## Abstract

A method of 2-dimensional (2-D) space-scanned (in the *x-y* plane) spatiotemporal double-slit interference is used to reconstruct the 2-D pulse-front (in the *x-y-t* domain) of a femtosecond pulsed beam. While comparing with recent other methods, the method possesses two advantages: no reference pulse/beam is required anymore, and an arbitrarily distorted pulse-front, not just pulse-front tilt and pulse-front curvature, could be detected. Meanwhile, the influence of different factors of unknown pulsed beams and optical elements on the measurement reliability is also analyzed for engineering applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Femtosecond ultra-intense lasers, which are widely used in the fields of high-field laser physics, astrophysics, material science etc [1], have experienced a rapid development in the past three decades. With the help of the technologies of chirped pulse amplification (CPA) [2] and optical parametric CPA (OPCPA) [3], the highest peak power has already reached to 10 Petawatt (PW, 10^{15} W) [4], and meanwhile, several 100 PW lasers are under construction [5] and/or consideration [6–8]. The recently available 1-10 PW lasers generally possess large-aperture beams of around 200 to 500 mm and ultra-short pulses of around 15 to 50 fs. In this case, such lasers need to be described in both space and time, e.g., pulsed beams or beam pulses. In engineering, because of huge scale and element imperfection, the spatiotemporal distortion, i.e., interdependence of temporal and spatial distortions, cannot be avoided [9–12], which challenges the performance and the further development of such lasers. The spatiotemporal distortion relevant to the traditional first-order temporal distortion (group delay) is the pulse-front distortion, which is a major distortion in a femtosecond ultra-intense laser and would seriously degrade the focused peak intensity. For example, if 30 fs pulses at different optical rays in a pulsed beam arrive at the focal spot at slightly different times, the focused pulse would be much longer than 30 fs. The pulse-front distortion in the *x-y-t* domain is the transverse 2-D space (the *x-y* plane) dependent group delay (*t* axis defined by *t = z/c*) or the deviation of the pulse-front to the phase-front. In a PW-class laser, because large amount of transmission lenses and dispersion elements (gratings or prisms) are used, the most general forms of the pulse-front distortions are pulse-front tilt and curvature, i.e., the pulse-front possessing a tilted and/or curved profile in the *x-y-t* domain [13,14]. Although several methods have already been proposed for measurement of these two forms [15–18], the real pulse-front of a femtosecond ultra-intense laser usually is much more complex and does not only contain linear tilt and/or spherical curvature. The measurement of an arbitrarily distorted pulsed beam actually is not a new technique. For example, the spatiotemporal interference [19,20] and the spatio-spectral interference [21–26] have already been proposed. In the method of spatiotemporal interference, a near-ideal reference pulsed beam and an unknown pulsed beam interfere in a Michelson or Mach–Zehnder interferometer, and the 2-D pulse-front could be reconstructed by scanning the time delay along the *z*-axis (propagation axis) and meanwhile recording the corresponding 2-D interference patterns in the *x-y* plane. In the method of spatio-spectral interference, a near-ideal reference pulsed beam and an unknown pulsed beam interfere with a small non-collinear angle, and their spectrum are spatially dispersed by a diffractive element along the orthogonal direction. The 2-D (e.g., *y-t*) spatiotemporal field could be reconstructed by analyzing the interference pattern. And if a scan along *x*-axis is introduced, the 3-D *I*(*x, y, t*) spatiotemporal field could also be reconstructed [24–26]. Besides that, some other methods of space-scanned spectral interference have also been proposed [27]. Here, we can find that in above methods a near-ideal reference pulsed beam is required, which is a key point. Even though some methods of self-referencing interferometry [28,29] and self-referencing frequency-resolved optical gating (FROG) [30] have been proposed, the measurement methods still suffer from complex beamlines and data processing. Recently, a new method of multispectral imaging using a custom multispectral camera combined with two different wave-front sensors has been successfully demonstrated [31], which recently is challenged by relatively low spectral resolution. In this paper, in order to propose a simple and easy-operation method to meet the engineering requirement of femtosecond PW lasers, we use a 2-D (*x-y*) space-scanned spatiotemporal double-slit interference to detect an arbitrarily distorted 2-D pulse-front *T*(*x, y*). Here, no reference pulsed beam is required anymore, and the data processing is also very simple. We demonstrated this method theoretically and experimentally, and the influences of factors on the measurement result have also been carefully analyzed and discussed.

