Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be \(470 \mathrm{nm}\) and the index of refraction of oil to be 1.40.

Thin film interference is a phenomenon that occurs when light waves are reflected and refracted by a thin layer of material and produce a spectrum of colors as a result. Imagine this thin layer as a sort of stage where light performs a complex dance. As light strikes the film, some of it bounces back off the top surface, while some travels through and reflects off the bottom surface before emerging out again.

This creates two waves that travel different paths and, once they reemerge, can either amplify or cancel each other out. That's where the vibrant colors come from, like seeing a rainbow in a soap bubble. The colors that appear depend on the thickness of the film and the wavelength of light that's involved. So, engineers and designers can harness this knowledge to coat materials in ways that can influence the colors that we perceive, without using pigments.

Constructive interference arises when waves align in a way that they boost each other. This creates a wave with a larger amplitude than either of the individual waves. Think of it like a perfectly timed high-five between light waves; when their peaks hit just right, they clap together to make a bigger wave.

In the context of thin film interference, this happens when the path difference between the two reflected light waves is a multiple of the light's wavelength. For constructive interference to cause a color, like blue, to appear, the thickness of the film plays a crucial part. When calculating the minimum thickness for constructive interference to occur, we assume ideal conditions where light hits the film head-on and the waves inside the film walk in step to emerge strong and united.

The index of refraction, often designated as 'n', is a measure of how much a material can bend, or refract, light. It tells us how fast light can travel through the material compared to the speed of light in a vacuum. When light enters a new medium, like oil, its speed changes and this bending of light is a key player in thin film interference. The higher the index, the more the light slows down and bends inside the material.

This bending process reshapes the path of light inside the thin film, influencing how it interferes with itself when it exits. It's similar to when swimmers change speed as they move from water to land. In our case, the refraction index of oil affects how it works with the wavelength to create that brilliant interference pattern.

The wavelength of light is the distance between successive peaks of a light wave, typically measured in nanometers (nm) or meters (m). Each color of light has its own unique wavelength, and that's what enables us to see the world in such a vivid array of colors. For example, blue light typically has a wavelength of about 470 nanometers.

When we talk about thin film interference, the wavelength is critical to determining what color appears. The thickness of the film determines which wavelengths are amplified through constructive interference. By adjusting the film's thickness, you effectively fine-tune what color emerges. So, in the case of an oil film appearing blue, we would calculate the minimum thickness needed to best amplify that specific wavelength of blue light.