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Phys 322 Chapter 8 Lecture 22 Polarization Reminder: Exam 2, October 22nd (see webpage) Dichroism = selective absorption of light of certain polarization Linear dichroism - selective absorption of one of the two P-state (linear) orthogonal polarizations Circular dichroism - selective absorption of L-state or R-state circular polarizations Using dichroic materials one can build a polarizer Dichroic crystals Anisotropic crystal structure: one polarization is absorbed more than the other Example: tourmaline Elastic constants for electrons may be different along two axes Polaroid 1928: dichroic sheet polarizer, J-sheet long tiny crystals of herapathite aligned in the plastic sheet Edwin Land 1938: H-sheet 1909-1991 Attach Iodine molecules to polymer molecules - molecular size iodine wires Presently produced: HN-38, HN-32, HN-22 Birefringence Elastic constants for electrons may be different along axes Resonance frequencies will be different for light polarized Refraction index depends on along different axes polarization: birefringence Dichroic crystal - absorbs one of the orthogonal P-states, transmits the other Optic axis of a crystal: the direction of linear polarization along which the resonance is different from the other two axes (assuming them equal) Calcite (CaCO3) Ca C O Image doubles Ordinary rays (o-rays) - unbent Extraordinary rays (e-rays) - bend Calcite (CaCO3) emerging rays are orthogonaly polarized Principal plane - any plane that contains optical axis Principal section - principal plane that is normal to one of the cleavage surfaces Birefringence and Huygens’ principle Crystal structure and birefringence NaCl - cubic crystal Four 3-fold symmetry axes - optically isotropic Anisotropic, uniaxial birefringent crystals: hexagonal, tetragonal, trigonal Uniaxial crystal: atoms are arranged symmetrically around optic axis O-wave - E everywhere is perpendicular to the optic axis, no c/v When E is parallel to optic axis: ne c/v|| Birefringence ne - no principal indices of refraction Calcite: (ne - no) = -0.172 (negative uniaxial crystal) Crystal structure and birefringence Negative uniaxial crystal Positive uniaxial crystal Biaxial crystals Two optic axes and three principal indices of refraction Orthorhombic, monoclinic, triclinic Example: mica, KH2Al3(SiO4)3 Birefringent polarizers Nicol prism Glan-Thompson Wollaston prism polarizer 1828, William Nicol Polarization by scattering Scattering can polarize light Scattering can depolarize light Polarization by reflection Reminder: Fresnel equations show r||=0 at B - Brewster angle 1 B tan n2 n1 At Brewster angle reflected and transmitted rays form right angle B Reflected light will be fully polarized at B Fresnel equations 2 2 tan i t R r R|| 2 Fresnel equations tan i t 2 sin sin i t i t R r 2 sin i t sini t tan r i t || tan … i t Reflectance of unpolaized light: R R R || 2 Fresnel equations Multiple plate polarizer Degree of polarization: V I p Itot analyzing polarizer I I V max min Imax Imin detector Glare is horizontally polarized Puddle reflection viewed Puddle reflection viewed through polarizer that through polarizer that transmits only horizontally transmits only vertically polarized light polarized light Light reflected into our eyes from the puddle reflects at about Brewster's Angle. So parallel (i.e., vertical) polarization sees zero reflection. Polarizer sunglasses transmit only vertically polarized light. Polarizers are very useful in photography. Without a polarizer With a polarizer The effect of a polarizer is probably the one “filter” effect that you can’t reproduce later using Photoshop! Retarders The two orthogonal P-states develop mutual phase shift as they pass through a retarder After propagating through: E E|| E polarized light phase shift: E E|| E birefringence extraordinary 2 in-phase ordinary dne no 0 in vacuum direction of polarization: E|| E0|| coskx t fast axis - the one that propagates faster slow axis - the one that propagates slower E E0 coskx t Full-wave plate 2 dn n e o = m.2, m=…,-1,0,+1,… 0 E|| E0|| coskx t Alternatively: dne no m0 E E0 coskx t full wave - no observable effect Note: n=n(), and it is generally true only for one wavelength Half-wave plate = +m.2, m=…,-1,0,+1,… 2 dne no 0 dne no 0 / 2 m0 E|| E0|| coskx t E E0 coskx t Polarization is rotated around the optic axis Special case: =45o Polarization is rotated by 90o Quarter-wave plate . 