## 2. Method

The diffraction of an ultrashort laser pulse by a young’s double-slit (YDS) has already been reported and analyzed by R. Netz *et al*. [32], and the pattern in the far field (at the Fourier plane) contains the information of the time delay between two sub-pulses through the YDS. Thereby, by scanning a YDS across the beam aperture and meanwhile detecting the spatiotemporal interference fringes, R. Netz *et al* measured the pulse-front of a pulsed beam. In this paper, we will use this method to measure the 2-D pulse-front of a pulsed beam in the *x-y-t* domains and, importantly, give necessary error analysis, reliability analysis and discussion for applications.

Figure 1(a) shows the mechanism of the spatiotemporal double-slit interference, while the input pulsed beam possesses a pulse-front distortion. Two sub-pulses spatially filtered by a YDS experience diffraction and are focused onto the Fourier plane by a focal lens. If the input pulsed beam has no any pulse-front distortion, as illustrated by the dash line, the central bright fringe of the far-field pattern would locate at the focal spot position, i.e., the intersection of the optical axis (*z*-axis) and the Fourier plane, which is exactly the same with the case of a monochromatic wave. However, if the input pulsed beam has a pulse-front distortion, which would induce an original time delay of *Δt* between two sub-pulses. And then, due to temporal separation, at the focal spot two sub-pulses cannot perfectly overlap with each other in time. In this case, the central bright interference fringe would be shifted to the position of *Δξ*. According to the first-order geometrical approximation, the shift *Δξ* equals to *ΔL/Λ·f*, where *ΔL* is the optical path difference, *Λ* is the slit distance of the YDS, and *f* is the focal length. The optical path difference *ΔL* can also be described by the original time delay *Δt*, i.e., *ΔL = Δt·c*, where *c* is the light velocity in vacuum. Then, the relationship between the original time delay *Δt* and the shift *Δξ* of the far-field central bright fringe could be written as

*x*axis for example, when the first slit of

*x*

_{1}locates at 0,

*Λ*, ··· and

*(N-1)Λ*and then the second slit of

*x*

_{2}locates at

*Λ*, 2

*Λ*, ··· and

*NΛ*, the corresponding shift of the central bright fringe in the far-field of

*Δξ*,

_{1}*Δξ*, ··· and

_{2}*Δξ*is respectively detected. By substituting the shifts of

_{N}*Δξ*,

_{1}*Δξ*, ··· and

_{2}*Δξ*into Eq. (1), we can get the time delays of

_{N}*Δt*,

_{1}*Δt*, ··· and

_{2}*Δt*, which correspond to the spatial scan of the YDS at (

_{N}*x*

_{1}

*= 0*,

*x*

_{2}

*= Λ*), (

*x*

_{1}

*= Λ*,

*x*

_{2}

*= 2Λ*), ··· and (

*x*

_{1}

*= (N-1)Λ*,

*x*

_{2}

*= NΛ*). If the initial pulse-front of

*T*(

*x*= 0) is assumed to be 0, the pulse-front

*T*(

*x*) can be reconstructed by

*x*is a discrete spatial coordinate with a step of

*Λ*. If the YDS is further scanned across the beam in the 2-D space (

*x-y*plane), the 2-D pulse-front of

*T*(

*x, y*) could be reconstructed. Figure 1(b) shows the procedure, firstly the pulse-front of

*T*(

_{x = 0}*y*) in the

*y-t*plane for

*x = 0*is measured, which is also the initial pulse-front value for the third step; secondly the pulse-front of

*T*(

_{y}*x*) in the

*x-t*plane for different

*y*are measured; finally by combining the two results, the real pulse-front of

*T*(

*x, y*) in the

*x-y-t*domain could be achieved. Here, a cylindrical lens instead of a spherical lens is used.

