Przemyslaw Wachulak
@wat.edu.pl
Institute of Optoelectronics
Military University of Technology
Scopus Publications
Scopus Publications
Karol Adam Janulewicz, Tomasz Fok, Bartosz Bartosewicz, Andrzej Bartnik, Henryk Fiedorowicz, and Przemysław Wachulak
MDPI AG
This paper discusses the results of the micro-Raman analysis performed on paper-like graphene oxide (GO) samples consisting of many functionalised graphene layers and annealed at moderate temperatures (≤500 °C) under vacuum conditions (p ≃ 10−4 mbar). The analysis of the standalone samples revealed that the obtained material is characterised by a noticeable disorder level but still stays below the commonly accepted threshold of high or total disorder. GO formed in a simple way showed two spectral bands above 1650 cm−1 recorded very rarely or not at all and their origin has been discussed in detail. The results also confirmed the metastable character of multilayer GO after the annealing process at moderate temperatures as the C/O ratio was kept between 2 and 3 and the spectral features were stable within the annealing temperature range.
Wiktoria Kasprzycka, Wiktoria Szumigraj, Przemysław Wachulak, and Elżbieta Anna Trafny
Wiley
AbstractFluorescence microscopy is a powerful tool used in scientific and medical research, but it is inextricably linked to phototoxicity. Neglecting phototoxicity can lead to erroneous or inconclusive results. Recently, several reports have addressed this issue, but it is still underestimated by many researchers, even though it can lead to cell death. Phototoxicity can be reduced by appropriate microscopic techniques and carefully designed experiments. This review focuses on recent strategies to reduce phototoxicity in microscopic imaging of living cells and tissues. We describe digital image processing and new hardware solutions. We point out new modifications of microscopy methods and hope that this review will interest microscopy hardware engineers. Our aim is to underscore the challenges and potential solutions integral to the design of microscopy systems. Simultaneously, we intend to engage biologists, offering insight into the latest technological advancements in imaging that can enhance their understanding and practice.
Andrzej Bartnik, Karol Jach, Robert Świerczyński, Tomasz Fok, Łukasz Węgrzyński, Przemysław Wachulak, and Henryk Fiedorowicz
AIP Publishing
Laser plasma produced using a double-stream gas puff target is an intense source of soft x-ray (SXR) and extreme ultraviolet (EUV) radiation, however, without the harmful emission of debris associated with a solid target. Debris-free laser plasma x-ray and EUV sources have been applied in many various applications, including metrology, imaging in a nanoscale, tomography, processing materials, emission and absorption spectroscopy, laboratory astrophysics and astrochemistry, radiobiology, and radiochemistry. In this work, the results of the experimental and theoretical studies on the spatial and temporal evolution of laser plasma produced as a result of irradiation of an argon/helium gas puff target with laser pulses of 1.3 or 6 ns time duration generated with an Nd:YAG laser system are presented. Imaging and spectral measurements of SXR emission from the plasma, created in the double-stream gas puff target, have been performed with the use of an x-ray streak camera. The analysis of the results of spectral measurements, supported by numerical simulations of plasma x-ray emission, allowed the estimation of the plasma electron temperature and its changes over time. Experimental data were compared with the results of theoretical studies performed using a computer model of plasma hydrodynamics. It was shown that plasma expansion is fast enough to reduce the plasma density in the laser focus area during the laser–plasma interaction.
