Fig. 1

Continuous SUSY transformation optics and its demonstration on a Si chip. a Two-dimensional refractive index distribution \(n(x,z)\) designed by SUSY transformation with continuously varying parameter \({\alpha }_{i}\) in Eq. 1. \(n(x)\) is shown at five different z coordinates: \(z=0, 0.25L, 0.5L, 0.75L\), and \(L\), where \(L=150\) µm, where the index profile is discretized in the \(x\) direction for the design implementation. b GRIN metamaterial designed to substantiate \(n(x,z)\), where different optical states (blue, red, green) with different propagation constants can propagate with spatial characteristics and directions dictated by the SUSY transformation. c Schematic of the on-chip subwavelength Si metamaterial for local index tuning, with physical parameters: \(h=300\) nm and \(p=225\) nm, and its numerically simulated relationship between the effective refractive index (\({n}_{eff}\)) and the size of the air gap (\(g)\) at the wavelength of 1550 nm (Supplementary Additional file 1: Fig. S2). d Scanning electron microscope images of the fabricated Si GRIN metamaterial and its prescribed gap distribution at three different \(z\) coordinates: \(z=0.25L, 0.5L,\) and \(0.75L\), corresponding to \(n(x,z)\) in a