Application of the Hilbert Transform Method for the Retrieval of the Phase Characteristics of Plasmonic Metal Bragg Gratings

Abstract

We demonstrate, with detailed examples, a method to retrieve the phase and group-delay responses of plasmonic metal Bragg gratings (MBGs) from their reflection and transmission spectra. The method is based on the Hilbert transform (HT) that relates the phase and the amplitude spectrum of a minimum-phase system. For the retrieval of the reflection phase, if the propagation loss of the grating is negligible, the method cannot tolerate any asymmetry in the grating profile and thus fails to operate for practical low-loss fiber or dielectric-waveguide Bragg gratings, where the fabrication errors are sufficient to destroy the symmetry requirement. The large propagation loss of the surface plasmon mode, however, makes possible the retrieval of the reflection phase of an MBG that contains a certain degree of asymmetry in the grating profile. For the retrieval of the transmission phase, the symmetry requirement is not an issue, while a large propagation loss can introduce significant errors in the retrieved phase spectrum. It is possible, however, to largely reduce such errors by applying the HT method to the spectrum with the background propagation loss artificially removed. The HT method provides a simple and effective way to obtain the phase and group-delay characteristics of MBGs, which are difficult to measure directly, and can find applications in the development of grating-based plasmonic devices.