1 / 41

Prof. David R. Jackson ECE Dept.

ECE 6341 . Spring 2014. Prof. David R. Jackson ECE Dept. Notes 6. Leaky Modes. v . TM 1 Mode. R . SW. u . Splitting point. f = f s. ISW. f > f s . We will examine the solutions as the frequency is lowered. v . SW. u . ISW. Leaky Modes (cont.).

gomer
Download Presentation

Prof. David R. Jackson ECE Dept.

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ECE 6341 Spring 2014 Prof. David R. Jackson ECE Dept. Notes 6

  2. Leaky Modes v TM1 Mode R SW u Splitting point f = fs ISW f > fs We will examine the solutions as the frequency is lowered.

  3. v SW u ISW Leaky Modes (cont.) a) f > fc SW+ISW TM1 Mode The TM1 surface wave is above cutoff. There is also an improper TM1 SW mode. Note: There is also a TM0 mode, but this is not shown

  4. Im kz ISW SW k0 k1 Re kz Leaky Modes (cont.) a) f > fc SW+ISW The red arrows indicate the direction of movement as the frequency is lowered. TM1 Mode

  5. Leaky Modes (cont.) v u b) f = fc TM1 Mode The TM1 surface wave is now at cutoff. There is also an improper SW mode.

  6. Leaky Modes (cont.) Im kz k0 k1 Re kz b) f = fc TM1 Mode

  7. Leaky Modes (cont.) v u c) f < fc2ISWs TM1 Mode The TM1 surface wave is now an improper SW, so there are two improper SW modes.

  8. Leaky Modes (cont.) c) f < fc2ISWs Im kz k0 k1 Re kz TM1 Mode The two improper SW modes approach each other.

  9. Leaky Modes (cont.) v u d) f = fs TM1 Mode The two improper SW modes now coalesce.

  10. Leaky Modes (cont.) d) f = fs Im kz Splitting point k0 k1 Re kz TM1 Mode

  11. Leaky Modes (cont.) v u e) f < fs The wavenumberkz becomes complex (and hence so do u and v). The graphical solution fails! (It cannot show us complex leaky-wave modal solutions.)

  12. Leaky Modes (cont.) e) f < fs2 LWs This solution is rejected as completely non-physical since it grows with distance z. Im kz LW k0 k1 Re kz LW The growing solution is the complex conjugate of the decaying one (for a lossless slab).

  13. Leaky Modes (cont.) Proof of conjugate property (lossless slab) TMx Mode TRE: Take conjugate of both sides: Hence, the conjugate is a valid solution.

  14. Leaky Modes (cont.) a) f > fc Here we see a summary of the frequency behavior for a typical surface-wave mode (e.g., TM1). b) f = fc c) f < fc Exceptions: • TM0: Always remains a proper physical SW mode. • TE1: Goes from proper physical SW to nonphysical ISW; remains nonphysical ISW down to zero frequency. d) f = fs e) f < fs

  15. Leaky Modes (cont.) A leaky mode is a mode that has a complex wavenumber (even for a lossless structure). It loses energy as it propagates due to radiation. x Power flow 0 z

  16. Leaky Modes (cont.) One interesting aspect: The fields of the leaky mode must be improper (exponentially increasing). Proof: Notes: z > 0 (propagation in +z direction) z > 0 (propagation in +z direction) x > 0 (outward radiation) Taking the imaginary part of both sides:

  17. Leaky Modes (cont.) x Region of weak fields Power flow 0 Region of exponential growth z Leaky mode Source For a leaky wave excited by a source, the exponential growth will only persist out to a “shadow boundary” once a source is considered. This is justified later in the course by an asymptotic analysis: In the source problem, the LW pole is only captured when the observation point lies within the leakage region (region of exponential growth). A hypothetical source launches a leaky wave going in one direction.

  18. Leaky Modes (cont.) A leaky-mode is considered to be “physical” if we can measure a significant contribution from it along the interface (0 = 90o) . A requirement for a leaky mode to be strongly “physical”is that the wavenumber must lie within the physical region (z=Re kz < k0) where is wave is a fast wave*. (Basic reason: The LW pole is not captured in the complex plane in the source problem if the LW is a slow wave.) * This is justified by asymptotic analysis, given later.

