Dynamical oscillations in nonlinear optical media

The spatial dynamics of pulses in Kerr media with parabolic index profile are examined. It is found that when diffraction and graded-index have opposite signs propagating pulses exhibit an oscillatory pattern, similar to a breathing behavior. Furthermore, if the pulse and the index profile are not aligned the pulse oscillates around the index origin with frequency that depends on the values of the diffraction and index of refraction. These oscillations are not observed when diffraction and graded-index share the same sign.     

Ultrashort pulse propagation under manipulated resonance conditions

The propagation of a polarized ultrashort laser pulse is analyzed by the inverse scattering method under initial conditions including a spatial pulse profile, a state of the medium, and a “switched-on” resonant atom-field interaction. Magnetic degeneracy of atomic levels is taken into account. The Maxwell-Bloch equations are rewritten in Hamiltonian form without redefining the spatial and temporal variables. The inverse scattering method is based on an analysis of a new spectral problem. Gelfand-Levitan-Marchenko-type equations are derived, a soliton solution is obtained, and the changes in parameters of two solitons after their collision are calculated. A possible experimental setup for implementing the system under analysis is discussed.     

反WKB方法的改进

改进了用于确定渐变平面波导折射率分布的反WKB方法。这种渐变折射率分布是从在WKB近似的条件下得到的本征方程用数值方法求得的。用改进的反WKB方法计算3种(指数、高斯和阶跃函数)折射率分布，结果验证了该方法的有效性。先用最小二乘法拟合测量得到的有效折射率。进而求出有效折射率函数，再利用改进的算法分布计算出各自的折射率分布。计算结果同精确值吻合得很好。计算出的波导表明折射率同精确值的绝对误差约为0．1%。     

Low-pass filter characteristics of multimode graded index fibres

On the basis of ray model solutions, the response of a lossless multimode graded-index fibre to a modulated source is calculated. The results demonstrate great sensitivity of the bandwidth of the fibre to small variations in index profile.     

Multiple-beam interference fringes applied to investigate the index profile of the optical fiber dip

A numerical-based phase shift method is presented to study optical fibers having a dip in the refractive index profile at the center of the core. Mathematical expressions for the shape of multiple-beam Fizeau fringes crossing a graded-index optical fiber (GRIN) comprising a central graded-index dip or GRIN fiber with a constant refractive index dip, immersed in a silvered liquid wedge, are derived. The index profile parameters of the central dip are determined by comparing the theoretical fringe shape across the fiber cross-section and the experimental fringe shape obtained from micro-interferograms of the wedge interferometer.     

Effet de l'irradiation par des neutrons sur les caractéristiques temporelles d'une fibre optique à gradient d'indice

The evolution of the temporal characteristics of a graded-index optical fibre during neutron irradiation has been performed. Narrowing of the transmitted pulse width has been clearly obtained. The irradiation effect on the index profile of the fibre is reported and a calculation based on the WKB method is derived which explains the experimental results.     

Unusual bound modes in asymmetrically graded planar waveguides

 We show analytically that for certain asymmetrically graded-index planar waveguides at the cutoff point where the effective refractive index of a mode coincides with the index of the substrate, there exist unusual bound modes. We have calculated the refractive-index profile that makes these modes possible. Received: 20 July 1996     

Heuristic derivation of quasi-geometrical dispersion equations for coupled graded-index waveguides

Correspondence rules between step and graded-index profiles are derived under heuristic arguments (quasi-geometrical approximation) in such a way that the celebrated WKB dispersion regime is directly obtained without using the connection formulae. Dispersion equations for coupled waveguides represented by index profiles, containing step, graded and truncated profile regions, with several turning points, are calculated. An application to both planar and non-planar graded index couplers with four turning points is presented. This revised version was published online in November 2006 with corrections to the Cover Date.     

Wave Resonance in Media with Modular,Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.     

Direct observation of the space-time transformation of a femtosecond laser pulse in fused quartz

The transformation of a femtosecond laser pulse propagating in fused quartz before and after its transition to a filamentation regime has been investigated by femtosecond time-resolved optical polarigraphy. The spatial periodicity of a light field along and across the propagation axis has been detected and its nature has been attributed to the interference of the conical and plane components of the wave packet of the filament. The “supraluminal” motions of the observed filament intensity maximum are due to the longitudinal transformation of the pulse profile.     

