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Birefringence Birefringence Outline • Polarized Light (Linear & Circular) • Birefringent Materials • Quarter-Wave Plate & Half-Wave Plate Reading: Ch 8.5 in Kong and Shen 1 True / False 1. The plasma frequency ω p is the frequency above which a material becomes a plasma. 2. The magnitude of the E -field of this wave is 1 E =(ˆx +ˆy)ej(ωt−kz) 3. The wave above is polarized 45o with respect to the x-axis. 2 Microscopic Lorentz Oscillator Model k ω2 = spring o m ωo ω p 3 Sinusoidal Uniform Plane Waves Ey = A1cos(ωt − kz) EEyx==AA12coscos((ωtωt−−kzkz)) 4 ˜ j(ωt−kz) 45° Polarization Ex(z,t)=ˆxRe Eoe E ˜ j(ωt−kz) Ey(z,t)=ˆyRe Eoe Ey E E x yˆ Ey E Ex Ex E xˆ The complex Ey E amplitude, ˜ Ex E o ,is the E same for Ey both components. E Therefore Ex E Ex zˆ and y are always in E Ey phase. Ex Where is the magnetic field? E Ey 5 Superposition of Sinusoidal Uniform Plane Waves Can it only be at 45o ? 6 Arbitrary-Angle Linear Polarization E-field variation over time y (and space) Here, the y-component is in phase with the x-component, x but has different magnitude. 7 Arbitrary-Angle Linear Polarization Specifically: 0° linear (x) polarization: E y/Ex =0 90° linear (y) polarization: E y/Ex = ∞ 45° linear polarization: Ey/Ex =1 Arbitrary linear polarization: Ey/Ex = constant 8 Circular (or Helical) Polarization yˆ ˜ Ex(z,t)=ˆxEosin(ωt − kz) Ey ˜ E Ey(z,t)=ˆyEocos(ωt − kz) xˆ … or, more generally, Ex ˜ j(ωt−kz) Ex(z,t)=ˆxRe{−jEoe } ˜ j(ωt−kz) Ey(z,t)=ˆyRe{jEoe } The complex amplitude of the x- E zˆ component is -j times the complex amplitude of the y-component. The resulting E-field rotates counterclockwise around the Ex and Ey are always propagation-vector 90 (looking along z-axis). If projected on a constant z plane the E-field vector would rotate clockwise !!! 9 Right vs. Left Circular (or Helical) Polarization ˜ Ex(z,t)=−xˆEosin(ωt − kz) E-field variation ˜ Ey(z,t)=ˆyEocos(ωt − kz) over time (at z = 0) yˆ … or, more generally, ˜ j(ωt−kz Ex(z,t)=ˆxRe{+jEoe } xˆ ˜ j(ωt−kz) Ey(z,t)=ˆyRe{jEoe } Here, the complex amplitude of the x-component is +j times the complex amplitude of the y-component. The resulting E-field rotates clockwise around the So the components are always propagation-vector 90 (looking along z-axis). If projected on a constant z plane the E-field vector would rotate counterclockwise !!! 10 Unequal arbitrary-relative-phase components yield elliptical polarization Ex(z,t)=ˆxEoxcos(ωt − kz) E-field variation over time yˆ Ey(z,t)=ˆyEoycos(ωt − kz − θ) (and space) where xˆ … or, more generally, j(ωt−kz) Ex(z,t)=ˆxRe{Eoxe } j(ωt−kz−θ) Ey(z,t)=ˆyRe{Eoye } The resulting E-field can rotate clockwise or counter- clockwise around the k-vector … where are arbitrary (looking along k). complex amplitudes 11 Sinusoidal Uniform Plane Waves Left IEEE Definitions: Right 12 A linearly polarized wave can be represented as a sum of two circularly polarized waves 13 A linearly polarized wave can be represented as a sum of two circularly polarized waves ˜ E (z,t)=ˆxE˜ sin(ωt − kz) Ex(z,t)=−xˆEosin(ωt − kz) x o ˜ ˜ Ey(z,t)=ˆyEocos(ωt − kz) Ey(z,t)=ˆyEocos(ωt − kz) CIRCULAR LINEAR CIRCULAR 14 Polarizers for Linear and Circular Polarizations CASE 1: CASE 2: Linearly polarized light with magnitude E o oriented 45o Circularly polarized light with E with respect to the x-axis. magnitude o . S = E2/(2η) 2 in o Sin = Eo /η Wire grid polarizer 2 2 Sout = Eo /(2η) Sout = Eo /(4η) What is the average power at the input and output? 15 Todays Culture Moment Image is in the public domain. vision 16 Anisotropic Material The molecular "spring constant" can be different for different directions If , then the material has a single optics axis and is called uniaxial crystal 17 Microscopic Lorentz Oscillator Model In the transparent regime … yi xi xr yr 18 Uniaxial Crystal Optic axis Uniaxial crystals have one refractive index for light polarized along the optic axis (ne) Extraordinary and another for light polarized in polarizations either of the two directions perpendicular to it (no). Ordinary Light polarized along the optic axis polarizations is called the extraordinary ray, and light polarized perpendicular to it is called the ordinary ray. Ordinary… These polarization directions are the crystal principal axes. Extraordinary… 19 Birefringent Materials o-ray e-ray no ne Image by Arenamotanus http://www.flickr.com/photos/ arenamontanus/2756010517/ on flickr All transparent crystals with non-cubic lattice structure are birefringent. 20 Polarization Conversion Linear to Circular inside Polarization of output wave is determined by… 21 Quarter-Wave Plate Circularly polarizedyˆ output left circular output E xˆ 45o Ey λ/4 Ex fast axis linearly polarized input Example: If we are to make quarter-wave plate using calcite (no = 1.6584, ne = 1.4864), for incident light wavelength of = 590 nm, how thick would the plate be ? dcalcite QWP = (590nm/4) / (no-ne) = 858 nm 22 Half-Wave Plate The phase difference between the waves linearly polarized parallel and perpendicular to the optic axis is a half cycle Optic axis 45o LINEAR IN LINEAR OUT 23 yˆ Key Takeaways Ey EM Waves can be linearly, circularly, or E elliptically polarized. xˆ Ex A circularly polarized wave can be represented as a sum of two linearly polarized waves having π/ 2 phase shift. A linearly polarized wave can be represented as a sum of two circularly polarized waves. E Circular Polarization In the general case, waves are elliptically polarized. Waveplates can be made from birefringent materials: Quarter wave plate: λ/4=( no − ne)d (gives π/ 2 phase shift) Half wave plate: λ/2=( no − ne)d (gives π phase shift) 24 MIT OpenCourseWare http://ocw.mit.edu 6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms..
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