Saturday , June 19 2021

X-ray Fourier ptychography | Scientific Achievements


To a large extent, the performance of imaging systems is determined by their goals, which affect properties ranging from collection efficiency, power resolving, and image distortion. Such limitations can be solved by the so-called aperture synthesis, a technique used, for example, in radar, astronomy and, increasingly, microscopy. Here we apply X-ray imaging techniques and show how Fourier Phyography can be used in X-ray transmission microscopes to increase resolution, provide quantitative absorption and phase contrast, and adjust lens aberrations. We expect that such methods will find common and frequent applications alleviating a number of constraints imposed by X-ray optical elements offering an alternative approach to depict phase contrast and provide new possibilities for reducing radiation damage.


In addition to the lack of distortion, the ideal imaging system should have three typical features: high resolution, high contrast and quantitative representation of the sample. Achievable resolution is fundamentally related to the digital aperture (NA) of the target, which invariably acts as a low-pass filter. While essentially ideal lenses allow microscopes to work near the theoretical limits in the case of visible light, there are not as powerful x-ray imaging targets. Therefore, the resolution of X-ray microscopes differs considerably less favorably than a drastic reduction in wavelengths may imply, and in somewhat paradoxical, instruments operating on shorter wavelengths for which thicker and dense samples are translucent, typically give lower resolutions than those that work at longer X-ray wavelengths.

X-ray optics can be classified into three categories: diffraction optics such as Fresnel zones (FZPs), reflecting optics, i. E. mirrors and refractive optics such as composite refractive lenses (CRLs). Despite the recent progress made (1, 2), mirrors and refractive optics often introduce distortions that make use of them as extremely challenging. Thus, X-ray transmission microscopes (TXMs) most commonly use FZPs as targets that allow high NA and are less prone to aberrations, but at high X-rays, in particular, they have limited efficiency.

This has led to a great interest in "flat-screen" X-ray images where imaging is not achieved by optical elements, but by mathematical algorithms for "phasing out" (3). From this set of techniques, petitography has become particularly popular over the past decade (4). The method requires that overlapping subsections of the sample are illuminated with coherent beam and diffraction pictures to be acquired at each position. Then iterative algorithms are used to recover the complex representation of the sample and the illuminating wave field (57).

Recently, a similar approach has been demonstrated to depict lenses based on visible light. In the so-called Future Pythograph (8, 9), scanning the illumination direction effectively scans the spatial frequency content of the sample through the target aperture. The use of the same phase extraction techniques results in a combination of the resolving capability of high NA targets with large field visions typical of low-NA systems; furthermore, both quantitative absorption and phase contrast can be reconstructed simultaneously. Finally, similar to extracting the probe into a "standard" petition (5, 6), Fourier's Phytography is able to reconstruct the function of the pupil, which allows a number of aberrations to be reliably corrected (10).

All of these properties are highly desirable for X-ray microscopy, and in 2016 Simons and colleagues suggested the use of X-ray imagery (11). Yet, as far as we know, so far has not been reported the successful realization of X-ray Fourier ptychography.


Direct adaptation of the most commonly used Fourier Pithography approach requires a change in the direction of illumination while keeping any other change of illumination to a minimum. For this task, we took advantage of the special capacitor structure found in TXM, in which the standard FZP is replaced by a number of linear arrays, and in which the constant height of each subfield keeps the subfield shape in diffraction to a common "focus"12, 13). While this geometry is usually used to achieve a uniform illumination similar to the topha, with respect to the field of view, in order to change the direction of illumination, we have inserted a 20 μm aperture through which we could illuminate individual subfields individually. Moving the hole, thus choosing different subfields, actually acts as a translation of the spectrum into the Fourier space (Figure 1A). As a lens, we used FZP with the outermost bandwidth of 70 nm corresponding to the NA of the condenser lens. Using X-rays with 8.7 keV, i. with a wavelength of 0.14 nm, we created an Au test model with a structure height of 1.5 μm.

Fig. 1 X-Ray Fourier Phytography by scanning the direction of illumination.

(A.) Experimental setup sketch. By inserting a movable bore (not shown) near the fixed capacitor, a standard TXM setting can be modified for the footed dielectric as the change in direction of illumination can be achieved by selecting individual subfields of the capacitor. (B to EThe comparison of the flat-plane full-field image (D) and the X-ray Fourier (E) dimension highlights the image quality improvement and resolution. Tape on the scale, 2 μm. The last row of the test pattern shows a line width of 150 nm, which is enabled by both techniques. Yet, the innermost lines, though washed in the full field image, are dealt with in the Fourier Revolution (B) (respectively blue and red). Even smaller features with a width of 85 nm are seen in the Fourier Revolution (C). (FFourier ring correlation (FRC) between two subsets of the Fourier Pythographic Scan confirms the resolution improvement. The dashed line marks the relay boundary (RRL).

