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Abstract
Ultrafast pulse-beam characterization is critical for diverse scientific and industrial applications from micromachining to generating the highest intensity laser pulses. The four-dimensional structure of a pulse-beam, $\widetilde {E}(x,y,z,\omega )$, can be fully characterized by coupling spatiospectral metrology with spectral phase measurement. When temporal pulse dynamics are not of primary interest, spatiospectral characterization of a pulse-beam provides crucial information even without spectral phase. Here we demonstrate spatiospectral characterization of pulse-beams via multiplexed broadband ptychography. The complex spatial profiles of multiple spectral components, $\widetilde {E}(x,y,\omega )$, from modelocked Ti:sapphire and from extreme ultra-violet pulse-beams are reconstructed with minimum intervening optics and no refocusing. Critically, our technique does not require spectral filters, interferometers, or reference pulses.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High intensity ultra-fast pulses of light are useful in fields such as micromachining [1,2], filamentation [3,4], wakefield acceleration [5,6], and high harmonic generation (HHG) [7–10]. A fully characterized pulse of light can be represented by its four dimensional complex electric field $\widetilde {E}(x,y,z,\omega )$, or $E(x,y,z,t)$. Pulses of light are fundamentally non-monochromatic. Pulses with greater fractional spectral bandwidth, $\Delta \omega /\omega _{0}$, allow for shorter pulse durations, where $\Delta \omega$ is the bandwidth, and $\omega _{0}$ is the central frequency of the spectrum. The complex spatial profile of the $j^{th}$ constituent spectral component, $\widetilde {E}(x,y,z,\omega _{j})$, also impacts pulse characteristics like peak power, duration, and intensity profile. Unknown chromaticity in optical systems leads to degradation of expected pulse characteristics. On the other hand, well-characterized spatiospectral couplings such as spatial chirp are often leveraged as an advantage [11,12]. Even when spatial chirp is intentionally introduced, it is challenging to ensure that other spatiospectral aberrations are not included due to misalignment. In either case, full pulse-beam characterization is necessary and remains a significant challenge.
One method for full pulse-beam characterization is spatiospectral metrology coupled with a measurement of spectral phase at one common spatial position. Spatiospectral metrology provides the complex spatial profile of each spectral component, $\widetilde {E}(x,y,z_{0},\omega _{j})$, in a plane transverse to the propagation of the pulse. This result can then be propagated to generate $\widetilde {E}(x,y,z,\omega _{j})$, showing the evolution of the pulse-beam. Accurately transforming to the temporal domain requires a measurement of the spectral phase at one transverse position, $\phi (x_{0},y_{0},\omega )$, which we do not present here. There are a number of techniques capable of spectral phase measurement including frequency resolved optical gating (FROG) [13], spectral-phase interferometry for direct electric-field reconstruction (SPIDER) [14], and dispersion scan [15]. These methods, and recent adaptations, provide multiple options for robust spectral phase determination [16,17]. However, spatiospectral characterization remains a challenge. The number of techniques that have been proposed for partial or full spatiospectral pulse characterization in recent years demonstrates the necessity of this diagnostic [18]. Some of these approaches include: total E-field reconstruction using a Michelson interferometer temporal scan (TERMITES) [19], Hartmann-Shack-assisted, multidimensional, shaper-based technique for electric-field reconstruction (HAMSTER) [20], spatially and temporally resolved intensity and phase evaluation device: full information from a single hologram (STRIPED FISH) [21,22], and INSIGHT [23,24]. Here we present spatiospectral characterization of pulse-beams by multiplexed broadband ptychography, which has several advantages that make it complementary to existing methods.
Ptychography is an advanced implementation of coherent diffractive imaging in which probing illumination is scanned transverse to an object and diffraction patterns associated with overlapping regions of the object are recorded [25,26]. A phase retrieval algorithm leverages redundant information from overlapping diffraction patterns to reconstruct the phase and amplitude of both the object and the probing illumination. Additionally, ptychography allows for various forms of multiplexing, including wavelength multiplexing [27–30]. Recent demonstrations illustrate that imaging of objects with ptychographic multiplexing can be extended from discrete wavelengths (multiwavelength), where probing illumination consist of continuous wave (CW) lasers of different wavelength, to broadband ptychography (continuum) where the probing illumination is a continuum of wavelengths [31]. In contrast to what we present here, these implementations of broadband ptychography rely on a pinhole upstream of the object or assume that all the spectral components of the broadband illumination share the same intensity and wavefront profile, both of which prevent beam metrology [32,33].
Ptychography was originally developed as a microscopy technique, however, shortly after its inception it was extended to characterize the incident illumination [25,26]. Broadband ptychography represents a paradigm shift where the difficulty in measuring the full electric field of a pulse-beam is shifted from hardware to software. This is especially important because it removes any optics that could introduce unknown spatiospectral aberration. Our table-top experimental setup for broadband ptychography is incredibly simple. It consists of an object mounted on a two-dimensional translation stage, oriented transverse to the average beam propagation and a monochromatic, intensity-only detector. Critically, broadband ptychography does not require any interferometers [34], spectral filters [35], or reference beams. Since broadband ptychography does not require any lenses, it can be implemented across the EM spectrum [36,37]. Additionally, ptychography can be implemented in a variety of modalities that may provide further advantages for beam metrology, such as single-shot ptychography [38,39] and three-dimensional ptychography [40,41]. Broadband multiplexed ptychography represents an extremely promising avenue towards single-shot pulse-beam characterization, since single-shot ptychography already exists.
