Optimization of optical inspection systems requires studying many illumination and collection modes. There are various parameters that can be adjusted, such as input and output light polarization, illumination and collection numerical apertures, defocusing, etc. In principle, for each configuration of the inspection system, a fully-vectorial simulation of the interaction of light with the analyzed structure is required. Optimization routines are usually composed of many of these simulations; therefore, the process can be very time consuming and demanding in terms of computing resources. By implementing a two-step simulation process, where the light-structure interaction and the optical system simulation are done separately, the optimization process can be done much more efficiently. This simulation workflow is described in this page.

## Simulation workflow

The proposed workflow is composed of two main steps:

- Fully-vectorial simulations of the interaction between light and a particular structure using FDTD. This step requires choosing an appropriate set of incident and outgoing plane waves and running a FDTD simulation for each incident plane wave. The results of these simulations are combined in a S-matrix connecting input and output states.
- Optical system simulation using optical transfer functions (OTFs) for illumination and collection conditions. A combination of these functions with the S-matrix provides the fields required to generate the image expected for the imaging system under consideration. Unless the wavelength or the structure (including the defect) are modified, the same S-matrix can be used to simulate different configurations of the optical system, allowing to run optimizations very efficiently.

In the following sections we describe these steps in more detail. For a simple example that illustrates the methodology please refer to Defect on A Metal Surface.

## FDTD simulation

### Supercell approach

Typical structures of interest are formed by many unit cells of a periodic structure. The presence of defects in these structures breaks the periodicity, and so it is not possible to simulate only one unit cell with periodic boundary conditions. Instead, we consider a supercell formed by many unit cells of the underlying periodic structure and the defects. This supercell is assumed to be repeated periodically; therefore, we use Bloch periodic boundary conditions, which are also consistent with plane wave illumination at oblique incidence.

### Choosing a plane wave basis

The main assumption behind the simulation workflow is that the interaction between light and structure can be correctly described by decomposing the incoming and outgoing light using a discrete set of plane waves at different angles. Therefore, **this approach is valid for linear interactions only**.

Wavevectors associated with the supercell grating orders are a convenient choice. They describe the possible directions for plane waves scattered by the periodic structure, depending on the incident wavevector of the incoming plane wave. Grating orders for normal incidence have a very useful property: light incident at one of these directions is scattered to the same set of wavevectors. Therefore, the same set of wavevectors can be used for input and output plane waves. Although it is not the only possible choice, we will use this basis for convenience.

The S-matrix provides the connection between the input and ouput plane waves, as explained in S Parameter Extraction. Once we have calculated the S-matrix we can store it in a Lumerical data file (.ldf), which can be included in the optical system simulation to construct the images of the structure.

### Simulation tips

The scattering signal associated with defects is usually many orders of magnitude weaker than the scattering to the grating orders of the periodic underlying structure. For this reason, some care is necessary to avoid artificial periodicity breaking caused by the simulation setup itself. The following tips can help minimizing simulation artifacts:

- The supercell must be large enough to provide the required resolution in k-space and avoid near-field coupling between defects.
- Use a uniform mesh in the plane of the structure. This can be achieved by including a mesh override region covering the entire area of the supercell. Avoid using a mesh override for the defect region only since this will cause an additional periodicity breaking and result in an artificial contribution to the scattering. Since we are typically not interested in resolving the exact form of the defect (we just want to know it is present), a particularly fine mesh is not required for the defect. Conformal mesh technology provides additional subcell accuracy as well.
- Extend the structure through the boundary condition region (the one mesh cell thick boundary region drawn in a light blue color). This ensures that the material properties are correctly set at the boundary regions, avoiding any artificial periodicity breaking there.
- For periodic structures it is important to make sure the supercell contains an integer number of unit cells of the underlying periodic structure and that each unit cell is meshed in the same way.

Some additional tips for running simulations:

- Since the S-matrix approach requires a large number of simulations, it is important to use your computational resources as efficiently as possible. Fortunately, in this approach simulations are independent of each other so it is possible to use concurrent computing by setting up your Resource configuration as explained here.

## Optical system simulation

Given a set of input plane waves with amplitudes **E**_{i,j}^{in}, the amplitude of the output plane waves that will be collected to construct the image can be found from

$$ {\mathbf{E}_{n, m}^{\text {out }}=\sum_{i, j} H_{\text {out }}\left(\mathbf{k}_{n, m}^{\text {out }}\right) S_{i, j ; n, m} H_{\text {in }}\left(\mathbf{k}_{i, j}^{\text {in }}\right) \mathbf{E}_{i, j}^{\text {in }}} $$

where H_{in} and H_{out} are the illumination and collection OTFs, and S_{i,j;n,m} is the S-matrix calculated from the FDTD simulations. In the amplitude **E**_{i,j}^{in} we can include the phase factor associated with focusing at a spot position **r**_{spot} in the object plane and also a vector describing the polarization of incident light:

$$ {\mathbf{E}_{i, j}^{\text {in }}=\hat{\mathbf{u}}_{p} E_{\text {in }} e^{i \mathbf{k}_{i, j}^{\text {in }} \cdot \mathbf{r}_{\text {spot }}}} $$

The amplitudes **E**_{i,j}^{out} can be used to generate the image according to the particular imaging technique; for example, a standard imaging system used in microscopy is described here. A different technique used in scanning-spot microscope imaging is described in the SRAM structure example.

Using this approach we can simulate the optical system without including it explicitly in the FDTD simulations. Therefore, any changes in the optical system (for example, the numerical aperture) do not require rerunning any FDTD simulation; we can reuse the calculated S-matrix and adjust the OTFs accordingly. As long as the wavelength and structure (including any defects) do not change, the same S-matrix can be used in the optical system simulation.

### Related publications

- O. Golani, I. Dolev, J. Pond, and J. Niegemann, "Simulating semiconductor structures for next-generation optical inspection technologies", Optical Engineering 55 (2), 025102 (2016)

### See also

Grating projections, Optical Defect Metrology (S-Matrix), Simplified Microscopic Imaging, Defect on A Metal Surface