Minimum-phase calibration of sampling oscilloscopes

We describe an algorithm for determining the minimum phase of a linear time-invariant response function from its magnitude. The procedure is based on Kramers-Kronig relations in combination with auxiliary direct measurements of the desired phase response. We demonstrate that truncation of the Hilbert transform gives rise to large errors in estimated phase, but that these errors may be approximated using a small number of basis functions. As an example, we obtain a minimum-phase calibration of a sampling oscilloscope in the frequency domain. This result rests on data obtained by an electrooptic sampling (EOS) technique in combination with a swept-sine calibration procedure. The EOS technique yields magnitude and phase information over a broad bandwidth, yet has degraded uncertainty estimates from dc to approximately 1 GHz. The swept-sine procedure returns only the magnitude of the oscilloscope response function, yet may be performed on a fine frequency grid from about 1 MHz to several gigahertz. The resulting minimum-phase calibration spans frequencies from dc to 110 GHz, and is traceable to fundamental units. The validity of the minimum-phase character of the oscilloscope response function at frequencies common to both measurements is determined as part of our analysis. A full uncertainty analysis is provided.     

Reflection and transmission properties of a metafilm: with an application to a controllable surface composed of resonant particles

In recent work, we derived generalized sheet transition conditions (GSTCs) for the average (or "macroscopic") electromagnetic fields across a metafilm, which, when properly designed, can have certain desired reflection and transmission properties. A metafilm is the two-dimensional equivalent of a metamaterial, and is essentially a surface distribution of electrically small scatterers characterized by electric- and magnetic-polarizability densities. In this paper, the GSTC is used to calculate the reflection and transmission coefficients of the metafilm. These coefficients are derived for both TM and TE polarized plane waves with arbitrary incidence angles. We show that the reflection and transmission properties of the metafilm are expressed in terms of the electric and magnetic polarizabilities of the scatterers themselves, and we derive conditions on the polarizabilities of the scatterers required to obtain total transmission and/or total reflection. We show various examples to illustrate the validity of the GSTC for the analysis of a metafilm. By controlling the polarization densities of the scatterers in the metafilm, a "smart" and/or "controllable" surface can be realized. We propose a metafilm composed of spherical magneto-dielectric particles for achieving such a controllable surface. To validate the results for the spherical particle metafilm, we show comparisons with a full-wave computation obtained from a mode-matching technique applied to the doubly infinite array of spherical scatterers. The results in this paper are in principle scalable; that is, the dimensions of the scatterers can range from relatively large to relatively small depending on the frequencies of interest.     

Calibration of sampling oscilloscopes with high-speed photodiodes

We calibrate the magnitude and phase response of equivalent-time sampling oscilloscopes to 110 GHz. We use a photodiode that has been calibrated with our electrooptic sampling system as a reference input pulse source to the sampling oscilloscope. We account for the impedance of the oscilloscope and the reference photodiode and correct for electrical reflections and distortions due to impedance mismatch. We also correct for time-base imperfections such as drift, time-base distortion, and jitter. We have performed a rigorous uncertainty analysis, which includes a Monte Carlo simulation of time-domain error sources combined with error sources from the deconvolution of the photodiode pulse, from the mismatch correction, and from the jitter correction.     

Traceable Waveform Calibration With a Covariance-Based Uncertainty Analysis

We describe a method for calibrating the voltage that a step-like pulse generator produces at a load at every time point in the measured waveform. The calibration includes an equivalent-circuit model of the generator that can be used to determine how the generator behaves when it is connected to arbitrary loads. The generator is calibrated with an equivalent-time sampling oscilloscope and is traceable to fundamental physics via the electro-optic sampling system at the National Institute of Standards and Technology. The calibration includes a covariance-based uncertainty analysis that provides the uncertainty at each time in the waveform vector and the correlations between the uncertainties at the different times. From the calibrated waveform vector and its covariance matrix, we calculate pulse parameters and their uncertainties. We compare our method with a more traditional parameter-based uncertainty analysis.     

A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials

A metafilm (also referred to as a metasurface) is the surface equivalent of a metamaterial. More precisely, a metafilm is a surface distribution of suitably chosen electrically small scatterers. Metafilms are becoming popular as an alternative to full three-dimensional metamaterials. Unfortunately, many papers in the literature present incorrect interpretations and mischaracterizations of these metafilms. In fact, some of the characterizations presented in the literature result in non-unique parameters for a uniquely defined metafilm. In this paper we discuss an appropriate interpretation and characterization of metafilms and present a correct manner to characterize a metafilm. Additionally, we illustrate the error that results from an incorrect characterization of metafilms. We present various examples to emphasize these points. Finally we present a retrieval approach for determining the uniquely defined quantities (the electric and magnetic susceptibilities of its constituent scatterers) that characterize a metafilm.