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Abstract
We propose a single-shot, high-repetition rate measurement scheme of the carrier-envelope phase offset of ultrashort laser pulses. The spectral fringes resulting from f-2f nonlinear interferometry, encoding the carrier-envelope-phase, are evaluated completely optically via an optical Fourier transform. For demonstration, the carrier-envelope-phase of a 200 kHz, few-cycle optical parametric chirped-pulse amplification (OPCPA) laser system was measured employing an interferometer as a periodic optical filter. The proposed method shows excellent agreement with simultaneous measurement of the spectral fringes by a fast line-scan camera.
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The carrier-envelope-phase (CEP) is a fundamental parameter in strong-field physics, when intense few-cycle laser pulses interact with matter, such as in high-order harmonic generation (HHG), strong field ionization, and the generation of light wave currents [1–3]. In such experiments, the precision and the overall signal-to-noise ratio (SNR) of a measurement can be greatly improved using CEP-stabilized laser pulses or by tagging, i.e., measuring the CEP of every individual pulse and using that information in data processing. This requires real-time, single-shot CEP measurement as fast as the laser repetition rate.
The standard technique for measuring the CEP offset of amplified ultrashort pulses is based on nonlinear interferometry, usually f-2f, where spectral interference fringes that encode the CEP, are recorded with a grating spectrometer and analyzed via a digital Fourier transform [4,5]. Due to the limited readout speed of standard spectrometers, this approach is often used for low-repetition-rate (sub-100 $\textrm {Hz}$) lasers or slow-loop feedback systems where the measurements are often non-consecutive and averaged. Using advanced camera technologies, this CEP measurement scheme has also been demonstrated in single-shot at a few kilohertz up to a few hundreds of kilohertz laser repetition rate [6–8]. Alternatively, the f-2f spectral fringes can be converted to the temporal domain via a dispersive Fourier transform and subsequently sampled via a fast photodiode and an oscilloscope, which has been demonstrated up to 100 $\textrm {kHz}$ laser repetition rate [9]. However, with the ongoing development of high-repetition-rate few-cycle sources [10,11], including post-compressed Yb-based, ultrashort pulse lasers [12], not only the speed of detectors and data transfer for recording the spectral fringes is a limiting factor but also the real-time numerical processing becomes increasingly challenging, which makes traditional approaches less attractive. Simplified measurement approaches that are easier to implement at high-repetition rates, such as detecting the shift of single fringes or the contrast of the interference pattern have been developed [13,14]. These approaches suffer, however, from low SNR and reduced sensitivity for particular CEP values. Alternative single-shot CEP detection schemes, not based on nonlinear interferometry, involve strong field phenomena and have been demonstrated at high repetition rates [15–17]. However, the highly nonlinear phenomena require at least a few microjoules of pulse energy for reliable measurements. Low pulse energy ultrashort pulse lasers or systems at even higher repetition rates, i.e., megahertz, require approaches that can perform robust single-shot CEP measurements with low pulse energy input.
In this work, we present an optical scheme for analyzing f-2f spectral fringes and measuring the CEP offset via an optical Fourier transform. The technique is realized via an interferometric setup, only made of standard optical components. For demonstration, the CEP of a 200 $\textrm {kHz}$ optical parametric chirped-pulse amplification (OPCPA) laser system [10] is measured and the method is benchmarked against conventional detection employing a fast line-scan camera [8].
Traditionally, f-2f spectral fringes acquired by a spectrometer, as illustrated in Fig. 1(a), are analyzed by a numerical Fourier transform, as
where $x, y$ are the real and imaginary amplitudes of the Fourier transform, $I(\omega )$ is the spectral intensity of the signal, and $\tau$ is a time delay. In the complex Fourier domain, the fringe oscillation can be identified as a peak at a given $\tau _0$, as shown in Fig. 1(b), which corresponds to the time delay between the fundamental and the second harmonic pulses in the f-2f setup. The phase of the fringes is subsequently determined by the argument of the complex amplitude at this peak, i.e., $\varphi ={\textrm{atan2}}(x_0,y_0)$. Often, the delay $\tau _0$ is a characteristic parameter of the f-2f setup. Therefore, the fringe analysis can be simplified from a complete Fourier transform to performing two overlap integrals only for a given delay $\tau _0$, i.e.,
Fig. 1. Basic concept of an optical Fourier transform. (a) Fringes from the f-2f interferometer and (b) absolute value of the Fourier transform of the signal; panels (c) and (d) illustrate sine- and cosine-like filters that are used to perform the partial Fourier transform from which the phase $\varphi$ can be obtained as the arctan between $x_0$ and $y_0$ as illustrated in panel (e), with background $c$ removed.
Our implementation of an optical CEP measurement, which we call an analog detector, is presented at the top of Fig. 2. The output of a standard f-2f interferometry setup is split equally with a polarization beam splitter (PBS) and sent to the analog detector [Fig. 2(a)] and a digital detector [Fig. 2(b)], based on a grating spectrometer with a fast line-scan charged-coupled device (CCD), which is used for benchmarking.
