Tight focusing of azimuthally polarized optical vortex produced by subwavelength grating

Focusing of an optical vortex with sectorial azimuthal polarization is numerically investigated. An effect of sector numbers on the results of focusing is estimated. It is shown that the focal spot produced by a beam with six sectors does not differ from the ideally azimuthally polarized optical vortex; a difference in the focal spot diameter does not exceed 0.001 of the wavelength. For a four-sectoral beam, the difference does not exceed 0.028 of the wavelength. We have investigated a four-Sector transmission Polarization Converter for a wavelength of 633 nm, that enables the conversion of a linearly polarized incident beam into a mixture of linearly and azimuthally polarized beams. It was experimentally shown that light propagated through the four-Sector transmission Polarization Converter and focused by Fresnel zone plate with a focal length of 532 nm produces focal spot with diameters 0.46 and 0.57 of wavelength.Publisher PD

Dependence of the γ³He → pd cross sections on the nuclear wave functions in the giant resonance region

The differential cross section and the asymmetry coefficient for the two-body ³He break up by linearly polarized photons are calculated with the wave functions for Reid soft core, Paris and Bonn potentials

Vortex-free laser beam with an orbital angular momentum

We show that if one cylindrical lens is placed in the Gaussian beam waist and another cylindrical lens is placed at some distance from the first one and rotated by some angle, then the laser beam after the second lens has an orbital angular momentum (OAM). An explicit analytical expression for the OAM of such a beam is obtained. Depending on the inter-lens distance, the OAM can be positive, negative, or zero. Such a laser beam has no isolated intensity nulls with a singular phase and it is not an optical vortex, but has an OAM. By choosing the radius of the beam waist of the source Gaussian beam, the focal lengths of the lenses and the distance between them, it is possible to generate a vortex-free laser beam equivalent to an optical vortex with a topological charge of several hundreds.This work was funded by the Russian Science Foundation grant # 17-19-0118

Calculation of the cross section for the reaction γ³He → pd at intermediate photon energies

The differential cross section and the asymmetry coefficient for the two-body photodisinteration of ³He by linearly polarized photons are calculated with wave functions for Bonn potential. Dependences of the observables on the components of the ³He wave function with the orbital angular momenta L, l ³ 1 are studied at photon energies Eg up to 300 MeV

Controlling the orbital angular momentum of Gaussian vortices by shifting the point of phase singularity

A simple formula is obtained to describe the normalized orbital angular momentum (OAM) of a Gaussian beam after passing through a shifter spiral phase plate (SPP). The formula shows that while being equal to the topological charge at the zero off-axis shift, the OAM becomes fractional with increasing shift and it is tending to zero exponentially. Analytic expressions of the complex amplitude of the Gaussian beam having passed through the off-axis SPP show that as the beam propagates, the isolated intensity null moves from the initial point defined by the vector of the SPP's center shift along a straight line perpendicular to the said vector. Using a liquid crystal light modulator, crescent-shaped beams are experimentally generated.This work was supported by the Federal Agency of Scientific Organizations (agreement No 007-ГЗ/Ч3363/26) and funded by the Russian Science Foundation (RSF), grant No. 17-19-01186

