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leakage constant of uniform leaky wave antenna

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aminloooti

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Hi all.
I want to obtain the relationship between leakage constant (alpha) and offset x0 (fig.4 from the article "mm-wave SIW waveguide long slot leaky-wave antennas and two 2D multibeam applications").
This is a uniform leaky wave antenna that has a straight long slot with the offset x0.
I simulate this antenna in rectangular form with the width of w=3.5mm,height of h=1.575mm and Length of L=155mm and then use the formula " exp(-2*alpha*L)=(abs(s21)^2) / (1- abs(s11)^2))".

Can anybody help me to obtain the relation between alpha and x0? 1-LWA.png2-S12.png3-Alpha1.png
 

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