1. Introduction

Ultrafast spectroscopy attempts to achieve a high resolution in time domain characterization of physical processes. As is well known from other fields, such as frequency metrology (see Ref. [1] and references therein) or sub-diffraction limited imaging [2–4], both physical and practical experimental limitations on the resolution can be overcome by the toolbox of precision measurement, which basically relies on the notion that resolution essentially depends on the signal to noise ratio rather than on the typical response width. Yet, ultrafast measurements based on these ideas have failed, so far, to compete with advanced nonlinear optical methods. Here we attempt to transfer these well-known concepts to the problem of ultrafast pulse characterization, which is the first step in any ultrafast measurement.

Complete characterization of an optical pulse implies the knowledge of its time profile, or, equivalently, its spectrum and the spectral phase. Despite the rapid progress of detection electronics [5], direct measurement of a subpicosecond time profile remains a difficult task. Usually, in the absence of a coherent well-characterized reference, pulse shape retrieval requires a self-referencing technique [6, 7]. Self-referenced characterization of an ultrashort pulse is usually performed by interacting it with its replica in a nonlinear optical medium. During the last two decades nonlinear optical characterization techniques such as FROG [8] and SPIDER [9] have become a standard tool in ultrafast research. Employing optical nonlinearities, however, imposes substantial limitations on power, spectral range and complexity of the characterized pulse [10].

These difficulties stimulated attempts to develop spectral phase retrieval methods using only linear optics and fast electronics. Several approaches have been developed that allow for reliable picosecond waveform reconstruction [7, 11–13]. Although the lack of fast enough electronics makes the direct characterization of femtosecond pulses challenging, the system resolution can be leveraged to femtosecond level using special measurement schemes [14–18]. The ability of a technique to resolve a pulse shape with a resolution significantly better than the system response depends critically on the signal to noise ratio. It is thus essential for linear optical femtosecond pulse characterization to maximize the measurement sensitivity, which is ultimately limited by the shot noise. While shot noise limited operation is common in pulse plus reference methods [19–21], the self referenced linear optical characterization techniques have only recently surpassed [14] the nonlinear optical methods [22] in sensitivity, remaining far from the shot noise limit.

Here we demonstrate a linear self-referencing ultrashort pulse characterization technique with shot noise limited temporal resolution. Similarly to the above mentioned femtosecond linear optical methods, we measure the derivative of the spectral phase ϕ(ω) with respect to frequency ω. This can be thought of as the arrival time of a spectral slice of a pulse, provided that the slice is narrow enough so that the phase within it can be approximated by a linear function [23]. The technique relies on precise timing of spectral slices using photon counting instrumentation. We take advantage of the fact that, depending on the signal to noise ratio, the timing error can be much smaller than the detector response time. For a train of identical pulses, the signal to noise ratio can be improved by averaging over many pulses. The temporal resolution thus increases as $\sqrt{N}$, where N is the number of photons detected (shot noise resolution scaling). Utilizing a differential measurement scheme with the signal and the reference pulses coupled into the same detector, we demonstrate the shot-noise limited uncertainty scaling down to few femtoseconds.

2. Experimental demonstration of shot-noise limited characterization

We implemented this method using a fast single photon avalanche photodiode (idQuantique id100) for signal detection and a time correlated single photon counting (TCSPC) module (PicoHarp 300) for electric pulse timing. The ability of TCSPC electronics to detect delays as large as few nanoseconds allowed for very large temporal dynamic range. The layout of our setup is presented in Fig. 1. The FWHM instrument response was measured to be about 60 ps. Spectral slices were selected by a 4f pulse shaper [24] with a mobile slit in the Fourier plane. At every slit position, the data was collected for 0.5 s; the spectrum was scanned several times to achieve the desired integration time. For every slice, we generated a histogram of arrival times of the signal relative to the trigger pulse. The arrival time was obtained by finding the centroid of the histogram peak.

 

Fig. 1 Experimental setup. The characterized pulse is passed through a 4f pulse shaper selecting a spectral component, and is detected by a single photon counting APD. One replica of the original pulse is used as a trigger signal for the TCSPC module, while the other, serving as an additional reference, is directed into the signal detector. The inset shows a sample histogram with the two peaks corresponding to the signal and the reference pulses.

