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In present-day economic conditions, extrusion is one of the advanced feedstuffs and food production processes involving intensive manifold heat and force action. During extrusion, the main function is performed by a compression mechanism which includes a forcing screw unit built into the cylinder (cowling). The forcing mechanism performs the task of transferring material while concurrently compressing it up to a required pressure and increasing the material’s temperature due to compression and friction against the cowling's sides. In our view, the first technique is appropriate when a material which is moving within the compression area is in viscous-flow state. The second technique is optimal for theoretical description of areas of loading, transportation, and compression (melting) where the working pressure is maintained.

The method of calculating the strength of fruit tree trunks during operation of a pneumatic fruit shaker is presented. Taking into account the technological and design parameters of the pneumatic shaker, the law of changing the bending moment along the length of the tree trunk is derived, the required angle of rotation of the fan nozzle of the pneumatic shaker is determined.

The article considers the theoretical calculation of a volumetric liquid dispenser. As a result of the calculations, a formula was obtained that relates all the main parameters of the dispenser to the angular speed of rotation of the metering tank, which allows calculating design and operating parameters of the metering devices.

The article describes a method for an axial fan selection for a pneumatic fruit shaker. Based on the design of the pneumatic shaker and the input parameters of its operation (air flow rate at the nozzle exit and the crown size of the treated trees), the fan characteristics (capacity, generated pressure, and power consumption) are determined. The brand of suitable commercially available fan is recommended.

We develop a thin phase screen model for the transmission of a surface plasmon polariton through a curved boundary between two different co-planar metal surfaces. The result is illustrated by applying it to the determination of the intensity distribution of the electric field of a surface plasmon polariton that has been transmitted through a quadratic (parabolic) boundary, through a cubic boundary, and through a periodic boundary.

An approach is presented for the inversion of experimental in-plane, co-polarized light scattering data in p and s polarization to obtain the normalized surface-height autocorrelation function and the rms-roughness of a two-dimensional randomly rough metal surface. The approach is based on an expression, obtained on the basis of second-order phase perturbation theory, for the contribution to the mean differential reflection coefficient from the light scattered diffusely by the rough surface. The inversion scheme is validated by using several sets of computer generated scattering data for rough silver surfaces defined by Gaussian surface height correlation functions. The reconstructions obtained by this approach are found to be rather accurate for weakly rough surfaces illuminated by p- and s-polarized incident light; this is also true in cases where the contributions to the input data from multiple scattering of surface plasmon polaritons is not insignificant. Finally, the inversion scheme is applied to experimental scattering data obtained for characterized two-dimensional randomly rough gold surfaces, and the results obtained in this way, compare favorably to what is obtained by directly analyzing the surface morphology. Such results testify to the attractiveness of the computationally efficient inversion scheme that we propose., Comment: 24 pages, 11 figures

The article provides a way to increase the productivity of a screw press variety – the expander. The use of an expander makes it possible to produce compound feed in the form of non-solid whole meals, which has a beneficial effect on the process of digestibility by animals. To correct the existing drawback of the expander – the reverse extrusion of the working mass through the gap between the screw and the body, the authors proposed a combined profile of the screw spiral groove: the left lateral (pushing) surface of the groove has the shape of a semicircle, and the right lateral surface has a semicircular shape in the lower part, and in the middle and upper – the shape of a sloping straight line. Theoretical studies have shown that this profile contributes to the appearance of a difference in lateral pressure forces from the side of boundary particles on the bulk of the material, in contrast to screws with a standard spiral groove made in the form of a triangle, trapezium, or semicircle. In turn, the uneven distribution of lateral pressure will prevent backward extrusion of the processed material through the gap between the auger and the body, thereby increasing the performance of the expander. Theoretical dependencies have been obtained that allow, on the basis of the basic geometric parameters of the screw, to describe geometrically the configuration of the screw of the proposed design for its subsequent manufacture., Volga Region Farmland, Выпуск 2 (6) 2020

The diffraction of a scalar plane wave from a doubly-periodic surface on which either the Dirichlet or Neumann boundary condition is imposed is studied by means of a rigorous numerical solution of the Rayleigh equation for the amplitudes of the diffracted Bragg beams. From the results of these calculations the diffraction efficiencies of several of the lowest order diffracted beams are calculated as functions of the polar and azimuthal angles of incidence. The angular dependencies of the diffraction efficiencies display features that can be identified as Rayleigh anomalies for both types of surfaces. In the case of a Neumann surface additional features are present that can be attributed to the existence of surface waves on such surfaces. Some of the results obtained through the use of the Rayleigh equation are validated by comparing them with results of a rigorous Green's function numerical calculation., Comment: 16 pages, 5 figures