Łukasz Wȩgrzyński, Tomasz Fok, Mirosław Szczurek, Andrzej Bartnik, Przemysław Wachulak, Karol Adam Janulewicz, and Chul Min Kim
Springer Science and Business Media LLC
The advent of very intense, short-pulse lasers changed dramatically the field of laser-matter interaction. The interest in new forms of target brought clusters to the forefront as a very promising and unique medium. Here, extremely large Xe or CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\text{CO}}_2$$\\end{document} clusters embedded in an abundance of smaller ones were formed and subsequently irradiated by intense laser pulses (≲1019W/cm2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lesssim 10^{19}\\,{\\text{W}}/{\\text{cm}}^2$$\\end{document}) to demonstrate the specific character of such a medium and its suitability for intense laser-matter interaction. Emission of short-wavelength radiation quantified in the spectral range known as the “water window” constituted the reference for the target performance. The clusters were formed in a double-stream gas-puff equipped with a gas reservoir cooled down to 245 K and backed by a low-to-moderate pressure of ≤12bar\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le 12 \\,{\\text{bar}}$$\\end{document}. The obtained atomic/molecular compounds, mostly of an irregular shape and of an average diameter ≃2.4±0.5μm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\simeq 2.4 \\pm 0.5 \\,\\upmu {\\text{m}}$$\\end{document} in the case of Xe and ≃2.0±0.4μm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\simeq 2.0\\pm 0.4 \\,\\upmu {\\text{m}}$$\\end{document} for CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\text{CO}}_2$$\\end{document}, have been imaged by optical microscopy while the size was determined by the standard scattering method. A scenario explaining the formation of such clusters has been proposed. The interaction results show that a photon/particle source can be copious when based on the developed target technology with an abundance of the extremely large clusters. These clusters enable also the experiments with a single-cluster target.
Joanna Czwartos, Angelika Zaszczyńska, Agata Nowak-Stępniowska, Tomasz Fok, Bogusław Budner, Andrzej Bartnik, Przemysław Wachulak, Dorota Kołbuk, Paweł Sajkiewicz, and Henryk Fiedorowicz
Elsevier BV
Karol A. Janulewicz, Łukasz Węgrzyński, Tomasz Fok, Andrzej Bartnik, Henryk Fiedorowicz, Sławomir Skruszewicz, Martin Wünsche, Erich Eckner, Silvio Fuchs, Julius Reinhard,et al.
Optica Publishing Group
The development of the broad-bandwidth photon sources emitting in the soft X-ray range has attracted great attention for a long time due to the possible applications in high-resolution spectroscopy, nano-metrology, and material sciences. A high photon flux accompanied by a broad, smooth spectrum is favored for the applications such as near-edge X-ray absorption fine structure (NEXAFS), extended X-ray absorption fine structure (EXAFS), or XUV/X-ray coherence tomography (XCT). So far, either large-scale facilities or technologically challenging systems providing only limited photon flux in a single shot dominate the suitable sources. Here, we present a soft, broad-band (1.5 nm - 10.7 nm) soft X-ray source. The source is based on the interaction of very intense laser pulses with a target formed by a cluster mixture. A photon yield of 2.4 × 1014 photons/pulse into 4π (full space) was achieved with a medium containing Xe clusters of moderate-size mixed with a substantial amount of extremely large ones. It is shown that such a cluster mixture enhances the photon yield in the soft X-ray range by roughly one order of magnitude. The size of the resulting source is not beneficial (≤500 µm but this deficit is compensated by a specific spectral structure of its emission fulfilling the specific needs of the spectroscopic (broad spectrum and high signal dynamics) and metrological applications (broad and smoothed spectrum enabling a sub-nanometer resolution limit for XCT).
Andrzej Bartnik, Karol Jach, Robert Świerczyński, Mateusz Majszyk, Tomasz Fok, Łukasz Węgrzyński, Przemysław Wachulak, and Henryk Fiedorowicz
AIP Publishing
In this work, the time development of plasmas produced by interaction of laser pulses, with a nitrogen gas, was investigated. The interaction took place inside a small portion of dense nitrogen gas injected temporarily into a chamber filled with the gas under low pressure. High-temperature plasmas produced directly by the laser pulse were a source of soft x rays and charged particles, ionizing and exciting the surrounding gas. In this way, low-temperature plasmas were produced. The formation of high-temperature plasmas was studied using soft x-ray spectroscopy and x-ray streak imaging. Low-temperature plasmas formed at various distances from the laser focus were investigated using an optical streak camera. Interpretation of the experimental data was supported by numerical modeling of the laser-produced plasma hydrodynamics. It was concluded that depending on the distance from the focal spot, the formation of the low-temperature plasmas was dominated by ion streams or by soft x-ray radiation.