## 3. Demonstration

#### 3.1 Theoretical demonstration

In order to verify the performance of the method, first of all we demonstrate it in theoretical simulation. A pulsed beam with a pulse-front distortion is generated by

*A*(

*ω*) is the spectral amplitude,

*ϕ*(

*ω*) is the spectral phase,

*A*(

*x, y*) is the 2-D spatial amplitude, and

*T*(

*x, y*) is the 2-D pulse-front in the

*x-y-t*domain. The spatial filtering of the YDS is given byWhere,

*x*

_{1}and

*x*

_{2}are positions of the YDS, and

*x*

_{2}-

*x*

_{1}=

*Λ*. The field at the Fourier plane in the frequency-space domain is given bywhere

*FT*denotes the 1-D Fourier transform in space (i.e., focusing by a cylindrical lens in the

*x*-

*z*plane). And the corresponding spatiotemporal distribution can be calculated bywhere

*iFT*denotes the inverse Fourier transform in frequency. Then, the time-integral detection result (e.g., by a CCD camera) could be calculated by

Here, the spectrum is assumed to have a Gaussian profile with a center wavelength of 800 nm and a bandwidth of 50 nm (Full width at half maximum, FWHM), and the spectral phase is zero, i.e., without any temporal dispersion. The beam also possesses a Gaussian distribution with a diameter of 15 mm (FWHM). The YDS has a slit distance *Λ* of 1 mm and a slit width *D* of 0.5 mm, and the focal length *f* is 1 m. Figure 2 shows the spatiotemporal simulation results, the measurement procedure and the reconstructed pulse-front of the input pulsed beam. In the near-field (before the YDS), by using Eq. (3) the pulse-front of the input pulsed beam is arbitrarily distorted. By using Eq. (4), the YDS is scanned across the beam aperture. For each position of the YDS, the spatiotemporal and time-integral far-field patterns are simulated by using Eqs. (5)-(7), and the shift of the central bright fringe is recorded. After the scan across the beam aperture (the scan step is *Λ*), the blue circle spot calculated by using Eqs. (1) and (2) illustrates the reconstructed pulse-front, and the red solid line is the introduced one [Eq. (3)]. Two results agree very well with each other, which shows the measurement method works well in the theoretical simulation.

#### 3.2 Experimental demonstration

In the experimental demonstration, a light source with a broadband spectrum from 660 nm to 1180 nm is used, and the measured pulse duration (FWHM) is around 10 fs. As shown in Fig. 3(a), the beam is expanded by a confocal telescope of *f*_{1} and *f*_{2} and collimated by another confocal telescope of *f*_{3} and *f*_{4}. The focal length of *f*_{1}, *f*_{2}, *f*_{3} and *f*_{4} is 50, 1150, 150 and 100 mm, respectively, and the final beam diameter is around 12 mm. The pulse-front distortion is due to the difference between the phase and group velocities in medium [14]. In the second telescope, while comparing with the first one, because the beam diameter is big and the focal lengths are short, the thickness of the lenses *f*_{3} and *f*_{4} varies seriously across the beam aperture. Consequently, according to the Eq. (1) in [14], the pulse-front distortion is mainly introduced by the second telescope. In the detector part, a home-made YDS with a 1 mm slit distance *Λ* and a 0.5 mm slit width *D*, a 1 m focal length cylindrical transmission lens and a CCD camera are used. The measurement follows the procedure illustrated in Figs. 1(b) and 2: firstly, the pulse-front of *T*(*y*) in the *y-t* plane for *x = 0* is measured; secondly, the pulse-fronts of *T*(*x*) in the *x-t* plane for different *y* are measured; and finally, the combined pulse-front of *T*(*x, y*) is obtained in the *x-y-t* domain. The 2-D spatial scan of YDS is in an around 10 × 10 mm^{2} square region in the *x-y* plane, and the scan step is 1 mm. The CCD camera has a ½ inch optical sensor format and a 1280 x 1024 pixel resolution.

