Leaky modes of optical waveguides with varied refractive index for microchip optical interconnect applications – Asymptotic solutions

In two-dimensional optical waveguides with the varied refractive index, approximate analytic solutions of the leaky modes are derived by using an asymptotic analysis for both transverse electric (TE) and transverse magnetic (TM) modes. Numerical simulations show that the asymptotic solutions of the leaky modes are quite close to the exact ones. The results are useful in the eigenmode method, where the leaky modes appear if a perfectly matched layer (PML) is used to terminate the transverse directions of optical waveguides.     

Local field analysis of bent graded-index planar waveguides

A local field analysis is proposed for bent planar waveguides with arbitrary refractive index profiles. Exact vector wave equations that include the gradient index, or polarization correction, term are derived for both transverse electric and transverse magnetic modes from Maxwell's equations in a local bent coordinate system. The approximate local field and correction to the propagation constants are obtained by perturbation analysis. As an example of the method, an infinitely extended parabolic index profile is studied     

A joint finite-difference and FFT full vectorial beam propagationscheme

A full vectorial beam propagation scheme is developed and it is applied on 3-dimensional waveguide structures. The formulation is based on the coupled wave equations for the transverse electric field. Each propagation step is performed by utilizing both the FFT and a finite-difference implementation. Under this perspective the offered advantages of FFT and finite-differences are exploited within a single propagation step resulting in a joint propagation scheme. The scheme is applied on a step-index circular fiber where analytical solutions are readily available for cross-checking. Moreover, the dependence of the phase constant on the reference refractive index is discussed. The polarized modes and the effective mode indices are derived in the case of rib waveguides by performing propagation along imaginary distance. Further, the rib waveguide coupler is examined and the energy transfer is simulated     

Propagation Constants for TE and TM Surface Waves on an Anisotropic Dielectric Cylinder

Maxwell's equations for wave propagation in a cylindrical anisotropic dielectric rod have been solved for various values of the longitudinal and transverse dielectric constants with the help of an IBM 7090 computer. The solutions are limited to modes having no rotational dependence about the direction of propagation. Families of curves for various ratios of longitudinal to transverse dielectric constants are given, showing the relationship between the guided wavelength and the diameter of the rod. Equations for the cutoff and asymptotic behavior are also given.     

Hybrid ray-mode parametrization of high frequency propagation in anopen waveguide with inhomogeneous transverse refractive index:formulation and application to a bilinear surface duct

The authors discuss a general complex spectral approach to representing high-frequency source-excited fields in open guiding environments characterized by a continuously varying refractive index in the cross section transverse to the propagation direction. The authors use problem parameters that are motivated by electromagnetic wave propagation in the Earth's atmosphere, in particular, a tropospheric surface duct adjacent to the Earth's surface. The main objective is the development of an observable-based algorithm that self-consistently incorporates various wave objects that synthesize the high-frequency propagation process in a numerically efficient and physically incisive manner. The authors concentrate on ray fields and mode fields, either individually or, most generally, in a rigorous self-consistent hybrid combination. Explicit solutions are generated for a bilinear refractive index profile that generates a leaky surface duct. For numerical results see ibid., vol.39, no.6, p.789-97 (1991)     

Infrared surface waves in circular hollow waveguides with smallcore diameters

An asymptotic form of the characteristic equation that describes wave propagation at near-infrared wavelengths in small core hollow circular waveguides is developed. Analytic solutions for the transverse and axial propagation constants are obtained. These demonstrate the transition of the TE₁₁ and TM₀₁ modes to surface waves as the guide radius is increased to values much greater than at cutoff. Relative power density distributions illustrating these mode transitions are shown     

Measurement of depolarization ratio and ultimate limit ofpolarization crosstalk in silica-based waveguides by using a POLCR

