Chapter 9 - Polarization

Transcription

1 Chapter 9 - Polarization Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC

2 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

3 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

4 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

5 Polarization b) Circular polarization: A A y clockwise clockwise y E E A A A c) Straight ellipse: y Chapter 9: Polarization 5

6 Jones Calculus for Birefringent Optical Systems Principles of Optical Imaging Electrical and Computer Engineering, UIUC

7 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

8 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

9 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

10 Retardation Plates (Waveplates) E.g. polarized V 1 V y j V y polarized 0 V 1 y 1 1 j Jones Calculus 10

11 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

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13 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

14 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

15 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

16 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

17 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

18 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

19 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

20 ½ Wave Plate Jones Calculus 19

21 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

22 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

23 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 e W R( 45) W0 R(45) 1 1 j e j j j e 4 1 j 1 j Jones Calculus

24 ¼ Wave Plate j j V 1 ' 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 ' 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

25 ¼ Wave Plate Jones Calculus 4

26 Waveplates and Polarizers Jones Calculus 5

27 Waveplates and Polarizers j 1 1 o o 1 e W R( 45 ) W0 R(45 ) j 0 e cos jsin cos jsin jsin cos jsin cos cos jsin P W P jsin cos cos cos jsin Jones Calculus 6

28 Waveplates and Polarizers Case I: unpolarized light 1 1 V V y 1 V ' 1 cos 0 1 cos V y ' T y linearly polarized: obvious!! V' Vy' 1 ½ lost at first polarizer! cos V V Jones Calculus 7

29 Waveplates and Polarizers V Case II: 1 V y 0 V ' cos 0 1 cos V y ' T cos Jones Calculus 8

30 Waveplates and Polarizers V Case III: 1 1 V y j V ' 1 cos cos Vy ' j T 1 cos For ½ wave plates, T 0 Jones Calculus 9

31 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 Py W P jsin 0 jsin cos Jones Calculus 30

32 Birefringent Plate between Crossed Polarizers Case I: unpolarized light 1 1 V V y 1 ' V Vy ' jsin 0 1 jsin 1 T sin Jones Calculus 31

33 Birefringent Plate between Crossed Polarizers V Case II: 1 V y 0 ' V 1 Vy ' jsin 0 0 jsin T sin Jones Calculus 3

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