Dispersive Temporal Interferometry toward Single‐Shot Probing Ultrashort Time Signal with Attosecond Resolution

With the intensive research on ultrafast dynamics in physics and chemistry, the single‐shot measurement of ultrashort time change has become increasingly important. The most advanced oscilloscope with the time‐lens technique possesses subpicosecond resolution, while the interferometer can single‐shot probe time event within one optical cycle. However, the single‐shot measurement of time difference beyond optical cycle but less than time‐lens’ resolution is still inaccessible. Herein, a technique, the dispersive temporal interferometer (DTI), to break through the gap between interferometer and time lens on single‐shot time measurement, is reported. The concept is an analogy to the spatial interference: a pulse passes two paths to assemble into a pulse pair, and temporal interference arises after large dispersion, which can single‐shot record the imposed time change. Experimentally, it is measured that its resolution is sub‐20 as (0.004 optical cycle) over a range of larger than 2 ps (400 optical cycle), with a range‐to‐resolution ratio of 105 and sampling rate of 42.8 MHz. Moreover, a DTI‐based gyroscope with sensitivity of 348 as/(deg s−1) is fabricated to demonstrate its applications in optical sensors. This technique can precisely single‐shot monitor ultrashort time change in attosecond resolution and has a prosperous application prospect in capturing ultrafast process.

DOI: 10.1002/adpr.202100303 With the intensive research on ultrafast dynamics in physics and chemistry, the single-shot measurement of ultrashort time change has become increasingly important. The most advanced oscilloscope with the time-lens technique possesses subpicosecond resolution, while the interferometer can single-shot probe time event within one optical cycle. However, the single-shot measurement of time difference beyond optical cycle but less than time-lens' resolution is still inaccessible. Herein, a technique, the dispersive temporal interferometer (DTI), to break through the gap between interferometer and time lens on single-shot time measurement, is reported. The concept is an analogy to the spatial interference: a pulse passes two paths to assemble into a pulse pair, and temporal interference arises after large dispersion, which can single-shot record the imposed time change. Experimentally, it is measured that its resolution is sub-20 as (0.004 optical cycle) over a range of larger than 2 ps (400 optical cycle), with a range-to-resolution ratio of 10 5 and sampling rate of 42.8 MHz. Moreover, a DTI-based gyroscope with sensitivity of 348 as/(deg s À1 ) is fabricated to demonstrate its applications in optical sensors. This technique can precisely single-shot monitor ultrashort time change in attosecond resolution and has a prosperous application prospect in capturing ultrafast process.

