1. Prof. David R. Jackson ECE Dept. Spring 2014 Notes 34 ECE 6341 1

2. Example By using a Fourier-transform method, the solution is where, for y > 0, (Please see the appendix.) Line current 2

3. Example (cont.) The vertical wavenumber is The wavenumber k y is interpreted as A convenient change of variables is the “steepest-descent transformation” 3 (This follows from the radiation condition at infinity.)

4. Example (cont.) Then The path C in the complex  -plane is not unique until we choose either + or – here. This is because the path is not uniquely determined by only To see this in more detail, write 4

5. Example (cont.) Because k x is real, Hence or 5

6. Example (cont.) There are four possible paths. 6

7. Example (cont.) k x will vary from -  to  along each of these paths. The path must be chosen so that along the path Assume we choose the + sign (an arbitrary choice): 7

8. Example (cont.) Correct path C : 8 C

9. Example (cont.) Now proceed with the change of variables: Hence, we have 9

10. Example (cont.) Next, let 10

11. Example (cont.) The integral then becomes Ignoring the constant in front, we can identify Hence 11

12. Example (cont.) SDP: so (SDP or SAP) Hence 12

13. Example (cont.) Using 13 we also see that This will help us determine which curve is the SDP and which is the SAP.

14. Example (cont.) (SDP or SAP) SDP SAP 14

15. Example (cont.) Examination of the original path allows us to determine the direction of integration along the SDP. 15 SDP

16. Example (cont.) Calculate : From the figure we see that the correct choice is so or 16

17. Example (cont.) We then have 17 or Recipe:

18. Example (cont.) It can easily be verified that The exact solution is: for 18

19. Appendix Derivation of formula TM z : 19

20. Appendix (cont.) We then have Define: 20

21. Appendix (cont.) Choose - sign for Boundary Conditions at y = 0 : (satisfied automatically) 21

22. Appendix (cont.) Hence We then have 22

23. Appendix (cont.) Hence And then 23