This section will detail the mathematical formulation of TO by first introducing the concept of conformal coordinate transformation mapping. Then, TO will be described by showing that a coordinate transformation that can be represented analytically can be reproduced through the use of spatially varying material parameters. If the transformation is conformal, the resulting media is isotropic. Finally, a numerical scheme for generating a nearly conformal map for an arbitrary coordinate transformation, called qTO, will be discussed.

Conformal Mapping

Laplace’s equation is a very important general second-order partial differential equation with applications in electrostatics, low-speed fluid flows, and gravitational fields [4]. Given a scalar potential f, Laplace’s equation is represented as:

where Д = V^{2} is the Laplace operator. Analytical functions are functions that can be represented by convergent power series, are infinitely differentiable, and can be used to map one complex space z = x + jy to another complex space f (z) = f (x + jy) = x' (x, y) + jy' (x, y). These functions also obey the Cauchy-Riemann equations:

This implies thatf (z) has the same derivative df/dz independent of which direction in the complex plane dz is oriented. For two different orthogonal orientations of dz in the complex plane, we have:

The Cauchy-Riemann equations directly follow by setting these two expressions equal to each other. The real and imaginary parts of any complex analytic function are automatically harmonic functions and thus are twice continuously differentiable functions, which satisfy Laplace’s equations [42]. Conformal mappings use analytic complex functions and ensure angles and aspect ratios are preserved through coordinate transformation. This can be proved by decomposing dx and dy into its differential components in the transformed space, dx' and dy', using the coordinate systems shown in Fig. 6.1:

In order to satisfy the Cauchy-Riemann equations, the following conditions must be satisfied:

Figure 6.1 Coordinate systems used in transformations.

This proves that a conformal map leads to a locally orthogonal coordinate system that ensures the scaling of x is the same as the scaling of y. An important thing to note about conformal maps as they relate to TO is that they lead to media that are locally isotropic and non-magnetic, which is crucial for broadband performance [36]. Quasi-conformal transformation optics is a numerical algorithm that attempts to minimize anisotropy for a general transformation by solving equation 6.1 with carefully selected boundary conditions [6].