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Thin film interference occurs when light waves reflect off the surfaces of a thin layer, such as a soap bubble or an oil slick, and interfere with one another. When light hits this thin film, part of it reflects off the top surface, and another part travels through the film, reflects off the bottom surface, and then exits the film.
These two reflected light waves can combine in a way that either amplifies the light (constructive interference) or diminishes it (destructive interference). In the specific case of **destructive interference**, the path difference between two reflected light waves should be such that they are out of phase by an odd integral multiple of half-wavelengths.

• This often results in certain colors in white light being diminished or eliminated based on the thickness of the film and the refractive index of the film material.
• Thin film interference is what causes the vivid colors in soap bubbles and oil films on water.

In our specific case, the problem involves calculating the minimum thickness of a thin film coating on glass that causes destructive interference for a particular wavelength of red light.

The refractive index, denoted as **n**, is a measure of how much the speed of light is reduced inside a medium compared to the speed of light in a vacuum. It gives an indication of how much the path of light is bent, or refracted, when entering a material.

• The refractive index for the film is a crucial factor in determining interference conditions because it affects both the speed and the wavelength of light as it passes through the film.
• In this exercise, the coating has a refractive index of 1.42 while the glass beneath has a refractive index of 1.52.
• This difference means that as light passes from air (with an index of approximately 1.00) into the coating, and then into the glass, it experiences multiple reflections and refractions.

These variations in speed and wavelength due to differing refractive indices can lead to the interference phenomena we are calculating, where precise thickness adjustment causes destructive interference at specific wavelengths.

When solving problems involving thin film interference, a key step is to adjust for the changed wavelength of light within the film. The wavelength of light inside the film is shorter than it is in air due to the refractive index of the material.
The general formula used to calculate the wavelength of light inside a material is given by:\[\lambda_{film} = \frac{\lambda_{air}}{n}\]where \(\lambda_{film}\) is the altered wavelength within the material, \(\lambda_{air}\) is the original wavelength in air, and \(n\) is the refractive index of the medium.
In our problem with a refractive index of 1.42 for the coating, if the original wavelength of the red component is 650 nm in air, this wavelength becomes shorter when it enters the coating:\[\lambda_{film} = \frac{650\, \mathrm{nm}}{1.42}\approx 457\, \mathrm{nm}\]Remember, the reduced wavelength is critical when calculating interference patterns since the light waves need to align precisely for minimal thickness that causes destructive interference. This formula is pivotal in finding the specific thickness where destructive interference occurs, corresponding to the calculated thickness of approximately 114.44 nm.