FWLec23

FWLec23 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #23 Normally-incident Reflection from Half Spaces Objectives: This note considers the problem of uniform plane waves reflected from homogeneous half spaces, when the wave is incident from the normal direction. Maxwell’s equations and the boundary conditions at the surface of the medium are used to obtain a solution. The concept of standing wave ratio is introduced, and a number of examples are presented. Reflection from penetrable half spaces Figure 1 depicts a more general situation than that considered in Note #22, involving an interface ( z = 0) between a lossless half space (Region 1) and a half space of a general material with permeability m 2 , permittivity e 2 , and conductivity s 2 (Region 2). We wish to consider the response of this interface to a normally incident wave from Region 1. For an incident wave of the form Ez xEe ii j z () ˆ = - 0 1 b (23.1) Hz y E e i i jz ˆ = - 0 1 1 h (23.2) the general solution in Region 1 can be written as Ee ij z rj z 10 0 11 ˆ =+ -+ bb (23.3) E e E e i r 1 0 1 0 1 ˆ =- Ê Ë Á ˆ ¯ ˜ hh (23.4) In Region 2, we expect only a wave propagating in the + ˆ z direction. The general form of this wave, given the possibly lossy nature of the medium, is Ez xE e e tz j z 20 22 ˆ = -- ab (23.5) E ee t zj z 2 0 2 ˆ = (23.6) where a 2 and 2 can be determined from the medium parameters. The coefficients E r 0 and E t 0 are unknowns to be determined, in order to satisfy the boundary conditions at the interface: ˆ nEE ¥- = 12 0 interface (23.7)

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 ˆ () nE E -= ee 11 2 2 0 interface (23.8) ˆ nHH ¥- = 12 0 interface (23.9) ˆ nH H mm 0 interface (23.10) where ˆˆ nz =- . Since there are no normal components of electric or magnetic fields at the interface, (23.8) and (23.10) are automatically satisfied by (23.3)–(23.6). The other two boundary conditions will be used to find E r 0 and E t 0 . Imposing (23.7) at z = 0 leads to the constraint EEE irt 000 += (23.11) while imposing (23.9) leads to 0 1 0 1 0 2 hhh (23.12) Solving these two equations yields EE ri 0 21 0 = - + hh (23.13) ti 0 2 0 2 = + h (23.14) It will prove convenient to define a reflection coefficient G as G= - + (23.15) and a transmission coefficient t as = + 2 2 (23.16) G and are dimensionless quantities, which may be complex valued if s 2 > 0. It should be apparent from an inspection of (23.15) that if 1 and 2 are both real-valued parameters, G must be in the range from –1 to +1. As long as 1 is real-valued, the magnitude of G may be no greater than 1.

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Using G and t , the total fields in Region 1 can be written as Ez xEe e ij z j z 10 11 () ˆ =+ -+ bb G (23.17) Hz y E ee i jz 1 0 1 ˆ =- h G (23.18) while those in Region 2 can be written as Ez xE e e iz j z 20 22 ˆ = -- ab (23.19) E i zj z 2 0 2 ˆ = (23.20) The coefficient E i 0 is determined by the source that produced the incident wave in (23.1)–(23.2); we assume that this value is known. The continuity of the electric field at the interface implies the relation 1 += G (23.21)
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FWLec23 - A F Peterson Notes on Electromagnetic Fields...

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