Two-wave Approximation Research Articles

Coherent X-ray radiation excited by a relativistic electron in a periodic layered medium

The work is devoted to the development of the theory of coherent X-ray radiation near the direction of Bragg scattering of a beam of relativistic electrons crossing a target with a periodic layered structure with three different amorphous layers per period. The coherent X-ray radiation is considered within the two-wave approximation of the dynamic theory of diffraction as a sum of parametric X-ray radiation and diffracted transition radiation excited in the target by a relativistic electron. Expressions describing the spectral-angular and angular densities of PXR and DTR have been obtained and studied.

Numerical and Analytical Investigation of Shock Wave Processes in Elastoplastic Media

The Wilkins model for an elastoplastic medium is considered. A theoretical analysis of discontinuous solutions under the assumption of one-dimensional uniaxial strain is performed. In this approximation, the material equations for the deviator stress tensor components are integrated exactly, and only the conservative part of the governing equations remains, which makes it possible to derive a class of exact analytical solutions for the model. To solve the full nonconservative system of equations (without assuming the uniaxial strain), a Godunov-type numerical method is developed, which uses an approximate Riemann solver based on integrating the system of equations along a path in the phase space. A special choice of path is proposed that reduces the two-wave HLL approximation to the solution of a linear equations. Numerical and exact analytical solutions are compared for a number of problems with various regimes of shockwave processes.

Is it possible to consider a thin hologram as a slide?

A thin phase hologram with sinusoidal modulation of the refractive index is considered. The applicability of the approach is discussed, in which it is assumed that light passes through a hologram, as through a slide, which leads to a sinusoidal modulation of the wave phase. A consequence of this consideration are expressions for the diffraction orders amplitudes, which coincide with the expressions obtained in the Raman-Nath theory of light scattering by ultrasonic waves. The paper compares expressions for the amplitude of the first order of diffraction, obtained in the framework of several theoretical approaches, including perturbation theory in a rigorous formulation of the boundary value problem. As the thickness of the hologram tends to zero, the expression obtained in this way coincides with the expression obtained in the two-wave approximation of the coupled-wave theory. Both expressions satisfy the electrodynamic reciprocity theorem. Under the same conditions, the expression obtained as a result of the slide-like consideration does not satisfy the reciprocity theorem and differs from the indicated expressions by a factor equal to the ratio of the cosines of the diffracted and incident waves. It is concluded that considering a thin hologram as a slide can lead to noticeable errors.

Coherent X-ray radiation excited by a beam of relativistic electrons in a layered periodic structure

The dynamic theory of coherent X-ray radiation, excited in a periodic layered medium by a divergent beam of relativistic electrons in the Bragg scattering geometry, is developed. In framework of the two-wave approximation of the dynamic theory of diffraction, expressions describing the spectral-angular and angular distributions of parametric X-ray radiation (PXR) and diffracted transition radiation (DTR) are obtained taking into account the multiple scattering of the relativistic electrons by the atoms of the target. Based on the expressions obtained, the possibilities of the manifestation of the effects of dynamical diffraction in coherent X-rays were investigated. The influence of the asymmetry of the electron Coulomb field reflection with respect to the target surface (the reflecting layers situate nonparallel to target surface) on the spectral-angular characteristics of the PXR and the DTR under conditions of the electron multiple scattering is estimated. It is shown that at a fixed Bragg angle the width of the PXR spectrum increases when the angle of incidence of the electron on the target decreases, which leads to an increase in the PXR angular density. In the same conditions the width of the frequency domain of total external reflection and the amplitude of the DTR spectrum also increase, which leads to a significant increase in the angular density of the DTR. The obtained analytical expressions can be used to determine the optimal parameters of the experiment on confirmation of the predicted dynamic effects.

Coherent X-ray Radiation Excited by a Beam of Relativistic Electrons in a Single Crystal in the Direction of Beam Axis

In the framework of the two-wave approximation of the dynamic diffraction theory, a theory of coherent X-ray radiation of a relativistic electron beam in a single crystal in the direction close to the direction of the beam axis in the Bragg scattering geometry is developed for the general case of asymmetric reflection of the electron field relative to the target surface. Expressions describing the spectral-angular characteristics of parametric X-ray radiation near the velocity direction, transition radiation, and their interference term are obtained and studied.

Interpretation of the results of the experiment on generation of parametric X-radiation by relativistic electrons in a single-crystal target

The coherent X-ray radiation generated by a relativistic electron beam in a monocrystalline target of a limited thickness is investigated in the framework of two-wave approximation of the dynamical theory of diffraction in the conditions of asymmetric reflection of the radiation waves in relation to the target surface. The calculations of the radiation angular density for the conditions of the recent experiment on generation of parametric X-ray radiation (PXR) have been carried out. It has been shown, that adequate description of such experiments is possible only with use of the dynamical theory which takes into account the effects of asymmetric reflection of the radiation.