2. Methods
When multiple wavelengths of light simultaneously illuminate an object, the diffracted intensity measured on a detector is the incoherent sum of the diffraction intensities from each spectral component or wavelength. Assuming that the object is thin such that it satisfies the multiplicative projection approximation [42], the measured diffraction intensity at one position in the ptychographic scan can be written:
where $I_{j}$ is the measured intensity at scan position $j$, $u$ and $v$ are the spatial coordinates of the detector, $\lambda = 2 \pi c/\omega$ is the wavelength of light, $c$ is the speed of light, $\mathscr {P}$ represents our normalized Fast Fourier Transform based propagator that uses appropriate Fresnel phase, $P_{\lambda }$ is the probe for each wavelength, $O_{\lambda ,j}$ is the object for each wavelength and position, and $x$ and $y$ are the spatial coordinates at the object plane. This model assumes paraxial propagation and mapping of the detector spatial coordinates $u$ ($v$) to spatial frequencies $f_{x}$ ($f_{y}$) where $u = \lambda zf_{x}$ and $z$ is the detector-to-object distance. Ultrafast pulses of light are composed of a spectrum of wavelengths that can be characterized by a central wavelength, $\lambda _{0}$, and bandwidth, $\Delta \lambda$. For beam metrology with broadband ptychography the goal is to reconstruct the quasi-monochromatic, complex beam-profile, $P_{\lambda }(x,y)$, from a set of measurements of $I_{j}(u,v)$. Accounting for all the constituent wavelengths in a pulse-beam is intractable due to the roughly linearly increasing computational demand with reconstruction of additional wavelengths and limited experimental signal to noise. Instead of attempting to account for all of the wavelengths in a pulse-beam, we reconstruct wavelength bins, or channels, each of which represents the sum over the sub-bandwidth of the binned spectrum [31,32]. By increasing the number of reconstructed wavelength channels (decreasing the sub-bandwidth) this approximation becomes more accurate.To enable spatiospectral ultrafast pulse-beam characterization by broadband ptychography we implemented a parallel version of the Ptychographic Information Multiplexing (PIM) [27] algorithm modified to perform updates based on the relaxed averaged alternating reflections (RAAR) formulation [43]. Additionally, we have adapted our algorithm to allow for application of total variation based denoising (TV) using the TV fista package [44], probe modulus enforcement [45], and object averaging (which will be explained below). A flow-chart and detailed explanation of the algorithm is shown in supplemental information Fig. S1. We have performed successful broadband ptychographic reconstructions without relying on TV, but denoising improves the quality of the reconstructions with relatively low computational expense. Since TV is not strictly necessary we only apply it to the component(s) of the reconstruction that exhibit significant noise. We found that probe modulus enforcement and object averaging are critical to effective broadband reconstructions.
Probe modulus enforcement was introduced in [45]. The technique requires an intensity image of the probe illumination measured at the detector when there is no object in the system, which we refer to as the probe image, $I_{p}$. In single wavelength ptychography, the probe image should be equal to the intensity of the reconstructed probe propagated to the detector. The probe image can be used to constrain the probe reconstruction, just as each diffraction pattern is used to constrain the exit surface wave reconstruction at each position in the modulus enforcement. In broadband ptychography, the probe modulus enforcement must be modified as:
Since we are focused on beam-metrology by broadband ptychography we have freedom to use any object. To exploit this degree of freedom, we use a non-dispersive specimen and algorithmically enforce that all wavelength bins reconstruct the same complex object. We call this technique object averaging. Object averaging is similar to the spectral smoothing technique used in Ref. [30] and to the probe replacement technique used in Ref. [32], but it is not equivalent to either. In our algorithm, object averaging is executed after each iteration of the object update by interpolating all of the objects onto the same grid, then averaging all of the objects with weights set to the relative intensities of the corresponding reconstructed probes, and then interpolating the averaged object back to their original grids. We observe that the strong constraint provided by object averaging significantly improves convergence of the algorithm, specifically the accuracy of the relative intensities of the probes.
It has been shown that a single-wavelength reconstruction of a pulse-beam provides reasonable results [46]. It is also known that the quality of the initial estimates for the probe and object given as inputs to the reconstruction algorithm (guesses) can dramatically affect convergence. To improve convergence, we can first perform a single wavelength reconstruction using the central wavelength in the pulse. The results of the single wavelength reconstruction can then be rescaled appropriately and used as guesses for the broadband reconstruction. Though this process is not critical for broadband reconstruction (and in some cases simply does not work), in some cases we observe improvements by using a probe guess, object guess, or both, from a single-wavelength reconstruction of the same data.
