Single-shot CEP drift measurement at arbitrary repetition rate based on dispersive Fourier transform

: This paper presents a single-shot technique for measuring CEP. The Temporal dispersion based One-shot Ultrafast Carrier envelope phase Analysis method (TOUCAN) is an arbitrary repetition rate single-shot CEP drift measurement technique based on dispersive Fourier transformations and has been experimentally tested at 100 kHz. TOUCAN was validated by a direct comparison of decimated data with an independent traditional CEP drift measurement technique. The impact of a temporal jitter on the CEP drift measurement is investigated and a new mitigation technique is shown to produce high accuracy jitter-free CEP drift extraction.

Also few cycle pulses with at least 30 µJ energy are required for CEP retrieval [9], which makes this diagnostic method unsuitable for a significant proportion of current laser systems.
A different approach for high repetition rate single-shot CEP measurement has been taken which is based on Dispersive Fourier Transform (DFT) [12][13][14][15]. The temporal dispersion based one-shot ultrafast carrier envelope phase analysis method (TOUCAN) allows singleshot CEP drift extraction up to gigahertz repetition rates from f-to-2f spectral interference. The concept is demonstrated by recording the CEP drift of 4 cycles pulses at 3.2 µm from an OPCPA laser operating at 100 kHz.

Concept
Linear optical interferometry cannot provide information about the CEP so a nonlinear optical effect has to be employed [3]. In f-to-2f self-referential interferometry, a fraction of the laser pulse is frequency doubled while another fraction is spectrally broadened until the blue part reaches the doubled frequency (i.e. broadened over one octave). The interference signal is created by overlapping these two pulses in the temporal, spatial, and spectral domain. The one dimensional temporal evolution of the fundamental laser pulse is given by where A(t) is the temporal envelope of the pulse, 0 ω is the central angular frequency and φ stands for CEP. In the frequency domain the field reads: Under perfect conditions, the second harmonic (SH) pulse is centered at twice the frequency of the original. The resultant phase is twice that of the fundamental pulse and also shifted by an offset. The complex spectrum of the second harmonic electric field is expressed as: The above equation describes a process, where dispersion is negligible. In reality material dispersion introduces different constant phase shift for the fundamental and SH pulses. The spectra of these pulses may overlap if the spectrum is either sufficiently broad or if it is broadened by supercontinuum generation. The overlapping region will exhibit a spectral interference pattern, where the fringe period depends on the relative delay between the fundamental and the SH pulse. The resulting signal as detected by an integrating device is described by: where the Δφ D is the phase difference between pulses introduced by the material dispersion. The third term in this equation creates a spectral intensity modulation where the phase of this pattern is in fact linearly related to φ. Thus extracting the phase of the spectral modulation provides the CEP value of the original pulse shifted by an unknown constant value. Conventional techniques would read the spectrum with an optical spectrometer with a limiting reading frequency of 1 or 10 kHz. The TOUCAN technique, which is based on dispersive Fourier transform, can reach higher data acquisition rate, because there is no dependence on an optical spectrometer. The DFT itself is performed by propagating the laser through a dispersive optical element with very high group delay dispersion (GDD). When the laser pulse enters a dispersive medium, different frequency components travel at different phase velocities, accumulating different time delays. Consequently, the pulse duration is stretched by orders of magnitude above its original value. The resulting temporal shape will only be the function of the original spectral shape and the dispersion. Dispersion can be considered as a transform, where each spectral component is mapped to the temporal domain. The relation between optical frequency and the corresponding time delay can be expressed in the far-field approximation [13] as: where ω is the angular frequency, ω 0 is the central angular frequency, T is accumulated time delay of the laser pulse traveling through the medium. The phase derivatives present in the equation are abbreviated as GD: group delay (first order), GDD: group delay dispersion (second order), TOD third order dispersion, FOD fourth order dispersion, etc…The effect of the third and even higher order dispersion can be neglected, if its effect is marginal compared to the effect of the first order GD and the second order GDD. Thus, the frequency-time mapping becomes linear in frequency as: Consequently, the spectral and temporal shape become congruent and by dispersing the output signal of an f-to-2f interferometer, the spectral interference pattern is mapped to the temporal domain. The stretched pulse duration can reach hundreds of nanoseconds, which allows for reading the temporal modulation with a relatively slow photodetector. The produced electric signal can then be digitized by an analog-to-digital (A/D) converter with a bandwidth at least twice the modulation frequency. The CEP drift can be extracted from the phase of the digital waveform, provided the dispersion is fully characterized.

Experimental setup
The concept has been demonstrated on a high-repetition rate system producing CEP stable few-cycle pulses equipped with a conventional f-to-2f measurement device for comparison and validation. The experimental setup comprises a commercial state-of-the-art mid-infrared (MIR) laser system operated at ELI-ALPS [16] and the MIR laser setup is shown in Fig. 1(a). The system produces passively CEP stabilized 150 µJ, 4 optical-cycle (42 fs) pulses centered at 3.2 µm at a repetition rate of 100 kHz. This OPCPA system uses a Dazzler (Fastlite) [17] to accurately shape the spectral amplitude and control the phase, including the CEP. Therefore, the CEP of the output pulses can be set to an arbitrary value in a very simple way. A sampled portion of the compressed output is sent to a commercial CEP measurement device based on a f-to-2f interferometer and a conventional grating spectrometer (Fringeezz from Fastlite) [18]. This device is able to record single shot CEP values at an under sampled repetition rate of 10 kHz, that is one pulse out of 10 on the 100 kHz laser system, and it is used for calibration and cross-checking purposes.
Another fraction of the laser output is steered towards the TOUCAN setup. The experimental arrangement of this single-shot CEP drift measurement device is shown in Fig.  1(b). The device consists of a f-to-2f interferometer and a DFT measuring system. Dispersion is achieved in the current case by propagation in a dispersion compensation fiber (DCF, FSC-DCM-014D, OFS/Lucent). The spectral modulations originating from the f-to-2f interferometer (described in the concept section) are mapped into the temporal domain and detected by an InGaAs photodiode (DET01CFC, Thorlabs). The temporal waveform is then digitized and recorded with a 600 MHz oscilloscope (RTO2004, R&S). The oscilloscope was operated in segmented memory mode (ultra segmentation) in order to record every waveform at the full repetition rate of 100 kHz [19].

