I was asked to explain the method I used to simulate and find extinction cross section of the antennas in my paper. I thought I write a blog post for someone who might be looking for the same problem on the internet.
Well, there are basically two different methods. One method is to find the scattered fields by the scatterer. Then, one should calculate the power of scattered fields over the steradian of the space. The power dissipated in the object should be also calculated. Then, one can find the extinction cross section by converting the sum of the absorbed and scattered powers into the cross section.
The second method, which is also simpler, is to use the optical theorem (see Wiki for further details). It readily relates extinction cross section with scattering amplitude of the object in the forward direction.
Therefore, we are down to finding scattering amplitude of the object. The method that I found simple to implement was introduced by Gustafsson et al (2012). This was introduced for the measurement purposes. I also used it to model the extinction cross section in full-wave EM packages like FEKO, CST, Ansys, etc. Based on this method, we place the scatterer in the origin. A spherical source is used to illuminate the scatterer, and the field is also recorded in the forward direction (). We also measure(or simulate) without the presence of the scatterer. This is basically to get an account of the path loss and phase delay between the source and observation points. Then, we can have:
Then, by the optical theorem, we can find the extinction cross section as:
Note 1: Time dependence
In the above discussions, a time dependence of is assumed. If one prefers $latex \exp(-j\omega t) dependence, he/she should change the sign in (1) and omit the conjugate operator in (2). I found this worth to mention since I was used to get negative cross sections while I measured my antennas in . I spent almost two months, to measure, and re-measure, and re-measure the cross sections, and I was getting the negative cross sections. I started asking people about the issue. Finally, one of the authors in  pointed the time dpendence issue to me,
Note 2: On the modelling of an antenna
Extinction cross section of an antenna can be found the same way as a general object. The only difference is one should add a load into the antenna terminals to setup the model or apparatus properly.
The imaginary part of stands for the extinction cross section, while I am not aware of any physical interpretation for the , though both of them have dimensions of the area. I believe intuitively one might somehow relate to the stored energy around the scatterer.
 M. Shahpari and D. V. Thiel, “The Impact of Reduced Conductivity on the Performance of Wire Antennas,” in IEEE Transactions on Antennas and Propagation, vol. 63, no. 11, pp. 4686-4692, Nov. 2015. doi: 10.1109/TAP.2015.2479241 arXiv: 1509.06709 RG: Priprint
 M. Gustafsson, J. B. Andersen, G. Kristensson and G. F. Pedersen, “Forward Scattering of Loaded and Unloaded Antennas,” in IEEE Transactions on Antennas and Propagation, vol. 60, no. 12, pp. 5663-5668, Dec. 2012. doi: 10.1109/TAP.2012.2214191 RG: Preprint