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FWLec24

# FWLec24 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #24 Oblique Reflection from Half Spaces Objectives: Analyze the problem of plane wave reflection from half spaces of general lossless material, when the wave is incident from an oblique direction. The field is represented in both regions of the problem and boundary conditions at the interface are used to obtain a solution. Fresnel reflection and transmission coefficients are introduced. Snell’s Law and its implications are discussed, and special cases such as the critical angle and the Brewster’s angle are considered. Waves propagating at oblique angles A uniform plane wave that is propagating at an oblique angle with respect to the coordinate axes has a phase behavior that can be depicted by the equation e jk r - (24.1) where k k k = ˆ (24.2) and r x x y y z z = + + ˆ ˆ ˆ (24.3) The vector ˆ k is a unit vector that points in the direction of propagation. The position vector r points from the origin to the point ( x , y , z ). For example, Figure 1 shows the propagation direction in terms of an angle q in the x - z plane. For such an angle, ˆ ˆ sin ˆ cos k x z = + q q (24.4) The phase behavior can be expressed as e e jk r j k x k z x z - - + = ( ) (24.5) where k k x = sin q (24.6) k k z = cos q (24.7) A similar notation will be used in the following analysis.

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Oblique reflection from half spaces Consider a problem involving two non-conducting regions, separated by an interface at z = 0 (Figure 2). Region 1 consists of a material with permeability m 1 and permittivity e 1 . Region 2 has permeability m 2 and permittivity e 2 . For phasor analysis, Region 1 can be characterized by its phase constant b 1 and its intrinsic impedance h 1 , while Region 2 can be characterized by b 2 and h 2 . For a wave incident at an oblique angle in the x - z plane, there are two situations that must be treated separately, the case of perpendicular polarization and the case of parallel polarization . The plane of incidence (the plane containing the normal vector to the interface, ˆ ˆ n z = - , and the ˆ k -vector of the incident wave) is the x - z plane. Perpendicular polarization is the situation when the electric field vector is perpendicular to the plane of incidence (in other words, in the ˆ y direction). Parallel polarization is the case when the electric field lies parallel to the plane of incidence, and the magnetic field is perpendicular ( H has only a ˆ y component). In the following, these two polarizations will be treated separately. In the event that the incident wave consists of a mixture of these two situations, the results can be combined together using the principle of superposition. Perpendicular polarization A uniform plane wave is incident from the Region 1 side of the interface, at an oblique angle q i with respect to the normal vector to the interface, ˆ ˆ n z = - . The incident part of the wave for perpendicular polarization can be expressed as E x z y E e i i j x z i i ( , ) ˆ ( sin cos ) = - + 0 1 b q q (24.8) H x z j E E x z e i i i i i j x z i i ( , ) ( ˆ cos

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