1. Spring 2015 Notes 32 ECE 6345 Prof. David R. Jackson ECE Dept. 1

2. Overview In this set of notes we extend the spectral-domain method to analyze infinite periodic structures. Two typical examples of infinite periodic problems:  Scattering from a frequency selective surface (FSS)  Input impedance of a microstrip phased array 2

3. FSS Geometry Incident plane wave Reflected plane wave Metal patch Transmitted plane wave Dielectric layer x y z a L b W 3

4. FSS Geometry (cont.) 4 Note: We are following “plane-wave” convention for k x0 and k y0, and “transmission-line” convention for k z0. Note:  denotes any field component of interest.

5. Microstrip Phased Array Geometry Probe current mn : z L W Metal patch Dielectric layer x b y a Ground plane Probe 5 The wavenumbers k x0 and k y0 are impressed by the feed network.

6. Microstrip Phased Array Geometry (cont.) 6 Note: If the structure is infinite, a plane wave get launched. Antenna beam

7. Floquet’s Theorem Fundamental observation: If the structure is infinite and periodic, and the excitation is periodic except for a phase shift, then all of the currents and radiated fields will also be periodic except for a phase shift. This is sometimes referred to as “Floquet’s theorem.” x y z 7

8. Floquet’s Theorem (cont.) From Floquet’s theorem: x y z a b L W Layered media 8

9. x y z a b L W 9 If we know the current of field at any point within the (0,0) unit cell, we know the current and field everywhere. Floquet’s Theorem (cont.)

10. Floquet Waves Let  denote any component of the surface current or the field (at a fixed value of z ). 10 where

11. Floquet Waves (cont.) Hence we have From Fourier-series theory, we know that the 2D periodic function P can be represented as 11

12. Hence, the surface current or field can be expanded in a set of Floquet waves: 12 Floquet Waves (cont.) Incident part Periodic part

13. Note: Each Floquet wave repeats from one unit cell to the next, except for a phase shift that corresponds to that of the incident wave. 13 Similarly, Hence Floquet Waves (cont.)

14. Periodic SDI The surface current on the periodic structure is represented in terms of Floquet waves: To solve for the unknown coefficients, multiply both sides by and integrate over the ( 0, 0 ) unit cell S 0 : 14 Use orthogonality:

15. Periodic SDI (cont.) 15 Hence, we have: Therefore we have: The current J s 00 is the current on the ( 0,0 ) patch. so

16. We now calculate the Fourier transform of the 2D periodic current J s (x, y) (this is what we need in the SDI method): Hence the current on the 2D periodic structure can be represented as Periodic SDI (cont.) 16

17. Next, we calculate the field produced by the periodic patch currents: Hence Periodic SDI (cont.) 17

18. Hence, we have Periodic SDI (cont.) 18

19. Therefore, integrating over the delta functions, we have: Periodic SDI (cont.) 19 The field is thus in the form of a double summation of Floquet waves.

20. Compare: Single element (non-periodic): Periodic array of phased elements: Periodic SDI (cont.) 20

21. Conclusion: The double integral is replaced by a double sum, and a factor is introduced. where Periodic SDI (cont.) 21

22. kxkx kyky Periodic SDI (cont.) Sample points in the (k x, k y ) plane 22

23. Example Microstrip Patch Phased Array x y z a b L W Find E x (x,y,0) 23

24. Single patch: Phased Array (cont.) 24

25. 2D phased array of patches: where Phased Array (cont.) 25

26. The field is of the form The field is thus represented as a “sum of Floquet waves.” Phased Array (cont.) where 26

27. Scan Blindness in Phased Array This occurs when one of the sample points ( p,q) lies on the surface-wave circle (shown for (-2, 0) ). kxkx kyky (E-plane scan) Scan blindness from (-2,0) 27

28. Scan Blindness (cont.) The scan blindness condition is: The field produced by an impressed set of infinite periodic phased surface-current sources will be infinite. 28

29. Scan Blindness (cont.) Physical interpretation: All of the surface-wave fields excited from the patches add up in phase in the direction of the transverse phasing vector: Proof: Start with the surface-wave array factor: N elements x y 29

30. Scan Blindness (cont.) x y Hence we have, in this direction, that N elements 30

31. Scan Blindness (cont.) x y N elements In the direction , the surface fields from each patch add up in phase. 31 Note: There is also an element pattern as well, with the field decaying as 1/  1/2, but this is ignored here.

32. Scan Blindness (cont.) Example x y 32 E-plane scan kxkx kyky

33. Grating Lobes  Grating lobes occur when one or more of the higher-order Floquet waves propagates in space.  For a finite-size array, this corresponds to a secondary beam that gets radiated. 33

34. Grating Lobes (cont.) 34 kxkx kyky Grating wave for (-1,0) k0k0

35. Pozar Circle Diagram E-plane scan H-plane scan k 0 sin  0 Define 35 Visible space circle

36. Pozar Circle Diagram (cont.) Grating Lobes So, we require that The first inequality gives us or 36

37. or Pozar Circle Diagram (cont.) 37 Summary of grating lobe condition: Part of the interior of the (p,q) circle is also inside the visible space circle.

38. Pozar Circle Diagram (cont.) 38 Grating lobe region +

39. E-plane scan H-plane scan Pozar Circle Diagram (cont.) 39 This diagram shows when grating lobes occur in the principal scan planes.

40. or Hence To avoid grating lobes for all scan angles, we require: or Pozar Circle Diagram (cont.) 40 (The circles do not overlap.)

41. Pozar Circle Diagram (cont.) Scan Blindness The equation gives us or 41 So, we require that

42. or where Pozar Circle Diagram (cont.) 42 Summary of scan blindness condition: Part of the boundary of the (p,q) circle is inside the visible space circle.

43. Pozar Circle Diagram (cont.) 43 Scan blindness curve +

44. or Hence To avoid scan blindness for all scan angles, we require or Pozar Circle Diagram (cont.) 44 (The circles do not overlap.)