Agata Nowak-Stępniowska, Wiktoria Kasprzycka, Paulina Natalia Osuchowska, Elżbieta Anna Trafny, Andrzej Bartnik, Henryk Fiedorowicz, and Przemysław Wachulak
MDPI AG
Soft X-ray microscopy is a powerful technique for imaging cells with nanometer resolution in their native state without chemical fixation, staining, or sectioning. The studies performed in several laboratories have demonstrated the potential of applying this technique for imaging the internal structures of intact cells. However, it is currently used mainly on synchrotrons with restricted access. Moreover, the operation of these instruments and the associated sample-preparation protocols require interdisciplinary and highly specialized personnel, limiting their wide application in practice. This is why soft X-ray microscopy is not commonly used in biological laboratories as an imaging tool. Thus, a laboratory-based and user-friendly soft X-ray contact microscope would facilitate the work of biologists. A compact, desk-top laboratory setup for soft X-ray contact microscopy (SXCM) based on a laser-plasma soft X-ray source, which can be used in any biological laboratory, together with several applications for biological imaging, are described. Moreover, the perspectives of the correlation of SXCM with other super-resolution imaging techniques based on the current literature are discussed.
T. Fok, K.A. Janulewicz, P. Wachulak, A. Bartnik, B. Nasiłowska, J. Kostecki, B. Budner, H. Fiedorowicz, M. Djas, P. Kuźmiuk,et al.
Elsevier BV
A. Arikkatt, L. Węgrzyński, A. Bartnik, H. Fiedorowicz and P. Wachulak
Optical coherence tomography (OCT) with the use of soft x-rays (SXR) and extreme ultraviolet (EUV) has been recently demonstrated [1-3]. This new imaging technique, termed XCT, enables the obtaining of cross-sectional and tomographic images of samples with nanometer spatial resolution. The article presents a newly developed laboratory system for XCT using a compact laser plasma light source operating in the SXR and EUV spectral ranges. Schematic od the setup and its view are shown in Fig. 1 and 2, respectively.
Antony Jose Arikkatt, Przemysław Wachulak, Henryk Fiedorowicz, Andrzej Bartnik, Piotr Nyga, and Karol Janulewicz
SPIE
Optical coherence tomography (OCT) is a well-established non-destructive imaging technique providing high-resolution cross-sectional views of objects. The axial resolution of OCT is limited to single micrometers when using infrared and optical wavelengths. Recently, optical coherence tomography using broadband soft X-rays and extreme ultraviolet has been proposed to improve axial resolution. This variant of OCT, known as X-ray coherent tomography (XCT), enables axial resolution of a few nanometers. The paper presents OCT with the use of extreme ultraviolet in the wavelength range of 10–20 nm generated using a compact laser-produced plasma (LPP) source based on a double-stream gas puff target. The use of a gas puff target enables efficient extreme ultraviolet emission without producing target debris by laser ablation from a solid target. Two axisymmetric ellipsoidal grazing-incidence mirrors were used to focus the radiation from the source to the sample, and then to focus the radiation reflected from the sample to the spectrometer. We also present measurements on Silver/Zirconium multilayer periodic structures with a periodicity of 60 nm.
Martyna Wardzińska, Henryk Fiedorowicz, Andrzej Bartnik, Tomasz Fok, Przemysław Wachulak, Łukasz Węgrzyński, Karolina Szamota-Leandersson, Jacek Krzywiński, Paweł Krawczyk, Mateusz Majszyk,et al.
