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

Leaky Modes 2 f = f s v u TM 1 Mode ISW SW f > f s We will examine the solutions as the frequency is lowered. Splitting point R

Leaky Modes (cont.) a) f > f c SW+ISW v u ISW SW 3 TM 1 Mode The TM 1 surface wave is above cutoff. There is also an improper TM 1 SW mode. Note: There is also a TM 0 mode, but this is not shown

Leaky Modes (cont.) a) f > f c SW+ISW Im k z k1k1 SW Re k z k0k0 ISW The red arrows indicate the direction of movement as the frequency is lowered. 4 TM 1 Mode

b) f = f c Leaky Modes (cont.) v u 5 TM 1 Mode The TM 1 surface wave is now at cutoff. There is also an improper SW mode.

b) f = f c Leaky Modes (cont.) 6 Im k z k1k1 Re k z k0k0 TM 1 Mode

c) f < f c 2 ISWs Leaky Modes (cont.) v u 7 The TM 1 surface wave is now an improper SW, so there are two improper SW modes. TM 1 Mode

c) f < f c 2 ISWs Im k z k1k1 Re k z k0k0 Leaky Modes (cont.) 8 The two improper SW modes approach each other. TM 1 Mode

d) f = f s Leaky Modes (cont.) v u 9 The two improper SW modes now coalesce. TM 1 Mode

d) f = f s Leaky Modes (cont.) 10 Im k z k1k1 Re k z k0k0 Splitting point TM 1 Mode

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

Leaky Modes (cont.) 12 Im k z k1k1 e) f < f s 2 LWs Re k z k0k0 LW This solution is rejected as completely non-physical since it grows with distance z. The growing solution is the complex conjugate of the decaying one (for a lossless slab).

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

Leaky Modes (cont.) a) f > f c 14 b) f = f c c) f < f c d) f = f s e) f < f s Here we see a summary of the frequency behavior for a typical surface-wave mode (e.g., TM 1 ).  TM 0 : Always remains a proper physical SW mode.  TE 1 : Goes from proper physical SW to nonphysical ISW; remains nonphysical ISW down to zero frequency. Exceptions:

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. 15 z Power flow x 00

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

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). 17 00 z Power flow Leaky mode x Region of exponential growth Region of weak fields Source A hypothetical source launches a leaky wave going in one direction.

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

f) f < f p Physical LW Physical leaky wave region ( Re k z < k 0 ) Leaky Modes (cont.) 19 Im k z k1k1 Re k z k0k0 LW f = f p Physical Non-physical Note: The physical region is also the fast-wave region.

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

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

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

Leaky Modes (cont.) Phased-array analogy z I0I0 x I1I1 InIn ININ d 00 23 Note: A beam pointing at an angle in “visible space” requires that k z < k 0. Equivalent phase constant:

Leaky Modes (cont.) 24 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*). 00 z Power flow Leaky mode x Fields are decreasing radially Source Fields are increasing radially  *Please see one of the homework problems.

The aperture field may strongly resemble the field of the leaky wave (creating a good leaky-wave antenna). Leaky Modes (cont.) Line source Requirements: 1)The LW should be in the physical region. 2)The amplitude of the LW should be strong. 3)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). Excitation problem: 25

Leaky Modes (cont.) Summary of frequency regions: a) f > f c physical SW (non-radiating, proper) b) f s < f < f c non - physical ISW (non-radiating, improper) c) f p < f < f s non - physical LW (radiating somewhat, improper) d) f < f p physical LW (strong focused radiation, improper) The frequency region f p < f < f c 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.) 27 Im k z k1k1 Re k z k0k0 LW f = f p Physical Non-physical ISW f = f c f = f s Spectral-gap region f p < f < f c

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

z / 0 x / x / Radiation occurs at 60 o.

z / 0 x / x / Radiation occurs at 60 o.

z / 0 x / x / The LW is nonphysical.

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

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

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

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

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

x / 0 Leaky-Wave Antenna (cont.) (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 Note: 38 W. W. Hansen, “Radiating electromagnetic waveguide,” Patent, 1940, U.S. Patent No y This is the first leaky-wave antenna invented. Slotted waveguide

x / 0 Leaky-Wave Antenna (cont.) The slotted waveguide illustrates in a simple way why the field is weak outside of the “leakage region.” 39 Top view Source Slot Region of strong fields (leakage region) Waveguide a TE 10 mode

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

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