MIT OpenCourseWare http ocw mit edu 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Please use the following citation format Markus Zahn 6 013 ESD 013J Electromagnetics and Applications Fall 2005 Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution Noncommercial Share Alike Note Please use the actual date you accessed this material in your citation For more information about citing these materials or our Terms of Use visit http ocw mit edu terms 6 013 Electromagnetics and Applications Fall 2005 Lecture 9 Oblique Incidence of Electromagnetic Waves Prof Markus Zahn October 6 2005 I Wave Propagation at an Arbitrary Angle From Electromagnetic Field Theory A Problem Solving Approach by Markus Zahn 1987 Used with permission z x sin z cos kz kx x kz z kx k sin kz k cos k j t kz i y Re Ee j t kx x kz z i y E x z t Re Ee j H H 1 E 1 i x Ey iz Ey E j z x j 1 j k Ei x j kz Ei z e j kx x kz z H z j H x z t Re E cos i x sin i z e j kx x kz z E cos i x sin i z ej t kx x kz z In general k kx i x ky i y kz i z is the wave vector r xi x y i y z i z is a position vector e jk r e j kx x ky y kz z jk r e jk r j kx ix ky iy kz iz e jk r j ke j k 1 j H E j H jk E H k E j E jk H E k H 0 jk E k E 0 jk H k H j E H 0 E 0 H k k E 0 E 2 E k k E k k k H k 2 kx2 ky2 kz2 2 C B A C C A B A B H 1 k E H 1E S 2 0 1 1 1 k E E E E S k E k E 2 2 2 k E S 2 in the direction of k S II Oblique Incidence Onto a Perfect Conductor Field Parallel to Interface TE Transverse Electric A E E i Re E i ej t kxi x kzi z i y E i H i Re cos i i x sin i i z ej t kxi x kzi z kxi k sin i kzi k cos i k E r Re E r ej t kxr x kzr z i y E r j t kxr x kzr z H r Re cos r i x sin r i z e kxr k sin r kzr k cos r Boundary conditions require that 2 From Electromagnetic Field Theory A Problem Solving Approach by Markus Zahn 1987 Used with permission E y x z 0 0 E yi x z 0 E yr x z 0 E i e jkxi x E r e jkxr x 0 H z x z 0 0 H zi x z 0 H zr x z 0 1 jkxi x Ei e sin i E r e jkxr x sin r 0 angle of incidence kxi kxr sin i sin r i r angle of re ection E r E i E i Ei real Ey x z t Re E i e jkz z e jkz z ej t kx x 2Ei sin kz z sin t kx x E i cos e jkz z e jkz z i x sin e jkz z e jkz z i z H x z t Re ej t kx x 2Ei cos cos kz z cos t kx x i x sin sin kz z sin t kx x i z 3 Ky x z 0 t Hx x z 0 t 2Ei cos cos t kx x 2 1 2Ei sin sin2 k z i H S Re E z x 2 Field Parallel to Interface TM Transverse Magnetic B H E i Re E i cos i i x sin i i z ej t kxi x kzi z E i j t kxi x kzi z e H i Re i y E r Re E r cos r i x sin r i z ej t kxr x kzr z E r j t kxr x kzr z H r Re e i y Ex x z 0 t 0 E i cos i e jkxi x E r cos r e jkxr x 0 kxi kxr sin i sin r i r E i E r Re E i cos e jkz z e jkz z i x sin e jkz z e jkz z i z ej t kx x E i Ei real E 2Ei cos sin kz z sin t kx x i x sin cos kz z cos t kx x i z i E Re e jkz z e jkz z ej t kx x i y H 2Ei cos kz z cos t kx x i y 2Ei cos t kx x s x z 0 Ez x z 0 2 Ei sin cos t kx x Kx x z 0 Hy x z 0 Check Conservation of Charge s 0 Kx s 0 K t x t surface divergence 2 1 2Ei sin cos2 k z i H S Re E z x 2 4 III Oblique Incidence Onto a Dielectric From Electromagnetic Field Theory A Problem Solving Approach by Markus Zahn 1987 Used with permission Interface Waves A TE E E i Re E i ej t kxi x kzi z i y E i j t kxi x kzi z H i Re cos i i x sin i i z e 1 E r Re E r ej t kxr x kzr z i y E r j t kxr x kzr z H r Re cos r i x sin r i z e 1 E t Re E t ej t kxt x kzt z i y E r H t Re cos t i x sin t i z ej t kxt x kzt z 2 5 kxi k1 sin i kxr k1 sin r kxt k2 sin t kzi k1 cos i kzr k1 cos r kzt k2 cos t k1 c 1 1 1 k2 c 2 2 2 c1 11 1 c2 12 2 1 11 2 22 Ey z 0 Ey z 0 E i e jkxi x E r e jkxr x E t e jkxt x 1 Hx z 0 Hx z 0 E i cos i e jkxi x E r cos r e jkxr x 1 1 E t cos t e jkxt x 2 kxi kxr kxt k1 sin i k1 sin r k2 sin t i r k1 c2 c2 sin t sin i sin i sin i Snell s Law k2 c1 c1 c0 Index of refraction n r r c …
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