One important aspect of our algorithm, as in the conventional PIM algorithm, is that the only inputs are the scan positions, the diffracted intensity measurements, the probe image, and guesses for the initial probe and object. In some cases we can use a spectrometer to record the relative intensities of the spectral components in a pulse-beam. Theoretically these measured relative intensities, or other a priori knowledge about the beams or object could be used as constraints to further improve reconstructions of pulse-beams. We found that our algorithm performed well without the use of such additional constraints. However, we still make use of a priori knowledge by employing it as a qualitative metric of the fidelity of the reconstructed pulse-beams. Specifically, where possible, we compare the reconstructed relative intensities to those measured by a spectrometer, and we compare the reconstructed object transmission function to its known binary spatial structure. Since the spectrum of a pulse-beam can have dramatic spatial dependence, in all cases, the reconstructed and measured relative intensities refer to the spatially averaged relative intensities of the spectral components of a pulse-beam. When comparing the reconstructed and measured spectra for an experiment we normalize the most intense spectral component to one.
3. Results and analysis
3.1 Simulation
We performed a set of simulations to investigate how broadband ptychography (continuum) differs from multiwavelength ptychography (discrete wavelengths). In these simulations we used a detector of 64x64 square pixels of width 15.9 $\mu m$. We used an image of a sector star as our object which was 51 $mm$ away from the detector, and simulated a scan pattern of 49 positions in a Fermat spiral with a central spacing of 200 $\mu m$ between positions, which gives an approximate overlap of 80% at the center of the spiral [41,47]. The first simulation models a multiwavelength ptychography experiment. We used three wavelengths, 775, 800, and 825 $nm$, to model illumination from three CW lasers, and constructed a different probe geometry for each wavelength. The simulated probe geometries are shown as insets in Fig. 1(i). Each probe’s maximum extent is less than 1.2 mm which gives minimum oversampling $\approx$ 2 [48]. The second simulation mimicked a broadband ptychography experiment. For this simulation we used 75 wavelengths from 763-837 $nm$ with Gaussian relative intensities as shown in Fig. 1(i)). We used three different probe geometries, one for each third of the wavelengths. From 763-787 $nm$ the simulated probe geometries were inverted triangles, from 788-812 $nm$ the simulated probe geometries were circles, and from 813-837 $nm$ the simulated probe geometries were triangles. At each of the 49 positions, diffraction patterns were simulated for each wavelength and then the patterns were summed in intensity to emulate a camera exposure. This was done for both the diffraction patterns and the probe image. As such, each simulation outputs a collection of 49 diffraction patterns and a single probe image which we fed into the PIM-RAAR reconstruction algorithm described in the supplemental information.
Fig. 1. shows results from simulated multiwavelength and broadband ptychography experiments. The first (second) column shows the results from the multiwavlength simulation reconstructed without (with) object averaging. The third (fourth) column shows the results from the broadband simulation reconstructed with two (three) wavelength bins. i) shows the relative intensities of the 75 simulated probes and the three simulated geometries. The first row shows the reconstructed objects. The second row shows the simulated (yellow) and reconstructed (red) relative intensities of the probes. The remaining rows show the intensities of the reconstructed probes which are vertically aligned by wavelengths. The inset in each of the two-wavelength reconstructed probes is the expected probe intensity, which is calculated as the sum of the first (second) half of the simulated probes.
We performed two reconstructions on each of the simulated datasets. All of the reconstructions used the simulated probes as guesses and complex images with amplitude of one and zero phase as a free-space object guess. The probe modulus constraint was used for each of them, there was no noise applied to the guesses or the diffraction patterns, and the reconstructions all ran for 500 iterations. As expected, the multiwavelength data reconstructs to machine precision, $\sim$2e-16 root-mean-square convergence error [26] (first column of Fig. 1). Reconstructing the same data using object averaging (second column of Fig. 1) gives reasonable results but with noticeable imperfections that are due to interpolation error as expected. We reconstructed the broadband data with two wavelengths, 787.5 and 812.5 $nm$, (third column of Fig. 1) and with three wavelengths, 775, 800, 825 $nm$, (fourth column of Fig. 1). The reconstructed wavelengths are chosen to equally divide the simulated spectrum. The three-wavelength reconstruction mirrors the multiwavelength reconstruction because each wavelength bin only includes one probe geometry, but the two-wavelength reconstruction is more complicated. The two reconstructed probes must account for all of the simulated illumination, so they each look like a sum of the triangular and circular probes. Comparing these reconstructed probes to the sum of the first (second) half of the simulated probes, which are displayed as insets in the corresponding reconstructions, shows that our broadband ptychographic algorithm reconstructs the probe as the sum of the illumination over the sub-bandwidth.