2 = /2 +m 2, m=…,-1,0,+1,… dne no 0 dne no 0 / 4 m0 E|| E0|| coskx t E E coskx t Linear polarization is converted into 0 circular/elliptical and vice versa Reflective retarders Total internal reflection: phase shift between the two components. Glass - n=1.51, and 45o shift occurs at incidence angle 54.6o. Fresnel rhomb Unlike devices that use birefringent materials this is achromatic Circular polarizer Can you make a polarizer that transmits right circularly polarized light but not the left circularly polarized light? Phys 322 Chapter 8 Lecture 23 Polarization Reminder: Exam 2, October 22nd (see webpage) Optical activity Discovered in 1811, Dominique F. J. Arago Optically active material: any substance that cause a plane of polarization to appear to rotate Dextrorotatory (d-rotatory) - polarization rotates right (CW) Levorotatory (l-rotatory) - polarization rotates left (CCW) Latin: dextro - right levo -left Optical activity Depending on the structure, quartz can be d-or l-rotatory Optical activity 1895, Fresnel: linearly polarized light can be represented at the sum of R- and L-states that propagate at different speeds (circular birefringence): E0 ˆ ˆ ER i coskR z t jsin kR z t 2 E0 ˆ ˆ EL i coskL z t jsin kL z t 2 kR kL ˆ kR kL ˆ kR kL E E0 cos z ti cos z jsin z 2 2 2 Optical activity kR kL ˆ kR kL ˆ kR kL E E0 cos z ti cos z jsin z 2 2 2 ˆ z = 0: E E0i cost Optical activity kR kL ˆ kR kL ˆ kR kL E E0 cos z ti cos z jsin z 2 2 2 ˆ ˆ z =d (kR > kL): E E0i cos jsin cost d nL nR 0 nL > nR : d-rotatory nL < nR : l-rotatory Nature of optical activity Optical activity and biology -helix DNA (protein) Induced optical effects: Photoelasticity = stress birefringence Faraday effect = magnetic field induced birefringence Faraday isolator /4 /4 Induced optical effects: Kerr effect = birefringence induced by electric field (n~ E2) Pockels effect = birefringence induced by electric field (n~ E) Pockels cell Laser amplifier Liquid crystals - possesses properties of liquid and solid Nematic liquid crystal: molecules spontaneously arrange in parallel manner Optical modulator: Prearrange crystals using windows with microgroves http://www.cs.sandia.gov/~sjplimp/lammps/movies.html Modeling a nematic liquid crystal, S. S. Patnaik, S. J. Plimpton, R. Pachter, W. W. Adams, Liquid Crystals, 19, 213-220 (1995). Liquid crystal display Screen Liquid crystal display Liquid crystal modulators Mathematical description of polarization 1852: Stokes parameters Assume we have a set of 4 filters, each changes the intensity of unpolarized incident light by 1/2: George G. Stokes 1819-1903 1. Isotropic 2. Linear polarizer at 0o 3. Linear polarizer at +45o 4. Circular polarizer (transmits R) Stokes parameters Position each of the filters one by one in front of I0 any beam and record transmitted irradiance I1 Stokes parameters: I S0 2I0 incident radiation I2 S1 2I1 2I0 I S2 2I2 2I0 state of polarization 3 S3 2I3 2I0 S0 Typically is normalized to set S0=1 S1 2 2 2 2 Stokes vector: S0 S1 S2 S3 S2 S3 Stokes parameters I0 unpolarized polarized polarized polarized light: at 0o at +45o at -45o I1 1 1 1 1 I 0 1 0 0 I2 0 0 1 -1 0 0 0 0 I3 polarized S 2I at 90o R -state L -state 0 0 S 2I 2I 1 1 1 1 1 0 S 2I 2I -1 0 0 2 2 0 0 0 0 S3 2I3 2I0 0 1 -1 Stokes parameters I0 Mixing two noncoherent beams: ' '' S0 S0 S0 ... I1 I I2 I unpolarized polarized 3 light: at 90o S0 2I0 2 1 3 S 2I 2I 0 -1 -1 1 1 0 += S 2I 2I 0 0 0 2 2 0 0 0 0 S3 2I3 2I0 Jones vectors applicable only to polarized waves i x Ex t E e E Jones vector: ~ 0x E i Ey t y E0 ye Normalize: E ei x 1 Horizontal polarization: ~ 0x ~ Eh Eh 0 0 ~ 0 ~ 0 Vertical polarization: E v i y Ev E0 ye 1 ~ E ei x 1 ~ 1 1 Sum of two coherent 0x i x E E0xe E45 i x beams: E0xe 1 2 1 R-state: E ei x ~ 0x i 1 1 ~ 1 1 x i x ER E0xe E e ER i y / 2 ei / 2 0x i E0xe i 2 Jones vectors: superposition ~ 1 Eh 0 ~ 0 Ev 1 twice the amplitude Mixing waves: ~ 1 1 E45 2 1 ~ ~ 1 1 1 1 2 1 ER EL 2 i 2 i 2 0 ~ 1 1 ER 2 i horizontal ~ 1 1 EL 2 i Jones vectors: optical elements ~ 1 Eh A 0 ~ ~ ~ a11 a12 Ei E A E A ~ 0 t i E a21 a22 v 1 1 0 1 Example: horizontal polarizer A ~ 1 0 0 E45 2 1 ~ ~ 1 01 1 ~ 1. E ~ 1 1 Et A Eh h 0 0 0 0 ER 2 i ~ ~ 1 00 0 2. ~ 1 1 Et A Ev 0 0 1 0 EL 2 i ~ ~ 1 0 1 1 1 1 3. E A E 1 ~ t R Eh 0 0 2 i 2 0 2 Jones vectors: multiple optical elements ~ E ~ ~ ~ i Et A 3A 2A 1Ei A Ei A1 A2 A2 See table 8.6 (page 378) for matrices.
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