Fig. 2. Schematic of the experimental setup: PBS, polarizing beam splitter; PD, photodiode; ADC, analog-to-digital converter. The PBS and PD assemblies in the analog detector are rotated by 45$^{\circ }$ relative to the plane of the figure.
The analog detector consists of a modified Michelson interferometer, designed to integrate both the periodic sine- and cosine-like filters in one device. The signal with spectral amplitude $A_0(\omega )=\sqrt {I(\omega )}$ enters the analog detector at 45$^{\circ }$ polarization and is split by another PBS into the two arms of the Michelson interferometer (the amplitude in each arm is $A_0(\omega )/\sqrt {2}$). The reflection, or the s-polarization part, hits a retro-reflector and is rotated with a $\lambda {/} 2$ plate to obtain equal spectral amplitude in s- and p-polarization,
where the superscript $(1)$ implies a constant delay $\tau _1$ introduced by the first interferometer arm.In the second interferometer arm, the transmitted p-polarization part is reflected by another retro-reflector on a translation stage to introduce a variable time delay $\tau _2$. The delay between the interferometer arms controls the periodicity of the filters and must be adjusted to match the periodicity of the f-2f spectral fringes, i.e., $\left |\tau _2-\tau _1\right |=\tau _0$. A $\lambda {/} 4$ plate is used to change the polarization to circular, introducing an additional $\pi {/}2$ phase delay between the p- and s-polarization components. This is the essential step to implement a cosine and a sine periodic filter in the same device. In practice, the phase delay may deviate from $\pi {/}2$ by a small offset $\Delta \phi$. Therefore, the spectral amplitudes of the two polarization components are
To demonstrate and benchmark the proposed method with relevant laser parameters, the CEP of a high repetition rate, few-cycle, visible–near-infrared OPCPA laser system is measured. The digital detector serves as reference (see Fig. 2). The OPCPA laser system is actively CEP-stabilized and provides 15 $\mathrm {\mu }\textrm {J}$, 6.2 fs pulses at 200 kHz repetition rate [7]. An energy of 0.5 $\mathrm {\mu }$J per pulse is split and sent to a homemade f-2f interferometer (see Supplement 1). A 10-nm wide interference filter selects CEP-dependent spectral fringes around 532 $\textrm {nm}$. An iris is placed in the beam path to avoid saturation of the detectors.
We present three different measurements performed by the analog detector, as shown in Fig. 3. In each case, the markers show the voltage readouts of the $x$ and $y$ channels from 100,000 consecutive laser shots (0.5 s) with the constant bias voltages subtracted from each channel. The result in blue, noted as “correlated”, is obtained using the setup described in Fig. 2, allowing a comparison with the digital detector. The results in yellow and red are obtained in a stand-alone mode, where the first PBS is replaced by a mirror, thus sending all the pulse energy to the analog detector. Between the two measurement modes, we did not reoptimize the analog detector in order to keep the measurement conditions as similar as possible. The stand-alone mode is based on direct interference between the fundamental and SH, resulting in a higher signal and higher contrast (see Supplement 1). The results in blue and red are obtained when the CEP is unlocked, while for the result in yellow the CEP is locked. The dashed cyan lines on top of the raw data present fitted ellipses based on Eq. (6), from which the parameters $a$, $b$, and $\Delta \phi$ are determined. The blue data points are located around a nearly perfect circle, with $a/b=1.04$ and $\Delta \phi =31.5\, \textrm {mrad}$. The red data points enclose a larger area compared with the previous measurement, indicating an overall higher signal. In this case, the ellipse has an aspect ratio of 1.19 with a slight tilt $\Delta \phi =45.0\,\textrm {mrad}$.
Fig. 3. CEP measurements from the analog detector, in a mode where it is compared with the digital detector (blue) or as a stand-alone device (red and yellow). The red and blue results are obtained with unstabilized CEP, whereas the yellow makers correspond to a stabilized CEP. Fitted ellipses, according to Eq. (6), are plotted as cyan dashed lines. The insert illustrates how the phase uncertainty is determined from the intensity fluctuations.
When the CEP from the oscillator is stabilized, the (yellow) scatter points are concentrated within an arc, corresponding to a phase instability of 146.8 $\textrm {mrad}$. As indicated in the inset of Fig. 3, the phase measurement uncertainty is estimated by comparing the signal uncertainty and mean value in the polar direction [15]. To eliminate the influence of a possible intensity-dependent phase shift from the f-2f interferometer [6], only a phase slice around the mean phase value ($\pm 50\,\textrm {mrad}$) is used for statistics. To avoid the varying radius in the ellipse, the actual evaluation is based on the relative standard deviation of the retrieved intensity, $\arcsin {(\delta I_0 / \bar {I}_0)}$, yielding a phase uncertainty of 4.2 $\textrm {mrad}$ (see Supplement 1).
