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This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincare sphere on polarization optics.

Interference, diffraction, and polarisation of light are the basic phenomena of wave optics. According to the findings of previous studies on wave optics, these phenomena are the source of many students’ conceptual difficulties. The goal of the present study was to qualitatively investigate and compare Croatian and Austrian secondary school students’ understanding of the main concepts and phenomena of wave optics as a first step that will serve as a preparation for a large-scale study on secondary school students understanding of wave optics. Here we present the preliminary results of six semi-structured demonstration interviews conducted in Croatia and Austria with secondary school students (age 18-19), after regular school instruction on wave optics.

In this work, the optical theory of physics helps to understand, analyze, and calculate the received power in a fifth-generation mobile telecommunications receiver. In these communication systems there are statistical models and deterministic physical models. We investigated how diffuse scattering in reflection contributes to specular reflection in received power at a receiver due to materials in the propagating environment in a communication channel using millimeter waves. The dimensions of building construction materials have sizes comparable to wavelength at millimeter wave frequencies. The power transmitted by communication equipment can be reflected in the roughness of these apparently smooth materials or spread out in multiple directions. The results show that the contribution due to diffuse scattering must be considered and that deterministic physical models using optical theory are valid and improve predictive analysis with good fit. With these results, the physical theory allows to make software with high precision and that improves the current applications.

Effects of the quantum gravity under Minkowski space-time background are considered. It is shown that despite the absence of the complete theory of quantum gravity, some concrete predictions could be made for the influence of the quantum gravitational fluctuations on the propagation of the massive particles. We demonstrate that although the gravitational potential fluctuations do not produce particle scattering, they cause decoherence of the matter waves due to off-shell effects. For point-like massive particles of the Planck mass order, the effect is considerable. However, this type of decoherence is beyond the measurable possibility for the real particles of the finite size., Comment: 14 pages, minor changes compared v3, but substantial changes compared v2

The textbook description of multiple slit interference employs a standard path difference formula that was originally calculated for the purpose of analyzing Young’s double slit experiment. It was arrived at on the basis of a pair of assumptions that help simplify the geometry of the slit-barrier-screen arrangement and the ensuing formalism needed to estimate the positions of interference fringes on the detection screen. According to the conventional approach, convergent rays of light that emanate from different slits can be treated as effectively parallel in the far field limit. Such a parallel ray approximation-based analysis was shown by Thomas (2019; 2020) to be redundant, paradoxical and limited in accuracy. In this paper, the task of reformulation that began previously with the elementary two-slit experiment is now extended to encompass the N-slit scenario, diffraction at a single-slit and the formation of circular fringes. In order to derive one of the key results (the generalized hyperbola theorem for N-point sources), a novel computational technique called analytical induction is introduced. The procedure involves a synthesis of concepts from Cartesian geometry and discrete mathematics, namely coordinate transformation by Euclidean translation and a method of proof called the principle of mathematical induction, respectively. The final goal attained are two distribution functions that succinctly and precisely describe the variation of intensity of interference fringes on the distant screen, when it is oriented parallel and perpendicular relative to the line joining the sources. A comparison is drawn between the predictions of the conventional and the new analyses by means of numerical-graphical simulation with MATLAB. The theoretical methods and results presented herein may have a significant bearing in areas of applied optics like interferometry and diffraction crystallography. On the pedagogic front, it is suggested that the current geometrical program being visually more intuitive, be incorporated into the standard curriculum of an advanced undergraduate/graduate level course in physical optics, to complement the conventional approach.

The presented article deals with the Conceptual Test in Wave Optics. It describes main steps of its development and structure and properties of the final version. Two sets of pilotage testing have been done so far. In the first part of the research 105 high school students were tested with a test reliability described by Cronbach’s alpha on the level 0.2. Due to this low value another testing was carried out, this time with university students. It has showed reliability on the level 0.72. This article deals with results from the analysis of these two data sets and shows examples of possible wave optics misconceptions detected by this testing. The test is attached.

Single slit diffraction is part of many high-school physics curricula throughout the world. In this study, we aimed to investigate whether high-school physics teachers from Bosnia and Herzegovina, Serbia and North Macedonia are adequately prepared to discuss with their students about various aspects of the single slit diffraction pattern, particularly about vertical length of diffraction fringes. To that end we conducted a written survey which included 57 high-school physics teachers. Besides asking teachers about students’ difficulties in learning about single slit diffraction, we also asked them to specify their own difficulties with this topic and required them to solve a conceptual task. Almost every second teacher believed that vertical length of diffraction fringes can be increased by changing shape or width of the slit and only 1 out of 57 teachers managed to correctly answer why diffraction fringes become shorter when we move away from the central maximum. We concluded that physics teacher education programmes should be changed to develop understanding of both, horizontal and vertical aspects of the diffraction pattern. To that end it is useful to provide learning opportunities which encourage combining ray and wave model of light.

