Quadratic-index Medium Research Articles

Propagation properties of Hollow higher-order cosh-Gaussian beams in quadratic index medium and Fractional Fourier transform

Hollow higher-order cosh-Gaussian (HhCG) beam is a new mathematical model to describe the dark-hollow beam. An analytical expression for HhCG beams propagating through a paraxail ABCD optical system is derived and used to investigate its propagation properties in a quadratic index medium (QIM) and a Fractional Fourier transform (FRFT) optical system. By using the derived expressions, the properties of HhCG beams of both systems are graphically illustrated with numerical example. Several influence parameters on the evolution of HhCG in the both systems QIM and FRFT, respectively, are discussed in detail. Results show that the intensity distribution of HhCG evolves periodically during propagation in the QIM. In FRFT system, it is found that the normalized intensity distribution of HhCG beams plane depends not only on the fractional order but also on the beam order and the truncation parameter. The results obtained in this work are valuable to provide a convenient way for HhCG beams shaping, optical fibers, lenses, and other fields.

Propagation properties of partially coherent anomalous hollow beams in quadratic-index media

Based on the generalized diffraction integral formula, analytical propagation expressions for a partially coherent anomalous hollow beam (AHB) propagating through a quadratic-index medium are derived. The distributions of intensity and Poynting vector for a partially coherent AHB through the quadratic-index medium are numerically and theoretically studied. It is found that the intensity and Poynting vector of the partially coherent AHB take on a periodical change. The evolution properties of the partially coherent AHB in a quadratic-index medium are closely relevant to its transverse coherence width and the quadratic-index parameter of media.

Propagation properties of self-accelerating rotating elliptical vortex wave packets in a quadratic index medium

The self-accelerating rotating elliptical vortex (SREV) wave packets which carry orbital angular momentum have been introduced and studied for the first time, to the best of our knowledge. The propagation properties of SREV wave packets in a quadratic index medium are investigated analytically. The numerical experiments on SREV wave packets have a good agreement with analytical solutions. Furthermore, the generation and manipulation of different SREV wave packet modes, which show temporal self-accelerating and transversal periodically rotating properties, are obtained by adjusting the cross phase coefficient and the elliptical factor. In addition, the evolutions of phase, energy flow and angular momentum density of SREV wave packets for different propagation distances are demonstrated to study more about the dynamic rotating properties. The SREV wave packets have numerous potential application in novel optical communications.

Paraxial propagation of the first-order chirped Airy vortex beams propagating in a quadratic index media

ABSTRACT We investigate the propagation of the first-order chirped Airy vortex (FCAiV) beams in a quadratic index media analytically by applying the Huygen-Fresnel integral. The intensity, the phase, the energy flow density, and the angular momentum density are investigated for different chirped parameters, decay factors and parameters related to the gradient refractive index. There is an inversion of the self-acceleration of the FCAiV beams during the propagation. The FCAiV beams propagating in the quadratic index media hold a periodic evolution. The focusing intensity, the intensity distribution, the centre of mass of the FCAiV beams can be controlled by choosing different parameters.

Spatiotemporal autofocused chirped Pearcey Pearcey Gaussian wave packets with an adjustable focus in a quadratic-index medium

Here we investigate the propagation properties of spatiotemporal autofocused chirped Pearcey Pearcey Gaussian (PePeG) wavepackets by solving (3 + 1) D Schro¨dinger equation in a quadratic index medium. When the spatial distribution factor p and the temporal distribution factor f are the same, PePeG wavepackets can simultaneously autofocus in spatial and temporal domain and the peak intensity at the focus is more 28 times than that at the initial plane. With the increase of distribution factor p, the scope of the radius of trapped particles decreases.

Propagation properties of hypergeometric-Gaussian type-II beams through the quadratic-index medium.

The analytical expressions of hypergeometric-Gaussian type-II (HyGG-II) beams propagating in the quadratic-index medium are deduced based on the ABCD optical transformation matrix method. The propagation properties of HyGG-II beams in the quadratic-index medium are numerically and theoretically analyzed in detail. The intensity distribution and the second-order moment-based beam width vary periodically when the HyGG-II beams pass through the quadratic-index medium. The interactions of two HyGG-II beams are also discussed graphically.

