Numerical field solution for an arbitrary asymmetrical graded-index planar waveguide

We present a numerical solution for the field amplitudes in an asymmetrical graded-index planar waveguide with an arbitrary index profile. For small index differences between the surface and bulk indexes and for large differences between the cover and surface indexes, the modes of the diffused waveguides can be described entirely in terms of normalized mode index, diffusion depth, effective modal width, and theVnumber. The results show a high degree of accuracy when checked against exact published results for the parabolic and exponential index profiles. Universal charts for the modal fields in terms of the normalized quantities are presented for profiles of practical interest, namely, Gaussian and complimentary error function index profiles. We show that the modal width, while somewhat sensitive to theVnumber, is surprisingly insensitive to the index profile. Tailoring of the index profile, therefore, does not seem important for the efficient fiber-waveguide endfire coupling. Error involved in the estimation of the phase shift at the cover-surface boundary as function of the asymmetry parameter is shown to be rather small for practical cases. Also, the discrepancy in the field distribution between an almost symmetrical and a highly asymmetrical waveguide at the cover-surface boundary is illustrated with a practical example.     

Strip-loaded diffused optical waveguides

A strip-loaded diffused optical waveguide consists of a one-dimensional diffused guide with a narrow loading strip of dielectric deposited on the surface. Two-dimensional modes can exist in the diffused material, confined in depth by the diffusion and in width by the loading strip. The propagation constants of these two-dimensional modes are calculated. The requirements for the index of refraction and width of the loading strip are discussed along with the rate of decay of electric-field amplitude away from the strip. A numerical example typical of Ti-diffused LiNbO3is given.     

Space-charge layer width in diffused junctions

This paper outlines a calculation of space-charge layer width in a planar junction made by diffusing an n or p impurity (assumed to follow a Gaussian or a complementary error function distribution) into a uniformly doped crystal of opposite conductivity type. The collector junction of most drift transistors conforms closely to this model. An exponential approximation to the impurity distribution permits curves to be drawn of the space-charge layer penetration in each direction from the junction as a function of applied reverse voltage, and of the electric field distribution. The quantities involved are normalized in terms of the initial doping level N1, the impurity diffusion lengthL = 2 sqrt{Dt}, and the junction depth xj. The curves should be useful in calculating depletion-layer capacitance, transistor punch-through voltage and junction breakdown voltage.     

Modal analysis of step discontinuities in graded-index dielectricslab waveguides

The modal solutions of the TE- and TM-modes of an arbitrary graded-index dielectric slab waveguide have been derived by applying the generalized telegraphist's equations to the equivalent inhomogeneous parallel-plate waveguide model with electric or magnetic walls. These modal solutions have been employed in a mode-matching procedure to calculate the transmission properties of step discontinuities in arbitrarily graded-index dielectric slab waveguides such as diffused optical waveguides having exponential, Gaussian and complementary error-function index profiles. For slab waveguides containing an abrupt offset, the radiated power is found to increase smoothly with increasing displacement. Power loss calculations for an abrupt change in diffusion depth for which the dominant mode is in the vicinity of cutoff, exhibit a sharp transition from almost zero loss to nearly total radiation loss     

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     

On the variation of cut-off frequency with emitter end concentration of a diffused base transistor

The effect of mobility degradation at high impurity concentrations on the cut-off frequency of a diffused base transistor with exponential impurity profile in the base has been considered analytically. It has been shown that the cut-off frequency exhibits a maximum with increase in emitter and concentration. The results are compared with those of complementary error function and Gaussian base impurity profiles.     

Finite-element solution of planar arbitrarily anisotropic diffused optical waveguides

The finite-element method for propagation in planar anisotropic diffused optical waveguides with arbitrary permittivity tensor is presented. A Galerkin procedure has been introduced to the finite-element formulation, to study both the nonleaky and leaky surface waves. The complex propagation constants are determined as a function of frequency for possible modes of propagation. The accuracy of the method has been checked by calculating the nonleaky and leaky surface waves of Ti-diffused LiNbO3waveguides with Gaussian index profiles. The numerical results of Ti-diffused LiNbO3waveguides with dielectric overlays are also presented and the effects of dielectric overlays on the propagation characteristics for the nonleaky and leaky surface waves are examined.     

