Two-Dimensional Topology Optimization of a Horn Antenna

A two-dimensional horn antenna is used as a model for topology optimization. In order to employ the topology optimization, each point in the domain is controlled by a function which is allowed to take values between 0 and 1. Each point’s distinct value then gives it an effective permittivity, either close to that of polyimide or that of air, two materials considered in this study. With these settings, the optimization problem becomes finding the optimal distribution of materials in a given domain, and is solved under constraints of reflection and material usage by the Method of Moving Asymptotes. The final configuration consists of two concentric arcs of air while polyimide takes up the rest of the domain, a result relatively unsensitive to the choice of constraints and initial values. Compared to the unoptimized antenna, a slimmer main lobe is observed and the gain boosts.


Introduction
Antennas are devices acting as a transition between the free space and the power source [1] [2]. They play an important role in wireless communication, where these devices transmit and receive signals from afar. In their long history, many types of antennas have been developed and constructed over nearly a century [3]- [8]. Among them, horn antennas are built to achieve high gain in frequencies above very high frequency.
Topology optimization is a method that optimizes material layout within a given design space, under a set of constraints. In order to generate the optimal topologies, microstructures, composites of material and void, are employed to form the domain. Therefore, the topology optimization problem in fact turns into a material distribution problem, whose calculation requires considerably less computing costs. Typical algorithm employed is either gradient-based methods such as the optimality criteria algorithm and the method of moving asymptotes (MMA) or non-gradient-based algorithms such as particle swarm or genetic algorithms. Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering [9] [10].
In this work, we adopt topology optimization method for designing a two-dimensional horn antenna.
Optimization results show that a highly directive antenna with non-intuitive materials distribution can be obtained with this powerful method, opening new ways for future antenna design.

Background and Model Definition
In electromagnetics, Maxwell's equations govern. Together with proper boundary conditions, the radiation pattern of antennas can be solved.
The simulation implemented in this paper seeks to find an optimal material distribution for a two-dimensional horn antenna in order to reach a high gain.
Leaving the geometry unchanged, the material configuration can be another angle for antenna design and whose optimization might improve the antenna's performances, particularly its gain as being discussed in this paper.
In this model, every point in the domain of the horn is assigned a η variable, whose value is to be changed by the optimization. η determines the permittivity, or equivalently what material is used, at one point according to the ε ramp function. η ranges from 0 to 1; when η decreases from 0.5, the value of ε ramp reduces rapidly to 1, suggesting a usage of air; when η increases from 0.5, the value of ε ramp rises sharply to 3.5, implying a usage of polyimide (PI). ε ramp as a function of η is shown in Equation (1). When the η value at one point equals to 0.5, it is saying that a composite of 50% air and 50% polyimide is used. However, in practice this sort of composite may be difficult to comprehend or implement. A definite usage of material is preferred. To ensure that most points are assigned a η value away from ambiguity, a lower bound of the average of all points' Weight function's value is applied to the domain in the optimization. Weight function yields a large value when η at that specific point is very different from 0.5, whereas when η is close to 0.5, it contributes negligibly to the congregation. The graphs of the ε ramp and Weight functions are shown in Figure 1 and Figure 2, respectively. The reason why only permittivity changed is that both permeability and conductivity of air and polyimide are nearly equal. All parameters used in this work are listed in Table 1.      The layout of the antenna and its corresponding simulation region is shown in Figure 3. A perfectly matched layer with circular shape surrounds the entire simulation region to eliminate any reflection that could occur at the outer boundary.
The optimization objective expressed in Equation (3)

Results and Discussions
The final electric field norm and η distribution given by the topology optimization are shown in Figure 5. Particularly, on the right side, dark color denotes a low η value, indicating at that point, the permittivity is the permittivity of air.

Conclusion
The topology optimization shows that a distribution of strips of air and polyimide occurring alternatingly provides a high gain for the horn antenna in this study. The materials distribution is non-intuitive, opening new ways for antenna Open Journal of Optimization performance optimization. This method could be developed to more complicated scenarios, such as phase arrays. Further researches can be conducted to find more combination of materials which may yield even better results.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.