Self-repeating properties of four-petal Gaussian vortex beams in quadratic index medium

In this paper, we investigate the propagation properties of four-petal Gaussian vortex (FPGV) beams propagating through the quadratic index medium, obtaining the analytical expression of FPGV beams. The effects of beam order n, topological charge m and beam waist ω0 are investigated. Results show that quadratic index medium support periodic distributions of FPGV beams. A hollow optical wall or an optical central principal maximum surrounded by symmetrical sidelobes will occur at the center of a period. At length, they will evolve into four petals structure, exactly same as the intensity distributions at source plane.

Propagation characteristics of controllable dark-hollow beams in a quadratic-index medium.

In this paper, the propagation of controllable dark-hollow beams (CDHBs) passing through a quadratic-index medium is studied. An analytical expression for the propagation of CDHBs in a quadratic-index medium is presented based on the ABCD matrix method, and the effect of the beam parameters on the evolution of a CDHB is investigated. It is found that the intensity distribution of CDHBs evolves periodically during propagation in the quadratic-index medium. The results indicate that the distance in which the dark-hollow center of a CDHB disappears can be increased by the beam parameters.

Propagation of Airy beams in the quadratic-index medium based on matrix optics

We investigate the propagation of the Airy beam in the quadratic-index medium based on matrix optics. It shows there is a periodic beam profile change induced by the inhomogeneity of medium, which displays the periodic change of the field distribution from the Airy profile to the Gaussian profile. We also know the duration time of the Gaussian profile of the beam is determined by the decay factor of the Airy beam. Yet, the accelerating trajectory of the Airy beam is independent of decay factor of the Airy beam. Moreover, we show that the beam size of the Gaussian profile is different for the Airy beam with the different decay factor.

Propagation properties of spatiotemporal chirped Airy Gaussian vortex wave packets in a quadratic index medium.

A type of chirped Airy Gaussian vortex (CAiGV) localized wave packets in a quadratic index medium are studied by solving the paraxial differential equation. For the first time, the propagation properties of spatiotemporal CAiGV light bullets in the quadratic index medium are demonstrated. Some typical examples of the obtained solutions are based on the temporal and spatial chirp parameters, the initial velocity, the distribution factor, and the topological charge. The radiation force of the spatial CAiGV wave packet on a Rayleigh dielectric particle has the periodically reversion and recovery abilities due to the quadratic potential. What we report here can obtain different radiation force trajectory and may have potential application in optical tweezing and bio-medical field.

Propagation properties of hollow sinh-Gaussian beams in quadratic-index medium

Based on the Collins integral formula, the analytical expression for a hollow sinh-Gaussian (HsG) beam propagating through the quadratic-index medium is derived. The propagation properties of a single HsG beam and their interactions have been studied in detail with numerical examples. The results show that inhomogeneity can support self-repeating intensity distributions of HsG beams. With high-ordered beam order n, HsG beams could maintain their initial dark hollow distributions for a longer distance. In addition, interference fringes appear at the interactional region. The central intensity is a prominent peak for two in-phase beams, which is zero for two out-of phase beams. By tuning the initial beam phase shift, the distribution of the fringes can be controlled.

Propagation of Airy-Gaussian beams in a quadratic-index medium

The propagation properties of the Airy-Gaussian beams are introduced in a quadratic-index medium analytically and numerically. The linear momentum, the beam width, the beam center of the Airy-Gaussian beam are analyzed. The Airy-Gaussian beam shape and curvature repeats itself at intervals proportional to \hbox{$1/\sqrt{az_{R}}$}1/azR due to the potential barrier. The bend (the transversely acceleration) and the periodicity of the Airy-Gaussian beam depend on the parameters χ0 and azR.

