Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling

Abstract

We demonstrate time-domain sampling of mid-infrared electric field transients and their conjugate counterparts exploiting the dynamical Pockels effect. To this end, the complete polarization change of few-femtosecond probe pulses is studied. An intuitive picture based on a phasor representation is established before gaining quantitative understanding in experiment and theory. In the standard version of electro-optic sampling, the electric field is determined by analyzing the change of ellipticity of the probe polarization. Beyond this, we find that a temporal gradient of the input electric field manifests itself in a rotation of the polarization ellipsoid of the probe. The relative contribution of sum-and difference-frequency mixing processes and their spectral distribution over the near-infrared probe bandwidth are identified as key aspects. If one of these processes dominates, detecting ellipticity changes and polarization rotation as a function of time delay results in two wave forms which are Hilbert transforms of each other. Such conditions may be achieved by angle phase matching in birefringent materials or spectral filtering of the probe after the nonlinear interaction. In this case, a static phase introduced by birefringence or reflection at metallic mirrors results in a specific phase shift of both time traces with respect to the input electric field. Contributions from sum-and difference-frequency generation are found to be equivalent when using electro-optic sensors with isotropic refractive index. Polarization rotations in the lowand high-frequency parts of the probe then tend to cancel out. In this limit, spurious additional phase shifts do not change the phase of the detected transients. This fact leads to a robust recovery of the carrier-envelope phase of the input wave form. Clarifying the role of imperfections of superachromatic phase retarders completes our survey on proper determination of the electric field and its conjugate variable.

A Bayesian view on the Hilbert transform and the Kramers-Kronig transform of electrochemical impedance data: Probabilistic estimates and quality scores

“…[1] Instead, we are going to focus on the HT, which, in the context of the transfer functions, is equivalent to the KK relations. [32] The HT is used in many applications, including fluid mechanics, [33] aerodynamics, [34] optics, [35] and geophysics. [36] Relative to the KK relations, the HT has far richer mathematical literature, with the availability of fast HT methods [37,38] and many theoretical results.…”

Minimally invasive optical sensors for microwave-electric-field exposure measurements

“…Since the 1960s and the development of the first electro-optical (EO) E-field sensors, 19,20 Pockels-based probes have experienced great success due to their intrinsic wide BW extending from DC to several THz, 21,22 their associated subpicosecond temporal resolution, 23 and their dielectric-and therefore minimally invasive-nature. By inducing a linear variation of the refractive index as a function of electric field components, the Pockels effect offers the additional advantage of enabling vector measurements, even in the most severe environments such as plasmas, 24,25 lightning, 26 gas insulation substations, or ultra-high voltage transmission towers.…”

Analysis of Subcycle Electro-Optic Sampling Without Background

“…Nevertheless, also the concepts for detecting electric field amplitudes with few-femtosecond resolution [6,7] include an element of homodyne amplification where a new electric field generated by mixing with the longwave electric field is generated with a polarization direction perpendicular to the near-infrared (NIR) probe [8]. An ellipsometric analysis of the relative optical phase of this polarization state gives direct access to either the electric field or its conjugate variable in the time domain, namely, the Hilbert transform [9]. In these schemes, technical excess noise of the pulse train used for probing may be suppressed by a balanced detector, but the shot noise of the flux of coherent NIR photons remains as the limiting factor for quantum detection.…”