A polarization optical low coherence reflectometer (POLCR) is described that enables us to excite only the transverse electric (TE) mode of a test waveguide and to measure its Rayleigh backscatter signal distributions in the TE and transverse magnetic (TM) modes at a spatial resolution of 0.3 mm. The depolarization ratios of silica-based waveguides with relative refractive index differences of Δ=0.45 and 0.75% are obtained as 0.14 and 0.10, respectively, by measuring the bias in the ratio between the distributions in the TM and TE modes of each waveguide. By using the depolarization ratios and the Rayleigh backscatter signal levels, we calculate the ultimate polarization crosstalks to be -53 and -51 dB over 1 km, respectively. The actual polarization crosstalks of previously fabricated waveguides are about 50 dB higher than their ultimate limits for the same length of fiber     

Singularities of ray and caustic structures of electromagnetic waves in metamaterials

Ray and caustic structures formed during propagation of electromagnetic waves from a point source of electromagnetic radiation in a layer of a metamaterial with a negative refractive index are studied. Propagation of waves in a homogeneous layer bounded by curvilinear surfaces with circular and parabolic shapes, i.e., in lenses of metamaterials, is analyzed. Propagation of waves in an inhomogeneous layer is considered in the cases of longitudinal and transverse variations in the refractive index. Caustic structures corresponding to wave catastrophes of caspoid and ombilic series are found. The possibility of backward waveguide propagation in a metamaterial is shown     

Nonlinear self-guiding and switching in an all-fiber device

We study the nonlinear propagation of light in optical fibers in the strong nonlinearity limit. In this limit, the transverse guiding properties are altered and cannot be decoupled from the longitudinal part of the wave equation. We consider a refractive index profile with no bound modes at low intensities and show that stable self-guiding occurs at a critical intensity. This critical intensity is a property of the guide and can be made much lower than the threshold for unstable self-focusing. We consider a particular design for which self-guiding occurs at three orders of magnitude lower than the self-focusing threshold.     

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.     

Leaking modes of guiding W-fibers with a large difference in the refractive index profiles

For a planar three-layer guiding fiber and a guiding W-fiber, comparative analysis of the characteristics of leaking modes is performed on the basis of numerical solution of the dispersion equation. It is shown that the complex plane distributions of solutions to the dispersion equation for the transverse wave numbers of leaking modes substantially depend on the geometrical and optical parameters of such structures. In particular, it is found out that a guiding W-fiber with a large step in the refractive index ensures the most effective filtering of leaking modes and the highest concentration of the guided mode field.     

Analysis of planar waveguides formed by nonuniform nonlinearity

A planar waveguide containing a nonlinear film whose thickness varies in the transverse direction can be used to simulate propagation in a one (transverse)-dimensional medium with constant refractive index but a spatially varying nonlinearity. This idea is illustrated by finding the steady-state solutions for several examples.<>     

Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity

This work is a theoretical study of waves in a circular-cylindrical radially inhomogeneous guiding medium. A vector theory based upon Maxwell's equatious is used to derive linear homogeneous fourth-order equations satisfied by the longitudinal electric and magnetic field components for a medium in which the permittivity decreases monotonically from the propagation axis. The percentage change of permittivity from the guide axis to some radius a is assumed small. For modes with propagation constants approximately equal to the wave number at guide center, all field components are shown to satisfy second-order differential equations. In particular, all transverse field components are proportional to a single scalar function. In a Iossless system with no containing boundary, a new class of polynomial-Gaussian solutions describes the longitudinal fields for the case of a quadratically decreasing permittivity, while the transverse fields are Gaussian-Laguerre. Mode patterns, propagation constants, and orthogonality relations are given. It is shown analytically that the modes tend to TE or TM as the mode order increases. Moreover, the transverse fields become dominant at large wave numbers, and the fields become tightly bound to the guide axis as the wave number and/or inhomogeneity increases. Studies of more general permittivity variations and wall effects will be reported shortly.     

A vectorial boundary element method analysis of integrated optical waveguides

In this paper, a new vectorial boundary element method is introduced and applied to the modal analysis of dielectric waveguides with piecewise homogeneous refractive indexes. The procedure, which is free of spurious modes, determines the full field distribution from the longitudinal fields at the refractive index boundaries. Singular kernels are evaluated through series solutions while the electric field discontinuity at corners is accommodated through either a grid refinement technique or a semianalytic approach. Our formalism generates propagation constants and modal field distributions for several representative refractive index profiles with far higher accuracy than standard finite-difference or finite-element procedures.     

Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking

Inhomogeneous waves behave locally like A(r) exp[ikS(r)], where A and S are spatially dependent complex amplitude and phase functions, and k is the (large) free-space wavenumber. A previously developed asymptotic theory for high-frequency propagation and scattering of such waves is here applied to the propagation and scattering of paraxial Gaussian beams. Attention is given to Gaussian beams in free space, to beams in a lens-like medium with parabolic variation of the refractive index, and to beam reflection by a cylindrical obstacle. In the latter instance, the obstacle size may be comparable to the incident beamwidth, thereby introducing substantial distortion into the reflected beam. The results obtained from the asymptotic theory are verified by comparison with rigorously derived solutions, thereby confirming the validity of the theory, which can also be applied to more general medium and obstacle configurations.     

Finite-difference method without spurious solutions for thehybrid-mode analysis of diffused channel waveguides

Diffused dielectric channel waveguides with an arbitrarily varying refractive index profile in the cross-sectional plane are analyzed with a rigorous finite-difference method. The method is formulated in terms of the wave equation for the transverse components of the magnetic field. This 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, and allows the immediate inclusion of large index difference levels as well as the two-dimensional continuously varying index profile function without the usual staircase approximation. As an example, dispersion characteristics are calculated for structures suitable for millimeter-wave and optical integrated circuits, such as channel waveguides with refractive index variations having stepped, linear, Gaussian, and exponential function profiles. The theory is verified by comparison with results available from other rigorous methods     

Variational analysis of matched-clad optical fibers

A variational analysis of weakly guiding optical fibers is described. A series expansion of Laguerre-Gauss functions and a modified Bessel function have been used to describe the field in the fiber core and in the uniform cladding, respectively. A numerical procedure optimizing the field parameters (the expansion coefficients, the Gaussian spot-size and the normalized transverse propagation constant) has been developed which allowed highly accurate field representations to be obtained through few-term expansions. Low-order LP modes have been analyzed for refractive index profiles with a power-law variation in the core for which reference solutions (either exact or approximate) are available. A modified profile exhibiting a high index ring around the core has also been analyzed. For power-law profiles, the results are in excellent agreement with the reference solutions and show that the proposed variational analysis also appears to be appropriate for determining the field at wavelengths near to cutoff. Moreover, the comparison with other field representations which use series expansions, shows that our solution needs a noticeably lower number of terms. The analysis of the modified profile, for which only numerical solutions are available, highlights that the developed method provides very accurate field representations.     

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     

Synthesis of guided wave optical interconnects

A guided wave optical interconnect has been designed which reduces or eliminates clock skew by ensuring simultaneous delivery of clock pulses to chips mounted on a wafer. The interconnect consists of a multimode trunk waveguide and a set of branch waveguides, one per chip, each of which couples one mode out of the trunk waveguide. The elimination of clock skew is accomplished by taking advantage of the different group velocities of the modes inherent in multimode waveguides and suitably tailoring the propagation constants of the trunk waveguide according to the location of the respective chip on the wafer. Inverse scattering theory, specifically, the method of Darboux transformations, is used to design the refractive index profiles of the trunk waveguides, using the set of propagation constants selected during the first stage of the design as input data. It is shown that, by using transverse coupling and suitable design of the trunk and branch waveguides, efficient coupling from the trunk to the branch waveguides can be ensured. Techniques for ensuring a symmetric trunk refractive index profile are investigated     

The effect of the ▿n2term on the modes of an optical square-law medium

Solutions of the mode problem of an optical waveguide with a square-law medium are usually based on the scalar wave equation. An exact derivation of the wave equation from Maxwell's equations results in an additional term containing the gradientnablan^{2}of the square of the refractive indexn. It is the purpose of this correspondence to evaluate the change of the propagation constants of the modes of the square-law medium that is caused by thenablan^{2}term. We find that the change is significant only for TM modes of low order, but insignificant for all other modes of a dielectric slab with parabolic index distribution.