By the use of phase perturbation theory we show that if a single realization of a one-dimensional randomly rough interface between two dielectric media is illuminated at normal incidence from either medium by a broadband Gaussian beam, it produces a scattered field whose differential reflection coefficient closely matches the result produced by averaging the differential reflection coefficient produced by a monochromatic incident beam over the ensemble of realizations of the interface profile function., 10 pages, 7 figures

We study the optical properties of metallic lamellar gratings with sub-wavelength period as a function of the groove depth and width, in situations in which there is only one propagating order (order zero or specular). The reflected light is calculated with an integral equation method for periodic surfaces. Under p-polarized illumination, different kinds of anomalies are observed as the angle of incidence or the grating parameters are changed. These are due to the excitation of surface plasmon polaritons (SPPs) and their interaction with the plasmonic and photonic modes supported by the grating grooves. We find that the propagation constant of the SPPs supported by shallow gratings is affected by the presence of the corrugations. As the groove depth increases, the formation of an energy gap inhibits the excitation of SPPs. However, for grooves deeper than about half a wavelength, new resonances appear. These are due to the coupling of the SPP excitations with the photonic and plasmonic modes supported by the grating grooves.

We measure and calculate the optical response of a structure consisting of a square array of subwavelength silicon posts on a silicon substrate at telecommunication wavelengths. By the use of the reduced Rayleigh equations and the Fourier modal method (rigorous coupled wave analysis) we calculate the reflectivity of this structure illuminated from vacuum by normally incident light. The calculated reflectivities together with experimentally determined ones, are used to test the accuracy of effective medium theories of the optical properties of structured silicon surfaces, and to estimate the effective refractive index of such surfaces produced by a homogeneous layer model.

A periodically corrugated interface between vacuum and a high-index dielectric medium supports a p-polarized leaky surface electromagnetic wave whose sagittal plane is perpendicular to the generators of the interface. This wave is bound to the surface in the vacuum region, but radiates into the high-index dielectric medium. We study the excitation of this wave by p-polarized light incident from a prism on whose planar base the highindex dielectric medium in the form of a film is bonded. The unilluminated surface of the film is periodically corrugated, and is in contact with vacuum. Peaks and dips in the dependence of several low-order diffraction efficiencies on the angle of incidence (Wood anomalies) are the signatures of the excitation of the surface wave.

In this research work, we have studied and compared the main parameters of the rheological state of dough made from triticale flour (a variety of the promising line by the FSBIS Agricultural Research Institute of the South-East), dough made from flour of white and red bean seeds, as well as parameters of dough from flour of composite mixtures based on them. It has been found that the water absorption capacity of flour highly correlates with moments of force during the liquefaction phase and with moments of force characterizing the minimum and maximum consistency of the dough during the phase of starch retrogradation. The type of beans significantly influenced the correlation of the water absorption capacity of flour with the dough stability, the moment of force characterizing the stability of the dough during the gelatinization phase, as well as with the total energy consumption for kneading the dough. Taking into account the research results of the rheological state of the dough, bread with various mass fractions of components was baked experimentally. The results obtained confirmed the improving effect of bean flour, which is in the fact that even with a high mass fraction (60-100%) of bean flour in the composite mixture, it was possible to obtain a complete rheological profile of the studied system, and, consequently, preservation of the optimal dough structure.

Department of Food Technology, at Saratov State Vavilov Agrarian University, studied and compared the main parameters of the rheological state of the dough from whole grain wheat flour and straight white wheat flour, dough from white beans and red beans. Dough parameters from flour composite mixtures based on beans were also studied and compared with each other. It was found that the water absorption ability of flour highly correlates with dough stability time, with a force moment characterizing the minimum consistency of the dough during the starch retrograde phase and with a force moment characterizing the maximum consistency of the dough during the end of the starch retrograde phase, as well as with the total energy consumed during dough mixing. To a lesser extent, the water absorption ability of flour correlates with the time of dough development, the force moment during the liquefaction phase, and the force moment characterizing the stability of the dough during the gelatinization phase. When comparing the main rheological parameters of the dough from the studied types of flour, it was found that both the degree of grinding and the morphological characteristics of the beans used as a donor or basis for composite mixtures influence the correlation between rheological parameters. The water absorption ability correlates to a greater extent with the main rheological indices of the dough of composite mixtures based on straight white wheat flour rather than composite mixtures based on whole grain flour.