Concept of Dispersive Temporal Interferometry
Owing to the equivalence of light propagations between spatial diffraction and temporal dispersion, most spatial optical components have their own temporal counterparts, such as time lens and time prism. [7,17] An emerging time-stretch technique, which stretches the ultrashort pulse into giant-chirp nanosecond pulse to write the spectral information into the temporal signal and thus is called Dispersive Fourier Transform (DFT), [18][19][20][21][22] can be seen as an analogue of spatial single-slit diffraction.
Spatially, two parallel slits can split a light beam off two coherent portions, and interferogram arises on the screen far from them ( Figure 1a). [23] Temporally, an ultrashort pulse is split off two portions and reassembled into a pulse pair, which works as a temporal double-slit; after dispersive propagation, the time interferogram arises on the oscilloscope (Figure 1b). Similar to spatial interferogram, the time interferogram encodes both the time separation and the relative phase of the two pulses, from which the time event imposed on the slits (Figure 1b) can be retrieved from a single interferogram.
An ultrashort pulse with the envelop 2E 0 ðtÞ and the carrier frequency ω 0 passes through two different paths to reassemble a pulse pair, as shown in Figure 1b. The Fourier transform of the initial pulse can be expressed as 2E $ 0 ðω À ω 0 Þ. The time separation of the pulse pair are determined by the time difference of the two paths. An event (time event), like refractive index change, distance change, or motion of the reflective mirror, can make the total time to pass one path change but leave the other fixed. To read out τ, the pulse pair is stretched with large dispersion to map the frequency spectrum of the pulse pair into time domain. Then, the time interferogram can be captured by oscilloscope. The corresponding relation among the frequency ω and the time location in the interferogram t is ω À ω 0 ¼ t=β, note that frequency ω 0 corresponds to t ¼ 0 and β is the total dispersion exerted into the pulse pair. Therefore, the time interferogram that captured by oscilloscope can be represented as In Equation (1), the cos item introduces interference fringe with period T ¼ 2πβ=τ, and the phase of interferogram, corresponding to the relative phase of the pulse pair, φ ¼ ω 0 τ þ 2kπ with k an integer to make 0 ≤ φ < 2π. The relative phase of the two pulse changes with their separation (Figure 1c), resulting in the movement of fringe location (interferogram phase) synchronously, as shown in Figure 1d. For a slight motion in separation, the change of τ and φ (δτ and δφ, respectively) have the relation δφ ¼ ω 0 δτ. Therefore, the time separation can be retrieved from both the fringe period and interferogram phase two-dimensionally by www.advancedsciencenews.com www.adpr-journal.com where T o ¼ 2π=ω 0 is the optical cycle at ω 0 . From Equation (2), the value β can be measured from the change of phase (δφ) and the change of the reciprocal of fringe period [δð1=TÞ] by slightly changing the time separation τ, δφ ¼ T o 4π 2 β δ 1 T À Á . Then, we illustrate how to retrieve separation from Equation (2). On the one hand, Equation (2a) shows the time separation can be retrieved from the fringe period. The resolution and range can be expressed as with t re the resolution of the oscilloscope. On the other hand, with τ the radius and φ the angle, Equation (2b) constitutes a continuous curve in the polar coordinates ( Figure 1e). For each phase φ, the τ that can match it constitute arithmetic progression with step of optical cycle, illustrated as olive points in Figure 1e. Therefore, the time separation can be retrieved only when k is a constant or measurable. If k is known, the resolution and range of retrieved time separation is For most ultrafast laser, τ c ≫ T o , and thus, τ re b ( τ re a , which means the resolution of retrieved τ from phase is much higher than that from fringe period. Nevertheless, the measurement range with Equation (2b) is several femtoseconds for 1550 nm while the measurement range with Equation (2a) is in picoseconds if t re ¼ 100 ps and β > 100 ps 2 . Thus, the measurement range to retrieve τ from fringe period is much longer than that from phase. To retrieve τ with long range and high resolution, the time separation should be retrieved in both the two dimensions. First, from the fringe period (Equation (2a)), the time separation can be retrieved (denoted as τ 1 ) with weaker resolution to determine the approximate location, corresponding to the gray ring in Figure 1e. Second, the phase can determine the direction (the olive points in Figure 1e). If τ re a < T o , only one point (the red point) locates at the ring, and this point corresponds to the time separation retrieved from the time interferometer (denoted as τ 2 ). Because the interference arises in time domain, and is implemented by large dispersion, this technique is named as the dispersive temporal interferometry.

Fixed Time Signal
To carry out the time measurement experimentally, we first measure the resolution of DTI. Figure 2a displays a typical DTI with a fixed time difference between the two parts. Two 1:1 optical couplers (OC), with one arm a little longer (%2 ps) than the other, assemble the pulse pair. The dispersion is implemented by 1 km dispersion compensating fiber (DCF) with total dispersion β ¼ 195 ps 2 at 1570 nm. The laser source is a home-made ultrafast laser (UL) that works at 1570 nm with pulse duration of 83 fs. [24] 95% of the laser is injected into the DTI, which is detected via a high-speed photodetector (PD, 40 GHz), and visualized by a real-time oscilloscope (OSC, 120 GHz). The other 5%, detected by another PD, serves as the timing trigger. The reference pulse location t p , detected by the trigger port, has a fixed separation t separation with the interferogram. Hence, the time of interferogram is t ¼ t raw À t p À t separation , where t raw is the raw data detected by oscilloscope. Figure 2b shows an exemplary interferogram detected by oscilloscope. The interferogram is monitored over 4 s, as shown in Figure 2c, which displays no obvious variation on fringe period and phase. We retrieve τ 1 and τ 2 from the time interferograms (see Supporting Information) in Figure 2c, as shown in Figure 2d,e. Each data is retrieved from a single interferogram, ensuring the single-shot measurement. The average time separation τ is 2008.85 fs, indicating 410 μm optical path difference. τ 1 and τ 2 fluctuate randomly with no apparent trajectory. The statistics histograms of τ 1 and τ 2 are shown in Figure 2f,g. Figure 2f displays a clear Gaussian profile with full width at half maximum (FWHM) 1.13 fs in the fitting (blue line). Moreover, most of the 4000 shots locate between τ AE 1.5 fs. Therefore, the resolution is 1.13 fs, much bellower than one cycle (5.27 fs). From the Gaussian profile in Figure 2f and the random evolution in Figure 2d, it can be deduced that the resolution of τ 1 is limited by the time resolution of the oscilloscope, because other noise, like setup instability, will introduce a trace to the evolution of Figure 2d rather than the random evolution. The statistical result of τ 2 is presented in Figure 2f, which shows a plateau in the center of Figure 2g. This can be explained by the deviation of the real pulse location t p real and the detected t p due to the limit of oscilloscope resolution: t p real has uniform probability in the range t p À t re =2, t p þ t re =2. The FWHM in Figure 2g is 76 as, close to the calculated value of 71.5 as from Equation (4). Using the interpolation method (see Supporting Information), the resolution of DTI is improved to sub-20as (Figure 2h, FWHM 15.0 as), smaller than 0.004 optical cycle. We'd like to point out that this resolution is much smaller than the pulse duration of initial pulse. This can be analogy to spatial dual-slit interference: although the wavelength of light source is much smaller than the slit separation, the wavelength is also measurable from the interferogram. Additionally, from Equation (4), it can be inferred that the resolution can be further improved by reducing time separation and/or adding dispersion.