Evolution of an optical vortex with initial fractional topological charge

В ряде теоретических и экспериментальных работ было показано, что для оптического вихря с начальным дробным топологическим зарядом его топологический заряд равен целому числу с недостатком, если дробная часть меньше 0,5 и целому числу с избытком, если дробная часть больше 0,5. В этой работе с помощью численного моделирования мы показали, что топологический заряд начального дробного оптического вихря равен целому числу с недостатком, если дробная часть меньше 0,12, и целому числу с избытком, если дробная часть больше 0,12. Дело в том, что дополнительный центр сингулярности формируется на периферии пучка, где интенсивность почти равна нулю (примерно миллионная часть от максимума), и экспериментально его нельзя обнаружить, а численно можно. In a number of theoretical (J. Opt. 6, 259 (2004)) and experimental (Opt Express 19, 5760 (2011)) works, an original fractional-TC optical vortex (with TC standing for topological charge) was shown to evolve into an integer optical vortex whose TC is equal to the nearest (i) smaller integer, if the fractional part is smaller than 1/2, and (ii) larger we show that the initial fractional optical vortex evolves into an integer optical vortex with TC equal to the nearest (i) smaller integer, if the fractional part is smaller than 0.12, and (ii) larger integer, if the fractional part is larger than 0.12. This can be explained by the fact that the additional center integer, if the fractional part is larger than 1/2. In this work, using numerical simulation, of singularity is generated on the beam periphery characterized by near zero-intensity (a millionth of the maximum), thus prohibiting the experimental detection, but allowing a numerical assessment.Работа выполнена при поддержке Российского фонда фундаментальных исследований (грант 18-29-20003), а также Министерства науки и высшего образования РФ в рамках выполнения работ по Государственному заданию ФНИЦ «Кристаллография и фотоника» РАН

Toroidal polarization vortices in tightly focused beams with singularity

В данной работе с помощью формул Ричардса–Вольфа промоделирована фокусировка цилиндрических векторных пучков второго порядка. Было показано, что кольца, на которых вектор Пойнтинга равен нулю, возникают не только в плоскости острого фокуса, но и в плоскостях, удалённых от фокуса. В частности, при фокусировке света линзой с числовой апертурой NA = 0,95 на расстоянии примерно 0,45 мкм от оси периодически возникают тороидальные вихри (с периодом по оси z – 0,8 мкм). Вихри возникают попарно: ближайший к фокальной плоскости вихрь закручен по часовой стрелке, а следующий от него – против часовой стрелки. Вихри сопровождаются седловыми точками. При фокусировке пучка, ограниченного узкой кольцевой апертурой, тороидальных вихрей не наблюдается. In this paper, we numerically investigated tight focusing of cylindrical vector beams of the second order using Richards-Wolf formulae. It was shown that intensity rings where the Poynting vector was equal to zero appeared not only in the focal plane but also in nearby planes. For example, a lens with numerical aperture NA=0.95 was shown to generate periodical toroidal vortices with a 0.8-m period along the z-axis at a distance of about 0.45 m from the axis. The vortices were generated pairwise, with the closest-to-focus vortex having clockwise helicity and the subsequent being anticlockwise. The vortices were also characterized by saddle points. When focusing an optical beam passed through a narrow annular aperture, no toroidal vortices were observed.Работа выполнена при поддержке Российского научного фонда (грант 18-19-00595) в части «Формулы Ричардса–Вольфа для случая узкой кольцевой апертуры», Российского фонда фундаментальных исследований (гранты 18-07-01122, 18-07-01380, 18-29-20003) в части «Результаты моделирования фокусировки поляризационного вихря» и Министерства науки и высшего образования РФ в рамках выполнения работ по Государственному заданию ФНИЦ «Кристаллография и фотоника» РАН в части «Введение»

A transverse energy flow at the tight focus of light with higher-order circular-azimuthal polarization