Edge localized modes of cold neutrons in periodic condensed media

It is found that for certain energies of discreet cold neutrons, quasi-stationary eigen solutions of the corresponding Schrodinger equation, which are localized in the layer of a periodic medium, exist. The localization time of these solutions is strongly dependent on the layer thickness, being finite for a finite layer thickness and increasing indefinitely upon a infinite growth of the layer thickness as the third power of the layer thickness. The problem has been solved in the two-wave approximation of the dynamic diffraction theory for the neutron propagation direction coinciding with the periodicity axes (normal incidence of the neutron beam on the layer). The expressions for neutron eigenwave functions in a periodic medium, the reflection and transmission coefficients, and the neutron wavefunction in the layer as a function of the neutron energy incident on the layer have been determined. It turns out that for the certain discrete neutron energies, the amplitudes of the neutron wavefunction in the layer reach sharp maxima. The corresponding energies are just outside of the neutron stop band (energies forbidden for neutron propagation in the layer) and determine the energies of neutron edge modes (NEMs) localized in the layer, which are direct analogs of the optical edge modes for photonic crystals. The dispersion equation for the localized neutron edge modes has been obtained and analytically solved for the case of thick layers. A rough estimate for the localization length L is L ~(db N)–1, where b is the neutron scattering length, d is the crystal period, and N is the density of nuclei in the crystal. The estimates of the localized thermal neutron lifetime show that acheaving of a lifetime close to the free neutron lifetime seems nonrealistic due to absorption of thermal neutrons and requires a perfect large size crystal. Nevertheless, acheaving the localized neutron lifetime exceeding by ~104 times the neutron time of flight through the layer appears as experimentally attainable. The perspectives of the NEM observation are briefly discussed. It is proposed to use NEM for ultrahigh thermal neutron monochromatization by means of NEM excitation in perfect single crystals.

Coherent X-ray radiation by relativistic electron in a structure “amorphous layer-vacuum-periodic layered medium”

The dynamic theory of coherent X-ray radiation by relativistic electron crossing a three-layer structure consisting of an amorphous substance layer, a layer of vacuum and a layer with artificial periodic structure has been developed. The process of radiation and propagation of X-ray waves in an artificial periodic structure have been considered based on two-wave approximation of dynamic diffraction theory in Laue scattering geometry.

Observation of optical second-harmonic generation in porous-silicon-based photonic crystals in the Laue diffraction scheme

Second-harmonic generation (SHG) in the Laue scheme of the dynamical Bragg diffraction in one-dimensional photonic crystal (PhC) is studied. The experiments are performed for partially annealed porous-silicon PhC containing 250 periods of the structure. Our measurements confirm that the phase-matched optical SHG is observed under the Bragg conditions, which is evidenced by a narrow angular and spectral distribution of the diffracted SHG outgoing the PhC. This is confirmed by both the analytical description of the SHG process performed in the two-wave approximation, and by direct calculations of the PhC dispersion curves for the fundamental and SHG wavelengths by the revised plane wave method. Possible types of phase- and quasi-phase-matching realized in the studied PhC under the Laue diffraction scheme are discussed.

X-ray topography using the forward transmitted beam under multiple-beam diffraction conditions

X-ray topographs are taken for a sapphire wafer with the [0001] surface normal, as an example, by forward transmitted synchrotron x-ray beams combined with two-dimensional electronic arrays in the x-ray detector having a spatial resolution of 1 μm. They exhibit no shape deformation and no position shift of the dislocation lines on the topographs. Since the topography is performed under multiple-beam diffraction conditions, the topographic images of a single diffraction (two-wave approximation condition) or plural diffractions (six-wave approximation condition) can be recorded without large specimen position changes. As usual Lang topographs, it is possible to determine the Burgers vector of each dislocation line. Because of high parallelism of the incoming x-rays and linear sensitivity of the electronic arrays to the incident x-rays, the present technique can be used to visualize individual dislocations in single crystals of the dislocation density as high as 1 × 10(5) cm(-2).

Expressions for reflection and transmission coefficients for one-dimensional photonic quasicrystals

We give general expressions for the light reflection and transmission coefficients for one-dimensional (1D) photonic quasicrystals in the framework of the so-called two-wave approximation. A special attention is paid to quasicrystals composed of dielectric layers, where the reflection and transmission coefficients as a function of the length of the structure take a simple form. The expression for the diffraction vectors of a 1D quasicrystal with an arbitrary number of different constituent segments is discussed as well.

Evidence for a dispersion relation of optical modes in the cuticle of the scarab beetle Cotinis mutabilis

Variable angle Mueller matrix spectroscopic ellipsometry is used to study the properties of light reflected from the exoskeleton (cuticle) of the scarab beetle Cotinis mutabilis. For unpolarized incident light, the ellipticity and degree of polarization of the reflected light reveal a left-handed helical structure in the beetle cuticle. Analysis of the spectral position of the maxima and minima in the interference oscillations of the Mueller-matrix elements provides evidence for a dispersion relation similar to that of optical modes in chiral nematic liquid crystals calculated within a two-wave approximation. Additionally, a structural model for the cuticle of C. mutabilis is derived from the properties of the optical modes for non-attenuated propagation or selective reflection.

