Andrew Dienstfrey

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

Covariance-Based Vector-Network-Analyzer Uncertainty Analysis for Time- and Frequency-Domain Measurements

We develop a covariance-matrix-based uncertainty analysis for vector-network-analyzer (VNA) scattering-parameter measurements. The covariance matrix not only captures all of the measurement uncertainties of the scattering-parameter measurements, but also the statistical correlations between them. This allows the uncertainties of VNA scattering-parameter measurements to be propagated into the uncertainties of other quantities derived from scattering parameters, including temporal waveforms and circuit model parameters.

A Physical Explanation of Angle-Independent Reflection and Transmission Properties of Metafilms/Metasurfaces

In this letter, we illustrate that a metafilm (the two-dimensional equivalent of a metamaterial, also referred to as a metasurface) can be designed to have transmission and reflection properties that are independent of the angle of the incident wave. We show theoretically and discuss physically why this behavior occurs in certain metafilms. We show that by choosing an inclusion with sufficiently strong resonances, the angle dependence of the metafilm becomes negligible. Metafilms operating at microwave frequencies and composed of both lossless and lossy resonant spherical inclusions as well as electrical resonators are investigated. Numerical and spherical-harmonic mode-matching approaches are used to investigate the angular dependence of the reflection properties of these metafilms. Such angular-independent properties can have applications in extending the modes supported in a metafilm waveguide and have direct applications to photonics where, due to fabrication obstacles, optical metamaterials are often limited in construction to single and multiple stacked two-dimensional arrays of plasmonic structures.

Covariance-based uncertainty analysis of the NIST electrooptic sampling system

We develop a covariance matrix describing the uncertainty of mismatch-corrected measurements performed on the National Institute of Standards and Technologys electrooptic sampling system. This formulation offers a general way of describing the uncertainties of the measurement system in both the temporal and frequency domains. We illustrate the utility of the approach with several examples, including determining the uncertainty in the temporal voltage generated by the photodiode

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.

Lattice sums and the two-dimensional, periodic Green's function for the Helmholtz equation

Many algorithms that are currently used for the solution of the Helmholtz equation in periodic domains require the evaluation of the Greens function, G(x,x0). The fact that the natural representation of G via the method of images gives rise to a conditionally convergent series whose direct evaluation is prohibitive has inspired the search for more efficient procedures for evaluating this Greens function. Recently, the evaluation of G through the ‘lattice‐sum’ representation has proven to be both accurate and fast. As a consequence, the computation of the requisite, also conditionally convergent, lattice sums has become an active area of research. We describe a new integral representation for these sums, and compare our results with other techniques for evaluating similar quantities.

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

A Statistical Study of De-Embedding Applied to Eye Diagram Analysis

We describe a stable method for calibrating digital waveforms and eye diagrams by use of the measurement system response function and its regularized inverse. The function describing the system response includes the response of the oscilloscope and any associated cables and test fixtures. We demonstrate the effectiveness of the method by performing a statistical analysis of the calculated eye height and eye width obtained from a controlled experiment consisting of multiple cable lengths, bit rates, and oscilloscope samplers. We also demonstrate our approach by measuring the transmission through a test device consisting of a short length of cable, a ball-grid array socket, and complicated circuit board.

Spectroscopic phase-dispersion optical coherence tomography measurements of scattering phantoms

We demonstrate a novel technique to determine the size of Mie scatterers with high sensitivity. Our technique is based on spectral domain optical coherence tomography measurements of the dispersion that is induced by the scattering process. We use both Mie scattering predictions and dispersion measurements of phantoms to show that the scattering dispersion is very sensitive to small changes in the size and/or refractive index of the scatterer. We also show the light scattered from a single sphere is, in some cases, non-minimum phase, and therefore the phase of the scattered light is independent of the intensity. Phase dispersion measurements may have application to distinguishing the size and refractive index of scattering particles in biological tissue samples.

Electric field metrology for SI traceability: Systematic measurement uncertainties in electromagnetically induced transparency in atomic vapor

We investigate the relationship between the Rabi frequency (ΩRF, related to the applied electric field) and Autler-Townes (AT) splitting, when performing atom-based radio-frequency (RF) electric (E) field strength measurements using Rydberg states and electromagnetically induced transparency (EIT) in an atomic vapor. The AT splitting satisfies, under certain conditions, a well-defined linear relationship with the applied RF field amplitude. The EIT/AT-based E-field measurement approach derived from these principles is currently being investigated by several groups around the world as a means to develop a new SI-traceable RF E-field measurement technique. We establish conditions under which the measured AT-splitting is an approximately linear function of the RF electric field. A quantitative description of systematic deviations from the linear relationship is key to exploiting EIT/AT-based atomic-vapor spectroscopy for SI-traceable field measurement. We show that the linear relationship is valid and can be...