A front-end optics system has been developed for the EAST microwave imaging reflectometry for 2D density fluctuation measurement. Via the transmitter optics system, a combination of eight transmitter beams with independent frequencies is employed to illuminate wide poloidal regions on eight distinct cutoff layers. The receiver optics collect the reflected wavefront and project them onto the vertical detector array with 12 antennas. Utilizing optimized Field Curvature adjustment lenses in the receiver optics, the front-end optics system provides a flexible and perfect matching between the image plane and a specified cutoff layer in the plasma, which ensures the correct data interpretation of density fluctuation measurement.

We have developed a numerical simulation method for diode-pumped alkali lasers (DPALs). We previously reported the use of a simplified gas-flow model, whose adaptability was restricted to forced gas-flow DPALs. In the present study, we have employed an incompressible, three-dimensional computational fluid dynamics (CFD) model. The natural convective flow is modeled using a modified Boussinesq approximation. The results of the simulations were found to be in good agreement with those for a compressible CFD model for a heat-induced flow problem. As a result, good agreement with the experimental results was achieved for a sealed DPAL. On the other hand, the results for the forced gas-flow DPAL were the same as those for the simplified gas-flow model, which confirms the validity of the latter model. The dependence of the beam quality on the pump power for the sealed DPAL was consistent with previously reported results.

Detailed knowledge of the image of the point spread function (PSF) is necessary to optimize astronomical coronagraph masks and to understand potential sources of errors in astrometric measurements. The PSF for astronomical telescopes and instruments depends not only on geometric aberrations and scalar wave diffraction but also on those wavefront errors introduced by the physical optics and the polarization properties of reflecting and transmitting surfaces within the optical system. These vector wave aberrations, called polarization aberrations, result from two sources: (1) the mirror coatings necessary to make the highly reflecting mirror surfaces, and (2) the optical prescription with its inevitable non-normal incidence of rays on reflecting surfaces. The purpose of this article is to characterize the importance of polarization aberrations, to describe the analytical tools to calculate the PSF image, and to provide the background to understand how astronomical image data may be affected. To show the order of magnitude of the effects of polarization aberrations on astronomical images, a generic astronomical telescope configuration is analyzed here by modeling a fast Cassegrain telescope followed by a single 90° deviation fold mirror. All mirrors in this example use bare aluminum reflective coatings and the illumination wavelength is 800 nm. Our findings for this example telescope are: (1) The image plane irradiance distribution is the linear superposition of four PSF images: one for each of the two orthogonal polarizations and one for each of two cross-coupled polarization terms. (2) The PSF image is brighter by 9% for one polarization component compared to its orthogonal state. (3) The PSF images for two orthogonal linearly polarization components are shifted with respect to each other, causing the PSF image for unpolarized point sources to become slightly elongated (elliptical) with a centroid separation of about 0.6 mas. This is important for both astrometry and coronagraph applications. (4) Part of the aberration is a polarization-dependent astigmatism, with a magnitude of 22 milliwaves, which enlarges the PSF image. (5) The orthogonally polarized components of unpolarized sources contain different wavefront aberrations, which differ by approximately 32 milliwaves. This implies that a wavefront correction system cannot optimally correct the aberrations for all polarizations simultaneously. (6) The polarization aberrations couple small parts of each polarization component of the light (∼10^(-4)) into the orthogonal polarization where these components cause highly distorted secondary, or “ghost” PSF images. (7) The radius of the spatial extent of the 90% encircled energy of these two ghost PSF image is twice as large as the radius of the Airy diffraction pattern. Coronagraphs for terrestrial exoplanet science are expected to image objects 10^(-10), or 6 orders of magnitude less than the intensity of the instrument-induced “ghost” PSF image, which will interfere with exoplanet measurements. A polarization aberration expansion which approximates the Jones pupil of the example telescope in six polarization terms is presented in the appendix. Individual terms can be associated with particular polarization defects. The dependence of these terms on angles of incidence, numerical aperture, and the Taylor series representation of the Fresnel equations lead to algebraic relations between these parameters and the scaling of the polarization aberrations. These “design rules” applicable to the example telescope are collected in § 5. Currently, exoplanet coronagraph masks are designed and optimized for scalar diffraction in optical systems. Radiation from the “ghost” PSF image leaks around currently designed image plane masks. Here, we show a vector-wave or polarization optimization is recommended. These effects follow from a natural description of the optical system in terms of the Jones matrices associated with each ray path of interest. The importance of these effects varies by orders of magnitude between different optical systems, depending on the optical design and coatings selected. Some of these effects can be calibrated while others are more problematic. Polarization aberration mitigation methods and technologies to minimize these effects are discussed. These effects have important implications for high-contrast imaging, coronagraphy, and astrometry with their stringent PSF image symmetry and scattered light requirements.

