Introduction

Optical pulse analysis techniques have been at the very heart of the development of broadband photonics ever since the introduction of short-pulse lasers. Over time a whole “zoo” [1] of methods has evolved encompassing temporal pulse and spatiotemporal optical wavepackets characterization. Most of these techniques rely on the interaction of the unknown optical signal with a reference pulse or a delayed copy of the signal itself. The interaction of the signal with the reference may be accomplished through nonlinear gating or interference methods.

Interference methods are very sensitive but require full spectral overlap of the signal and reference, making them incompatible with many experiments in nonlinear optics, where the signal can have an arbitrarily broad spectrum, unless self-referencing is employed. This restriction also applies to generalizations of these methods to the characterization of spatiotemporal optical wavepackets, such as SEA-TADPOLE [2], STRIPED FISH [3,4], STARFISH [5–7], and techniques with subwavelength spatial resolution [8–10].

Many nonlinear gating techniques allowing the reconstruction of the temporal phase have been developed in the last two decades, such as SPIDER [11], SEA-SPIDER [12], the FROG family [13], or MIIPS [14]. They work even in the presence of a narrowband or unknown reference pulse; in this context usually called the gate pulse. These methods, however, do not characterize the spatial dependence of the pulse envelope and usually integrate over the pulse’s cross-section. They can be directly used for temporal stitching of spectrally resolved spatial phase measurement techniques, as demonstrated by the SHACKLED FROG [15,16] and HAMSTER [17] experiments.

Another route towards spatiotemporal reconstruction is the spatial multiplexing of temporal phase retrieval techniques, such as spatially resolved shearing interferometry techniques [18,19], CROAKS [20], spatial SPIDER [21] and spatial SEA-SPIDER [22]. All these demonstrations are, however, 1D space multiplexed and thus limited to the analysis of spatiotemporal pulses with cylindrical symmetry or independent variation along a single transverse dimension. As space-multiplexed versions of temporal phase retrieval techniques they require spatial phase stitching.

Here we present imaging cross-correlation frequency-resolved optical gating (ImXFROG), a nonlinear, ultrafast, spatiotemporal pulse-retrieval technique, based on a combination of an imaging cross-correlator (iXCorr) [23–25] with a Cross-Correlation FROG (XFROG) technique [26]. It is related to spatially and spectrally resolved correlation techniques [27–30] but increases the number of independent data axes, introduces gate pulse deconvolution, and reconstructs the optical phase. It is designed as a plug-in upgrade to iXCorr, which allows us to quickly select the measurement technique based on a tradeoff between the desired amount of information and measurement time. We show that ImXFROG can reconstruct pulses with a resolution of 10 fs and a number of 10⁷ independently measured variables, which can be distributed over the spatial or temporal axes in a tunable manner. It is a 2D space-multiplexed temporal phase retrieval technique and as such requires spatial phase stitching, which will be demonstrated in a later work.

The ImXFROG method

ImXFROG is based on the iXCorr technique [23–25], which is now routinely used to characterize spatiotemporal fields. An ultrashort, known reference pulse $A_{\mathrm{Re}f}(x,y,t-\tau )$ of carrier wavelength ${\lambda }_{\mathrm{Re}f}$ and tunable delay $\tau$ propagates collinearly with the unknown signal pulse $A_{Sig}(x,y,t)$ of carrier wavelength ${\lambda }_{Sig}$ through a thin ${\chi }^{(2)}$ crystal cut and oriented for broadband sum frequency generation at ${\lambda }_{SF}^{-1}={\lambda }_{Sig}^{-1}+{\lambda }_{\mathrm{Re}f}^{-1}$. Care must be taken that all three signals are spectrally separated, so that spectral filtering of the signal and reference pulse from the SF field is possible. Here we chose ${\lambda }_{\mathrm{Re}f}=800nm$, and ${\lambda }_{Sig}=1550nm$, such that ${\lambda }_{SF}=527nm$. The field is then imaged onto a camera, which records the SF field as a function of the delay . If none of the pump fields are depleted and the angular spectrum of the reference pulse is narrower than the angular spectrum of the signal pulse, which in turn is much smaller than the angular acceptance range of the ${\chi }^{(2)}$ crystal the SF signal recorded by the camera can be approximated as [24]:

Note that the former requirement can be achieved by using a spatially broad reference beam whereas the latter can be achieved by selection of a sufficiently thin crystal or a large magnification of the signal field as it is imaged onto the BBO crystal. We have also assumed that the ${\chi }^{(2)}$ crystal has sufficiently broad phase-matching, which can also be achieved by a thin a crystal, simultaneously mitigating pulse walk-off and other higher order effects [31]. $I_{SF}(x,y,\tau )$ is thus a measure of the spatiotemporal intensity of the signal field ${\left|A_{Sig}(x,y,t)\right|}^2$, gated with the reference pulse. From Eq. (1) one can easily deduce that iXCorr, being a cross-correlator based measurement scheme, is intrinsically limited to the observation of optical intensities and has a minimal temporal resolution in the range of the duration of the reference pulse, typically no better than a few tens of femtoseconds.

