1. ECE 6341
Spring 2016
Prof. David R. Jackson
ECE Dept.
Notes 34

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

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

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

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

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

7. Example (cont.) kx 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):

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

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

10. Example (cont.)
Next, let

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

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

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

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

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

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

17. Example (cont.)
Method of steepest-descent recipe:
We then have or

18. Example (cont.) The exact solution is:
It can easily be verified that the asymptotic result is correct, since
so that

19. Appendix
Derivation of formula
TMz :

20. Appendix (cont.) Introduce the Fourier transform pair: We then have
Define:

21. Appendix (cont.) Boundary conditions at y = 0:
(satisfied automatically)

22. Appendix (cont.)
Hence
We then have

23. Appendix (cont.)
Hence
And then