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HW4_Solution - ECE 201A Homework 4 solution 1 kr Er Ei ki...

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ECE 201A Homework 4 solution 1. i E G Beam splitter z x y i k G r E G r k G i H G r H G t E G t k G t H G (a) 0 ˆ jkz i x i E a E e = G , 0 ˆ jkz i i y E H a e η = G . 0 ˆ jkx r z i E a jrE e = − G , 0 ˆ jkx i i y jrE H a e η = G . 0 ˆ jkz t x i E a tE e = G , 0 ˆ jkz i t y tE H a e η = G . k ω εμ = and μ η ε = . (b) ( ) ( ) 0 2 * 0 0 1 1 ˆ ˆ ˆ Re Re 2 2 2 i i jkz jkz i i i x y i z E E P E H a a E e e a η η + = × = × = G G G ( ) ( ) ( ) 0 2 2 0 * 0 1 1 ˆ ˆ ˆ Re Re 2 2 2 i i jkx jky r r r z y i x r E jr E P E H a a jrE e e a η η = × = × = G G G ( ) ( ) 0 2 2 * 0 0 1 1 ˆ ˆ ˆ Re Re 2 2 2 i t jkz jkz i t t x y i z t E tE P E H a a tE e e a η η + = × = × = G G G Due to power conservation i r t P P P = + G G G . Hence 0 0 0 2 2 2 2 2 2 2 2 i i i E r E t E η η η = + or 2 2 1 r t + = as expected
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