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Polarized Light 1 EE485 Introduction to Photonics Polarized Light 1. Matrix treatment of polarization 2. Reflection and refraction at dielectric interfaces (Fresnel equations) 3. Polarization phenomena and devices Reading: Pedrotti3, Chapter 14, Sec. 15.1-15.2, 15.4-15.6, 17.5, 23.1-23.5 Polarization of Light Polarization: Time trajectory of the end point of the electric field direction. Assume the light ray travels in +z-direction. At a particular instance, Ex ˆˆEExy y ikz() t x EEexx 0 ikz() ty EEeyy 0 iixxikz() t Ex[]ˆˆEe00xy y Ee e ikz() t E0e Lih Y. Lin 2 One Application: Creating 3-D Images Code left- and right-eye paths with orthogonal polarizations. K. Iizuka, “Welcome to the wonderful world of 3D,” OSA Optics and Photonics News, p. 41-47, Oct. 2006. Lih Y. Lin 3 Matrix Representation ― Jones Vectors Eeix E0x 0x E0 E iy 0 y Ee0 y Linearly polarized light y y 0 1 x E0 x E0 1 0 Ẽ and Ẽ must be in phase. y 0x 0y x cos E0 sin (Note: Jones vectors are normalized.) Lih Y. Lin 4 Jones Vector ― Circular Polarization Left circular polarization y x EEe it EA cos t At z = 0, compare xx0 with x it() EAsin tA ( cos( t / 2)) EEeyy 0 y 1 1 yxxy /2, 0, E00 EA Jones vector = 2 i y Right circular polarization 1 1 x Jones vector = 2 i Lih Y. Lin 5 Jones Vector ― Elliptical Polarization Special cases: Counter-clockwise rotation 1 A Jones vector = AB22 iB Clockwise rotation 1 A Jones vector = AB22 iB General case: Eeix A 0x A B22C E0 i y bei B iC Ee0 y Jones vector = 1 A A ABC222 B iC 2cosEE00xy tan 2 22 EE00xy Lih Y. Lin 6 Superposition of Waves Superposition of waves → Summation of Jones vectors Exercises Superposition of left- and right-circularly polarized light: 11 2 ii 0 → Linearly polarized light of twice the amplitude Quiz: Does it have to be horizontally polarized? Superposition of vertically- and horizontally-polarized light in phase: 011 101 → Linearly polarized light at 45º inclination. Lih Y. Lin 7 Jones Matrix ― Linear Polarizer M ab Jones Polarization Jones matrix M vector J Jout MJ in cd in device Linear polarizer Vertical transmission axis (TA) x For all Jones vector y ab x000 ab cd y y cd 01 Horizontal transmission axis 10 Jones matrix = 00 TA inclined at 45º to the x-axis In general, a linear polarizer with TA at ab11 ab 1 0 and angle : cd11 cd 10 cos2 sin cos ab 1 11 M 2 cd 2 11 sin cos sin Lih Y. Lin 8 Jones Matrix ― Phase Retarder (Wave Plate) iixxx() Ee00xx Ee What we want: iiy ()yy Ee00yy Ee ix iixxx() e 0 Ee00xx Ee What can do it: i y iiyyy() 0 e Ee00yy Ee eix 0 Phase retarder M i 0 e y QWP, SA is y-axis ei /4 010 Mei /4 i/4 0 e 0 i i/4 10 QWP, SA is x-axis Me 0 i FA: Fast axis 10 SA: Slow axis i /2 HWP, SA is y-axis Me = 90º: Quarter-wave plate (QWP) 01 Same = 180º: Half-wave plate (HWP) i/2 10 HWP, SA is x-axis Me 01 Lih Y. Lin 9 Jones Matrix ― Polarization Rotator abcos cos( ) cdsin sin( ) Polarization rotator of angle cos sin M sin cos Coordinate transformation For Jones vectors, JRJ'() cos sin R() sin cos For Jones Matrices, TRTR'()() TR ()'() TR Lih Y. Lin 10 Special Actions of Phase Retarders 101 101 01i 011 1 1 i 1 10 1 10 1 0 ii 01 i 1 1 1 i Experiment: Given two linear polarizers, one