Time-Varying Time Signal
To verify that DTI can probe time difference over a long range, a delay line is used to provide tunable time difference (Figure 3a). By tuning the delay, the evolution of interferogram is tracked, as shown in Figure 3b. We retrieve retrieved τ 1 and φ from each interferogram and plot their phase map in Figure 3e, which displays no overlap in adjacent screws. τ 1,2 ( Figure 3c) shows a total time change of 211 fs, and their largest bias is 2.25 fs (Figure 3d), smaller than T o =2, which verifies that the time event can be single-shot probed over such range. Moreover, we also test a fast-varying process of delay line, the evolution of interferogram is presented in Supporting Information. The retrieved data shows that the time separation τ 1,2 decreases from 2.436 ps to 321 fs, and their relation is sketched in Figure 3f. These results indicate that the DTI can probe the time over a range of 2.1 ps, about 400 optical cycles, much longer than the range of conventional interferometers (one optical cycle). The range-to-resolution ratio is larger than 10 5 , which highlights that DTI is effective over a very long range compared with its resolution. Moreover, the time difference is retrieved with only one interferogram, which guarantees the single-shot measurement. Thus, its sampling rate is identical to the repetition frequency of source pulse, which is 42.8 MHz in our experiment. Such superior performance of DTI on time measurement makes it possible to fabricate time-measurement optical sensor.

