Lecture 36 Date: 5.04.04 Reflection of Plane Wave at Oblique Incidence (Snells Law, Brewster s Angle, Parallel Polarization, Perpendicular Polarization etc.) Introduction to RF/Microwave
Introduction One can t expect plane waves to be incident normally on a plane in all types of applications. Therefore one must consider the general problem of a plane wave propagating along a specified axis that is arbitrarily located relative to a rectangular coordinate system. The most general form of a plane wave in a lossless media is given by: ( r, t) o cos(. r t) Where: aˆ aˆ aˆ r xaˆ yaˆ zaˆ x x y y z z x y z x y z
Introduction (contd.) ( r, t) o cos(. r t) One can deduce Maxwell s equations in the following form: H H. H 0. 0 They show two things: (i), H and β are orthogonal, (ii) and H lie on the same plane. r x y y cons tant x y z The corresponding magnetic field is: H aˆ
Reflection at Oblique Incidence The plane defined by the propagation vector β and a unit normal vector a n to the boundary is called the plane of incidence. The angle between β and a n is the angle of incidence. cos( x y y t) i io ix iy iz i cos( x y y t) r ro rx ry rz r β r β t cos( x y y t) t to tx ty tz t β i Where: β i, β r and β t will have normal and tangential components to the plane of incidence.
Reflection at Oblique Incidence (contd.) β rz = β cosθ r β sinθ i β sinθ r β sinθ t β iz = β cosθ i β tz = β cosθ t From boundary condition we can write: the tangential component of must be continuous at z = 0. i tan z r tan z t tan z ( 0) ( 0) ( 0) This boundary condition can be satisfied if: i r t ix rx tx x iy ry ty y
Reflection at Oblique Incidence (contd.) First condition implies that the frequency remains unchanged. From second and third conditions we can write: β sinθ i = β sinθ r β sinθ i = β sinθ t Where, θ r is the angle of reflection and θ t is the angle of transmission. We know, for lossless media: θ i = θ sin r t u sin u i
Reflection at Oblique Incidence (contd.) sint u sin u i Snell s Law n sin n i sin t n and n are the refractive indices of the media After these general preliminaries on oblique incidence, let us consider two special cases (i) the field perpendicular to the plane of incidence, (ii) the field parallel to it. Any other polarization may be considered as a linear combination of these two cases.
Parallel Polarization Consider following figure: field lies in the xz-plane, the plane of incidence. It illustrates the case of Parallel Polarization. r t H t In medium the incident and reflected waves are: H r i i ( aˆ cos aˆ sin ) e j x z is io x i z i ( sin cos ) i r r ( aˆ cos aˆ sin ) e j x z rs ro x r z r ( sin cos ) H i io j x z ( sini cos i) H is e a ˆ y io j x z ( sinr cos r) H rs e a ˆ y
Parallel Polarization (contd.) The transmitted fields in medium are given by: t t ( aˆ cos aˆ sin ) e j x z ts to x t z t to j x z ( sint cos t) H ts e a ˆ ( sin cos ) We know: θ i = θ r and tangential components of electric and magnetic fields are continuous at the boundary z=0. Therefore: cos cos io ro i to t y io ro to
Parallel Polarization (contd.) Simplification gives: cos cos cos cos ro t i io t i to cosi cos cos io t i Fresnel s quations For θ i = θ t = 0, we get: ro io to io
Parallel Polarization (contd.) Furthermore, the expressions for reflection coefficient and transmission coefficient can be written in terms of angle of incidence. u cost sin t sin i u In addition: cos t cosi The reflection coefficient Γ equals zero when there is no reflection (only the parallel component is not reflected), and the incident angle at which this happens is called Brewster s Angle θ B. The Brewster s Angle is also known as polarizing angle. At this angle, the perpendicular component of will be reflected. Brewster s concept is utilized in laser tube used in surgical procedures.
