EME Solver

Solver Physics

The Eigenmode Expansion Method(EME) solver is a frequency-domain method particularly suited for simulating long-distance light propagation. It is more efficient than Finite-Difference Time-Domain(FDTD) method for designing long and periodic devices. The EME method primarily involves two key steps: mode expansion of the electromagnetic fields and transmission analysis. This approach offers advantages in both accuracy and computational efficiency compared to other methods, such as Beam Propagation Method (BPM) and FDTD, especially when dealing with long structures, such as MMI, tapered fibers, directional couplers and so on.

The EME method dividing the geometric structure into multiple cells and performing mode expansion within each cell using a set of eigenmode bases (as detailed in the FDE solver). It describes the electromagnetic field's transmission across the entire structure by calculating the transmission matrix for each cell and the mode coupling matrix between adjacent cells. The transmission matrix reflects the phase and amplitude changes of the electromagnetic field within a cell as a function of distance, while the mode coupling matrix represents the transition relationship of the electromagnetic field as it crosses the boundaries between cells. Ultimately, by cascading these matrices, the electromagnetic field distribution for long-distance transmission can be computed.

When computing simulation with high refractive index materials, the EME method, unlike the BPM based on the Slowly Varying Envelope Approximation (SVEA), better captures the complex light field distribution through mode decomposition. This makes the EME method more accurate for handling materials with high refractive index. Additionally, for simulating long structures, EME offers advantages over FDTD. The computational time and resource consumption for FDTD increase significantly when dealing with long structures. In contrast, the EME method efficiently handles long structures in the frequency domain through mode decomposition and transmission analysis. It allows for changes in transmission distance without repeating the initial decomposition steps, thus significantly improving computational efficiency.

In a structure where the optical propagation constant is invariant along the 𝑧 direction, the mode solution to Maxwell's equations can be expressed as:

$$\underline{E} (x,y,z)=\underline{e}_m(x,y)\cdot e^{-i\beta_m z}$$

Here, $\underline{e}_m(x,y)$ and $\beta_m$ represent the eigenvectors and eigenvalues of the solution, respectively. Consequently, a mode has a very simple harmonic z-dependence, which is crucial for the EME method to quickly and efficiently solve long, slowly varying structures.

In a waveguide, there are certain modes that propagate within the waveguide with minimal energy loss, referred to as guided modes or propagating modes. Additionally, there are modes that radiate energy outside the waveguide, leading to energy dissipation, known as radiation modes. Guided modes typically form a complete and orthogonal basis set. Consequently, any solution to Maxwell's equations within the waveguide can be expressed as a linear combination of these guided modes.

$$\mathbf{E} (x,y,z)= {\textstyle \sum_{k=1}^{M}} (a_ke^{-i\beta _kz }+b_ke^{+i\beta _kz })\mathbf{E_k} (x,y)$$

$$\mathbf{H} (x,y,z)= {\textstyle \sum_{k=1}^{M}} (a_ke^{-i\beta _kz }-b_ke^{+i\beta _kz })\mathbf{H_k} (x,y)$$

In the equation, $\beta_k$ denotes the propagation constant, $a_k$represents the amplitude of the forward-propagating wave, and $b_k$represents the amplitude of the backward-propagating wave.

Note the bidirectionality of this equation. It represents an exact solution to Maxwell's equations in a linear medium.

By applying the continuity conditions for the electromagnetic field, we obtain:

$${\textstyle \sum_{M}^{k=1}} (a_k^{(+)}e^{-i\beta _kL}-a_k^{(-)}e^{+i\beta _kL})E^{(a)}_{k,t}(x,y)= {\textstyle \sum_{M}^{k=1}} (b_k^{(+)}e^{-i\beta _kL}-b_k^{(-)}e^{+i\beta _kL})E^{(b)}_{k,t}(x,y)$$

Based on the aforementioned relationships and the orthogonality and normalization of the modes:

$$\int (E_{x,j}\cdot H_{y,k} -E_{y,j}\cdot H_{x,k})\cdot dS=\delta _{j,k}$$

Derive the coefficient relationships at the interface:

$$\binom{\underline{a}^{(-)} }{\underline{b}^{(+)}} = S_J\binom{\underline{a}^{(+)} }{\underline{b}^{(-)}}$$

$S_J$ is the scattering matrix at the interface.

$$\begin{pmatrix} e^{i\beta _1z}& 0 & 0 & \cdots \\ 0& e^{i\beta _2z} & 0 & \cdots \\ 0 & 0 & e^{i\beta _3z} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}$$

Feature Description

Adds or sets EME simulation area and boundary conditions.

