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Knowledge Base

Lumerical provides a number of mesh refinement options which can give sub-cell accuracy from a simulation. This section explains the various options and how to choose the appropriate one for your simulation.

In this topic

How to choose which mesh refinement method to use

Summary of each method

Conformal mesh technology details

See also

Mesh refinement

How to choose which mesh refinement method to use

The default, Conformal Variant 0 setting can be used for most FDTD and Propagator simulations. In this setting, Conformal Mesh Technology (CMT) is applied to all materials except metals and Perfect Electrical Conductors (PEC). For the purpose of this discussion, we define metals (or plasma materials) as materials with real(ε) < 1 over the bandwidth of the simulation and ε is the relative permittivity. The material properties in your simulation can always be viewed in the Materials Explorer.

 

Exceptions can be determined based on your material properties, but if options other than Conformal Variant 0 is to be used, careful convergence testing is highly recommended:

If your simulation involves metals, then you may want to consider using Conformal variant 1. In this variant, CMT is applied to all materials, including metals. CMT will give better convergence than staircasing for small enough mesh sizes, but at larger mesh sizes it can sometimes give much worse results. Unfortunately, the size of a 'small enough' mesh is highly dependent on the simulation. In some cases, a <5nm mesh is sufficient, while in other cases a 1nm is not sufficient (optical wavelengths). Please do some careful convergence testing between Conformal variant 0 and Conformal variant 1 to test which method you should use for your particular application. Conformal variant 1 may give numerical artifacts when |εplasma|>> |εdielectric| and the mesh size is not sufficiently small.

If your simulation uses PEC instead of metals, then you will get better convergence with Conformal variant 1.

 

For the Eigenmode Solver in MODE, the default setting is Conformal variant 1. Since unphysical modes resulting from the possible numerical artifacts can be easily detected in modal analysis.

Note on meshing time

Any conformal meshing technique will increase the time it takes to mesh the structure prior to the simulation itself. For extremely complex structures involving many objects and large simulations the meshing time can become significant compared to the simulation time. If this is the case, you may want to return to Staircase meshing for your initial simulations and only use Conformal meshing for final results. Advanced users may also want to learn how to re-use the simulation mesh between simulations.

Special note for dipole calculations

CMT can give much better convergence for many simulations involving calculations such as Mie scattering from spheres, or reflection and transmission from multilayer stacks. However, it can also modify the local density of states. If you are doing calculations involving the power radiated by dipoles placed very small distances (10s of nm) from metal interfaces, you should compare Conformal variant 1, or Conformal variant 2 with the standard Conformal setting, which disables CMT for metal interfaces, to be sure that your results have converged.

Summary of each method

Staircasing: The material at each position of the Yee cell is evaluated to determine which material it is in, and the E field at that location uses only that single material property. The resulting discretized structure is unable to account for structure variations that occur within any single Yee cell, resulting in a "staircase" permittivity mesh that coincides with the Cartesian mesh Furthermore, any layers are effectively moved to the nearest E field position on the Yee cell, meaning that layer thickness cannot be resolved to better than dx.

 

Conformal variants: Lumerical's Conformal Mesh Technology (CMT) uses a rigorous physical description of Maxwell's integral equations near interfaces between two materials that is able to incorporate Lumerical's Multi-Coefficient Materials. The CMT can handle interfaces between arbitrary dispersive media. In general, this provides greater accuracy for a given mesh size, or make it possible to run jobs much faster without sacrificing accuracy. Due to the 1/dx4 dependence of the simulation time on the mesh size, results can often be achieved in roughly 1/10 the time. See Conformal Mesh Technology details below for more information. If more than two materials are found in a single cell, the method reverts to Staircasing for that cell. The Conformal meshing comes in three flavors:

Conformal variant 0

CMT is not applied to interfaces involving metals or PEC material.

Conformal variant 1

CMT is applied to all materials, including PEC and metals.

Conformal variant 2

The Yu-Mittra method 1 (see description below) is applied to interfaces involving metals and PEC, while CMT is applied to all other interfaces.

 

Dielectric Volume Average: The dielectric volume average method can only handle interfaces between one or more dielectric materials in a Yee cell. In this case, an average permittivity is created that is the simple volume average of all the dielectric permittivities in the given cell. The volume average is calculated by dividing the cell into NxN subcells in 2D and NxNxN subcells in 3D, where N is the "meshing refinement" value specified by the user. If non-dielectric materials are present at the cell center, then the method reverts to the Staircase method for that particular cell. This method has no real physical basis, but can be used for very low index contrast dielectric structures. In particular, if there is a low index contrast spatial variation in the permittivity, such as 1.4+0.01sin(x), then this method can give good results because it will average the permittivity over the entire Yee cell.

