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

Uniform plane wave (UPW) power transmission from W/m2 RS

field in the slab is given by Example 6.7: Consider an electric field incident on a copper slab such that the field in the slab is given by V/m We want to find the average power density. Since copper is a good conductor, we can use The intrinsic impedance is Therefore, At the surface (z = 0) the power density is 300 W/m2. But after only 1 skin depth, in this case 21 m, the wave’s power density drops to e-2 (13.5%) of its surface value, or 41 W/m2 in this case. 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. Specifying only the electric field direction is sufficient since magnetic field is readily found from using Maxwell’s equation RS

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

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. RS

Linear polarization The tilt angle  (tau) is the angle the line makes with the x-axis The axial ratio is the ratio of the long axis of an ellipse to the short axis RS

More generalized linear polarization More generalized of two linearly polarized waves, Linear polarization occurs when two linearly polarized waves are Linear polarization is a special case of elliptical polarization that has an infinite axial ratio in phase out of phase RS

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

Circularly polarized wave occurs when Exo and Eyo are equal and Right hand circularly polarized (RHCP) wave Left hand circularly polarized (LHCP) wave Left and right are referred to as the handedness of wave polarization RS

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

Ex1 Given ,determine the polarization of this wave RS

Ex2 The electric field of a uniform plane wave in free space is given by , determine The magnetic field intensity RS

d) Describe the polarization of the wave RS

Reflection and transmission of UPW at normal incidence 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 RS

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

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

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

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

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

Reflection and transmission coefficients (3) Use Eqns. (1) and (2) to eliminate , we’ll get Reflection coefficient Transmission coefficient RS

Types of boundaries: perfect dielectric and perfect conductor (1) From  . Since 2 = 0 then  = -1 and Ex10+= -Ex10-  RS

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

Standing waves (1) When t = m, Ex1 is 0 at all positions. and when z = m, Ex1 is 0 at all time. Null positions occur at RS

Standing waves (2) Since and , the magnetic field is or . Hy1 is maximum when Ex1 = 0 So, E and H are said to be 90o out of phase. There will be no power transmission on either side of the media Poynting vector 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 RS

Ex3 Let medium 1 have 1 = 100  and medium 2 have 2 = 300 , given Ex10+ = 100 V/m. Calculate average incident, reflected, and transmitted power densities 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 RS

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

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

Wave impedance w (2) Since from B.C. at z = -l we may write Using eqns (1a) and (2a) to eliminate , we’ll get … RS

Input impedance in solve to get RS

Power Transmission and Reflection Pin Pt Pr The power in region 2 stays constant in steady-state; power leaves that region to form the reflected and transmitted waves, but is Immediately replenished by the incident wave (from region 1) If  = 0, then there’ll be total transmission. And  = 0 when in = 1 , or the input impedance is matched to that of the incident medium. So how do we achieve this? RS

Half-wave matching method Suppose and Therefore, and so Hence, the 2nd region thickness is the multiple “half-wavelength” as measured in that medium Using eqn: We’ll get when The general effect of a multiple half-wave is to render the 2nd region immaterial to the results on reflection and transmission. Equivalently we have a single interface problem involving 1 and 3 RS

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Refractive index Under lossless conditions, RS

Homework 6.32: Given V/m, find the polarization and handedness. 6.38: Suppose medium 1 (z < 0) is air and medium 2 (z > 0) has r = 16. The trans- mitted magnetic field intensity is known to be Ht = 12cos(t – β2z)ay mA/m. Determine the instantaneous value of the incident electric field. (b) Find the reflected time-averaged power density 6.48: A 100-MHz TE polarized wave with amplitude 1.0 V/m is obliquely incident from air (z < 0) onto a slab of lossless, nonmagnetic material with r = 25 (z > 0). The angle of incidence is 40o. Calculate (a) the angle of transmission, (b) the reflection and transmission coefficients, and (c) the incident, reflected and transmitted fields. RS