Applications in Optical Sensor
The DTI can monitor the time difference of two optical paths with resolution of sub-20 as and range of 2 ps, indicating that it can be used to fabricate optical sensor, based on the time measurement. As a typical example, we fabricate a gyroscope based on the DTI, as depicted in Figure 4a. The light in counterpropagating paths have a travel time difference δt when rotation due to the Sagnac effect. [25][26][27] Moreover, the linear relation between δt and Ω makes the gyroscope an ideal test bed for DTI because the ultrashort time difference change is measurable based on the angular velocity.
The pulse source is identical to that in Figure 2a, with 95% of the laser injected into gyroscope and the other 5% serving as trigger. The pulse is split into two portions: one portion (red) is injected into the fiber reel after the lower circulator (CIR), and propagates in clockwise (CW) direction, after which passes the upper CIR; the other portion (green) propagates in counter clockwise (CCW) direction in the fiber reel after the upper CIR, and then passes the lower CIR. The two portions encounter after another OC, which is detected by the oscilloscope (real time, www.advancedsciencenews.com www.adpr-journal.com 10 GHz). The length of the fiber reel is 10 km with total dispersion of 230 ps 2 , which can stretch the pulse into nanosecond. Therefore, time interference arises when the two portions encounter, and the time interferogram is caught by oscilloscope.
Here, the two CIRs is used to make the two paths have an initial difference t 0 . Hence, the relation of τ and Ω can be expressed as where L and r are the length and the radius of the path, and c is the light speed in vacuum. First, the gyroscope was placed on a rotation stage without motor. Given an initial velocity, the gyroscope can run freely for several minutes before stopping, during which the rotation velocity declines uniformly because of the resistance. The angular velocity is monitored with a camera, and simultaneously, the time interferogram is captured by oscilloscope. Figure 4b displays the dependence of the time interferogram on rotation velocity. The retrieved δt from the interferograms as a function of Ω over seven experiments are plotted in Figure 4d. All the results locate almost at the same line, proving the stability and repeatability of the apparatus. The line slope is 348 as/ (deg s À1 ), conforming well to the prediction from Equation (4) of 346 as/(deg s À1 ), which proves the retrieved time separation exactly indicates the exerted time event.
As a high-data-rate gyroscope, its transient angular velocity can be measured. Then, the gyroscope is placed on a motorized rotation stage. As presented in Figure 4c,e, when the angular velocity is set 24 deg s À1 , a transient angular velocity drift is revealed. This is not due to the thermal drift because no such drift is observed when the gyroscope is placed on a manual rotation stage (Figure 4b and Figure S6, Supporting Information). Therefore, it is attributed to the angular velocity drift: the work the electromotor does is not uniform, leading to the rotation velocity change over time. Meanwhile, because the gyroscope is on an optical flat with a large mass, the inertance of the flat can retard the angular velocity change. The interaction between flat and electromotor will introduce vibration along rotation direction. As shown in Figure 4e, the drift is 23 deg s À1 , near 100% of the average velocity. The vibration frequency is about 3 kHz, corresponding to the vibration frequency of aluminum alloy. Meanwhile, these results highlight that the DTI can be used to monitor transient angular velocity of gyroscope. In some areas, like the rocket launching, the vibration will seriously impact the performance of device. To observe the temporal evolution of the vibration can help human to find out the source of vibration, and to reduce or restrain it, which will help to improve the device performance. The vibration in Figure 4e present that the DTI can temporally unveil such vibration.
The concept of DTI is a novel method to probe the time difference. With only a single interferogram, a rough time difference is retrievable from the fringe period, and a more precision signal is available after the assistance of interferogram phase. Such two-dimensional measurement endows the DTI the ability to probe time difference with range-to-resolution ratio larger than 10 5 with a single interferogram. The single-shot measurement makes the sampling rate of DTI is the pulse repetition rate, 42.8 MHz, which is inaccessible for interferometric or ring- . The black and red lines correspond to τ 1 and τ 2 , respectively. d) Difference of τ 1 and τ 2 in (b). e) φ versus τ 1 (both retrieved from (b)). Here, δτ ¼ τ 1 À 1.45 ps. f ) Relation of τ 1 and τ 2 from 0.32 to 2.44 ps. The olive line is τ 2 versus τ 1 , and the wine scatters are their difference.
www.advancedsciencenews.com www.adpr-journal.com cavity-based sensors. [28,29] The short-term thermal drift, ubiquitous in interferometric sensors, [15,30] is not observed in our experiments (Figure 2b and 4b), which highlights their high stability. Hence, by probing the time difference, the effective optical path difference can be accurately measured using DTI. In this way, if the parameter to be measured can change the effective optical path, it is retrievable from the signal. Therefore, DTI can be used to fabricate various optical sensors, like displacement sensor (sub-nanometer resolution, millimeter range). Moreover, the interferogram measurement is much faster than the wavelength or frequency measurement, which is a prominent advantage of DTI-based sensor. Additionally, DTI can also serve for the observation of dynamical evolution in ultrafast optics like breather soliton and soliton interaction, [31][32][33] which may spur a new course of ultrafast dynamics in this area. Furthermore, injecting a closely-spaced pulse train into the target ( Figure S7 and Vid. 2, Supporting Information), some ultrafast phenomena that occurs in femtosecond can be recorded. [34]

Conclusion
In conclusion, we have proposed and demonstrated a new technique, the DTI, to single-shot probe the time event from attosecond to picosecond. As an analogy to spatial doubleslit interference, this technique can encode the time difference into interferogram, and the time difference can be decoded from both the fringe period and the phase of the interferogram. Such measurement can determine the cycle number when phase change is beyond optical cycle, which make it possible to single-shot probe time difference that traditional interferometer is inaccessible. Experimentally, we show that such time detector has resolution of sub-20 as (0.004 optical cycle), range of more than 2 ps (400 optical cycle), range-toresolution ratio of 10 5 . Moreover, its sampling rate is 42.8 MHz. As a typical example of the application in sensors, we fabricated a novel gyroscope and recorded its transient angular velocity drift, proving that this technique may achieve a breakthrough on high-data-rate sensors. This technique may be used to probe ultrafast physical or chemical reaction, and some other ultrafast dynamical phenomena.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author. www.advancedsciencenews.com www.adpr-journal.com