Straight long slots have high side-lobes in the far-field amplitude patterns, which reduces their use as high-performance antennas. To reduce these side-lobes, a long slot may be tapered to produce the desired radiation patterns. The theory of control of the aperture distribution to reduce side-lobes has been already reported in some works and well known for already some decades. It is, however, shown in this paper that it may not be good enough to achieve ultra-low side lobes. The theory to analyze and design tapered leaky-wave antennas is described in this paper. Since it is very challenging to achieve a mathematical equation in this regard, some parameters will be calculated using simulation in the first step and the shape of the antenna field is obtained based on these parameters. In the next step, a differential equation is derived for the first step parameters. The solution of this differential equation which is the main motivation of this paper will be expressed in three ways where each part is more accurate than the previous one. According to the measurement results, the structure has a side-lobe level more than −45 dB.
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