Do solitonlike self-similar waves exist in nonlinear optical media?

We show analytically that bright and dark spatial self-similar waves can propagate in graded-index amplifiers exhibiting self-focusing or self-defocusing Kerr nonlinearities. The intensity profiles of the novel waves are identical with those of fundamental bright or dark spatial solitons supported by homogeneous passive waveguides with the same type of nonlinearity. Thus, we reveal a previously unnoticed connection between spatial solitons and self-similar waves. We also suggest that the discovered self-similar waves can be used in a promising scheme for the amplification and focusing of spatial solitons in future all-optical networks.     

偏离束腰入射对非局域非线性介质中高斯光束演化的影响

从1+1维强非局域模型出发，讨论了偏离束腰入射的高斯光束在非局域非线性介质中的传输 特性，得到了精确的解析解.结果表明，在聚焦介质中偏离束腰入射时，不论入射功率多大 ，光束束宽将发生周期性波动，光孤子不复存在，这与从高斯光束束腰入射的情况有本质的 不同；入射功率决定了光束平均束宽的大小，入射位置决定了光束初始的演化趋势.比较了在入射位置相同的条件下，聚焦介质、散焦介质和线性均匀介质中光束的演化.给出了“空 间啁啾”的定义，偏离束腰入射的物理本质是光束的不同入射位置对应不同的初始空间啁啾 .空间啁啾的概念， 关键词： 非局域非线性介质 空间光孤子 高斯光束     

A simple way of understanding some exact results on critical phenomena in non-homogeneous and finite Ising systems II: Cooperativity in periodically layered systems

By means of a continuum approximation introduced in a preceding paper, we present a more transparent treatment of the critical phenomena in the specific heat of a two-dimensional Ising model with a “layer” type spatial inhomogeneity of its exchange coupling. This calculational procedure (which will be tested by comparison with an earlier exact treatment by Becker and Hahn) enables us to develop a physical picture of cooperative effects in real (spatially non-homogeneous) systems with quenched impurities. To be more concrete: By examination of the one-particle eigenvalues and eigenfunctions of the transfer matrix, it becomes apparent how the critical behaviour is dominated by i) “local” correlation lengths, describing the spatial decay of correlations within homogeneous subdomains of the system and becoming large near certain temperatures which indicate transitions from disorder to spin alignment within each of these domains, and ii), by the spatial average of the inverse of these local decay lengths, termed “global” reciprocal correlation length, which describes correlations between different subdomains on a scale on which these domains look like fluctuating “block spins”. This “global” decay length diverges at a true phase transition of the system as a whole. Correspondingly, for each sort of subdomain a sharp, but finite logarithmic peak in the specific heat and, as long as the layering of the inhomogeneous Ising system is periodic, a true logarithmic singularity are found.     

迭代法重建平板光波导折射率分布

提出了一种用于重建渐变折射率平板光波导折射率分布的方法,这种方法通过数值法求解亥姆霍兹方程．从光波导的低阶模开始采用迭代法顺次对各个模式对应的“转折点”位置进行校正．并多次重复迭代以消除“转折点”外折射率分布的变化对相应模式“转折点”位置的影响。借助WKB近似原理说明了这种方法迭代过程中“转折点”坐标的收敛性。对折射率按照指数规律变化的平板光波导的折射率分布重建结果证明这种方法具有较高的精度。分析表明：这种方法克服了WKB法固有的缺陷,即忽略模式“转折点”外折射率分布对相应模式传输常鞋的影响和“转折点”处相移的不确定性,因而比逆WKB法具有更好的自洽性和精确性。     

Influence of temperature on the optical and structural properties along the diameter of optical fibres

Multiple-beam Fizeau fringes technique with an opto-thermal device is used to study the effect of temperature on the optical properties of Philips graded-index optical fibres. The refractive index profile of the optical fibre is measured at different temperatures. From these profiles the opto-thermal coefficient, the profile shape parameter α, the cladding/core maximum refractive index difference Δn and some guidance parameters of the optical fibre that play an important role in communication are determined. The variation of oscillation and dispersion energies along the diameter of the optical fibre (energy profile) are calculated at different temperatures. An empirical formula of the energy profile is obtained. Microinterferograms are given for illustrations.     

