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

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 low-and 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.