Mode Solvers

Modes are the fields that maintain the same transverse distribution and polarization at all distances along the waveguide axis. For a limited number of ideal, simple waveguide configurations, modes and corresponding propagation constants can be found analytically. However, for a majority of realistic waveguides, numerical simulations are essential. Even the waveguides with relatively simple cross-sections and refractive index profiles in reality may suffer from anisotropy, inhomogeneities due to fabrication tolerances, and material losses that affect their modal properties. Depending on the refractive index profile and other waveguide characteristics, various types of modes may be supported, including antiguiding, leaky, lossy, or radiating modes.

This section contains several mode solvers based on numerical techniques of different complexity and specialization. Some of the tools can simultaneously be used as mode solvers and as wave propagators (Transfer-Matrix Method, Eigenmode Expansion Method, Beam Propagation Method, Finite Element Method), while others are Mode Solvers only (Plane Wave Expansion Method, Multipole Method, Source Model Technique). While some of the numerical methods, such as the Finite Element Method, the Plane Wave Expansion Method, the Beam Propagation Method-based mode solvers, the Film Mode Matching Method, and the Finite-Difference Method, can be used for finding modes of the arbitrary cross-section and refractive index profile waveguides, others are more specialized. For instance, the Multipole Method (CUDOS MOF Utilities) and the Source Model Technique (developed by Professor Leviatan’s group at Technion) are techniques that are particularly useful mode-finding modes of microstructured or photonic crystal fibers.

The Other Method subsection includes several software packages that comprise more than one mode-solving technique (FIMMWAVE Mode Solvers by Photon Design, MODE Solutions by Lumerical, OlympIOs Mode Solver Modules by C2V, OptiFIBER by Optiwave). These packages include various techniques, ranging from approximate analytical methods, including the Effective Index Method and the Marcatili’s Method (OlympIOs Mode Solver Modules by C2V), and the Gaussian Mode Fiber Solver (FIMMWAVE Mode Solvers), to advanced numerical methods, such as the semi-vectorial and full-vectorial Finite-Difference Method (MODE Solutions, OlympIOs Mode Solver Modules, OptiFIBER by Optiwave), the Film Mode Matching Method (FIMMWAVE Mode Solvers by Photon Design, OlympIOs Mode Solver Modules), the Transfer-Matrix Method (OptiFIBER by Optiwave), and the Finite Element Solver (FIMMWAVE Mode Solvers by Photon Design). Depending on the problem, required accuracy and simulation speed, the user can choose an appropriate method. In some cases it may be useful to run fast approximate techniques first in order to build intuition and narrow the parameter range, and then refine calculations using more advanced and accurate numerical approaches. In addition, there are two other methods in this section that have not been placed out into separate categories, since only a single numerical tool has been found so far for each method. These are the Beam Propagation Method-based mode solvers, utilizing the iterative method and the correlation method (BeamPROP Mode Solvers by RSoft), and the Wave Matching Technique (WMM developed by Dr. Hammer at the University of Osnabrück).

The finite element method (FEM) is a method used for finding the approximate solution of partial differential equations (PDE) that handle complex geometries (and boundaries), such as waveguides with arbitrary cross-sections, with relative ease. The field region is divided into elements of various shapes, such as triangles and rectangles, allowing the use of an irregular grid. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques, such as finite differences. In a context of optical waveguides, the FEM can be used for mode solving and propagation problems. Two approaches to solve waveguide problem include the variational method and the weighted residual (Galerkin) method. Both methods lead to the same eigenvalue equation that needs to be solved.

The multipole expansion method is a numerical method that can be used for full-vector modal calculations of microstructured fibers and other photonic crystal structures. It yields both the real and the imaginary parts of the mode propagation constant (thus providing information about losses) and achieves high accuracy and rapid convergence with modest computational resources. In freely available software the multipole method is implemented for circular inclusions; however, it can be extended to noncircular inclusions. It can deal with two types of MOF: a solid core MOF, surrounded by air holes, and air core MOF. Systems with large numbers of inclusions can be modeled, and for structures with discrete rotational symmetries the computational overhead is further reduced by exploitation of the symmetry properties of the modes. The method is limited to nonintersecting circular inclusions, and convergence problems arise as the spacing between the inclusions decreases. In the multipole method each dielectric boundary in the system is treated as a source of radiating fields. The essential feature of a multipole method is the application of a certain field identity that relates the regular field in the vicinity of any scatterer to fields radiated by other scatterers and external sources. The multipole method can analytically preserve symmetry and exploit the natural basis of functions for the scatterers (e.g., cylindrical harmonic functions for cylindrical scatterers). As a result, it can yield important physical insight into the scattering processes not otherwise possible with purely numerical methods. An important feature of the method is that frequency can be used as an input parameter, whereas the propagation constant follows from the calculation. This is a significant advantage when one is dealing with dispersive media: Because is fixed, the appropriate refractive indices are known from the outset. Applications of the multipole methods include the modeling of photonic crystals, microstructured optical fibers, the study of the radiation properties of sources embedded in such structures, and the design of complex/composite devices, e.g., splitters, couplers, interferometers, among others.