A planar surface on which a scalar wave satisfies the Neumann boundary conditions does not support a surface wave. However, a structured Neumann surface can support such a wave. By means of a Rayleigh equation for a scalar field in the region above the two-dimensional rough surface of a semi-infinite medium on which the Neumann boundary condition is satisfied, we derive the dispersion relation for surface waves on both doubly periodic and randomly rough surfaces. Dispersion curves for these waves on doubly periodic surfaces with three forms of the surface profile function are presented together with dispersion curves for surface waves on a two-dimensional randomly rough surface.

The scattering of electromagnetic waves from films with randomly rough surfaces and/or volume disorder is discussed in this chapter. One or both of the surfaces of the film are rough. The surface roughness considered is either one-dimensional or two-dimensional, while the volume disorder is three-dimensional. Advances in computational approaches to the solution of these scattering problems, with an emphasis on multiple-scattering phenomena are described, and results obtained by these approaches are presented. Examples of applications of the results of calculations of scattering from rough surfaces of, and volume disorder in, films are presented with a focus on remote sensing.

The reduced Rayleigh equation for the scattering of a surface plasmon polariton incident non-normally on a one-dimensional ridge or groove on an otherwise planar metal surface is solved by a purely numerical approach. The solution is used to calculate the transmission, reflection, and out-of-plane scattering coefficients of the surface plasmon polariton. The angular dependence of the out-of-plane scattering is found to have a conical nature.

Author(s): Iurov, A; Huang, D; Gumbs, G; Pan, W; Maradudin, AA | Abstract: The effects of induced optical polarization by Dirac electrons in graphene on the hybridization of radiative and evanescent fields are found. Such effects result in a localized polarization field which significantly modifies an incident surface-plasmon-polariton (SPP) field. This yields a high sensitivity to local dielectric environments and provides an investigative tool for molecules or proteins selectively bound with carbons. A scattering matrix is utilized with varied frequencies in the vicinity of the surface-plasmon (SP) resonance for the increase, decrease, and even a full suppression of the polarization field, which enables accurate effective-medium theories to be constructed for Maxwell-equation finite-difference time-domain methods. Moreover, double peaks in the absorption spectra for hybrid SP and graphene-plasmon modes are significant only with a large conductor plasma frequency, but are overshadowed by a round SPP peak at a small plasma frequency as the graphene is placed close to the conductor surface. These resonant absorptions facilitate polariton-only excitations, leading to polariton condensation for a threshold-free laser.

A back action from Dirac electrons in graphene on the hybridization of radiative and evanescent fields is found as an analogy to Newton's third law. Here, the back action appears as a localized polarization field which greatly modifies an incident surface-plasmon-polariton (SPP) field. This yields a high sensitivity to local dielectric environments and provides a scrutiny tool for molecules or proteins selectively bounded with carbons. A scattering matrix is shown with varied frequencies nearby the surface-plasmon (SP) resonance for the increase, decrease and even a full suppression of the polarization field, which enables accurate effective-medium theories to be constructed for Maxwell-equation finite-difference time-domain methods. Moreover, double peaks in the absorption spectra for hybrid SP and graphene-plasmon modes are significant only with a large conductor plasma frequency, but are overshadowed by a round SPP peak at a small plasma frequency as the graphene is placed close to conductor surface. These resonant absorptions facilitate the polariton-only excitations, leading to polariton condensation for a threshold-free laser., 14 pages and 6 figures

We use phase perturbation theory to design a one-dimensional randomly rough interface between two dielectric media that produces a specified mean differential transmission coefficient for light transmitted through it.

We show how to design a one-dimensional randomly rough interface between two dielectric media that, when illuminated at normal incidence by a Gaussian beam, produces a transmitted beam whose spectral density in the mean interface has a Gaussian form, while its spectral degree of coherence in this plane has a nearly arbitrary form.