Рассмотрена острая фокусировка света с циркулярно-азимутальной поляризацией m-го порядка. Это новый тип неоднородной гибридной поляризации, объединяющий свойства цилиндрической поляризации m-го порядка и круговой поляризации. С помощью формализма Ричардса–Вольфа получены аналитические выражения в фокусе для проекций векторов напряженности электрического и магнитного полей, распределения интенсивности, проекций вектора Пойнтинга и спинового углового момента. Теоретически и численно показано, что интенсивность в фокусе имеет 2(m+1) локальных максимумов, расположенных вдоль замкнутого контура с нулевой интенсивностью в центре (на оптической оси). Показано, что в фокусе имеется 4m вихрей поперечного потока энергии, центры которых расположены между локальными максимумами интенсивности. Также показано, что поперечный поток энергии меняет направление вращения 2(2m+1) раза при обходе в плоскости фокуса вокруг оптической оси. Интересно, что продольная проекция спинового углового момента в фокусе меняет знак 4m раза. В тех областях плоскости фокуса, где поперечный поток энергии вращается против часовой стрелки, продольная проекция спинового углового момента положительная, а вектор поляризации вращается в плоскости фокуса против часовой стрелки. И наоборот, где поток энергии вращается по часовой стрелке, там и вектор поляризации вращается по часовой стрелке, а продольная проекция спинового углового момента отрицательная. Моделирование подтверждает выводы теории. Tight focusing of light with mth-order circular-azimuthal polarization was investigated. This is a new type of inhomogeneous hybrid polarization that combines the properties of mth order cylindrical polarization and circular polarization. Using the Richards-Wolf formalism, we obtained analytical expressions in the focal spot for the projections of the electric and magnetic field, the intensity distribution, the projections of the Poynting vector, and the spin angular momentum. It was shown theoretically and numerically that at the focus, the intensity has 2(m+1) local maxima located on a circle centered on an on-axis intensity null. It was shown that 4m vortices of a transverse energy flow were produced at the focus, with their centers located between the local intensity maxima. It was also shown that in the focal plane, the transverse energy flow changes the direction of rotation 2(2m+1) times around the optical axis. It is interesting that the longitudinal projection of the spin angular momentum at the focus changes sign 4m times. In those areas of the focal plane where the transverse energy flow rotates counterclockwise, the longitudinal projection of the spin angular momentum is positive, and the polarization vector rotates counterclockwise in the focal plane. Conversely, if the energy flow rotates clockwise, the polarization vector rotates clockwise, and the longitudinal projection of the spin angular momentum is negative. Numerical simulations are in agreement with the theoretical investigation.Работа выполнена при поддержке Министерства науки и высшего образования в рамках выполнения работ по Государственному заданию ФНИЦ «Кристаллография и фотоника» РАН в частях «Введение» и «Заключение», Российского научного фонда (грант № 18-19-00595) в части «Результаты моделирования фокусировки света с гибридной поляризацией», Российского фонда фундаментальных исследований РФФИ (грант № 18-29-20003) в части «Интенсивность света с гибридной поляризацией в фокусе»)

Topological charge of a superposition of two Bessel-Gaussian beams

В работе теоретически показано, что у суперпозиции двух пучков Бесселя–Гаусса с разными топологическими зарядами и разными масштабными множителями (радиальными проекциями волновых векторов) топологический заряд равен топологическому заряду того пучка Бесселя–Гаусса, у которого больше масштабный множитель. Если у пучков Бесселя–Гаусса масштабные множители равны, то топологический заряд суперпозиции равен топологическому заряду того пучка Бесселя–Гаусса, у которого больше модуль весового коэффициента (больше мощность). Если и мощности пучков одинаковы, то топологический заряд суперпозиции равен среднему арифметическому от топологических зарядов каждого пучка Бесселя–Гаусса в суперпозиции. При условии, что сумма топологических зарядов обоих пучков нечётная, топологический заряд суперпозиции будет полуцелым числом. Но на практике из-за конечного радиуса окружности, на котором рассчитывается топологический заряд, полуцелого топологического заряда для вырожденного случая не получается. Вместо полуцелого топологического заряда, получается целый топологический заряд, меньший из двух. Моделирование показывает, что при небольшой разнице в весовых коэффициентах топологический заряд суперпозиции не сохраняется: в ближней зоне и зоне Френеля топологический заряд равен большему из двух, а в дальней зоне – меньшему. Причем переход топологического заряда от большего к меньшему происходит не скачком, а непрерывно на некотором расстоянии. В переходной зоне топологический заряд дробный. Here we show theoretically that a superposition of two Bessel-Gaussian (BG) beams with different topological charges (TC) and different scaling factors (radial components of the wave vectors) has the TC equal to that of the BG beam with the larger scaling factor. If the scaling factors of the BG beams are equal, then TC of the whole superposition equals TC of the BG beam with the larger (in absolute value) weight coefficient in the superposition (i.e. with larger power). If the constituent BG beams are also same-power, TC of the superposition equals the average TC of the two BG beams. Therefore, if the sum of TCs of both beams is odd, TC of the superposition is a half-integer number. In practice, however, TC is calculated over a finite radius circle and, hence, the half-integer TC for the degenerated case cannot be obtained. Instead of the half-integer TC, the lower of the two integer TCs is obtained. Numerical simulation reveals that if the weight coefficients in the superposition are slightly different, TC of the superposition is not conserved on propagation. In the near field and in the Fresnel diffraction zone, TC is equal to the highest TC of the two BG beams, while in the far field it is equal to the lower TC. What is more, TC changes its value from high to low not instantly, but continuously at some propagation distance. In the intermediate zone TC is fractional.Работа выполнена при поддержке Российского фонда фундаментальных исследований (грант 18-29-20003 в частях «Расчет топологического заряда суммы двух пучков БГ» и «Топологический заряд суперпозиции двух пучков Бесселя–Гаусса с одинаковыми весовыми и масштабными коэффициентами»), Российского научного фонда (грант 18-19-00595 в частях «Моделирование» и «Моделирование в случае примерного равенства весовых коэффициентов»), а также Министерства науки и высшего образования РФ в рамках выполнения работ по Государственному заданию ФНИЦ «Кристаллография и фотоника» РАН в части «Введение» и «Заключение»