The resolution in this selected acquisition scheme remains limited by the combined HC of the condenser and the object; i. the frequency range over which the imaging system has some final answer does not increase fundamentally. In addition, this approach is quite sensitive to any structure in the illumination that can be caused by imperfections before the capacitor or by unwanted variations between the capacitor's subfield. However, as shown in Figure 1, the pitographic reconstruction of 142 acquisitions of 3 s allows for increased high frequency content compared to a total field TXM image obtained for 10 seconds in the absence of an aperture. As a total resolution metric, a Fourier Ring Correlation (FRC) is used between two independent Fourier measurement images to determine the reproducibility of the sample spectrum frequency (14). For this task, we split the scan of two sets of 71 acquisitions each, which we reconstructed independently. By choosing a threshold corresponding to an average information content of ½ bits per pixel, FRC shows an approximate resolution of 78 nm (Figure 1F), well maintained by linear scans (Figures 1, B and C). There is no indication that any image or reconstruction (Figures 1, D and E, or even individual FRC input images that are not shown) should be considered dose-limited, but note that due to an opening, the flow over time of the Pictograph scan is reduced by more than three orders of magnitude.

An alternate experimental conversion, also suggested in11), includes scanning the lens of the lens and a detector through the diffracted beam (Figure 2A)15). This approach not only avoids the complexity of keeping a constant illumination when changing the drop angle, but the maximum resolution is now limited by the maximum diffraction angle captured by scanning rather than by optical elements. The scanning range expands NA to the target, thus forming a synthetic lens and exceeding the NA for any available X-ray optics.

Fig. 2 X-ray Fourier ptychography by scanning a lens.

(A.) Experimental setup sketch. Both lenses and detectors are scanned perpendicular to the optical axis. (B) Reconstructed phase image of ASIC. Tape on the scale, 5 μm. (° C) The reconstruction resolution using FZP with the outermost zone width of 70 nm, a diameter of 100 μm and a scanning range of 80 μm was evaluated by means of a Fourier Ring Cortex (FRC) between two independent scans to 47 nm, i. , considerably lower than the 85 nm RRL (RRL) for a conventional TCM marked with the dashed line.

Flat-wave illumination is determined by a 30 μm hole near the sample, which in this case is an application-specific integrated circuit (ASIC) (16). Due to the limited size of the detector, the movement of the lens of the lens had to be matched by the positioning of the detector. In order to account for position uncertainty and variance variations for different directions, raw data frames obtained had to be aligned in the post-processing step (17). A scanning range of 80 μm for FZP corresponding to a synthetic lens with the outermost zone width of 39 nm and 180 μm in diameter was selected, thus improving the Rayleigh range from 85 to 47 nm.

The ability of Fourier to generate phase information in the absence of phase-shifting elements is shown in Fig. 2B. Not only the halo artefacts, a common Zernike phase contrast problem, are missing in the reconstructed image, but the contrast is quantitative, as can be confirmed in comparison with earlier, independent high resolution measurements of the same microelectronic sample (Figure 3 ). ) (16, 18). FRC gives an approximate X-ray resonance resolution of 47 nm (Figure 2C).

Fig Confirmation of quantity.

(A. and ° C) Reconstruction of the same ASIC, measured with a frieze periphery (A) and conventional pituitary (C) (16). Tape on the scale, 2 μm. The resolution resolution for (A) is 47 nm and that for (C) is 41 nm (16). (B) The X-ray X-ray X-ray reconstruction phase is compared to its conventional pictographic analogy. (° C) The cut lines of the two reconstructed phases show that the phase profiles are combined and thus quantitative reconstruction can be provided (red for the piographical reconstruction of Fourier and blue for the "conventional" petition).

The reconstructed pupil function shows a significant phase contribution, which can be interpreted as an aberration of the image system (Figure S3). They are conveniently expressed through the Chernike polynomials, Embedded image, Concerning Freedom of Gauge, Polynomials of Radial Degree n = 0 or 1 does not correspond to image degradation and can not be attributed uniquely to the student's function or sample spectrum. The contribution of Embedded imagehowever, indicates that the pattern of the image is not perfectly focused on the detector. Although this would reduce image quality and inhibit the quantitative interpretation of contrast in the standard TXM mode, Fourier X-ray reconstructions do not suffer from such adverse effects (Figure S4).


These measurements indicate that relatively simple modifications in the data capture strategy can be used in conventional TXMs to improve imaging capabilities in terms of resolution, contrast and resilience against image distortion. The experimental complications we encountered during these measurements with proof of the principle can mostly be considered quite easily. The most cumbersome are the limitations caused by detectors, i. their limited size, which requires them to move to lens scanning, slow CCD reporting or low flow requirements when interpolation of subpixel data is desired (19). However, faster and larger detectors are generally available and / or developed. Specialized optical elements and additional control over the X-ray source can increase the reliability and speed of light scanning. Along with further resistance to lighting fluctuations (20ptyhographic scan can be part of standard acquisition schemes unobtrusive to the experiment.