By the use of Green's second integral identity we determine the field scattered from a two-dimensional randomly rough isotropic or anisotropic Dirichlet or Neumann surface when it is illuminated by a scalar Gaussian beam. The integral equations for the scattering amplitudes are solved nonperturbatively by a rigorous computer simulation approach. The results of these calculations are used to calculate the full angular distribution of the mean differential reflection coefficient. For isotropic surfaces, the results of the present calculations for in-plane scattering are compared with those of earlier studies of this problem. The reflectivities of Dirichlet and Neumann surfaces are calculated as functions of the polar angle of incidence, and the reflectiveties for the two kinds of surfaces of similar roughness parameters are found to be different. For an increasing level of surface anisotropy, we study how the angular intensity distributions of the scattered waves are affected by this level. We find that even small to moderate levels of surface anisotropy can significantly alter the symmetry, shape, and amplitude of the scattered intensity distributions when Gaussian beams are incident on the anisotropic surfaces from different azimuthal angles of incidence., Comment: 23 pages, 9 figures

We study properties of a p-polarized surface plasmon polariton propagating circumferentially around a portion of a cylindrical interface between a vacuum and a metal, a situation investigated earlier by Berry (J. Phys. A: Math. Gen. 8 (1975) 1952). When the metal is convex toward the vacuum this mode is radiative and consequently is attenuated as it propagates on the cylindrical surface. An approximate analytic solution of the dispersion relation for this wave is obtained by an approach different from the one used by Berry, and plots of the real and imaginary parts of its wave number are presented. When the metal is concave to the vacuum, the resulting dispersion relation possesses a multiplicity of solutions that have the nature of waveguide modes that owe their existence to the curvature of the interface.

The transmission of polarized light through a two-dimensional randomly rough interface between two dielectric media has been much less studied, by any approach, than the reflection of light from such an interface. We have derived a reduced Rayleigh equation for the transmission amplitudes when p- or s-polarized light is incident on this type of interface, and have obtained rigorous, purely numerical, nonperturbative solutions of it. The solutions are used to calculate the transmissivity and transmittance of the interface, the mean differential transmission coefficient, and the full angular distribution of the intensity of the transmitted light. These results are obtained for both the case where the medium of incidence is the optically less dense medium and in the case where it is the optically more dense medium. Optical analogues of Yoneda peaks observed in the scattering of x-rays from metal surfaces are present in the results obtained in the former case. For p-polarized incident light we observe Brewster scattering angles, angles at which the diffuse transmitted intensity is zero in a single-scattering approximation, which depend on the angle of incidence in contrast to the Brewster angle for flat-surface reflection., 23 pages, 15 figures

The rise and sustainable development of mining enterprises are inextricably linked with the creation of competitive technology for advanced technologies in construction, mining, engineering and geological surveys. In the improvement of characteristic technologies, an important role is played by power discrete drives with linear electromagnetic engines, which make it possible to abandon motion converters and especially successfully compete with traditional electric engines in machines and installations where discrete translational motion of the working body is necessary. The kinematic separation of the armature and the output shaft and the use of an axial channel in a linear electromagnetic engine made it possible to create electromagnetic machines with new properties in which the mechanical power is derived using a toothed bar of any desired length. In the work, the force analysis of the interaction of the working body with the locking mechanism in a stepper linear electromagnetic drive engine of the conveyors is carried out. The degree of influence of the angle of inclination of the locking elements on the amount of effort required to drive the working body is revealed.

A model of high-angle general-type grain boundaries is developed and grain-boundary diffusion of niobium in a Nb–Cu system is investigated using the method of molecular dynamics. The results of predictions of the effect of a copper impurity on grain-boundary self-diffusion of niobium are compared with the experimental data on diffusion annealing of multi-layered composites containing Cu/Nb boundaries. Recrystallization of polycrystalline niobium is investigated under the conditions of grain-boundary diffusion of copper.

By the use of the reduced Rayleigh equation for the amplitude of a surface plasmon polariton on a one-dimensional randomly rough metal surface that is in contact with vacuum, we calculate the dispersion and damping of the surface electromagnetic wave to the lowest nonzero order in the rms height of the surface. It is found that the frequency of the surface plasmon polariton is depressed by the surface roughness. The attenuation of the surface plasmon polariton in the long wavelength limit is due primarily to its scattering into other surface plasmon polaritons, while in the short wavelength limit it is due primarily to its roughness-induced scattering into volume electromagnetic waves in the vacuum. The energy mean free path of the surface plasmon polariton is shorter on a randomly rough metal surface than it is on a lossy planar metal surface, and the surface plasmon polariton is more tightly bound to a rough surface than to a planar one.

