1. Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 17 ECE 6340 Intermediate EM Waves 1

2. General Plane Waves General form of plane wave: where Helmholtz Eq.: so Property of vector Laplacian: Hence The wavenumber terms may be complex. 2

3. General Plane Waves This gives or (separation equation or wavenumber equation) 3

4. General Plane Waves (cont.) Denote Then and (wavenumber equation) Note: For complex k vectors, this is not the same as saying that the magnitude of the k vector is equal to k. 4

5. General Plane Waves (cont.) We can also write The  vector gives the direction of most rapid phase change (decrease). The  vector gives the direction of most rapid attenuation. To illustrate, consider the phase of the plane wave: Similarly, so 5

6. Hence General Plane Waves (cont.) Next, look at Maxwell’s equations for a plane wave: 6

7. General Plane Waves (cont.) Summary of Maxwell’s curl equations for a plane wave: 7

8. General Plane Waves (cont.) Gauss law (divergence) equations: Reminder: The volume charge density is zero in the sinusoidal steady state for a homogeneous source-free region. 8

9. General Plane Waves (cont.) Furthermore, we have from Faraday’s law 9 Dot multiply both sides with E.

10. General Plane Waves (cont.) Summary of dot products: Note: If the dot product of two vectors is zero, we can say that the vectors are perpendicular for the case of real vectors. For complex vectors, we need a conjugate (which we don’t have) to say that the vectors are orthogonal. 10

11. Power Flow 11

12. Power Flow (cont.) so Use so that Note:  is assumed to be real here. and hence 12

13. Assume E 0 = real vector. (The same conclusion holds if it is a real vector times a complex constant.) Hence Power Flow (cont.) 13 (All of the components of the vector are in phase.) Note: This conclusion also holds if k is real, or is a real vector times a complex constant.

14. so Recall Power Flow (cont.) (assuming that  is real) The power flow is: 14

15. Then Power flows in the direction of . Power Flow (cont.) Assumption: Either the electric field vector or the wavenumber vector is a real vector times a complex constant. (This assumption is true for most of the common plane waves.) 15

16. Direction Angles First assume k = real vector (so that we can visualize it) and the medium is lossless ( k is real): The direction angles  are defined by: y x   z Note: 16

17. Direction Angles (cont.) Even when ( k x, k y, k z ) become complex, and k is also complex, these equations are used to define the direction angles, which may be complex. From the direction angle equations we have: 17

18. Homogeneous Plane Wave  are real angles where Definition of a homogenous (uniform) plane wave: 18 In the general lossy case:

19. Homogeneous Plane Wave (cont.) Hence we have The phase and attenuation vectors point in the same direction. The amplitude and phase of the wave are both constant (uniform) in a plane perpendicular to the direction of propagation. Note: A simple plane wave of the form  = exp (-j k z) is a special case, where  = 0. y x z 19 Note: A homogeneous plane wave in a lossless medium has no  vector:

20. Example Plane wave An infinite surface current sheet at z = 0 launches a plane wave in free space. y x z Assume The vertical wavenumber is then given by 20

21. Example (cont.) Part (a): Homogeneous plane wave y x z Power flow We must choose Then 21

22. Example (cont.) Part (b) Inhomogeneous plane wave We must choose Then 22 The wave is evanescent in the z direction.

23. Example (cont.) Power flow (in xy plane) so Note: The inverse cosine should be chosen so that sin  is correct (to give the correct k x and k y ): sin  > 0. 23 z y x 

24. Propagation Circle kxkx kyky Propagating (inside circle) Evanescent (outside circle) k0k0 Free space acts as a “low-pass filter.” 24 Propagating waves:

25. Radiation from Waveguide z y WG PEC EyEy x y b a 25

26. Radiation from Waveguide (cont.) Fourier transform pair: 26

27. Radiation from Waveguide (cont.) Hence 27

28. Radiation from Waveguide (cont.) Hence Next, define We then have 28

29. Hence Radiation from Waveguide (cont.) Solution: y Power flow (Homogeneous) (Inhomogeneous) 29 Note: We want outgoing waves only.

30. Radiation from Waveguide (cont.) Fourier transform of aperture field: 30

31. Radiation from Waveguide (cont.) Note: Hence For E z we have 31 This follows from the mathematical form of E y as an inverse transform.

32. Radiation from Waveguide (cont.) Hence 32 In the space domain, we have

33. Radiation from Waveguide (cont.) Summary (for z > 0 ) 33

34. Some Plane-Wave “  ” Theorems Theorem #1 Theorem #2 If PW is homogeneous: If medium is lossless: Theorem #3 (lossless) (general lossy case) 34

35. Example Find H and compare its magnitude with that of E. Note: It can be seen that Plane wave in free space 35 Given:

36. Example (cont.) so 36

37. Example (cont.) Note: The field magnitudes are not related by  0 ! Hence 37

38. Example (cont.) At the origin (  = 1 ) we have: Verify: (Theorem #1) 38

39. Example (cont.) Hence Verify: (Theorem #3) 39