SPIE
The PolFEL free electron laser, currently under construction at the National Centre for Nuclear Research in Poland, will generate a beam of coherent electromagnetic radiation in the ultraviolet (UV) spectral range with a wavelength of about 150 nm to 300 nm, in the form of several hundred fs pulses, energy up to 50 μJ, and repetition rate of 50 kHz. Vacuum ultraviolet (VUV) radiation beam in the wavelength range from 50 nm to 100 nm will be obtained by selecting the third harmonic using an absorption filter. The optical system of the UV/VUV beamline consists of two plane M1 and M2 mirrors and one focusing ellipsoidal M3 mirror. The radiation produced in the laser hits on the M1 mirror at a grazing incidence angle of 5°. After reflection from the M1 mirror, the beam falls on the M2 mirror at an angle of 17°, which directs the beam to the ellipsoidal M3 mirror, focusing the beam at the image plane at the second focal point of the ellipsoid. The M1 mirror is placed behind the 3 m-thick concrete wall in a hutch separated from the experimental hall by a 1.6 m-thick concrete wall. The optical properties of the beamline were tested by ray-tracing simulations using the RAY-UI software, the results of which are presented in the paper.
Michał P. Nowak, Bogusz Stępak, Mateusz Pielach, Yuriy Stepanenko, Tomasz Wojciechowski, Bartosz Bartosewicz, Urszula Chodorow, Przemysław Wachulak, and Piotr Nyga
SPIE
Plasmonic nanostructures can efficiently absorb and scatter light, and thus exhibit vibrant colors. Colors originate from the excitation of plasmon resonances – collective oscillations of free electrons in metallic nanoparticles. Semicontinuous metal films (SMF) are a special type of disordered plasmonic structures that can be mass-produced in a rather inexpensive physical vapor deposition process. SMFs are comprised of random in their nature nano-island structures of various sizes and shapes resonating at different wavelengths. When irradiated with high-intensity laser radiation, the nanostructures can be locally modified. Spatially local restructuring originates from highly localized SMF absorption of light in hotspots, regions of a high local electric field. Thus, the optical response of the film can be altered in a spectrally selective manner. Hence, locally different colors are obtained. In this work, we demonstrate the generation of structural plasmonic colors through femtosecond laser-induced modification of thin semicontinuous aluminum films deposited on an aluminum mirror coated with an isolator layer. The structures show vivid colors in reflection. The effects of laser parameters on final color are explored. In addition, we report laser-induced control of diffuse reflectance of aluminum SMF structures. Self-passivation effect of aluminum results in the long-term stability of generated colors.
Tomasz Fok, Przemysław Wachulak, Łukasz Węgrzyński, Andrzej Bartnik, Michał Nowak, Piotr Nyga, Jerzy Kostecki, Barbara Nasiłowska, Wojciech Skrzeczanowski, Rafał Pietruszka,et al.
MDPI AG
A near 1-keV photons from the Xe/He plasma produced by the interaction of laser beam with a double stream gas puff target were employed for studies of L absorption edges of period 4 transitional metals with atomic number Z from 26 to 30. The dual-channel, compact NEXAFS system was employed for the acquisition of the absorption spectra. L1–3 absorption edges of the samples were identified in transmission mode using broadband emission from the Xe/He plasma to show the applicability of such source and measurement system to the NEXAFS studies of the transition metals, including magnetic materials.
A Bartnik, W Skrzeczanowski, P Wachulak, T Fok, Ł Węgrzyński, M Szczurek, and H Fiedorowicz
IOP Publishing
Abstract In this work, low-temperature plasmas, induced in a gaseous CO2 by intense extreme ultraviolet (EUV) pulses were investigated with a purpose to determine their ionic/molecular composition and the resulting, potentially reactive species. Two laser-produced plasma EUV sources based on a xenon gas puff target were used to irradiate and ionize of the CO2 gas. The sources, driven by Nd:YAG lasers of different parameters, delivered EUV beams created using reflective, focusing collectors. The CO2-based, low-temperature plasmas induced using both systems, emitted radiation in a wide wavelength range, from vacuum ultraviolet (VUV) to visible light (VIS). The radiation was measured using spectrometers and a streak camera operating in these spectral ranges. In the VUV range, multiple emission lines corresponding to ionic and atomic species together with the CO molecular bands were acquired. Spectra from the UV–VIS range were mainly composed of the CO2 + molecular bands. Numerical simulations of the molecular spectra allowed us to estimate rotational and vibrational temperatures of the EUV induced plasmas. As could be expected, plasmas created in both experimental systems were characterized by different temperatures and intensity ratios of the ionic–atomic spectral lines. The spatio-temporal measurements performed using the streak camera indicated a few times longer lifetime of the EUV induced plasmas, compared to the driving, EUV pulses.