3.2 Experimental
3.2.1 Spatially chirped pulse-beam
We performed a number of experiments that demonstrate spatiospectral pulse-beam characterization by multiplexed broadband ptychography. In the first experiment we reconstructed a Ti:sapphire pulse-beam that passed through a pair of zinc selenide (ZnSe) prisms which provide spatiospectral aberration in the form of linear spatial chirp (Fig. 2(a)). We conducted this experiment with and without a knife edge placed after the prism pair to partially block the chirped beam as shown in Fig. 2(e). In the second experiment we imaged a Ti:sapphire pulse-beam that passed through an m = 1 vortex plate (Fig. 4). In the third experiment we sampled an EUV pulse-beam from HHG (Fig. 5). These experiments demonstrate that multiplexed broadband ptychography accurately reconstructs complicated pulse-beams across the electromagnetic (EM) spectrum, and that it provides quantitative spatiospectral aberration characterization. Critically, our experimental system uses minimal intervening optics, does not rely on refocusing, and does not require spectral filters, interferometers, or reference pulses.
The set up for the two Ti:sapphire-based experiments was comparable, and schematics are shown in Fig. 2(a) and Fig. S2. We used a home-built modelocked Ti:sapphire oscillator to provide the pulsed-beam for the experiments. The repetition rate of the Ti:sapphire oscillator ($\sim$80 MHz) was many orders of magnitude higher than the data collection rate (1 kHz for the minimum exposure time), so each exposure of the detector collected the sum of many pulses. As such, our technique averages over any pulse-to-pulse fluctuations. The pulse-beam passed through a spatial filter and was collimated to provide roughly uniform input illumination for the experimental system. This illumination passed through either the prism pair and knife edge or the m = 1 vortex plate. Next, the modified illumination passed through a focusing lens and impinged on a 1951 USAF resolution test pattern (object) that was placed on a two-dimensional translation stage downstream of the focus of the lens. The focusing lens resizes the probes and makes an image of the knife edge at the object plane, but it is not necessary for broadband ptychography generally. Finally, a 14-bit monochromatic detector (Thorlabs S805MU1) was placed further downstream of the object to collect diffraction patterns. In addition to collecting diffraction patterns and a probe image for each experiment, we also collected the spectrum of the pulse-beam at the focus of the lens using an Ocean Insight spectrometer (Flame-S-VIS-NIR-ES) fiber coupled to a custom designed 3D printed integrating sphere [49]. Note that while the measured relative spectral intensities were not used in the reconstruction algorithm, comparison of the reconstructed and measured relative spectral intensities along with the fidelity of the reconstructed object provide metrics to ensure that the algorithm accurately reconstructs the pulse-beam.
Fig. 2. experimental system and results from the prism experiments with and without the knife edge (knife edge shown as dotted line in bottom row). a) prisms imparting linear spatial chirp on the beam which passes through the lens and impinges on the object. b), d) reconstructed object intensities, which are the same for all wavelengths via object averaging, along with the ground truth, inset. c), f) reconstructed and measured relative intensities for each experiment. e) relative alignment of the knife edge and the chirped beam. Bottom two rows display reconstructed probe intensities for each wavelength bin, with each probe individually normalized to highlight the spatial chirp. Complex plots showing the phase and amplitude of the reconstructed probes are shown in supplemental information Fig. S4.
The results from the two experiments that used the prisms are shown in Fig. 2. For these experiments we attempted to place the object at the image plane of the removable knife edge. The object alignment method is described in the supplemental information. We used an anti-reflection (AR) coated 40 $mm$ focal length singlet (Thorlabs LA1422-B), as the focusing lens downstream of the knife edge, and positioned our detector 57.5 $mm$ downstream of the object, which gives an oversampling of about 25 for our probe which had $1/e^{2}$ diameters of 330 (V) x 215 (H) $\mu m$. Note that the overall probe asymmetry is due to spatial chirp. We scanned the object in a Fermat spiral pattern using 75 positions with a central step size of about 50 $\mu m$ which gives an approximate overlap of 80% at the center of the spiral. To collect diffraction patterns, and probe image, we used a high dynamic range (HDR) algorithm which stitched together four images with exposure times of 1, 16, 256, and 4096 $ms$ at each position. HDR increases our acquisition time to about an hour, but it ensures that the high spatial frequency content of the diffraction has an adequate signal to noise ratio. HDR would not be necessary if we used a detector with higher bit depth.
We performed reconstructions on both datasets, with and without the knife edge, using the same parameters. We started the algorithm with identical probe and object guesses. The probe guesses were Gaussians with a hard edge of 200 $\mu m$ diameter and defocusing phase, and the object guesses were free space. We employed the PIM-RAAR reconstruction algorithm described in the supplemental information with modulus enforcement on the probes, object averaging, and TV on the averaged object. Further leveraging the freedom of object choice, we intentionally chose an object that had minuscule spatially varying phase and that was thin such that it satisfied the assumptions of an amplitude-only object and the projection approximation. We enforced an amplitude-only object reconstruction by taking the absolute value of the objects after each iteration of the object update. Negligible object tilt relative to the beam propagation axis assures the object is amplitude only. A full account of the parameters for these reconstructions is given in table S1. After termination of the algorithm, we applied additional TV to the reconstructed probes to provide cleaner visualizations, which are presented in the bottom two rows of Fig. 2. We attempted to improve the broadband reconstructions by performing single-wavelength reconstructions of each dataset to provide better guesses, but we found that those single-wavelength reconstructions did not converge. The inability to perform single wavelength reconstructions indicates that the spatial chirp on the pulse-beam significantly alters the diffracted intensity such that a single wavelength cannot account for the increased complexity of the pattern. This further motivates the use of broadband ptychography.
