Report

ECE 6341 Spring 2014 Prof. David R. Jackson ECE Dept. Notes 6 1 Leaky Modes v TM1 Mode v R 1 r u tan u R (k0 h) n12 1 SW u Splitting point ISW f > fs f = fs We will examine the solutions as the frequency is lowered. 2 Leaky Modes (cont.) v a) f > fc SW+ISW TM1 Mode SW The TM1 surface wave is above cutoff. There is also an improper TM1 SW mode. u ISW Note: There is also a TM0 mode, but this is not shown 3 Leaky Modes (cont.) a) f > fc SW+ISW The red arrows indicate the direction of movement as the frequency is lowered. Im kz ISW SW k0 k1 Re kz v TM1 Mode kz k0 v h (k z2 k02 )1/2 k z2 k02 SW u kz k0 k z k0 ISW 4 Leaky Modes (cont.) v b) f = fc TM1 Mode u The TM1 surface wave is now at cutoff. There is also an improper SW mode. 5 Leaky Modes (cont.) b) f = fc Im kz k0 k1 Re kz v TM1 Mode u 6 Leaky Modes (cont.) c) f < fc 2 ISWs v TM1 Mode u The TM1 surface wave is now an improper SW, so there are two improper SW modes. 7 Leaky Modes (cont.) c) f < fc 2 ISWs Im kz k0 k1 Re kz v TM1 Mode The two improper SW modes approach each other. u 8 Leaky Modes (cont.) d) f = fs v TM1 Mode u The two improper SW modes now coalesce. 9 Leaky Modes (cont.) d) f = fs Im kz Splitting point k0 k1 Re kz v TM1 Mode u 10 Leaky Modes (cont.) v e) f < fs u The wavenumber kz becomes complex (and hence so do u and v). The graphical solution fails! (It cannot show us complex leaky-wave modal solutions.) 11 Leaky Modes (cont.) e) f < fs 2 LWs This solution is rejected as completely non-physical since it grows with distance z. kz z j z Im kz k0 LW k1 Re kz LW The growing solution is the complex conjugate of the decaying one (for a lossless slab). 12 Leaky Modes (cont.) Proof of conjugate property (lossless slab) TRE: 1 2 2 z 1 2 2 0 1 2 2 2 r tan (k1 k z ) h (k z2 k ) (k12 k ) TMx Mode Take conjugate of both sides: 1 *2 2 z 1 2 2 0 2 *2 12 r tan (k1 k z ) h (k z*2 k ) (k12 k ) Hence, the conjugate is a valid solution. 13 Leaky Modes (cont.) Im kz SW ISW k0 k1 a) f > fc Re kz Here we see a summary of the frequency behavior for a typical surface-wave mode (e.g., TM1). Im kz k0 k1 b) f = fc Re kz Im kz k0 c) f < fc k1 Exceptions: Re kz Im kz splitting point k0 d) f = fs k1 Re kz Im kz e) f < fs k0 LW LW k1 TM0: Always remains a proper physical SW mode. TE1: Goes from proper physical SW to nonphysical ISW; remains nonphysical ISW down to zero frequency. Re kz 14 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. ˆ x zk ˆ z xˆ x zˆ z Re k Re xk x tan0 z / x Power flow 0 z kz z j z 15 Leaky Modes (cont.) One interesting aspect: The fields of the leaky mode must be improper (exponentially increasing). Proof: 1 2 2 z k x 0 (k k ) 2 0 Notes: z > 0 (propagation in +z direction) z > 0 (propagation in +z direction) x > 0 (outward radiation) kx20 k02 kz2 x j x k02 z j z 2 2 x2 x2 j 2x x k02 z2 z2 j 2 z z Taking the imaginary part of both sides: x x z z 16 Leaky Modes (cont.) 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). x 0 Power flow z Source Region of weak fields Leaky mode Region of exponential growth kz z j z A hypothetical source launches a leaky wave going in one direction. 17 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. 18 Leaky Modes (cont.) f) f < fp Physical LW Im kz k0 Physical k1 LW Re kz Non-physical f = fp Note: The physical region is also the fast-wave region. Physical leaky wave region (Re kz < k0) 19 Leaky Modes (cont.) If the leaky mode is within the physical (fast-wave) region, a wedgeshaped radiation region will exist. This is illustrated on the next two slides. 