Quadratic Refractive Index Profile Research Articles

Influence of a variable Rayleigh scattering-loss coefficient on the light backscattering in multimode optical fibers.

The recently developed diffraction technique of analytical investigation of the Rayleigh backscattering produced by an incident fundamental mode in a multimode optical fiber with an arbitrary refractive index profile is supplemented by taking into account the Rayleigh scattering-loss coefficient, which could be variable within the fiber cross section. The relative changes in various radial and azimuthal modes' excitation levels, due to some typical radial dependences of this coefficient, are computed for the quadratic- and step-index fibers. It is stated that the excitation efficiency could either rise or decay for different modes. The effect of the variable Rayleigh scattering-loss coefficient is shown to be more noticeable in the fibers with a quadratic refractive index profile, whereas it is negligible in actual multimode step-index fibers.

Rayleigh backscattering from the fundamental mode in multimode optical fibers.

Rayleigh backscattering produced by an incident fundamental mode in a multimode optical fiber is analyzed using a diffraction technique, with a full set of backward-propagating modes being taken into consideration. Explicit formulas are derived for mode excitation efficiency via radial distributions of the mode fields, and it is proved that only half-azimuthal modes are backscattered by the incident wave of a fixed polarization. Advanced analytical expressions are developed for fibers with a quadratic refractive-index profile, and mode groups of even numbers, composed of modes with equal propagation constants, are stated to be excited with equal efficiencies.

Microwave propagation over the Earth: image inversion

An extremely accurate but simple asymptotic description for the path of a light ray propagating over a curved Earth with steady radial variations in refractive index is derived using simple scaling arguments. It is used to determine effectively exact analytic solutions for the path of rays through refractive-index profiles described in terms of patched quadratics. Such patched quadratics can be used to accurately describe almost all refractive-index profiles of practical interest. The results show that images generated by rays passing through a quadratic refractive-index profile are uniformly magnified in the vertical direction, and magnification and displacement observations can be used to determine the refractive-index profile parameters. For patched quadratics, observations of critical rays can be used to determine the thicknesses of the quadratic layers. An effectively exact solution is also obtained for exponential index profiles and this is used to determine the path of rays through a thin boundary layer attached to the Earth; an inferior mirage situation

Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems.

We analyze the evolution of the vortex and the asymmetrical parts of orbital angular momentum during its propagation through separable first-order optical systems. We find that the evolution of the vortex part depends on only parameters a(x), a(y), b(x), and b(y) of the ray transformation matrix and that isotropic systems with the same ratio b/a produce the same change of the vortex part of the orbital angular momentum. Finally, it is shown that, when light propagates through an optical fiber with a quadratic refractive-index profile, the vortex part of the orbital angular momentum cannot change its sign more than four times per period.

Analytical models of optical refraction in the troposphere.

An extremely accurate but simple asymptotic description (with known error) is obtained for the path of a ray propagating over a curved Earth with radial variations in refractive index. The result is sufficiently simple that analytic solutions for the path can be obtained for linear and quadratic index profiles. As well as rendering the inverse problem trivial for these profiles, this formulation shows that images are uniformly magnified in the vertical direction when viewed through a quadratic refractive-index profile. Nonuniform vertical distortions occur for higher-order refractive-index profiles.

Reconstruction of the optical correlation function in a quadratic refractive index medium

The propagation of a partially coherent light beam through a weakly inhomogeneous medium with quadratic refractive index profile is described within the framework of the generalized Fresnel transform. Using this formalism, the general problem of the reconstruction of the correlation function of an input optical field from intensity measurements is discussed. As a particular application, the reconstruction of a field having a Gaussian correlation function is carried out. Moreover, a method is proposed to obtain the parameters of the refractive medium, e.g. a GRIN fiber.

