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An analysis of guided modes in graphene waveguides in the presence of magnetic-electric barrier is presented. The graphene waveguide is controlled by the gate voltages and the magnetic field by depositing two parallel ferromagnets on a dielectric layer. Both in the classical motion and Klein tunneling cases, the fundamental modes exist with the applied magnetic field. The electron wave propagation can be modulated in two different ways by changing the magnetic intensity or the incident energy. We hope that these characteristics can provide potential applications in the graphene-based waveguide devices.

We study the bistable and negative lateral shifts of the reflected light beam from Kretschmann configuration containing left-handed metamaterials with self-focusing and self-defocusing Kerr-type nonlinearity. It is shown that the lateral shifts can be large and negative when the thickness of the middle metal film is smaller than the critical value. Taking the nonlinear effect into account, there exists a hysteretic response between the beam shift and the intensity of the incident light beam. These results suggest that the bistable and negative beam shifts can be modulated by nonlinear coefficients and intensity of incident beam, which might be used in integrated photonics and optical switches.

In this Letter, we investigate giant and negative bistable lateral shifts for a light beam transmitted through a one-dimensional photonic crystal containing a nonlinear metamaterial defect. It is shown that there exists a typical S-shaped curve of the normalized input–output intensity. The hysteresis of lateral shift, resulting from the reshaping effect (constructive and destructive interferences of each plane wave components undergoing different phase shifts) is dependent on the nonlinear defect, the incident intensity and incidence angle. Numerical simulations are also made for the validity of stationary phase approach. All these phenomena lead to potential applications in integrated optics and optical switches.

We present a solution to the problem of reflection and transmission of a polarized paraxial light beam at an interface between two homogeneous media by using a two-form amplitude and an extension matrix to represent the vectorial angular spectrum of a three-dimensional (3D) light beam. We derive general formulas for the Imbert–Fedorov (IF) shift of the reflected and transmitted beams of a polarized paraxial light beam. The IF shift of two different types of polarized beams is calculated, and the influence of the polarization state and the polarization feature of the vectorial angular spectrum on the IF shift is discussed.

Since the Goos-Hänchen (GH) effect is the displacement of the totally reflected beam at a dielectric interface from the position prediction by geometrical reflection, the concept of GH displacement is applicable only when the reflected beam retains the shape of the geometrically reflected or incident beam. The necessary and sufficient condition has been advanced for the totally reflected beam to retain the shape of the incident beam. Numerical simulations have been performed to confirm this condition. It has been shown that the GH displacement results from the mechanism of beam reshaping due to the linear dependence of the reflection phase shift upon the parallel component of the wave vector, in the interval in which the angular spectral distribution function of the incident beam is appreciable.

Photon tunnelling in a metamaterial photonic crystal composed by positive index material (PIM) and negative index material (NIM) layers is discussed. First, a single period of this photonic crystal is considered as the barrier in a frustrated total internal reflection (FTIR) structure, and the transmittance, the post-tunnelling position shift and the group delay time of different frequency incident waves through the barrier are calculated. In the case of a barrier of more periods of photonic crystal, the transmittance and the group delay time also vary with layer number, which are analysed in detail.

The propagation of a Gaussian-shaped beam through a slab of optically denser dielectric medium in the air, a two-dimensional optical counterpart of the quantum potential well, is investigated in order to elucidate the negative lateral shifts of transmitted and reflected beams. It is shown that the locus of the peak of the beam inside the slab consists of two parts. One propagates backwards and seems to be the geometric reflection of the other that propagates forwards. But they are not continuous with the loci of the peaks of the incident and reflected beams. A restriction to the thickness of the slab is obtained that is necessary for the shape of the beam to remain unchanged in the travelling.

A parameter, called the degree of paraxiality, is introduced upon examining the property of energies originating from the longitudinal as well as the transverse components of both the electric and magnetic fields. It is shown that this degree of paraxiality is determined by the magnitude of the plane-wave spectrum, without depending on the vectorial feature of the plane-wave spectrum. Its value ranges from 0 to 1. A larger value stands for a more paraxial beam. As an application, two examples are considered.

It is observed that there exist two different classes of vector light beams that have the same first-order paraxial approximation. They are characterized by the axial and transverse orientations of a constant unit vector that comes from the constraint of transversality condition. Their vectorial structures in the nonparaxial regime and angular momentum properties are compared.

This is a reply to the Comment by Bliokh on our paper that appeared in Opt. Commun. 281 (2008) 3427. After a brief introduction of the representation theory of vector electromagnetic beams advanced in a recent paper, I point out that the Imbert–Fedorov effect is the evidence of the change of the beam parameter Θ and the polarization ellipticity σ caused by the reflection or transmission process in the linear approximation. Then I explain that it is because the linear approximation of the incident beam that we used in our paper had been assumed in previous works that we reproduced their results.

