We analyze the high-order above-threshold ionization (HATI) process of a small polyatomic molecule with C3 symmetry, which is induced by a bicircular strong laser field. This field consists of two coplanar, counter-rotating, circularly polarized components with frequencies rω and sω where r and s are integers. In our study, we use an improved molecular strong-field approximation to obtain electron energy-angle-resolved and momentum spectra of the BF3 molecule. We analyze the contributions of individual atoms as well as the impact of molecular symmetries on these spectra. We find that these spectra are significantly affected by the characteristics of the molecule and the laser-field parameters. Furthermore, we observe pronounced interference minima in the HATI spectra. We demonstrate that these minima result from the destructive interference of rescattered wave packets from different atomic centers, and we determine the conditions under which they occur, including two-, three-, and four-center interference.

Abstract The quantum-mechanical transition amplitudes for atomic and molecular processes in strong laser fields are expressed in the form of multidimensional integrals of highly oscillatory functions. Such integrals are ideally suited for the evaluation by asymptotic methods for integrals. Furthermore, using these methods it is possible to identify, in the sense of Feynman’s path-integral formalism, the partial contributions of quantum orbits, which are related to particular solutions of the saddle-point equations. This affords insight into the physics of the problem, which would not have been possible by only solving these integrals numerically. We apply the saddle-point method to various quantum processes that are important in strong-field physics and attoscience. The special case of coalescing or near-coalescing saddle points requires application of the uniform approximation. We also present two modifications of the saddle-point method, for the cases where a singular point of the subintegral function exactly overlaps with a saddle point or is located in its close vicinity. Particular emphasis is on the classification of the saddle-point solutions. This problem is solved for the one-dimensional integral over the ionization time, relevant for above-threshold ionization (ATI), while for two-dimensional integrals a classification by the multi-index ( α , β , m ) is introduced, which is particularly useful for the medium- and high-energy spectrum of high-order harmonic generation (HHG) and backward-scattered electrons (for high-order ATI). For the low-energy structures a classification using the multi-index ( ν , ρ , μ ) is introduced for the forward-scattering quantum orbits. In addition to laser-induced processes such as ATI, HHG and high-order ATI, we consider laser-assisted scattering as an example of laser-assisted processes for which real solutions of the saddle-point equation exist. Particular attention is devoted to the quantum orbits that describe and visualize these processes. We also consider finite laser pulses, the semiclassical approximation, the role of the Coulomb field and the case of laser fields intense enough to lead into the relativistic regime.

We present a modified saddle-point method and apply it to the high-order above-threshold ionization (HATI) and high-order harmonic generation (HHG) processes. When we use the Slater-type orbitals to describe the ground-state wave function of the valence electron in a noble gas atom, we cannot apply the ordinary saddle-point method (SPM) because the singular points of the ionization matrix element, as well as the recombination matrix element for HHG, are in the vicinity of the corresponding saddle points. Therefore, we present a modification of the SPM and show that the obtained results are in excellent agreement with the spectra calculated by numerical integration. Furthermore, we compare the results obtained by the modified SPM with those obtained by the ordinary SPM with the asymptotic ground-state wave function. We conclude that both methods work for HATI, but for HHG we show that it is necessary to use the modified SPM because the recombination happens close to the atomic core, where the electron ground state is well described only by the Slater-type orbitals.

In the present paper, we study the high-order above-threshold ionization of noble-gas atoms using a bi-elliptic orthogonal two-color (BEOTC) field. We give an overview of the SFA theory and calculate the differential ionization rate for various values of the laser field parameters. We show that the ionization rate strongly depends on the ellipticity and the relative phase between two field components. Using numerical optimization, we find the values of ellipticity and relative phase that maximize the ionization rate at energies close to the cutoff energy. To explain the obtained results, we present, to the best of our knowledge, for the first time the quantum-orbit analysis in the BEOTC field. We find and classify the saddle-point (SP) solutions and study their contributions to the total ionization rate. We analyze quantum orbits and corresponding velocities to explain the contribution of relevant SP solutions.

When exposed to strong laser fields, atoms or molecules can absorb more photons from the laser field than is necessary for ionization. This process is called above-threshold ionization (ATI). In analyzing this process, the strong-field approximation (SFA) turns out to be a very useful theoretical tool. In the SFA the differential ionization rate, which is an observable quantity, can be expressed as an integral over the ionization time and can be calculated by numerical integration (NI) or using the saddle-point method (SPM). When we use the Slater orbitals to describe the ground-state wave function of the valence electron, the results obtained using the SPM and NI do not agree. We find the reasons for this disagreement and introduce a modified SPM that leads to excellent agreement between the SPM and NI results for various strong laser fields.