Dispersio
The dispersiv technique req content from The DFT e spectral easure the dispersion based on interferometric techniques [20]. An ideal dispersion measurement technique would propagate a frequency comb in the fiber and detect on the oscilloscope the delays between each successive frequency line. Such a comb was not available, so the modulated spectrum from the f-to-2f, mimicking a periodic spectral etalon, was used. The ~100 ns temporal waveform was measured with the photodiode and oscilloscope. The spectral measurements were made using a high resolution optical spectrum analyzer (AQ6375 B, Yokogawa). Both measurements were low-pass filtered during processing to remove the high frequency noise. The modulation, as predicted by the preliminary calculation, is resolvable with the photodiode and the A/D converter of the oscilloscope. The fiber adds a positive chirp to the signal so the temporal waveform is reversed when compared to the spectrum, if plotted as a function of wavelength. Superposition is thus visualized with the time axis reversed (Fig. 3.). An excellent overlap is obtained when the frequency dependent time delay function (Eq. (5).) is fully retrieved. The different dispersion orders of the equation are determined by fitting the dispersed spectral signal over the whole temporal signal according to Eq. (5) using an iterative algorithm, shown in Fig. 4. In the first step, the dispersion parameters are given initial values and the resultant temporal profile is calculated. The difference between the measured and the calculated temporal signals are used as a fitting error, which is subsequently minimized via the iteration process. The most intense peaks have the highest weight in the fitting process, making local difference smaller for the main peaks when compared to the side peaks. The comparative measurement of the interference pattern was done for 13 data sets with different CEP values. The retrie includes dispe manufacturer, ps/nm calcula time delay me and a FOD o coefficients a insufficient to contribution.

CEP drift
The CEP drif at the full rep The two data pulses for a memory. How statistical ana sets was achie are cut by the post-processin In order to TOUCAN CE coincides wit data sets, acq properties of C (7). α i and β i are means [22] of vary between mean is small Figure 6. the ten possib compared to consistently, point on, the d as the decima  ated from difference is found as decimating a data set is not equivalent to low-pass filtering the data [23]. After decimation, high-frequency noise contributions are aliased into the lower frequency region, which means that the overall noise level does not change significantly. The measurements with the conventional technique gives a smaller CEP noise standard deviation than the TOUCAN method. The reported uncertainty of the Fringeezz measurement originate from the spectral resolution of the grating spectrometer, provided by the manufacturer. The uncertainty of the TOUCAN method originates from the temporal resolution of the recorded waveforms and partly on time jitter affecting the measurement. The measurement resolution is set by the time resolution of the oscilloscope. The frequency to time mapping is not perfectly linear so the CEP resolution differs at different time in the waveform. For instance, the CEP resolution ranges from 33 to 43 mrad along the recorded waveform and is 38 mrad at the center where the CEP value is extracted. Multiple data points are used for the CEP extraction so this value should only be considered as an upper bound for the uncertainty. The standard deviation of the time jitter is 126 ps, which translates to an additional 48 mrad CEP noise. The combined random error can be calculated by taking the root sum of squares of these two uncertainties, which is also displayed in Table 1. The validation of TOUCAN with the Fringeezz measurements is not straightforward as data set is being compared with another undersampled data set. The CEP standard deviation calculated on both data sets can be identical although the data may be uncorrelated i.e., the measured Fringeezz 10 kHz data set is shifted in time by several pulses compared to the decimated set. In this case, a correlation evaluation should identify the corresponding 10 kHz data sets. However, the correlation coefficient is insufficient to warrant the correct calibration of the device because multiplying one of the data set by an arbitrary small number would still produce almost the same correlation value. Then, in order to certify the CEP calibration, a linear fitting was performed on the scatter plot of the corresponding data sets (Fig. 7.). The slope indicates the relative magnification of the measured CEP between the two measurement techniques. A slope of 1 would be a high correlation (all points scattered along a narrow line) would confirm a proper calibration, which was the case in almost all experiments. The record integrated pha for the decima (on Fig. 8(b). both methods However, the attributed to frequency for Noise con kHz. One big feedback sign the Nyquist f reaches 88% noise could on CEP detection 7. The correlation by TOUCAN tech line.
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The energy needed for the CEP detection with the TOUCAN method was 2.5 µJ minimum with the current setup. This can be significantly lowered as f-to-2f interferometers with specialized optics require only nJ energies, and avalanche photodetectors sensitive at the telecom wavelength are readily available. With further development the energy requirements can be decreased orders of magnitude below that of the single-shot stereo-ATI phase meter.