Diffused waveguides: mode perturbation by thin-film cladding

Normalised parameter plots are given enabling prediction of propagation-constant perturbation in diffused optical waveguides due to a thick dielectric cladding. Refractive-index profiles of Gaussian and complementary-error-function form are treated. The variation of the number of propagating modes with time of diffusion is derived.     

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.     

Analytic evaluation of diffused impurity layers in silicon

An analytic method is outlined for the calculation of average conductivity of diffused layers in silicon for a complementary error function and Gaussian impurity profiles and the results are found to be in good agreement with those obtained by the numerical integration method. It is shown that the results for Gaussian distribution are obtained directly from the derivations for an error-function distribution. The method is also useful for the estimation of the temperature coefficient of the diffused layers.     

The feasibility of obtaining general impurity distributions by diffusion

Although it is possible to optimize selected characteristics of semiconductor devices by controlling the net impurity distribution, present methods used for fabricating planar devices by diffusion techniques invariably produce profiles which are approximately either Gaussian or complementary error functions. The purpose of this paper is to examine the feasibility of obtaining a general impurity profile within intrinsic material using diffusion techniques. It is shown that by treating the problem as an optimum control problem, one can, in principle, obtain a best-fit approximation to a given profile by controlling the gas stream impurity concentration. A first-order model is developed to characterize the surface interactions between the gas ambient and the solid; the boundary value problem for diffusion is then solved. A computational routine using linear programming techniques is developed which, for a given diffusion time, determines the necessary control function producing a best-fit approximation to the desired profile over the spatial range of interest. The optimum control functions governing the diffusion of boron in silicon for three given impurity profiles are synthesized: a constant profile, an exponential profile, and the minimum transit time profile. The computational results clearly establish the feasibility of obtaining an approximation to a given impurity profile.     

The statistics of HF sea-echo Doppler spectra

Several important statistical properties of the HF sea echo and its Doppler power spectrum, which are useful in optimizing the design of radar oceanographic experiments, are established. First-and second-order theories show that the echo signal (e.g., the voltage) should be Gaussian; this is confirmed with experimental surface-wave data i) by comparison of the normalized standard deviation of the power spectrum at a given frequency with its predicted value of unity, and ii) by cumulative distributiun plots of measured spectral amplitudes on Rayleigh probability charts. The normalized standard deviation of the dominant absolute peak amplitudes of the power spectrum (which wander slightly in frequency) are shown from experimental data to besim 0.7for the first-order peaks andsim 0.5for the second-order peaks. The autocorrelation coefficient of the power spectra is derived from measured data and interpreted in terms of the spectral peak widths; from this information, the correlation time (or time between independent power spectrum samples) is shown to besim 25-50s for radar frequencies above 7 MHz. All of these statistical quantities are observed to be independent of sea state, scattering cell size, and relatively independent of radar operating frequency. These quantities are then used to establish the statistical error (and confidence interval) for radar remote sensing of sea state, and it is shown, for example, that 14 power spectral samples result in a sample average whose rms error about the true mean is 1.0 dB.     

Characteristics of diffused P-N junctions in epitaxial layers

A one-dimensional analysis has been made to determine properties of diffused p-n junctions in epitaxial layers with nonuniform impurity concentration. Impurity diffusion from the surface and from the substrate is assumed to have complementary error function distribution. The transcendental equations obtained by analytical integration of Poisson's equation were evaluated numerically with the IBM 7090/94. Junction depth, impurity gradient and impurity level at the junction are given for a variety of diffusion parameters and impurity concentrations. In addition, graphs are presented, showing the relationship between reverse voltage and depletion layer thickness, capacitance per unit area, and peak electric field for the case of silicon. A comparison between the actual impurity profile and the usual linear approximation using the impurity gradient at the junction gives the range of depletion layer thickness or reverse voltage in which such an approximation is justified. Further, examples are presented of the electric field distribution in the depletion layer for several impurity concentration profiles. Calculated and experimentally determined values of some readily accessible junction characteristics show reasonably good agreement.     