Control of the radiation parameters in a longitudinally inhomogeneous nonlinear medium with a quadratic refractive-index profile

The paraxial approximation is used in an investigation of nonlinear propagation of an axial Gaussian light beam in a longitudinally inhomogeneous medium with a quadratic refractive-index profile. Explicit expressions are obtained for the width and radius of curvature of the beam wavefront. It is shown that variation of the beam power and of the law governing the longitudinal variation of the gradient parameter can be used to control the width and radius of curvature of a beam in a finite longitudinally homogeneous section of a nonlinear medium. Explicit expressions are obtained for the coefficients describing excitation of Gaussian–Hermitian modes in a finite region of a linear medium with a quadratic refractive-index profile. The results obtained can be useful in the design of variable-sensitivity power sensors and of various matching devices.

Influence of the Rayleigh backscattering on the mode composition of radiation in multimode graded-index waveguides with a quadratic refractive-index profile

A method of coherent states is used to describe the process of Rayleigh scattering in a multimode graded-index waveguide with a quadratic refractive-index profile. Explicit expressions are obtained for the coefficients representing excitation of Gaussian–Hermite backscattering modes in two cases of practical importance: excitation of a waveguide by an extended noncoherent light source and selective excitation of different modes at the entry to a waveguide. An analysis is also made of the coefficients of coupling between forward and backward modes. Explicit expressions for the coefficients representing capture of backscattered radiation by a waveguide are obtained for two special cases of excitation (extended light source and zeroth mode).

Depolarization of quasimonochromatic radiation in randomly inhomogeneous fiber waveguides with a quadratic refractive index profile

A theoretical analysis is made of the depolarization of quasimonochromatic radiation in isotropically random inhomogeneous fiber waveguides with a quadratic profile of the refractive index. An analysis is made of the characteristic features of the process and an estimate is obtained of the depolarization coefficient of radiation in the case of typical single-mode fiber waveguides. A method is developed for calculating the depolarization coefficient which can be applied to fiber waveguides with an arbitrary distribution of the refractive index over the cross section of the fiber, and also to multimode fiber waveguides. The theoretical predictions are in good agreement with experimental results and are of practical importance for coherent fiber-optic communication and measuring systems.

Wave propagation in media with three-dimensional quadratic refractive index profile

This article presents a new method for studying wave propagation in an unbounded medium where the square of the refractive index is of a general three-dimensional quadratic form. By this method, the Green’s function can be expressed exactly by a onefold integral, for which the integrand is derived from a path integral. The numerical results for one-dimensional Green’s functions in linearly stratified media are first checked with those obtained from the well-known Airy function expressions. The method is then applied to obtain the three-dimensional Green’s functions for linear stratified media and for parabolic profiles with range-dependent (horizontal) variations. Some interesting phenomena caused by the horizontal variations (i.e., range dependence) of the medium will be shown and interpreted by simple ray pictures whenever appropriate. For the special case of a parabolic profile with symmetric turning points, a wave duct is formed and the Green’s function can be expanded as a series of wave modes. It will be shown that an evanescent mode in a purely stratified medium may switch to a propagating mode due to the range dependence of the medium property. Application of the results to acoustic wave propagation in the atmosphere and in the ocean will be discussed.

Spatial coherence of optical fields in longitudinally inhomogeneous media with quadratic refractive index profiles

The paraxial approximation for the scalar wave equation is used to study the spatial coherence of optical fields in media with an overall quadratic refractive index profile and an arbitrary regular longitudinal inhomogeneity. Analytic expressions are obtained for the parameters describing the spatial coherence of fields in such media, particularly for the correlation radius and the width of a partly coherent beam generated by a variety of light sources. It is shown that for certain distributions of the longitudinal inhomogeneity there is no change in the coherent properties of the field because of a matching effect. This effect is proposed as a method for ensuring a matched contact between two different graded-index waveguides. A study is made of the influence of the degree of coherence on the focusing region of the beam and it is shown that, in principle, it is possible to focus a paraxial beam with a finite initial width in a region smaller than the wavelength. It is shown that variation of the longitudinal inhomogeneity distribution can be used to control coherent properties of the radiation. The results obtained can be used in fiber-optic communication lines, integrated optics, and coherent optical data processing.