However, these shortcomings are overcome by the addition of a tunable wavelength-resolving element for the SF field, with a free spectral range (FSR) of $\Delta \lambda$ and a resolution of $\delta \lambda$, transmitting only a certain wavelength $\lambda$, corresponding to an angular frequency $2\pi c{\lambda }^{-1}$. If this wavelength $\lambda$ and the delay $\tau$ are varied independently this allows us to measure a spatially resolved cross-correlation FROG trace

which can be used to reconstruct the sought-after spatially resolved, complex amplitude of the signal field $A_{Sig}(x,y,t)$ in a point-by-point manner with standard XFROG algorithms. Note that XFROG cannot retrieve a pulse’s absolute phase ${\varphi }_0(\tau =0)$. Thus ImXFROG in its current implementation is complete, but for a spatial phase front ${\varphi }_0(x,y,\tau =0)$. This unknown phase front can, however, be easily measured for a particular delay and a particular wavelength using a simple phase detector, such as a Shack-Hartman sensor [32] or reconstructed from one or more measurements at different positions, using the phase-diversity technique [33].

The maximal length of the measurement window $\Delta T$ and the temporal resolution $\delta T$ are determined by the FSR $\Delta \lambda =100$ and the spectral resolution $\delta \lambda =1nm$ of our setup. Correspondingly we have:

$F=\Delta \lambda /\delta \lambda$ is the finesse of the system. Note that the temporal resolution equates to less than two optical cycles at the signal wavelength with a spectral sensitivity ranging from just above 1300 nm to 2400 nm The FSR limitation of $\Delta \lambda =100nm$ is of purely technical nature and is related to the availability of broadband dielectric coatings for the interferometer described below.

The measureable number of independent variables along the time axis, $N_T=\Delta T/\delta T$ seems moderate, however, we need to keep in mind that we reconstruct an individual complex signal per spatial pixel, such that the total number of independent variables $N=N_T{N}_X^2$ must take the number $N_x$ of independently measurable spatial points into account. The adding of spatial points and thus an angular spectrum, imposes further constraints on the construction of the wavelength-selective element. It must have a spatial diameter and angular acceptance range that is compatible with the desired spatial resolution.

Consequently we chose a combination of 10 nm interference filters ranging from 500 nm to 600 nm and a tunable Fabry-Perot-Interferometer (FPI) with an FSR $\Delta {\lambda }_{FPI}=20nm$ and a resolution of $\delta \lambda =1nm$ as the wavelength selective element. The FPI consists of two identical parallel 1” mirrors with a reflectivity of 85% over the complete spectral range and thus a finesse of $F_{FPI}=\Delta {\lambda }_{FPI}/\delta \lambda =20$. The desired FSR of the FPI imposes a mirror separation d of roughly $7\mu m$, such that the FPI is working in $n={\lambda }_{SF}/\Delta {\lambda }_{FPI}\approx {25}^{th}$ order. The angular acceptance cone of the FPI can be approximated as the half-angle $\alpha$ under which the transmitted wavelength shifts by $\delta \lambda /2$ and is given by $\alpha ={(2n{F}_{FPI})}^{-1/2}\approx 1.8$, which is related to a cut-off in the angular spectrum of $\Delta k\approx 2\pi \alpha /{\lambda }_{SF}$. The finite diameter $D=5mm$ imposed by mechanical constraints of the setup gives a lower bound on the angular resolution $\delta k=2\pi /D$, such that we find for the spatial $N_X$ and correspondingly for the total number of independent points $N$

This means that the experimental setup is, in principle, able to measure ${10}^7$ independent complex quantities of an ultrashort, spatiotemporal pulse of almost arbitrary shape, making ImXFROG a very detailed space-time pulse-analysis technique if compared to the ${10}^5$ independent points reported for STRIPED FISH [4], which is, on the other hand, a faster retrieval scheme.

One fundamental point to note is that the total number of independently measurable points _N can be further approximated as the second factor $2F/(n{F}_{FPI})\Delta \lambda /{\lambda }_{SF}$ is smaller than unity but not much less so. We therefore arrive at

which implies that the FROG part of the setup (iXCorr plus FPI) does not increase the number of measurable quantities but instead introduces temporal degrees of freedom at the expense of spatial ones. The limiting case is that of single-shot FROG devices, such as the Grenouille design [34], where all spatial degrees of freedom acquired by the recording of a two-dimensional image are invested to retrieve a temporal pulse shape.