Fluorescence characterization and gain studies on a dye-doped graded index polymer optical-fiber preform

Preparation of an appropriate optical-fiber preform is vital for the fabrication of graded-index polymer optical fibers (GIPOF), which are considered to be a good choice for providing inexpensive high bandwidth data links, for local area networks and telecommunication applications. Recent development of the interfacial gel polymerization technique has caused a dramatic reduction in the total attenuation in GIPOF, and this is one of the potential methods to prepare fiber preforms for the fabrication of dye-doped polymer-fiber amplifiers. In this paper, the preparation of a dye-doped graded-index poly(methyl methacrylate) (PMMA) rod by the interfacial gel polymerization method using a PMMA tube is reported. An organic compound of high-refractive index, viz., diphenyl phthalate (DPP), was used to obtain a graded-index distribution, and Rhodamine B (Rh B), was used to dope the PMMA rod. The refractive index profile of the rod was measured using an interferometric technique and the index exponent was estimated. The single pass gain of the rod was measured at a pump wavelength of 532 nm. The extent of doping of the Rh B in the preform was studied by axially exciting a thin slice of the rod with white light and measuring the spatial variation of the fluorescence intensity across the sample.     

Length-dependent effects due to leaky modes on multimode graded-index optical fibres

The radiation losses of tunnelling leaky modes in graded-index optical fibres are calculated theoretically, and it is shown that the near-field intensity profile has a length dependence. Consequently measurements of the near-field intensity distribution do not give the refractive index profile directly, and a correction factor must be applied. We have investigated this factor and find that it depends only on a single normalisation parameter involving fibre length, core radius and normalised frequency. A further use of the correction factor is to determine the total power attenuation due to the loss of leaky modes.     

Interpretation of a layer mechanical impedance measured using a hard round die

Theoretical expressions for the impedance characteristics of a layer bound to a rigid base are obtained for various profiles of the normal pressure under a die that vibrates on the layer surface without producing any shear stresses. The frequency dependences of the impedance characteristics of a homogeneous gelatin layer and their variation with changes in the die diameter are measured by means of a specialized software-hardware system. The impedance characteristics are calculated for the models with “uniform,” “parabolic,” and “hyperbolic” pressure profiles under the die, and the results are compared with the experimental data. The model with a uniformly distributed pressure under the die is found to be the most adequate one.     

Nonlinear processes in media with an acoustic hysteresis and the problems of dynamic interaction between piles and earth foundation

Evolution of a pulsed disturbance in a nonlinear medium whose properties irreversibly vary in the course of wave propagation is studied. Equations describing the propagation process are obtained. It is demonstrated that the waveform distortion and the dynamics of the field and energy characteristics of a signal noticeably differ from those observed in conventional nonlinear media. New nonlinear equations describing a pulse in a medium with relaxation of its nonlinear properties are derived. A finite “delay time” for irreversible processes is introduced in the defining equation. The shape of a pulse reflected from the boundary between an ordinary medium and a nonlinear hereditary medium is calculated. It is demonstrated that, in the case of a fixed relation between the peak pressure in the incident pulse and the ratio of linear impedances of the two media, a total transmission of the trailing edge of the pulse into the compressed medium occurs. Possible applications of the results to topical construction problems are discussed.     

On rainbows of inhomogeneous spherical droplets

The paper is devoted to the study of the intensity distributions and the angular spectra of the second and fifth rainbows of homogeneous and inhomogeneous spherical particles predicted by Lorenz-Mie theory. The results show that the distribution around the second rainbow angle for a homogeneous sphere of refractive index between 1.32 and 1.33 is due to the interference of the light after two or five internal reflections. The structure of the scattering diagram and the angular spectrum for homogeneous and radially inhomogeneous spheres are studied. For a homogeneous sphere we show that the second and fifth rainbows can be independently reconstructed by filtering the calculated spectrum. Since each order of rainbow penetrates the particle to different depths, such methods could be used to provide information about the refractive index profile or the temperature gradient of an inhomogeneous sphere. The Airy-like peaks of the second and fifth rainbows are closely connected with the refractive index profile, which is beneficial to the measurement of its refractive index profile or temperature gradient.