Chapter 9 - Polarization Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical Imaging Electrical and Computer Engineering, UIUC
Polarization The y components of the field can be epressed via a D Jones vector: J i E Ae E i y y Ae y Time variation: (9.1) E Acos( t ) (9.) E A cos( t ) y y y Chapter 9: Polarization
Polarization Eliminate time: E A cos t cos sin t sin E A sin t sin cos t cos y y y y (9.3) Take as eercise to prove that the trajectory of the E field is: E A Ey EE y A y A A y where cos sin (9.4) y Chapter 9: Polarization 3
Polarization This is an ellipse: y In general, it s rotated by angle θ This is what we would see looking straight at the incoming beam There are few particular cases: a) Linear polarization: 0, E E y 0 line A A y θ Chapter 9: Polarization 4
Polarization b) Circular polarization: A A y clockwise clockwise y E E A A A c) Straight ellipse: y Chapter 9: Polarization 5
Jones Calculus for Birefringent Optical Systems Principles of Optical Imaging Electrical and Computer Engineering, UIUC
Introduction ABCD matrices told us about amplitude distribution on propagation Jones calculus deals with polarization changes for polarization sensitive optical elements input output? polarizes ais along lines Direfringent crystal c-ais along vector (uniaial) Jones Calculus 7
Introduction Jones calculus developed around 1940 Based on matrices (similat to ABCD) Need to describe Z polarization Light travels 1 of tranverse normal modes Birefringent crystals play role because two modes travel at different phase velocities Uniaials simple c ais in plate surface Jones Calculus 8
Retardation Plates (Waveplates) Change polarization state Assumption: no reflections at surface of elements (can use anti reflection cratings) Use Jones matrices discussed before 1 ( ) E VE (, y, z ) e j tkz cc describes polarization Slow variation O.K. V V y ; V Vy 1 Jones Calculus 9
Retardation Plates (Waveplates) E.g. polarized V 1 V y 0 1 1 j V y polarized 0 V 1 y 1 1 j Jones Calculus 10
Retardation Plates (Waveplates),y,z laboratory frame of reference fsz f,s,z crystal frame of reference f fast ais s slow ais n s n f Jones Calculus 11
Retardation Plates (Waveplates) From before V s cos sin V V f sin cos Vy just inside the crystal R( ( ) just outside the crystal V s & V f propagate independently in the crystal After a distance L: jnskl 0 Vs' Vse Rewrite: jnf kl 0 Vf ' Vfe 1 1 j ( n n ) k L j ( n n ) k L s f 0 s f 0 V ' V e e s s 1 1 j ( n n ) k L j ( n n ) k L s f 0 s f 0 V ' V e e f f Jones Calculus 1
Retardation Plates (Waveplates) 1 Define ( n ) n n k L s nf k0l ( s f ) 0 j V ' s j e 0 Vs V j s e e W 0 Vf ' j Vf V f 0 e W describes birefringent medium W 0 To get output, return to lab frame of reference Jones Calculus 13
Retardation Plates (Waveplates) V' cos sin Vs' Vs' R( ) Vy ' sin cos Vf ' Vf ' V ' V R( ) W0 R( ) Vy ' Vy order of doing products of matrices Note: each matri is unitary R( ) 1 Polarization remain orthogonal Jones Calculus 14
Polarizers ais y ais j 1 0 p P e 0 0 p phase change through polarizer j 0 0 p P e y 0 1? V' V R( ) P0 R( ) Vy ' Vy Jones Calculus 15
L ( ns n f ) s e j y j Half (1/) Wave Plate W R ( ) W R ( ) 0 cos sin j 0 cos sin f sin cos 0 j sin cos cos sin cos sin j sin cos sin cos cos sin sin cos j sin cos sin cos Jones Calculus 16
½ Wave Plate W cos sin rotates a plane of j sin cos Polarization by Ψ angle between lab and crystal frames Another important factor is the transmission through the element E ' V' Vy' T E V V y Jones Calculus 17
E.g. 1 0 input ½ Wave Plate V ' cos sin 1 cos 0 j j V ' o y sin cos 0 sin if 45 1 Just get net polarization rotation cos sin 1T 1 Jones Calculus 18
½ Wave Plate Jones Calculus 19
E.g. 1 1 j input ½ Wave Plate V ' cos sin 1 1 cos jsin Vy ' sin cos j j sin j cos 1 T (cos jsin )(cos jsin ) (sin jcos )(sin jcos ) 1 (cos sin cos sin ) 1 Is the output still circulary polarized? Depends on Ψ Jones Calculus 0
Choose 4 V ' j j 1 1 V ' 1 y j (1) still circulary polarized ½ Wave Plate () reverses sense of rotation (3) for, get elliptical polarization! 4 Jones Calculus 1
L ( ns n f ) 4 j 4 1 ¼ Wave Plate e cos jsin (1 j) 4 4 Assume Ψ= 45 0 : j 4 o o 1 1 1 e 0 1 1 W R( 45) W0 R(45) 1 1 j 1 1 0 e 4 1 1 1 j j j e 4 1 j 1 j Jones Calculus
¼ Wave Plate j j V 1 ' 4 1 1 1 4 V e j j e 1 j E.g. ; ' Vy 0 Vy 1 j 1 j 0 1 j 1 j j 1 e 4 j RCP j 1 ' 4 1 1 1 4 0 Eg. V V e j j 1 e ; V ' 1 1 y j V y j j j j 1 j j 0 4 y polarized 1 e j Jones Calculus 3
¼ Wave Plate Jones Calculus 4
Waveplates and Polarizers Jones Calculus 5
Waveplates and Polarizers j 1 1 o o 1 e 0 1 1 W R( 45 ) W0 R(45 ) 1 1 1 1 j 0 e cos jsin cos jsin jsin cos jsin cos cos jsin 1 0 1 0 P W P 0 0 0 0 jsin cos cos 0 1 0 cos 0 0 0 jsin 0 0 0 Jones Calculus 6
Waveplates and Polarizers Case I: unpolarized light 1 1 V V y 1 V ' 1 cos 0 1 cos V y ' 1 0 0 0 T y linearly polarized: obvious!! V' Vy' 1 ½ lost at first polarizer! cos V V Jones Calculus 7
Waveplates and Polarizers V Case II: 1 V y 0 V ' cos 0 1 cos V y ' 0 0 0 0 T cos Jones Calculus 8
Waveplates and Polarizers V Case III: 1 1 V y j V ' 1 cos 0 1 1 cos Vy ' j 0 0 0 T 1 cos For ½ wave plates, T 0 Jones Calculus 9
Birefringent Plate between Crossed Polarizers Ψ= 45 0 : cos jsin o o W R( 45 ) W0 R(45 ) jsin cos cos sin 0 0 j 0 0 1 0 Py W P 0 1 0 0 jsin 0 jsin cos Jones Calculus 30
Birefringent Plate between Crossed Polarizers Case I: unpolarized light 1 1 V V y 1 ' 0 0 0 V 1 1 1 Vy ' jsin 0 1 jsin 1 T sin Jones Calculus 31
Birefringent Plate between Crossed Polarizers V Case II: 1 V y 0 ' 0 0 0 V 1 Vy ' jsin 0 0 jsin T sin Jones Calculus 3