Joanna Czwartos, Bogusław Budner, Andrzej Bartnik, Przemysław Wachulak, Beata A. Butruk-Raszeja, Adam Lech, Tomasz Ciach, and Henryk Fiedorowicz
MDPI AG
Polyetheretherketone (PEEK), due to its excellent mechanical and physico-chemical parameters, is an attractive substitute for hard tissues in orthopedic applications. However, PEEK is hydrophobic and lacks surface-active functional groups promoting cell adhesion. Therefore, the PEEK surface must be modified in order to improve its cytocompatibility. In this work, extreme ultraviolet (EUV) radiation and two low-temperature, EUV induced, oxygen and nitrogen plasmas were used for surface modification of polyetheretherketone. Polymer samples were irradiated with 100, 150, and 200 pulses at a 10 Hz repetition rate. The physical and chemical properties of EUV and plasma modified PEEK surfaces, such as changes of the surface topography, chemical composition, and wettability, were examined using atomic force microscopy (AFM), scanning electron microscopy (SEM), X-ray photoelectron spectroscopy (XPS), and goniometry. The human osteoblast-like MG63 cells were used for the analysis of cell viability and cell adhesion on all modified PEEK surfaces. EUV radiation and two types of plasma treatment led to significant changes in surface topography of PEEK, increasing surface roughness and formation of conical structures. Additionally, significant changes in the chemical composition were found and were manifested with the appearance of new functional groups, incorporation of nitrogen atoms up to ~12.3 at.% (when modified in the presence of nitrogen), and doubling the oxygen content up to ~25.7 at.% (when modified in the presence of oxygen), compared to non-modified PEEK. All chemically and physically changed surfaces demonstrated cyto-compatible and non-cytotoxic properties, an enhancement of MG63 cell adhesion was also observed.
Paulina Natalia Osuchowska, Przemysław Wachulak, Wiktoria Kasprzycka, Agata Nowak-Stępniowska, Maciej Wakuła, Andrzej Bartnik, Henryk Fiedorowicz, and Elżbieta Anna Trafny
MDPI AG
Understanding cancer cell adhesion could help to diminish tumor progression and metastasis. Adhesion mechanisms are currently the main therapeutic target of TNBC-resistant cells. This work shows the distribution and size of adhesive complexes determined with a common fluorescence microscopy technique and soft X-ray contact microscopy (SXCM). The results presented here demonstrate the potential of applying SXCM for imaging cell protrusions with high resolution when the cells are still alive in a physiological buffer. The possibility to observe the internal components of cells at a pristine and hydrated state with nanometer resolution distinguishes SXCM from the other more commonly used techniques for cell imaging. Thus, SXCM can be a promising technique for investigating the adhesion and organization of the actin cytoskeleton in cancer cells.
Przemysław Wachulak, Tomasz Fok, Łukasz Węgrzyński, Andrzej Bartnik, Piotr Nyga, Karol Janulewicz, and Henryk Fiedorowicz
The Optical Society
Alfio TORRISI, Przemysław WACHULAK, and Lorenzo TORRISI
IOP Publishing
S. Skruszewicz, S. Fuchs, J. J. Abel, J. Nathanael, J. Reinhard, C. Rödel, F. Wiesner, M. Wünsche, P. Wachulak, A. Bartnik,et al.