In this study, our aim was to identify high school and university students’ postinstruction conceptions about the law of conservation of mechanical energy (LCME). A cross-sectional survey design was used. Firstly, a test consisting of 14 multiple-choice questions was developed. 23 physics teachers analysed the test and concluded that our questions are useful for measuring understanding about the LCME. Next, the test was administered to a convenient sample of high school and university students who had already received conventional instruction about the energy concept. The sample consisted of 138 students from the University of Zagreb (Croatia), and 115 high school students from two different schools in Sarajevo (Bosnia and Herzegovina). While the distribution of item difficulties was good, the reliability of test scores proved to be barely acceptable. Therefore, we primarily focused on analysing how frequently the students chose the individual distractors and drew corresponding conclusions about students’ conceptions. It has been shown that many students associate conservation of mechanical energy with certain surface features of physical problems (e.g., “pulley problems”), instead of reasoning about the processes a chosen system undergoes over time. Students often believe that mechanical energy is conserved even for phenomena in which air resistance cannot be neglected. Similarly, they sometimes do not recognize that most collisions of everyday objects necessarily include the conversion of mechanical energy into thermal energy. We could conclude that many students from all educational levels in Bosnia and Herzegovina and Croatia still fail to apply a system-based approach to energy analysis.

A theory of high-order above-threshold ionization (HATI) of atoms is presented and applied to the ionization of inert gases by an orthogonal two-color (OTC) field. The transition matrix element is derived within the strong-field approximation and calculated by numerical integration and the saddle-point method. The atomic bound state is expressed as an expansion in terms of the Slater-type orbitals as well as an asymptotic wave function. The energy and momentum distributions of the HATI electrons are calculated for the ω – 2ω and ω – 3ω OTC fields. A detailed analysis of the saddle-point solutions and their contribution to the overall differential ionization rate for these two cases is presented. Optimal values of the OTC field parameters for which the ionization rate is maximal in the high-energy region of the photoelectron spectra are found. Special attention is devoted to the symmetry consideration of the photoelectron momentum distributions. All these results are analyzed in detail using the quantum-orbit formalism, which gives a better physical insight into the HATI process.

The purpose of this study was to explore the effectiveness of the conventional high school instruction about conservation of mechanical energy in Canton Sarajevo. To that end we tested 441 high school students from six different schools in Sarajevo (Bosnia and Herzegovina) for their competence to apply the law of conservation of mechanical energy. Concretely, students were expected to solve 5 open-ended tasks that covered conceptually different situations. In each task we asked a set of sub-questions to check whether the students possess all the prerequisite sub-competencies for systematic reasoning about conservation of mechanical energy. In addition, we investigated how students’ ideas about conservation of mechanical energy were affected by the choice of the physical system, as well as by the choice of the observed time interval. Data analysis was performed on the level of individual tasks. The students’ written answers were analyzed and the frequencies of most prominent student responses were reported. Generally, it has been shown that most high school students from Sarajevo fail to identify and distinguish internal, external, conservative and non-conservative forces. Also, many students think that applicability of the conservation law does not depend on the chosen physical system and its evolution over time. We could conclude that high school students’ use of the conservation law is mostly based on remembering similar problem solving experiences, rather than on relevant strategic knowledge.

Conventional teaching about the law of conservation of mechanical energy (LCME) often results with students trying to solve problems by remembering similar problems they already covered in classes. Consequently, many students fail to transfer their knowledge to simplest real-life problems. Therefore, a pre-test – post-test quasi-experiment was conducted to evaluate the effects of an alternative, system-based approach to teaching about LCME. The study included 70 upper-secondary students from the First Bosniak Gymnasium Sarajevo, Bosnia and Herzegovina. Firstly, all students learned about energy in a conventional way. Then they wrote a test on LCME and had three additional hours of teaching about this topic, where one group of students learned in line with the forces-variant of the system approach (e.g., discussing conservative and non-conservative forces) and the other group with the process-variant of the same approach (e.g., discussing system’s states and processes like in thermodynamics). For both variants, only three hours of system-based teaching proved to substantially improve the students’ level of LCME understanding compared to the level of understanding they had after conventional teaching. It follows that the system approach may work well at the upper-secondary level, if it is introduced through the scaffolding-and-fading technique. Keywords: quasi-experiment, mechanical energy, teaching materials, teaching strategies

Maintaining item banks that continually reflect the measured construct can be achieved through periodically removing obsolete items and adding validated items.