QWP and one HWP, find a way to distinguish the QWP from the HWP. Lih Y. Lin 11 Summary of Jones Matrices Cascaded polarization devices Jin M1 M2 Mn Jout JMMMJMJout n 21 in in MM n MM21 Lih Y. Lin 12 Reflection and Refraction at a Dielectric Interface Transverse electric (TE) polarization Transverse magnetic (TM) polarization Boundary conditions: Tangential components of E and B are continuous. EE rt E BBrt B BBcosrrtt cos B cos EEcos rrtt cos E cos c EvBB n Lih Y. Lin 13 Fresnel Equations Define relative refractive index nnn 21/ TE wave E cosn22 sin Reflection coefficient r r E cosn22 sin E 2cos Transmission coefficient t t E cosn22 sin or tr1 TM wave E nn222cos sin Reflection coefficient r r E nn222cos sin E 2cosn Transmission coefficient t t E nn222cos sin or ntr 1 For both cases, rt22 1 ! Lih Y. Lin 14 Power Reflectance and Transmittance Consider the cross-sections IAiirrtt(cos) IA (cos) IA (cos) IEHnE|||| ||2 22n2 cost 2 ||||EEir || E t n1 cosi 2 cos 2 1 rnt || t cosi 2 PI E 2 R rr r r Reflectance PIii E i Energy conservation cos 2 Tn t || t Transmittance Power Pirt PP cosi P P RTr , t PPii 1 RT Lih Y. Lin 15 Characteristics of Reflection and Transmission nnn 21/1.50 Transmission: No phase change. Reflection: Possible phase change, possible going through “zero” transition. Let’s focus on reflection … Lih Y. Lin 16 External and Internal Reflections Lih Y. Lin Phase Changes on Reflection Lih Y. Lin 18 Example: Fresnel Rhomb n Incident light: Linearly polarized at 45⁰ to 1 the plane of incidence n2 What is the polarization of the output light? nnn 12/1/1.5 Lih Y. Lin 19 Total Internal Reflection Fluorescence (TIRF) Microscopy (www.olympusmicro.com) Application: Near-field excitation and detection of cells/molecules Principle: Evanescent wave in frustrated total internal reflection dP sin 2 21 n2 Lih Y. Lin 20 Dichroism: Polarization by Selective Absorption Example: Polaroid H-sheet containing aligned hydrocarbon molecules. Lih Y. Lin 21 Polarization by Selective Reflection Brewster window Recall the phenomenon of Brewster angle. (perfect window for TM light) p Brewster-angle polarizer Lih Y. Lin 22 Birefringence Refractive index depends on polarization directions, due to asymmetry of the molecular structure with respect to light propagation direction. Lih Y. Lin 23 Optical Activity Rotate the polarization direction of linearly-polarized light by circular birefringence. Refractive indices for left- and right-circular polarized are nL and nR, respectively. Decompose the linear-polarized light into left- and right-circular polarized lights. cos11ii 1 1 ee sin 22ii After the material, Specific rotation or Rotatory power (rotation angle per unit 11iiii11 i cos eeLR ee e0 length): 22ii sin ()nnLR 1 (rad/length) () 0 2 LR 0 1 dn()LR n () LR 2 0 Lih Y. Lin 24 Example: Optical Activity in Quartz Determine the thickness required, in mm, to achieve -45º rotation at 762 nm. d(1.53920 1.53914) 4 762 106 d 3.175 (mm) Reversibility of optical activity Lih Y. Lin 25 Liquid Crystal Plane of polarization rotates in alignment with the molecular twist. z Application: Optical switch and Liquid crystal display (LCD) Lih Y. Lin 26.
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