In this method, the spatial shift of the central bright fringe is the only and direct measurement parameter, which determines the correction of the measurement. In this case, the origin position at the Fourier plane, i.e., the intersection of the optical axis (*z*-axis) and the Fourier plane, should be carefully determined in advance. In our experiment, as shown in Figs. 3(b)(i) and 3(c)(i), before the measurement we remove the YDS and then the focal line formed by the cylindrical lens at the CCD camera just is the origin position. Because, without the spatiotemporal interference, the spatial focusing process is not sensitive to the temporal distortion, i.e., the pulse-front distortion, and the location of the focal line is the same for the cases with and without any pulse-front distortions. Then, the YDS is inserted and scanned along the *y*- and *x*-axis respectively, Figs. 3(b)(ii)-(iv) and 3(c)(ii)-(iv) show some captured far-field fringes during the experiment. Figure 3(d)(i) shows the pulse-front of *T _{x = 0}*(

*y*) in the

*y-t*plane for

*x*= 0, Fig. 3(d)(ii) shows the pulse-front of

*T*(

_{y}*x*) in the

*x-t*plane for different

*y*, and Fig. 3(e)(i) shows the final combined pulse-front of

*T*(

*x*,

*y*) in the

*x-y-t*domain. In Fig. 3(e)(ii), the red and blue lines show the measured pulse-front curves at the

*x*- and

*y*-axes, and the black line is the calculated one according to the experimental setup and the Eq. (1) in [14]. For the general profile, the measured and calculated curves agree well with each other, and, for the details, some small deviations could also be found. Because, in the calculation, only the pulse-front curvature induced by ideal plano-convex spherical lenses are considered. However, in the real case, small pulse-front tilt and/or other pulse-front distortions cannot be completely avoided, and meanwhile the used lenses are not ideal plano-convex spherical lenses, either.

## 4. Error analysis

In this part, we will discuss the measurement error of the method, which is determined by two factors: the error transfer formula and the reading error of the direct detection parameter. In the near-field, from Eq. (1), the pulse-front tilt between two sub-pulses after the YDS is given by

Then, the error transfer formula can be written aswhere,*δT*

_{tilt}is the measurement error, and

*δ*(

*Δξ*) is the reading error of the direct detection parameter, i.e., the reading error of the central bright fringe shift

*Δξ*in the far-field. From diffraction optics, the YDS Fraunhofer diffraction with a monochromatic plane wave

*λ*satisfies

*α*=

*πΛξ*/(

*λf*), and

*β*=

*πDξ*/(

*λf*). cos

^{2}

*α*is interference, and (sin

*β*/

*β*)

^{2}is diffraction. The width

*W*of the central bright fringe is determined by interference, which satisfies

*W*=

*λf*/

*Λ*. If the reading error

*δ*(

*Δξ*) is

*sW*, where

*s*is a coefficient, Eq. (9) can be written asThe measurement error is determined by wavelength

*λ*, YDS slit distance

*Λ*, and coefficient

*s*, which has nothing to do with the focal length

*f*.

Figure 4(a) shows the evolution of the measurement error *δT*_{tilt} for various coefficients *s* between −0.2 and 0.2, when the YDS slit distance *Λ* is 1 and 2 mm, respectively. For the case of *Λ* = 1 mm, if the coefficient *s* is controlled within ± 0.1, i.e., the reading error *δ*(*Δξ*) is within ± 10% of the width of the central bright fringe, the measurement error of the pulse-front would be within around ± 0.3 fs/mm. Figure 4(b) shows when the YDS slit distance *Λ* is 1 mm, the width *W* of the central bright fringe is round 0.8 mm, which is slightly different for different wavelengths (spectral effect). In this case, the reading error *δ*(*Δξ*) should be within ± 80 μm, and it is not a very high requirement for a CCD camera. However, the key influence factor is the profile of the interference fringe, and taking experimental imperfection into account, a fit profile based on the directly measured one should be used. Figures 4(a) and 4(b) also show that, if *Λ* is increased, both the measurement error and the spectral effect could be reduced, however the spatial resolution of the measurement would be reduced. Consequently, a 1 mm YDS slit distance *Λ* is suitable, here. In our experimental demonstration, if the reading error is one pixel, the measurement error is around ± 0.03 fs/mm.

## 5. Reliability analysis

In this part, to discuss the reliability of the measurement, we will analyze the influence of different parameters of input pulsed beams and optical elements on the spatiotemporal interference and then the measurement result.

