Modes in buried planar optical waveguides with graded-index profiles

The refractive index profile of buried graded-index planar optical waveguides fabricated by ion migration can be described by a buried secant hyperbolic function. Using such a profile, we have obtained the exact propagation characteristics for such buried optical waveguides. We have also shown that profiles which deviate slightly from the above profile can be treated as perturbations and a first-order perturbation correction gives very accurate results.     

Metal-clad graded-index planar optical waveguides: accurate perturbation analysis

We present a perturbation analysis of propagation constants and attenuation coefficients of TE and TM modes in a metal-clad linear index planar optical waveguide. The imaginary part N″ of the complex modal index N=N′ + iN″ is given by N″ = (∂N′ / ∂′), where =′ + i″ is the complex dielectric constant of the metal cladding and ∂N′ / ∂′ is obtained by numerical differentiation of the solution of the real eigenvalue equation. The cumbersome solution of a complex transcendental equation is thus completely eliminated. The results are in good agreement with those obtained by solving the eigenvalue equation in the complex plane. By taking the metal-clad linear planar waveguide as a preselected waveguide, we can use our RWKB method to solve metal-clad planar waveguides with parabolic, exponential, gaussian and complementary error function index profiles.     

Critical ray-like propagation modes along graded-index planar optical waveguides

Graded-index planar optical waveguides and surface compression layers were formed simultaneously in the surface layer of glass plates by ion exchange. A change in the refractometer patterns was observed. Before ion exchange, only a critical ray fringe was observed, but with ion exchange a guided wave fringe appeared on the high effective refractive index side of the ‘critical ray’ fringe, and the number of guided wave fringes increased. The guided wave fringe or fringes were birefringent, whereas the ‘critical ray’ fringe was kept nonbirefringent. It was concluded that the ‘critical ray’ propagated along the bottom of a waveguide, ie at the foot of the refractive index distribution.     

Propagation of Gaussian beam in an absorbing waveguide

Propagation of a Gaussian beam in an absorbing waveguide is analyzed for cubic-quintic and saturable media taking into account both linear and nonlinear absorption. A “collective variable approach” technique, based on trial functions, is used to solve the general nonlinear Schrodinger equation. In the absence of losses, we construct a diagram which defines regions of oscillatory and diffractive beam propagation for both types of media, and also a diagram that compares bistable behavior in such media. We show that if the linear and nonlinear absorption coefficients are small, the behavior of the oscillations of the beam width on propagation allows one to distinguish between cubic-quintic and saturable media. By reversing the sign of the linear absorption, we analyze the behavior of the beam propagation in media with gain and nonlinear absorption. In cubic-quintic media, the energy reaches a plateau for certain ratios of gain to losses, whereas for saturable media, the energy increases throughout the beam propagation.     

Self-similar optical beam in nonlinear waveguides

By means of the similarity transformation, we obtain exact bright and dark similariton-pair solutions in nonlinear waveguides for the generalized nonlinear Schr?dinger equation exhibiting spatial inhomogeneity, inhomogeneous nonlinearity and gain or loss at the same time. Then we investigate the interaction behaviors of these solitonic similaritons in a periodic distributed amplification system.     

On the accuracy of WKB analysis of TE and TM modes in planar graded-index waveguides

The TE and TM mode spectra in inhomogeneous optical waveguides with various refractive index profiles have been calculated in the WKB approximation and compared with data obtained numerically from rigorous theory. The effective index errors better than 3 × 10^-4 and 1.5 × 10^-3 have been found for unburied profiles, respectively. The results obtained may help to take full advantage of the WKB analysis for practical purpose.     

Lateral displacement of a Gaussian beam reflected from absorbing media

Lateral displacements of well-focused beams reflected from absorbing media are demonstrated in this paper. The incident beam is modeled as a tapered wave with a Gaussian spectrum. The field solutions are obtained on both sides of the interface by solving analytically from Maxwell's equations and by matching the boundary conditions at the interface. Numerical simulations are presented and the field values as well as the time-averaged power densities are computed. The influences of the polarization, the angle of incidence and the beam width on the displacement are discussed. The origin of the displacement lies in the absorption of the media, for which there is a phase shift between the incident and reflected waves. Because of this phase shift, the center of the reflected beam does not coincide with that of the incident one, similar to the Goos-Hänchen shift.     

Matching of a Gaussian beam into hollow oversized circular waveguides

A Gaussian beam entering an hollow oversized circular waveguide of dielectric or low-conductivity wall material is considered: a simple formula giving the optimal beam charac-teristics for the lowest losses in the guide is derived from an optical theory. This optimization depends on the geometry and the wall material of the guide. The formula is valid for small beam diameters at the guide entrance and for low losses inside the guide, which are conditions corresponding to usual cases. Physical interpretations are given. Comparisons with some experimental results show a good agreement with the theory. These results are of interest for waveguides lasers and waveguide transmission lines.     

Cutoff behaviour of graded-index slab waveguides

The cutoff behaviour of graded-index planar waveguides is analysed in terms of the normalized waveguide parameters by using the multilayer technique and the WKB approach. The results show a very high degree of accuracy when checked against exact known values for exponential and hyperbolic index profiles. A set of universal plots showing the general features is given for the cutoff values of both fundamental and first-order modes. The TM polarization is also studied by comparison with the TE one.     

