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Absolute and relative determination of CEP from the interference of multiple low-order harmonics

It is generally considered that rigorous measurement of the carrier envelope phase (CEP) requires a fieldsensitive nonlinear process such as optical field ionization by a particular half-cycle of a few-cycle field, which results in well-known spectro-temporal signatures in the cut-off regions of above-threshold ionization (ATI) [1] and higher-order harmonic generation (HHG) [2]. Conversely, methods based on multiple quantum pathway interference (QPI) [3,4] and techniques based on f-to-n·f spectral interferences of harmonic orders [5] are only appropriate for tracking the CEP drift, because they are based on direct or indirect observation of an interference term containing the information on a relative phase shift and do not provide access to the CEP value itself unless the nonlinear phase meter has been previously calibrated by a “time-domain-sensitive” technique. In this contribution we provide simple considerations that the absolute CEP calibration does not actually require access to the temporal envelope because, following from the Fourier phase shift theorem, the very same phase is the common phase offset of all spectral components of the pulse. Therefore, the challenge is to find a way to calibrate this common (spectral) phase via nonlinear spectral interference. This is a seemingly difficult task because for a field described by equation where ‘∼’ denotes the amplitude and phase of the Fourier transformed field, the frequency-independent phase offset ϕ0 can be chosen arbitrarily, unless it is pegged to the phase delay of an optical fringe in the time domain. Nevertheless, the freedom of choosing ?0 as an arbitrary parameter is constrained if more than one spectral interference term of different nonlinearity orders is detected simultaneously. For example, the interference of 1st, 2nd, and 3rd-order nonlinear polarizations in the region of a mutual spectral overlap will yield three interference cross-terms that will depend on the cosines of two differential phases equation and equation. All these terms will form a peak of constructive interference at equation, which means that the phase offset is now defined. Therefore, unlike in the case of a single-term interference between the nearest harmonic orders, which reveals the common phase drift from one laser shot to another, simultaneous interference of several terms gives access to full phase calibration (within the cyclic ambiguity of 2πn.) We illustrate this multiple-order interference technique experimentally by measuring the interference of low-order optical harmonics of a mid-IR (λ0=3.9 µm) few-cycle-pulse generated in an OPCPA system with a free-running CEP phase. Because of the low carrier frequency in the mid-IR, it becomes straightforward to achieve spectral overlap of not just adjacent orders but also more distant harmonics (Fig.1a). As shown in Fig.1b, taken in the single-shot acquisition mode, spectrally broadened 5th, 7th and 9th harmonics can be brought into overlap around 500 nm resulting in a three-way interference that uniquely defines ϕ0. Figure 1c shows the distribution of ϕ0 obtained from 1500 consecutive OPCPA shots.