Analytic expressions for the dispersion relation and electric field profiles of guided waves supported by an asymmetric graded-index dielectric waveguide are derived. The system studied consists of vacuum in the region , and a dielectric medium in the region whose dielectric permittivity decreases continuously with increasing distance into it from the interface according to . Here is the (real) index of refraction of the medium at , and it is assumed that . It is found that for frequencies below a certain critical value given by , where is the speed of light in vacuum, the dispersion curve consists of a single branch that exists in a narrow spectral range. Its electric field decays exponentially with increasing distance into each medium. For frequencies above this critical value, the dispersion curve possesses several branches. The corresponding electric field decays exponentially with increasing distance into the vacuum, and decays in an oscillatory fashion with increasing distance into the grad...

The scattering of polarized light from a dielectric film sandwiched between two different semi-infinite dielectric media is studied experimentally and theoretically. The illuminated interface is planar, while the back interface is a two-dimensional randomly rough interface. We consider here only the case in which the medium of incidence is optically more dense than the substrate, in which case effects due to the presence of a critical angle for total internal reflection occur. A reduced Rayleigh equation for the scattering amplitudes is solved by a rigorous, purely numerical, nonperturbative approach. The solutions are used to calculate the reflectivity of the structure and the mean differential reflection coefficient. Optical analogues of Yoneda peaks are present in the results obtained. The computational results are compared with experimental data for the in-plane mean differential reflection coefficient, and good agreement between theory and experiment is found. Published by Optical Society of America. Open access.

928 (1993)) measured and calculated the independent elements of the Stokes matrix for in-plane scattering from a one-dimensional, randomly rough, metal surface. They found that the agreement between the computer simulation results and the experimental results for these matrix elements was significantly improved if the statistical properties of the surface profi

The electromagnetic coupling of surface-plasmon-polariton (SPP) modes, which are localized around the surface of a conductive substrate, to quantum plasmons in a graphene sheet above the surface is investigated and their hybrid quantum-plasmon modes are analyzed. For a double-layer graphene structure, on the other hand, the interplay between the electromagnetic couplings of SPPs to each graphene sheet is explored. An effective- polarizability tensor for a combined system, including coupled double-layer graphene and conductive substrate, has been derived, which consists of the retarded nonlocal Coulomb interactions between electrons in different graphene sheets and the conductive substrate. Additionally, this calculated effective-scattering tensor can be used for constructing an effective-medium theory to study optical properties of inserted nanorods between the graphene sheets and metallic surface.

The contribution to the mean differential reflection coefficient from the in-plane, co-polarized scattering of p- polarized light from a two-dimensional randomly rough dielectric surface is used to invert scattering data to obtain the normalized surface height autocorrelation function of the surface. Within phase perturbation theory this contribution to the mean differential reflection coefficient possesses singularities (poles) when the polar scattering angle θs equals ±θB= ± tan-1√E, where E is the dielectric constant of the dielectric medium and θB is the Brewster angle. Nevertheless, we show in this paper that if the mean differential reflection coefficient is measured only in the angular range |θs| < θB, these data can be inverted to yield accurate results for the normalized surface height correlation function for weakly rough surfaces. Several parameterized forms of this correlation function, and the minimization of a cost function with respect to the parameters defining these representations, are used in the inversion scheme. This approach also yields the rms height of the surface roughness, and the dielectric constant of the scattering medium if it is not known in advance. The input data used in this minimization procedure consist of computer simulation results for surfaces defined by exponential and Gaussian surface height correlation functions, without and with the addition of multiplicative noise. The proposed inversion scheme is computationally efficient.

We describe a surface structure consisting of a metal-air interface where the metallic part consists of two metallic segments with a periodic modulation of the interface between them. Such a structure possesses a different transmissivity for a surface plasmon polariton incident on it from one side of it than it has for a surface plasmon polariton incident on it from the opposite side. This asymmetric transmission of a surface plasmon polariton is based on the suppression of the zero-order Bragg beam which, for a certain value of the modulation depth, is not transmitted through the structure, while the diffraction efficiencies of the +1 and -1 Bragg beams can be modified by varying the period of grating and/or the angle of incidence. For a certain range of the incidence angle one can observe asymmetry in transmittance for the -1 mode while the +1 mode is completely suppressed. By varying the material and geometrical parameters of the diffractive structure one can control the contrast transmission that characterizes the degree of the asymmetry. This property of the structure is demonstrated by the results of computer simulation calculations., 9 pages, 13 Postscript figures

