1. INTRODUCTION

Ptychography is a scanning coherent diffractive imaging (CDI) technique [1–3] that has recently gained remarkable momentum in optical microscopy in the visible [4,5], extreme ultraviolet [6], and x ray spectral regions [7,8], as well as in electron microscopy [9] and other applications, e.g., optical encryption [10,11]. Ptychography is attractive for bio-imaging [12,13] and phase imaging [14] because it can provide label-free, high-contrast quantitative amplitude and phase information. In a typical ptychographic microscope, a complex-valued object is scanned in a step-wise fashion through a localized beam. In each scanning step, the intensity diffraction pattern from the illuminated region in the object is measured in a Fraunhofer plane. The set of typically hundreds of diffraction patterns is used for reconstructing the complex field describing the object, and possibly also the probe beam [15–18]. Critically, the illumination spot in each step overlaps substantially with neighboring spots, resulting in significant redundancy in the measured data. This redundancy makes ptychography a very powerful CDI technique that offers several advantages over “conventional” coherent diffraction imaging techniques [2,3]. These strengths include significant improvement in the robustness to noise, no requirement for prior information (e.g., support) on the object and probe beam, less sensitive to the loss of information due to beam stops, and generally faster and more reliable reconstruction algorithms. However, the redundancy is obtained through scanning, which results in several limitations: first, the temporal resolution is relatively low (the overall acquisition time is typically longer than a second), precluding the application of ptychography to imaging of fast dynamics. Second, even tiny imprecisions in the scanning steps reduce the resolution of ptychographic microscopes [19] (an algorithmic approach addressing this limitation was recently demonstrated [20]). Third, the space-bandwidth product is limited by the fact that the available step motors cannot exhibit both a large dynamic range that yields a large field of view (FOV) [21] and, at the same time, the very short steps that are crucial for high resolution [19]. Thus, it would be highly beneficial to have ptychographic microscopes that benefit from the large redundancy of ptychography and can work in a single shot, i.e., without scanning. In a pioneering work, ptychography was demonstrated with 16 partially overlapping beams that illuminated the object simultaneously by using a transmission grating [22]. However, as explained in [22], this scheme results in poor resolution. Extending this technique to ptychography with the simultaneous illumination of a hundred beams is a formidable task. Implementing this grating-based method with a short wavelength (e.g., extreme UV and x rays) or multispectral [23,24] radiation is very challenging.

Here, we propose single-shot ptychography: robust ptychographic microscopes in which tens or hundreds of intensity diffraction patterns from an array of partially overlapping illuminating spots are recorded in a single exposure. Notably, our single-shot ptychography can be applied across the electromagnetic spectrum, up to the x ray region. We analyzed the performances of single-shot ptychographic microscopy, showing that diffraction-limited resolution and a large field of view are accessible simultaneously. We also demonstrate single-shot ptychography experimentally. Finally, we also propose various schemes for single-shot ptychography as well as single-shot Fourier ptychography. We anticipate that the combination of the celebrated power of ptychography with the possibility for fast acquisition will lead to new possibilities in optical and electronic microscopy, e.g., in bio-imaging.

2. METHODS

Before presenting single-shot ptychography, we briefly discuss scanning ptychography. In a typical scanning ptychographic microscope, a plane wave illuminates a pinhole that is located in front of the object [Fig. 1(a)]. The CCD image sensor is located in the focal plane of a lens or in the Fraunhofer plane of the object. In both cases, each measured intensity pattern is proportional to the magnitude square of the Fourier transform of the part of the object that was illuminated. After each acquisition, the pinhole is shifted a distance $\mathbf{R}$ with respect to the object. In this way, multiple diffraction patterns are measured from a set of partially overlapping spots on the object. In ptychography, the measured intensity patterns correspond to [2]

 

Fig. 1. Schematic setups for conventional (scanning) and single-shot ptychography. (a) Conventional ptychographical setup with scanning. (b) Single-shot ptychographical setup with array of pinholes and plane wave illumination. (c) Ray tracing in single-shot ptychography. Lens L1, with focal distance f1, focuses the light beams that diffracted from the pinholes into the object, which is located distance $d$ before the back focal plane of lens L1. Lens L2, with focal distance f2, focuses the diffracted light from the object to the CCD.

