Arbitrary Refractive Index Profile Research Articles

A search-and-track algorithm for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles

A search-and-track algorithm is proposed for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles. The algorithm starts with an initial guess point in the parameter space that supports a specific number of guided modes. Then, it searches for, and tracks, the boundaries of this space or another space supporting different number of modes. It does so by monitoring the sign of a unified cutoff dispersion function. The algorithm is applied to both symmetric and asymmetric silicon-based parabolic-index waveguides. It shows that unlike asymmetric waveguides, the single-mode condition of symmetric waveguides is controlled by TM-, as opposed to TE-, polarization. This abnormal polarization control is strongest for high index contrast waveguides of sub-micrometer core sizes. The results are verified by full-vectorial beam propagation method.

A vector boundary matching technique for efficient and accurate determination of photonic bandgaps in photonic bandgap fibers

A vector boundary matching technique has been proposed and demonstrated for finding photonic bandgaps in photonic bandgap fibers with circular nodes. Much improved accuracy, comparing to earlier works, comes mostly from using more accurate cell boundaries for each mode at the upper and lower edges of the band of modes. It is recognized that the unit cell boundary used for finding each mode at band edges of the 2D cladding lattice is not only dependent on whether it is a mode at upper or lower band edge, but also on the azimuthal mode number and lattice arrangements. Unit cell boundaries for these modes are determined by mode symmetries which are governed by the azimuthal mode number as well as lattice arrangement due to mostly geometrical constrains. Unit cell boundaries are determined for modes at both upper and lower edges of bands of modes dominated by m = 1 and m = 2 terms in their longitudinal field Fourier-Bessel expansion series, equivalent to LP0s and LP1s modes in the approximate LP mode representations, for hexagonal lattice to illustrate the technique. The novel technique is also implemented in vector form and incorporates a transfer matrix algorithm for the consideration of nodes with arbitrary refractive index profiles. Both are desired new capabilities for further explorations of advanced new designs of photonic bandgap fibers.

A semi-analytical calculation method of bending loss for large-mode-area fibers with arbitrary refractive-index profile

Predicting mode-dependent bending loss is important for large-mode-area (LMA) fibers to strip out unwanted modes. In order to seek a reasonable calculation method, a new bending loss formula that allows for both mode field distortion and weaker mode confinement was derived in detail. Based on the formula, a semi-analytical method was introduced to calculate the mode bending loss for LMA fibers. By combining the applications of the finite element method, discrete Fourier transforms and numerical integral, the method is valid for LMA fibers with arbitrary refractive index profile. The obtained loss characteristics are basically consistent with the previous results. This accordance demonstrates the feasibility and generality of the semi-analytical method.

Using the method of B splines to design an integrated-optics X splitter fabricated by diffusing titanium into a lithium niobate substrate

This paper presents a universal technique for calculating the parameters of channel optical waveguides. The method of finite elements using the Hermitian set of B splines is proposed as an algorithm. A calculation is carried out for a titanium-diffusion channel optical waveguide on a substrate made from single-crystal lithium niobate. Results are presented of the numerical modelling of radiation propagation by the proposed method in an X-type splitter fabricated using titanium-diffusion technology. It is shown that the B-spline technique makes it possible to design waveguides with an arbitrary refractive-index profile and various geometries for constructing a lightguide structure. The modelling results agree well with the data of an experimental sample of an X splitter.

Modeling Brillouin Gain Spectrum of Solid and Microstructured Optical Fibers Using a Finite Element Method

We report the first detailed implementation of a 2-D finite-element method (FEM) through a commercial FEM solver that can be universally applied to calculate the Brillouin gain characteristics of optical fibers with arbitrary refractive index profiles and material composition, including radially asymmetric and microstructured optical fibers (MOFs). Experimental results on various fabricated solid and MOFs are presented that demonstrate the widespread applicability and high accuracy of the proposed technique, which should prove to be an invaluable tool for designing novel optical fibers with tailored Brillouin response for a wide range of applications.

Analytical expression of giant Goos–Hänchen shift in terms of proper and improper modes in waveguide structures with arbitrary refractive index profile

We analytically relate the giant Goos-Hänchen shift, observed at the interface of a high refractive index prism and a waveguide structure with an arbitrary refractive index profile, to the spatial resonance phenomenon. The proximity effect of the high refractive index prism on modal properties of the waveguide is discussed, and the observed shift is expressed in terms of proper and improper electromagnetic modes supported by the waveguide with no prism. The transversely increasing improper modes are shown playing an increasingly important role as the high refractive index prism comes closer to the waveguide.

Bending Loss Evaluations of Holey Fibers Having a Core Consisting of an Elliptical-Hole Lattice by Various Approaches

In general, an accurate evaluation of the bending loss of an optical fiber with an arbitrary refractive index profile is a cumbersome task, especially for holey fibers (HFs). In this paper, various approaches are applied and compared for the bending loss evaluations of single-polarization elliptical-hole core circular-hole HFs (EC-CHFs), which are circular-hole HFs having a core with an elliptical-hole lattice structure. Our simulation results show that when bending loss formulas for weakly guiding circular fibers are applied to a HF by regarding it as an approximate circular fiber having an equivalent step-index (SI) profile, the definition of the core region is the most crucial issue. Moreover, we also show that the results obtained by the approximate approach based on the bending loss formulas for the equivalent SI circular fiber are in good agreement with direct approaches with numerical simulations for the real EC-CHF structure, if the equivalent core radius is suitably defined for the real cross section.