Elegant Ince-Gaussian beams in a quadratic-index medium

Elegant Ince—Gaussian beams, which are the exact solutions of the paraxial wave equation in a quadratic-index medium, are derived in elliptical coordinates. These kinds of beams are the alternative form of standard Ince—Gaussian beams and they display better symmetry between the Ince-polynomials and the Gaussian function in mathematics. The transverse intensity distribution and the phase of the elegant Ince—Gaussian beams are discussed.

Virtual dielectric waveguide mode description of a high-gain free-electron laser. I. Theory

A set of mode-coupled excitation equations for the slowly growing amplitudes of dielectric waveguide eigenmodes is derived as a description of the electromagnetic signal field of a high-gain free-electron laser (FEL), including the effects of longitudinal space charge. This approach of describing the field basis set has notable advantages for FEL analysis in providing an efficient characterization of eigenmodes, and in allowing a clear connection to free-space propagation of the input (seeding) and output radiation. The formulation describes the entire evolution of the radiation wave through the linear gain regime, prior to the onset of saturation, with arbitrary initial conditions. By virtue of the flexibility in the expansion basis, this technique can be used to find the direct coupling and amplification of a particular mode. A simple transformation converts the derived coupled differential excitation equations into a set of coupled algebraic equations and yields a matrix determinant equation for the FEL eigenmodes. A quadratic index medium is used as a model dielectric waveguide to obtain an expression for the predicted spot size of the dominant system eigenmode, in the approximation that it is a single Gaussian mode.

CLASSICAL OPTICAL TRANSFORMS STUDIED IN THE CONTEXT OF QUANTUM OPTICS VIA THE ROUTE OF DEVELOPING DIRAC'S SYMBOLIC METHOD

Via the route of developing Dirac's symbolic method and following Dirac's assertion: "⋯ for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory", we find the generalized Fresnel operator (GFO) corresponding to the generalized Fresnel transform (GFT) in classical optics. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in quantum optics. With the aid of entangled state representation the entangled Fresnel transform is proposed; new eigenfunctions of the complex fractional Fourier transform and fractional Hankel transform are obtained; the two-variable Hermite eigenmodes of light propagation are used in studying the Talbot effect in quadratic-index media; the complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Moreover, quantum optical version of classical z-transforms is obtained on the basis of the eigenvector of creation operator. Throughout our discussions, the coherent state, squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used.

Ince–Gaussian beams in a quadratic-index medium

The propagation of Ince-Gaussian beams in media where the refractive index varies quadratically with the distance from the optical axis is studied. Explicit expressions for the complex beam parameter and the longitudinal phase shift are derived and discussed. Ince-Gaussian eigenmodes with constant width can be obtained by satisfying a relation between the beam width and the quadratic-medium coefficient. The derivation has included the possibility of propagation of Ince-Gaussian beams in complex lenslike media having quadratic transverse variations of the index of refraction and the gain or loss.

Talbot effect in a quadratic-index medium studied with two-variable Hermite polynomials and entangled states.

We have found the new eigenmodes, two-variable Hermite polynomials, exist in propagating plane waves in quadratic-index media and that a two-dimensional Talbot effect can be demonstrated with these modes.

Optical wave propagation of fractal fields

The evolution of the complex amplitude of a fractal optical field during propagation through free space (Fresnel transform) and through a quadratic refractive index medium (fractional Fourier transform) is discussed. To that end, the two above transforms have been introduced within the framework of the generalized Fresnel transform (GFT) and their scaling properties formulated. These scaling properties are shown to be suitable to reveal the self-similarity behaviour of the fractal field. A general method to determine the fractal dimension and other fractal features from the diffraction patterns is proposed. As an application of the method, a numerical experiment has been carried out for the diffraction of a triadic Cantor set in free space and in a quadratic index medium. The determination of the fractal dimension is illustrated in the two cases.

Talbot effect in a quadratic index medium

A generalization of Talbot effect to the case of graded index medium is considered. An initial periodic distribution is shown to repeat, however its period is reduced by an amount depending on the strength of the graded index.

Propagation of higher-order intensity moments in quadratic-index media

Propagation of higher-order intensity moments in quadratic-index media