Experimental setup

A schematic overview of the experimental setup is given in Fig. 1(a). A signal pulse (red) propagates collinearly with a reference pulse through a thin BBO crystal. The reference pulse is delayed with respect to the signal pulse before combination with the signal pulse on a low-dispersion beam combiner. The BBO crystal has a thickness of $25\mu m$ and is cut and oriented for efficient, broadband SF generation of the oo-e type. As both signal and reference are polarized horizontally we obtain a vertically polarized SF pulse. Immediately after the BBO crystal the residual signal, reference, and its second harmonic are filtered with appropriate bandpass filters.

 

Fig. 1 – Artist’s impression of the ImXFROG setup. A known reference pulse (green) with delay is used to gate an unknown signal pulse (red) using sum-frequency generation in a BBO crystal. Only a certain wavelength (dark blue) of the sum-frequency pulse (light blue) can pass a spectral filter and is imaged onto a CCD camera. The filter (inset) consists of two mirrors forming a tunable Fabry-Perot-Interferometer. Cross-polarized white light and a spectrometer form a feedback loop, used to tune mirror parallelity and transmitted wavelength.

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The SF pulse (blue) is imaged onto a CCD camera using an imaging lens. The wavelength filtering is achieved by the placement of an FPI and optical interference filters along the path of the imaging optics. The FPI consists of two dielectrically coated, partially reflective broadband mirrors. To achieve sufficient positional accuracy in terms of parallelity and distance one of the two mirrors is mounted on an open-loop, clear-aperture, three-axis piezo stage.

We achieve closed-loop operation of the setup by monitoring the spectral response of the FPI with respect to white light of crossed (horizontal) polarization. This light is coupled into the beam path before the FPI and coupled out after the FPI using polarization beam splitters and analyzed with a spectrometer. A computer program automatically tunes the piezo stage to achieve and maintain maximum peak contrast and thus mirror parallelity. Mirror separation is tuned to transmit the desired wavelength with an appropriate FSR. Coarse wavelength filtering is achieved by IF-filters with 10 nm bandwidth, which are not shown in Fig. 1.

The complete FPI setup is placed on a single-piece steel mount to maintain stability and rigidity. One main advantage is that ImXFROG is a plug-in to iXCorr, simply removing the FPI mirrors switches from ImXFROG to iXCorr mode.

In an experiment we record a series of images with the camera, while scanning the delay line. Image acquisition and delay line motion are synchronized to minimize measurement times to as little as a few seconds for a single iXCorr run. Wavelength sweeping is carried out step-by-step afterwards; a complete measurement therefore takes in the order of a few tens of minutes to an hour, depending on measurement conditions. The resulting four dimensional data set of multi-gigabyte size is then fed into a software program, which carries out XFROG retrieval for every camera pixel using a simple generalized projection algorithm. Utilizing optimized algorithms, parallelization and GPU computing we expect this step to finish within a few minutes. After retrieval we end up with a complex (intensity and phase) data set of three dimensions $(x,y,\tau )$ of some megabytes of data. This data set represents the optical pulse without further approximation except for the technical limitation of the unknown phase front ${\varphi }_0(x,y)$ discussed above and can then be used to calculate any compatible pulse property, such as pulse front tilt, spatially resolved chirp and local spectra, etc.

Experimental results

To demonstrate the power of ImXFROG we ran two sets of experiments, serving as case studies. The first experiment analyses the pulse shape of a discrete Light Bullet (LB), a spatiotemporally, non-dispersing, non-diffractive wave-packet. In previous works we have made extensive use of the iXCorr method to characterize LB properties and dynamic features. One core property, however, remained out of reach of direct characterization by iXCorr: we predicted that the LB has, for fundamental reasons, a temporal duration of less than 25 fs and does not exhibit chirp. We therefore revisited this experiment with the new ImXFROG technique to close this gap in the demonstration of the temporal properties of the LB. As such we could have achieved the same results with an ordinary FROG or SPIDER. However, we use the LB as a temporal test object with largely known features close to the resolution limit of the ImFROG. True spatiotemporal operation of the ImXFROG is demonstrated below.

We launched an input pulse with 50 fs duration into a two-dimensional fiber array with a length of 30 mm and used the ImXFROG method to analyze the pulse leaving the central core of the array. Results are depicted in Fig. 2. Figure 2(a) shows that the LB consists of a main lobe of 25 fs duration, which is trailing a much weaker pre-pulse. The main lobe has a linearily decreasing temporal phase, consistent with a redshift of 30 nm, whereas the pre-pulse has a linearily increasing temporal phase, consistent with a blueshift of 90 nm. These findings are consistent with earlier results, which show that the LB is redshifted due to stimulated Raman scattering and thus separates from blue shifted dispersive waves, which propagate through the array at higher velocity. As expected for an LB the pulse is almost transform-limited with a time-bandwidth product of roughly 1.2 and a measured bandwidth of 180 nm as seen in Fig. 2(b). Linear broadening over the length of the sample would lead to a duration of approximately 90 fs and considerable pulse chirp, which is clearly not present in the spectrographic representation in Fig. 2(c) of the retrieved field.