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In Eq. (1), $\mathit{\nu }$ and $\mathbf{r}$ are the spatial vectors in the CCD and object planes, respectively, $m=\mathrm{1,2},3\dots N^2$ is the scanning index, and $N^2$ is the total number of steps. $\mathrm{F}$ stands for the two-dimensional spatial Fourier operator, $\mathrm{O}$ is the complex transmission function of the object, $\mathrm{P}$ is the (possibly unknown) complex envelope of the localized probe beam that illuminates the object, and ${\mathbf{R}}_m$ is the center of the illuminated spot in step $m$. Efficient algorithms have been developed for reconstructing the object, and possibly also the probe beam, from the set of measurements that are described by Eq. (1) [15–18].

Next, we move to presenting and analyzing an example of single-shot ptychography with coherent illumination. Figures 1(b) and 1(c) show an exemplary scheme for single-shot ptychography. Here, a coherent monochromatic plane wave illuminates a square array of $N\times N$ pinholes positioned before or at the input face of an asymmetric 4f system (with lenses L1 and L2 with focal lengths f1 and f2, respectively). We assume that the pinholes are circular with diameter $D$ and that the distance between consecutive pinholes is $b$. The object is located at distance $d\ne 0$ before (or after) the Fourier plane of the 4f system and the CCD is located at the output plane of the 4f system. As shown in Figs. 1(b) and 1(c), the object is illuminated simultaneously by multiple ($m=\mathrm{1,2},3\dots N^2$) partially overlapping probe beams, ${\mathrm{\Sigma }}_m\mathrm{P}(\mathbf{r}-{\mathbf{R}}_m)\exp (\mathrm{i}{\mathbf{k}}_m\mathbf{r})$, where ${\mathbf{k}}_m$ is the transverse k vector of beam $m$ (every probe beam has a different transversal k vector). The complex envelope of each probe beam is quasi-localized (it decays more slowly than exponentially). For $d=0$, it is the Airy disk function, $\frac{D^2{J}_1(\pi D\mathit{r})}{2D\mathit{r}}$ (Fourier transform of a circular aperture), where $J_1$ is the first-order Bessel function. For a nonzero yet small $d$, the probe beam still resembles the Airy disk function (an approximate analytic scaling for the width of the probe beam, $W=\frac{\lambda f_1}{\pi D}\sqrt{1+d\frac{\pi D^2}{\lambda f_1^2}}$, is obtained by replacing the aperture pinholes array with an array of Gaussians with an FWHM of $D$). As can be easily deduced from Fig. 1(c), the distance between the centers of adjacent illuminating spots in the object plane (which is the analog to the scanning distance in scanning ptychography) is $R=bd/f_1$. Thus, for arrays of Gaussians, the overlap between neighboring spots is $\frac{W-R}{W}$. While for most type of arrays, the overlap cannot be described by an analytical formula, it is clear from Fig. 1(c) that the overlap is 100% for $d=0$ and that it decreases with increasing $|d|$. Similarly to the operation of a single lens in scanning ptychography, lens L2 transfers the field after the object to the k-space domain at the CCD plane with coordinate transformation $\mathit{\nu }=\frac{\mathbf{r}}{\lambda f_2}$ (the fact that the object is located distance $d+f_2$ before lens L2 merely adds a phase, which is not detected by the CCD) [25]. Thus, the detected intensity pattern in coherent single-shot ptychography is given by

Can the measured intensity pattern in Eq. (2) be used for the ptychographic reconstruction of the object? In other words, can each probe beam (approximately) give rise to a separate diffraction pattern in a known region in the CCD such that Eq. (2) can be approximated by Eq. (1)?

