Electromagnetic cloaking in convex and concave media with surface modelled as a parameterised function.

Prior research has shown that transformation optics enables the manipulation of electromagnetic waves through engineered materials. It was already known that anisotropic media can guide waves in non-traditional ways. However, no prior work had resolved the challenge of designing cloaks with arbitrary geometries. This gap motivated the need for a generalized algorithm. Existing methods rely on orthogonal coordinate systems, which limit the range of possible shapes. The complexity of non-orthogonal systems remains a barrier for practical applications. No prior work had addressed the analytical difficulty of ray tracing in arbitrary geometries. That uncertainty drove the development of numerical alternatives to finite difference methods. This paper aims to bridge the gap between theoretical models and practical implementation.

This paper proposes an algorithm for designing three-dimensional electromagnetic cloaks with arbitrary geometries. The goal is to overcome limitations in existing methods that assume orthogonal coordinate systems. The study focuses on convex and concave surfaces, which are common in real-world applications. The purpose is to provide a flexible framework for cloaking arbitrary shapes. The motivation stems from the need for cloaks that can conform to complex geometries. The algorithm uses coordinate transformations and piecewise functions to model surfaces. The study also addresses the computational challenges of ray tracing in non-orthogonal systems. The aim is to improve the accuracy of derivative calculations for electromagnetic simulations.

The algorithm uses coordinate transformations to model arbitrary geometries. Surfaces are represented as parameterized functions. The method assumes conformal mapping between the cloak and the body. Non-orthogonal coordinate systems are used for arbitrary shapes. A ray tracing process is implemented to verify the algorithm. The Hamiltonian equation is solved numerically for ray propagation. Forward differences are replaced with a numerical method for better accuracy. The method is tested on an ellipsoid and a concave surface with axial symmetry.

The algorithm successfully models cloaks with arbitrary geometries. Ray tracing simulations confirm the effectiveness of the method. The numerical approach outperforms conventional finite difference methods. The method handles non-orthogonal coordinate systems effectively. The ellipsoid and concave surface tests show consistent results. The numerical method provides better approximation of partial derivatives. The results suggest improved accuracy in ray propagation simulations. The approach is suitable for complex shapes like convex and concave media.

The proposed algorithm enables the design of cloaks with arbitrary geometries. The method uses coordinate transformations and parameterized functions. The numerical approach improves derivative calculations for ray tracing. The results validate the algorithm for convex and concave surfaces. The study demonstrates the feasibility of non-orthogonal systems. The method is suitable for practical applications in transformation optics. The findings suggest that the algorithm can be extended to other complex shapes. The study confirms the effectiveness of the numerical method over traditional approaches.