#### 5.1 Influence of YDS and laser bandwidth

First, the results of two YDS with different parameters are compared. The first YDS has a 1 mm slit distance *Λ* and a 0.5 mm slit width *D*, and the second one has a 2 mm slit distance *Λ* and a 1 mm slit width *D*. Here, the slit distance *Λ* is fixed twice as large as the slit width *D* (i.e., *Λ = 2D*), because in this condition the interference/diffraction pattern in the far-field would only contain three bright fringes, and then it is very convenient to detect the central one. The parameters of the input laser are Gaussian-Gaussian pulsed beam, 800 nm center wavelength, 400 nm (FWHM) bandwidth and 15 mm diameter (FWHM). The time delay between two sub-pulses passing through the YDS is 1 fs, and the focal length of the cylindrical lens is 1 m. Figures 5(a) and 5(b) show the spatiotemporal interference fringes and the corresponding time-integral energy distributions at the Fourier plane for the first and the second YDS, respectively. When the slit distance *Λ* is a little bit larger, the diffraction effect is reduced, and then it is not easy to get clear interference fringes at the Fourier plane [see Fig. 5(b)]. Thereby, a YDS with a relatively small slit distance *Λ* is suggested, and besides that, it could also improve the spatial resolution of the pulse-front measurement.

Second, to study the influence of the laser bandwidth (or the pulse duration), we define a parameter *R* to describe the sharpness of the central bright fringe of the far-field pattern, which equals to the intensity/energy ratio between the dark fringe and the central bright fringe. Figure 5(c) shows the evolution of the parameter *R* for various laser bandwidths from 50 to 600 nm (i.e., pulse durations from around 19 to 1.6 fs). *R* increases (i.e., the sharpness decreases) with increasing the laser bandwidth, and the sensitivity and the value for the case of *Λ* = 2 mm is higher than those for the case of *Λ* = 1 mm. Consequently, apart from the above advantage, choosing a YDS with a relatively small slit distance *Λ* could also reduce the adverse influence of a broad bandwidth. Therefore, in the following sections, all of the discussion is based on a YDS with a 1 mm slit distance *Λ* and a 0.5 mm slit width *D*. Figure 5(d) shows the evolution of the location *ξ*_{p} of the central bright fringe for various laser bandwidths from 50 to 600 nm (i.e., pulse durations from around 19 to 1.6 fs), which remains unchanged. Then, the laser bandwidth (or the pulse duration) would not influence the measurement reliability.

#### 5.2 Influence of residual angular dispersion

In a femtosecond PW-class laser, because of the CPA or OPCPA configuration, the residual angular dispersion within the output pulsed beam usually cannot be completely avoided. Unfortunately, the interference and diffraction in theory is very sensitive to the wave-vector, and then the measurement of the method here would be definitely affected by the residual angular dispersion. Holding all parameters to the same used in the previous section, we compare the sensitivities of a 50 nm and a 400 nm (FWHM) bandwidth pulsed beams to the residual angular dispersion. Figures 6(a) and 6(b) show the spatiotemporal interference fringes and the corresponding time-integral energy distributions at the Fourier plane for two pulsed beams. In Figs. 6(a)(i) and 6(b)(i), there is no residual angular dispersion, while in Figs. 6(a)(ii) and 6(b)(ii) a 1 μrad/nm residual angular dispersion is introduced. For the narrow bandwidth of 50 nm, the spatiotemporal interference fringes and the time-integral energy distribution almost remain unchanged [see Fig. 6(a)(ii)]. However, for the broad bandwidth of 400 nm, both of them distort a lot [see Fig. 6(b)(ii)]. Because of the asymmetric distribution of the spatiotemporal interference fringes and the time-integral energy distribution, we improve the definition of the parameter *R* to *R*_{+/−}, and *R*_{+} and *R*_{-} is the intensity/energy ratio of the positive and negative side dark fringe to the central bright fringe, respectively. Figure 6(c) shows both *R*_{+} and *R*_{-} increase (i.e., the sharpness of the central bright fringe decreases) with increasing the residual angular dispersion, and the sensitivity and the value for the broad bandwidth of 400 nm is much higher than those for the narrow bandwidth of 50 nm. And comparing with *R*_{-}, *R*_{+} is even sensitive. Figure 6(d) shows the evolution of the detected location *ξ*_{p} of the central bright fringe for various residual angular dispersion. In the case of the narrow bandwidth of 50 nm, the 1 fs time delay induced shift is 0.3 mm, which remains unchanged for various residual angular dispersion from 0 to 2 μrad/nm. However, in the case of the broad bandwidth of 400 nm, when the residual angular dispersion is larger than around 0.7 μrad/nm, the location *ξ*_{p} of the central bright fringe begins to shift toward the negative axis direction, which would reduce the reliability of the measurement result. Consequently, to improve the measurement reliability, the residual angular dispersion should be carefully removed in advance by observing the far-field focal spot [33,34], especially for a broadband (e.g., > 50 nm) pulsed beam. In our experimental demonstration, it is checked before the measurement.