The answer to this question is yes, since the effect of $k_m$ is to shift the diffraction pattern laterally in the CCD plane. From straightforward geometry [see Fig. 1(c)], $|{\mathbf{k}}_m-{\mathbf{k}}_{m-1}|\approx 2\pi b/\lambda f_1$. Thus, assuming that the power spectra of all the regions in the object (the regions illuminated by the multiple probe beams) are largely confined to a low-frequency region with cutoff frequency ${\mathit{\nu }}_{\max }\approx \mathit{b}/{\lambda f}_1$, the intensity pattern of Eq. (2) consists of clearly distinguished $N^2$ diffraction patterns that are located in $N^2$ blocks on the CCD [e.g., see Fig. 2(c)]. Moreover, we can associate each block and its diffraction pattern with the scattering of a beam that originated from a specific pinhole and illuminated the object at a specific given spot. Mathematically, this assumption allows us to transfer the sum in Eq. (2) outside the absolute value, and then divide the pattern into separate blocks and retrieve a set of intensity patterns in the form of Eq. (1). Thus, we can employ the ordinary reconstruction algorithms of scanning ptychography in single-shot ptychography. Still, in a sharp contrast to scanning ptychography, here the multiple diffraction patterns from all the beams actually interfere. Thus, when applying the ptychographic reconstruction algorithm using the diffraction patterns in the blocks, contributions to a diffraction pattern from beams that originated from other pinholes are regarded as noise. In order to estimate the resolution of single-shot ptychography, we calculate the maximum (i.e., the cutoff) spatial frequency that can be detected under the blocks assumption. The side of each square block is bM, where $M=f_2/f_1$ is the magnification of the 4f system (the side of the CCD should therefore be larger than $L_{\mathrm{CCD}}>\mathrm{NbM}$). Thus, taking into account the coordinate transformation $\mathit{\nu }=\frac{\mathbf{r}}{{\lambda f}_2}$, the cutoff frequency is ${\mathit{\nu }}_{\max }=\frac{\mathit{bM}}{2{\lambda f}_2}=\frac{\mathrm{b}}{2{\lambda f}_1}$. Importantly, this means that in some spectral regions (e.g., the visible), the resolution in single-shot ptychography can get close to the Abbe resolution limit by using $f_1\sim b$. Moreover, ptychography was demonstrated to yield a somewhat higher resolution than the measured bandwidth in a single diffraction pattern (which in our case is $\frac{1}{2{\mathit{\nu }}_{\max }}$) [26]. In the limit of a very large $N$, the field of view (with the dimension of length) of single-shot ptychography is given by $\mathrm{FOV}=\mathrm{NR}=\mathrm{Nbd}/f_1$. It is also instructive to take the product of ${\mathit{\nu }}_{\max }$ and the FOV (which is proportional to the space-bandwidth product [20]): $\mathrm{SBP}\propto \frac{{\mathrm{Nb}}^{2}d}{\lambda f_1^2}$. Single-shot ptychography includes many parameters (e.g., $N$, $b$, $d$, $f_1$, $f_2$, and the location of the pinhole array) that can be used for optimization according to specific requirements. For example, using the following parameters that are available in the visible spectral region (by using objective lenses), $N=10$, $b=1.5\mathrm{mm}$, $d=0.5\mathrm{mm}$, “$f_1=1.5\mathrm{mm}$ and $f_2=1.5\mathrm{mm}$, we get that the cutoff frequency is ${\mathit{\nu }}_{\max }=\frac{1}{2\lambda }$, yielding the diffraction-limited resolution, and $\mathrm{FOV}=5\mathrm{mm}$. Finally, it is worth noting that combining single-shot geometry with scanning is clearly possible and may be used for further optimization.

 

Fig. 2. Numerical demonstration of single-shot ptychography using the scheme in Fig. 1(b) with $f_1=f_2=75\mathrm{mm}$, $d=18.75\mathrm{mm}$, $b=1.4\mathrm{mm}$, $D=25\mathrm{\mu m}$, $N=12$, and $\lambda =405\mathrm{nm}$. Amplitude (a) and phase (b) of the original object. Black dashed square marks the region confining the centers of the 144 illuminating probe beams. (c) Amplitude of the incident field, i.e., the interference of the 144 probe beams, on the object plane. (d) Measured diffraction pattern from the object. 144 diffraction patterns are clearly distinguishable. (e) Zoom in on plot d, showing nine diffraction patterns. (f) Reconstructed probe beam (note the different scale in this plot with respect to the other plots). Reconstructed amplitude (g) and phase (h). Reconstruction is good within the illuminating region (the black square) and degrades outside of it.