20 Leaky Modes (cont.) x 0 Power flow z Source Leaky mode xˆ x zˆ z kz z j z z sin 0 2 x2 z2 k x2 k z2 k02 Hence z k0 sin 0 (assuming small attenuation) Significant radiation requires z < k0. 21 Leaky Modes (cont.) x 0 Power flow z Source Leaky mode kz z j z 0 sin1 z / k0 As the mode approaches a slow wave (z k0), the leakage region shrinks to zero (0 90o). 22 Leaky Modes (cont.) Phased-array analogy x 0 d z I0 I1 In j k0d sin0 n I n I0 e IN n k0d sin 0 n Equivalent phase constant: e jkz z z nd e j k0 d sin 0 n kz k0 sin 0 Note: A beam pointing at an angle in “visible space” requires that kz < k0. 23 Leaky Modes (cont.) The angle 0 also forms the boundary between regions where the leakywave field increases and decreases with radial distance in cylindrical coordinates (proof omitted*). x Fields are increasing radially Power flow 0 Fields are decreasing radially Source z Leaky mode kz z j z *Please see one of the homework problems. 24 Leaky Modes (cont.) Excitation problem: Line source The aperture field may strongly resemble the field of the leaky wave (creating a good leaky-wave antenna). Requirements: 1) 2) 3) 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). 25 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 < fs non-physical LW (radiating somewhat, improper) d) f < fp physical LW (strong focused radiation, improper) The frequency region fp < f < fc is 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. 26 Leaky Modes (cont.) Spectral-gap region f = fc fp < f < fc Im kz k0 ISW f = fs k1 Re kz Physical LW Non-physical f = fp 27 Field Radiated by Leaky Wave x TEx leaky wave Aperture z Line source For x > 0: 1 Ey x, z 2 Ey 0, kz e jkx x e jkz z dkz , k x k02 k z2 1/2 Assume: E y 0, z e jk zLW z LW k z Then E y 0, k z 2 j k 2 k LW 2 z z Note: The wavenumber kx is chosen to be either positive real or negative imaginary. 28 x x/ /0 k LW z 3 / k0 j 0.02 2 Radiation occurs at 60o. 0 10 10 5 5 z / 0 00 -10 -10 0 0 10 10 29 x x/ /0 k LW z 3 / k0 j 0.002 2 Radiation occurs at 60o. 0 10 10 5 5 z / 0 00 -10 -10 0 0 10 10 30 x x/ /0 kzLW / k0 1.5 j0.02 The LW is nonphysical. 0 10 10 5 5 z / 0 00 -10 -10 0 0 10 10 31 Leaky-Wave Antenna x x / 0 Aperture z Line source x x/ / 0 0 10 10 Near field Far field 5 5 z / 0 00 -10 0 10 32 Leaky-Wave Antenna (cont.) x x / 0 z line source Far-Field Array Factor (AF) AF E y 0, z e j k0 sin z dz e jk zLW z e j k0 sin z dz LW k z AF 2 j 2 k 2 sin 2 k LW z 0 33 Leaky-Wave Antenna (cont.) z j z AF 2 j 2 2 2 k sin j z z 0 z j z AF 2 2 2 k0 sin z2 z2 j 2 z z 1/2 2 2 z z AF 2 2 k 2 sin 2 2 2 2 2 z z z z 0 A sharp beam occurs at k0 sin 0 z 34 Leaky-Wave Antenna (cont.) Two-layer x / (substrate/superstrate) structure excited by a line source. 0 x r2 t r1 z b Far Field r 2 r1 b / 1 0.5 t / 2 0.25 Note: Here the two beams have merged to become a single beam at broadside. 35 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) 36 Leaky-Wave Antenna (cont.) x / 0 (E-plane shown on one side, H-plane on the other side) 37 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. y This is the first leaky-wave antenna invented. y Slotted waveguide x z k02 a z b a Note: 2 z k0 38 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 TE10 mode Waveguide Source Top view 39 Leaky-Wave Antenna (cont.) x / 0 Another variation: Holey waveguide y y x z r b a 2 z k02 a p p 0 40 Leaky-Wave Antenna (cont.) x / 0 Another type of leaky-wave antenna: Surface-Integrated Waveguide (SIW) s ls w ws via d p slot h εr 41