Characterization of annealed proton exchanged LiNbO3waveguides for nonlinear frequency conversion

A thorough and detailed characterization of annealed proton-exchanged (APE) waveguides in Z-cut LiNbO₃ is described. The mode index measurements in planar waveguides as a function of wavelength and annealing time are reported, including useful analytical relations for the refractive index change, its dispersion, and the depth profile as a function of annealing parameters. Analytical expressions for the mode propagation characteristics are presented and experimentally verified with reasonable accuracy. It is shown that the planar waveguide characterization results can be used to model the channel waveguide characteristics accurately. The model provides closed-form expressions for the mode index and the mode field profile, and the theoretical results are in excellent agreement with the measured data. The technique is used to accurately predict the phase mismatch between the fundamental and second harmonic modes in frequency-doubling experiments using APE channel waveguides. An optimum waveguide geometry for which the phase mismatch is relatively insensitive to the waveguide nonuniformity was predicted and verified experimentally     

Accurate determination of the ordinary index profile ofproton-exchanged waveguides

A new method for an accurate characterization of the ordinary refractive index profile of proton-exchanged waveguides is presented and discussed. The method is based on the measurement of the power coupled into radiation modes of the waveguide as a function of the corresponding effective indexes, and allows the determination of the depth and of the film refractive index of the ordinary profile by a least squares fitting of the experimental points. The depth and the film refractive index have been experimentally obtained with an accuracy of 0.005 μm and 0.0001, respectively     

3-D analysis of LiNbO3: Ti channel waveguides and directional couplers

An efficient method of calculation of effective refractive indexes and field distribution of guided optical modes in diffused channel waveguides and directional couplers is presented. Refractive index distribution which follows from the theory of two-dimensional anisotropic diffusion is considered. Experiments which relate the refractive index profile with fabrication parameters are described. Then the coupling length of a directional coupler can be calculated from the fabrication parameters. The results correspond well to recently published experimental data.     

Rigorous finite-difference analysis of coupled channel waveguideswith arbitrarily varying index profile

A rigorous finite-difference formulation for the hybrid-mode analysis of coupled diffused dielectric channel waveguides is presented. The method includes the two-dimensional continuous index profile variations directly in the finite-difference form of coupled equations and avoids the shortcomings inherent in the usual staircase approximations. The formulation in terms of the wave equation for the transverse components of the magnetic field leads to an eigenvalue problem where the nonphysical, spurious modes do not appear. The analysis includes the complete set of hybrid-modes, takes mode-conversion effects and complex waves into account, if they exist, and allows the calculation of dielectric channel waveguides with large index difference levels. Dispersion characteristic examples are calculated for coupled structures suitable for optical integrated circuits. The theory verified by comparison with results available from other methods     

Crosstalk characteristics of Ti-LiNbO3intersecting waveguides and their application as TE/TM mode splitters

Crosstalk characteristics of an intersecting waveguide are presented. Two straight channel waveguides which intersect at an angle of a few degrees ony-cut LiNbO3were fabricated by in-diffusion of Ti. Experimental results show that the crosstalk characteristics are determined by the refractive index change profile and the geometry of intersection associated with guided wave modes. In a special case, a TE/TM mode splitter was obtained by using the intersecting waveguide which provides adequate anisotropy by the change in refractive indices. Splitting ratio was 17 and 14 dB for the TE and TM modes, respectively.     

Wavelength dispersion of effective refractive indices of Rb:KTiOPO/sub 4/ optical waveguides

Multimode planar optical waveguides in z-cut KTiOPO/sub 4/ (KTP) substrates have been fabricated using Rb/sup +/:K/sup +/ ion-exchange process. Variations in the refractive index profile and the surface index change with wavelength in these waveguides are approximated by a simple relation. The dispersion characteristics of the effective refractive indices for different guided modes before and after annealing are calculated based on this approximation and the WKB method. Furthermore, the wavelength dispersion can also be expressed using a sellmeier-like equation, and there is an excellent agreement between the calculated values and the measured data.     

Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides--II: Blazing effects

Grating-coupled radiation in lasers and waveguides is analyzed for trapezoidal-shaped grating teeth which includes all triangular shapes. A perturbation technique is used which involves calculating the modes in a geometry with an inhomogeneous refractive index profile in the grating region. Curves are presented for radiated power from traveling waves as a function of blazing angle, grating tooth height, refractive index, and grating period. It is shown that the ratio of power radiated into the outer media depends greatly on tooth shape and size. By judiciously selecting parameters this ratio can exceed 50:1.     