Arguments from scale physics, augmented by numerical and analytical investigations, are used to consider the probability and the detectability of superoscillations in generic functions. The detectability is defined as the fraction of the total power in the field that is located at regions of superoscillations. It is found that the probability for a superoscillation of a particular scale follows a quadratic power-law decay curve above the scale of the bounded support of the spectrum. The detectability is found to be even more severely suppressed, following a fourth-order power-law decay curve above the scale of the spectrum., 11 pages, 5 figures

We investigate how the near field affects partially coherent light scattered from an aperture in an opaque screen. Prior work on this subject has focused on the role of surface plasmons, and how they affect spatial coherence is well documented. Here, we consider other near-field effects that might impact spatial coherence. We do this by examining the statistics of the near-zone field scattered from an aperture in a perfect electric conductor plane—a structure that does not support surface plasmons. We derive the near-field statistics (in particular, cross-spectral density functions) by applying electromagnetic equivalence theorems and the Method of Moments. We find, even in the absence of surface plasmons, that near-field physics can affect the coherence of the scattered field. The analysis and findings presented herein complement the existing coherence-related surface plasmons literature, and could find use in the design of photonic devices built to engineer spatial coherence.

Students often have difficulties comprehending situations in which wave characteristics of light are manifested. After teaching wave optics, we studied the level of understanding of reflective diffraction of light with a group of secondary school students, and the phenomenon of diffraction in general. We analysed the possibility of using convenient reflexive diffraction gratings (compact discs and rulers with millimetre and half-millimetre markings) using suitable research school experiments which can be prepared. With appropriate methods of visualising the superposition of light waves, those experiments represent the basis for an appropriate conceptual change in the understanding the diffraction of light.

Recently, with the development of more powerful and accurate computational tools, the inclusion of new didactic materials in the classroom is known to have increased. However, the form in which these materials can be used to enhance the learning process is still under debate. Many different methodologies have been suggested for constructing new relevant curricular material and, among them, just-in-time teaching (JiTT) has arisen as an effective and successful way to improve the content of classes. In this paper, we will show the implemented pedagogic strategies for the courses of geometrical and optical physics for students of optometry. Thus, the use of the GeoGebra software for the geometrical optics class and the employment of new in-house software for the physical optics class created using the high-level programming language Python is shown with the corresponding activities developed for each of these applets.

We present an elementary analysis of the effects on light reflected from a uniformly moving mirror by using the photon picture of light and the conservation laws for energy and momentum of the system photon-mirror. Such a dynamical approach to the problem seems very suitable for introductory physics courses, not requiring any previous knowledge in wave optics or special relativity., 4 pages, 1 figure, LaTeX, v2: a slight error in the derivation corrected

This article presents a new technique for performing most well-known demonstrations of wave optics. Demonstrations which are normally very hard to show to more than a few people can be presented easily to very large audiences with excellent visibility for everyone. The proposed setup is easy to put together and use and can be very useful for high-school physics teachers and even university professors.

We present a derivation of the paraxial geometrical laws starting from a wave-optics approach, in particular by using simple continuity conditions of paraxial spherical waves at boundaries (discontinuities) between optical media. Paraxial geometrical imaging and magnification laws, under refraction and reflection at boundaries, are derived for several instructive cases and without using Fresnel diffraction theory. The primary aim is to provide a complementary insight into the standard axiomatic approach of paraxial geometrical optics and likewise to allow the introduction of some wave imaging concepts, such as the transmittance function, with a notable didactic interest for advanced subjects such as Fourier optics. This approach provides a more homogeneous vision of classical optics in which the use of the optical field continuity conditions at a boundary is a usual requirement as is clearly seen, for example, in the case of the derivation of Fresnel formulas. The work is particularly intended for university physics teachers and pregraduate and first year postgraduate students.

A sticky capture thread from the spiral element of spider orb-webs is formed of almost regularly spaced droplets that surround a supporting axial fibre. From the perspective of physical optics it represents a periodic linear array of scattering elements that acts as a diffraction grating. This is a novel aspect, which is of vital importance for the understanding of the overall scattering pattern. To demonstrate its significance, we present our experimental findings and compare them with results of a simplified model.

Excel spreadsheets are used as an excellent tool in physics laboratory experiments related to wave optics. This paper presents a simple way to simulate diffraction of waves through a slit using Excel spreadsheets without any tedious programming. These simulations are used to visualize the effect of variation of different parameters such as number of slits, wavelength of incoming light, slit width and slit separation on Fraunhofer diffraction pattern.

