An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How could she use this knowledge to make a nonreflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on the index of refraction and thickness of the coating? How might this be impractical?

Destructive interference occurs when two waves, typically of the same wavelength, combine in such a way that they cancel each other out. This phenomenon is an integral concept in creating nonreflective coatings for lenses. Such coatings are designed to reduce glare by ensuring that the reflected light waves from the surface interfere destructively. The condition for this to happen is when the path difference between the reflected light from the top and bottom of the coating is half a wavelength (λ/2). This results in the waves being out of phase and canceling each other.

Practically, a thin film of material with a particular thickness is applied to the lens. When light hits this coating, some of it reflects off the top layer, while some enters the film, reflects off the bottom layer, and exits out, potentially interfering with the initial reflected wave. If the thin film's thickness corresponds with the specific condition for destructive interference, the glare can be significantly reduced.

The index of refraction, often denoted by the symbol n, is a measure of how much a medium slows down light as compared to its speed in a vacuum. It determines how much light bends, or 'refracts,' when it enters a material from another medium.

This property is crucial when designing nonreflective coatings. For a given medium, the wavelength of light within it is inversely related to the index of refraction—the higher the index, the shorter the wavelength inside the material (λ' = λ/n). Since the coating's effectiveness is dependent on the relationship between its thickness, the wavelength of light, and its index of refraction, choosing a suitable material with an optimal index is pivotal. Different materials can be used to match the required index of refraction that will facilitate the desired destructive interference.

Snell's law is the fundamental rule that describes the relationship between the angles of incidence and refraction when light passes through the boundary between two different isotropic media. Mathematically, it can be expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 represent the indices of refraction of the first and second medium, respectively, while θ1 and θ2 are the angles to the normal of the surface.

When applied to lens coatings, Snell's law helps in determining how light will behave as it transitions from air (or another medium) into the coating. The law can also be used to relate the wavelength of light in different media, which is essential when designing coatings for multiple wavelengths.

Wavelength, denoted as λ, is the distance over which a wave's shape repeats. It is a critical factor in determining the behavior of waves, including light, and is typically measured in meters. In the visible spectrum, different wavelengths correspond to different colors, with violet having the shortest and red the longest wavelengths.

Lens coatings need to account for a range of wavelengths to be effective at reducing reflection across the visible spectrum. A nonreflective coating's performance depends on its capacity to cause destructive interference for the targeted range of wavelengths. However, achieving this for every wavelength simultaneously is challenging, as the thickness of the coating that produces destructive interference for one wavelength will not necessarily work for another. As such, there's a trade-off in designing coatings that are broadly effective but may not be perfect for every single wavelength.