Radiation characteristic of the periodic leaky wave structure and its application to leaky wave antenna design (2015)

For a leaky waveguide with periodic slots cut in the wall, the guided wavelength can be expressed as: λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> = λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> /√(ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> ). (1) Here, ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> is an equivalent dielectric constant of the leaky waveguide, and is expressed by ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> = ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> - (λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> /λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ') <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , (2) where εr is the dielectric constant of the material filled in the waveguide, and it equals 1 when no dielectric is filled; λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the wavelength in free space, and λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ' is the cut off wavelength of the fundamental mode in the leaky waveguide. λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ' is actually different from that in closed waveguide, but the difference is not significant. The propagation constant of the leaky waveguide is then expressed by β = k0 √(ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> ), (3) where k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is the wave number in free space. As we know, periodic structures can excite a lot of spatial harmonics, and some of them can leak away from the structures. The radiation condition and the beam angle of the mth harmonic are -1 <; √ ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> + m λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> / P <; 1, (4) φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> = cos <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (√(ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> ) + mλ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> / P), (5) where P is the period of the slots cut in the waveguide wall. From the analysis we find that the radiation beam directions of the spatial harmonics of leaky waveguide are determined by both the equivalent dielectric constant ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> and the period of the slots P, so either changing of ε <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> or changing of P will lead to changing of radiation beam angle. Based on these theory and formulas, the transmission and radiation characteristics of spatial harmonics of the leaky waveguide are studied first, and the relationships between the radiation field and structure parameters are studied as well. It can be found that for a leaky waveguide, both the harmonics with negative order (m <; 0) and positive order (m ≥ 0) can generate leaky waves. This is different from that of the leaky coaxial cable, in which only the harmonics with negative order (m <; 0) can generate leaky wave. An approach based on numerical method for analyzing the propagation constant of leaky waveguide is also given, and the relationship between the propagation constant and waveguide parameters is analyzed. The radiation property of the spatial harmonics of periodic leaky wave structures is then used in the design of antennas with beam-formed radiation patterns, for low power consumption wireless communication application. The basic principle of the design is to superpose the radiations from the -1th order spatial harmonics of a number of leaky wave structures with different slot periods in a weighting way, and the weighting factors are determined by the desired radiation pattern. Since different periodic structures generate radiation beams in different directions, and different weighting factors lead to different radiation strengths, so by properly selecting the periodic structures and weighting factors, the desired radiation pattern can be formed. But for realistic application, these leaky structures with different slot periods should be mapped into one structure that can be fabricated. So the functions describing different periodic slot structures are also superposed together in a weighting way, and the combination slot structural function for the realistic leaky wave antenna is finally obtained. The weighting factors used in the radiation field superposition and the weighting factors used in the structural function superposition are two different sets of weighting functions, and they should be mapped into each other, the key is to find the relationship between the radiation field and the structure parameters. While the weighting factors for structural function superposition is found, we obtain the final antenna structure. Using this method, two kinds of new leaky wave antennas with beam-formed radiation patterns are designed based on rectangular waveguide and SIW structures.