The results in Fig. 2 demonstrate spatiospectral characterization of the pulse-beam, including accurate reconstruction of the expected linear spatial chirp from the ZnSe prisms. For both data sets, the reconstructed objects (Figs. 2(b), 2(d)) and relative intensities (Figs. 2(c), 2(f)) match the ground truth and measured relative intensities well. These aspects of the reconstruction provide fidelity checks that instill confidence in the accuracy of the reconstructed probes. Moreover, we observe that the spatial resolution of the reconstructed objects is not significantly degraded relative to the reconstruction pixel size. The reconstruction pixel size is calculated as $dx = \lambda z/(dX N)$, where $dX$ is the detector pixel size (5.5 $\mu m$), and N is the number of detector pixels in one dimension (3296x2472). The detector asymmetry leads to asymmetric spatial resolution, and the natural wavelength dependence gives slightly different resolution for each spectral component. Taking the maximum reconstruction pixel size (maximum wavelength and smaller detector dimension) gives a theoretical spatial resolution of 3.58 $\mu m$. Group 6 element 3, which has features of 6.20 $\mu m$ gives a very conservative estimate of the smallest features that are resolved in the object reconstructions. Note that high resolution imaging was not the goal of these experiments, but the high resolution achieved provides further confidence in the accuracy of probe reconstructions.
The high fidelity of the reconstructed objects and relative probe intensities provide confidence in the accuracy of the probe reconstructions, which are key for beam metrology. Comparison of the reconstructed probes with and without the knife edge shows that the knife edge stays stationary relative to the probe reconstructions, as expected, and that there is negligible crosstalk between the probes. Here crosstalk indicates information from one wavelength channel influencing another wavelength channel. Due to the spatial chirp, crosstalk between the probes would manifest in these reconstructions as blurring of the vertical edge of the knife. While that edge is not perfectly sharp, it is clearly distinguishable between different probe reconstructions. The blurring is not so significant that the vertical positions of the horizontal knife-edge between neighboring wavelength channels smear together, which would be characteristic of crosstalk. Rather than crosstalk, we suspect that the lack of a perfectly sharp knife edge in the probe reconstructions is due to not placing the object exactly at the correct image plane. The TV based denoising we performed when post processing the probe reconstructions does sharpen the edges somewhat, but that denoising is not strong enough to negate the effect of crosstalk if it had been present. Only the reconstructed probe intensities, $|P_{\lambda }(x,y)|^{2}$, are shown in Fig. 2, but one significant advantage of beam metrology by broadband ptychography is that we reconstruct the complex field for each wavelength bin. Complex plots showing the phase and amplitude of the reconstructed probes are shown in supplemental information Fig. S4. Since the algorithm reconstructs the amplitude and phase of the field, we can propagate these results to obtain the spatiospectral characteristics of the pulse-beam at other axial locations. In other words, we can obtain $P_{\lambda }(x,y,z)$, and we can recombine the individual spectral components to visualize the full pulse-beam intensity. In Visualization 1, we show the spatiospectral profile of the pulse-beam as it propagates around the focus of the focusing lens. During this propagation, we observe the expected dynamics of the lateral and angular spatial chirp near focus. We can also identify the image plane of the spatial filter in our system about 2.5 $mm$ upstream of the focus of the lens. Interestingly, at this plane we find that the spatial profiles of the beams are clipped differently by wavelength, indicating that the spatial filter selects only the portions of the beams propagating in the same direction. Thus, the spatial filter effectively removes spatial chirp from the Ti:sapphire pulse-beam. Images of the sagittal and tangential spatial profiles of the beams, colored by their local wavelengths, are presented in Figs. 3(a), 3(b). Based on these images and Visualization 1, we can separately identify the beamlet (i.e. the spatial profile for a given wavelength bin) crossing plane and the focal plane of the individual beamlets. Additionally, comparison of Fig. 3(a) and 3(b) indicates that, while the majority of the spatial chirp is in the y-direction, there is also non-negligible spatial chirp in the x-direction. We suspect the slight spatial chirp in the x-direction is from imperfect leveling of the ZnSe prisms. There is also evidence of chromatic focal shift from the focusing lens and astigmatism.
