1. Prof. David R. Jackson ECE Dept. Spring 2014 Notes 37 ECE 6341 1

2. Line Source on a Half Space 2 There are branch points at

3. Steepest-Descent Transformation  There are no branch points in the  plane from k y0 (cos  is analytic).  There are still branch points in the  plane from k y1 : 3 Steepest-descent transformation:

4. Line Source on a Grounded Substrate 4 There are branch points only at (even function of k y1 )

5. Steepest-Descent Path Physics  There are no branch points in the  plane (cos  is analytic). Both sheets of the k x plane get mapped into a single sheet of the  plane. 5 Steepest-descent transformation: We focus on the grounded substrate problem for the remaining discussion.

6. Steepest-Descent Path Physics Examine k y0 to see where the  plane is proper and improper: 6

7. SDP Physics (cont.) P: proper I: improper 7 II I I P P PP C

8. SDP Physics (cont.) Mapping of quadrants in k x plane Non-physical “growing” LW pole (conjugate solution) 8 1 2 3 4 I LWP SWP I I I P P P P 4 1 1 4 3 2 2 3 C (symmetric about  /2 line)

9. SDP Physics (cont.) SDP: A leaky-wave pole is considered to be physical if it is captured when deforming to the SDP (otherwise, there is no direct residue contribution). 9 SDP LWP SWP C

10. SDP Physics (cont.) LWP: 10 SDP: (exists if pole is captured) The leaky-wave field is important if: 1)The pole is captured (the pole is said to be “physical”). 2)The residue is strong enough. 3)The attenuation constant  is small. Comparison of Fields:

11. SDP Physics (cont.) LWP captured: Note: The angle  b represents the boundary for which the leaky-wave field exists. 11 SDP LWP

12. SDP Physics (cont.) Behavior of LW field: In rectangular coordinates: where 12 (It is an inhomogeneous plane-wave field.)

13. SDP Physics (cont.) Examine the exponential term: Hence since 13

14. Radially decaying: SDP Physics (cont.) LW exists: Also, recall that 14 LW exists LW decays radially Line source

15. Power Flow 15 Power flows in the direction of the  vector.

16. Power Flow (cont.) Hence Note that Also, 16 Note: There is no amplitude change along the rays (  is perpendicular to  in a lossless region).

17. ESDP (Extreme SDP) ESDP Fast Slow Set We can show that the ESDP divides the LW region into slow-wave and fast-wave regions. The ESDP is important for evaluating the fields on the interface (which determines the far-field pattern). 17

18. ESDP (cont.) (SDP) (ESDP) Recall that To see this: Hence 18

19. ESDP (cont.) Fast-wave region: Slow-wave region: Hence Compare with ESDP: 19

20. ESDP (cont.) ESDP Fast Slow The ESDP thus establishes that for fields on the interface, a leaky-wave pole is physical (captured) if it is a fast wave. LWP captured LWP not captured SWP 20

21. SDP in k x Plane SDP: We now examine the shape of the SDP in the k x plane. The above equations allow us to numerically plot the shape of the SDP in the k x plane. so that 21

22. SDP in k x Plane (cont.) (Please see the appendix for a proof.) LW SDP SW 22

23. Fields on Interface The leaky-wave pole is captured if it is in the fast-wave region. LW ESDP SW fast- wave region 23 The SDP is now a lot simpler!

24. Fields on Interface (cont.) The contribution from the ESDP is called the “space-wave” field or the “residual-wave” (RW) field. (It is similar to the lateral wave in the half-space problem.) 24 LW ESDP SW

25. Asymptotic Evaluation of “Residual-Wave” Field “Residual-Wave” Field Use 25 - +

26. Define 26 Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.)

27. Then Assume Watson’s lemma (alternative form): 27 We then have Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.)

28. It turns out that for the line-source problem at an interface, Hence Note: For a dipole source we have 28 Asymptotic Evaluation of “Residual-Wave” Field (cont.) “Residual-Wave” Field (cont.)

29. Discussion of Asymptotic Methods We have now seen two ways to asymptotically evaluate the fields on an interface as x   for a line source on a grounded substrate: 1) Steepest-descent (  ) plane 2) Wavenumber ( k x ) plane There are no branch points in the steepest-descent plane. The function f (  ) is analytic at the saddle point  0 =  =  /2, but is zero there. The fields on the interface correspond to a higher- order saddle-point evaluation. The SDP becomes an integration along a vertical path that descends from the branch point at k x = k 0. The integrand is not analytic at the endpoint of integration (branch point) since there is a square-root behavior at the branch point. Watson’s lemma is used to asymptotically evaluate the integral. 29

30. Summary of Waves LW SW RW Continuous spectrum 30

31. Interpretation of RW Field The residual-wave (RW) field is actually a sum of lateral-wave fields. 31

32. Proof of angle property: Hence Appendix: Proof of Angle Property The last identity follows from or 32

33. As Hence Proof (cont.) On SDP: or (the asymptote) SDP SAP 33

34. ESDP: Hence To see which choice is correct: In the k x plane, this corresponds to a vertical line for which Proof (cont.) 34