Transformation of modal power distribution in multimode fiber with abrupt inhomogeneities

A transformation operator is constructed for the calculation of the modal power distribution changes caused by abrupt inhomogeneity in multimode fibers. Simplification and practical applicability of the analysis are achieved by using the mode-continuum approximation. Thus, the model enables taking into account the fiber inhomogeneities of the following types: fiber lateral offset, core diameter mismatch, numerical aperture difference, and joints of fibers with different index profiles. The admission of an arbitrary refractive index profile, input modal power distribution, and parameters of fiber mismatch is an advantageous feature of the model. The operator is applied to calculations of power loss at joints between different multimode fibers, and the results of the calculations are shown to be in a good agreement with experimental data.

Refractive index profile optimisation for the design of optical fibres

Owing to advanced manufacturing techniques, it is possible to produce cylindrical single-mode fibres with nearly arbitrary refractive index profiles. For the design of optical fibres automated optimisation schemes have yet to be exploited. We have employed deterministic local, and stochastic optimisation schemes for the minimisation of a cost function based on dispersion, dispersion slope, macro-bending losses and mode-field diameter, on the space of continuous piecewise linear dopant concentration profiles. For the local schemes (modified and quasi Newton), it appears possible to select a few initial profiles, such that the optimisation results are close to the global optima (within 8%), found using schemes (simulated annealing and differential evolution), while reducing computation times significantly (minutes instead of days). For the local schemes, the cost function gradient is required. Frechet derivatives are more efficient than finite-difference approximations. A sensitivity analysis provides useful information for manufacturers regarding the required profile accuracy. A comparison of our optimised fibre designs with commercially available optical fibres demonstrates that existing fibres can be improved.

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.

Coordinate stretching for finite difference optical waveguide mode solvers

Numerical methods are necessary to calculate propagating modes in optical waveguides. The finite difference method is widely used because it is applicable to waveguides with arbitrary refractive index profiles and it is easy to implement. To improve the efficiency and to reduce the size of the resulting large sparse matrix, the finite difference method is often used with a variable grid size strategy. This is related to the technique of coordinate stretching. In this paper, we develop a technique for optimizing the coordinate stretching function based on discrete reflection coefficients. We demonstrate our method using a scalar model which is valid for weakly-guided optical waveguides.

Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile

Circular slab waveguides are conformally transformed into straight inhomogeneous waveguides, whereupon electromagnetic fields in the core are expanded in terms of Legendre polynomial basis functions. Thereafter, different analytical expression of electromagnetic fields in the cladding region, viz. Wentzel-Kramers-Brillouin solution, modified Airy function expansion, and the exact field solution for circular waveguides, i.e., Hankel function of complex order, are each matched to the polynomial expansion of the transverse electric field within the guide. This field matching process renders different boundary conditions to be satisfied by the set of orthogonal Legendre polynomial basis functions. In this fashion, the governing wave equation is converted into an algebraic and easy to solve eigenvalue problem, which is associated with a matrix whose elements are analytically given. Various numerical examples are presented and the accuracy of each of the abovementioned different boundary conditions is assessed. Furthermore, the computational efficiency and the convergence rate of the proposed method with increasing number of basis functions are briefly discussed.

Synthesis of symmetric and asymmetric planar optical waveguides

A novel and accurate refractive index profile synthesis method for planar optical waveguides is presented and demonstrated using the transmitted near-electric-field-data. This method is based on the inverse transmission-line (TL) technique. From Maxwell's equations, a TL equivalent circuit (electric T-circuit) for the refractive index profile of a planar optical waveguide is derived. The authors demonstrate how to use this model to carry out the inverse problem and synthesise the exact refractive index profile numerically from near-field-data. The TL method can reconstruct arbitrary refractive index profiles for planar optical waveguides that support singlemode or multimodes. The cases of both symmetric and asymmetric arbitrary refractive index profile planar waveguides are discussed. The accuracy of the reconstructed waveguides is examined numerically.

Methods for analysis of optical fibers with arbitrary refractive index profile

Two fast and accurate algorithms based on the orthogonal polynomial approximation and the Chebyshev polynomial interpolation for obtaining the actual field solutions of a fiber with arbitrary refractive index profile are presented. These solutions in turn can be used to evaluate the dispersion parameters of the fiber. The numerical methods are compared in terms of their performances, viz., complexity and accuracy. The method based on the orthogonal polynomial approximation (OPA) is simple in principle and easy to implement on a computer, and it has several advantages compared to the method based on the Chebyshev polynomial interpolation (CPI).

