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ch05 - 5 Reflection and Transmission 5.1 Propagation...

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5 Reflection and Transmission 5.1 Propagation Matrices In this chapter, we consider uniform plane waves incident normally on material inter- faces. Using the boundary conditions for the fields, we will relate the forward-backward fields on one side of the interface to those on the other side, expressing the relationship in terms of a 2 × 2 matching matrix . If there are several interfaces, we will propagate our forward-backward fields from one interface to the next with the help of a 2 × 2 propagation matrix . The combination of a matching and a propagation matrix relating the fields across different interfaces will be referred to as a transfer or transition matrix. We begin by discussing propagation matrices. Consider an electric field that is lin- early polarized in the x -direction and propagating along the z -direction in a lossless (homogeneous and isotropic) dielectric. Setting E (z) = ˆ x E x (z) = ˆ x E(z) and H (z) = ˆ y H y (z) = ˆ y H(z) , we have from Eq. (2.2.6): E(z) = E 0 + e jkz + E 0 e jkz = E + (z) + E (z) H(z) = 1 η E 0 + e jkz E 0 e jkz = 1 η E + (z) E (z) (5.1.1) where the corresponding forward and backward electric fields at position z are: E + (z) = E 0 + e jkz E (z) = E 0 e jkz (5.1.2) We can also express the fields E ± (z) in terms of E(z), H(z) . Adding and subtracting the two equations (5.1.1), we find: E + (z) = 1 2 E(z) + ηH(z) E (z) = 1 2 E(z) ηH(z) (5.1.3) Eqs.(5.1.1) and (5.1.3) can also be written in the convenient matrix forms:
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5.1. Propagation Matrices 151 E H = 1 1 η 1 η 1 E + E , E + E = 1 2 1 η 1 η E H (5.1.4) Two useful quantities in interface problems are the wave impedance at z : Z(z) = E(z) H(z) (wave impedance) (5.1.5) and the reflection coefficient at position z : Γ(z) = E (z) E + (z) (reflection coefficient) (5.1.6) Using Eq. (5.1.3), we have: Γ = E E + = 1 2 (E ηH) 1 2 (E + ηH) = E H η E H + η = Z η Z + η Similarly, using Eq. (5.1.1) we find: Z = E H = E + + E 1 η (E + E ) = η 1 + E E + 1 E E + = η 1 + Γ 1 Γ Thus, we have the relationships: Z(z) = η 1 + Γ(z) 1 Γ(z) Γ(z) = Z(z) η Z(z) + η (5.1.7) Using Eq. (5.1.2), we find: Γ(z) = E (z) E + (z) = E 0 e jkz E 0 + e jkz = Γ( 0 )e 2 jkz where Γ( 0 ) = E 0 /E 0 + is the reflection coefficient at z = 0. Thus, Γ(z) = Γ( 0 )e 2 jkz (propagation of Γ ) (5.1.8) Applying (5.1.7) at z and z = 0, we have: Z(z) η Z(z) + η = Γ(z) = Γ( 0 )e 2 jkz = Z( 0 ) η Z( 0 ) + η e 2 jkz This may be solved for Z(z) in terms of Z( 0 ) , giving after some algebra: Z(z) = η Z( 0 ) tan kz η jZ( 0 ) tan kz (propagation of Z ) (5.1.9) 152 5. Reflection and Transmission The reason for introducing so many field quantities is that the three quantities { E + (z), E (z), Γ(z) } have simple propagation properties, whereas { E(z), H(z), Z(z) } do not. On the other hand, { E(z), H(z), Z(z) } match simply across interfaces, whereas { E + (z), E (z), Γ(z) } do not. Eqs. (5.1.1) and (5.1.2) relate the field quantities at location z to the quantities at z = 0. In matching problems, it proves more convenient to be able to relate these quantities at two arbitrary locations.
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