or retarder is an optical device that alters the polarization state of a light wave travelling through it. Two common types of waveplates are half-wave plate: rotates the polarization direction of linearly polarized light quarter-wave plate: converts linearly polarized light into circularly polarized light and vice versa 18 9

10 Characteristics of Waveplate (aka Retarders) In principle, waveplates are wavelength specific. Zero Order Waveplate: the total retardation is the desired value without excess. True zero order waveplates for visible light are made from a single crystalline birefringent material that has been processed into a fragile ultra-thin plate only a few microns thick. Multiple Order Waveplates: total retardation is the desired value plus an integer number of wavelengths. In principle, the excess integer portion has no effect on performance. Both zero order and multiple order waveplates require precise control of the thickness of the plate. A ¼-wave plate converts linearly polarized light to circularly polarized light. 19 BASIC IDEA: E LP zt, E coskztˆ x More Generally: Linearly polarized wave, standard notation: E Phase Shifting Light z, t E coskzt ˆ E coskzt LP ox oy Phase shifted wave: E' o glass, n air x+ yˆ zt, E coskztˆ E coskzt LP ox oy d E LP 2 2 = E ˆ o cos d t n1 dx air ai r 2 zt, E cos ˆ o dtx air x+ yˆ E 2 2 zt, E cos d t ˆ E cos d t x= xˆ glass air n LP o o Example: for d=1 m, n=1.5, air =545 nm, then =1.83π 20 10

11 Extension to Birefringent Material E E BASIC IDEA: Birefringent slab, n fast & n slow LP LP zt, E coskztˆ x o zt, E coskzt yˆ o air In calcite: n slow = , n fast =1.486 d n slow n fast E 2 2 zt, E cos dt x= ˆ E cos d t ˆ x glass air n slow LP o o E 2 2 = E cos 1 ˆ o d t nslow d x air air 2 2 zt, E cos dt y= ˆ E cos dt ˆ y glass air n fast LP o o 2 2 = E cos 1 ˆ o d t n fast d y air a ir html 21 Arrows show E-fields n slow linearly polarized n fast optical axis The ¼-wave plate (converts linearly polarized light into circular polarized light) d Horizontal polarization goes through plate slower phase shifted Vertical polarization goes through plate faster Tracing the tip of the total E-field reveals a helix, with a period of precisely one wavelength. This represents circularly polarized light. What is typical time delay for ¼-plate? Let s say the wave plate is designed to operate at 560 nm plate = air /n 560 nm/1.5 = 375 nm One-quarter of 375 nm is d = 95 nm. How long does it take light to travel 95 nm? Time delay Δt ~ d/(c/n) ~ 0.5 fs fast slow 2 2 OPL n n d n n d air OPL fast slow air air If 1 OPL wave plate 4 4 air 1 then d 4 n n fast slow 22 11

12 The ¼-wave vs. the ½-wave plate ¼-wave plate Converts linear to circular polarization ½-wave plate Changes plane of linear polarization 2 45 o Fast axis Fast axis Operation Required Configuration Operation Required Configuration Change linear polarization to circular Change circular polarization to linear Insert a 1/4 plate with axis at 45 to the input polarization. Insert a 1/4 plate with axis at 45 to the desired output polarization. Rotate linear polarization to different orientation Change the handedness of circular polarization Insert a 1/2 plate with axis at ½ the desired rotation angle. Insert a 1/2 plate, orientation unimportant 23 Equipment Na Lamp Polarizer #2 ¼-wave plate Polarizer #1 Dial for azimuthal angle of ¼-wave plate and polarizers ¼-wave mica plate VERY FRAGILE 24 12

13 The ¼-wave plate experiment a) b) c) 1. Adjust Polarizer 2 discharge lamp slit Polarizer 2 discharge lamp slit Polarizer 2 discharge lamp slit Collimator Collimator Collimator ¼ wave plate 3. Adjust ¼-wave plate 5. Rotate 45 o Polarizer 1 Telescope Polarizer 1 Telescope 6. Rotate Polarizer 1 thru 360 o Telescope 2. Extinction, no light 4. Extinction, no light 7. Always bright! 25 The Viking Sun Stone Sunstones could have helped the Vikings in their navigation from Norway to America, well before a magnetic compass was introduced in Europe