Accurate Magnified Near-Field Measurement of Optical Waveguides Using a Calibrated CCD Camera

A fast and accurate magnified near-field (NF) measurement technique that employs a charge-coupled device (CCD) camera is described to measure the two-dimensional (2-D) intensity distributions of optical waveguides with arbitrary refractive-index profiles. Calibration of the InGaAs CCD camera used in the system is essential in terms of correction for bad pixels, linearity, and uniformity in order to obtain accurate results. The magnification factor of the system is obtained using a calibrated optical dimensional standard. The NF measurement system with the calibrated CCD camera is validated from the mode field diameter (MFD) measured for six single-mode fibers with different refractive-index profiles. The MFD results from the magnified NF technique exhibit good agreement with those obtained from the far-field scanning technique to within 2.3%. An error of about 15% in the MFD would result if corrections to the CCD camera were not applied to the raw measured data. The refractive-index profiles computed from the measured data for a single-mode fiber and a Ti:LiNbO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> waveguide are also presented. The radial-index profile of the fiber exhibits good agreement with the profile measured from the refracted NF technique

Oxidation of evaporated porous silicon rugate filters

Rugate filters are thin-film optical interference coatings with sinusoidal variation of the refractive index. Several of these filters were fabricated with glancing angle deposition, which exploits atomic competition during growth to create nanoporous materials with controllable effective refractive index. This method enables the fabrication of devices with almost arbitrary refractive index profiles varying between the ambient, 1.0, and the index of the film material, in this case silicon with an index of 4.0 (at 600 nm). As these filters are inherently porous, oxidation of the silicon can occur throughout the device layer, and here we study the intentional oxidation of silicon filters by high-temperature reaction with gaseous oxygen. We find that a significant portion of the silicon filter oxidizes in approximately 10 min when heated to 600 degrees C-650 degrees C in an oxygen environment; oxidation then continues slowly over several hours. The presence of water vapor has little apparent effect on the oxidation reaction, and attempts to oxidize with ozone at room temperature were unsuccessful. As silicon filters oxidize to become silica, spectral blueshifts and increased short-wavelength transmittance are observed. Measured and calculated transmittance spectra generally agree, although the lack of absorption and dispersion in the theoretical model limits detailed comparison.

Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals

A new approach is developed to calculate diffraction efficiency for a dielectric grating with an arbitrary refractive index profile. By treating a one-dimensional grating as a segment of a virtual two-dimensional (2D) photonic crystal, we exploit a rigorous theory of photonic crystal refraction and calculate the diffraction efficiencies. We expand, analytically in many cases, the dielectric function of the grating into 2D Fourier series. We find the eigenmodes for the virtual photonic crystal, and then use these eigenmodes to match the boundary conditions by solving a set of linear equations. In two such simple steps, the diffraction efficiencies can be computed rigorously without slicing the grating into thin layers.

Mode dispersion and delay characteristics of optical waveguides using equivalent TL circuits

A new analysis leading to an exact and efficient algorithm is presented for calculating directly and without numerical differentiation the mode dispersion characteristics of cylindrical dielectric waveguides of arbitrary refractive-index profile. The new algorithm is based on the equivalent transmission-line (T-L) technique. From Maxwell's equations, we derive an equivalent T-L circuit for a cylindrical dielectric waveguide. Based on the TL-circuit model we derive exact analytic formulas for a recursive algorithm which allows direct calculation of mode delay and dispersion. We demonstrate this technique by calculating the fundamental mode dispersion for step, triangular, and linear chirp optical fiber refractive index profiles. The accuracy of the numerical results is also examined. The proposed algorithm computes dispersion directly from the propagation constant without the need for curve fitting and subsequent successive numerical differentiation. It is exact, rapidly convergent, and it results in savings for both storage memory and computing time.

Calculation of losses in micro-ring resonators with arbitrary refractive index or shape profile and its applications

A perturbative approach is discussed for the calculation of the radiation and scattering losses in cylindrical microresonators with arbitrary refractive index or shape profile. A new method for inducing losses in the cavity using the electrooptic effect with periodic modulation of the refractive index is introduced. The perturbative method is used for the calculation of the losses due to periodic modulation of the refractive index. The applications of this approach for the calculation of scattering losses due to surface roughness of microresonators and scattering due to deviation from ideal cylindrical shape are discussed.

Modeling of Photonic Crystal Fiber Raman Amplifiers

An accurate model has been developed to study photonic crystal fiber Raman amplifiers. The proposed approach calculates the Raman gain efficiency starting from the knowledge of the fiber cross-sectional characteristics, i.e., the geometric parameters and the germania concentration in the doped area. The model allows the study of the Raman amplification process in a photonic crystal fiber with any air-hole dimension and disposition, with or without a central doped area, as well as in standard single-mode fibers with an arbitrary refractive-index profile. This flexibility allows identification of the optimum values of the physical and geometrical photonic crystal fiber parameters, and thus it guarantees an efficient design for Raman amplifiers. For example, a Raman gain of 13.7 dB at 1554.8 nm has been obtained with a 6-km-long low-loss germania-doped triangular photonic crystal fiber with d//spl Lambda/=0.44 and /spl Lambda/=4.2 /spl mu/m.