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The rigorous way of determining the arrival time with the least possible error is to use the maximum likelihood estimation of the delay. The variance of such an estimate is determined by the Cramer-Rao bound [25] (corresponding to the shot-noise limit for this particular detector response), which in this case is given by

In this work, instead of using maximum likelihood estimation, we utilized nearly as efficient but more economic and robust approach of multiplying the histogram by a gaussian ’cap’ with a root-mean-square width of 32 ps and finding the centroid of the product. The cap position was adjusted iteratively to coincide with the centroid. This procedure is widely used for emitter localization in super-resolution microscopy [17]. As shown below (see Fig. 2), this results in a 15% worse temporal resolution, while being less sensitive to small variations in the detector response.

 

Fig. 2 Allan variance. The plot shows the standard deviation of the measured pulse arrival time as a function of the integration time. Two replicas of the same pulse separated by a constant delay were used as the signal and the reference, each attenuated to 1 MHz detection rate. The arrival time of the signal pulse was measured by finding the absolute position of the peak in the histogram (magenta) and by finding its position relative to the reference pulse (blue). The dashed black line represents the shot noise limit as given by Eq. (1). At short integration times, the relative timing standard deviation scales as $\sqrt{2/N}\cdot 45\text{ps}$, which is 15% larger than the shot noise limit.

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The femtosecond relative delays between the spectral components are much smaller than the systematic shifts induced by the intensity dependence of the detector response and the drifts of the centroid due to temperature fluctuations. To overcome this difficulty, we introduced another reference beam: a delayed replica of the original pulse was directed into the detector along with the signal beam. Since the photon flux was lower than the saturation threshold, the detector was randomly activated by photons coming from one of the two beams. The arrival time of each spectral component was then given by the relative delay of the two corresponding peaks in the histogram. An Allan variance plot [26], demonstrating the measurement timing error as a function of the integration time for a single peak and for the interval between the two peaks is shown in Fig. 2. While the unreferenced measurement deviates from the Rao-Cramer bound by an order of magnitude already at 1 s integration time, the relative delay measurement remains nearly shot noise limited up to almost 1000 seconds of integration, reaching a minimal sub-cycle standard deviation of 2 fs.

We first tested our setup by characterizing pulses from a Ti:Sapphire mode-locked oscillator. We retrieved the 85 fs transform limited laser output, as well as a chirped pulse obtained by passing it through a 152 mm slab of F3 glass. For comparison, the same pulses were characterized by a second harmonic FROG setup utilizing a 100 μm BBO crystal. For the linear measurement, the characterized beam was attenuated prior to entering our characterization setup to about 1 photon per pulse (approximately 10 pW), leading to ∼ 10⁶ detector counts per second (corresponding to ∼ 0.01 detection events per pulse). The integration time was 40 s per data point. Prior to the measurements, a calibration curve accounting for any uncompensated dispersion in the setup was recorded using a transform limited pulse.

The measured spectral component delays with and without the glass slab are presented in Fig. 3. The resolution of the measurements, estimated by calculating the standard deviation in the range 784 to 789 nm after subtraction of the linear trend, was 9.3 fs. for the glass measurement and 10.6 fs for the transform limited pulse. These values are in excellent agreement with the scaling of resolution with integration time as given in Fig. 2. As expected, the plots show linear delay dependence on wavelength, with the slope of −41.14 fs/nm. The 10% systematic deviation from the results of the FROG measurements (−46.4 fs/nm) and from Sellmeier equation for the F3 glass [27] (−47.4 fs/nm), which are also shown in Fig. 3 can be attributed to a dependence of the calibration curve on the exact alignment of the beam into the detector. This coupling sensitivity is probably due to the fact that in our detector the entrance aperture is connected to the actual APD by a short length of a multimode fibre.

 

Fig. 3 Characterization of a 85 fs transform limited pulse and a pulse chirped by a 152 mm long F3 glass slab. (a) Density plot of a typical raw measurement trace featuring two peaks, the reference and the signal. The inset shows a cross section of the signal peak with FWHM of about 70 ps. (b) Measured delays of the spectral components for the transform limited and the chirped pulse (represented by the blue and red dots, respectively). The dashed black and cyan lines show the delays derived from the FROG measurements, and the dashed green line represents Sellmeier equation. The gray solid curve in the lower part of the plot shows the spectrum of the pulses in arbitrary units.

