ENE 428 Microwave Engineering Lecture 3 Polarization Reflection

ENE 428 Microwave Engineering Lecture 3 Polarization, Reflection and Transmission at normal incidence 1 RS

Uniform plane wave (UPW) power transmission from W/m 2 2 RS

Polarization • UPW is characterized by its propagation direction and frequency. • Its attenuation and phase are determined by medium’s parameters. • Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation. 3 RS

Linear polarization • Consider in free space, • At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave” RS 4

Linear polarization • A pair of linearly polarized wave also produces linear polarization At z = 0 plane At t = 0, both linearly polarized waves have their maximum values. 5 RS

More generalized linear polarization • More generalized of two linearly polarized waves, • Linear polarization occurs when two linearly polarized waves are in phase out of phase 6 RS

Elliptically polarized wave • Superposition of two linearly polarized waves that • If x = 0 and y = 45 , we have 7 RS

Circularly polarized wave • occurs when Exo and Eyo are equal and • Right hand circularly polarized (RHCP) wave • Left hand circularly polarized (LHCP) wave 8 RS

Circularly polarized wave • Phasor forms: from for RHCP, for LHCP, Note: There also RHEP and LHEP RS 9

Ex 1 Given , determine the polarization of this wave 10 RS

Ex 2 The electric field of a uniform plane wave in free space is given by , determine a) f b) The magnetic field intensity 11 RS

c) d) Describe the polarization of the wave 12 RS

Reflection and transmission of UPW at normal incidence 13 RS

Incident wave • Normal incidence – the propagation direction is normal to the boundary Assume the medium is lossless, let the incident electric field to be or in a phasor form since then we can show that 14 RS

Transmitted wave • Transmitted wave Assume the medium is lossless, let the transmitted electric field to be then we can show that 15 RS

Reflected wave (1) • From boundary conditions, At z = 0, we have and 1 = 2 are media the same? 16 RS

Reflected wave (2) • There must be a reflected wave and This wave travels in –z direction. 17 RS

Reflection and transmission coefficients (1) • Boundary conditions (reflected wave is included) from therefore at z = 0 (1) 18 RS

Reflection and transmission coefficients (2) • Boundary conditions (reflected wave is included) from therefore at z = 0 (2) 19 RS

Reflection and transmission coefficients (3) • Solve Eqs. (1) and (2) to get Reflection coefficient Transmission coefficient 20 RS

Types of boundaries: perfect dielectric and perfect conductor (1) From Since 2 = 0 . then = -1 and Ex 10+= -Ex 10 - 21 RS

Types of boundaries: perfect dielectric and perfect conductor (2) This can be shown in an instantaneous form as Standing wave 22 RS

Standing waves (1) When t = m , Ex 1 is 0 at all positions. and when z = m , Ex 1 is 0 at all time. Null positions occur at 23 RS

Standing waves (2) Since and , the magnetic field is or . Hy 1 is maximum when Ex 1 = 0 Poynting vector 24 RS

Power transmission for 2 perfect dielectrics (1) Then 1 and 2 are both real positive quantities and 1 = 2 = 0 Average incident power densities 25 RS

Ex 3 Let medium 1 have 1 = 100 and medium 2 have 2 = 300 , given Ex 10+ = 100 V/m. Calculate average incident, reflected, and transmitted power densities 26 RS

Wave reflection from multiple interfaces (1) • Wave reflection from materials that are finite in extent such as interfaces between air, glass, and coating • At steady state, there will be 5 total waves 27 RS

Wave reflection from multiple interfaces (2) Assume lossless media, we have then we can show that 28 RS

Wave reflection from multiple interfaces (2) Assume lossless media, we have then we can show that 29 RS

Wave impedance w (1) Use Euler’s identity, we can show that 30 RS

Wave impedance w (2) Since from B. C. at z = -l we may write 31 RS

Input impedance in solve to get 32 RS

Refractive index Under lossless conditions, 33 RS