#### 5.3 Influence of time delay

In above two sections, the time delay between two sub-pulses passing through the YDS is fixed to be 1 fs. In this section, we analyze its influence on the far-field pattern and accordingly the measurement reliability. Figures 7(a) and 7(b) show the spatiotemporal interference fringes and the corresponding time-integral energy distributions at the Fourier plane, when the laser bandwidth is 50 and 400 nm (FWHM), respectively. In (i), the time delay is 1 fs, and in (ii) which is increased to 3 fs. In Figs. 7(a)(i) and 7(a)(ii), the shift of the central bright fringe increases from 0.3 mm to 0.9 mm, and the distribution has no obvious change. However, when the time delay is increased to 3 fs, in Figs. 7(b)(i) and 7(b)(ii), apart from the shift of the central bright fringe, both the spatiotemporal interference fringes and the time-integral energy distributions are changed [see Fig. 7(b)(ii)]. The parameter *R* increases, i.e., the sharpness of the central bright fringe degrades. Figure 7(c) shows the parameter *R* increases with increasing the time delay. For a broad bandwidth of 400 nm, the increasing velocity is very high, while for a narrow bandwidth of 50 nm, it is very small and can be neglected. Figure 7(d) shows the evolution of the location *ξ*_{p} of the central bright fringe for various time delays, which increases linearly with increasing the time delay and agrees well with Eq. (1) for both narrow and broad bandwidths of 50 and 400 nm. In this case, the location of the central bright fringe (i.e., the measurement correction) cannot be affected by different values of the time delay, although the detailed profile of the far-field pattern would be affected. In engineering, because the slit distance *Λ* of the YDS is very small (1 mm here), the time delay between two sub-pulses usually is very small, and the distortion of the profile of the far-field pattern can be well controlled.

#### 5.4 Influence of spectral amplitude and phase distortion

In this section, we further vary the spectrum and the spectral phase (i.e., temporal dispersion) to analyze the influence on the measurement. Figure 8(a) shows a 400 nm bandwidth (FWHM) 15 mm diameter (FWHM) Gaussian-Gaussian pulsed beam with 1 fs/mm pulse-front tilt is incident at a YDS (*Λ* = 1 mm and *D* = 0.5 mm). The time delay between two sub-pulses is 1 fs, and Fig. 8(a)(ii) shows the far-field central bright fringe has a 0.3 mm shift. When the spectrum is distorted and deviates from the ideal Gaussian profile [see Fig. 8(b)(iii)], Figs. 8(b)(i) and 8(b)(ii) show the near-field pulsed beam and the far-field spatiotemporal interference fringes almost remain unchanged. If we further introduce a 10 fs^{2} residual group-velocity dispersion (GVD), Figs. 8(c)(i) and 8(c)(ii) show although the near-field pulsed beam and the far-field spatiotemporal interference fringes are stretched and distorted in time, the time-integral energy distribution still has no change [see the red solid line in Fig. 8(c)(ii)]. It actually is the very information detected by the CCD camera, and thereby the measurement would not be affected by spectral amplitude and phase distortions. Here, the comparison simulation is based an ultra-broad bandwidth of 400 nm. For a narrow bandwidth, the sensitivity to spectral amplitude and phase distortions would be even low. In this case, we can conclude that the spectral amplitude and phase distortions of an input pulsed beam would not influence the measurement reliability.

