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Prof. David R. Jackson Dept. of ECE

ECE 6340 Intermediate EM Waves. Fall 2013. Prof. David R. Jackson Dept. of ECE. Notes 17. General Plane Waves. General form of plane wave:. The wavenumber terms may be complex. where. Helmholtz Eq.:. Property of vector Laplacian:. Hence. so. General Plane Waves. This gives. or.

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Prof. David R. Jackson Dept. of ECE

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

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

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

  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.

  5. General Plane Waves (cont.) We can also write so 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,

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

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

  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.

  9. General Plane Waves (cont.) Furthermore, we have from Faraday’s law 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.

  11. Power Flow

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

  13. Power Flow (cont.) Assume E0=real vector. (The same conclusion holds if it is a real vector times a complex constant.) (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. Hence

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

  15. Power Flow (cont.) Then Power flows in the direction of  . 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.)

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

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

  18. Homogeneous Plane Wave Definition of a homogenous (uniform) plane wave: (, f) are real angles In the general lossy case: where

  19. Homogeneous Plane Wave (cont.) z y x 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. Note: A homogeneous plane wave in a lossless medium has no  vector:

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

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

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

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

  24. Propagation Circle Propagating waves: ky k0 kx Evanescent (outside circle) Propagating (inside circle) Free space acts as a “low-pass filter.”

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

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

  27. Radiation from Waveguide (cont.) Hence

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

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

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

  31. Radiation from Waveguide (cont.) Note: Hence For Ezwe have This follows from the mathematical form of Ey as an inverse transform.

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

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

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

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

  36. Example (cont.) so

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

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

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

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