top of page

## SOFTWARE → EXACT SOLUTION

EXACT SOLUTION (GALERKIN PROCEDURE)

Exact - Overview

The Exact solution is based on the Bubnov-Galerkin procedure which apply a global expansion solution on Maxwell equations. The solution solves the 3D full-vector Maxwell equation with H  being the magnetic field (μ=μ0), ε=ε0n²  the electric permittivity, and ω the angular frequency. The equation is solved for Hx and Hy to every mode, where after, the components Hz, Ex, Eand Ez are found directly from Maxwell equations. The middle term in the equation accounts for polarization effects and the corners effect of the waveguide shape. The solution has the form where Nx,Ny are the number of terms per axis and Cuv are the amplitude constants. Substituting the solution into the vector Maxwell equation and integrating over the entire space can give This equation leads to an eigenvalue matrix problem which represents the solution of guided modes supported by the waveguide.

The basis, ψuv(x,y),  had been cleverly chosen and optimized such that the integrals are solved analytically. Consequently, the solution requires only basic matrix handling to provide the effective index of the modes and their amplitude constants.

In a similar manner, the 3D full-vector Maxwell equation for the electric field is with E  being the electric field. The equation is solved for Ex and Ey to every mode, where after, the components Ez, Hx, Hand Hz are found directly from Maxwell equations. The middle term in the equation accounts for polarization effects and the corners effect of the waveguide shape. The solution has the form where Nx,Ny are the number of terms per axis, Cuv are the amplitude constants and ψuv(x,y) is the spatial basis. Substituting the solution into the vector Maxwell equation and integrating over the entire space then gives This equation leads to an eigenvalue matrix problem which represents the solution of guided modes supported by the waveguide.

The eigenvalues obtained in either electric or magnetic field equations are identical. However, the distribution in space of the various field components will be more accurate depending on the field equations: for Ex, Ey and Ez field components, the electric field equations, and for Hx, Hy and Hz field components, the magnetic field equations.

Exact - Basis

Basis Strategy

The software currently supports three basis, ψuv(x,y),  that can either be chosen for solving the integral equation of the method:

• Sine basis: Conventional Sine basis in a bounded domain with zero value at the boundaries. To assess the size of the domain, Effective Index Method is used to estimate the effective width of the mode's field according to various optimization strategies.

• Slab Modes (Fourier) basis: The 1D solutions for the guided, substrate and radiated modes of a slab waveguide are tied together and form an orthonormal basis. For a parallel waveguides structure, the 1D modes solutions of a parallel slab waveguides are used. To make discrete set of slab modes solutions, the 1D domain is bounded in size depending on structure parameters.

• Hermite Gauss basis: Conventional Hermite-Gauss basis with scaling and offset factors depending on structure parameters or determined by optimization to a specific mode. For the latter, Effective Index Method is used to estimate the effective width of the mode's field which is then considered with Hermite's spectrum for assess the factors (see T. Tang, J. Sci. Comput., 1993).

The Fourier basis is inherently optimized per structure, since the 1D modes are found by setting parameters of the structure itself.

Because the basis functions in each axis represent pretty close the actual modes of the structure, the solutions (neff) converge at least in the 4 digit even for waveguide with high refractive index contrast, e.g Silicon Photonics platform. Such is the case for both

single-guide and parallel-guides structures.

Despite the impressive convergence, the field discontinuity is rarely observed when plotting the field distribution, mainly because the radiation modes are not similar to guided modes in shape. The result is a continues field where the discontinuity is averaged.

In the Sine basis, the discontinuity is clearly revealed for 30 terms and more, mainly since the sine functions resemble the shape of the guided modes. In general, due to the decay behavior of sine function, which is not exactly exponent in nature but similar, the convergence be achieved is between 3 to 4 digit for high RI.

For the Hermite Gauss basis, however, due to the decay behavior of exp(-x²)  and the "static" shape of its functions, the convergence be achieved is only in the 2 digit for high RI and may be worsen for parallel-guides structure. In some cases, the Hermite basis will provide a better description of the field distribution than the Slab Modes basis will, since its basis functions resemble those of guided modes only.

Below, screenshots of the settings dialog for each method.   Exact - Prons and Cons

Nowadays, commercial mode solvers can typically be divided into two groups; one that deploy a mesh over the waveguide's domain, e.g Finite Difference (FDM), Finite Elements (FEM) and Beam Propagation (BPM), and another, that decompose the waveguide's domain into several sub-domains (Domain-Decomposition), e.g Mode Matching (FMM), Spectral Method (MSM). Employing the Galerkin method with our algorithm holds several significant advantages compared to other commercial software from either groups.

Some advantages of our software are:

• Run Time: Solving one eigenvalue problem of a single domain is much faster and sometimes more accurate than dealing with a tight mesh or solving extra-large eigenvalue matrix. The time scale of our algorithm is that of a few seconds.

• Memory: Memory required for operation is significantly lower since our method, inherently, produces much smaller matrix.

• Profiles: For curved shapes, such as circle, ellipse or trapeze, where segmentation may be needed, the run time and memory requirement keep roughly the same (matrix size unchanged), while the other methods drastically increase their run time and resources since their matrices are expanding.

• Reliability: Spurious solutions are not generated.

• Efficiency: General operations, like generating field distribution in 2D or 3D space, is done faster since the total amount of coefficients required per mode is much smaller.

These advantages yield a very fast and reliable simulations without missing supported modes and can be carried for every waveguide profile with any refractive index contrast, even at cutoff conditions.

Some disadvantages of the software are:

• The software simulates some typical waveguide profiles, however, for additional profiles a software update is required.

• The Galerkin technique may not be applicable for structures that vary in the Z direction.

Case Study (Method Validation)

Exact - Case Study

To validate the performances of the Galerkin method with our Fourier basis algorithm, we compare Doctor Modes™ solutions with other solutions obtained by methods from either groups: mesh-based group and domain-decomposition group. The comparison is done over widely used waveguide structures which are commonly accepted for numerical methods validation and comparison.

In addition, profiles of modes produced by Doctor Modes™ were used in Lord PICS™ to simulate and design various optical circuits comprising directional couplers. Over 15 different circuits that include about 40 couplers were fabricated and characterized in Silicon Photonics platform and showed a very good match with simulations [Link]. Normalized frequency, b, of the fundamental mode E1,1 (TE)  Effective index, neff, and number of unknown coefficients  Normalized frequency, b, of various fiber optic modes bottom of page