Fig. 3. shows images of the propagated probes around focus and spatial chirp analysis. a) and b) show the propagated probes around the focus of the beams in the sagittal and tangential planes. The spatial chirp is mostly in the y-direction (horizontal in the inset) but there is some chirp observable in the x-direction as well. c) shows quantitative comparison of the measured and expected linear spatial chirp from the zinc selenide prism pair. The bottom left inset shows the summed intensity of the central five reconstructed probes, with the centroids of each individual beam as white circles (intensity only); the inset just to the right shows the same but with the intensity of each beamlet colored by their wavelength and then superimposed (spectrally resolved). The top right inset shows the ray tracing model of the prisms which was used to calculate the form of the expected linear spatial chirp as a function of the input angle, $\theta _{1}$. The measured linear spatial chirp, calculated as a shift from the centroid of the 825 nm probe after correcting for the demagnification (5.33) from the imaging lens, is shown in red. The black line is the result of the one parameter fit which gives the incident angle as $\theta _{1} = 72.76 \pm 0.018^{\circ }$.
The expected linear spatial chirp from the ZnSe prisms is evident in the reconstructed probes (bottom rows of Fig. 2) and in the images of the propagated pulse-beam (Figs. 3(a), 3(b)). For better comparison, we quantified the reconstructed spatial chirp and compared it to the predicted spatial chirp from the prism pair as calculated from a ray tracing model. The top right inset in Fig. 3(c) shows the prism system and traced rays for the central five reconstructed spectral components. The calculated chirp is highly sensitive to the incident angle, $\theta _{1}$, and does not account for the finite beam size. Nevertheless, we find that after accounting for the demagnification from the focusing lens, a factor of 5.33, our data precisely matches the functional form of the expected spatial chirp (Fig. 3(c)). For quantitative comparison we only considered the central five wavelength bins which had high relative spectral intensities and therefore best signal to noise. We performed a one parameter fit to the reconstructed data with $\theta _{1}$ as the fit parameter using Mathematica’s "NonLinearModelFit" function, and we extracted an incident angle of $72.76 \pm 0.018^{\circ }$. This is consistent with the laboratory-measured angle, $72 \pm 1.7^{\circ }$. To emphasize the utility of this result, we show intensity only and spectrally resolved images of the pulse-beam in the bottom left inset of Fig. 3(c). The overlaid white circles on each image represent the centroid of each beamlet. Note that the centroids cannot be determined from an intensity only image, which would be collected with a traditional detector. This quantitative comparison demonstrates that broadband ptychography reconstructs spatiospectral aberrations, like linear spatial chirp, with high quantitative accuracy.
3.2.2 Vortex pulse-beam
To provide further validation of the technique we imaged a vortex pulse-beam. In this experiment we placed an m = 1 vortex plate (RPC Photonics VPP-800-E1) in the collimated pulse-beam far upstream of an AR coated 60 $mm$ focal length plano-convex singlet focusing lens (LA1134-B). We placed the same air force test pattern object 8.7 $mm$ downstream of the focus of the lens, where the size of the beam as measured by the $1/e^{2}$ diameter on a camera (Thorlabs UI-3240CP-NIR-GL-TL) was about 300 $\mu m$ in each direction. The same monochromatic detector described in section 3.2.1 was used to collect the data, and it was placed 57.8 $mm$ downstream of the object. Again, the object was scanned in a Fermat spiral pattern with 75 positions and a central step size of about 50 $\mu m$ which gives an approximate overlap of 80% in the center of the spiral. We used the same HDR algorithm and the same exposure times as previously stated to collect data in this experiment, and the collection took about an hour.
To reconstruct the vortex beam data, we used slightly different methods. We employed the same PIM-RAAR reconstruction algorithm described in the supplemental information, with modulus enforcement on the probes and object averaging, but instead of using TV on the averaged object we used TV on the individual probe reconstructions for each wavelength. Additionally, we performed single wavelength reconstructions of the data using $\lambda _{0}$ = 790 $nm$ and used those results as guesses for all of the probes and objects, after appropriate rescaling. Unlike the prism experiments, the pulse-beam imaged here had negligible spatial chirp, so single wavelength reconstructions provide reasonable initial guesses. Additionally, we reconstructed the vortex data with larger, 20 $nm$, spectral binning to cover the bandwidth of the illumination, which happened to be larger than that of the illumination in the previous experiments. A full account of the reconstruction parameters for this reconstruction is given in table S1. The results of the broadband vortex pulse-beam reconstructions are presented in Fig. 4, along with the results of the single-wavelength reconstruction which were used as guesses. The reconstructed object for the single-wavelength and broadband reconstructions, Fig. 4(a) and 4(d) respectively, both compare well with the ground truth, but there are significant differences. The broadband reconstruction shows more uniform transmission across the bars of the air force test pattern, which is more accurate, but the single wavelength reconstruction yields slightly better resolution as evidenced by the element number labels of group 6. We suspect that interpolation errors from object averaging are responsible for the slight degradation of the resolution for the broadband reconstruction. The complex spatial profiles of the reconstructed probes are presented in the third row of Fig. 4, where the brightness indicates the amplitude scaled by the square root of the relative spectral intensities, and the hue indicates the relative phase. The relative spectral intensities of these reconstructed probes agree well with the measured spectral intensities, as shown in Fig. 4(e).