#### 5.5 Influence of amplitude difference between two sub-pulses after YDS

According to the interference theory, the amplitude difference between two waves would affect the coherent combination. In this section, we will analyze this influence and define a parameter of the amplitude ratio *η* between two peak amplitudes of two sub-pulses passing through the YDS. Figures 9(a) and 9(b) show the spatiotemporal interference fringes and the corresponding time-integral energy distributions at the Fourier plane, when the bandwidths of pulsed beams are 50 and 400 nm (FWHM). In (i), the amplitude ratio *η* is 0.5, and in (ii) which is reduced to only 0.1. By comparing Figs. 9(a)(i) and 9(b)(i) (same amplitude ratio *η* of 0.5), for a broad bandwidth of 400 nm, the parameter *R* increases, i.e., the sharpness of the far-field central bright fringe degrades. When the amplitude ratio *η* is reduced to 0.1, Figs. 9(a)(ii) and 9(b)(ii) show that, for both narrow and broad bandwidths of 50 and 400 nm, the parameter *R* increases, i.e., the sharpness of the far-field central bright fringe degrades. And the degradation for the case of the broad bandwidth of 400 nm is even more serious. Figure 9(c) shows the evolution of the parameter *R* for various amplitude ratios *η* from 1 to 0.1 for narrow and broad bandwidths of 50 and 400 nm. The parameter *R* increases (i.e., the sharpness of the central bright fringe decreases) with reducing the amplitude ratio *η*, and the value for the case of a broad bandwidth of 400 nm is always higher than that for the case of a narrow bandwidth of 50 nm. However, Fig. 9(d) shows that, for both narrow and broad bandwidths of 50 and 400 nm, the location *ξ*_{p} of the central bright fringe remains unchanged for various amplitude ratios *η*, and consequently, the amplitude ratio *η* would not influence the measurement reliability. In engineering, because the slit distance *Λ* of the YDS is very small (1 mm here), the amplitude ratio *η* generally is closed to 1, which induced distortion of the far-field pattern usually is negligible.

## 6. Discussions

As shown in Figs. 1 and 3(a), transmission lenses are used in our experiment. In theory, the pulse-front distortion is induced by different group velocity delays corresponding to different path lengths of light in medium [14]. In this case, it is better to choose a reflection focusing optics instead of a transmission one. Here, we choose a transmission focusing optics for three reasons. First, the slit distance *Λ* of our home-made YDS is very small (only 1 mm), and the path length difference of two sub-pulses in the lens is very small and can be neglected. Second, from the section 5.4, the small amount of the residual temporal dispersion (spectral phase distortion) would not influence the measurement reliability. And third, compared with a reflection focusing optics, a transmission one is much easier for alignment. And, in this paper, we only focus on the measurement of a distorted pulse-front, however, if the measurement includes both the pulse-front and the pulse duration, transmission optics should be avoided and it is necessary to choose reflection ones.

In our theoretical model, which is obtained based on the first-order approximation, the diffraction propagation between the YDS and the focusing optics is neglected [see Fig. 1(a)]. In experiment, it is necessary to reduce this distance to avoid some unnecessary influences.

One major disadvantage of this method is that the cylindrical lens and the YDS should be rotated 90° for the 2-D (*x*- and *y*-axes) spatial scan. In the next-step work, we will try to find a new 2-D modulator/filter (YDS here) to avoid this procedure of rotation. And, another major disadvantage is that, because the measurement is based on a 2-D spatial scan, and then it cannot operate on single-shot. In this case, it can only be applied in a repetition femtosecond PW laser or a femtosecond PW laser operating on low-energy mode. In the latter case, the thermal (full energy operation) induced dynamic pulse-front distortion cannot be included in the measurement. Fortunately, comparing with the dynamic pulse-front distortion, the static pulse-front distortion usually is the majority [11–14]. And next step, a single-shot measurement method would also be studied to measure both static and dynamic pulse-fronts for a single-shot femtosecond PW laser.

## 7. Conclusion

In conclusion, by using a 2-D space-scanned (in the *x-y* plane) spatiotemporal double-slit interference, a 2-D distorted pulse-front (in the x-y-t domain) of a femtosecond pulsed beam could be measured. Comparing with previous methods, not only the pulse-front tilt and curvature but also an arbitrarily distorted pulse-front could be detected. Besides that, in this measurement, no reference pulse/beam is required anymore. We demonstrated this method theoretically and experimentally, and importantly, we also analyzed the measurement error and the measurement reliability. The measurement is only sensitive to the residual angular dispersion, which recently could be easily removed by using far-field methods. We believe it provides an optional way to measure the arbitrarily distorted 2-D pulse-front of a femtosecond PW laser under the repetition mode.

## Funding

JST-Mirai Program, Japan (JPMJMI17A1).

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