 

Volume Average: The permittivity of each cell is the simple volume average of the permittivities of the 2 materials in the cell. Each material can be fully dispersive. If more than two materials are found in a single cell, the method reverts to Staircasing for that cell.

 

Yu-Mittra method 1: This method was introduced by Yu and Mittra to provide greater accuracy when modeling PEC/dielectric interfaces. Lumerical's formulation is a slight extension of the original Yu-Mittra formulation that can be used with arbitrary dispersive media. In the figure below, we see the original formulation where the presence of a perfect electrical conductor (PEC) is taken into account by reducing the contour integral C to include only the region outside the PEC where the electric field is non-zero (C1). In our implementation, when updating the B field, the contour C1 is evaluated in material 1, while the contour C2 = (C-C1) is evaluated in material 2. In the event that one of the materials is PEC (where E=0), this is identical to the original Yu-Mittra formulation for PEC. If more than two materials are found in a single cell, the method reverts to Staircasing for that cell.

ref_yu_mittra1_zoom24

 

Yu-Mittra method 2: This method was introduced by Yu and Mittra to provide greater accuracy when modeling dielectric interfaces. Lumerical's formulation is a slight extension of the original Yu-Mittra formulation that can be used with arbitrary dispersive media. An effective permittivity is assigned each permittivity component in the Yee cell, that is weighted by the fraction of the mesh step that is inside material 1 or material 2. If more than two materials are found in a single cell, the method reverts to Staircasing for that cell.

 

 

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For more information on the Yu-Mittra method and other conformal mesh algorithms, see the following references:

Yu, W., and R. Mittra, "A conformal finite difference time domain technique for modeling curved dielectric surfaces," IEEE Microwave Components Lett,, Vol. 11, 2001, pp. 25-27.

Allen Taflove, Computational Electromagnetics: The Finite-Difference Time-Domain Method. Boston: Artech House, (2005).

Y. Zhao and Y. Hao, “Finite-difference time-domain study of guided modes in nano-plasmonic waveguides,” IEEE Trans. Antennas Propag. 55, 3070–3077 (2007).

A. Mohammadi, H. Nadgaran, and M. Agio, “Contour-path effective permittivities for the two dimensional finite-difference time-domain method,” Opt. Express 13, 10367–10381 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10367.

Conformal Mesh Technology details

Motivation

In recent years, photonic design has focused on ever smaller devices expected to operate over broad wavelength ranges, and that make use of highly dispersive materials. Emerging photonic technologies including plasmonics and silicon photonics exploit dispersive material characteristics and modern semiconductor manufacturing techniques to provide new functionality to the optical designer. While such technologies open the door to greater innovation, the combination of dispersive, high index contrast materials and nanoscale feature sizes have in turn place extreme demands on commercially available photonic design tools.

 

Background: Conformal Mesh FDTD

While the finite-difference time-domain (FDTD) technique is ideally suited in principle to provide broadband performance data for nanoscale photonic devices, the standard Yee-cell FDTD algorithm relies on discretizing the underlying structure onto a Cartesian mesh. The resulting discretized structure is unable to account for structure variations that occur within any single Yee cell, resulting in a "staircase" permittivity mesh that coincides with the Cartesian mesh.

In general, conformal mesh methods try to account for subcell features by solving Maxwell’s integral equations near structure boundaries. For example, the Yu-Mittra method , shown in Figure 1, can be used to improve accuracy when modeling curved PEC surfaces. While this method is simple to implement, and accurately describes the interaction of electromagnetic radiation at microwave and radio frequencies with most metallic materials, it will not provide accurate results for many types of simulations conducted at optical frequencies, such as propagation through multilayer stacks of dispersive materials. Nevertheless, the Yu-Mittra method is illustrative of the approach used by all conformal meshing methods.        

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FIGURE 1. In the Yu-Mittra model, the presence of a perfect electrical conductor (PEC) is taken into account by reducing the contour integral C to include only the region outside the PEC where the electric field is non-zero (C1).

 

Lumerical’s Conformal Mesh Technology

At optical frequencies, the dispersive nature of commonly-used materials must be taken into account. This can be done accurately via Lumerical’s Multi-coefficient Materials (MCMs), which makes it possible to simulate highly-dispersive materials used in applications ranging from solar cells through biosensors and CMOS image sensors. Figure 2 shows common materials employed in such designs, and the types of fits that can be accomplished using MCMs.

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FIGURE 2. Complex permittivity of typical materials employed in current photonic designs over wavelengths ranging from the ultraviolet to near infrared, together with the fits generated by Lumerical’s MCMs.

