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.     

A refined WKB method for symmetric planar waveguides withtruncated-index profiles and graded-index profiles

An accurate method is proposed, based on a modified version of the WKB method in quantum mechanics, for evaluating the propagation constants and field distribution of symmetric planar waveguides with a truncated-index and graded-index profile. Simple formulas are given for calculating the dispersion characteristics including those for the truncated parabolic and exponential profile, and results are much more accurate than those of the ordinary WKB method, especially in the near cutoff regions. This method also can be extended to other more complicated cases     

Refractive-Index Profiling of Buried Planar Waveguides by an Inverse Wentzel–Kramer–Brillouin Method

We propose an inverse Wentzel-Kramer-Brillouin (WKB) method for the construction of the refractive-index profiles of buried graded-index planar waveguides from the measured effective indexes of the guided modes. Our method involves experimental determination of the location of the peak index in the buried profile, measurement of the effective indexes in air, and an index-matching liquid, and application of a searching algorithm to find the best-fit profile through inverting the WKB equations. We present numerical examples and experimental results to demonstrate the performance of the method.     

Uniform Asymptotic Technique for Analyzing Wave Propagation in Inhomogeneous Slab Waveguides

The guided modes of inhomogeneous dielectric slab waveguides are analyzed by a uniform asymptotic technique based on the related equation method. This technique gives highly accurate solutions in the sense of asymptotic expansion. The algorithm for calculating the guided modes of slab waveguides with an even polynomial refractive-index medium is presented. As an example, we calculate the third-order approximate solutions for the guided modes in an analytic form. The results show that the WKB solutions for higher order modes are more accurate than for the lower order modes and the correction to the WKB solutions is significant for the lower order modes. The numerical result for eigenvalues and modal fields confirms that the third-order asymptotic solution is accurate for all the guided modes of the near-parabolic profile waveguides and for higher order modes in the case of the quasi-Gaussian profile.     

Refractive-index profiling of graded-index planar waveguides from effective indexes measured with different external refractive indexes

We extend the well-known inverse Wentzel-Kramer-Brillouin (WKB) method for refractive-index profiling of graded-index planar waveguides. We demonstrate, with numerical examples and experimental results, that the refractive-index profile of a graded-index planar waveguide can be calculated from effective indexes measured with different external refractive indexes. With this technique, single-mode and two-mode waveguides can be profiled easily to a good accuracy.     

Losses in optical waveguide formed by silver-sodium ion exchange

Propagation losses of optical waveguides formed by ion exchange from molten salts of silver nitrate are discussed. The refractive index in pyrex glass substrates is shown to obey an exponential profile and the resultant mode functions are also examined for the graded-index surface waveguide in order to understand the mechanism of a scattering loss. The attenuation of the sample waveguides is found to range from 2.0 to 2.5 dB/cm for TE0mode excitation and to be decreasing with increasing both the mode number of the guided beam and the waveguide thickness that is proportional to a square root of diffusion time. These results indicate that the predominant loss origin is silver ion concentration diffused in the glass.     

Analysis of optical fibers with graded-index profile by acombination of modified Airy functions and WKB solutions

An almost exact eigenvalue equation for optical fibers with graded-index profile is derived mathematically based on a combination of the modified Airy functions and the Wentzel, Kramers, Brillouin (WKB) trial solution. By applying proper boundary conditions, a phase shift correction term δ is found out which improves the inherent error problems of the conventional WKB method. It is shown through computer simulations that results of the derived eigenvalue equation are in excellent agreement with those of the finite-element method     

Refractive-index profiling of graded-index planar waveguides fromeffective indexes measured for both mode types and at differentwavelengths

It is well known that, under the WKB approximation, the refractive-index profile of a graded-index planar waveguide can be calculated from the corresponding effective-index function, which can be found approximately by least-squares fitting of a set of measured effective indexes. In the previous work, effective indexes measured for a particular type of modes (the TE or TM modes) at a single wavelength are used. This requires that the waveguide supports at least three modes of the same type. In this paper, techniques of combining the effective indexes measured for both mode types and at different wavelengths are proposed and demonstrated with examples. With these techniques, single-mode and two-mode waveguides can be profiled     

General eigenvalue equations for optical planar waveguides witharbitrarily graded-index profiles

Accurate eigenvalue equations for planar waveguides with arbitrarily graded-index profile are derived and expressed in closed forms. A combination of the modified Airy functions and the Wenzel-Kramers-Brillouin (WKB) solutions are employed as field solutions, which turn out to represent almost exact field profiles. The use of new trial solutions enables us to calculate phase shifts at turning points very precisely, allowing us almost exact eigenvalues. It is demonstrated that the results obtained by the proposed method are in excellent agreement with those by the finite element method, achieving significant improvement over the conventional WKB method     

Modes in optical waveguides formed by silver-sodium ion exchange

Exact solutions of the scalar wave equation for the guided modes of planar optical waveguides formed by silver-sodium ion exchange have been obtained. It has been shown that there is good agreement between the exact solution and the solution obtained by using the Wentzel-Kramer-Brillouin (WKB) approximation.     