Settings Description

1 General

1) Algorithm: The type of mesh algorithm used (Currently,it s only supports Finite Difference methods).

2) Solver Type: Default setting is 3D: X Prop; other propagation directions are not supported at the moment.

3) Background Material: The combo box allow user to set the background material from drop down menu. Project, Object Defined Dielectric, and Go to Material Library can be operated.

• Project: Materials that have been imported into the project can be selected and used as background media in simulation. We can also select "Object Defined Dielectric" to customize the background material index in the refractive index.

  • Object Defined Dielectric: The object defined dielectric material, a default setting if user forgets to set background material, is defined for the current object background material setting. Once users chooses this option, they does not need to set any material from the standard, user, or project material database. And the object-defined dielectric will not be loaded into any material database.
    • Refractive Index: specify background index manually in stead of choosing form library or project. (Default :1).

• Go to Material Library: If selected, user can go to standard material database to set background material according to needs. The selected material will be imported into the project automatically.

4) Frequency/Wavelength: Sets the simulation frequency (Default: 193.414 THz) or wavelength (Default:1.55 μm) to calculate.

5) Use Wavelength Sweep: If enabled, the Wavelength Sweep feature is activated and can be performed in the EME Analysis window; if disabled, Wavelength Sweep cannot be selected in EME Analysis.

2 EME Setup

1) Y, Z: The center position of the simulation region.

2) Y/Z Min, Y/Z Max: Y/Z min, Y/Z max position.

4) Y Span, Z Span: Y, Z span of the simulation region.

5) Cell geometry：

• X Min: Minimum x position of solver region. The first cell group starts from this position.

• Number of Modes for All Cell Group: When the switch “Allow Custom Eigensolver Settings” is off, determine the number of modes for all cell group.

• Allow Custom Eigensolver Settings: if the button switch on, which allows users to set a different number of modes to solve for different cell groups in the Cell Group Definition table.

• Cell Group Definition: Specify the parameters of each cell group.

  • Span: Specify the span of each cell group.

  • Cell Number: The number of cells in each cell group.

  • Number of Modes: The number of modes in each cell group.

  • Sub Cell Method: Decide which sub-cell method to employ. (None or Sub Cell)

  • Custom: When “Custom” is Default, the value in “Number of Modes” is equal to the “Number of Modes for All Cell Group”.

  • Delete: Right Click and choose the Delete button, will delete the selected cell group.

  • Add: Right Click and choose the Add button, will add one cell group.

• Clear Settings for Cell Group: After selecting the specified Cell Group, this option will clear the custom settings for the chosen Cell Group.

• Custom Settings for Cell Group: After selecting the specified Cell Group, clicking the "Custom Settings for Cell Group" button will open the Select Mode window, allowing the user to customize parameters such as the number of modes to solve, solving methods, mode indices, etc. If this feature is used, the "Default" in the custom column will change to "Custom".

• Notes:The availability of clear settings for cell groups and custom settings for cell groups is contingent upon enabling the “Allow Custom Eigensolver Settings” button.

• Display Groups: Displays cell boundaries in the CAD.

• Display Cells: Displays cells in the CAD.

3 Mesh Settings

Since the propagation in the EME solver is along the x-direction, the mesh settings are applicable merely along the y and/or z axes, depending on the solver type (3DX2D).

1) Mesh Refinement: Select an approach to calculate refined mesh properties.

