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Transformation optics provides a geometry-based tool to create new components taking advantage of artificial metamaterials with optical properties that are not available in nature. Unfortunately, although guided electromagnetic waves are crucial for optical circuitry, transformation optics is not yet compatible with two-dimensional slab waveguides. Indeed, after determining the propagation of confined waves along the waveguide with a two-dimensional coordinate transformation, the conventional application of transformation optics results in metamaterials whose properties are insensitive to the coordinate perpendicular to the waveguide, leading to bulky, and therefore impractical, designs. In this contribution, we formulate an alternative framework that leads to feasible coordinate-based designs of two-dimensional waveguides. To this end, we characterize a guided transverse-magnetic light mode by relevant electromagnetic equations: a Helmholtz equation to account for wave propagation and a dispersion relation to impose a continuous light profile at the interface. By considering how two-dimensional conformal transformations transform these equations, we are able to materialize the coordinate-designed flows with a nonmagnetic metamaterial core of varying thickness, obtaining a two-dimensional device. We numerically demonstrate the effectiveness and versatility of our equivalence relations with three crucial functionalities, a beam bender, a beam splitter and a conformal lens, on a qualitative and quantitative level, by respectively comparing the electromagnetic fields inside and the transmission of our two-dimensional metamaterial devices to that of their three-dimensional counterparts at telecom wavelengths. As a result, we envision that one coordinate-based multifunctional waveguide component may seamlessly split and bend light beams on the landscape of an optical chip.
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