Method of refractive index profile reconstruction from effectiveindex of planar optical monomode waveguides: Application to potassiumion-exchanged waveguides

A simple nondestructive method for determination of refractive index profile of single-mode waveguides using the measured effective refractive indexes is proposed. The method is based on the extraction of the additional data about the waveguide profile from the investigation of its evolution with the duration of fabrication process. The profile reconstructions are carried out using an appropriate algorithm in the frames of the WKB approximation. The method is applied for few- and single-mode waveguides, obtained by K⁺-Na⁺ field assisted ion exchange. The results are corroborated with the reconstruction from the effective refractive indexes measured at a shorter wavelength, as well as with profile study by means of step-by-step etching     

Impedance boundary method of moments for accurate and efficientanalysis of planar graded-index optical waveguides

An impedance boundary method of moments (IBMOM) is proposed to accurately and efficiently compute the propagation characteristics including the number of guided modes of general graded-index dielectric slab waveguide structures. The method is based on Galerkin's procedure in the method of moments and employs the exact impedance boundary condition at the interfaces between the graded-index region and constant-index cladding. Legendre polynomials are utilized in the field expansion. Computational results are shown for waveguides with various inhomogeneous refractive index profiles. The results indicate that typically five Legendre polynomials are sufficient for accurate solutions of the dominant TE and TM modes in optical waveguides having a finite region of inhomogeneous refractive index. Diffused optical waveguides with untruncated index profiles as well as coupled dielectric waveguides can be accurately analyzed using ten Legendre polynomials     

Direct index measurements of diffused channel waveguides

A nondestructive method is described for obtaining information about the refractive index profiles of diffused channel waveguides of the type used in integrated optics. The method assumes that the index profile can be approximated as a product of two functions each of which depends on one of the two transverse coordinates. By observing the focusing effect, which the waveguide has on collimated light passing through it transversely to its axis, the shape of the index profile in a direction parallel to the surface of the substrate can be obtained. From measurements of the reflection coefficients we obtain the refractive index at the upper surface of the waveguide. By combining the two measurements the effective depth of the waveguide can be calculated. Results of measurements on multimode and single-mode waveguides formed in glass and lithium niobate are presented.     

A simple and accurate model analysis of strip-loaded opticalwaveguides with various index profiles

A simple first-order perturbation approach has been developed to study the propagation characteristics of strip-loaded diffused waveguides with various refractive index profiles. Propagation constants of the guided modes of rib waveguides and strip-loaded waveguides with exponential and Gaussian refractive index profiles are obtained. The results are in good agreement with those reported in the literature that were obtained by variational and numerical techniques. The presented technique provides analytical expressions for the modal field profile that should be useful in the design of various integrated optical devices     

Iterative simplex-finite difference method for the characterizationof optical waveguides

A new method is presented for reconstructing smooth refractive index profiles of optical waveguides from measured effective indexes. It is based on the semivectorial finite difference method to solve the polarized wave equation for a given refractive index profile. An iterative simplex algorithm is used to find the best refractive index parameters that give, as a solution, effective indexes close to the measured ones. The method is applied successfully to Ag⁺-Na ⁺ ion-exchanged glass slab waveguides and to diffused Mg/Ti:LiNbO₃ slab waveguides. Dopant concentration profiles are obtained by using secondary ion mass spectrometry. The relationship between the refractive index change and the dopant concentration is determined for both cases. The iterative simplex algorithm-finite difference method (ISA-FDM) is compared to other index profile reconstruction methods, and the advantages with respect to WKB-based methods are pointed out     

Dispersion characteristics of diffused channel waveguides

A variational method is used to calculate dispersion curves for channel waveguides fabricated by masked diffusion. Composition profiles for the waveguides are obtained by numerical solution of the diffusion equation, assuming that the diffusant concentration in the unmasked region of the substrate surface is constant during the diffusion process. A linear dependence of refractive index change on diffusant concentration is assumed. The dependence of the number of guided modes on mask gap width and diffusion depth is determined as an aid to the design of modulators and switches for integrated optics.     