The quantum Goos-H\"{a}nchen shift for ballistic electrons is investigated in a parabolic potential well under a uniform vertical magnetic field. It is found that the Goos-H\"{a}nchen shift can be negative as well as positive, and becomes zero at transmission resonances. The beam shift depends not only on the incident energy and incidence angle, but also on the magnetic field and Landau quantum number. Based on these phenomena, we propose an alternative way to realize the spin beam splitter in the proposed spintronic device, which can completely separate spin-up and spin-down electron beams by negative and positive Goos-H\"{a}nchen shifts., Comment: 6 pages, 6 figures

Motivated by the realization of the Dirac point (DP) with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM), the reflection, transmission, and Goos-H\"{a}nchen (GH) shifts inside the NZPIM slab are investigated. Due to the linear Dirac dispersion, the transmission as the function of the frequency has a gap, thus the correspond reflection has a frequency or wavelength window for the perfect reflection, which is similar to the Bragg reflection in the one-dimensional photonic crystals. Near the DP, the associated GH shifts in the transmission and reflection can be changed from positive to negative with increasing the wavelength. These negative and positive shifts can also be enhanced by transmission resonances, when the frequency is far from that at the DP. All these phenomena will lead to some potential applications in the integrated optics and optical devices., Comment: 5 pages, 5 figures

It is investigated that the lateral shifts of the ballistic electrons transmitted through a semiconductor quantum slabs can be negative as well as positive, which are analogous to the anomalous lateral shifts of the transmitted light beam through a dielectric slab. The necessary condition for the shift to be negative is advanced. It is shown that the lateral shifts depend not only on the structure parameters of semiconductor quantum slab, but also on the incidence angle and the incident energy. Numerical calculations further indicate that the lateral shifts can be tuned from negative to positive by the external applied electric field. The voltage-tunable lateral shifts may lead to potential applications in quantum electronic devices., Comment: 6 pages, 6 figures, submitted to J. Appl. Phys

We report that the transmitted evanescent light beam under total reflection will experience a lateral displacement similarly to that of the reflected beam, called Goos-Hanchen (GH) effect. By a dielectric thin film coated onto the dielectric surface, which is usually considered as the near-field enhanced configuration in atom optics, we indicate that the displacement of the transmitted evanescent beam can be greatly enhanced to several tens of wavelengths at transmission resonance under total internal reflection. Numerical simulations have been performed and it is shown that the transmitted evanescent beam maintains well the shape of the incident beam when the thickness of the thin film is required to be of the order of wavelength. Further, it is demonstrated that the stationary-phase approach is also applicable to transmitted evanescent beam and the GH displacement of the transmitted beam is just half the GH displacement of the reflected beam. The discussions presented here may arouse interest in atom optics and near-field microscope.

A unified theory is advanced to describe both the lateral Goos-H\"{a}nchen (GH) effect and the transverse Imbert-Fedorov (IF) effect, through representing the vector angular spectrum of a 3-dimensional light beam in terms of a 2-form angular spectrum consisting of its 2 orthogonal polarized components. From this theory, the quantization characteristics of the GH and IF displacements are obtained, and the Artmann formula for the GH displacement is derived. It is found that the eigenstates of the GH displacement are the 2 orthogonal linear polarizations in this 2-form representation, and the eigenstates of the IF displacement are the 2 orthogonal circular polarizations. The theoretical predictions are found to be in agreement with recent experimental results., Comment: 15 pages, 3 figures

It is investigated that the Goos-Hanchen-like displacement of light beams transmitting through periodical multiple-layered structures can be negative as well as positive. The numerical simulations for Gaussian-shaped beam are made to demonstrate the validity of the stationary phase approach.

It is predicted that large and opposite generalized Goos-Hänchen (GGH) shifts may occur simultaneously for TE and TM light beams upon reflection from an asymmetric double-prism configuration when the angle of incidence is below but near the critical angle for total reflection, which may lead to interesting applications in optical devices and integrated optics. Numerical simulations show that the magnitude of the GGH shift can be of the order of beam's width.

It is found that when a light beam travels through a slab of optically denser dielectric medium in air, the lateral shift of the transmitted beam can be negative. This is a novel phenomenon that is reversed in comparison with the geometrical optic prediction according to Snell's law of refraction. A Gaussian-shaped beam is analyzed in the paraxial approximation, and a comparison with numerical simulations is made. Finally, an explanation for the negativity of the lateral shift is suggested, in terms of the interaction of boundary effects of the slab's two interfaces with air.

An amendment is made to the nonparaxial beam solution advanced by Barnett and Allen for the purpose of exploring the problem of angular momentum.