Fourier's Phyography allows high-resolution optics to be replaced by more efficient elements, even if they were themselves limited to lower SCs or subjected to aberrations (1, 21). The technique can be useful for in situ or operando measurements that require large working distances between the sample and objective lenses. Whenever there is a need for scanning, we do not see any fundamental obstacles that exclude the Fourier Pithography in order to improve images or modalities of the image (22) for multiple full-field X-ray microscopes. Given that the brightness of the available sources is expected to increase in the next years (23Fourier's piracy may be a promising way of taking advantage of the dramatically increased coherence. At the same time, limited coherence requirements (9) allow Fourier acquisition schemes also to be used in less brilliant sources, including laboratory systems. In terms of both development and use, we expect the availability of the technique to allow for choice between target microscopes (1, 12, 24, 25) and techniques based on distribution (26, 27) and be a valuable addition to existing imaging capabilities.


Experimental setup

The experiments were conducted in the cSAXS beam (X12SA) of the Swiss light source, Paul Scherrer Institut, Switzerland. During the experiments, the photon energy was chosen to be 8.7 keV with the bandwidth ΔE/E ≈ 2 × 10-4, Source size ~ 200 x 20 μm2 (horizontally × vertically) is approximately 34 m before the sample, resulting in a coherent patch of ~ 24 × 240 μm2 (horizontally × vertically).

Measuring Equipment I

To obtain a full TXM image field, the beam illuminates a 1 mm diameter condenser lens, a 50 μm subfield and a 70 nm outermost zone (Figure S1A) (13), resulting in a wavelength of 0.14 nm in NA of 10-3To block the direct beam, a central stop 150 μm is mounted at the same stage, near the condenser. An order sort aperture is introduced to limit sample light to diffraction on the first row of the capacitor. As a lens, the 100 μm diameter FZP and the outermost zone width of 70 nm corresponding to the NA of the condensation lens is 49.5 mm after the sample. Each optical element is mounted on a set of stepping motors to allow three-dimensional (3D) movements.

At a distance of 7.3 meters, behind a He-filled 7-meter flight tube, the photos were collected with a Photonic Science VHR Image Star X-ray camera based on a full-frame Kodak CCD with an optical resolution of 4 μm. This geometry resulted in an increase of 150 and an effective pixel size of 27 nm in the sample plane. From 3056 × 3056 pixels, we chose a reading resolution of 470 × 392 pixels, which corresponds to the highlighted area of ​​interest. For the full field image shown in Fig. 1D, 10 acquisitions with an exposure time of 1 s have been taken and summarized. To take into account the unevenness of the illumination or the detection efficiency, the images were divided by acquisitions without a sample.

To select single subfields for a Fourier Measurement Capacitor, place a 20 μm hole at about 80 mm before the condenser. Because the scan pattern corresponded to the subfield centers, the movement of the hole had to be carefully aligned with the capacitor. For this task we received a transmitter card by scanning the hole through the capacitor (Figure S1B). Assuming that the highest intensities occur in the center of the subfield, at each such local maximum we have acquired a 3 s frame. A total of 142 acquisitions were used to rebuild Fourier.


In order to reduce the artefacts in the raw data associated with optical camera connection (Figure S1C), acquisitions were intensely threshold and pixels below that threshold were replaced by interpolation of the nearest neighbor.

Реконструкцията е извършена с помощта на MATLAB-базиран пакет и високопроизводителни двигатели за реконструкция, написани на C ++, които са разработени от Coherent X-Ray Scattering Group, Paul Scherrer Institute, Швейцария. Алгоритъм на картата на разликите (5) се сближиха след 2000 повторения и последваха 150 повторения на максимизиране на вероятността (7). Реконструираният спектър се разпространява обратно, за да насочи пространството.

За да се оцени разделителната способност на изображението с помощта на FRC, данните се разделят на два разединени множества, които се анализират независимо. След като разделихме потока с две, сравнихме FRC с 1/2-битови критерий (14).