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3. RESULTS

A. Single-Shot Ptychography: Numerical Demonstration

In this section, we demonstrate an example of single-shot ptychography numerically, using the system depicted in Fig. 1(c) with the following parameters: $f_1=f_2=75\mathrm{mm}$, $d=18.75\mathrm{mm}$, $b=1.4\mathrm{mm}$, $D=25\mathrm{\mu m}$, $N=12$, and $\lambda =405\mathrm{nm}$. The Fresnel–Kirchhoff diffraction formula [27] is used for free propagation, while the lenses are modeled by phase elements: $\exp (i\pi \frac{x^2+y^2}{\lambda f})$. The amplitude and phase transmissions of the object are displayed in Figs. 2(a) and 2(b), respectively. The black square marks the region confining the centers of the 144 illuminating probe beams. Figure 2(c) shows the amplitude of the incident field, i.e., the interference of all the probe beams, on the object plane. We first calculated the intensity pattern at the exit of the 4f system without the object and added 35 dB of white Gaussian noise. This measurement is used for locating the centers of the blocks (i.e., the $\mathit{\nu }=0$ in each block). We then calculated the intensity pattern at the exit of the 4f system with the object and added 35 dB of white Gaussian noise [Figs. 2(d) and 2(e)]. Clearly, the $N^2$ diffraction patterns are well distinguished. After dividing the measured intensity pattern into $N^2$ separate diffraction patterns, we applied the extended ptychographical iterative engine (ePIE) reconstruction algorithm [18] and reconstructed the probe beam [Fig. 2(f)] as well as the object [amplitude in Fig. 2(g) and phase in Fig. 2(h)]. As shown, the reconstruction is good within the illuminating region (the black square), but degrades outside of it. This numerical example indicates that single-shot ptychography is applicable even with complex objects.

B. Single-Shot Ptychography: Experimental Demonstration

Next, we demonstrate single-shot ptychography experimentally. The experimental setup is shown schematically in Fig. 3(a). A sub-millwatt diode laser ($\lambda =405\mathrm{nm}$) is spatially filtered and collimated by a $10\times$ objective, a 25 μm pinhole, and lens L1 with a focal distance of 50 mm. The spatially coherent light illuminates an $N^2=49$ square array with $b=1.4\mathrm{mm}$ and $D=75\mathrm{\mu m}$ circular pinholes that is located at the input plane of a 4f system with $f_1=f_2=75\mathrm{mm}(M=1)$. The object is placed at $d=18.75\mathrm{mm}$ before the Fourier plane of the 4f system. We measured the intensity patterns with [Fig. 3(b)] and without [Fig. 3(c)] the object. The overall acquisition time of the data in Fig. 3(b) was 180 ms (limited by the CCD minimal acquisition time).

 

Fig. 3. Experimental single-shot ptychography. (a) A scheme of the setup. Laser diode ($\lambda =405\mathrm{nm}$ and 1 mW power) is spatially filtered and collimated. The beam illuminates a $7\times 7$ square array of pinholes with $b=1.4\mathrm{mm}$ and $D=75\mathrm{\mu m}$ that is located at the input face of a symmetric 4f system with $f=75\mathrm{mm}$. The object is located 18.75 mm before the Fourier plane of the 4f system. The CCD is located at the output face of the 4f system. Measured diffraction patterns with (b) and without (c) the object.

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The 49 diffraction patterns are clearly distinguishable in both plots. We use the calibration measurement (i.e., the one without the object) for locating the centers of the blocks and for the power normalization of each block in Fig. 3(b) (because the laser illumination on the pinhole array is not completely uniform).

Next, we apply the ePIE reconstruction algorithm and obtain the results shown in Fig. 4. Figures 4(a) and 4(b) display the reconstructed amplitude and phase of the object, respectively, while Fig. 4(c) shows the reconstructed probe beam. For comparison, Fig. 4(d) shows an image of the object using ordinary microscope with $10\times$ magnification.

 

Fig. 4. Experimental demonstration of single-shot ptychography. Reconstructed amplitude (a) and phase (b) of the object and amplitude (c) of the probe beam from measured diffraction patterns [Figs. 3(b) and 3(c)]. (d) An image of the object measured by conventional microscope with $10\times$ magnification.

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The agreement between the two images [Figs. 4(a) and 4(b)] is good. The smallest features in this image—two opaque 40 μm squares at the bottom left—are clearly observable in the reconstructed image. This section demonstrated the applicability of single-shot ptychography in real experiments.

