Tutorial

Basic usage

This section assumes that the Lua frontend is used. The program may be run with a Lua script as an argument, in which case the script is run and then the program terminates, or without an argument, in which case it enters interactive mode with a Lua shell.

The basic flow of a script is listed below. To find more information about these functions and for a full listing, see the Lua API reference.

1. Obtain a new simulation object:

   S = S4.NewSimulation()
   

   S now contains a simulation object with a blank specification, and no solutions.

2. Set the lattice and number of basis functions:

   S:SetLattice({ux,uy}, {vx,vy})
   S:SetNumG(100)
   

   This sets the periodicity based on the lattice vectors (ux,uy) and (vx,vy). The number of basis functions is set to 100.

3. Define all materials:

   S:AddMaterial('Vacuum', 1)
   S:AddMaterial('Silicon', {12, 0.01})
   

   This adds a material called “Vacuum” with a dielectric constant of 1, and a material called “Silicon”, with a dielectric constant of 12+0.01i.

4. Add all layers:

   S:AddLayer('top', 0, 'Vacuum')
   S:AddLayer('slab', 0.5, 'Silicon')
   S:AddLayerCopy('bottom', 0, 'top')
   

   Here we are adding a semi-infinite Vacuum layer (the thickness is largely irrelevant), a middle slab layer made of silicon with a thickness of 0.5, and finally a bottom semi-infinite layer that is a copy of the top layer. It is always preferable to copy layers if possible since it reduces the computational cost.

5. Add patterning to layers:

   S:SetLayerPatternCircle('slab',   -- layer to pattern
                           'Vacuum', -- material inside circle
                           {0,0},    -- center of circle
                           0.2)      -- radius of circle
   

6. Specify the excitation mechanism:

   S:SetExcitationPlanewave(
           {0, 0}, -- phi in [0,180), theta in [0,360)
           {1, 0}, -- s-polarization amplitude and phase in degrees
           {0, 0}) -- p-polarization
   

   This sets the excitation to a normally incident planewave with amplitude 1.

7. Specify the operating frequency:

   S:SetFrequency(0.4)
   

8. Obtain desired output:

   transmission = S:GetPowerFlux('bottom')
   print(transmission)
   

   This obtains the forward power flux in the bottom layer, which is also the transmitted power.

Typical usage

The only fully supported front end is the Lua API. The Python API is largely supported by the user community. Most users are interested in computing frequency (or wavelength) spectra, in which case SetFrequency() is called in a loop over frequency. Putting GetPowerFlux() afterwards in the same loop allows obtaining reflection/transmission/absorption power spectra. If the materials used are dispersive, then SetMaterial() must also be called within the loop before the call to GetPowerFlux() to update material constants for each new frequency.

When running parameter sweeps for optimizing structures, a number of structural parameters are typically varied, such as lattice constants, layer thicknesses, and patterning dimensions. Due to the way in which S⁴ performs lazy computation (things are not computed or recomputed until they are required), the order of nested loops matters greatly. In general, all output quantities for a given structure should be computed at once before modifying the structure, since structure modification changes the solution completely. Next, layer thicknesses can be changed without requiring recomputation of layer eigenmodes. A slightly more computationally expensive modification is changing a layer’s patterning, which requires recomputing the layer’s eigenmodes. Changing material properties invalidates eigenmodes of any layer using the material. Finally, changes in lattice constant, frequency, and incidence angles require a complete recomputation of the solution.

Fourier Modal Method formulations

There has been extensive literature on the best way to generate the Fourier series coefficients for the in-plane dielectric profiles of each layer. S4 implements a number of different formulations. The following functions determine which formulation is selected:

• UseDiscretizedEpsilon
• UsePolarizationDecomposition
• UseSubpixelSmoothing
• UseJonesVectorBasis
• UseNormalVectorBasis

In addition, the following functions control accuracy and the lattice truncation:

• SetResolution
• SetLatticeTruncation

To simplify the choice for users, the table below summarizes the recommended settings. It is recommended to always use circular truncation unless there is a good reason to do otherwise. Speed indicates the speed of the Fourier coefficient generation, which is usually not the dominant part of the simulation time.

Options	Can handle Anisotropic?	Recommended resolution	Speed	Accuracy
none	yes	N/A	fast	poor
Disc	yes	8	medium	poor
Subpixel	yes	4	medium	medium
Pol	no	8	slow	good
Pol+Normal	no	8	slow	good
Pol+Jones	no	8	slow	good
Disc+Pol	no [1]	4	slow	medium
Disc+Pol+Normal	no [1]	4	slow	medium
Disc+Pol+Jones	no [1]	4	slow	medium

It is recommended that users use the default (no options) despite the poor accuracy, since the polarization basis settings can introduce spurious regions of gain into the material patterning. This may or may not be a problem for some users using extremely lossy metals, but for low loss metals, typically energy conservation is violated.

Footnotes

[1]	(1, 2, 3) The formulation does not strictly work correctly for anisotropic media however it may still work. Proper support for anisotropic materials is in principle possible. There are currently no plans for implementing generation of the proper basis fields for this feature.

Examples

The source distribution of S⁴ includes numerous fully working didactic examples as well as examples replicating published results. You can find these examples in the examples/ directory of the source distribution.