Coherent X-Ray Radiation of a Relativistic Electron in a Bilayer Amorphous Layer – Single Crystal Target

UDC 537.8 A theory of coherent X-ray radiation is proposed, which is generated when a relativistic electron is transversing a hybrid target consisting of two parallel plates: amorphous and crystalline. Within a two-wave approximation of the dynamic diffraction theory, expressions describing the spectral-angular density of coherent radiation from this target are obtained.

A two-dimensional theory of a carcinotrode developed with allowance for the finite intensity of the focusing magnetic field

A kinematic theory of a backward-wave tube with self-modulated emission (carcinotrode) is formulated with allowance for the effect of the focusing magnetic field. Using the method of successive approximations and the dispersion equation technique, the starting conditions for the carcinotrode in the general case and in the two-wave approximation (the synchronous regime and the new cyclotron resonance regime) are obtained.

Dynamical theory of the resonant X-ray diffraction in coplanar Bragg geometry

A dynamical theory of the resonant diffraction of X-ray synchrotron radiation within the two-wave approximation in coplanar Bragg geometry at large grazing angles in perfect crystals has been developed. The predictions of the kinematical and dynamical diffraction theories are compared by the example of Ge crystals.

Borman effect in resonant diffraction of X-rays

A dynamic theory of resonant diffraction (occurring when the energy of incident radiation is close to the energy of the absorption edge of an element in the composition of a given substance) of synchronous X-rays is developed in the two-wave approximation in the coplanar Laue geometry for large grazing angles in perfect crystals. A sharp decrease in the absorption coefficient in the substance with simultaneously satisfied diffraction conditions (Borman effect) is demonstrated, and the theoretical and first experimental results are compared. The calculations reveal the possibility of applying this approach in analyzing the quadrupole-quadrupole contribution to the absorption coefficient.

Fermi-edge polaritons in Bragg multiple-quantum-well structures

The polariton propagation in one-dimensional photonic crystals and quasicrystals based on highly doped quantum-well structures has been theoretically studied. The superradiant and photonic crystal regimes have been considered and expressions for band gap edges for light waves in the Bragg structures have been obtained. The reflection and absorption spectra of such systems have been calculated and the so-called two-wave approximation to their study has been applied. The optical properties of the doped multiple-quantum-well structures are compared with those of undoped ones.

Modification of the theory of diffracted channeling radiation for axial electron and positron channeling

A new type of combinational channeling radiation induced by subbarrier (interband) transitions for the transverse motion of relativistic electrons (positrons) is studied. It is known as diffracted channeling radiation (DCR). The formula describing the DCR angular distribution in the case of axial channeling is obtained by taking into account the band structure of energy levels for the transverse motion of electrons (positrons). It is shown that, in the two-wave approximation of the wave function A(r) of virtual photons, the DCR matrix elements in the dipole approximation for axial and plane channeling coincide formally (with the dimension of the problem taken into account). However, the formulas for DCR angular distributions in the cases of axial and plane channeling differ considerably.

Two-wave approximation in surface effects in asymmetric Laue crystals

This paper is a study of surface effects, e.g. roughness or asymmetrical cut, in the Laue diffraction of X-rays by crystals, based on the Takagi-Taupin equations. By means of Riemann-Green integrals, first a formal solution has been obtained when the entrance and the exit surfaces are arbitrary. Then a coordinate transformation mapping a propagation domain with arbitrary boundaries into a rectangular domain with straight boundaries is given. Potential measurement errors in \gamma-ray wavelength and silicon lattice-parameter measurements by double-crystal diffractometry and X-ray interferometry, respectively, are outlined and anticipated by studying, in the two-wave approximation, the reflection peak shift and extra phase originating from an asymmetrically cut crystal. A relationship between analyser displacement, interferometer-signal phase and relative uncertainty in lattice-parameter measurement is also given.

Manifestation of dynamic effects in coherent X-radiation from relativistic electron in Bragg scattering geometry

Analytic expressions describing the spectral-angular distribution of parametric X-radiation (PXR) and diffracted transient radiation (DTR) of a relativistic electron traversing a single-crystal plate in the Bragg scattering geometry are derived on the basis of the two-wave approximation of the dynamic diffraction theory. The expressions are obtained in the general case of asymmetric reflection of a particle’s field relative to the surface of the crystal plate. It is shown that in the given geometry, a significant increase of the PXR angular density is possible due to the dynamic effect of PXR spectral broadening. The conditions for an increase in the DTR spectrum width, which causes a considerable increase of the DTR angular density, are derived. The possibility of the most pronounced manifestation of the Bormann dynamic effect in PXR is demonstrated for a thick absorptive crystal.