Electromagnetic Waves Analysis and Solutions

phy 712 electrodynamics 9 9 50 am olin 105 n.w
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"Explore the properties of plane polarized electromagnetic waves, reflectance, transmittance, and solutions to Maxwell's equations without sources. Understand wave equations, wave solutions, and Poynting vector in the context of anisotropy and complexity."

  • Electromagnetic waves
  • Maxwells equations
  • Wave solutions
  • Polarization
  • Reflectance
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  1. PHY 712 Electrodynamics 9-9:50 AM Olin 105 Plan for Lecture 16: Read Chapter 7 1. Plane polarized electromagnetic waves 2. Reflectance and transmittance of electromagnetic waves extension to anisotropy and complexity 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 1

  2. 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 2

  3. 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 3

  4. = = D E B H For linear isotropic media and no sources: Coulomb's law: ; = E 0 E = B Ampere-Maxwell's law: 0 t 0 B + = E Faraday's law: 0 t = B No magnetic monopoles: 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 4

  5. Analysis of Maxwells equations without sources -- continued: : law s Coulomb' E = 0 E t = B Ampere - Maxwell' law s : 0 B t + = E Faraday' law s : 0 = B magnetic No monopoles : 0 ( ) t E E = 2 B B t 2 B = + = 2 B 0 2 t ) ( t B B + = + 2 E E t 2 E = + = 2 E 0 2 t 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 5

  6. Analysis of Maxwells equations without sources -- continued: Both E and B fields are solutions to a wave equation: 2 B 1 v = 2 B 0 2 2 t 2 E 1 v = 2 E 0 2 2 t 2 c 2 2 0 0 where v c wave to 2 n equation Plane wave solutions : ( ) ( ) ( ) ( ) = = k r k r i i t i i t B r B E r E , , t e t e 0 0 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 6

  7. Analysis of Maxwells equations without sources -- continued: wave to solutions wave Plane = e t B r B equation = t r : ( ) ( ) ( ) ( ) i i k r k r i t i t E E , , e 0 0 2 2 v c n 2 = = k where n 0 0 Note: , n, k can all be complex; for the moment we will assume that they are all real (no dissipation). E B Note that and are not independent; + = k = 0 0 k nc B = E from Faraday's law: 0 t k E k E n = B 0 0 0 c k E k B = = also note: 0 and 0 0 0 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 7

  8. Analysis of Maxwells equations without sources -- continued: electromag plane of Summary c v netic waves : k c ( ) E n ( ) ( ) i i = = k r k r i t i t B r E r E 0 , , t e t e 0 2 2 n k 2 = = = k E where and 0 n 0 0 0 Poynting vector and energy density: 2 E n 1 2 k k 2 = = 0 c S E 0 2 avg E0 1 2 2 = E u 0 avg B0 k 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 8

  9. Reflection and refraction of plane electromagnetic waves at a plane interface between dielectrics (assumed to be lossless) k kR ki i R 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 9

  10. Reflection and refraction -- continued medium In ' : ' ( ' ' k ) ( ) ( ( ) ) ' ' k r ic n ct = E r E ' , ' t e k 0 ki n kR ( ) ( ) t , ' ' k = = i R B r E r E r ' , ' , ' t t c medium In : ( ) ( ) ( ( ) ) k r i n ct = E r E , t e i c 0 i i n ( ) ( ) k k = = B r E r E r , , , t t t i i i i i c ( ) ( ) ( ( ) ) k r i n ct = E r E , t e R c 0 R R n ( ) ( ) t , k k = = B r E r E r , , t t R R R R R c 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 10

  11. Reflection and refraction -- continued z Snell s law matching phase factors at boundary plane z=0. ( ) ( ) k k r k r ' ' i n ct i n ct x = e e i c c ki kR i R = = 0 0 z z ( ) k r i n ct = e R c = 0 z = + + r x y z matching plane: ' sin sin ' ' Snell's law : 'sin 0 x y = = r k r x k = = r = k sin R x i x i R i R k = = k r r ' sin sin n sin n n n x = nx i i n i 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 11

  12. Reflection and refraction -- continued z k x ki kR i R Continuity D = equations boundary w at B = sources no ith : 0 D t 0 B t = + = H E 0 0 Matching D amplitudes field boundary at plane : , z B continuous z H , z E z PHY 712 Spring 2019 -- Lecture 16 continuous 02/19/2019 12