 

Fig. 2 – ImXFROG reconstruction of LB in a 30 mm fiber array sample. The excitation pulse is a 50 fs Gaussian pulse at 1550 nm. (a) Temporal, (b) spectral, and (c) spectrographic representation of the pulse. The measured pulse is composed of an advanced, blue-shifted, dispersive pulse, trailed by a red-shifted LB. The pulse duration of 25 fs is accordance with predictions and essentially transform-limited. Linear dispersion would broaden the pulse to more than 50 fs over the sample length.

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This result does not, however, harness the potential of the ImXFROG method for full spatiotemporal retrieval. As a second demonstration we have therefore analyzed a spatiotemporal Airy pulse. The pulse has a transform limited minimum width of 30 fs, upon which we impose a third order chirp (TOC) of $27\cdot {10}^{3}f{s}^3$, using a spatial-spectral pulse shaper in a classic 4f setup [14,35]. The pulse has a high degree of spatiotemporal distortion due to self-phase-modulation and due to imaging aberrations introduced by the pulse-shaping setup.

Experimental results of the ImXFROG retrieval of the Airy pulse are displayed in Fig. 3. Figure 3(a) shows an isointensity map of the pulse. One can immediately see the pulse-front with the trailing ripples that are characteristic for Airy pulses. Note that the ripples are of less than 25 fs duration and yet are fully resolved in the reconstruction. An in-depth analysis of the pulse is possible from the following subfigures.

 

Fig. 3 – ImXFROG reconstruction of an Airy pulse with complex spatiotemporal distortion (see also Media 1 and Media 3). (a) Isointensity plot of the spatiotemporal intensity (see also Media 2). (b) Locally resolved third order chirp superimposed on map of the spatiotemporal intensity. (c) Local mean wavelength. (d) Local mean time of arrival. (e) Local width of the spectrum. (c-e) The (white line) marks the region which contains 90% of the pulse energy; (black regions) denote areas with insufficient power for pulse retrieval.

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Figure 3(b) shows a surface representation of the locally resolved TOC superimposed on a time-integrated image of the pulse. It is evident that the pulse as a considerable variation of the TOC over its cross-section. In particular the TOC at the fringes of the pulse is somewhat larger than in the center and its absolute value is in the order of $20\pm 5\cdot {10}^{3}f{s}^3$. We attribute the deviation from the value imposed by the SLM to some undetected initial TOC of roughly -$7\cdot {10}^{3}f{s}^3$.

Figures 3(c)-3(e) give a good understanding of the general spatiotemporal distortions, which are superimposed on the pulse. It is immediately evident that the pulse is considerably smaller in x-direction, than in the y-direction, with some level of redshift in the center of the pulse over the fringes, as can be seen from Fig. 3(c). The center of the pulse also appears considerably earlier than the pulse fringes, as can been seen in Fig. 3(d) and is also obvious by the comparatively brighter features of the pulse’s center in Fig. 3(a). We attribute this pulse front bending to the greater spectral bandwidth at the center seen in Fig. 3(e) and the interaction of the spectrum with TOC imposed on the pulse.

Conclusions

In this paper we have presented the imaging cross-correlator FROG (ImXFROG), a method for the spatiotemporal reconstruction of complex, optical fields with femtosecond resolution. The reconstruction of the electrical field is complete but for a spatial reference phase ${\varphi }_0(x,y)$, which can be easily determined externally. ImXFROG is based on the established XFROG and imaging Cross-correlation method (iXCorr), by using the imaging properties of the latter to add spatial resolution to the former. At the heart of the method is a plug-in wavelength-filter for the iXCorr setup, adding sufficient spectral resolution to enable FROG retrieval of the complex electric field. The wavelength-filtering, which must not perturb the spatial spectrum, is implemented with a Fabry-Perot Interferometer (FPI) and a set of IF filters; although the specific implementation of the tunable wavelength filter is arbitrary. We show that the introduction of the FPI trades spatial resolution for temporal resolution. We demonstrate that the FROG allows for field reconstruction with up to ${10}^7$ independently determined variables, a spatial resolution of a few $\mu m$ and temporal resolution in the order of 10 fs.

ImXFROG allowed us to give definite proof of the non-dispersive nature of Light Bullets in waveguide arrays. Moreover we have shown that we can characterize pulses with massive, nonlinear, spatiotemporal distortions and extreme phase variations, such as nonlinearly distorted Airy pulses.

These findings underline the ability of ImXFROG to analyze complex spatiotemporal pulses and show its potential for the application as a versatile and powerful space-time pulse measurement tool.