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ls1_unit_8 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 74...

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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 74 R. Victor Jones, February 29, 2000 VIII. G UIDED W AVES IN P LANAR S TRUCTURES C HARACTERISTICS OF P LANE W AVE S OLUTIONS : For the record, let us restate the frequency domain, macroscopic Maxwell's equations which are valid in the high frequency or o ptical regime for a linear, local, isotropic medium -- viz. r × r E r r , ω ( ) = r ×ε 1 r r , ω ( ) r D r r , ω ( ) = i ω r B r r , ω ( ) = i ω μ 0 r H r r , ω ( ) [ VIII- 1a ] r × r B r r , ω ( ) 0 r × r H r r , ω ( ) = μ 0 r J r r , ω ( ) + i ω μ 0 ε r r , ω ( ) r E r r , ω ( ) 0 r J r r , ω ( ) + i ω μ 0 r D r r , ω ( ) [ VIII- 1b ] r ⋅ε r r , ω ( ) r E r r , ω ( ) = r r D r r , ω ( ) = ρ r r , ω ( ) [ VIII- 1c ] r r B r r , ω ( ) 0 r r H r r , ω ( ) = 0 [ VIII- 1d ] Further, in regions free of explicit sources of current and charge we may write r × r E r r , ω ( ) = i ω μ 0 r H r r , ω ( ) [ VIII- 2a ] r × r H r r , ω ( ) = i ωε eff r r , ω ( ) r E r r , ω ( ) [ VIII- 2b ] r ⋅ε r r , ω ( ) r E r r , ω ( ) = 0 [ VIII- 2c ] r r H r r , ω ( ) = 0 [ VIII- 2d ] where ε eff r r , ω ( ) ≡ε eff r r , ω ( ) i σ r r , ω ( ) ω . In this set of lectures it is our intention to explore in some depth plane wave propagation within a uniform medium -- i.e. ε eff r r , ω ( ) ≡ ε eff ω ( ) . To that end we consider a plane wave solution in the form
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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 75 R. Victor Jones, February 29, 2000 r E r r , ω ( ) = r E ω ( ) exp i r r r k ( ) = r E ω ( ) exp i xk x + y k y + z k z ( ) [ ] [ VIII- 3 ] which may pictorially represented as Therefore r r E r r , ω ( ) = r r E ω ( ) exp i r r r k ( ) [ ] = i r k r E r r , ω ( ) [ VIII- 4a ] r × r E r r , ω ( ) = r × r E ω ( ) exp i r r r k ( ) [ ] = i r k × r E r r , ω ( ) [ VIII- 4b ] and the Maxwell's equations formulated in Equation [VIII-2 ] become
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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 76 R. Victor Jones, February 29, 2000 i r k × r E r r , ω ( ) = i ω μ 0 r H r r , ω ( ) [ VIII- 5a ] i r k × r H r r , ω ( ) = i ωε ω ( ) r E r r , ω ( ) [ VIII- 5b ] i r k r E r r , ω ( ) = 0 [ VIII- 5c ] i r k r H r r , ω ( ) = 0 [ VIII- 5d ] Operate through on both sides of Equation [ VIII- 5a ] with the operator " r k × " we obtain r k × r k × r E r r , ω ( ) [ ] = ω μ 0 r k × r H r r , ω ( ) [ VIII- 6a ] Using the "bac-cab" rule 30 and Equation [ VIII- 5b ] this becomes r k r k r E r r , ω ( ) [ ] r k r k [ ] r E r r , ω ( ) = −ω 2 μ 0 ε eff ω ( ) r E r r , ω ( ) [ VIII- 6b ] or finally r k r k [ ] r E r r , ω ( ) = ω 2 μ 0 ε eff ω ( ) r E r r , ω ( ) k 2 = ω 2 μ 0 ε eff ω ( ) [ VIII- 6c ] Substituting these results into Equation [ VIII- 5a ] we obtain r H r r , ω ( ) = ω μ 0 ( ) 1 k ˆ k × r E r r , ω ( ) [ ] = ε eff ω ( ) μ 0 ˆ k × r E r r , ω ( ) [ ] [ VIII- 7 ] so that the wave impedance is given by ηω ( ) = r E r r , ω ( ) r H r r , ω ( ) = μ 0 ε eff ω ( ) . [ VIII- 11 ] 30 That is r a × r b × r c ( ) = r b r a r c ( ) r c r a r b ( ) .
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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 77 R. Victor Jones, February 29, 2000 Thus, the complete expression for an electromagnetic plane wave propagating in a direction ˆ k in a uniform medium is given by r E r r , t ( ) = r E ω ( ) exp j r r r k − ω t ( ) [ ] [ VIII- 9a ] r H r r , t ( ) = η ω ( ) [ ] 1 ˆ k × r E r r , ω ( ) [ ] [ VIII- 9b ] E LECTROMAGNETIC I NTERFACIAL C ONTINUITY C ONDITIONS : The previous section gives a
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