Antiturcica Iterata – another Look at Croatian anti-Turkish Writings during the Renaissance

Otpor »Turcima«, održavanje granice prema Osmanskome Carstvu, jedna je od dominantnih tema u ranonovovjekovnoj Hrvatskoj. Sukladno tome, »turska tema« prisutna je i u književnosti. Problematizirana je u velikom vremenskom rasponu, na različitim jezicima, u različitim žanrovima, s različitim intencijama. Važnu podskupinu tekstova ove tematike – izravne i neizravne pozive na otpor osmanskim napadima i osvajanjima u razdoblju 1436-1600 – ovdje smo pokušali popisati što potpunije i što konzistentnije. Uvodno dajemo pregled istraživanja i interpretacija hrvatskoga protuturskog korpusa od 1945. do 2014. Potom upozoravamo na probleme vezane uz konstituiranje korpusa i iznosimo načela po kojima smo korpus uspostavili. Zatim opisujemo korpus prema očuvanosti popisanih tekstova, prema pisanom mediju u kojem su prenošeni (rukopisi ili tiskovine), prema jeziku (latinski, hrvatski, talijanski), načinu izražavanja (proza, poezija), književnim vrstama, kronologiji (upozoravamo i na diskontinuitete, na godine kad se javlja manje protuturskih tekstova), te prema prostorima koji su u tekstovima posebno isticani. Razmatramo i autore, prema broju tekstova, prema porijeklu i društvenom statusu. Naposljetku skiciramo sinoptičku interpretaciju korpusa, posebnu pozornost poklanjajući tekstovima koji se pojavljuju istodobno ili su vremenski bliski. Bibliografska skica hrvatskog protuturskog korpusa prilog je radu, a dostupna je i kao baza podataka na internetu.In the 15th and 16th centuries, Croatian writing about the Ottoman threat was both a reaction to the continuous crisis of several states, their societies, and culture, and an attempt to overcome this crisis by developing new ideas and new ways of consolidating the society. It has been claimed that this desperate attempt laid the foundations of modern Croatia. To be able to identify continuities and discontinuities of the so-called Croatian anti-Turkish writings, we have compiled a list of as many such works as possible. In Croatian literary history, the anti-Turkish writings seem to have first been presented as a group by Mihovil Kombol in 1945. In 1974, Marin Franičević tried to consider together texts from this group written in different languages (Latin and Croatian). In 1983, Vedran Gligo selected and published a canonical collection of sixteen (Latin) »anti-Turkish speeches« from the Renaissance. At the same time, Tomislav Raukar interpreted the anti-Turkish theme as a stimulus to Early Modern Croatia. In 2004, Davor Dukić proposed an imagological interpretation of the anti-Turkish corpus. All researchers, however, worked with a restricted number of texts, and there was not even an estimate of the actual size of the corpus. To construe a corpus with some pretence to comprehensiveness, we had to establish whether (and to what degree) a text addresses the anti-Turkish theme, whether it could be considered a »Croatian« text, whether it could be considered a literary and a public as opposed to a documentary or a private text). Such considerations led us to exclude some famous texts (such as Judita by Marko Marulić),while including e. g. an Oratio contra Turcam which Ivan Pergošić reprinted from the book by Johannes Avenarius. The fuzzy boundaries between confidentiality and publicity have been demonstrated by the fate of reports by the papal nuntio Antonius Fabregues, whose official dispatches about the Battle of Krbava field have appeared also in print as propagandistic broadsheets. In our corpus, a Croatian anti-Turkish work is a text urging Christian action against the Turks in defence of Dalmatia, Croatia, Hungary, and Christian Europe as a