Измервателна техника II

Във втори експеримент, кондензаторната леща, централният ограничител и сортиращата апертура на поръчката бяха отстранени и заменени с 30 μm pinhole близо до равнината на обекта, определяйки зрителното поле. Същият обектив, както е използван в предишния експеримент, създава увеличено изображение на пробата на разстояние 7,9 m, където директният лъч е блокиран от 1.5 mm конична централна стомана от неръждаема стомана, монтирана на 13 μm Kapton фолио. Тъй като изискваната прецизност на обективните движения се увеличава с разделителната способност, обектът е монтиран на 3D пиезоелектричен етап с максимален обхват от 100 μm. Данните бяха събрани с помощта на прототип MOENCH детектор с размери 400 × 400 пиксела с физически размер на пиксела 25 μm (19, 28). Като се използват процепи, разположени на 22 m по-нагоре от пробата, потокът се намалява, така че еднофотонните събития се откриват в рамките на 0.25 ms придобивания, за да се даде възможност за интерполация на данни до размер на пиксела 6.25 μm. Между другото, използването на прорези води до приблизително същия поток в позицията на пробата, както при измервателната техника I, която използва по-малък отвор и кондензатор, и увеличава дължината на хоризонталната кохерентност до около 150 цт.

За точка на сканиране са взети 15 000 придобивания, което е довело до общо време на придобиване от около 15 s на точка, включително и режийни. FZP се сканира след спирала на Fermat (29) със среден размер на стъпката от 3,5 μm и диаметър 80 μm, което води до 522 точки. За да се проследи образното изображение, което се движи по време на сканирането на обективния обектив, детекторът се монтира на хексапод и синхронно се превежда с микронна точност в зависимост от увеличението 160.

За да се избегнат систематични артефакти в реконструкцията, които биха могли да нарушат оценката на разделителната способност от FRC, беше извършено второ сканиране на Fermat, със среден размер на стъпката от 5 μm за FZP, като същевременно се запази същият общ сканиращ обхват от 80 μm. При всяко от 255-те точки на сканиране са взети 30 000 придобивания от 0,25 ms. Тъй като можем да използваме две независими измервания, сравнихме FRC с 1-битния критерий (14).


Интерполацията на данните на детектора (19) въведени периодични артефакти (фиг. S2), които бяха адресирани чрез определяне на приноса на съответните честоти на нула. За да се отчете зависимостта на увеличението от посоката, трябваше да се осъществи подреждане. Тъй като тази регистрация е възпрепятствана от директния лъч, попадащ върху детектора, ние отхвърляме вътрешната част на сканиращия модел, т.е. в рамките на 25 μm от центъра. В случая на първото сканиране това намалява броя на точките за сканиране, използвани за анализ, до 362; в случая на второто сканиране, броят на точките за сканиране се намалява до 177.

Тъй като кадрите се различават в пространствено-честотното съдържание, подравняването (17) се извършва последователно на рамки със значително "припокриване" в честотната област. По-конкретно, ние дефинирахме поредица от изображения с бавно изменящо се честотно съдържание и подравняваме всеки кадър с предходния. Натрупването на грешки беше противодействано чрез повтаряне на процеса в обратен ред, което беше достатъчно за сближаване. В литературата могат да се намерят по-общи схеми за препозициониране за петикографски сканирания[напр[eg([напр[eg(20)]но не бяха намерени за по-нататъшно подобряване на качеството на реконструкцията. В случай на големи детектори или достатъчно прецизно движение на детектора, този процес на подравняване може да бъде сведен до търсене в афинитетна матрица и по този начин би бил независим от параметрите на сканиране.

Птихографската реконструкция на Фурие отнема 600 итерации на картата на разликата, за да се сближи (5). След това решението бе прецизирано с 1800 повторения на оптимизацията за вероятност (7), преди реконструираният спектър да се разпространи обратно в пространството. Реконструирани са два кохерентни режима за функцията на зеницата (30), първият от които е показан на фиг. S3. Отхвърлихме втория режим, съдържащ приблизително 33% от енергията и показвайки по същество никаква структура, като фон.

Това е статия с отворен достъп, разпространявана съгласно условията на лиценза Creative Commons Attribution, който позволява неограничено ползване, разпространение и възпроизвеждане във всеки носител, при условие че оригиналното произведение е правилно цитирано.

Благодарности: Благодарим на г-н Lebugle за производството на централната спирка, използвана в първия експеримент, и X. Donath за техническата поддръжка на линията cSAXS. финансиране: Признаваме финансовата подкрепа на швейцарската Национална научна фондация (SNF) [grant numbers 166304 (K.W.) and 153556 (J.I.)], Авторски вноски: K.W., A.D., A.Bo., и A.M. замисли експериментите с подкрепата на M.S. K.W., A.D., A.Bo, A.Be., J.I., и A.M. проведени експерименти. K.W., A.Be., и A.M. анализира данните с подкрепата на M.G.-S. K.W. и А.М. написал ръкописа с участието на всички автори. Конкуриращи се интереси: Авторите заявяват, че нямат конкурентни интереси. Наличност на данни и материали: Всички данни, необходими за оценка на заключенията в доклада, се съдържат в статията и / или в допълнителните материали. Изображенията, използвани за реконструкцията, са качени на [DOI: 10.5281/zenodo.2537927], Допълнителни данни, свързани с този документ, могат да бъдат поискани от авторите.

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