A different way of producing the magnetic waveguide by loading gradually varied magnetic fields on graphene is proposed. The vector potential induced by the magnetic fields is connected by two square magnetic barriers and one magnetic well with linear variation. We investigate the guided modes and probability current density distribution within the graphene waveguide. It provides channels freely propagating and enables strong confinement for the modes by the gradually varied magnetic fields. The guide modes are caused by the electron reflection at the interfaces of the magnetic barriers. At the specific incident energy of electrons, there exist stable waveguide modes analogous to those in an optical waveguide. The hierarchy of guided modes decreases with the increasing incident angles. We hope that these results may lead to the potential applications in graphene-based waveguide structure.

It is observed that a constant unit vector denoted by $\mathbf I$ is needed to characterize a complete orthonormal set of vector diffraction-free beams. The previously found diffraction-free beams are shown to be included as special cases. The $\mathbf I$-dependence of the longitudinal component of diffraction-free beams is also discussed., 8 pages and 2 figures

We investigate guided modes in the asymmetric waveguide structure with a left-handed material (LHM) layer surrounded by air and metal. A graphical method is proposed to determine the guided modes. New properties of the oscillating and surface guided modes, such as absence of the fundamental mode, coexistence of the oscillating and surface guided modes, fast attenuation of the surface guided modes, and mode degeneracy, are analyzed in detail. We also investigate dispersive characteristics of the metal-LHM-air optical waveguide. The propagation constant increases with decreasing slab thickness for the first-order oscillating mode, which is different from that in traditional metal-cladding waveguides.

The origin of negative and positive lateral shifts of the light beams in a dielectric slab configuration is investigated from the viewpoint of the interference between multiple light beams, due to the multiple reflections and transmissions inside the slab. It is shown that the whole reflected and transmitted beams with negative shifts can be obtained from the successively reflected and transmitted constituents, respectively, with positive displacements predicted from geometrical optics. When the slab's thickness is much smaller in comparison with the width of the beam waist, the reshaping reflected and transmitted beams retain the profile of the incident beam. The physical mechanism presented here could also be applicable in investigating other counterintuitive phenomena, such as the superluminal traversal time in quantum tunneling and dispersive media.

We have investigated the tunable lateral shift and polarization beam splitting of the transmitted light beam through electro-optic crystals, based on the Pockels effect. The positive and negative lateral shifts could be easily controlled by adjusting the permittivity tensor, which is modulated by the external applied electric field. An alternative way to realize the polarization beam splitter was also proposed by the polarization-dependent lateral shifts. Numerical simulations for Gaussian-shaped incident beam have demonstrated the above theoretical results obtained by stationary phase method. All these phenomena have potential applications in optical devices.

The guided modes are investigated in a symmetric five-layer slab waveguide with a core of negative refractive index material surrounded by four normal dielectrics. The graphical method is applied to determine the guided optical modes. The left-handed material waveguide supports both oscillating and surface guided modes. For oscillating guided modes the fundamental mode can be supported. Mode double-degeneracy appears and some modes are absent in the five-layer waveguide. High-order surface guided modes exist due to the oscillating electric field in the inner claddings. Energy flux for the oscillating and surface guided modes is also discussed in detail.

We investigate the reflection of a TM-polarized light beam from a Kretschmann configuration with a Kerr nonlinear dielectric. It is found that there exists a hysteretic response between the lateral shift of the reflected beam and the intensity of the incident beam. In contrast to the lower switch threshold, the higher switch threshold of optical bistability is more sensitive to the variations of the angle of incidence and the thickness of metal film. It is also found that the peak value of the lateral shift is strongly dependent on the thickness of metal film.

It is predicted that the Goos-Hanchen displacement in the usual frustrated total internal reflection configuration can be resonantly enhanced greatly by coating a dielectric thin film onto the surface of the first prism when the angle of incidence is larger than the critical angle for total reflection at the prism-vacuum interface and is smaller than but close to the critical angle for total reflection at the prism-film interface. Theoretical analysis shows that the displacement of transmitted beam is about half the displacement of reflected beam in the thick limit of the vacuum gap between the two prisms. This is to be compared with the relation in the usual symmetric double-prism configuration that the displacement of transmitted beam is equal to the displacement of reflected beam. Numerical simulations for a Gaussian incident beam of waist width of 100 wavelengths reveal that when the dielectric thin film is of the order of wavelength in thickness, both the reflected and transmitted beams maintain well the shape of the incident beam in the thick limit of the vacuum gap. So largely enhanced displacements would lead to applications in optical devices and integrated optics.

The authors demonstrated the lasing emission of the cavity constructed by the allyl derivative of fluorescein doped dendrimer/polymer film and two opal photonic crystals. A dendrimer encapsulated the structure modified laser dye and increased its concentration up to 5.3wt% in the polymer film with high optical gain. Laser emission of the cavity was observed, which is located at 589nm with a full width at half maximum of 1.7nm above the lasing threshold of 15μJ∕pulse. The lasing output was observed at an unordinary angle owing to the superprism effect of opal photonic crystals.