Graded-index Planar Optical Waveguides Research Articles

Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand–Levitan–Marchenko integral equation

The leapfrogging technique solving the Gel'fand–Levitan–Marchenko integral equation is a viable numerical tool for producing the refractive index profile associated with the design of single-mode graded index planar optical waveguides. This method is seen here to work also for the design of multi-mode waveguides of the same type, a case not examined previously in the literature. Truncated potentials are employed for modeling the behavior of guided and unguided modes of the waveguide. The accuracy of the method is checked against analytical expressions and new designs are produced for different waveguide material structures and optical frequencies.

Simple algebraic recursion formulas for calculation of propagation characteristics of graded-index waveguides

Using the perturbation method with the aid of the hypervirial theorem and Hellmann–Feynman theorem in quantum mechanics, simple algebraic recursion formulas for calculating propagation constants of graded-index planar optical waveguides in successive orders of approximation are derived. Algebraic formulas for calculation of the modal field functions are also presented. Numerical results for typical graded-index waveguides (Gaussian, exponential, complementary error function and symmetric Epstein profiles) demonstrate that the present method can be applied to an accurate calculation of the propagation characteristics to an arbitrary order of perturbation approximation.

Waveguide luminescence and Raman spectroscopy: Characterization of an inhomogeneous film at different depths

In graded-index planar optical waveguides, different guided modes propagate in layers with different thickness. By comparison of spectra (Raman, luminescence) taken by waveguide excitation in different modes, it is possible to characterize the guide at different depths. The method has been applied to waveguides obtained by ion exchange of silver in a soda-lime substrate. Luminescence from silver ions and Raman scattering from the optical vibrations of the glass and from the acoustic vibrations of silver nanoclusters depend on the silver concentration, providing different spectra for excitation in different modes of the guide.

Coupling efficiency of butt-joined planar waveguides with simultaneous tilt and transverse offset

An efficient method has been proposed for evaluating the junction loss between two nonidentical arbitrarily graded-index planar optical waveguides caused by simultaneous transverse and angular misalignments. The salient feature of the method is an exact representation of the modal. Fields which is particularly adapted to loss evaluation at imperfect junctions. The power transmission coefficient at the junction is expressed via the overlap integral method as a weighted sum of simple Laguerre-Gaussian functions, the weight factors being determined by solving a linear matrix eigenvalue problem. The present formulation has a very broad scope of validity and can be used for the modal analysis of integrated optical planar components such as waveguide lenses and to estimate the coupling efficiency of more practical fiber-to-channel butt-joints. The method can also he used to test the accuracies of other approximate methods such as those based on the Gaussian approximation of the lowest order planar waveguide mode.

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.

Characterization of asymmetric graded-index planar optical waveguides from the knowledge of TE/sub 0/-TE/sub 1/ mode cutoff wavelengths

The authors have proposed (1991) a simple method for predicting the index profile parameter (the profile exponent q in case of a power law profile or the aspect ratio S in case of trapezoidal index profile) of a graded-index optical fiber which is single moded in the 1.3-1.55 mu m wavelength range from a measurement of the LP/sub 11/ and LP/sub 02/ cutoff wavelengths. They extend the method with appropriate modification to determine the asymmetry parameter sigma , the maximum refractive index n/sub f/ and the characteristic thickness d of the guiding layer of a few-moded graded-index asymmetric planar optical waveguide from a measurement of the TE/sub 0/ and TE/sub 1/ cutoff wavelengths. The method is simple and can give firsthand information about these parameters with reasonably good accuracy.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Guided wave characteristics of metal-clad graded-index planar optical waveguides: Analytical approach

The complex eigenvalue equations of metal-clad graded-index waveguides are generally solved by numerical methods. The role of waveguide and material parameters in determining mode dispersion and attenuation becomes difficult to elucidate. We describe a perturbative method of solution which is suited to the class of waveguide in which the guided field is described analytically. Mode dispersion and attenuation are explicitly obtained. Applications to three commonly used waveguide models demonstrate the accuracy of our approach in comparison to numerically obtained results.

Piecewise linear approximation for analyzing TM modes in graded-index planar optical waveguides

A piecewise linear approximation is used to evaluate the propagation properties of TM modes in a planar optical waveguide with arbitrary index profile. This method is based on the vector wave equation and its solution can approach the exact solution by increasing the number of sublayers. The convergence properties of the propagation constant are examined numerically for two typical index profiles. The piecewise linear approximation is superior in convergence to the conventional step approximation method.

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.

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.

Planar optical waveguides formed by silver-ion migration in glass

A detailed study has been carried out on the propagation-mode characteristics of graded-index planar optical waveguides formed by silver-sodium ion exchange. It has been found that the index profile can be accurately modeled by a monotonic decrease from a maximum value at the surface, following a second-order polynomial distribution. Definition of an effective diffusion constant leads to a universal mode-dispersion diagram applicable over a wide range of fabrication conditions. Interferometric observations of the refractive-index profile, together with electron microprobe analysis, give further support to the second-order polynomial distribution. The index profile has also been compared with silver-concentration profiles predicted by interdiffusion theory. Results obtained are also relevant to methods of coupling between planar and circular fiber-optical waveguides, where sections of optical fiber undergo ion exchange.