Improved inverse WKB procedure to reconstruct refractive index profiles of dielectric planar waveguides

A new method to reconstruct smooth refractive-index profiles from measured effective indices is presented. The profile form near the surface and its approach towards the bulk value are explicitely taken into account. Well behaved profiles such as Gaussian or exponential can be reconstructed within experimental accuracy from three modes only, and even two modes may provide useful information. The method is tested for numerical stability.     

Light-induced refractive index changes in LiNbO3:Ti waveguides

Holographic measurements of light-induced refractive index changes in LiNbO₃Ti waveguides are reported. The results demonstrate that Fe²⁺ centers are stabilized against oxidation in the region of large Ti concentration near the surface thus increasing the sensitivity to optical damage considerably. This undesired effect may be largely abolished by additional in-diffusion of protons.     

Losses in diffused optical waveguides

It has been shown that the attenuation characteristics of TE modes in diffused waveguides are different from those of a conventional slab. A simple semiempirical analysis is performed on the base of a ray-optics picture assuming a parabolic index distribution. Analytical expressions for surface scattering and volume loss are derived. The analysis precisely describes the modal loss behaviour of Ag⁺ ion-exchanged waveguides in glass (new results) and Nb diffused waveguides in LiTaO₃ (data reported by other authors).     

Second harmonic generation in planar optical waveguides

We study optical second harmonic generation (SHG) in planar waveguide structures composed of several layers with different dielectric constants. We develop a general formalism for the calculation of mode generation by a planar antenna embedded in the waveguide. As an application we consider a monolayer of high second-order susceptibility adsorbed at the interface between two layers of the waveguide structure. Periodic modulation of the nonlinear susceptibility allows phase matching leading to dramatically enhanced second harmonic intensities. We investigate the SHG-efficiency of various experimentally realizable geometries.     

Bisection algorithm to calculate hybrid modes of birefringent planar graded index waveguides

Guided modes of a planar dielectric waveguide which encounter a nondiagonal permittivity tensor are neither TE nor TM, but hybrid. They are described by a pair of coupled second-order differential equations for the transversal electric and magnetic field components. We construct a real-valued function which plays the role of the transversal electric or magnetic field in the uncoupled Sturm-Liouville differential equation for TE or TM modes. The number of zeroes, or nodes, of this function labels the modes. The nodes increase with the prospective propagation constant. This fact is proven by constructing suitable self-adjoint operators and referring to the minimax principle. The nodal properties allow to formulate an efficient bisection algorithm for effective indices and field distributions of guided hybrid modes.     

Determination of refractive index profiles of gradient optical waveguides by ellipsometry

Ag⁺-Na⁺ and K⁺-Na⁺ ion-exchanged optical waveguides in soda-lime glass are characterised by ellipsometry. Refractive index profiles of the waveguides are calculated from ellipsometric multiple angle of incidence data using the Newton-Kantorovitch type iterative procedure and compared with those reconstructed by inverse WKB method. It is demonstrated that such continuous profiles with relatively small index gradient (of the order of 0.1 and 0.01), extending to few micrometers in depth, can be determined by ellipsometric measurements. A good agreement is found between results obtained by ellipsometry and by the inverse WKB method at depths above 500–600 nm, while there is a difference in the subsurface region, where ellipsometry is more sensitive to the quality of the surface. The profiles obtained by the two methods are consistent if the surface thin layer is etched.     

Ti:LiNbO3 optical waveguides

Various experiments on Ti diffused optical waveguides in LiNbO₃ have been carried out in order to determine precisely the character of the diffusion process. The required guide parameters and the effective mode indices could be controlled by adjusting only the diffusion time under fixed temperature and film thickness. Therefore the dependence of the guide characteristics on the diffusion time has been investigated in detail. On the basis of the data obtained, a two-stage diffusion model is proposed. In the first stage, the Ti diffusant profile is described by a erfc-function, and the second stage is characterized by a modified Gaussian form.     

Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O ion implantation into lithium niobate

A method named intensity calculation method (ICM), which is based on beam propagation method (BPM) and image processing, was carried out to reconstruct the extraordinary refractive index profile (RIP) of single-mode planar waveguide in lithium niobate (LiNbO₃), which was fabricated by multi-energy megaelectron-volt (MeV) O²⁺ ion implantation. In addition, it has been proved reasonable that the alternation of extraordinary refractive index induced by ion implantation into LiNbO₃ is mainly due to the degradation of polarization and reduction of material physical density. As a result, the possible extraordinary RIP of the double-mode planar waveguide could be reconstructed using BPM according to such a hypothesis and the calculated guiding mode values. The end-fire coupling and m-line arrangements were carried out to obtain the near-field modal patterns and dark-mode spectra of waveguides, respectively.     