A simple Python-based open source software library for the numerical simulation of the linear or nonlinear time-dependent Schrödinger equation in one and two dimensions is presented. The integration is performed using a first-order split-step pseudospectral method, relying on the fast Fourier transform. The software library could be useful for undergraduate courses in elementary quantum mechanics, wave optics and computational physics. It could also be of interest for graduate students working with nonlinear waves, in frameworks such as laser beam propagation in nonlinear optical materials, matter waves within ultracold gases, dark matter or superfluid dynamics, among others. The discussion is complemented by solved examples and suggestions for educational applications of the code. Ministerio de Economía y Competitividad | Ref. FIS2014-58117-P Ministerio de Economía y Competitividad | Ref. IS2017-83762-P Xunta de Galicia. Consellería de Cultura, Educación e Ordenación Universitaria | Ref. GPC2015/019

Factors influencing the quality of lensless ghost imaging are investigated. According to the experimental results, we find that the imaging quality is determined by the number of independent sub light sources on the imaging plane of the reference arm. A qualitative picture based on advanced wave optics is presented to explain the physics behind the experimental phenomena. The present results will be helpful to provide a basis for improving the quality of ghost imaging systems in future works.

By combining Fourier optics with classical radiometry, a simple, compact formula is derived for computing the absolute irradiance of diffracted and aberrated images in optical systems. Within appropriate limits, the formula reduces to the familiar equations of classical radiometry and of physical optics. It is argued that the approach presented is pedagogically appealing as it combines the principal results of geometrical optics, physical optics and radiometry into a single equation, thus, providing a convenient and simple means of describing imaging phenomena at the level of an advanced undergraduate or introductory graduate course on radiometry. A practical example concerning the image irradiance of stars is discussed.

The optical microscope is a standard optical instrument that is used as an example to introduce the principles of optics. However, some of the assumptions that are used to describe image formation are highly suspect. Take the paraxial approximation, for example, which assumes angles are small when in practice they can be greater than 70°. Research into new microscope techniques and improvements in understanding quantitatively the image formation process has led to investigations in various areas of optics including diffraction theory, pupil filters, non-paraxial focusing, vectorial effects, finite Fresnel number, partial coherence, three-dimensional image formation, beam propagation, ultra-short pulse propagation and scattering theory. Some of these topics are reviewed, with particular emphasis on their applications in microscopy. Also, in addition to their intrinsic scientific interest, some of these areas have other applications, in optical information storage and lithography for example. This paper is based on a plenary presentation at Photonics06, Manchester UK, on the award of the Institute of Physics Optics and Photonics Division Prize 2006.

An extremely simple and inexpensive modification of the Michelson interferometer is presented together with the outlines of a curriculum of physical optics. The most important experiments of optics such as diffraction, interference of light originated by one, or alternatively, by two independent sources and interference of polarized light can be easily arranged for classroom demonstrations without changing instrumentation. A simple and effective experiment to show how the resolution limit of an optical system is diffraction-limited is reported.

The resolution concept in connection with the Fabry–Perot interferometer is difficult to understand for undergraduate students enrolled in physical optics courses. The resolution criterion proposed in textbooks for distinguishing equal intensity maxima and the deduction of the resolving power equation is formal and non-intuitive. In this paper, we study the practical meaning of the resolution criterion and resolution power using a computer simulation of a Fabry–Perot interferometer. The light source in the program has two monochromatic components, the wavelength difference being tunable by the user. The student can also adjust other physical parameters so as to obtain different simulation results. By analysing the images and graphics of the simulation, the resolving power concept becomes intuitive and understandable.

We present terahertz performance of quasioptical front-ends consisting of a hotelectron bolometer imbedded in a planar feed antenna and integrated with an immersion lens. The impedance and radiation pattern of the log-spiral and double-slot planar feeds are evaluated using the method of moments; the collimating action of the lens is modelled using the physical optics. The total efficiency of the front-ends is computed taking into account frequency dependent impedance of the bolometer. Measured performance of the front-ends qualifies the simulation technique as a reliable tool for the design of terahertz receivers.

The geometrical and wave optics are explored for light emerging from a slab of transparent biaxial crystal with optical activity (chirality), for an incident beam directed along the optic axis. The geometrical optics, here derived from Hamilton's principle, is dominated by a circularly symmetric cusped caustic surface ('spun cusp') threaded by an axial focal line. In wave optics, formulated exactly in the paraxial approximation in terms of integrals previously obtained by Belsky and Stepanov and here derived using a unitary evolution operator, the field is determined by two dimensionless parameters. The geometrical features are decorated by interference, here explored in the focal image plane (where the Poggendorff rings of the non-chiral case are in sharpest focus) and along the axis. Asymptotic approximations are derived in terms of the geometrical optics rays (including interference and evanescent waves), near the spun cusp, and uniformly across the caustic surface far from the cusp.