A fast and simple design methodology for transformation optics (TO) is described for devices having completely arbitrary geometries. An intuitive approach to the design of arbitrary devices is presented which enables possibilities not available through analytical coordinate transformations. Laplace's equation is solved using the finite-difference method to generate the arbitrary spatial transforms. Simple techniques are presented for enforcing boundary conditions and for isolating the solution of Laplace's equation to just the device itself. It is then described how to calculate the permittivity and permeability functions via TO from the numerical spatial transforms. Last, a modification is made to the standard anisotropic finite-difference frequency-domain (AFDFD) method for much faster and more efficient simulations. Several examples are given at the end to benchmark and to demonstrate the versatility of the approach. This work provides the basis for a complete set of tools to design and simulate TO devices of any shape and size.
Photonic crystals can be engineered so that the flow of optical power and the phase of the field are independently controlled. The concept is demonstrated by creating a self-collimating lattice with an embedded cylindrical lens. The device is fabricated in a photopolymer by multi-photon lithography with the lattice spacing chosen for operation around the telecom wavelength of 1550 nm. The lattice is based on a low-symmetry rod-in-wall unit cell that strongly self-collimates light. The walls are varied in thickness to modulate the effective refractive index so light acquires a spatially quadratic phase profile as it propagates through the device. Although the phase of the field is altered, the light does not focus within the device because self-collimation forces power to flow parallel to the principal axes of the lattice. Upon exiting the device, ordinary propagation resumes in free space and the curved phase profile causes the light to focus. An analysis of the experimentally observed optical behavior shows that the device behaves like a thin lens, even though the device is considerably thick.
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