Parallel Polarization (contd.) For Brewster s Angle, set Γ = 0: cos cos sin t sin B t B sin B For a lossless and nonmagnetic medium: There is a Brewster Angle for any combination of ε and ε. sin B sin B
Perpendicular Polarization The field is perpendicular to the plane of incidence (the xz-plane). In this situation we get Perpendicular Polarization. Here, H field is parallel to the plane of incidence. H r t e a j ( x sin iz cos i is io ) ˆ y r H t e a j ( x sin rz cos r rs io ) ˆ y i H a a e io ( sin i cos i) is ( ˆ cos ˆ sin ) j x z x i z i H i H a a e ro ( sin r cos r) rs ( ˆ cos ˆ sin ) j x z x r z r
Perpendicular Polarization (contd.) The transmitted fields in medium are given by: e a j ( x sin tz cos t ts to ) ˆ y H a a e to ( sin t cos t) ts ( ˆ cos ˆ sin ) j x z x t z t Again, θ i = θ r and tangential components of electric and magnetic fields are continuous at the boundary z=0. Therefore: io ro to cos cos io ro i to t
Perpendicular Polarization (contd.) Simplification gives: cos cos cos cos ro i t io i t to cosi cos cos io i t Fresnel s quations for perpendicular polarization For θ i = θ t = 0, we get: ro io to io
Perpendicular Polarization (contd.) Simplification for Brewster s Angle in Perpendicular Polarization gives: cos cos B t sin B sin t sin B For nonmagnetic media, μ = μ = μ 0 and therefore: sin B Brewster s Angle doesn t exist as sine of an angle is never greater than unity
Perpendicular Polarization (contd.) If μ μ and ε = ε then: sin B sin B Theoretically possible but rarely occurs in practice
Applications of RF/Microwaves The use of RF/microwaves has greatly expanded. xamples include telecommunications, radio astronomy, land surveying, radar, meteorology, UHF television, terrestrial microwave links, solid-state devices, heating, medicine, and identification systems. Features that make microwaves attractive for communications include wide available bandwidths (capacities to carry information) and directive properties for short wavelengths. Currently, there are three main techniques to carry energy over long distances: (a) microwave links, (b) coaxial cables, and (c) fibre optics. A microwave system normally consists of a transmitter (including a microwave oscillator, waveguides, amplifiers, and transmitting antenna) and a receiver subsystem (including a receiver antenna, transmission line or waveguide, and amplifiers). A microwave network is usually an interconnection of various microwave components and devices.
Applications of RF/Microwaves (contd.) Common microwave components include: Coaxial cables, which are transmission lines for interconnecting microwave components. Waveguide sections, which may be straight, curved, or twisted. Antenna, which transmit or receive M waves efficiently. Terminators, which are designed to absorb the input power and therefore acts as one port network. Attenuators, which are designed to absorb some of the M power passing through the device, thereby decreasing the power level of the microwave signal. Directional couplers, with a mechanism to couple between different ports. Isolators, which allow energy flow in only one direction. Circulators, which are designed to establish various entry/exit points where power can be either fed or extracted. Filters, which suppress unwanted signals and/or separate signals of different frequencies.
RF/Microwave Circuit A microwave circuit consists of microwave components such as sources, transmission lines, waveguides, attenuators, circulators, and filters. One way of analyzing, such circuits, are to relate the input and output variables of each component. At RF/microwave frequencies, where current and voltage are not well defined, it is a common practice to use S-parameters for analysis. S-parameters are defined in terms of wave variables which are more easily measurable at high frequencies than voltage and current.
RF/Microwave Circuit (contd.) Let us consider following -port network: The traveling waves are related to the S-parameters as: b Sa Sa b Sa Sa where, a and a are incident waves at port and respectively; while b and b represent the reflected waves.
RF/Microwave Circuit (contd.) In matrix form: b S S a b S S a The off-diagonal terms represent transmission coefficients, while the diagonal terms represent reflection coefficients. If the network is reciprocal, it will have the same transmission characteristic in either direction. S S If the network is symmetric, then: S S For matched two port network: S S 0
RF/Microwave Circuit (contd.) Γ L The input reflection coefficient in terms of the S-parameters and the load Z L : b S S L i S a S L ZL Zo L ZL Zo
RF/Microwave Circuit (contd.) Similarly, the output reflection coefficient (with V g = 0) can be expressed in terms of the generator impedance Z g : b S S g o S a S g Zg Zin g Zg Zin
xample S-parameters are obtained for a microwave transistor operating at.5 GHz: S = 0.85 < 30, S = 0.07 < 56, S =.68 < 0, S = 0.85 < 40. Determine the input reflection coefficient when Z L = Z o = 75Ω. L Z Z L L Z Z o o 0 S S S S 0.85 30 i L SL