Extended Data Fig. 2: Interface component of the spectra extracted by finding the minimum difference between the measured spectrum and all possible linear combinations of two bulk spectra.

The fitting was performed by minimizing \(\Vert S(\omega )-{a}_{1}{S}_{{\rm{cBN}}}(\omega )-{a}_{2}{S}_{{\rm{diamond}}}(\omega )\Vert \), where S(ω) is the measured spectrum (Fig. 2b), \(S(\omega )\) with subscripts means the bulk spectra, and \({a}_{1},{a}_{2}\) are adjusted coefficients. a, Line profile of the fitting residual. Since the fitting gives the linear combination that is closest to the measured spectrum, the residual represents the interface component that cannot be obtained from bulk modes. Near the interface, three red peaks correspond to three interfacial modes in Fig. 2g. The blue region at 160 meV is due to the isolated mode with reduced vibration at the interface. b, Norm (root sum squared) of residuals as a function of position (left axis), and the fitting coefficients \({a}_{1}\) and \({a}_{2}\) (right axis). The residual is sharply peaked at the interface (FWHM = 1.8 nm), indicating new vibrational features are highly localized at the interface.