Optical Second Harmonic Generation in Anisotropic Multilayers with Complete Multireflection of Linear and Nonlinear Waves using #SHAARP.ml Package

Light has long been a tool for seeing the world, but it is also a tool for changing it. When a beam of light strikes certain materials, the atoms inside can act like tiny amplifiers, catching two photons of one color and merging them into a single photon with twice the energy and half the wavelength. This process, known as second harmonic generation, turns red light into blue, or infrared into visible green, creating new colors from old ones. Scientists have used this trick for decades to build lasers, analyze the structure of crystals, and even peer into the hidden magnetic domains of materials. However, as researchers push into new territories with thinner films and complex stacks of different materials, the math required to predict exactly how this light will behave has become a tangled knot. Traditional methods often rely on simplifications that work for simple, single crystals but fail when light bounces back and forth between layers, interfering with itself in complex ways.

A team of researchers at Pennsylvania State University has untangled this knot by developing a new, open-source software package called ♯SHAARP.ml. This tool allows scientists to calculate the behavior of light in complex, multi-layered structures without relying on the rough approximations that have plagued the field for years. Instead of assuming that light travels in a straight line or that reflections are weak, the software tracks every single bounce of the light waves as they travel through a stack of materials. It accounts for how the light waves interfere with one another, creating patterns of bright and dark spots that depend on the precise thickness of the layers and the angle at which the light hits them. By testing this software against real-world experiments with materials like quartz, lithium niobate, and zinc oxide, the researchers proved that their method is accurate enough to measure the fundamental properties of new materials with high precision.

The core challenge the team addressed is that light does not simply pass through a material like a car driving down a straight road. When light hits a boundary between two different materials, some of it reflects back. In a thin film or a stack of layers, this reflected light bounces again and again, creating a complex web of waves traveling in opposite directions. These waves interfere with each other, sometimes canceling out and sometimes reinforcing each other. For decades, scientists analyzing second harmonic generation often ignored these backward-traveling waves or assumed the material was perfectly transparent and uniform. While this worked for simple, thick crystals, it led to significant errors when applied to modern devices, which are often made of very thin films or complex heterostructures where these reflections are strong. The new software, ♯SHAARP.ml, was built to handle this complexity by calculating the exact path and interaction of every wave, including the subtle interference patterns that arise from multiple reflections.

To verify their tool, the researchers put it through a series of rigorous tests using a variety of materials. They started with a single crystal of quartz, a material whose optical properties are well known. They measured how the crystal responded to light at different angles and compared their experimental data with the software's predictions. The results matched perfectly, confirming that the software could reproduce the classic patterns seen in older, simpler models. But the real test came when they added a layer of gold to the back of the quartz. This metal layer acted like a mirror, causing the light to bounce back and forth intensely within the crystal. In this scenario, the older, simplified models failed to predict the correct intensity of the light, while the new software, which accounted for the full multiple reflections, matched the experimental data with high accuracy. This demonstrated that for materials with reflective layers, ignoring the bouncing waves leads to large errors in the calculated properties.

The team then moved on to more complex, real-world materials used in modern technology. They analyzed thin films of zinc oxide grown on a substrate with a platinum bottom layer, a structure that creates a near-perfect mirror effect for the light. In this case, the simplified models suggested that the material's ability to generate new light was ten times stronger than it actually was, simply because they failed to account for the light bouncing around inside the film. The new software corrected this, revealing the true, much lower value. They also tested the software on crystals of lithium niobate and potassium titanyl phosphate, which are widely used in lasers and telecommunications. By measuring how these crystals responded to light from different angles and polarizations, the researchers were able to extract precise values for their optical properties, matching known standards and proving the tool's reliability for characterizing new materials.

Beyond just measuring known materials, the software opened the door to understanding complex systems that were previously too difficult to model. The researchers used it to simulate a stack of two different crystals, one made of lithium niobate and the other of quartz, to see how they would interact. They found that the light generated by the two layers could interfere with each other in a way that revealed the direction of the internal electric polarization within the lithium niobate. This is a crucial capability because standard measurements often cannot tell the difference between two opposite directions of polarization if the light intensity is the same. By adding a reference layer, the interference pattern changes, allowing scientists to distinguish between the two states. This technique could be vital for studying ferroelectric materials, which are used in memory devices and sensors, where knowing the exact direction of the internal polarization is essential.

The researchers also applied their tool to the cutting edge of materials science: twisted bilayer molybdenum disulfide. This is a structure made of two atomic layers of a material stacked on top of each other, with one layer rotated slightly relative to the other. The angle of this twist changes how the material interacts with light. The software successfully simulated how the light intensity would change as the twist angle varied, matching experimental observations. It showed that the substrate underneath the material, which acts as a mirror, significantly affects the final signal. This level of detail is important for designing future electronic and optical devices based on two-dimensional materials, where the precise arrangement of atoms and layers dictates the device's performance.

The development of ♯SHAARP.ml represents a shift from relying on simplified assumptions to using a complete, rigorous description of light behavior in complex materials. By making this powerful tool open-source, the researchers have provided the scientific community with a way to accurately model and understand the optical properties of new materials, from simple crystals to intricate stacks of thin films. This capability is essential as the field moves toward more complex devices where light interacts with matter in increasingly sophisticated ways. The software does not just predict what will happen; it explains why, by tracing the path of every photon as it bounces, interferes, and transforms within the material. As scientists continue to discover new materials with unique optical properties, having a tool that can handle the full complexity of their behavior will be key to unlocking their potential for future technologies.