Electrodynamics Lecture Highlights: Reflectivity, Fresnel Equations & Wave Behavior

phy 712 electrodynamics 9 9 50 am olin 105 n.w
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Explore topics such as reflectivity of plane waves, complex response functions, Fresnel Equations, behaviors of s and p polarization, Brewster's angle, total internal reflection, electromagnetic plane waves in isotropic media, and reflection/refraction at interfaces between dielectrics. Gain insights into the behavior of electromagnetic waves in different scenarios.

  • Electrodynamics
  • Reflectivity
  • Fresnel Equations
  • Wave Behavior
  • Electromagnetic Waves
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  1. PHY 712 Electrodynamics 9-9:50 AM Olin 105 Plan for Lecture 18: Complete reading of Chapter 7 1. Comments on reflectivity of plane waves 2. Summary of complex response functions for electromagnetic fields 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 1

  2. 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 2

  3. Some comments on the Fresnel Equations 1. Different behaviors of s and p polarization 2. Brewster s angle 3. Total internal reflection 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 3

  4. Review: Electromagnetic plane waves in isotropic medium with real permeability and permittivity: . ( ( ) ( ) ,t c electromag plane for vector Poynting ) ( ) ( ) k r ic n ct = = 2 2 E r E , t e n c 0 n ( ) ,t r k k = = B r E r E ,t netic waves : 2 E 2 n 1 k k 2 0 c = = S E 0 2 avg electromag plane for density Energy netic waves : 1 2 = E u 0 2 avg 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 4

  5. Review: Reflection and refraction of plane electromagnetic waves at a plane interface between dielectrics (assumed to be lossless) = = = ' ' ' n n k i n R = sin 'sin i n kR ki i R k k = = nc i R k ' = ' n c 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 5

  6. Review: Reflection and refraction between two isotropic media z k x ki kR i R Reflectanc transmitt e, ance : 2 2 S z S z ' i ' ' cos E E n = = = = 0 0 R T R R S z S z ' cos E E n i 0 + 0 i i i = Note that R T 1 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 6

  7. For s-polarization (E perpendicular to plane of incidence) cos ' cos n i n ' 2 cos E E n i ' = = 0 0 R E E + + cos ' cos cos ' cos n i n n i n 0 0 i i ' ' = 2 2 2 Note that : ' cos ' sin n n n i For p-polarization (E in plane of incidence) 2 ' cos 'cos n i nn ' 2 'cos E E E E nn i ' = = 0 0 R + + 2 2 ' cos 'cos ' cos 'cos n i nn n i nn 0 0 i i ' ' = 2 2 2 Note that: 'cos ' sin n n n i 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 7

  8. Reflectance for s-polarization 2 2 2 2 cos ' sin n i n n i 2 E E ' = = 0 R R s + 2 2 2 cos ' sin n i n n i 0 i ' Reflectance for p-polarization 2 n n n n 2 2 2 'cos ' sin n i n n i 2 E E ' ' = = R 0 R p + 2 2 2 'cos ' sin n i n n i 0 i ' ' 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 8

  9. = = = Example for ; 1 and ' 1. 5 n n s-polarization p-polarization 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 9

  10. n=1 surface normal ki kR i n >1 Polarization due to reflection from a refracting surface = = Brewster's angle: for , R B i ( ) B i 0 i p 2 n n n n 2 2 2 'cos ' sin n i n n i 2 ' n n E E ' ' = = 1 For , t an i = = R 0 R B p + 2 2 2 ' cos ' sin n i n n i 0 i ' ' 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 10

  11. Reflection and refraction between two isotropic media -- continued z ( ( ) ,t r B each wave For : ( ) ( ) ) k r ic n ct = = 2 2 E r E , t e n c 0 n ( ) ,t r ( ) ,t r k k k = = E E x c ki kR i R Matching condition interface at : = 2 2 2 ' cos ' sin n n n i ' n 1 If , ' n for sin , n i i Total internal reflection: 0 n refracted longer no field propagates medium in ' ' 2 sin i = = 2 2 2 ' cos ( E sin ' 1 n i n i n i n 2 sin i 0 2 ) sin i n 1 z ( ) ( ) c 2 k sin i r i n ct = E r ' , ' t e e 0 || c 0 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 11

  12. Example of total internal reflection n =1 and n=1.5 i0 = sin-1(1/1.5)=41.81o Relative transmitted intensity i=42o i=50o i=90o z/ Transmitted illumination confined within a few wavelengths of the surface. 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 12

  13. TIRF (total internal reflection fluorescence) www.nikon.com/products/microscope-solutions/bioscience.../nikon_note_10_lr.pdf 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 13