whole, written by an author significantly connected with Croatia, and calling to arms either by direct appeals or by a more indirect strategy of reporting, celebrating and lamenting events in the Christian-Turkish struggle. The corpus currently comprises basic bibliographic data on 141 texts written between 1436 and 1600. It exists as a list (see the Appendix to the article) and as a database published by the TeMrežaH project (temrezah.ffzg.unizg.hr/antiturcica-biblio.html). We have included data on a few texts that are not preserved, and on texts by unknown authors. Ninety-four texts first appeared as manuscripts, 44 in print (two exist in roughly contemporaneous handwritten and printed versions). Two texts (poems in Croatian from 1565 and 1596) were first printed almost a hundred years after the events they describe (in 1655). One hundred and four texts are in Latin, 26 in Croatian, nine in Italian; 75 texts are in prose, 66 in verse. An especially active period of production falls in 1493-1548 (when three fifths of all texts in the corpus were written). There are three longer periods of silence, when just a few anti-Turkish texts appear: the years 1504-1509, 1546-1560, 1575-1591. Christian Europe is the central topic in 46 texts, the Kingdom of Hungary in 31, Croatia itself in 20, Dalmatia (and Dubrovnik) in 20, the Levant in 15, the Venetian Republic in 11; this distribution follows the geopolitical situation of Croatian regions during the Renaissance, but Croatia and Dalmatia are treated in about the same number of texts, while the Kingdom of Hungary was thematised significantly more often than Venice. The texts were written by 61 authors (seven are anonymous), 40 of which wrote just one text; fourteen are known as authors only because of their anti-Turkish works. The five authors with most anti-Turkish texts are Fran Trankvil Andreis, Damjan Beneša, Marko Marulić, Ludovik Paskalić, Ivan Vitez od Sredne. Beneša and Paskalić were absent from the Croatian »canon« of anti-Turkish authors. Four of the authors are not Croatian by birth (Tideo Acciarini, Antonio Fabregues, Francesco Marcello, Bernard Zane), but they were all connected to Dalmatia by office, as teachers, diplomats or Church officials. Most of the authors can be regarded as humanists (17) or priests (24, or 34 including priests serving as diplomats); there were 15 diplomats, five of them lay persons. The authors whose anti-Turkish writings appear during the longest chronological span are Fran Trankvil Andreis from Trogir (53 years), Mavro Vetranović from Dubrovnik (writing in Croatian over 45 years), and Frano Božićević Natalis from Split (35 years). A chronological analysis also reveals two main generations of anti-Turkish writers. The first one was active during 1500-1535, and the second during 1520-1570. Among the authors those from Dalmatia and Dubrovnik prevail; they belong mostly to the civic patriciate, to the lesser nobility, or even to the general citizenry. A synoptic analysis of dates, genres, authors and contexts suggests a discontinuity in the corpus of Croatian anti-Turkish writings: some time after 1530-1540 public speeches and appeals disappear, to be replaced by administrative reports to the authorities of the Habsburg empire, by sensational testimonies about travels or battles, and by highly individual »grassroots« entreaties, often, seemingly, without significant political support, often without the benefit of print or of larger print runs. The discontinuity may be connected with the agony of the Kingdom of Hungary and Croatia (after the Battle of Mohács), and with the gradual disappearance of the generation of Croatian humanists that had tried to play significant roles in this kingdom