 

By applying a more rigorous physical description of the conformal mesh approach, Lumerical has developed Conformal Mesh Technology (CMT) that is able to incorporate arbitrary dispersive MCMs. As an extension of general conformal mesh algorithms to handle dispersive materials, it is easily possible to generate more simple conformal mesh models like the Yu-Mittra model from Lumerical’s CMT.

CMT Capabilities

Greater Accuracy for a Coarse Mesh

Lumerical’s CMT is capable of generating significant accuracy improvements relative to staircase results. This can be illustrated by applying the CMT to a multilayer stack, which is a common element incorporated within various photonic designs. As a conceptually simple but challenging test case, we consider the reflection and transmission of non-normal incidence p-polarized light through a five layer stack which includes dielectrics, metals and semiconductors. As the analytic transmission from this multilayer stack can be easily computed with transfer matrix theory, it provides a good test case to demonstrate the ability of CMT to account for subcell features as shown in Figure 3.

 

a)

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b)

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c)

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FIGURE 3. Multi-layer stack composed of gold, a constant dielectric material of index 1.9, Si, GaAs, and Ge tested over a wavelength range of 400-1000nm using a mesh resolution of 10 points per wavelength. The staircase result in (b) shows significant deviations relative to the analytic response calculated from transfer matrix theory, while the CMT result in (c) demonstrates that subcell features can be accurately accounted for.

 

This simple test demonstrates the ability of CMT to resolve subcell features which, in this case, is the location of interfaces that do not necessarily match the discretized mesh. Testing shows that at a mesh resolution of 10 points per wavelength, CMT provides significantly greater accuracy than the staircase results obtained at a much higher mesh resolution of 34 points per wavelength. Examination of the average error calculated for the sum of the reflection and transmission signals over the 600nm bandwidth simulated in Figure 4 shows that the level of accuracy achieved with CMT at a coarse mesh of 10 points per wavelength could not be achieved with the staircase approach at any reasonable mesh size, and likely that more than 60 points per wavelength would be required.        

 

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 FIGURE 4. Average RMS error for reflection and transmission of five layer multilayer stack over 600nm bandwidth. The accuracy achieved by CMT at 10 points per wavelength would likely require more than 60 points per wavelength for the staircase approach.

 

The ability of CMT to produce much higher accuracy results with a more coarse mesh can also be demonstrated for 3D structures, like the organic solar cell device shown in Figure 5. While there is no analytic result that can be calculated for such a structure, 3D FDTD results obtained with a very fine mesh can be used in place of an analytic result. As computation time within FDTD varies with inversely with the mesh size to the fourth power (1/dx4) using a coarser mesh can result in significantly faster simulation times. For example, Figure 5 shows that the CMT can achieve the same accuracy at 14 points per wavelength as the staircase method at 26 point per wavelength, which translates into a speedup of more than 7 times.

 

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Points per wavelength

 

FIGURE 5. A 3D organic solar cell structure that is comprised of a periodic nanoscale pattern in a multilayer structure. By comparing staircase and CMT results against 3D FDTD results obtained using a very fine mesh, the RMS error of the two methods can be compared. To achieve a 1% RMS error, CMT requires a mesh size of less than 14 points per wavelength, while the staircase approach requires a mesh size of more than 26 points per wavelength.

 

Faster and Smoother Convergence

In most applications, the CMT shows faster and smoother convergence than the staircase method. This allows designers to work at coarser mesh sizes and realize the significant simulations speedups that can be obtained due to the 1/dx4 dependence of the FDTD simulation time. In Figure 6, we see the convergence of the green pixel response in a typical CMOS image sensor. The CMOS image sensor involves complex material properties, such as silicon and color filters which require the MCM to fit, AR coatings the thickness of which must be resolved to within a few nanometers, as well as curved microlens surfaces used to focus the light. The extremely quick convergence of the CMT method for this design demonstrates its robustness in a very complicated application.

 

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FIGURE 6. A 3D CMOS image sensor model composed of microlenses, color filters, and metallic interconnects on a multilayer substrate. The CMT allows for much faster and smoother convergence. At coarse mesh sizes, such as 6-14 points per wavelength, the CMT shows that 33% of the incident light is converted to electron/holes in the green photodiodes. The staircase method, on the other hand, may not yet be converged even at 26 points per wavelength.

Summary

The conformal mesh can enhance simulation accuracy for a given mesh size, or make it possible to run jobs much faster without sacrificing accuracy. Due to the 1/dx4 dependence of the simulation time on the mesh size, results can often be achieved in roughly 1/10 the time. Also, the CMT provides submesh sensitivity to changes in geometrical parameters, which greatly facilitates design optimization. Owing to its inherent compatibility with Lumerical’s MCMs, CMT allows designers to more efficiently prototype broadband, nanoscale photonic design concepts in high index contrast, dispersive materials.

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