An expression is obtained on the basis of phase perturbation theory for the contribution to the mean differential reflection coefficient from the in-plane co-polarized component of the light scattered diffusely from a two-dimensional randomly rough dielectric surface when the latter is illuminated by s-polarized light. This result forms the basis for an approach to inverting experimental light scattering data to obtain the normalized surface height autocorrelation function of the surface. Several parametrized forms of this correlation function, and the minimization of a cost function with respect to the parameters defining these representations, are used in the inversion scheme. This approach also yields the rms height of the surface roughness, and the dielectric constant of the dielectric substrate if it is not known in advance. The input data used in validating this inversion consists of computer simulation results for surfaces defined by exponential and Gaussian surface height correlation functions, without and with the addition of multiplicative noise, for a single or multiple angles of incidence. The reconstructions obtained by this approach are quite accurate for weakly rough surfaces, and the proposed inversion scheme is computationally efficient., Comment: 19 pages, 9 figures

The scattering of polarized light incident from one dielectric medium on its two-dimensional randomly rough interface with a second dielectric medium is studied. A reduced Rayleigh equation for the scattering amplitudes is derived for the case where p- or s-polarized light is incident on this interface, with no assumptions being made regarding the dielectric functions of the media. Rigorous, purely numerical, nonperturbative solutions of this equation are obtained. They are used to calculate the reflectivity and reflectance of the interface, the mean differential reflection coefficient, and the full angular distribution of the intensity of the scattered light. These results are obtained for both the case where the medium of incidence is the optically less dense medium, and in the case where it is the optically more dense medium. Optical analogues of the Yoneda peaks observed in the scattering of x-rays from metal surfaces are present in the results obtained in the latter case. Brewster scattering angles for diffuse scattering are investigated, reminiscent of the Brewster angle for flat-interface reflection, but strongly dependent on the angle of incidence. When the contribution from the transmitted field is added to that from the scattered field it is found that the results of these calculations satisfy unitarity with an error smaller than $10^{-4}$., 25 pages, 14 figures

By means of a modal method we have calculated the angular dependence of the reflectivity and the efficiencies of several other diffracted orders of a perfectly conducting lamellar reflection grating illuminated by p-polarized light. These dependencies display the signatures of Rayleigh and Wood anomalies, usually associated with diffraction from a metallic grating. The Wood anomalies here are caused by the excitation of the surface electromagnetic waves supported by a periodically corrugated perfectly conducting surface, whose dispersion curves in both the nonradiative and radiative regions of the frequency-wavenumber plane are calculated. © 2016 IOP Publishing Ltd & European Optical Society. This is the authors' accepted and refereed manuscript to the article.

By the use of the Rayleigh method we have calculated the angular dependence of the reflectivity and the efficiencies of several other diffracted orders when the periodically corrugated surface of an isotropic elastic medium is illuminated by a volume acoustic wave of shear horizontal polarization. These dependencies display the signatures of Rayleigh and Wood anomalies, usually associated with the diffraction of light from a metallic grating. The Rayleigh anomalies occur at angles of incidence at which a diffracted order appears or disappears; the Wood anomalies here are caused by the excitation of the shear horizontal surface acoustic waves supported by the periodically corrugated surface of an isotropic elastic medium. The dispersion curves of these waves in both the nonradiative and radiative regions of the frequency-wavenumber plane are calculated, and used in predicting the angles of incidence at which the Wood anomalies are expected to occur. This is the authors' accepted and refereed manuscript to the article. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. The following article appeared in Low Temperature Physics and may be found at http://aip.scitation.org/doi/pdf/10.1063/1.4948617

We study theoretically the propagation of a wave packet that is a superposition of three s-polarized guided waves with different frequencies in a planar waveguide consisting of a dielectric medium with a graded index of refraction, sandwiched between perfectly conducting walls. The electric field at each point within the waveguide is calculated, and it is shown that each of the constituent modes ceases to propagate at a specific distance along the waveguide that depends on its frequency and on the geometrical and material parameters defining the waveguide. This simple model displays the phenomenon of rainbow trapping of guided waves in an explicit fashion, without the use of a negative index metamaterial.

We study theoretically the negative refraction of a surface plasmon polariton. The system we consider consists of a vacuum that is in a contact with two metals that each fill a quarter space with a periodic boundary between them. The interface of each of the metals with the vacuum supports a surface plasmon polariton. When a surface plasmon polariton passes through the periodic boundary between the metals it transforms partly into the Bragg beams of the refracted and reflected surface plasmon polaritons and partly into volume waves in the vacuum. A suitable choice of the period can ensure that the only propagating Bragg beams of the refracted surface plasmon polariton are the specular beam and the (−1)-order beam, which is negatively refracted. With a suitable choice for the form of the profile function and its amplitude the specular beam can be made to vanish.