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, which 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 $(\alpha,\beta,m)$ 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 $(\nu,\rho,\mu)$ 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.

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 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.

Physics curricula around the world recognise energy as one of the core concepts in science (Duit, 2014), as it is fundamental in development of integrated scientific understanding of phenomena (Linn et al., 2006; National Research Council, 2012; Nordine et al., 2011). Importance of the energy concept is also recognised by PISA and TIMSS studies and is reflected in many science standards (National Research Council, 2012; Next Generation Science Standards, 2013). However, research about students’ energy conceptions keeps showing that students at all educational levels have significant difficulties with the concept of energy (Goldring & Osborne, 1994; Lawson & McDermott, 1987; Neumann et al., 2013; Pride et al., 1998). Concretely, students exhibit difficulties with understanding work-energy processes (Van Huis & Van den Berg, 1993), energy degradation and energy conservation (Goldring & Osborne, 1994; Liu & McKeough, 2005; Neumann et al., 2013). Thereby, a large number of studies detected conservation of energy as the most difficult aspect of the energy concept (Lindsey et al., 2012; Neumann et al., 2013; Van Heuvelen & Zou, 2001; Van Huis & Van den Berg, 1993). In fact, only a very few students develop deeper understanding of energy conservation until the time they finish secondary school (Herrmann-Abell & DeBoer, 2018). For purposes of improving the quality of teaching about energy, it is useful to identify possible sources of above-mentioned students’ difficulties. Firstly, it is important to note that many students’ difficulties with the energy concept may be related to students’ (mis)understanding of systems (Seeley et al., 2019; Van Heuvelen & Zou, 2001; Van Huis & Van den Berg, 1993). Consequently, students should be helped to recognize the importance of carefully choosing the physical system, if one wants them to gain a functional understanding of energy conservation (Lindsey et al., 2012; Seeley et al., 2019). Such system-based Abstract. 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 nonconservative 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-andfading technique.

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.