Arbitrary Index Profile Research Articles

Two dimensional gradient-index beam shapers fabricated using ultrafast laser inscription.

In this paper gradient-index beam shapers are fabricated using the ultrafast laser inscription method. This method enables the fabrication of two-dimensional refractive index profiles inside silica glass, resulting in highly robust and compact beam shapers. The magnitude of this refractive index change can be tailored by adjusting the laser pulse energy, enabling arbitrary two-dimensional refractive index profiles to be manufactured. The process is then demonstrated by fabricating planar waveguides with quadratic index profiles that predictably resize Gaussian beams. Then a more complex two-dimensional refractive index profile is fabricated to transform an input Gaussian beam into a super-Gaussian (flat-top) beam.

The Improved Definitions for Mode-Field Radius and Divergence Half-Angle of Optical Fiber

The improved definitions for mode-field radius and divergence half-angle of optical fiber with an arbitrary index profile are proposed in this paper, and their simple relationship as well as their relations to the second moment and differential operator definitions are derived. Analytical expressions of these definitions are obtained for the LP 01 mode in step-index optical fiber; furthermore, relations between the mode-field radius, divergence half-angle, and normalized frequency are presented. Numerical data show that they are more suitable as the beam spot size of the Gaussian approximation for the near-field and far-field, respectively.

A RAPID ACCURATE TECHNIQUE TO CALCULATE THE GROUP DELAY, DISPERSION AND DISPERSION SLOPE OF ARBITRARY RADIAL REFRACTIVE INDEX PROFILE WEAKLY-GUIDING OPTICAL FIBERS

This paper introduces a new numerical method to calculate the group delay, chromatic dispersion and dispersion slope of weakly-guiding optical fibers with arbitrary radial refractive index profiles. It is based on the analytic differentiation of the propagation coefficient up to the third order. The simulation results are compared to experimental data, with those calculated by other approaches and exact data where possible. Due to the analytical differentiation of the matrix equation, the method is more accurate compared to other approaches, it is also much faster than numerical differentiation as allows avoiding repeated solution of the eigenvalue problem to calculate the derivatives of the propagation coefficient. The precision of the method is limited only by the approximation errors of the mode solver. The Galerkin method with Laguerre-Gauss basis functions is used to determine the propagation coefficients of weakly-guiding structures. The new method enables fiber manufacturers to rapidly design dispersion characteristics of graded index, step index, single- and multiple-clad fibers, as well as few-mode and bend insensitive fibers.

Mode-field half-width and divergence half-angle definitions for a planar waveguide

Mode-field half-width and divergence half-angle definitions for a planar waveguide with an arbitrary index profile are presented based on the near- and far-field distributions and their relations are given. Then, analytical formulae of these definitions are derived for the TE0 mode in a symmetrical step-index planar waveguide and numerical results prove that they are more suitable as the mode-field half-width and divergence half-angle of the Gaussian approximation for the near- and far-fields respectively.

Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments

Gradient refractive index metamaterials are of interest for various applications of transformation optics. Wave propagation through gradient index metamaterials using an exact analytical approach is investigated. Composite materials containing constituents with negative real and positive real indexes of refraction are considered. An exact analytical solution for the field distribution is obtained for the sinusoidal spatial variation of complex effective permittivity and permeability along a fixed direction, under the assumption that the wave impedance remains spatially uniform across the structure. Loss factors in the constituent materials can be different from each other corresponding to the realistic situations. Temporal dispersion can be arbitrary subject to the physical limitations imposed by the Kramers-Kronig relations. A numerical model based on the Z -transform is developed to verify the analytical results. The approach can be applied to arbitrary periodic refractive index profiles using the Fourier series method.

Mode analysis of realistic optical waveguide structures and microstructured holey fibers by modal field evolution

In this article, we describe a useful technique for calculating modes of practical optical waveguides having two-dimensional arbitrary transverse refractive index profile. The method uses a finite difference platform for evaluating Helmholtz's equation in scalar and semivectorial forms through a field evolution algorithm. The method is straightforward, easy to handle and does not involve any complex analysis or matrix formulation. We tested the accuracy of our analysis approach by applying it on a large number of realistic waveguide problems having known results or results available in the literature. The formulation has facilitated us to study the modal properties, viz., field distribution, birefringence, dispersion and mode effective area, of a variety of practical two-dimensional structures namely, planar structure, coupler, semiconductor optical waveguides, optical fibers and arbitrary profile microstructured fibers which are uniquely important in photonics and guided-wave devices. The algorithm will therefore be very useful in designing and studying any arbitrary-structure waveguides, and to explore new geometry and properties.

Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes

A rigorous full-vector analysis based on the finite-difference mode-matching method is presented for three-dimensional optical wave propagation problems. The computation model is facilitated by a perfectly matched layer (PML) terminated with a perfectly reflecting boundary condition (PRB). The complex modes including both the guided and the radiation fields of the three-dimensional waveguide with arbitrary index profiles are computed by a finite-difference scheme. The method is applied to and validated by the analysis of the facet reflectivity of a buried waveguide and the power exchange of a periodically loaded dielectric waveguide polarization rotator.

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.

Bending loss in optical fibers—a full-wave approach

The radiated power emanating from a bent single-mode fiber is computed for various radii of curvature by a full-vectorial analysis. The only approximation is the truncation of a spectral series, the accuracy of which can be controlled. Hence, the complex propagation coefficient of the fundamental mode approaches the exact value and consequently, the bending loss does as well. Two widely accepted bending-loss formulas, based on asymptotic approximations to scalar-field theory, are compared with our full-vectorial results. Both have a limited region of validity. For simplicity, the comparison is performed on a step-index fiber with a cladding of infinite extent. However, the full-wave method is capable of dealing with arbitrary index profiles.

Form and structure factors for impedance and reflection from periodic layers

In an exact treatment of the Maxwell equations, we derive form and structure factors for reflection from periodic layers, and we show that these factors are significantly different from their analogs in kinematic x-ray diffraction. Quite generally, we show that reflection and impedance can be written precisely as the sum of an additive form factor and the product of a structure factor and a second form factor. This additive form factor does not have an analog in kinematic x-ray diffraction. It is demonstrated that the form factors are found by analytic continuation to an arbitrary wavelength of expressions for the impedance both at long wavelengths and at quarter wavelengths. A correction to the Bragg law relating fringe spacing to the total structure thickness is derived. We go beyond previous numerical work by deriving simple analytic exact expressions for reflection and impedance of periodic layers for all frequencies within the reflection passband, and for an arbitrary number of periods in the structure, an arbitrary index profile within each period, arbitrary layer thicknesses (not just quarter-wave layers), and for arbitrary sizes of the refractive index differences.

Modal analysis of circular Bragg fibers with arbitrary index profiles

A finite-difference approach based upon the immersed interface method is used to analyze the mode structure of Bragg fibers with arbitrary index profiles. The method allows general propagation constants and eigenmodes to be calculated to a high degree of accuracy, while computation times are kept to a minimum by exploiting sparse matrix algebra. The method is well suited to handle complicated structures comprised of a large number of thin layers with high-index contrast and simultaneously determines multiple eigenmodes without modification.

Improved differential-transfer-matrix method for inhomogeneous one-dimensional photonic crystals

A band structure of one-dimensional periodic media having an arbitrary inhomogeneous refractive index profile is extracted based on the modification committed on the differential-transfer-matrix method (DTMM). Furthermore, the most recently modified differential transfer matrix is improved by reshaping the formulation in terms of which the electromagnetic fields are expanded and closed form formulas, providing the allowed values of Bloch wavenumbers, which were not available before. Although the frequency gaps in previously published results that we derived by using the conventional DTMM were not in agreement with the well-known Bragg law at the edge of each Brillouin zone, the new results obtained by the proposed method are now matched with the Bragg condition. The final results are also justified by either employing conventional transfer-matrix method or comparing it with exact analytical solutions, wherever such exact solutions were available.

Full vectorial modal analysis of specialty fibers and their Bragg grating characterization

Optical fibers and specialty waveguides are the bases of the majority of today's telecommunication, biomedical, sensing, and light-delivery applications. Modal analysis plays an important role in optimizing the optical performance of these fibers when they are integrated with optical systems. We present a full vectorial modal theoretical analysis of specialty cylindrical symmetric fibers with arbitrary index profiles, using a staircase approximation and scattering matrix approach with no constraints on the refractive index profile. We demonstrate the generality of this method by investigating the modal characteristics of two specialty fibers: graded-index fiber and concentric-shell multicore fiber. The calculated modal effective indices for the graded-index fiber are compared with those calculated by the WKB method, stressing the main differences between the scalar and vectorial approaches. Using the same approach, we calculate the Bragg grating response of a holographic grating written in the guiding regions of a concentric-shell fiber and compared them with experimental measurements.