Fig. 4. shows the results of the vortex pulse-beam reconstructions. The top row shows the results of the single-wavelength reconstruction, with the transmission of the object on the left and the complex spatial profile of the probe in the middle. The top right panel shows the same probe reconstruction after propagating it to the image plane of the vortex plate and subtracting the quadratic phase. For the images of the probe reconstructions in this figure, the hue represents the phase, and the brightness represents amplitude scaled by the square root of the relative spectral intensities. The second row shows the transmission values of the reconstructed broadband object on the left and the measured and reconstructed relative intensities in the middle. The third row shows the reconstructed probes for each of the six wavelength bins. The bottom row shows the same reconstructed probes after propagating to the image plane of the vortex plate and subtracting the quadratic defocusing phase for the 790 $nm$ probe.
Based on the results presented in Fig. 4, we conclude that multiplexed broadband ptychography accurately reconstructs the highly structured spatial profile of this complicated vortex pulse-beam. While the spatial profiles of the reconstructed probes in the broadband reconstruction are all similar to that of the single-wavelength probe which was used as a guess, the agreement of the measured and reconstructed relative spectral intensities gives confidence that the reconstruction algorithm accounts for the broadband nature of the pulse-beam. Further evidence is provided by comparing the propagated reconstructed probes after subtracting quadratic defocusing phase. The bottom row of Fig. 4 shows the reconstructed probes propagated to the image plane of the vortex plate after subtraction of the quadratic defocusing phase for the 790 $nm$ probe. Figure 4(c) shows the same analysis performed on the 790 $nm$ single wavelength reconstructed probe. Since we subtracted defocusing phase for the 790 $nm$ probe, there is no radial dependence of the phase for that wavelength. However, the other probes still exhibit radial phase dependence that wraps a different direction for wavelengths higher and lower than 790 $nm$. This further indicates that the broadband nature of the pulse-beam is accurately reconstructed, as each probe exhibits a different amount of quadratic defocusing phase and the phase variation is qualitatively consistent with expected axial chromatic focal shift from the focusing lens.
3.2.3 HHG pulse-beam
The first two experiments show that broadband ptychography accurately reconstructs complicated pulse-beams. To demonstrate the applicability of multiplexed broadband ptychography for spatiospectral characterization across the EM spectrum we performed an additional experiment on an EUV pulse-beam created by HHG. Precise fabrication and characterization of broadband EUV optics remains a significant challenge. Imprecise characterization of the optics limits the ability to fully characterize the EUV pulse-beam, since the action of the optics cannot be separated from pulse-beam dynamics. Recall that our broadband ptychography imaging system requires minimal intervening optics, and does not rely on refocusing of the pulse-beam after generation, which are significant advantages.
Our EUV pulse-beam was created in argon with gas-jet HHG. Other than the object, the only components between the generation point and detector are two 350 $nm$ thick aluminum filters to block the fundamental Ti:sapphire beam. A slim-bar variable quadrant copper grid (SPI 2150C-XA) served as an amplitude-only, clear-optical-path, object (see the inset of Fig. 5(a)). We placed this object roughly 5 $cm$ downstream of the generation point and positioned our 16-bit detector (PIXIS-XO-1024B) 42.9 cm beyond that. A schematic of the HHG experimental system is shown in the supplemental information Fig. S3. For this data collection we scanned the object in a Fermat spiral pattern with 100 positions and a central step size of 35 $\mu m$ which gives approximate overlap of 80% at the center of the spiral. Since we used a detector with higher bit depth to collect this data, we did not need to employ our HDR algorithm. Instead we captured a single exposure of 450 $ms$ at each position. This reduced the time required for data collection to about 20 minutes. The repetition rate of the fundamental pulse-beam was $\sim$ 1 KHz, which is orders of magnitude faster than the camera exposure time. As such each exposure sums over many pulses and our reconstruction is insensitive to pulse-to-pulse fluctuations. In addition to the 100 diffraction patterns, we collected a probe image with the same exposure time, however, unlike the previous experiments we were unable to measure the spectrum of this beam. Based on the HHG process we expect odd harmonics of the central wavelength of the fundamental beam. So, despite the unknown relative intensities of the spectral components, we know which components to expect.
Fig. 5. shows the results of the EUV pulse-beam reconstruction. a) shows the reconstructed object, where the region inside of the white outline was illuminated with at least 5% of the peak photon flux. The ground truth is shown in the bottom left inset. b) shows the reconstructed relative intensities of the probes after correcting for the quantum efficiency of the detector. The transmission of the aluminum filters and various pressures-lengths products of argon are shown as solid colored curves. The bottom row shows the intensity of the reconstructed probes, which are scaled by their relative intensities. The three shortest wavelength probes ($25^{th}-29^{th}$) are in the cutoff region. The relative intensities of the longer wavelength probes qualitatively match the expected transmission of the aluminum filters and various pressure-length products of argon which are reasonable based on the known parameters of the gas jet and background gas in the vacuum chamber. Complex plots showing the phase and amplitude of the reconstructed probes are shown in supplemental information Fig. S5.
