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

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

3. RS 3 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/m 2. 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/m 2 in this case.

4. 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 4

5. RS Linear polarization Consider in free space, At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave” 5

6. RS A pair of linearly polarized wave also produces linear polarization Linear polarization At z = 0 plane At t = 0, both linearly polarized waves have their maximum values. 6

7. RS 7 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

8. RS 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 More generalized linear polarization in phase out of phase 8

9. RS Superposition of two linearly polarized waves that If  x = 0  and  y = 45 , we have Elliptically polarized wave 9

10. RS occurs when E xo and E yo 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 Circularly polarized wave 10

11. RS Phasor forms: for RHCP, for LHCP, Circularly polarized wave from Note: There are also RHEP and LHEP 11

12. RS Ex1 Given,determine the polarization of this wave 12

13. RS Ex2 The electric field of a uniform plane wave in free space is given by, determine a)f b)The magnetic field intensity 13

14. RS c) d) Describe the polarization of the wave 14

15. RS Reflection and transmission of UPW at normal incidence 15

16. RS Assume the medium is lossless, let the incident electric field to be or in a phasor form since then we can show that Normal incidence – the propagation direction is normal to the boundary Incident wave 16

17. Transmitted wave RS Assume the medium is lossless, let the transmitted electric field to be then we can show that Transmitted wave 17

18. RS At z = 0, we have and   1 =  2 are media the same? From boundary conditions, Reflected wave (1) 18

19. There must be a reflected wave RS and This wave travels in –z direction. Reflected wave (2) 19

20. Boundary conditions (reflected wave is included) RS from therefore at z = 0 (1) Reflection and transmission coefficients (1) 20

21. RS from therefore at z = 0 (2) Boundary conditions (reflected wave is included) Reflection and transmission coefficients (2) 21

22. Use Eqns. (1) and (2) to eliminate, we’ll get RS Reflection coefficient Transmission coefficient Reflection and transmission coefficients (3) 22

23. RS Types of boundaries: perfect dielectric and perfect conductor (1) From . Since  2 = 0 then  = -1 and E x10 + = -E x10 -  23

24. RS Types of boundaries: perfect dielectric and perfect conductor (2) This can be shown in an instantaneous form as Standing wave 24

25. RS Standing waves (1) When  t = m , E x1 is 0 at all positions. and when  z = m , E x1 is 0 at all time. Null positions occur at 25

26. RS Standing waves (2) Since and, the magnetic field is or. H y1 is maximum when E x1 = 0 So, E and H are said to be 90 o out of phase. There will be no power transmission on either side of the media Poynting vector 26

27. 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 27

28. RS Ex3 Let medium 1 have  1 = 100  and medium 2 have  2 = 300 , given E x10 + = 100 V/m. Calculate average incident, reflected, and transmitted power densities 28

29. 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 29

30. RS Wave reflection from multiple interfaces (2) Assume lossless media, we have then we can show that 30

31. RS Wave impedance  w (1) Use Euler’s identity, we can show that 31

32. RS Wave impedance  w (2) Since from B.C. at z = -l we may write 32 (1a) (2a) Using eqns (1a) and (2a) to eliminate, we’ll get …

33. RS Input impedance  in solve to get 33

34. RS 34 Power Transmission and Reflection  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? P in PrPr PtPt

35. RS 35 Half-wave matching method Suppose and Therefore, and so Hence, the 2 nd region thickness is the multiple “half-wavelength” as measured in that medium Using eqn: We’ll getwhen The general effect of a multiple half-wave is to render the 2 nd region immaterial to the results on reflection and transmission. Equivalently we have a single interface problem involving  1 and  3

36. RS 36

37. RS Refractive index Under lossless conditions, 37

38. RS 38 Homework 6.32: Given V/m, find the polarization and handedness. 6.38: Suppose medium 1 (z 0) has  r = 16. The trans- mitted magnetic field intensity is known to be H t = 12cos(  t – β 2 z)a y mA/m. (a)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). The angle of incidence is 40 o. Calculate (a) the angle of transmission, (b) the reflection and transmission coefficients, and (c) the incident, reflected and transmitted fields.