The quantum-mechanical transition amplitude of an ionization process induced by a strong laser field is typically expressed in the form of an integral over the ionization time of a highly oscillatory function. Within the saddle-point (SP) approximation this integral can be represented by a sum over the contributions of the solutions of the SP equation for complex ionization time. It is shown that, for the general case of an elliptically polarized polychromatic laser field, these solutions can be obtained as zeros of a trigonometric polynomial of the order n and that there are exactly n relevant solutions, which are to be included in the sum. The results obtained are illustrated by examples of various tailored laser fields that are presently used in strong-field physics and attoscience. For some critical values of the parameters two SP solutions can coalesce and the topology of the 'steepest descent' integration contour changes so that some SPs are bypassed. Around the critical parameters a uniform approximation should be used instead of the SP method.

Successful application of the Huygens-Fresnel principle often requires reasoning about the interplay of aperture and light beam dimensions for purposes of identifying the unobstructed part of the light beam which is the source of secondary waves. Therefore we decided to identify university students' ideas about the role of this interplay in the formation of diffraction patterns. We conducted a survey research with 191 first-year students from the Faculty of Chemical Engineering and Technology at the University of Zagreb, Croatia. They were administered six constructed-response questions in which aperture or laser beam dimensions were varied and students were expected to verbally and pictorially describe how these changes would affect the diffraction pattern. It has been shown that 63% of students think that a change in the length of the vertical slit necessarily results in a change of the diffraction pattern, even when the illuminated portion of the slit remains the same. In addition, it has been found that nearly 40% of students believe that in optical grating diffraction an increase of beam diameter leads to bigger diffraction fringes. A possible way to overcome some of these difficulties would be to insist on consistent application of the Huygens-Fresnel principle.

We investigate strong-field ionization of linear molecules by a two-color laser field of frequencies rω and sω having coplanar counterrotating or corotating elliptically polarized components (ω is the fundamental laser field frequency and r and s are integers). Using the improved molecular strong-field approximation we analyze direct above-threshold ionization (ATI) and high-order ATI (HATI) spectra. More precisely, reflection and rotational symmetries of these spectra for linear molecules aligned in the laser-field polarization plane are considered. The reflection symmetries for particular molecular orientations, known to be valid for a bicircular field (this is the field with circularly polarized counterrotating components), are valid also for arbitrary component ellipticities. However, specific rotational symmetries that are satisfied for HATI by a bicircular field, are violated for an arbitrary elliptically polarized field with counterrotating components. For the corotating case and the N2 molecule we analyze molecular-orientation-dependent interferences and plateau structures for various ellipticities.

Even graduate physics students have many misconceptions about basic wave optics phenomena. This suggests that there is much room for improvement of the traditional wave optics curriculum. An effective way for initiating a curriculum change is to reconsider and revise the expected learning outcomes and corresponding assessment instruments. By systematically enriching our wave optics instruction and assessment with conceptual tasks, we may increase the probability of students actively engaging in learning the conceptual aspects of wave optics. In this paper, we present the process of developing an item bank for measuring understanding of wave optics in typical introductory physics courses at universities. Thereby, the Rasch modeling approach has been used. The development of the item bank has been guided by results from multiple expert and student surveys, as well as from group interviews and think aloud interviews. Altogether 65 multiple-choice items with a single correct answer and three distractors have been prepared for field testing. Until now, 35 out of 65 items have been field tested by means of a paper and pencil survey which included 188 participants from five universities in Bosnia and Herzegovina, Croatia, and Slovenia. The field test showed that 32 out of 35 items have good psychometric characteristics and that they may be very useful for uncovering students' misconceptions in wave optics.

We investigate strong-field ionization of homonuclear diatomic molecules, exemplified with the N-2 molecule, by a bichromatic elliptically polarized laser field having corotating components. We assume that both the emitted electron momentum vector and the internuclear vector of the diatomic molecule lay in the laser-field polarization plane. Our analysis of the low-energy electron spectra caused by the direct above-threshold ionization (ATI) and of the high-energy rescattered electron spectra that can form an extended plateau (high-order ATI or HATI) is based on the improved molecular strong-field approximation. The photoelectron spectra obtained by (H)ATI of molecular targets are more complex and have a reacher structure in comparison to the analogous spectra for atomic targets. We explain the observed interference structures by the interference of two electron wave packets emitted from the two centers of the diatomic molecule. Particular attention is devoted to the HATI spectra. For small values of the ellipticity the photoelectron spectra exhibit a plateau whose length can be as high as 17U(p), with U-p the electron ponderomotive energy. The yield of high-energy electrons emitted nearly antiparallel to the semimajor axis of the laser-field polarization ellipse is one order of magnitude higher for perpendicular than for the parallel molecular orientation.