To reconstruct the data from the EUV pulse-beam experiment, we used the PIM-RAAR algorithm described in the supplemental information. We used identical probe and object guesses for each spectral component. The probe guesses were Gaussians with a hard edge of 50 $\mu m$ radius and defocusing phase, and the object guesses were free space. For this reconstruction we used modulus enforcement on the probes, and object averaging. We did not use TV on the object or probe during or after the reconstruction. A full account of the reconstruction parameters for these reconstructions is given in table S1. We reconstructed the $15^{th}-29^{th}$ harmonics of the EUV pulse-beam, which based on the reconstructed relative intensities represent the most prominent spectral components. The resulting reconstruction is presented in Fig. 5. The reconstructed object compares well with the ground truth which is shown in the bottom left inset of Fig. 5(a). The white outline on the object reconstruction represents the region that was illuminated with significant photon flux (>5%). The high fidelity of the object reconstruction ensures that the other aspects of the reconstruction are also accurate.
While we were unable to measure the spectrum of the EUV pulse-beam used in this experiment, the reconstructed relative intensities of the HHG modes (Fig. 5(b)) are similar to those published elsewhere [50–52]. Additionally, we can identify features in the spectrum that provide further confidence in the accuracy of the result. There is a maximum photon energy that can be produced by HHG, known as the cutoff [53]. The decreasing relative intensities of the harmonic modes higher than the $23^{rd}$ is consistent with those modes being in the cutoff region [54,55]. With the cutoff region identified, the remaining HHG modes, from the $15^{th} - 23^{rd}$, should be in the plateau region. However, we note that residual argon from the gas jet and the vacuum chamber, as well as the aluminum filters, will absorb a different amount of each HHG mode. To account for the Al filters we used the CXRO database to calculate the transmission of 10 $nm$ of Al$_{2}$O$_{3}$ on top of 700 $nm$ of Al, which includes the oxide layer that forms on the Al filters [56]. Accounting for the transmission of residual Ar is more complicated. The Ar gas jet used in the experiment had a backing pressure of about 1000 Torr and it traveled through a capillary of 150 $\mu m$ diameter into a vacuum chamber. Residual argon filled the vacuum chamber to a background pressure of about 3 mTorr. Since the exact gas density profile is not known, we calculated the expected transmittance of argon for various pressure-length products as a function of wavelength using the CXRO database [56]. Figure 5(b) shows the transmission curves for the Al filters and various pressure-length products of Ar that best matched the reconstructed relative intensities after correcting for the quantum efficiency of the detector. The range of pressure-length products of Ar are reasonable based on the backing pressure, size of the gas jet, and background pressure of the vacuum chamber.
4. Discussion and conclusions
Broadband spatiospectral characterization is critical for full pulse-beam characterization. Here we demonstrated that multiplexed broadband ptychography provides quantitative spatiospectral characterization of complicated pulse-beams across the EM spectrum. The attractiveness of saptiospectral characterization by multiplexed broadband ptychography is particularly evident when considering the simplicity of the table-top experimental setup which does not require any reference beams, spectral filters, or interferometric stability. It should be noted that since the technique is based on scanning ptychography it has the same stability requirements as conventional ptychography. The required stability may preclude application of the technique to sources that suffer severe shot-to-shot fluctuations. A future single-shot broadband ptychography technique would alleviate the stability requirement. Broadband ptychography generalizes multiwavelength ptychography to allow reconstruction of multiple wavelength bins from input illumination of myriad spectral components. Through numerical simulations we verified that a wavelength channel in our reconstruction algorithm accurately accounts for all of the light in its bin by reconstructing the sum of the illumination in the bin, and that our object averaging technique improves broadband reconstructions, particularly the relative intensities of the reconstructed probes. We demonstrated the technique in a number of experiments. First, we showed quantitative diagnosis of spatiospectral aberrations, specifically linear spatial chirp, second, we showed that it can handle complicated spatial profiles like vortex beams, third, we demonstrated that the technique is applicable across the EM spectrum. Significantly more information can be gleaned from the spatiospectral characterization of the pulse-beams that we performed here. In future work the technique will be applied to characterize behavior of individual high harmonics like generated spot sizes and divergences, as well as further characterization of the orbital angular momentum and polarization characteristics of vortex and vector pulse-beams. These promising potential applications and our current results indicate that multiplexed broadband ptychography is an attractive addition to the collection of existing spatiospectral characterization methods.
Ultrafast pulse-beams are used in diverse scientific and industrial applications. Coupling spatiospectral characterization by multiplexed broadband ptychography with a spectral phase measurement can enable full pulse-beam characterization which is critical for all ultrafast pulse-beam applications. Our foundational implementation of spatiospectral characterization by multiplexed broadband ptychography demonstrates that it is an attractive alternative to existing methods. The pertinence of the technique across the EM spectrum, its quantitative nature, and the simplicity of the experimental system are highly advantageous.
Funding
National Science Foundation (2010359); Air Force Office of Scientific Research (FA9550-16-1-0121, FA9550-18-1-0089, FA9550-20-1-0143).
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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