高斯光束通过非线性介质层的限幅效应

本文用数值分析的方法研究了两维高斯光束通过一定厚度的非线性折射介质后在线性空间传播的状态,分析了光束的近场和远场横向强度分布及通过一定孔径光阑的功率.结果表明,在考虑到光束经过非线性介质时,相位与横向分布都会发生变化后,仅在非线性折射效应作用下,在远场就存在限幅效应.在一定的强度范围之内,近场也存在限幅利用的可能性.     

The dynamics of nonautonomous soliton inside planar graded-index waveguide with distributed coefficients

We present a series of exact nonautonomous solitons inside the planar graded-index waveguide amplifier with Kerr nonlinearity by Darboux transformation. Especially, properties of the nonautonomous soliton such as typical width, peak intensity and trajectory of wave center are analytically investigated. We also study the trajectory of wave's center with different conditions in detail. Solitons in planar waveguide without graded-index are also demonstrated.     

Wave propagation in uniaxial graded-index optical waveguides

Propagation of guided-modes in uniaxial optical waveguides is investigated. When the optical axis lies in the plane of the waveguide, a coupling between TE and TM modes takes place. Based on the coupled-mode theory, the propagating constants of the hybrid modes, the coupling coefficient and the interaction length are determined. A simple model is proposed to describe the guided mode behaviour. The results are illustrated for the case of Ti-diffused LiNbO₃ waveguide.     

Optical levitation of absorbing particles with a nominally Gaussian laser beam

We use a Gaussian laser beam to study the levitation of absorbing Mie particles. Several metal oxide particles are stably levitated, and their movement over time is recorded. Our studies show that the position of each particle is highly dependent on the other particles' locations. The observations are explained by the phenomenon of thermal creep. The increased local pressure that is due to a temperature gradient along the particle's surface induces levitation. The particles rest close to minima in the intensity distribution near the optical axis. An experiment is suggested that can be used to locate these minima in a laser beam.     

Mode dispersion characteristics of graded-index optical waveguides

A method to compute the propagating modes of graded index waveguides with an arbitrary refractive index distribution is proposed, based on the variational method applied to the integral wave equation. The method is applied to two refractive index distributions: a Gaussian index distribution and a complementary error-function index distribution. The results are compared with the predictions of perturbation analysis and are experimentally verified by measuring the mode spectrum of a Ti-diffused LiNbO₃ waveguide.     

Thermally induced self-focusing and optical beam interactions in planar strontium barium niobate waveguides

We present an experimental study of thermally induced self-focusing effects and interactions of incoherent light beams in strontium barium niobate waveguides. Depending on the input power, a single parallel beam is strongly focused inside the sample up to diameters of several micrometers. For higher input power we observe the splitting of the beam in a sequence of several spots. We demonstrate that these thermally induced refractive-index patterns can be used to focus and deflect an incoherent guided probe beam in the waveguide with time constants below 1 ms.     

A propagating beam method analysis of nonlinear effects in optical waveguides

We extend the propagating beam method to nonlinear media and subsequently analyse several examples of three and four photon processes in optical waveguides.     

Modal attenuation in metal-clad graded-index slab optical waveguides

The paper presents the influence of refractive profile shape on the distribution of modal attenuation in planar metal-clad optical waveguide structures. A calculation method has been presented in which we applied the Snyder–Love expression on modal attenuation coefficients and matrix method 4 × 4 for the analysis of planar waveguides. The obtained results are almost identical with the exact results for refractive profiles: the parabolic, exponential, Gaussian and linear ones. It has been demonstrated that the applied calculation method can be used in waveguides of any profiles of the refractive index.     

Formation and characteristics of graded-index optical waveguides buried in glass

Low-loss optical waveguides have been prepared through thermally induced ion exchange, where sodium ions in glass are replaced by silver ions from an unstirred silver nitrate melt. The index distributions of the inhomogeneous guides produced in the glass are deduced by analysing the measured mode spectra with the WKB-method. The increase in the refractive index caused by the presence of the silver is found to have a nearly Gaussian form with a peak of about 0.082. The index profile is correlated to the two parameters of the preparation process, the diffusion time and the operating temperature. In a typical case (9 hrs and 221°C) the index takes on its maximum value 2.6 μm below the surface of the glass and one obtains an 8-mode guide with five modes buried within the glass and three modes extending to the air-glass interface.A single-mode guide is formed in less than 5 min. The measurements yield an activation energy for the diffusion of Ag⁺ in glass of 8.4·10⁴J/mole and a diffusion coefficient at 221°C of 0.67·10⁻¹² cm²/s, in good agreement with the experimental results reported by other authors. The present observations suggest that a potential gradient is set up in the glass when silver diffuses into it so that the silver ions drift into the glass with a velocity of about 0.3μm/hr.     

Formation of expanded smooth quasi-Bessel beam range due to the transformation of the Gaussian beam in absorbing axicon

The quasi-Bessel beam range produced by an absorbing axicon is investigated. It is shown that when an incident Gaussian beam is used, the absorbing axicon stretches (with respect to transparent axicon case) the uniform on-axis light intensity distribution length, as well as expanding the quasi-Bessel beam range. In addition, the peak of axial light distribution is shifted from the axicon tip. As a result any spatial modulation formed due to the bluntness of the tip is reduced close to the on-axis intensity peak, which becomes more smooth and uniform.