The digital detector [Fig. 2(b)] is based on a conventional CEP-detection method. The signal from the f-2f interferometer is first diffracted by a grating (1200 lines/mm) and subsequently focused by a lens with 150 $\textrm {mm}$ focal length. A fast line-scan CCD camera (SPL2048-50K, Basler), read at 200 $\textrm {kHz}$ (with 512 pixels) and placed in the focal plane of the lens, records the spectral fringes in single-shot at the full repetition rate of the laser. Figure 4(a) shows a single-shot spectrum of the recorded f-2f fringes. The Fourier transform is shown in figure 4(b). The peak, corresponding to the fringe oscillation, is with 20 $\textrm {dB}$ above the noise floor, implying a 10 $\textrm {mrad}$ uncertainty of the phase retrieved from the complex amplitude (see Supplement 1).
Fig. 4. Measurement results from the digital CEP detector. (a) A single-shot measurement of the spectral fringes from the f-2f interferometer. (b) Absolute value of the Fourier transform of the spectral fringes in logarithmic scale.
To perform simultaneous measurements by the analog and digital detectors, the data acquisition of the analog detector starts slightly before that of the digital detector and lasts longer (0.5 s instead of 0.25 s). The delay between the two measurements is determined by correlating the retrieved phase from the two detectors (see Supplement 1). After the synchronization procedure, the phase difference between the digital and analog measurements can be plotted for all shots [Fig. 5(a)]. This phase difference, with a standard deviation $\sigma =33.1 \textrm {mrad}$, contains the total measurement error from both detectors. Fourier analysis helps to identify different contributions to the phase difference. The Fourier spectrum is plotted in blue in Fig. 5(b). We distinguish three intervals identified by shaded areas: low- ($0$ to 200 $\textrm {Hz}$), medium- (200 $\textrm {Hz}$ to 2 $\textrm {kHz}$), and high-frequency (2 $\textrm {kHz}$ to 100 $\textrm {kHz}$). By numerical spectral filtering, the individual contributions to the total phase difference can be found as $\sigma _\mathrm {L}=24.3\,\textrm {mrad}$, $\sigma _\mathrm {M}=21.4\,\textrm {mrad}$, and $\sigma _\mathrm {H}=7.1\,\textrm {mrad}$. The low-frequency component is dominated by $1/f$ flicker noise, as indicated by the black dashed line. This spectral component shows a similar spectrum as the intensity ($\sqrt {x'^2+y'^2}$) indicated by the red curve. This observation shows that the phase retrieval is contaminated by intensity fluctuations, which can be caused optically by, e.g., imperfect components, or electronically by, e.g., unbalanced gain of TIAs and limited common mode rejection ratio (CMRR) of the homemade differential amplifiers. The medium-frequency range mainly contains discrete spectral peaks, which can be associated with mechanical vibrations from, e.g., turbo pumps. The high-frequency part exhibits a white noise floor, attributed to the digital detector, with an estimated uncertainty of 10 $\textrm {mrad}$ (see Fig. 4). In contrast, the estimated equivalent phase noise floor of the analog detector is with 0.9 $\textrm {mrad}$, corresponding to ${3.5\times 10^{-6}}\,{\textrm {rad}/{\sqrt {\textrm {Hz}}}}$ of spectral amplitude density, attributed to Johnson noise in the TIA (see in Supplement 1). The estimated shot noise only contributes an uncertainty of 0.2 $\textrm {mrad}$ or ${8.3\times 10^{-7}}\,{\textrm {rad}/{\sqrt {\textrm {Hz}}}}$. However, for lasers with higher repetition rate and lower pulse energy, the noise floor may also be limited by shot noise [13].
Fig. 5. Comparison between the measurement results of the digital and analog detector. (a) Phase differences at maximum overlap and (b) Fourier transform of the phase difference (blue), Fourier transform of the laser intensity (red), and fitted $1/f$ line (dashed black).
In this work, we suggest, implement, and benchmark a new scheme to detect the CEP offset of ultrashort laser pulses in single-shot at high repetition rate. The idea is based on analyzing the CEP-dependent fringes from f-2f interferometry via optical Fourier transform. The demonstrated implementation utilizes the periodic spectral transmission function of a Michelson interferometer built by using standard optical components. The method is benchmarked against a conventional detector at a laser repetition rate of 200 $\textrm {kHz}$. High signal-to-noise ratio is demonstrated. This proposed scheme opens up for new possibilities to measure, control, or tag the CEP offset of ultrafast lasers. It can potentially measure the CEP offset of individual pulses at MHz repetition rate, and ultimately at the full repetition rate of an ultrafast oscillator, introducing a time domain alternative to measuring its carrier-envelope offset frequency $f_\mathrm {CEO}$.
Funding
Vetenskapsrådet (2013-8185, 2016-04907, 2021-04691, 2022-03519); European Research Council (884900); HORIZON EUROPE European Innovation Council (101058075 - SISHOT); Knut och Alice Wallenbergs Stiftelse (WACQT).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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