Nd：MgO：LiNbO3和Er：LiNbO3波导激光器的阈值泵浦功率和斜效率的理论研究

本文从速率方程出发，考虑了泵浦光和激光模式的空间分布分别推导了Ｎｄ：ＭｇＯ：ＬｉＮｂＯ３和钛扩散Ｅｒ：ＬｉＮｂＯ３波导激光器的阈值泵浦功率和斜率效率的一般表达式。     

The effect of spatially varying diffusion constant on the transient decay of short circuit current in electron irradiated p−n junctions

A Gaussian doped p−n junction has an E field in the diffused layer which increases linearly from the surface toward the junction. It will also have a diffusion constant which varies with doping density and thus with position in the diffused layer.

In this paper these two effects are taken into account in calculating the transient decay of short-circuit current when such a p−n junction is irradient by an electron beam. The Rayleigh-Ritz method is used to approximate the eigenvalues which determine the decay time constants. The limiting cases of zero and infinite surface recombination velocities are treated.

It is found that for typical doping the decay time is about a factor of 2 longer than would be expected if the diffusion constant were taken to be spatially invariant and to have a value equal to its spatial average value taken over the diffused layer.     

Coupled mode theory of parallel waveguides

A new coupled mode formulation for parallel dielectric waveguides is described. The results apply to any guided modes (TE, TM, or hybrid) in waveguides of arbitrary cross-section, dissimilar index, and nonidentical shape. Additional index perturbations not included within the waveguides are encompassed by the theory. Propagation constants and mode patterns for the coupled modes computed according to this theory are shown to agree very well with numerical solutions for the system modes when the latter can be determined. Moreover, the new results are more accurate than those obtained from prior coupled mode formulations. It is shown that even for Iossless guides the coupling coefficients from waveguide"b"to"a"and from"a"to"b,"described by kaband kbarespectively, are not related by their complex conjugates if the guides are not identical.     

Performance of lattice codes over the Gaussian channel

We derive an upper bound on the error probability of lattice codes combined with Quadrature Amplitude Modulation (qam) over the additive white Gaussian noise channel. This bound depends on a lattice figure of merit and is readily put in exponential form by using Chernoff bound. An interesting lower bound is derived by a similar reasoning. We also examine the estimation of the average information rate based upon the continuous approximation of the average power normalized to two dimensions, and suggest to improve it by using the sphere packing idea. Examples of performance evaluation are given for a few lattices. Finally, we present upper and lower bounds on the best fundamental coding gains per dimension (due to both density and thickness) for an arbitrarily large number of dimensions. It is shown in the Appendix that, as the Ungerboeck codes, the lattice codes do not shape the signal power spectrum.     

Impurity-density distribution in the base region of drift transistors

The nature of the impurity grading in the base of drift transistors is studied by measuring the dependence of the base transit time on collector voltage. Provided that modulation of collector-depletion layer width caused by change of collector voltage occurs only in the base material, as in the case of an alloyed collector, it is possible to deduce the base-region field parameterm=Delta V/(kT/q)from such transit-time measurements. By this means, the validity of assumed distributions of impurity density may be verified; in particular, it may be established whether the impurity grading approximates an exponential or a complementary error function (erfc) form. Results are given for a number of drift-transistor samples, most of which are believed to have undergone impurity diffusion into the base material from a constant surface concentration during fabrication. In all cases, however, interpretation of measured data indicates a base impurity-density distribution approximating exponential rather than erfc form to be present.     

Base transport factor calculations for transistors with complementary error function and Gaussian base doping profiles

By taking account of the carrier mobility degradation at high impurity concentrations, the high-frequency base transport factor of an n-p-n germanium base transistor was computed numerically for different base doping levels. The doping profiles under consideration were Gaussian and complementary error functions. The base doping level adjacent to the emitter was optimized for minimum base transit time. The optimum values are 4×10¹⁷atom/ cm³for complementary error function profile and 2×10¹⁷atom/cm³for Gaussian profile. The effects of emitter barrier capacitance, base resistance, collector barrier capacitance, and the collector depletion layer on the overall frequency response of a junction transistor are also discussed.