  19. Leaky Modes (cont.) f) f < fp Physical LW Im kz k0 k1 LW Re kz Non-physical Physical f = fp Note: The physical region is also the fast-wave region. Physical leaky wave region (Re kz < k0)

  20. Leaky Modes (cont.) If the leaky mode is within the physical (fast-wave) region, a wedge-shaped radiation region will exist. This is illustrated on the next two slides.

  21. Leaky Modes (cont.) x Power flow 0 z Leaky mode Source (assuming small attenuation) Hence Significant radiation requires z<k0.

  22. Leaky Modes (cont.) x Power flow 0 z Leaky mode Source As the mode approaches a slow wave (zk0), the leakage region shrinks to zero (0 90o).

  23. Leaky Modes (cont.) Phased-array analogy x 0 d z I0 I1 In IN Equivalent phase constant: Note: A beam pointing at an angle in “visible space” requires that kz < k0.

  24. Leaky Modes (cont.) The angle 0 also forms the boundary between regions where the leaky-wave field increases and decreases with radial distance  in cylindrical coordinates (proof omitted*). x Fields are increasing radially Power flow 0 Fields are decreasing radially  z Leaky mode Source *Please see one of the homework problems.

  25. Leaky Modes (cont.) Line source Excitation problem: The aperture field may strongly resemble the field of the leaky wave (creating a good leaky-wave antenna). Requirements: • The LW should be in the physical region. • The amplitude of the LW should be strong. • The attenuation constant of the LW should be small. A non-physical LW usually does not contribute significantly to the aperture field (this is seen from asymptotic theory, discussed later).

  26. Leaky Modes (cont.) Summary of frequency regions: a) f > fc physical SW (non-radiating, proper) b) fs<f < fc non-physical ISW (non-radiating, improper) c) fp <f < fsnon-physical LW (radiating somewhat, improper) d) f < fp physical LW (strong focused radiation, improper) The frequency region fp <f < fcis called the“spectral-gap” region (a term coined by Prof. A. A. Oliner). The LW mode is usually considered to be nonphysical in the spectral-gap region.

  27. Leaky Modes (cont.) Spectral-gap region f = fc fp < f < fc Im kz ISW k0 k1 f = fs Re kz LW Physical Non-physical f = fp

  28. Field Radiated by Leaky Wave x z Line source TEx leaky wave Aperture For x > 0: Assume: Note: The wavenumber kx is chosen to be either positive real or negative imaginary. Then

  29. Radiation occurs at 60o. x / 0 x / 0 10 10 5 5 z / 0 0 0 -10 0 10 -10 0 10

  30. Radiation occurs at 60o. x / 0 x / 0 10 10 5 5 z / 0 0 0 -10 0 10 -10 0 10

  31. The LW is nonphysical. x / 0 x / 0 10 10 5 5 z / 0 0 0 -10 0 10 -10 0 10

  32. Leaky-Wave Antenna x z Line source x / 0 Aperture Near field Far field

  33. Leaky-Wave Antenna (cont.) x z line source x / 0  Far-Field Array Factor (AF)

  34. Leaky-Wave Antenna (cont.) A sharp beam occurs at

  35. Leaky-Wave Antenna (cont.) t b Two-layer (substrate/superstrate) structure excited by a line source. x / 0 x z Far Field Note: Here the two beams have merged to become a single beam at broadside.

  36. Leaky-Wave Antenna (cont.) x / 0 Implementation at millimeter-wave frequencies (62.2 GHz) r1 = 1.0, r2 = 55, h = 2.41 mm, t = 0.484 mm, a = 3.73 0 (radius)

  37. Leaky-Wave Antenna (cont.) x / 0 (E-plane shown on one side, H-plane on the other side)

  38. Leaky-Wave Antenna (cont.) • W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622. W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No. 2.402.622. This is the first leaky-wave antenna invented. y Slotted waveguide Note:

  39. Leaky-Wave Antenna (cont.) x / 0 The slotted waveguide illustrates in a simple way why the field is weak outside of the “leakage region.” Region of strong fields (leakage region) Slot a Waveguide TE10 mode Source Top view

  40. Leaky-Wave Antenna (cont.) Another variation: Holey waveguide x / 0 y

  41. Leaky-Wave Antenna (cont.) x / 0 Another type of leaky-wave antenna: Surface-Integrated Waveguide (SIW)

More Related