• Curve Mesh: The effective permittivity can be derived using the contour path formula, which effectively takes into account the shape of the dielectric interface as well as the weighting of the material filling ratio within the mesh.
• Staircase: It is possible to compute any point within the Yee grid to determine which material it is filled with, and the properties of that single material are used to describe the E-field at that point.As a result, the discrete structure can barely account for the structural variations within a single Yee grid, leading to a Staircase dielectric constant mesh that is fully consistent with the Cartesian grid. Moreover, all layers are effectively moved to the closest E-field position within the Yee grid, meaning that thicknesses cannot be resolved to finer than dx, and thus, cannot be resolved better than dx.

2) Grading: The Grading factor specifies the biggest ratio of the neighboring spatial grids. (Default: 1.2)

3) Maximum Mesh Step Settings dy/dz: Maximum mesh step settings in Y/Z direction. (The default value is 0.02 μm).

4) Minimum Mesh Step Settings: Minimum Mesh Step indicates the minimum mesh step in the whole simulation region.(Default: 1e-4 μm)

4 Boundary Conditions

1) Apply in All Directions: Turning on this switch ensures that the boundary conditions are consistent in different directions.

2) PML (Perfectly Matched Layer): Electromagnetic waves incident on the PML boundary will be completely absorbed, effectively simulating an ideal open (or non-reflective) boundary. Unlike traditional boundary conditions, PML boundaries occupy a finite volume around the simulation region and have a limited thickness where light absorption occurs. Important parameters for PML include:

• Layers: Number of layers in the PML region for discretization purposes.

• Kappa, Sigma, Alpha: Kappa and Sigma control the absorption performance of the PML boundary. According to reference, Kappa is dimensionless, while Sigma needs to be normalized to be input as a dimensionless value in the PML settings table. Both Kappa and Sigma are evaluated through polynomial variations to assess their geometric positioning within the PML region.

• PML Polynomial: Specifies the polynomial order used for grading Kappa and Sigma.

2) PEC(Perfect Electric Conductor): The PEC boundary condition is introduced to simulate boundaries that behave as perfect electric conductor. Metal boundaries reflect all electromagnetic waves, thus preventing any energy from passing through the simulation volume enclosed by the metal.

3) Symmetry/Anti-symmetry: Symmetric/Anti-Symmetric boundary condition is used to reduce the simulation time with electromagnetic fields that are symmetric with respect to a plane. The choice between Symmetric and Anti-Symmetric conditions is based on the relationship between the source polarization and the symmetry plane. Select the Symmetric option if the normal to the symmetry plane is tangent to the source polarization. Otherwise, choose the Anti-Symmetric option.

5) Periodic: Periodic BCs allow you to analyze the whole system by studying only one unit cell if the interested system is somewhat spatially periodic, and they are easily enabled by setting the simulation span identical to the length of one unit cell, plus choosing then “Periodic BCs” for that boundary. Upon doing so, the EM fields at one side of the unit cell (which is subjected to Periodic BCs) are always duplicated accordingly at the other side during the entire simulation.

Notes: The most important detail to remember is that when using Periodic BC's, everything in the system must be periodic: both the physical structure and the EM fields. A common source of error is to use periodic boundary conditions in systems where the structure is periodic, but the EM fields are not.

5 Advanced

1) EME settings:

Max Stored Modes: Maximum number of modes for each cell in the EME setup.(Default: 1000, the input is limited to [1,1000]).

2) Dispersion Settings:

Fractional Offset for Group Delay: Numerically, the group delay of the device is computed by means of a finite-difference approximation of diffentiating the phase with respect to frequency. The “fractional offset for group delay” refers to the fractional amount of the frequency used in the step size of finite difference. If this setting is too small, the phase change may be severely affected by noise, whereas a too large setting could result in an unrealistic group delay since the phase may change by more than 2π. For rather long devices (10000+ wavelengths) in which the phase varies quickly with frequency, the user is encouraged to reduce this setting from the default value. Otherwise the default setting is generally recommended. (Default:0.0003 μm)

$$n_{g}=n_{eff} -\lambda \frac{\Delta n_{eff}}{\lambda_{offset} }$$