The source model technique (SMT) is a fully vectorial method that can be used to determine the modes of a cylindrical structure of a piecewise-homogeneous cross section. In the SMT, the fields in each homogeneous region of the cross section are approximated by the fields due to a linear combination of elementary sources placed outside of it, on curves conformal with the region boundary. The sources radiate in a homogeneous medium with the same material parameters as those of the region they enclose in the PCF cross section. Their fields, therefore, have well-known analytic expressions. The amplitudes of the elementary sources are determined so as to satisfy the continuity conditions across the media interfaces at a set of testing points. The elementary sources used are electric and magnetic current filaments, carrying longitudinally varying currents, that vary with the z-coordinate as . For each pair of under consideration, the vector that yields the smallest least squares error can be found. Once a low error solution is found for some pair, the current amplitudes are inserted into the linear combinations that approximate the fields, yielding the mode field patterns and any other parameter of interest. The SMT has much in common with the multipole method, in which the basis functions are employed. Although the multipole method representation is considered very compact, it is directly applicable only to cross sections composed of circular homogeneous inclusions in a homogeneous medium, while in the SMT, the homogeneous regions in the cross section can be arbitrarily shaped. For more detail see:

Plane-Wave Expansion Method

The plane wave expansion (PWE) method is a frequency-domain approach based on the expansion of the fields as definite-frequency states in some truncation of a complete basis (e.g. plane waves with a finite cutoff) and then solution of the resulting linear eigenproblem. The method is applicable to optical waveguides with arbitrary cross-sections and resonators, photonic band-gap materials, and photonic crystal structures, or for calculations of optical dispersion relations and eigenstates for conventional optical fibers. One disadvantage of the PWE method is that the wavevector serves as the free parameter, whereas the frequency eigenvalues follow from the calculation. For certain applications, especially for finding waveguide modes, it is often more convenient to specify the frequency and solve for the required propagation constant, as is done in the multipole method. In particular, such an approach would make it much easier to incorporate the effects of material dispersion, which is invariably specified as a function of frequency. Also, for some problems, such as defect modes, the plane wave expansion method may be extremely time-consuming.

Transfer-Matrix Method

The transfer-matrix method (TMM) is a simple technique that can be used for modeling for obtaining propagation characteristics, including losses for various modes of an arbitrarily graded planar waveguide structure, which may have media of complex refractive indices. The method is applicable for obtaining leakage losses and absorption losses, as well as for calculating beat length in directional couplers. The waveguide is divided into a number of layers. At a fixed frequency, one computes the transfer matrix, relating field amplitudes at one end of a unit cell, with those at the other end (via finite-difference, analytical, or other methods). This yields the transmission spectrum directly, and mode wavevectors via the eigenvalues of the matrix. Transfer-matrix methods may be especially attractive when the structure is decomposable into a few more-easily solvable components, and also for other cases, such as frequency-dependent dielectrics. It was also implemented for analyzing modal properties of optical fibers with a layered cladding structure. A modified TMM method capable of direct leaky mode analysis by expressing field in the outermost layer in terms of Hankel functions was proposed. This method is extremely efficient in treating multi-clad optical fibers, which would otherwise be very complex, if not computationally forbidding, to all numerical methods. For more detail see:

Eigenmode Expansion Method

The eigenmode expansion method (EME) is based on a rigorous solution of Maxwell’s equations, representing the electromagnetic fields everywhere in terms of a basic set of local modes. In principle an exact solution can be obtained using an infinite number of modes in our expansion. Of course, in practice the number of modes is limited and there will be numerical errors in the implementation, as in any numerical technique. To obtain higher accuracy one can simply add more modes, enabling the method to accurately compute problems that cannot be computed with other techniques. The algorithm is inherently bi-directional and utilizes the scattering matrix (S-matrix) technique to join different sections of the waveguide or to model nonuniform structures. The scattering matrix technique relates the incoming waves, i.e. the forward wave at the beginning of the section and the backward wave at the end of the section, to the outgoing waves, i.e. the backward wave at the beginning and the forward wave at the end. All reflections are taken into account in the method. If one part of a device is altered, only the S-matrix of that part needs to be re-computed. The method can simulate light propagating at any angle, even 90 degrees to the propagation axis, simply by adding more modes. Applications include diffractive elements, directional couplers, tapers, MMIs, bend modeling, periodic structures and others. Structures with a very large cross-section are less suitable for the method since computational time typically scales in a cubic fashion, with, for example, cross-section width.

The Other Mode Solvers section includes the software packages containing multiple-mode solving techniques including analytical methods, such as the effective index method and Marcatili’s method, and numerical approaches, such as the finite-difference method and meshless techniques, so that depending on a problem, one of the available techniques can be used. It also includes some two-mode solvers based on the beam propagation method (imaginary distance and correlation method) and the wave-matching method (WMM).

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