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One of the most challenging tests for ultrashort pulse measurement is characterization of pulses with a large dynamic range or time-bandwidth product. Due to its inherent huge dynamic range our system is highly suitable for this task. To demonstrate this, we characterized ultrashort pulses after passing through a highly dispersive hot and dense Rubidium vapor cell. Rb has two narrow absorption lines in the relevant spectral range, at 780 and 795 nm. At large optical densities, the transmitted pulses are highly distorted by the resonant lines dispersion profile.

Each resonance modifies the electric field of the pulse multiplying it by a factor of f(ω) = exp [−αl (1/(1 + iωT₂))], where T₂ is the inverse pressure broadened linewidth, αl is the optical density and ω is the frequency detuning [28]. The corresponding spectral slice delay is then $t(\omega )=-\alpha l{T}_2\frac{1-{\omega }^2{T}_2^2}{{\left(1+{\omega }^2{T}_2^2\right)}^2}$. The results of the measurements are presented in Fig. 4. The spectral resolution of the system was set to 0.5 nm, thus the observed relative delays were averages of the true delays over that range, weighted by the intensity for every spectral component. The spectral delay measurements were complemented by a higher resolution absorbtion spectrum measurements shown in Fig 4(c). At room temperature, no effect is observed. At 100°C the delayed peaks around the two resonances are clearly visible, although the shape of the delay peaks is dominated by the instrument response. As the temperature of the cell is increased to 210 °C, the delay in the Lorentzian tails of the above profile becomes significant and is clearly observed. According to the equations above, the magnitude of the tails of the spectral delay curve is αL (T₂ω²)⁻¹, while the magnitude of the Lorentzian tails of absorbtion lines in the transmitted spectrum (shown in Fig 4(c)) can be written as 2αL (T₂ω)⁻². Extracting the two coefficients for the 780 nm resonance from the 210 °C data and comparing them gives an estimate of self-broadened resonance linewidth of Γ = 1/T₂ ≃ 10 GHz, which is in reasonable agreement with the spectroscopic data on Rb pressure broadening [29]. The maximal delays measured in this experiment exceeded 30 ps, being limited by the spectral resolution of our system and the fact that frequency components delayed by more than ∼ T₂ are strongly attenuated by absorption. This measurement demonstrates a temporal dynamic range of about 1000.

 

Fig. 4 Measurement of the spectral delay of pulses passed through a Rb cell. (a) Measured delays of spectral components at 100°C (blue) and 210°C (red). The integration time was 4s per point at 100°C and 8s per point at 210°C. (b) A magnified view of the same (markers) fitted with a resonance delay profile (lines). (c) Normalized spectrum of the transmitted pulse at 100°C (blue) and 210°C (red) measured at a resolution of 0.05 nm. (The dashed black line shows the resonant absorbtion profile corresponding to the fit of 210°C spectral delay data.)

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3. Conclusion

While the above experiments demonstrate the possibility of characterization of ultrashort pulses with 10 fs resolution with just a few picowatts of signal power, the performance of the method is not limited by these values. The maximal resolution and sensitivity achievable in this approach are determined by the integration time, which, in turn, is limited by the stability of the characterization setup and of the source itself. This potentially enables complete characterization of ultrashort scattering from microscopic sources, such as single molecules or nanoparticles, a regime relevant for coherent nonlinear microspectroscopy [30, 31].

The integration time necessary to achieve a given level of temporal resolution can be reduced by using an array of fast detectors, either proportional or single photon counting, for spectrally multiplexed measurements. Another possible direction that can be explored is the characterization of ultrashort pulses in the UV and XUV range. The accumulative nature of this method makes it potentially suitable for characterization of attosecond pulse sources with megahertz repetition rates [32].

In conclusion, we have adapted the well established tools of precision spectroscopy to ultrafast metrology. This enabled us to demonstrate experimentally a self-referencing ultrashort pulse characterization technique that combines the single-photon sensitivity and versatility of linear detection with temporal resolution previously achieved only by nonlinear optical methods. This technique can become a potent tool for characterization of ultrafast light sources and for the analysis of the fast dynamics of microscopic physical systems.