Springer Science and Business Media LLC
AbstractWe present an overview of recent results on optical coherence tomography with the use of extreme ultraviolet and soft X-ray radiation (XCT). XCT is a cross-sectional imaging method that has emerged as a derivative of optical coherence tomography (OCT). In contrast to OCT, which typically uses near-infrared light, XCT utilizes broad bandwidth extreme ultraviolet (XUV) and soft X-ray (SXR) radiation (Fuchs et al in Sci Rep 6:20658, 2016). As in OCT, XCT’s axial resolution only scales with the coherence length of the light source. Thus, an axial resolution down to the nanometer range can be achieved. This is an improvement of up to three orders of magnitude in comparison to OCT. XCT measures the reflected spectrum in a common-path interferometric setup to retrieve the axial structure of nanometer-sized samples. The technique has been demonstrated with broad bandwidth XUV/SXR radiation from synchrotron facilities and recently with compact laboratory-based laser-driven sources. Axial resolutions down to 2.2 nm have been achieved experimentally. XCT has potential applications in three-dimensional imaging of silicon-based semiconductors, lithography masks, and layered structures like XUV mirrors and solar cells.
John T Sheridan, Raymond K Kostuk, Antonio Fimia Gil, Y Wang, W Lu, H Zhong, Y Tomita, C Neipp, J Francés, S Gallego,et al.
IOP Publishing
Adam Lech, Beata A. Butruk-Raszeja, Tomasz Ciach, Krystyna Lawniczak-Jablonska, Piotr Kuzmiuk, Andrzej Bartnik, Przemyslaw Wachulak, and Henryk Fiedorowicz
MDPI AG
Recently, extreme ultraviolet (EUV) radiation has been increasingly used to modify polymers. Properties such as the extremely short absorption lengths in polymers and the very strong interaction of EUV photons with materials may play a key role in achieving new biomaterials. The purpose of the study was to examine the impact of EUV radiation on cell adhesion to the surface of modified polymers that are widely used in medicine: poly(tetrafluoroethylene) (PTFE), poly (vinylidene fluoride) (PVDF), and poly-L-(lactic acid) (PLLA). After EUV surface modification, which has been performed using a home-made laboratory system, changes in surface wettability, morphology, chemical composition and cell adhesion polymers were analyzed. For each of the three polymers, the EUV radiation differently effects the process of endothelial cell adhesion, dependent of the parameters applied in the modification process. In the case of PVDF and PTFE, higher cell number and cellular coverage were obtained after EUV radiation with oxygen. In the case of PLLA, better results were obtained for EUV modification with nitrogen. For all three polymers tested, significant improvements in endothelial cell adhesion after EUV modification have been demonstrated.
Alfio Torrisi, Przemysław W. Wachulak, Andrzej Bartnik, Łukasz Węgrzyński, Tomasz Fok, and Henryk Fiedorowicz
MDPI AG
Over the last decades, remarkable efforts have been made to improve the resolution in photon-based microscopes. The employment of compact sources based on table-top laser-produced soft X-ray (SXR) in the “water window” spectral range (λ = 2.3–4.4 nm) and extreme ultraviolet (EUV) plasma allowed to overcome the limitations imposed by large facilities, such as synchrotrons and X-ray free electron lasers (XFEL), because of their high complexity, costs, and limited user access. A laser-plasma double stream gas-puff target source represents a powerful tool for microscopy operating in transmission mode, significantly improving the spatial resolution into the nanometric scale, comparing to the traditional visible light (optical) microscopes. Such an approach allows generating the plasma efficiently, without debris, providing a high flux of EUV and SXR photons. In this review, we present the development and optimization of desktop imaging systems: a EUV and an SXR full field microscope, allowing to achieve a sub-50 nm spatial resolution with short exposure time and an SXR contact microscope, capable to resolve internal structures in a thin layer of sensitive photoresist. Details about the source, as well as imaging results for biological applications, will be presented and discussed.
P. Wachulak, A. Bartnik, M. Duda, T. Fok, Ł. Węgrzyński, D. Adjei, M. Ayele, L. Vysin, and H. Fiedorowicz
OSA