Scattering at an Interface:

Transcription

1 8/9/08 Course Istructor Dr. Raymod C. Rumpf Office: A 337 Phoe: (95) E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagetics Topic 3h Scatterig at a Iterface: Phase Matchig & Special Agles Phase Matchig These otes & Special may cotai Agles copyrighted material obtaied uder fair use rules. Distributio of these materials is strictly prohibited Slide Lecture Outlie Phase Matchig at a Iterface The Critical Agle Brewster s Agle Phase Matchig & Special Agles Slide

2 8/9/08 Phase Matchig at a Iterface Phase Matchig & Special Agles Slide 3 Illustratio of the Dispersio Relatio k k k k x z 0 Idex ellipsoid k k y k x x The dispersio relatio for isotropic materials is essetially just the Pythagorea theorem. It says a wave sees the same refractive idex o matter what directio the wave is travellig. Phase Matchig & Special Agles Slide 4 z

3 8/9/08 Idex Ellipsoid i Two Differet Materials Material (Low ) Material (High ) x, z, 0 x, z, 0 Phase Matchig & Special Agles Slide 5 Phase Matchig Whe < Material x, z, 0 Material x, z, 0 Phase Matchig & Special Agles Slide 6 3

4 8/9/08 Summary of Phase Matchig for < Material Material x, z, 0 x, z, Properly phased matched at the iterface. Phase Matchig & Special Agles Slide 7 Phase Matchig Whe > ic Material c k ic x, kz, k k0 c Material x, z, 0 Phase Matchig & Special Agles Slide 8 4

5 8/9/08 Summary of Phase Matchig for > Material Material x, z, 0 x, z, 0 ic c ic c 3 4 ic Properly phased matched at the iterface. c Phase Matchig & Special Agles Slide 9 ic c The Critical Agle Phase Matchig & Special Agles Slide 0 5

6 8/9/08 Aimatio of Sell s Law ( of ) Beam propagates from low idex medium to a high idex medium. si si i t Phase Matchig & Special Agles Slide Aimatio of Sell s Law ( of ) Beam propagates from high idex medium to a low idex medium. si si i t Phase Matchig & Special Agles Slide 6

7 8/9/08 The Critical Agle, c The critical agle c is the agle of icidece i that produces a agle of trasmissio t that is exactly 90. si i si t si c si 90 c si si 90 I order for there to be a critical agle c, the wave must be icidet oto a low idex medium from a high idex medium. c si > Phase Matchig & Special Agles Slide 3 Field at a Iterface Above ad Below the Critical Agle (Igorig Reflectios) No critical agle C C This called a evaescet field The field always peetrates material, but it may ot propagate. Above the critical agle, peetratio is greatest ear the critical agle. Very high spatial frequecies are supported i material despite the dispersio relatio. I material, power always flows i the trasverse directio, but ot ecessarily i the logitudial directio. Phase Matchig & Special Agles Slide 4 7

8 8/9/08 Simulatio of Reflectio ad Trasmissio at a Sigle Iterface ( > ) =.4, =.0 c =45 Phase Matchig & Special Agles Slide 5 Field Visualizatio for c =45 ic = 44 ic = 46 ic = 67 ic = 89 Phase Matchig & Special Agles Slide 6 8

9 8/9/08 Electromagetic Tuelig If a evaescet field touches a medium with higher refractive idex, the field may o loger be cutoff ad become a propagatig wave. This is a very uusual pheomeo because the evaescet field is cotributig to power flow. This is called electromagetic tuelig ad is aalogous to electro tuelig through thi isulators. Phase Matchig & Special Agles Slide 7 Brewster s Agle Phase Matchig & Special Agles Slide 8 9

10 8/9/08 Ca Reflectio Ever Be Zero? If we ispect the Fresel equatios log eough, we see that the reflectio coefficiets have a differece i their umerator. This meas there must exist special coditios where reflectio ca be zero. These are the Brewster s agles. r t TE, s, Polarizatio TE TE r cosi cost cos cos TE i t cosi cos cos i t t TE TM, p, Polarizatio r t TM TM r cost cosi cos cos TM t i cosi cos cos t i cost t cos i TM We do ot observe a similar coditio for trasmissio. Phase Matchig & Special Agles Slide 9 Brewster s Agle for TE Polarizatio ( of 3) We start with the Fresel equatio for reflectio of the TE polarizatio. cosi cost rte cos cos i t We set the umerator equatio to zero. cos cos 0 i t The Brewster s agle B is the agle of icidece i that satisfies this expressio ad makes reflectio go to zero. cosb cost 0 cos cos B t cosb cost cos B cos t We would like to elimiate this term. B si si t Phase Matchig & Special Agles Slide 0 0

11 8/9/08 Brewster s Agle for TE Polarizatio ( of 3) Solve Sell s law for si t. sib si t si t si B Substitute this result ito our previous expressio for the Brewster s agle. si B si t si B si B si B,TE Phase Matchig & Special Agles Slide Brewster s Agle for TE Polarizatio (3 of 3) Now write ad ad i terms of ad. r r 0 0 r r r r r r The expressio for the Brewster s agle becomes si r r r r r B,TE r Ispectig this equatio, we see that there is o Brewster s agle for the TE polarizatio uless the permeability is differet i each medium. Phase Matchig & Special Agles Slide

12 8/9/08 Brewster s Agle for TM Polarizatio ( of 3) We start with the Fresel equatio for reflectio of the TM polarizatio. cost cosi rtm cos cos t i We set the umerator equatio to zero. cos cos 0 t i The Brewster s agle B is the agle of icidece i that satisfies this expressio ad makes reflectio go to zero. cost cosb 0 B si si t Phase Matchig & Special Agles Slide 3 Brewster s Agle for TM Polarizatio ( of 3) Solve Sell s law for si t. sib si t si t si B Substitute this result ito our previous expressio for the Brewster s agle. si B si t si B si B si B,TM Phase Matchig & Special Agles Slide 4

13 8/9/08 Brewster s Agle for TM Polarizatio (3 of 3) We ow write the impedaces ad refractive idices i terms of r r 0 0 r r r r r r Our expressio for the Brewster s agle becomes si r r r r r r B,TM Ispectig this equatio, we see that we still have a Brewster s agle eve whe the materials do ot have a magetic respose. ta B,TM r r r r Phase Matchig & Special Agles Slide 5 Simulatio of Reflectio ad Trasmissio at a Sigle Iterface ( < ) =.0, =.73 B =60 Phase Matchig & Special Agles Slide 6 3