Expanding RCWA capabilities with advanced S-matrix algorithms

Light behaves in predictable ways when it travels through empty space, but the moment it encounters a stack of thin films or a surface patterned with tiny structures, the rules become complex. This is the realm of layered photonic structures, the backbone of modern optical devices like anti-reflective coatings, mirrors, and the intricate components inside fiber-optic networks. When light hits these layers, it bounces, interferes, and scatters at every boundary, creating a tangled web of energy that is difficult to predict. To design better devices, engineers need a way to calculate exactly how light will behave as it moves through these stacks. For decades, scientists have relied on a powerful mathematical tool called the scattering-matrix method to solve this problem. This method is prized because it remains stable and accurate even when dealing with very thick layers or light that fades away rapidly, a situation where older calculation methods often fail. However, while the tool worked well for standard problems, its full potential for exploring complex designs and understanding the deep physics of light flow had not yet been fully unlocked.

A team of researchers at the University of Bordeaux and the Institut d'Optique in France has now refined this tool, transforming it into a more flexible and powerful instrument for both design and discovery. They introduced a new way of defining the scattering matrix, shifting the focus from the connection between two layers to the properties of a single layer itself. In the traditional approach, the calculation for a specific layer was tied to the materials sitting immediately above and below it, meaning that if an engineer wanted to test that same layer in a different stack, they had to recalculate everything from scratch. The new method treats the layer as an independent entity with its own intrinsic behavior, described by how it handles light passing through it, regardless of its neighbors. The interfaces where layers meet are then handled separately. This separation allows the layer's properties to be computed once and reused endlessly in different combinations, much like a single brick that can be used in any wall without needing to be re-molded for each new design.

This shift in perspective does more than just speed up calculations; it reveals the hidden pathways of light inside these structures. By using this reformulated approach, the researchers can now directly calculate how external light waves couple with the internal waves that naturally exist within a periodic structure, known as Bloch modes. These modes are the specific ways light can travel through a patterned material, similar to how a guitar string can only vibrate at certain frequencies. The new method provides a set of generalized coefficients that act like a map, showing exactly how much energy moves from an incoming beam into these internal modes and how they interact with each other at the boundaries. To demonstrate this, the team analyzed a silicon grating designed to be highly transparent across a wide range of colors. While previous explanations focused on the resonances of individual nanowires, the new analysis showed that the transparency is actually driven by the excitation and propagation of just two specific internal light modes. One mode dominated the behavior at longer wavelengths, while a second, unusual mode with a very low effective speed was responsible for the anti-reflective effect at shorter wavelengths. This level of detail allows designers to see exactly which internal channels are carrying the energy, turning the simulation from a black box that outputs a number into a transparent window into the physics of the device.

The practical benefits of this reformulation are equally significant, particularly for the speed of design optimization. Because the new method treats the layer's thickness as a simple variable in an analytical formula, it can calculate the optical response for hundreds of different thicknesses in the time it used to take to calculate just one. The researchers tested this by simulating a structure with two silicon gratings separated by a silica spacer, varying the thickness of the upper grating across 400 different values. The entire sweep took only a fraction of the time required for a single calculation, a feat that would be impossible with traditional simulation methods that require a complete re-run for every new parameter. This capability also offers a clever solution for dealing with thick materials where light loses its coherence, a common issue in solar cells and other real-world applications. Instead of building complex new theories to handle this randomness, the team showed that simply averaging the results over many thickness values produces an accurate picture of the incoherent behavior, effectively smoothing out the rapid fluctuations caused by interference.

Furthermore, the new framework allows for the simultaneous analysis of light hitting the structure from many different angles at once. Conventional software typically calculates the result for a single beam of light, requiring separate runs for every angle of interest. The updated approach, however, computes the scattering coefficients for all possible incident and diffracted directions in a single step. This is particularly useful for analyzing large, disordered surfaces modeled with artificial periodicity, where a single calculation can reveal the behavior of the entire system under various lighting conditions. All these capabilities have been integrated into a free software package called RETICOLO, which the authors have made available to the scientific community. By providing the code and detailed scripts used to generate their results, the researchers have offered a practical toolkit that allows other scientists to perform these advanced modal analyses, accelerate their design workflows, and gain a deeper physical understanding of how light moves through the complex layered structures that power our optical technologies.