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.     

梯形截面介质光波导的模方程

本文简化了有效折射率法和WKB法对梯形截面介质光波导传输特性的分析过程,给出了计算E_(mn)~X模有效折射率的近似模方程,并用数值计算结果和实验结果检验了该模方程的精度。     

Exact analysis of intermodal dispersion compensation in spliced graded-index optical fibers

Intermodal dispersion compensation in spliced graded-index optical fiber is investigated. Group delay times of modes in individual and spliced fibers are exactly analyzed using the Wentzel-Kramers-Brillouin (WKB) method. It is found that in actual fibers whose refractive index profiles are not represented by the power-law function, the compensation effect is less remarkable than that predicted from the power-law approximation. The baseband bandwidth of spliced fiber, composed of under- and over-compensated index profile fibers, is calculated based on the measured index profile. As a result, the measured bandwidth is found to be well predicted with an error less than 5 percent.     

Electron waveguiding characteristics and ballistic current capacityof semiconductor quantum slabs

Semiconductor slab electron waveguides with and without spatially varying effective mass are analyzed using the single-band effective-mass equation. Starting with ballistic electron incidence on a potential energy/effective mass interface, expressions for the phase shift, the lateral shift, and the time delay upon total internal reflection are found. It is shown that heterostructure wells, homostructure voltage-induced wells, and heterostructure barriers can act as waveguides for ballistic electrons, and that the waveguiding is described by a single dispersion relation. The guided mode wave functions, dispersion curves, cutoffs, group velocity. effective mass, density of states, and ballistic guided current density are determined     

Guided-wave characteristics of optical graded-index planarwaveguides with metal cladding: a simple analysis method

The guided mode characteristics of metal-clad graded-index waveguides are generally determined by complex eigenvalue equations or perturbative methods. However, these methods either require several iterations in the complex plane or are only suited to the class of waveguides whose guided field can be described analytically. We describe a method of calculating complex propagation constants for metal-clad graded-index waveguides under very general but weakly guiding conditions. The method, based on Galerkin's formalism using trigonometric basis functions, allows arbitrary inhomogeneous and complex refractive index profiles. Applications to three generally used waveguide models show our approach to be in good agreement with other analytical or numerical methods     

Unified nonlinear waveguide dispersion equations without spuriousroots

Analysis is made of the behavior of transverse electric (TE) waves in an optical waveguide comprising a nonlinear film sandwiched between two semi-infinite linear media. Unified expression of dispersion equation is presented for all different nonlinear waveguides, which could have any possible linear dielectric profile, and comprise a nonlinear film with either self-focusing or self-defocusing nonlinearity. This new dispersion equation offers simple numerical calculations and avoids the appearance of spurious roots     

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     

Fabrication of planar optical waveguides by K+-ion exchange in BK7 and Pyrex glass

Planar graded index optical waveguides have been formed by K+ ion exchange in BK7 optical glass and Pyrex glass. Measurements of the mode indexes have been made and are well described by the WKB dispersion relation with a complementary error function refractive index profile for both glass types with refractive index changes (Delta n) of 0.0094 and 0.0073 for BK7 and Pyrex, respectively. A linear relation was found between the diffusion depth and the square root of the diffusion time for both materials, and effective diffusion coefficients were calculated. Diffusion times necessary to achieve a desired number of modes and diffusion depths have also been established.     

High spatial resolution diagnostics of optical waveguides using aphoton-scanning tunneling microscope

We propose a high-resolution diagnostics technique for optical waveguides using a photon-scanning tunneling microscope (P-STM). This technique has the advantageous capabilities of nondestructive measurement and informative analysis of guided mode. The capability of this technique was evaluated by measuring the characteristics of an LiTaO₃ waveguide. Scattered light spot of 500-nm diameter caused by defects was measured, and a normalized width was estimated to be 3.7 μm. Moreover, we obtained the power dispersion in Y-branch waveguides, which were in good agreement with the theoretical plots calculated by the beam propagation method (BPM)     

Dispersion in optical waveguides with graded refractive index

Dispersion in graded-index optical fibres can be calculated by ray-optical or wave-optical methods. This letter reports a seeming discrepancy between the results of both methods: in a fibre with a refractive profile that has been optimised by the WKB wave-optical method, helical rays may occur with a much larger transit-time spread than the modes computed with the WKB method. The question is what transit-time spread is relevant for the bandwidth of the fibre.