# Maximum Gain, Effective Area, and Directivity

## TL;DR

This paper establishes fundamental bounds on antenna gain and directivity through convex optimization, introduces methods for analyzing superdirectivity with minimal efficiency costs, and demonstrates practical applications across various antenna configurations.

## Contribution

It provides a unified convex optimization framework for antenna gain bounds, analyzes superdirectivity with efficiency considerations, and offers computational techniques for large and high-conductivity structures.

## Key findings

- Derived formulas for gain bounds applicable to arbitrarily shaped radiators.
- Efficient eigenvalue-based solutions for optimization problems.
- Practical examples demonstrating the theoretical framework's relevance.

## Abstract

Fundamental bounds on antenna gain are found via convex optimization of the current density in a prescribed region. Various constraints are considered, including self-resonance and only partial control of the current distribution. Derived formulas are valid for arbitrarily shaped radiators of a given conductivity. All the optimization tasks are reduced to eigenvalue problems, which are solved efficiently. The second part of the paper deals with superdirectivity and its associated minimal costs in efficiency and Q-factor. The paper is accompanied with a series of examples practically demonstrating the relevance of the theoretical framework and entirely spanning wide range of material parameters and electrical sizes used in antenna technology. Presented results are analyzed from a perspective of effectively radiating modes. In contrast to a common approach utilizing spherical modes, the radiating modes of a given body are directly evaluated and analyzed here. All crucial mathematical steps are reviewed in the appendices, including a series of important subroutines to be considered making it possible to reduce the computational burden associated with the evaluation of electrically large structures and structures of high conductivity.