Earlier research has shown that students have tremendous difficulties with understanding certain aspects of rolling without slipping, such as the zero-velocity at the contact point and plausibility of application of the law of conservation of mechanical energy despite action of the friction force. The aim of this research was to explore whether using analogies and reasoning about extreme cases can facilitate conceptualization of the above-mentioned phenomena. A pre-test – post-test quasi-experiment has been conducted, with 93 students in the control group (CG) and 91 students in the experimental group (EG). Whereas control group students received conventional teaching, in the experimental group rolling of a cylinder has been considered as a special case of a tumbling prism for which the number of prism surfaces tended to infinity. The results of analysis of covariance showed that students from the experimental group significantly outperformed their peers from the control group on the Rolling Motion Concept Test (RMCT). Between-group differences were greater on test items that required higher level of cognitive transfer. This research suggests that using analogies and extreme case reasoning can facilitate comprehension of certain seemingly counterintuitive aspects of rolling motion. Keywords: analogy-based teaching, energy conservation, extreme case reasoning, misconceptions, rolling motion.

Using the improved molecular strong-field approximation, we investigate (high-order) above-threshold ionization [(H)ATI] of various linear polyatomic molecules by a two-color laser field of frequencies r omega and s omega (with integer numbers r and s) having coplanar counter-rotating circularly polarized components (a so-called bicircular field). Reflection and rotational symmetries for molecules aligned in the laser-field polarization plane, analyzed for diatomic homonuclear molecules in Phys. Rev. A95, 033411 (2017), are now considered for diatomic heteronuclear molecules and symmetric and asymmetric linear triatomic molecules. There are additional rotational symmetries for (H) ATI spectra of symmetric linear molecules compared to (H) ATI spectra of the asymmetric ones. It is shown that these symmetries manifest themselves differently for r + s odd and r + s even. For example, HATI spectra for symmetric molecules with r + s even obey inversion symmetry. For ATI spectra of linear molecules, reflection symmetry appears only for certain molecular orientation angles +/- 90 degrees -jr180 degrees /(r + s) (j integer). For symmetric linear molecules, reflection symmetry appears also for the angles -jr180 degrees /(r + s). For perpendicular orientation of molecules with respect to the laser-field polarization plane, the HATI spectra are very similar to those of the atomic targets, i.e., both spectra are characterized by the same type of the (r + s)-fold symmetry.

With the development of intense femtosecond laser sources it has become possible to study atomic and molecular processes on their own subfemtosecond time scale. Table-top setups are available that generate intense coherent radiation in the extreme ultraviolet and soft-X-ray regime which have various applications in strong-field physics and attoscience. More recently, the emphasis is moving from the generation of linearly polarized pulses using a linearly polarized driving field to the generation of more complicated elliptically polarized polychromatic ultrashort pulses. The transverse electromagnetic field oscillates in a plane perpendicular to its propagation direction. Therefore, the two dimensions of field polarization plane are available for manipulation and tailoring of these ultrashort pulses. We present a field that allows such a tailoring, the so-called bicircular field. This field is the superposition of two circularly polarized fields with different frequencies that rotate in the same plane in opposite directions. We present results for two processes in a bicircular field: High-order harmonic generation and above-threshold ionization. For a wide range of laser field intensities, we compare high-order harmonic spectra generated by bicircular fields with the spectra generated by a linearly polarized laser field. We also investigate a possibility of introducing spin into attoscience with spin-polarized electrons produced in high-order above-threshold ionization by a bicircular field.

We investigate high-order harmonic generation of non-planar ammonia molecules by a bicircular field, which consists of two coplanar counter-rotating circularly polarised fields of frequencies. and 2 omega. This field possesses dynamical symmetries, which can be used together with the symmetry of the molecular Hamiltonian to obtain the selection rules for harmonic emission. We show analytically and confirm by numerical calculations that circularly polarised harmonics of order n = 3q +/- 1, with q integer, as well as linearly polarised harmonics of order n = 3q are emitted. The presence of well-separated both linearly and circularly polarised harmonics in the same spectrum is unique to nonplanar molecules. [GRAPHICS] .