  13. Reflection and refraction -- continued Matching field amplitudes at boundary plane: continuous: continuous: i i R n + k E k z D z k ( ) x B z + = z E E ' ' E z 0 0 0 i R ki kR i R ( ) = E z E z E continuous: + E 0 0 R ( H z ) k E z = ' ' n ' z E z ' 0 0 0 0 i R continuous: k E ( ) Known: , n k k + E E z 0 i E i 0 0 i i R R k E Unknown: ' , , ' 0 0 R ' ' n = k E z ' ' 0 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 13

  14. Reflection and refraction -- continued s-polarization Efield polarized perpendicular to plane of incidence continuous: ' continuous: n z E z E k ( H z ) + = z E E z x 0 0 0 i R ki kR i R ( ) ' ' n k k k + = E E z E z ' ' 0 0 0 i i R R 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 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 14

  15. Reflection and refraction -- continued p-polarization Efield polarized parallel to plane of incidence continuous: continuous: n z D z ( ) k H z + = z E E ' ' E z x 0 0 0 i R ki kR i R ( ) ' ' n k k k + = E E z E z ' ' 0 0 0 i i R R ' 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 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 15

  16. Reflection and refraction -- continued 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 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 16

  17. For s-polarization 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 ' 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 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 17

  18. 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 2019 -- Lecture 16 ' 02/19/2019 18

  19. Multilayer dielectrics (Problem #7.2) n1 n3 n2 ka ki kt kb kR d 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 19

  20. Extension of analysis to anisotropic media -- 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 20

  21. Consider the problem of determining the reflectance from an anisotropic medium with isotropic permeability and anisotropic permittivity where: 0 0 0 0 0 0 xx yy zz By assumption, the wave vector in the medium is confined to the x-y plane and will be denoted by + k x y ( ),where and are to be determine d. n n n n t x y x y c The electric field inside the medium is given by: n x n y + i t ( ) i = + + x y E x y z ( )e . E E E c x y z 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 21

  22. Inside the anisotropic medium, Maxwells equations are: 0 0 i = E H = = H 0 H E + = 0 i E 0 0 After some algebra, the equation for E is: n n n n n 2 y 0 0 n E E E xx x y n x = 2 x 0. x y yy y + 2 x 2 y 0 0 ( ) n zz z From E,H can be determined from 1 n x n y + i t ( ) i = ) ( + x y H x y z ( ) e . E n n E n E n c z y x y x x y c 0 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 22

  23. The fields for the incident and reflected waves are the same as for the isotropic case. = + k x y (sin cos ), i i i c = k x y (sin cos ). i i R c = sin xn i Note that, consistent with Snell s law: Continuity conditions at the y=0 plane must be applied for the following fields: H ( ,0, , ), x ( ,0, , ), E x ( ,0, , ), and E x z t ( ,0, , ). D x z t z t z t x z y There will be two different solutions, depending of the polarization of the incident field. 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 23

  24. Solution for s-polarization = = = 2 y 2 x 0 E E n n x y zz 1 n x n y + i t n x n y + i t ( ) ( ) i i = = x y x y E z H x y e ( ) e E E n n c c z z y x c 0 must be determined from the continuity conditions: + = E z = + = ( )cos ( ) in s E E E E E i E n E E i E n 0 0 0 0 0 0 z z y z x + cos cos i i n n E E y = 0 . 0 y 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 24

  25. Solution for p-polarization = = 2 y 2 x 0 ( ). xx E n n z yy yy n n n x n y + i t ( ) i = x y E x y e . xx x E c x yy y E n x n y + i t ( ) i = x y H e z . x c n xx c 0 y must be determi ned from the continuity conditions: + = E x n = + = ( )cos ( ) ( )sin . xx xx n x E E i E E E E E E i E 0 0 0 0 0 0 x x x n y y + cos cos i i n n E E xx y = 0 . 0 xx y 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 25

  26. Extension of analysis to complex dielectric functions = simplicity For assume that 0 Suppose dielectric the function complex is : ( ) 2 = + = + + i n in i R I R I 0 / 1 2 / 1 2 + + + 2 2 2 2 = = n n R I 2 2 ( ) ( ) ( ) ( ) ( ) k k k r r r i n ct i n ct n = = E r E E , t e e e R I c c c 0 0 02/19/2019 PHY 712 Spring 2019 -- Lecture 16 26

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