  14. Design of TIRF device using laser and high power lens 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 14

  15. 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 15

  16. Special case: normal incidence (i=0, =0) ' n n ' 2 E E n ' = = 0 0 R E E + ' + ' n n n n 0 0 i i ' ' Reflectanc transmitt e, ance : 2 ' n n 2 E ' = = 0 R R E + ' n n 0 i ' 2 2 ' ' 2 ' E n n n = = 0 T ' ' E n n + ' n n 0 i PHY 712 Spring 2013 -- Lecture 19 ' PHY 712 Spring 2019 -- Lecture 18 02/25/2019 16 16

  17. Extension to complex refractive index n= nR + i nI = = = + Suppose , ' real, ' ' ' n n n in R I Reflectanc normal at e incidence : 2 ' n n ( ( ) ) ( ( ) ) 2 2 2 + ' ' E n n n ' = = = 0 R R R I 2 2 + + E ' ' n n n + ' n n 0 i R I ' Note that for ' ' : n n n I R R 1 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 17

  18. Origin of imaginary contributions to permittivity -- Review: Drude model dielectric function: ( ) i i m 0 0 2 1 q = + 1 i N f i 2 2 i i i ( ) 0 ( ) 0 = + R I i ( ) 0 2 2 2 q i i = + 1 i R N f ( ) i 2 m 2 + 2 2 2 0 i i i ( ) 0 2 q i = i i I N f ( ) i 2 m 2 + 2 2 2 0 i i i 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 18

  19. Drude model dielectric function: ( ) R 0 ( ) I 0 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 19

  20. Drude model dielectric function some analytic properties: ( ) 2 1 q 1 = + i N f i i 2 i 2 m i 0 0 i i ( ) 2 1 q 1 For i N f i i 2 m i 0 0 i 2 P 1 2 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 20

  21. Analysis for Drude model dielectric function continued -- Analytic properties: ( ) 1 0 0 P i P z z z f ( ) 2 1 2 q z i = = i f z N f i 2 i m z iz i i ( ) 2 = 2 has poles at 0 iz P i 2 = 2 i i i z i P 2 2 ( ) z ) ( ) z ( ) z Note that ( 0 analytic is for 0 f P P pz ( ) analytic f z ( ) pz 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 21

  22. Because of these analytic properties, Cauchys integral theorem results in: Kramers-Kronig transform for dielectric function: ( ) 0 ( ) 0 1 ' 1 = d 1 ' R I P - ' ( ) with ( ) ; 1 ' 1 = d ' 1 I R P - = ' 0 0 ( ) ( ) ( ) ( ) = R R I I 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 22

  23. Further comments on analytic behavior of dielectric function between ip relationsh Causal" " and E fields D : ( ) ( ) ( ) ( ) 0 = + D r E r E r , , , t t d G t 0 ( ) ( ) 0 = ie 1 d G 0 Some details: Consider a convolution integral such as = ( ) ( ') ( g t h t ') ' where the functions ( ), ( ), and ( ) f t t dt f t g t h t are all well-defined functions with Fourier transforms such as 1 = = ' i t i t ( ) ( ') f t e ' ( ) ( ) f dt f t f e d 2 = ( ) ( It follows that: ( f ) ) g h 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 23

  24. Further comments on analytic behavior of dielectric function E D "Causal" relationship between and fields: = + ( ) ( ) ( ) ( ) d G D r E r E r , , , t t t 0 0 ( ) ( ) 1 ( ) = = d G i i ( ) 1 ( G ) = 1 G e d e 2 ( ) 0 0 0 2 1 q m N = For 1 f i i 2 i 2 i i 0 0 i i ( ) sin 2 q m N = /2 i ( ) ( ) G f e i i i i 0 i i 2 i 2 i where / 4 i 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 24

  25. Some details ( ) 1 1 = i iz ( ) 1 = ( ) f z e G e d dz 2 2 0 ( ) z 2 1 q m ( ) f z = = Let 1 i N f i = 2 i 2 z iz i 0 z 0 i i ( ) f z 2 i 2 has poles at 0 z iz P P P i 2 2 = = 2 i 2 i or i i i i z i z i P P 2 2 2 2 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 25

  26. 1 = = iz ( ) ( ) f z e Res( ) G dz i z P 2 P For >0, For <0, iz no 0 ok e iz e G = ( ) 0 for 0 zP iz For >0, For <0, 0 ok no e e iz 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 26

  27. 1 = = iz ( ) ( ) f z e Res( ) G dz i z P 2 P 2 i m ( ) z 1 q ( ) f z = = Let 1 N f i = 2 i 2 z iz i 0 z 0 i i ( ) f z 2 i 2 has poles at 0 z iz P P P i 2 2 = = 2 i 2 i or i i i i z i z i P P 2 2 2 2 ( ) sin 2 q m N = / 2 i ( ) ( ) G f e i i i i 0 i i 2 i 2 i 2 i 2 i where / 4 assuming / 4 0 i 02/25/2019 PHY 712 Spring 2019 -- Lecture 18 27

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