Conformal Boundary Transformation
We developed the concept of CONFORMAL BOUNDARY OPTICS, an analytical method - based on first principle derivations- that allows us to engineer transmission and reflection of light for any interface geometry and any given incident wave.
By resolving the boundary conditions between two materials at an interface of arbitrary geometry, this method addresses recent developments in nanophotonics with the general technique of differential geometry and coordinates transformation.
In many optics textbooks, the boundary conditions are derived by considering an Amperian loop and a Gaussian pillbox at an interface, often graphically represented -- for aesthetical reasons -- as a nonplanar surface. This, however, can be misleading because these conditions are originally derived in the coordinate system that conforms to the interface whereas the electromagnetic fields are expressed in the ambient -- typically Cartesian, spherical or cylindrical -- coordinate system. Because the surface normal of the nonplanar interface is changing, the boundary conditions vary depending on the position along the interface.
To circumvent this difficulty, we use the language of differential geometry to derive the electromagnetic boundary conditions in the surface coordinate from first principle derivations starting from the integral forms of Maxwell’s equations.
We treat the interface as volumetric boundary of sub-wavelength thickness -operation valid for optical metasurfaces of subwavelength thicknesses- and we define the tangent and normal vectors to be the coordinate system that conforms to the interface. We can then derive the relations between the surface susceptibilities and the desired fields written in any arbitrary coordinate system.
This concept provides a wide range of new design opportunities, for example, to hide objects behind an “optical curtain”, to create optical illusions by reflecting virtual images, or to suppress the diffraction generally occurring during light scattering at corrugated interfaces. Our approach is also significant for conventional interfaces that involve only regular dielectric materials. Prior to this work, boundary conditions existed only for cases where the interface -- which the coordinate system conforms to – has a basic geometry e.g., Cartesian, cylindrical, spherical. We believe that this work constitutes a first step towards the development of a general theory for waves in systems containing interfaces with complex geometries.
WORK in PROGRESS