14 Birefringent Properties of Calcite 1) Unpolarized light through hole a) hole 1 Calcite crystal paper b) 2 Rotate 1 4 calcite crystal Spot 1 remains stationary (ordinary) Two bright spots always observed as crystal is rotated 2) Polarized light through hole Insert polarizer beneath calcite a) hole 1 Calcite crystal paper b) 2 1 Rotate calcite crystal Spot 1 remains stationary (ordinary) spot disappears! Intensity of 2nd bright spot is modulated as crystal is rotated 27 Zenith Light is polarized by scattering from gas molecules Zenith max polarization Sky Point Viking Observer 90 o Sun Sun Horizon Horizon a) Scattered light is polarized perpendicular to the plane containing the sky point, the sun, and the Viking observer b) Maximum polarization occurs in a direction 90 o from the sun in a plane containing the sun, the zenith, and the Viking observer 28 14

15 Application: Viking Sun Stone Sun (hidden) Search for direction of maximum polarization 90 o Move screen around while rotating calcite crystal until intensity transmitted is a minimum Sun Stone (Icelandic Spar) Viking Observer KEY IDEA: If the calcite is oriented to block out the polarized component, the sky appears darker. For demo, see: 29 Altering the Polarization of Light Produces a Myriad of Optical Effects 1. Rotation of Plane Polarized Light by a Sugar Solution 2. Birefringent polymer filters 3. Photoelasticity etc. Light box 30 15

16 Regular corn syrup contains only glucose. Chromatographic separation High fructose corn syrup (HFCS) contains mainly fructose and is allegedly obesigenic. Some Facts about Sugar Glucose (aka dextrose) (C 6 H 12 O 6 ) and Fructose (C 6 H 12 O 6 ) have identical chemical formulas Glucose and Fructose are chiral molecules (cannot be superposed onto a mirror image). Molecules with a chiral structure exhibit circular birefringence. Circular birefringence: different refractive index forleft-handcircular(lhc)and right-hand circular (RHC) polarized light, i.e. n RHC (λ) n LHC (λ). No matter where the glucose comes from, it is always dextro-rotary, i.e. it rotates light clockwise (defined from the viewpoint when light approaches an observer). Fructose on the other hand always rotates polarized lightinacounterclockwisedirectionandisknownas levo-rotary. from the Latin: d=dextera right, l=laevus left. 31 Rotation of Plane Polarized Light by a Sugar Solution Linearly polarized light is equivalent to the superposition of LHC and RHC polarized light. The interaction of linearly polarized light with glucose (aka dextrose) molecules causes a rotation in the plane of polarization. If naturally produced sugar has equal amounts of left-handed and right-handed glucose molecules, the plane of polarization of linearly polarized light will not rotate while in transit. If naturally produced sugar is comprised solely of right-handed glucose molecules, then the plane of polarization of linearly polarized light will rotate while in transit. Experiment shows a linearly polarized beam passing through a solution rich in glucose is systematically rotated in only one direction. Thus nature preferentially makes one chirality glucose molecule over the other. scattered light transmitted light 32 16

17 Experiment Glass tube filled with sugar solution Polarized laser d Polarized laser Glass tube E-field What does it mean when the scattered light disappears? 33 Color by subtraction in thin polymer films Optically Isotropic Optically Anisotropic cellophane acts as a Stretched polymer strands birefringent filter for white light Unstretched polymer strands Optic Axis optic axis is parallel to plane of film s (,thickness) Linearly polarized white light film s thickness 34 17

18 Example: Cellophane wrappers Polarascope: Two Crossed Polarizers cellophane wrapper Cellophane acquires color by subtraction 35 Example: Cellophane wrappers viewed through crossed polarizers Brightest image when optic axis is at 45 o angle wrt two crossed polarizers 36 18

19 Photoelasticity Stress-induced birefringence (Coloration Indicates Stress Distribution) Polarizer #1 Polarizer #1 Crossed Polarizer #2 Molded Plastic Cuvette Cuvette Light Box Light Box 37 Up Next Wrap up 38 19

20 Appendix: Phase shifts between E and B Most of this lecture was about shifting the phase of E-fields in an EM wave. Don t mix this up with phase shifts between the E and B-fields. Example: When the E and B-fields are phase shifted, the fields become reactive, not radiative. The energy oscillates back and forth instead of actually propagating as it would in radiation. Such EM-fields are not produced by a dipole antenna