Sol-gel glass waveguide and grating on silicon

This paper reports on the fabrication and characterization of hybrid organic-inorganic glass sol-gel slab and channel waveguides by ultraviolet light imprinting in thin films deposited by a one-step dip-coating process. The adjustment of chemical composition of the materials provides precise selection of refractive index from 1.48 to 1.52 at the wavelength of 632.8 mn. The refractive index of the waveguides at 1.55 μm is similar to that of optical fiber, thus reducing the reflection loss between the two to less than 0.01 dB. The effect of ultraviolet light exposure and heat treatment on waveguide refractive index is studied. Fabrication parameters to produce ridge waveguides are optimized to achieve very smooth side walls. Propagation losses in these waveguides are ~0.1 dB/cm. Single mode buried waveguides, at 1.55 μm wavelength, with circular mode profile are demonstrated     

Differential-Transfer-Matrix Based on Airy's Functions in Analysis of Planar Optical Structures With Arbitrary Index Profiles

A novel analytical method for solution of planar optical structure with arbitrary refractive index profile is proposed. This new method is founded on differential-transfer-matrices, whose field solutions are based on Airy's trial functions. In contrast to conventional Wentzel, Kramers, and Brillouin (WKB) solutions, which diverge around the turning points, this approach can be successfully used for exact calculation of various functions, including eigenvalues of optical waveguides with arbitrary index profiles, and complex reflection and transmission coefficients, even at the presence of turning points. The method is rigorous and can be applied for both major polarizations.     

A comparison between finite element calculations and experimentalresults on InGaAsP/InP waveguides

Calculated and measured properties of InGaAsP/InP waveguides are compared. The spectral response of Bragg gratings was used to measure precisely the effective indexes of the guided modes of InGaAsP/InP waveguides. The shape of the near fields and the effective refractive indexes of waveguides, which support more than one guided mode, show an excellent agreement between theory and experiment. It is shown that rigorous calculation methods such as the finite-element method can be used to predict exactly the optical properties of integrated devices     

硅基二氧化硅阵列波导光栅相位误差数值分析

采用传输函数法对硅基二氧化硅阵列波导光栅(AWG)的相位系统误差和随机误差进行了详细的分析. 系统误差的模拟结果表明阵列波导的有效折射率和相邻阵列波导长度差ΔL的偏移将会对使中心波长λ0偏离设计值，平板波导有效折射率、阵列波导的间距、罗兰圆聚焦长度R的偏移会使通道间隔偏离设计值. 随机误差的模拟结果表明相邻阵列波导长度差、阵列波导中芯区折射率、芯区宽度、芯区厚度的随机波动对AWG的串扰影响较大，而波导上、下包层折射率的波动对AWG串扰影响较小.     

Calculation of the fundamental mode sizes in optical channelwaveguides using Gaussian quadrature

A fast numerical method using Gaussian quadrature, which takes only seconds on a microcomputer, is presented for calculating the fundamental mode sizes in optical channel waveguides. Variational expressions for the square of the propagation constant, β² , of the TE- and TM-like modes are derived using the vector wave equations. For channel waveguides with gradual refractive index distributions, these expressions approach the variational expression obtained using the scalar wave equation. To show the usefulness of the numerical technique the authors present the results for titanium indiffused lithium niobate channel waveguide which are commonly used in integrated optical circuits. Since these waveguides have gradual refractive index distributions, both types of expressions give the same results; however, it takes less time to compute the mode sizes when using the variational expression obtained from the scalar wave equation. The authors find the calculated mode sizes are in good agreement with published measurements. From the comparison process, best fit parameters are obtained, which give mode sizes close to the values published in the literature. For one special case the authors are able to obtain an analytical variational expression and they use it to test the accuracy of the numerical method. They find that the values of β² given by both methods agree to six significant figures     

Fiber-compatible K+-Na+ ion-exchanged channelwaveguides: fabrication and characterization

A systematic study of waveguides fabricated by K⁺-Na ⁺ exchange in soda-lime silicate and BK7 glasses is presented. The measured K⁺ concentration profile, the refractive index profile, and the diffusion profile obtained by solving the one-dimensional diffusion equation are correlated to explain the differences in the index profiles in the two glasses. The mobility of the potassium ions was measured by fabricating waveguides using electromigration. Surface waveguides formed by diffusion from a molten KNO₃ salt bath were buried by applying an electric field. Single-mode channel waveguides for operation at a wavelength of 1.3 μm that exhibit excellent mode matching with conventional optical fibers, achieving a fiber-waveguide insertion loss of less than 1 dB for a 20-mm-long waveguide, have been obtained     

Features of Propagation Characteristics of Ultrahigh Frequency Radio Waves in an Above-Water Duct as Functions of the Duct Parameters and Wavelength

A new numerical method for finding roots of the characteristic equation in problems of radio wave propagation in tropospheric ducts with a piecewise linear profile of the refractive index is proposed. For the selected model of the vertical profile of the refractive index of an surface duct, dynamics of propagation constants of modes and features of the behavior of the attenuation function of the electromagnetic field as functions of the troposphere parameters and the wavelength are studied. The contribution of the whispering gallery modes is also estimated.