Finite-difference-based mode-matching method for 3-D waveguide structures under semivectorial approximation

A finite-difference-based mode-matching method (FD-MMM) is developed and demonstrated for three-dimensional (3-D) optical waveguide structures with arbitrary index profiles and multiple scattering interfaces along the waveguide axis. The computation domain is enclosed by a perfectly conducting box coated by a perfectly matched layer (PML). Semivectorial approximation is employed to simplify the formulations and calculations. Tradeoff and guidelines for choice of the PML parameters in the mode-matching analysis are discussed by comparison with known benchmark solutions for a two-dimensional (2-D) structure. The method is subsequently applied to and validated for 3-D structures in which both single and multiple scattering interfaces are simulated.

Modeling of stimulated Brillouin scattering in optical fibers with arbitrary radial index profile

Stimulated Brillouin scattering (SBS) is an impairment seen in narrowband transmission over optical telecommunications fiber at high laser power. Accurate modeling is necessary to predict the strength of this phenomenon. One method of SBS control is the optimal design of the fiber radial index of refraction profile. Previous work has been limited to modeling optical fibers with one or more radial step changes in index. Here, the author derives and solves a general set of differential equations that allow the numerical solution of SBS spectral gain for an arbitrary radial index profile. Simulated and measured spectra are compared for several fibers with GeO/sub 2/-doped cores. It is found that radial material displacement plays a significant role in the SBS interaction in fibers with a complex radial index profile.

Finite Difference Calculations of InP/InGaAsP Optical Waveguides with Arbitrary Index Profile

A novel realization of non-uniform semivectorial finite difference method for the analysis of semiconductor optical waveguides is presented. By using non-uniform mesh structure computer time is saved while the accuracy is maintained. Also, the accuracy of the results increased by balancing the coefficient matrix without increasing computation time. Calculation of all eigenvalues (real and complex) is enabled by the method. The results of the calculation are compared for uniform and non-uniform methods and for two different optical waveguides. It is indicated that accuracy of the result is mainly dependent on the mesh refinement near the material interfaces. It is also indicated that for small mesh size using non-uniform method is necessary while uniform method is useless due to huge computer time and memory needed.

Determination of Field Configuration and Eigenvalues of PlanarOptical Waveguides with Arbitrary Index Profiles

We extend the analytic transfer matrix (ATM) method to solve thefield configuration of planar optical waveguides with arbitraryindex profiles including several turning points. The results arecompared with the predictions obtained from the finite element methodand the Wentzel–Kramer–Brillouin approximationtechnique. It is shown that the ATM method not only can produceaccurate eigenvalues, but also reach correct and meaningful fieldconfiguration for any type of optical planar waveguides.

A closed-form solution for wave propagation in optical waveguides of longitudinally invariant structures without using beam propagation algorithm

In this paper, we propose a different method to study wave propagation in longitudinally invariant waveguides with arbitrary index profile. In our method, both the electric field and the refractive index profile are expanded into two Fourier cosine series. With these series substituted into the wave equation, a differential matrix equation can then be obtained. We show here that such a matrix equation can be solved and an explicit expression for the wave field at any longitudinal position along an optical waveguide can be obtained. The solution proposed in this method can thus exclude the use of the beam propagation algorithm. This study demonstrates that our approach yields the same results as those obtained by using commercial softwares in which a beam propagation method with the Padé approximation is used.

Numerical analysis of Bragg fibers using a compact 1D finite-difference frequency-domain method

A simple full vectorial finite-difference frequency-domain method is presented for dispersion and loss analysis of Bragg fibers. In both axial and azimuthal directions, the derivatives of the Maxwell’s equations are calculated analytically, and the solution of Maxwell’s equations is reduced to a 1D eigenvalue problem. Standard step-index and graded-index fibers with parabolic-index cores are analyzed to verify the validity of this method to round symmetric fibers with arbitrary index profiles. Numerical results on the mode propagation characteristics of Bragg fibers are found to agree very well with those obtained by transfer matrix method and Chew’s method.

Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles

A rapidly converging numerical technique for the evaluation mode characteristics of circularly symmetric optical fibers with an arbitrary complex refractive index profile is presented. This method is based on transmission-line principles. From Maxwell's equations, we derive a transmission-line equivalent circuit for the optical fiber refractive index profile and we demonstrate how it can be used to determine the mode effective index (normalized propagation constant) of cylindrical dielectric waveguides. To illustrate the effectiveness of the procedure, we have applied it to circularly symmetric fibers with complex step, parabolic, and segmented optical refractive index profiles. We have used this method to evaluate and manipulate the gain in a typical 980-nm pumped erbium-doped fiber as well as for calculating attenuation of optical fibers when radial loss factors are present.