Non-Hermitian curved space via inverted wave equation

Inverting design method of solving passive graded materials from predefined amplitude and phase was developed along the line of transformation optics (TO), which however precluded the presence of source and sink in the pragmatic world. So in this Letter we extend such an inverse method to non-Hermitian media, offering more freedom to manipulate the wave flows. Our principle of a curved-space analogue picture powered with gain and loss, is exemplified by three types: amplitude controlling, phase conversion, and direction shunting. These examples showcase precise wave manipulation in a surprisingly simple manner, which goes beyond convectional paradigms of TO and is readily implementable in realistic photonic platform.