The Geometric Algebra Lift of Qubits and Beyond

The Geometric Algebra formalism opens the door to developing a theory upgrading conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions; unambiguous definition of states, observables, measurements bring into reality clear explanations of conventional weird quantum mechanical features, particularly the results of double split experiments where particles create diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of quantum mechanics.

Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves

The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates two-dimensional (2D) diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th & –1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and three-dimensional (conical) diffraction problems are discussed.

OPERATOR METHOD IN THE PROBLEM OF A PLANE ELECTROMAGNETIC WAVE DIFFRACTION BY AN ANNULAR SLOT IN THE PLANE OR BY A RING

Purpose: The problem of a plane electromagnetic wave diffraction by an annular slot in the perfectly conducting zero thickness plane is considered. As a dual problem, the problem of diffraction by a perfectly conducting zero thickness ring is also considered. The paper aims at developing the operator method for the axially symmetric structures placed in free space. Design/methodology/approach: The problem is considered in the spectral domain. The scattered field is expressed in terms of unknown Fourier amplitudes (spectral functions). The annular slot is given as a unity of two simple discontinuities, namely of a disk and a circular hole in the plane, which interact with each other. The Fourier amplitude of the scattered field is sought as a sum of two amplitudes, the Fourier amplitude of the field of currents on the disk and Fourier amplitude of the field of currents on the perfectly conducting plane with circular hole. The operator equations are written for these amplitudes, which take into account the electromagnetic coupling of the disk and the hole in the plane. The equations use the reflection operators of a single isolated disk and a single hole in the plane. They are supposed to be known and can be obtained for example by the method of moments.The reflection operators can have singularities. After transformations, the equations are obtained, which are equivalent to the Fredholm integral equations of second kind and they can be solved numerically. Findings: The operator equations relative to the Fourier amplitudes of the field scattered by the discussed structure are obtained. The far zone scattered field for an annular slot and a ring for different values of parameters are studied. Conclusions: The rigorous solution of the problem of the electromagnetic wave diffraction by an annular slot in the plane and by a circular ring is obtained. The problem is reduced to the Fredholm integral equations of second kind. The far field distribution for different parameters is studied. The developed approach is an effective instrument for a number of problems of antenna technique to be solved. Key words: circular hole; disk; annular slot; ring; operator method; diffraction

Theory of vector equations of electromagnetic wave diffraction on a rectilinear tubular cylinder

The vector equation for the diffraction of electromagnetic waves on the surface of a rectilinear circular cylinder without ends with respect to surface currents is considered. As a result of transformations from the original equation, one-dimensional systems of integral equations are obtained. For all four integral operators describing the systems, the main parts are highlighted. Using the remarkable properties of one-dimensional diffraction operators, the Fredholm equation of the second kind in Sobolev spaces is obtained.