Construction of the refractive index profiles for few-mode planar optical waveguides

We present a nondestructive technique to predict the refractive index profiles of isotropic planar waveguides, on which a thin gold film is deposited to as the cladding. The negative dielectric constant of the metal results in significant differences of effective indices between TE and TM modes. The two polarized modes and a surface plasmon resonance (SPR) with abundant information of the surface index can be used to construct the refractive index profiles of single-mode and two-mode waveguides at a fixed wavelength.     

Propagation characteristics of Ag+ ion-exchanged stripe waveguides

A detailed experimental study of stripe waveguides formed by Ag⁺/Na ion-exchange has been carried out. Propagation constants were precisely measured by excitation of single output pq-lines. The resulting data allowed us to approximate the transverse silver distribution by a rectangular profile. The effective index method is used for calculating theoretical propagation constants, assuming no sideways diffusion. A comparison between experiment and theory is made over the range of mask widths studied.     

Holography with guided optical waves

This paper deals with “waveguide holograms” recorded in a layer of storage material covering a planar dielectric waveguide. Reference and read-out waves are guided modes of the waveguide. Their field in the storage medium is evanescent. A theory of the diffraction efficiencies of these waveguide holograms is presented fors-polarized wavefields. To calculate the hologram structure the attenuation of the reference wave caused by absorption in the light-sensitive storage material is taken into account. Analytical expressions for the local and the overall diffraction efficiencies and for the intensity profiles of the diffracted fields are derived. The dependence of these quantities on experimental parameters (the waveguide thickness, the mode numbers of the reference and read-out waves, and the angle of incidence of the plane object wave) is presented graphically, i.e., by computer plots. Grating couplers for integrated optics can be made by waveguide holography. We consider this application to be interesting because incoupling efficiencies for Gaussian beams of up to 96% can be achieved theoretically.     

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.     

Spectral excess attenuation of monomode optical fibers with sinusoidally perturbed core radius

The spectral excess attenuation caused by coupling between the guided HE₁₁ mode and radiation modes is calculated with the aid of the formula derived by Marcuse for the radiation loss of a monomode optical fiber with step-index profile and sinusoidally perturbed core radius. The pronounced minima and maxima of the spectral excess attenuation curve converge as the spatial frequency of the perturbation of the core radius increases. Even perturbation amplitudes as small as 0.01 μm may lead to excess attenuations of the order of 10 dB/km if the radiation wavelength does not coincide with an attenuation minimum. The spectral behavior of the excess attenuation can be utilized to design fibers which act as optical filters.     

Improvement of photorefractive properties of proton-exchanged LiTaO3 waveguides

3 are fabricated by both a proton exchange and a successive copper exchange. The influence of different fabrication steps on the refractive index profile and optical absorption is investigated. The holographic efficiency is determined together with the dark- and photo-conductivity. We show that the photorefractive properties of proton-exchanged LiTaO₃ waveguides are considerably improved by the additional copper exchange. It is established that the holographic efficiency depends on the copper content, the phase of the proton-exchanged layer and the light intensity. Received: 3 March 1997     

Photorefractive waveguides in oxide crystals: fabrication, properties, and applications

3 , LiTaO₃, BaTiO₃, KNbO₃, Sr_xBa_1-xNb₂O₆ (0.25≤x≤0.75, SBN), and Bi₁₂(Si,Ti,Ge)O₂₀ (BSO, BTO, BGO) waveguides are discussed. Furthermore, the suitability of photorefractive waveguides for nonlinear optical components is demonstrated in some examples. Received: 21 April 1998/Revised version: 28 May 1998     

An analysis of optical waveguide tapers

The characteristics of slab waveguide tapers with finite-thickness cladding are investigated by approximating the tapered section as a series of uniform five-layer waveguides with successive changes in the core and cladding thickness. The behaviour of multimode coupling and transmission characteristics for typical tapers are studied in detail by computer simulation, employing parameter values appropriate for practical systems. Strong coupling and efficient conversion between core and cladding modes are demonstrated. In addition, it is shown that for tapers of moderate slopes, the nearest-neighbour coupling approximation is quite adequate for the calculation of the individual mode amplitudes. For steeper tapers, however, additional neighbouring modes have to be taken into account, and the modal energy tends to spread further towards the higher-order modes. It is also demonstrated that relatively large step-size can be used in the iterative numerical calculations. This, in conjunction with the choice of an appropriate coupling scheme (e.g., the nearest-neighbour approximation for mild tapers), makes it possible to reduce the cost of computer simulation. The relevance of the present analysis to fiber tapers is also discussed. Supported in part by the Department of Communications of Canada     

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.     

Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals

The use of surface-impedance and surface-admittance concepts for analyzing reflection and refraction at an isotropic dielectric interface (first developed about 1938) is extended to include an interface between uniaxial birefringent dielectrics. Total internal reflection and the polarizing (Brewster) angle at an anisotropic interface are shown to be naturally explainable in terms of surface impedance (for TM polarization) and surface admittance (for TE polarization). The allowable modes in an integrated optical uniaxial asymmetric dielectric slab waveguide are also shown to be directly obtainable using the surface impedance/admittance approach. Numerical examples are presented.