4. DISCUSSION

A. More Schemes for Single-Shot Ptychography

In this section, we propose and discuss more exemplary schemes for single-shot ptychography, as well as propose single-shot Fourier ptychography. In the scheme proposed in Fig. 1(b), the pinhole array was illuminated by a coherent wave. It is also possible to illuminate the array by a partially coherent beam with a coherence length much longer than the pinhole diameter and much smaller than the distance between consecutive pinholes. In this case (partially coherent single-shot ptychography), the detected intensity pattern is given by

 

Fig. 5. More schemes of single-shot ptychography. (a) Single-shot ptychography using LED array. (b) Single-shot ptychography in reflection mode. (c) Single-shot ptychography using microlens array.

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B. Scheme for Single-Shot Fourier Ptychography

Finally, we propose a scheme for single-shot Fourier ptychography. In Fourier ptychography [30,32], a set of low-resolution, high-FOV images are recorded, where the probe beam in each measurement propagates in a different direction. To date, Fourier ptychography is based on scanning the propagation angle (i.e., the scanning is in the Fourier domain). A ptychographic algorithm then reconstructs a high-resolution, large-FOV image of the object [30]. Figure 6 displays the setup for single-shot Fourier ptychography (exactly the same setup was used before, with the motivation to produce multiple identical images for image processing applications [33]). It consists of a 4f system and MLAs, with focal distance $f_{\mathrm{MLA}}$, before the input and after the output of the 4f system. The first MLA produces an array of localized spots at the input face of the 4f system. The second MLA transfers the multiple Fourier images at the output face of the 4f system to an array of non-identical images of the object (note that the object is located in the Fourier plane of the 4f system, i.e., $d=0$). As shown in the figure, each image is probed by a beam that propagates at a different angle. Similar to single-shot ptychography, the detected intensity pattern on the CCD can be divided into blocks, where each block approximately corresponds to an image of the object that was obtained by a probe beam that propagated at a different angle. These blocks form the required set of measurements for Fourier ptychography.

 

Fig. 6. Schematic setup for single-shot Fourier ptychography.

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5. CONCLUSION

In summary, we proposed, analyzed, and demonstrated, numerically and experimentally, single-shot ptychography. Several exemplary schemes for single-shot ptychography and Fourier ptychography were presented (of course, combinations of these schemes can give rise to more possibilities and advantages). Single-shot ptychographic microscopes should allow for the retrieval of the complex (i.e., amplitude and phase) structure of label-free objects within a very short exposure time with high (diffraction-limited) resolution and a large field of view. Notably, single-shot ptychographic microscopes can be obtained by slightly modifying commercial confocal microscopes (e.g., adding a pinhole array). Remarkably, single-shot ptychography can be implemented in every spectral region and for every type of wave for which lenses (or other focusing elements) are accessible (in some spectral regions it can be useful to replace the second lens with free propagation). These new capabilities will surely open up many new opportunities in optical and electronic microscopy. Moreover, it is worth noting some directions that can further improve and extend the scope of single-shot ptychographic microscopes: (1) improved reconstruction algorithms should be able to retrieve information that is currently lost by our assumption that the detected intensity pattern consists of non-interacting blocks [i.e., information that is contained in the intensity patterns of Eqs. (2) and (3), but not in Eq. (1)]; (2) a very exciting direction is to utilize structure-based prior knowledge of the object in order to enhance the resolution of single-shot ptychography algorithmically [34–36]. This direction may yield ultrafast sub-wavelength imaging; (3) in this work, we explored single-shot ptychography using monochromatic radiation. We believe that the method can be extended to broadband [37] and multi-spectral ptychography [23], which, for example, will allow the use of femtosecond laser pulses in single-shot ptychography; (4) the combination of single-shot ptychography with recent developments that yielded three-dimensional ptychography [38] and three-dimensional Fourier ptychography [39] should be straightforward; and (5) finally, we note that it will be interesting to explore the influence of other types of pinhole arrays (lattice structure and shape of pinholes) on the performance of single-shot ptychography. For example, an array with diverse pinhole shapes should lead to improved robustness to noise and faster algorithmic convergence [40]. All in all, we believe that single-shot ptychography will lead to many new opportunities and discoveries in fast and ultrafast microscopy.